Research ArticleDynamic Phasors Estimation Based on Taylor-FourierExpansion and Gram Matrix Representation
Predrag PetroviT and Nada DamljanoviT
Faculty of Technical Sciences Cacak University of Kragujevac Svetog Save 65 Cacak Serbia
Correspondence should be addressed to Predrag Petrovic predragpetrovicftnkgacrs
Received 20 September 2018 Revised 8 November 2018 Accepted 14 November 2018 Published 26 November 2018
Academic Editor Michele Brun
Copyright copy 2018 Predrag Petrovic and Nada Damljanovic This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
The paper presents a new approach to estimation of the dynamic power phasors parameters The observed system is modelled inalgebra of matrices related to its Taylor-Fourier-trigonometric series representation The proposed algorithm for determinationof the unknown phasors parameters is based on the analytical expressions for elements of the Gramrsquos matrix associated with thissystem The numerical complexity and algorithm time are determined and it is shown that new strategy for calculation of Gramrsquosmatrix increases the accuracy of estimation as well as the speed of the algorithm with respect to the classical way of introducingthe Gramrsquos matrix Several simulation examples of power system signals with a time-varying amplitude and phase parameters aregiven by which the robustness and accuracy of the new algorithm are confirmed
1 Introductions
The very intense and rapid development of electronic tech-nologies including the production of renewable energysources (commercial solar wind power plants and biomasspower plants) and power devices caused a change in thestructure of the traditional power system Electrical variablessuch as the basic harmonics amplitude and its frequencycan be significantly altered inmicronetworks weak networksand networks of the island type due to reduced capacity interms of short-circuit currents In addition the harmonicis injected into the system using electrical electronic equip-ment [1] Under these circumstances an ideal algorithm forassessing and estimation of the pharos must enable quickaccurate and steady monitoring of the changing parametersof the electrical signals that are contaminated by the presentharmonic components
In the last few yearsmany algorithms have been proposedin the literature for evaluating the phases in different dynamicstates and conditions that can be controlled in the networkIn general they can be classified into methods based on theapplication of discrete Fourier transform (DFT) [2ndash4] andnon-DFT-based methods Each of the algorithms requires
a harmonicpharos model and uses some of the specifictechniques thatmodel the parameters of the observed systemIn accordance with the modelling method the measurementmethod the algorithms can be divided into two main classesalgorithms that rely on the pure sinusoidal signal model(static model) and algorithms based on a nonsinusoidalmodel (dynamic model) [5]
A simultaneous estimation of phasor and frequencybased on the application of the fast recursive Gauss-Newtonalgorithm was proposed in [6] while the method based onthe modified Fourier transformation in order to eliminatethe DC offset was presented in [7] The paper [8] proposesa new method based on an adaptive band-pass filter inorder to evaluate the phasor On the other hand the authorsof [9] introduce a special angle-shifted energy operator toseparate the value of the instantaneous amplitude of thephasor The least square curve fitting approach was proposedin [10] in order to eliminate the unwanted effect of saturationof the current transformer Measurement of the phase andfrequency during the transient processes was considered in[11] while the dynamic estimation of the phase based on theapplication of maximally flat differentiators was proposed in[12] and the phasorlet in [13] The paper [14] introduces an
HindawiMathematical Problems in EngineeringVolume 2018 Article ID 7613814 17 pageshttpsdoiorg10115520187613814
2 Mathematical Problems in Engineering
approach to estimate the parameters of the phasor based onrecursive wavelet transformation Taylor-Fourier algorithm[15] approximates the dynamic phase with the second-orderTaylor expansion and the least square observer The Taylor-Kalman method based on the Kalman observer in order toachieve nondelayed dynamic phase estimation is described in[16] This method is further enhanced by the development ofthe state space for harmonic infiltration in [17] and is calledTailor Fourier Kalman The Prony algorithm [18] was used toestimate the phasor described by the exponential amplitudeand the linear phase with no Taylor expansion There areother methods that are modifications of earlier methods [1920] The Shank method is another method for estimating adynamic phasor using the least square and consecutive delaysof the unit response system [21] The algorithm [22] for TFT(Taylor-Fourier transform) harmonic analysis uses recursivemultiple-resonator-based computational techniques whichenable the reduction of both the computational cost and thememory requirements of the algorithm
In [23] a more precise dynamic model describing thecomplex trajectory of the dynamic phasor is derived owingto a revised state transition equation of the rotating pha-sor and its derivatives Based on the improved dynamicmodel [23] a modified Taylor-Kalman Filter for instan-taneous dynamic phasor estimation is developed whichis coincident with the self-adaptive nature of the Kalmanfilter principle An extension of the Taylor Weighted LeastSquares (TWLS) algorithm for the estimation of the pha-sor frequency and ROCOF Rate-Of-Change-Of-Frequencyparameters of an electric waveform has been analyzed in[24] Paper [25] introduces a dynamic phasor estimationmethod for PMUsIEDs based on a modified time domainhybridmethodDynamic phasor is estimated using theTaylorexpansion where frequency deviation is derived directlyfrom fitting parameters to avoid magnification of fittingerrors A double suboptimal-scaling factor-adaptive strong-tracking Kalman filter (DSTKF)-based phasor measurementunit algorithm which can meet the accuracy requirement ofthe IEEE standard C371181 under the dynamic condition wasproposed in [26] This method uses a kth Taylor polynomialto linearize the complex exponential of the signal modeland estimates the dynamic phasor using DSTKF In [27] acombination of the least square-based Prony analysis andTaylor expansion called TaylorndashProny is proposed to estimatethe dynamic phasor
In this paper a special transformation (representation) ofTaylor-Fourier expansion and corresponding Gram matrixis used to estimate the dynamic phasors from input pha-sorlets in real domain based on synchronously sampledvalues of the input signal The dynamic systemrsquos behaviouris described in algebra of matrices related to their Taylor-Fourier-trigonometric series representation The explicit for-mulas for calculation of elements of the associated Gramrsquosmatrix are given In this way we are in a position to improvethe computational performance of the proposed algorithm asregards precision and speed Namely the proposed algorithmis based on matrix multiplication It is a well-known factthat mathematical definition of multiplication of matricesgives an algorithm that takes 119901119902119903 time to multiply a matrix
of type 119901 times 119902 with a matrix of type 119902 times 119903 (in the case ofsquare 119901 times 119901-matrices it takes 1199013 time) If we use the explicitformulas for elements of Gramrsquos matrix in our algorithmthen one matrix multiplication is avoided and the task isreduced to 119901119903 consecutive calculations of the values of spe-cific polynomials Moreover each element of Gramrsquos matrixdepends on the number of samples119873 (it is expressed in termsof a polynomial of variable 119873) but the time necessary forthis calculation depends on the degree of that polynomialConsequently the time needed for calculation of Gramrsquosmatrix remains the same even if the number of samplesis increased Another important reason for using analyticalexpressions for elements of Gramrsquos matrix instead of theirinterpretation through the usual matrix multiplication can befound in the fact that rounding of exact numbers is almostunavoidable when reporting many computations and theserounding errors generally accumulate (explicit formulas forelements of Gramrsquos matrix induce smaller round-off errors inthe case of large number of samples)
The accuracy of the proposed method is compared withsome well-known methods from literature illustrating thecapability of tracking dynamic phasors especially in com-parison with other procedures based on the least squaresmethod Sinusoidal and step changes of the amplitude andphase harmonic condition frequency tracking test and com-putation time are different tests which are used to validate theproposed method according to the definition and test casesin the standard [28] The potential of the proposed approachis demonstrated by simulating various numerical signals inMATLAB The proposed algorithm is particularly suitablefor the integration of distributed generating sources withmicrogrids when fast detection of faults and the islandingcondition is required
2 Dynamic Signal Model andAlgorithm Description
The behaviour of a power system under oscillation is usuallymodelled by trigonometric series
119909 (119905) = 119872sum119898=0
119886119898 (119905) sin (119898120596119905 + 120595119898 (119905)) (1)
where 119886119898(119905) and 120595119898(119905) are the amplitude and the phasefunctions which represent variations of the mth dynamicharmonic over time and 120596 = 21205871198911 (1198911 is frequencyof fundamental component) In an equivalent form thebehaviour of such power system can be represented as
Let 119860119898(119905) and 119861119898(119905) be approximated with two Taylorseries within a short period of time near the reference time119905119903119890119891 = 0 ie
In order to perform the necessary and proposed calcu-lations for the estimation of unknown phasors parametersfrom input phasorlet samples of the processed signal arefirst transformed through the Fourier nonrecursive algorithm[29 30] After that they are introduced in the form of inputparameters to the algorithm proposed in this paper Themathematical model of the algorithm for digital filtering isdefined as
where 119909119898119891119894119897(119899) is nth filtered sample of the mth harmonicvoltage or current signal 119909((119896 + 119899)119879) is the processedsignalndashsample in the timemoment (119896+119899)119879119879 is the samplingperiod and 119873 is the number of samples 119873 = 1198911198911 (119891 =1119879) After the extraction we obtain the vector of the filteredsamples the length of 119873 on which to apply the proposedprocedure For the realization of this filter transformation theFIR structure described in [31] can be used but it was chosen(11) for the reason of obtaining a simpler and faster procedurefor estimating the processed phasors
Let us observe that for 119870 = 2 we obtain the followingsystem of equations
The traditional algorithm for estimating the value of thephasor is based on the subsystem of (12) in which onlytwo column vectors are taken into account in this waythe dynamic phasor is approximated by Tailors zero-orderpolynomial over the interval in which observations are madeThis generates a staircase function with a variable step fromone interval to another Such a model is accurate only whenthe input signal is in the stationary state This is certainly notenough in a situation where oscillations in the power systemoccur in which the first and second derivatives are as relevantas the constant term A matrix representation of this discretesystem (12) has the form
The matrix 119867119879 sdot 119867 is called the Gramrsquos matrix Thebest solution (in the least squares sense) exists providedthat the Grams matrix is invertible which is fulfilled whenthe column vectors H are linearly independent [32] TheGrammian inversion depends on the size of the interval Nand the order describing the model of the signal itself thatis the subject of processing In our approach we assume that2119898120596119873119879 = 2119904120587 for some 119904 isin N (in the case that 119904 = 1 thenumber of samples N covers the whole period)
It is clear that phasors are given by the inverse transformof the phasorlets and that the best solution is given by
Thepseudoinversematrix (119867119879 sdot119867)minus1 sdot119867119879 depends only onthe parameters of the adopted signal model The estimationof the phasors in the center of the evaluation interval whereTailors error is zero is correct if the input signal is welldescribed by the adopted model for which the least meansquared error (LMS) is also zero An LMS error would affectthe estimate if the signal was outside the projection subspaceof the LMS algorithm
3 Gramrsquos Matrix of Dynamic Signal Model
In this section we will investigate the Grams matrix of thedynamic signal model (12) and for each element of thatmatrix an explicit formula will be given
The Gramrsquos matrix 119867119879 sdot 119867 has the form
where119860 = [119886119894119895]3119894119895=1 119861 = [119887119894119895]3119894119895=1 and 119862 = [119888119894119895]3119894119895=1 and for all119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements 119886119894119895 119887119894119895 and 119888119894119895 arefinite trigonometric series
where 120577(119911) is Riemann zeta function [33 34] 120577(119911 119886) isHurvitz zeta function [35] and119867119899(119896) is generalized harmonicnumber [36] The Swiss mathematician Jacob Bernoulli
(1654-1705) derived the formula for the finite sum of powersof consecutive positive integers [37] representing 119878119901(119873) aspolynomial in 119873 of degree 119901 + 1 In the case that thepower119901 takes values in the set 0 1 4 the correspondingpolynomials are presented in Table 1
Now let us consider the finite sums involving trigonomet-ric functions
119877119901 (119873 119909) = Re 1(2i)119901 sdot 119892119901 (119890i2119909) 119868119901 (119873 119909) = Im 1(2i)119901 sdot 119892119901 (119890i2119909)
(38)
The series 119860119901(119873 119909) 119861119901(119873 119909) and 119862119901(119873 119909) can berepresented in the terms of 119878119901(119873) and 119892119901(119909) in the followingway
119860119901 (119873 119909) = 12 sdot 119878119901 (119873) minus 12 sdot Re 1(2i)119901 sdot 119892119901 (119890i2119909) (39)
Also these series can be represented in the terms of119878119901(119873) 119877119901(119873 119909) and 119868119901(119873 119909) in the following simple nota-tion
In the case that 119901 takes values in the set 0 1 4the explicit forms of functions 119892119901(119911) obtained by 119901 consec-utive derivations according to formula (35) are presented inTable 2
In the main section of this article the synchronous modelis observed ie the number 119873 is chosen with respect to therelation1198731199090 = 120587 For 119901 = 0 1 4 and 1199110 = 11989011989421199090 this leadsto the values 119864119901(119873 1199090) given in Table 3
If we represent each 1199110119904 in 119864119901(119873 1199110) in its expanded formas a polynomial in sin 1199090 and cos 1199090 of degree 119901 (the degree ofeach term in this polynomial in two variables is the sumof theexponents in each term) then 119877119901(119873 1199090) can be transformedinto the form given in Table 4
6 Mathematical Problems in Engineering
Table2Th
efun
ctions
119892 119901(119911)
119892 0(119911)
119911 119911minus1sdot(119911119873
minus1)119892 1(119911
)2i119911 (119911minus
1)2sdot(119873
119911119873+1minus(119873
+1)119911119873 +
1)119892 2(119911
)(2i)2
119911(119911minus
1)3sdot(1198732119911119873+2minus(2
1198732 +2119873
minus1)119911119873+1+(1198732+2119873
+1)119911119873 minus
119911minus1)
119892 3(119911)
(2i)3119911
(119911minus1)4
sdot(1198733119911119873+3minus(3
1198733+31198732minus3119873
+1)119911119873+2+(3
1198733+61198732minus4)
119911119873+1minus(119873
+1)3119911119873 +
1199112 +4119911+
1)119892 4(119911
)(2i)4119911
(119911minus1)5
sdot(1198734119911119873+4minus(4
1198734 +41198733
minus61198732+4119873
minus1)119911119873+3+(6
1198734+12
1198733minus61198732minus12
119873+11)119911119873+2minus(4
1198734 +121198733+6119873
2minus12
119873minus11)119911119873+1+(119873
+1)4119911119873 minus
1199113 minus111199112minus11
119911minus1)
Mathematical Problems in Engineering 7
Table 3 The values of functions 119864119901(119873 1199090)1198640(119873 1199090) 0
1198641(119873 1199090) 119902111199110122i sdot Im 119911012 11990211 = 119873
11990241 = 1198734119911119873+411990242 = minus41198734 minus 41198733 + 61198732 minus 411987311990243 = 61198734 + 121198733 minus 61198732 minus 1211987311990244 = minus41198734 minus 121198733 minus 61198732 + 12119873
In a similar way by representing each 1199110119904 in 119864119901(119873 1199110)in its expanded form as a polynomial in sin 1199090 and cos1199090 ofdegree 119901 119868119901(119873 1199090) can be transformed into the form givenin Table 5
Now by formula (42) we obtain values 119860119901(119873 1199090) 119901 =0 1 4 given in Table 6Using formula (43) we obtain values 119861119901(119873 1199090) for 119901 =0 1 4 given in Table 7Finally by (44) we obtain values 119862119901(119873 1199090) for 119901 =0 1 4 given in Table 8Now by (27)we obtain the elements ofGramsmatrix119867119879 sdot119867 Namely for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119886119894119895 are given by
11988611 = 1198732 (45)
11988612 = 11988621 = 11987911987324 (46)
11988613 = 11988622 = 11988631 = 1198792 sdot (16 (1198733 minus 119873) minus 14119873sdot ctg2119898120596119879)
(47)
11988623 = 11988632 = 1198793 sdot (18 (1198734 minus 21198732) minus 381198732 sdot ctg2119898120596119879) (48)
11988633 = 1198794 sdot ( 130 (31198735 minus 101198733 + 7119873) + 34119873sdot ctg4119898120596119879 minus 12 (1198733 minus 2119873) sdot ctg2119898120596119879)
(49)
8 Mathematical Problems in Engineering
Table 5 The values of functions 119868119901(119873 1199090)1198680(119873 1199090) 0
1198681(119873 1199090) minus1198732 sdot ctg11990901198682(119873 1199090) minus11987322 sdot ctg11990901198683(119873 1199090) 14 sdot (3119873 sdot ctg3 1199090 minus (21198733 minus 3119873) sdot ctg1199090)1198684(119873 1199090) 12 sdot (31198732 sdot ctg3 1199090 minus (1198734 minus 31198732) sdot ctg1199090)
Further for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119887119894119895 are given by
sdot (3119873 sdot ctg3119898120596119879 minus (21198733 minus 3119873) sdot ctg119898120596119879) (58)
11988833 = 11987944sdot (31198732 sdot ctg3119898120596119879 minus (1198734 minus 31198732) sdot ctg119898120596119879)
(59)
The presented representation of Gramrsquos matrix shows thatthe computation of pseudoinverse (119867119879 sdot 119867)minus1 sdot 119867119879 can beperformed more rapidly and with more precision
4 Simulation Results
The proposed phasor measurement scheme is employedto estimate coefficients of the power signal under nonsta-tionary scenarios during which we compared the resultsobtained in the proposed algorithm with some other well-known procedures Practically through the simulation testwe estimated the ability of the algorithms to track harmonicchanges Tests have been performed to show the performanceof the proposed technique with reference to the conditionsof the standard [28] for synchronized phasor measurementsystems in power systems The first is a proposed estimationprocedure verified on the basis of the following test signalwhich is characteristic for oscillating conditions
where 119886(119905) = 1198860 + 1198861 sin 2120587119891119886119905 120601(119905) = 1206010 + 1206011 sin 2120587119891120601119905 1198860 =1206010 = 1 1198861 = 1206011 = 01 119891119886 = 119891120601 = 5 and119873 = 24As can be seen from the above relations the test signal
is sampled with a frequency of 1200 Hz thus forming 24samples within a 20 ms wide window which is the period ofthe signal with a fundamental frequency of 50Hz Figures 1(a)and 1(b) show the size of the estimated amplitude and phaseof the fundamental dynamic phasor using the proposed esti-mation method It is clear that the main difference betweenthe proposed method and some other methods based on theapplication of the least square methods [15 21 27] and themethods based on the application of Kalman filters [16 1723 31] is in the occurrence of the delay in the estimation dueto the use of the data window
One of the established criteria for assessing the qualityof the estimation is based on the Total Vector Error (TVE)where we can estimate the error size in the estimation of thephasor magnitude and angle defining it as
119879119881119864 = 1003816100381610038161003816119883119903 minus 119883119890100381610038161003816100381610038161003816100381610038161198831199031003816100381610038161003816 (61)
where 119883119903 and 119883119890 are real and estimated values Figure 2shows the total vector error of the proposed method in thefirst ten cycles one cycle time delay of estimated phasor iscompensated as in [27] It can be concluded that the erroroccurring is above all the results of the introduced delaysimilar to that of [15 21] Owing to the modification madein calculating the parameters of the phasors the magnitudesof this calculated error are smaller than in other methodsbased on the least squares method with their size becomingcomparable to [16 17 23 38] therefore being much smallerthan [27 39ndash41] The proposed approach can calculate thederivatives of the phasor-phasor speed and accelerationwhich reduces the estimation error In addition it is possibleto calculate the frequency of the processed signal and detectfaults and power swings
Mathematical Problems in Engineering 9
Table 6 The values of 119860119901(119873 1199090)1198600(119873 1199090) 11987321198601(119873 1199090) 119873241198602(119873 1199090) 16 (1198733 minus 119873) minus 14119873 sdot ctg211990901198603(119873 1199090) 18 (1198734 minus 21198732) minus 381198732 sdot ctg211990901198604(119873 1199090) 130 (31198735 minus 101198733 + 7119873) + 34119873 sdot ctg41199090 minus 12 (1198733 minus 2119873) sdot ctg21199090
Table 8 The values 119862119901(119873 1199090)1198620(119873 1199090) 0
1198621(119873 1199090) minus1198734 sdot ctg11990901198622(119873 1199090) minus11987324 sdot ctg11990901198623(119873 1199090) 18 sdot (3119873 sdot ctg31199090 minus (21198733 minus 3119873) sdot ctg1199090)1198624(119873 1199090) 14 sdot (31198732 sdot ctg31199090 minus (1198734 minus 31198732) sdot ctg1199090)
41 Amplitude Oscillation Case The ability of the proposedapproach to track an arbitrary oscillating amplitude is givenin the following example where
119909 (119905) = 119886 (119905) infinsumℎ=0
119888ℎ cos 2120587ℎ1198911119905 (62)
is the input signal to the estimator with 1198911 = 50 Hz andnonzero Fourier coefficients (1198881 = 1 1198883 = 04 and 1198885 = 02)and where 119886(119905) represents an amplitude oscillation given bythe following second-order polynomial
with coefficients 119886 = minus4 119887 = 4 and 119888 = 0 Thecorresponding digital signal is sampled at 24 samplescycle(119891 = 1200Hz) The oscillation of main-original signal andits corresponding estimated signal components are shown in
Figure 3 Based on this representation it can be clearly seenthat the proposed method for estimation is fully conduciveto following the dynamics of all components of the processedinput signal Also we can see that the quality of estimationdecreases with increasing the number of harmonic which arethe results of relatively small and fixed-constant number ofsamples For higher harmonics components this number ofsamples cannot provide precise reconstruction If the numberof samples adaptively changed as the function of the orderof harmonics the accuracy of the estimation would be thesame as on the first-fundamental harmonic component Anadditional problem is certainly the nonideality of the filterfunction
42 Magnitude-Phase Step Test In this section the amplitudeand the phase step tests are performed using the proposedestimation procedureThe test signal is defined as [28] (in thestandard changes in this step are happening separately)
119909 (119905) = 119883119898 (1 + 119896119886119906 (119905 minus 119905119904)) cos (120596119905 + 119896120593119906 (119905 minus 119905119904)) (64)
where 119906(119905) is a unit step function 119896119886 is the amplitude stepsize 119896120593 is the phase step size and 119905119904 is the step instant In thistest a signal with the 10∘ phase angle step and the amplitudestep changes from 09 to 1 pu in the fundamental componentwhich is fed to the algorithm The parameters are chosen as119883119898 = radic2120596 = 2120587 sdot 50119867119911 119896119886 = +01119901119906 119896120601 = 12058718 whilethe measurement noise is set to be negligible The results forthe amplitude and phase step tests are shown in Figure 4 inwhich the upper subplot shows the transient response of theamplitude or the phase while in the lower plot the TVE areillustrated
It is observed that the proposed approach achievesrelatively long transient periods but with small overshootsBased on the results presented in Figure 4 the proposedmethod is capable of performing a precise estimation ofthe amplitude and phase after the expiry of the transitionperiod (the model is more appropriate for smooth amplitudeand phase changes) since it represents the modification ofTailorrsquos-Fourier expansion The duration of this transition inthe proposed approach is shorter than in some othermethodsthat have the same starting point [17 27] which is the resultof the position of the poles of the transfer function withrespect to the unit circle in the 119911 plane (as the poles aremore spaced) In addition there is no large overshoot in theresponse as opposed to [16 17 23] due to the amplitude andphase discontinuities at the step change since no previousassessment is performed concerning a sample window
In Table 9 the performance indices defined in the stan-dard [28] including response time delay time and overshootare listed for both amplitude and phase step change tests Theperformances achieved by the proposed algorithm fulfill theM-Class measurement requirements in the standard
43 Frequency Step Test During this simulation test a signalhaving a 5 Hz frequency step is used to evaluate the possi-bility of a transient response in a frequency step condition(although 5 Hz frequency step is not likely to happen in apower grid this test is designed based on IEEE standard [28])
119909 (119899) = cos 21205871198911119899Δ119905 0 lt 119899 lt 101198731119909 (119899) = cos 2120587 (1198911 + 5) 119899Δ119905 101198731 le 119899 (65)
Table 9 Performance indices for phasor estimation under stepchanges in amplitude and phase using the proposed estimationalgorithm
Response time Delay time OvershootAmplitude step 197ms 886ms 0008 [pu]Phase step 967ms 002 [rad]
The frequency was calculated as a ratio of roots of char-acteristic equation based on extracted unknown coefficients1198601198980 1198601198981 1198601198982 1198611198980 1198611198981 1198611198982 from (12) and 2120587 [15 27]The proposed method shows characteristics as well as theevaluation process similar to the previous type of signal(magnitude-phase step condition) Figure 5 shows that theproposed processing method can be successfully tracked as+5 Hz frequency step versus the traditional approach toprocessing the phasors
44 Harmonic Infiltration Test Input signal definedwith (66)was used to validate the proposed methods in harmoniccondition119909 (119905) = 119886 (119905) cos (1205961119905 + 120601 (119905)) + 005 cos 51205961119905 + 003 cos 71205961119905119886 (119905) = 1198860 + 1198861 sin 2120587119891119886119905120601 (119905) = 1206010 + 1206011 sin 21205871198911206011199051198860 = 1 1206010 = 05 1198861 = 01 1206011 = 005 119891119886 = 119891120601 = 5 119873 = 24
(66)
Figure 6 represents the output of the proposed method inthe harmonic condition According to Figure 6 fundamentalphasor estimation based on proposed modification is freefrom harmonic which shows the superiority of harmonicmodification of the proposed method in this paper Since119873 = 24 in this test it is possible to estimate the dynamicphasor of the first 119873 minus 1 = 23 harmonics To increase therange of harmonics the sampling number per cycle shouldbe increased
45 Frequency Response In order to evaluate the frequencycharacteristic of the proposed estimator its transfer functionis defined in the form
Figure 3 Measured and estimated waveforms in the case of a time-varying signal (62) (a) tracking the first phasors (b) tracking the 3rdphasors (c) tracking the 5th phasors and (d) measured and estimated waveforms of main input signal
12 Mathematical Problems in Engineering
0minus20 40 60 8020Time [ms]
realestim
095
1
105
11
115
12
Am
plitu
de [p
u]
(a)
0minus20 40 60 8020Time [ms]
realestim
minus01
0
01
02
03
Phas
e [ra
d]
(b)
0minus20 40 60 8020Time [ms]
0
05
1
15
2
25
3
TVE
[]
(c)
Figure 4 Estimation of (a) amplitude (b) phase and (c) TVE in percentage
minus10
0
10
Freq
uenc
y (k
Hz)
1510 205Normalized time (cycles)
Figure 5 Frequency estimation during frequency step test
where119883(119911) and 119901(119911) are 119911 transforms of the input (main sig-nal) and output (estimated dynamic phasor) The frequencyresponse of the proposed method is given in Figure 7 Themagnitude responses are unsymmetrical which is completelyunderstandable since it belongs to complex filters In thesurroundings of the fundamental frequency complete elimi-nation of the negative one and complete pass of the positiveone occur The flatter transfer characteristic causes a lowerdistortion in the estimation of the phasor and provides zero-gain in the nonfundamental component In this way the har-monics at the output are removed as opposed to [16 21 23]
46 Error Bounds In order to determine the boundary of thepossible error in the estimation of the phasor signals withvariable envelopes were used
119886 (119905) = 1 + 05 3sum119894=1
119890119905120591119894 cos 21205871198911119905 (68)
where the time constants (120591i) were generated by a uniformrandom process in the interval of [20 40] cycles In a similarway the three frequencies were randomly generated in theintervals of [1 3] [3 5] and [5 7]Hz The error is calculatedby
119903119898119904 minus 119890119903119903119900119903 = radicsum(119886 (119899) minus 119886 (119899)) (69)
where 119886(119899) and 119886(119899) are real and estimated amplitudesrespectively Figure 8 shows the histograms of the errorsattained by proposed method
The RMS error belongs to the range of 3times10minus3 and8times10minus3 for proposed estimation method This error is relatedto elimination of the third derivative term of the Taylorexpansion in phasor estimation process and it is much betterthan error estimated in [16 17 22ndash24 27]
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
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2 Mathematical Problems in Engineering
approach to estimate the parameters of the phasor based onrecursive wavelet transformation Taylor-Fourier algorithm[15] approximates the dynamic phase with the second-orderTaylor expansion and the least square observer The Taylor-Kalman method based on the Kalman observer in order toachieve nondelayed dynamic phase estimation is described in[16] This method is further enhanced by the development ofthe state space for harmonic infiltration in [17] and is calledTailor Fourier Kalman The Prony algorithm [18] was used toestimate the phasor described by the exponential amplitudeand the linear phase with no Taylor expansion There areother methods that are modifications of earlier methods [1920] The Shank method is another method for estimating adynamic phasor using the least square and consecutive delaysof the unit response system [21] The algorithm [22] for TFT(Taylor-Fourier transform) harmonic analysis uses recursivemultiple-resonator-based computational techniques whichenable the reduction of both the computational cost and thememory requirements of the algorithm
In [23] a more precise dynamic model describing thecomplex trajectory of the dynamic phasor is derived owingto a revised state transition equation of the rotating pha-sor and its derivatives Based on the improved dynamicmodel [23] a modified Taylor-Kalman Filter for instan-taneous dynamic phasor estimation is developed whichis coincident with the self-adaptive nature of the Kalmanfilter principle An extension of the Taylor Weighted LeastSquares (TWLS) algorithm for the estimation of the pha-sor frequency and ROCOF Rate-Of-Change-Of-Frequencyparameters of an electric waveform has been analyzed in[24] Paper [25] introduces a dynamic phasor estimationmethod for PMUsIEDs based on a modified time domainhybridmethodDynamic phasor is estimated using theTaylorexpansion where frequency deviation is derived directlyfrom fitting parameters to avoid magnification of fittingerrors A double suboptimal-scaling factor-adaptive strong-tracking Kalman filter (DSTKF)-based phasor measurementunit algorithm which can meet the accuracy requirement ofthe IEEE standard C371181 under the dynamic condition wasproposed in [26] This method uses a kth Taylor polynomialto linearize the complex exponential of the signal modeland estimates the dynamic phasor using DSTKF In [27] acombination of the least square-based Prony analysis andTaylor expansion called TaylorndashProny is proposed to estimatethe dynamic phasor
In this paper a special transformation (representation) ofTaylor-Fourier expansion and corresponding Gram matrixis used to estimate the dynamic phasors from input pha-sorlets in real domain based on synchronously sampledvalues of the input signal The dynamic systemrsquos behaviouris described in algebra of matrices related to their Taylor-Fourier-trigonometric series representation The explicit for-mulas for calculation of elements of the associated Gramrsquosmatrix are given In this way we are in a position to improvethe computational performance of the proposed algorithm asregards precision and speed Namely the proposed algorithmis based on matrix multiplication It is a well-known factthat mathematical definition of multiplication of matricesgives an algorithm that takes 119901119902119903 time to multiply a matrix
of type 119901 times 119902 with a matrix of type 119902 times 119903 (in the case ofsquare 119901 times 119901-matrices it takes 1199013 time) If we use the explicitformulas for elements of Gramrsquos matrix in our algorithmthen one matrix multiplication is avoided and the task isreduced to 119901119903 consecutive calculations of the values of spe-cific polynomials Moreover each element of Gramrsquos matrixdepends on the number of samples119873 (it is expressed in termsof a polynomial of variable 119873) but the time necessary forthis calculation depends on the degree of that polynomialConsequently the time needed for calculation of Gramrsquosmatrix remains the same even if the number of samplesis increased Another important reason for using analyticalexpressions for elements of Gramrsquos matrix instead of theirinterpretation through the usual matrix multiplication can befound in the fact that rounding of exact numbers is almostunavoidable when reporting many computations and theserounding errors generally accumulate (explicit formulas forelements of Gramrsquos matrix induce smaller round-off errors inthe case of large number of samples)
The accuracy of the proposed method is compared withsome well-known methods from literature illustrating thecapability of tracking dynamic phasors especially in com-parison with other procedures based on the least squaresmethod Sinusoidal and step changes of the amplitude andphase harmonic condition frequency tracking test and com-putation time are different tests which are used to validate theproposed method according to the definition and test casesin the standard [28] The potential of the proposed approachis demonstrated by simulating various numerical signals inMATLAB The proposed algorithm is particularly suitablefor the integration of distributed generating sources withmicrogrids when fast detection of faults and the islandingcondition is required
2 Dynamic Signal Model andAlgorithm Description
The behaviour of a power system under oscillation is usuallymodelled by trigonometric series
119909 (119905) = 119872sum119898=0
119886119898 (119905) sin (119898120596119905 + 120595119898 (119905)) (1)
where 119886119898(119905) and 120595119898(119905) are the amplitude and the phasefunctions which represent variations of the mth dynamicharmonic over time and 120596 = 21205871198911 (1198911 is frequencyof fundamental component) In an equivalent form thebehaviour of such power system can be represented as
Let 119860119898(119905) and 119861119898(119905) be approximated with two Taylorseries within a short period of time near the reference time119905119903119890119891 = 0 ie
In order to perform the necessary and proposed calcu-lations for the estimation of unknown phasors parametersfrom input phasorlet samples of the processed signal arefirst transformed through the Fourier nonrecursive algorithm[29 30] After that they are introduced in the form of inputparameters to the algorithm proposed in this paper Themathematical model of the algorithm for digital filtering isdefined as
where 119909119898119891119894119897(119899) is nth filtered sample of the mth harmonicvoltage or current signal 119909((119896 + 119899)119879) is the processedsignalndashsample in the timemoment (119896+119899)119879119879 is the samplingperiod and 119873 is the number of samples 119873 = 1198911198911 (119891 =1119879) After the extraction we obtain the vector of the filteredsamples the length of 119873 on which to apply the proposedprocedure For the realization of this filter transformation theFIR structure described in [31] can be used but it was chosen(11) for the reason of obtaining a simpler and faster procedurefor estimating the processed phasors
Let us observe that for 119870 = 2 we obtain the followingsystem of equations
The traditional algorithm for estimating the value of thephasor is based on the subsystem of (12) in which onlytwo column vectors are taken into account in this waythe dynamic phasor is approximated by Tailors zero-orderpolynomial over the interval in which observations are madeThis generates a staircase function with a variable step fromone interval to another Such a model is accurate only whenthe input signal is in the stationary state This is certainly notenough in a situation where oscillations in the power systemoccur in which the first and second derivatives are as relevantas the constant term A matrix representation of this discretesystem (12) has the form
The matrix 119867119879 sdot 119867 is called the Gramrsquos matrix Thebest solution (in the least squares sense) exists providedthat the Grams matrix is invertible which is fulfilled whenthe column vectors H are linearly independent [32] TheGrammian inversion depends on the size of the interval Nand the order describing the model of the signal itself thatis the subject of processing In our approach we assume that2119898120596119873119879 = 2119904120587 for some 119904 isin N (in the case that 119904 = 1 thenumber of samples N covers the whole period)
It is clear that phasors are given by the inverse transformof the phasorlets and that the best solution is given by
Thepseudoinversematrix (119867119879 sdot119867)minus1 sdot119867119879 depends only onthe parameters of the adopted signal model The estimationof the phasors in the center of the evaluation interval whereTailors error is zero is correct if the input signal is welldescribed by the adopted model for which the least meansquared error (LMS) is also zero An LMS error would affectthe estimate if the signal was outside the projection subspaceof the LMS algorithm
3 Gramrsquos Matrix of Dynamic Signal Model
In this section we will investigate the Grams matrix of thedynamic signal model (12) and for each element of thatmatrix an explicit formula will be given
The Gramrsquos matrix 119867119879 sdot 119867 has the form
where119860 = [119886119894119895]3119894119895=1 119861 = [119887119894119895]3119894119895=1 and 119862 = [119888119894119895]3119894119895=1 and for all119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements 119886119894119895 119887119894119895 and 119888119894119895 arefinite trigonometric series
where 120577(119911) is Riemann zeta function [33 34] 120577(119911 119886) isHurvitz zeta function [35] and119867119899(119896) is generalized harmonicnumber [36] The Swiss mathematician Jacob Bernoulli
(1654-1705) derived the formula for the finite sum of powersof consecutive positive integers [37] representing 119878119901(119873) aspolynomial in 119873 of degree 119901 + 1 In the case that thepower119901 takes values in the set 0 1 4 the correspondingpolynomials are presented in Table 1
Now let us consider the finite sums involving trigonomet-ric functions
119877119901 (119873 119909) = Re 1(2i)119901 sdot 119892119901 (119890i2119909) 119868119901 (119873 119909) = Im 1(2i)119901 sdot 119892119901 (119890i2119909)
(38)
The series 119860119901(119873 119909) 119861119901(119873 119909) and 119862119901(119873 119909) can berepresented in the terms of 119878119901(119873) and 119892119901(119909) in the followingway
119860119901 (119873 119909) = 12 sdot 119878119901 (119873) minus 12 sdot Re 1(2i)119901 sdot 119892119901 (119890i2119909) (39)
Also these series can be represented in the terms of119878119901(119873) 119877119901(119873 119909) and 119868119901(119873 119909) in the following simple nota-tion
In the case that 119901 takes values in the set 0 1 4the explicit forms of functions 119892119901(119911) obtained by 119901 consec-utive derivations according to formula (35) are presented inTable 2
In the main section of this article the synchronous modelis observed ie the number 119873 is chosen with respect to therelation1198731199090 = 120587 For 119901 = 0 1 4 and 1199110 = 11989011989421199090 this leadsto the values 119864119901(119873 1199090) given in Table 3
If we represent each 1199110119904 in 119864119901(119873 1199110) in its expanded formas a polynomial in sin 1199090 and cos 1199090 of degree 119901 (the degree ofeach term in this polynomial in two variables is the sumof theexponents in each term) then 119877119901(119873 1199090) can be transformedinto the form given in Table 4
6 Mathematical Problems in Engineering
Table2Th
efun
ctions
119892 119901(119911)
119892 0(119911)
119911 119911minus1sdot(119911119873
minus1)119892 1(119911
)2i119911 (119911minus
1)2sdot(119873
119911119873+1minus(119873
+1)119911119873 +
1)119892 2(119911
)(2i)2
119911(119911minus
1)3sdot(1198732119911119873+2minus(2
1198732 +2119873
minus1)119911119873+1+(1198732+2119873
+1)119911119873 minus
119911minus1)
119892 3(119911)
(2i)3119911
(119911minus1)4
sdot(1198733119911119873+3minus(3
1198733+31198732minus3119873
+1)119911119873+2+(3
1198733+61198732minus4)
119911119873+1minus(119873
+1)3119911119873 +
1199112 +4119911+
1)119892 4(119911
)(2i)4119911
(119911minus1)5
sdot(1198734119911119873+4minus(4
1198734 +41198733
minus61198732+4119873
minus1)119911119873+3+(6
1198734+12
1198733minus61198732minus12
119873+11)119911119873+2minus(4
1198734 +121198733+6119873
2minus12
119873minus11)119911119873+1+(119873
+1)4119911119873 minus
1199113 minus111199112minus11
119911minus1)
Mathematical Problems in Engineering 7
Table 3 The values of functions 119864119901(119873 1199090)1198640(119873 1199090) 0
1198641(119873 1199090) 119902111199110122i sdot Im 119911012 11990211 = 119873
11990241 = 1198734119911119873+411990242 = minus41198734 minus 41198733 + 61198732 minus 411987311990243 = 61198734 + 121198733 minus 61198732 minus 1211987311990244 = minus41198734 minus 121198733 minus 61198732 + 12119873
In a similar way by representing each 1199110119904 in 119864119901(119873 1199110)in its expanded form as a polynomial in sin 1199090 and cos1199090 ofdegree 119901 119868119901(119873 1199090) can be transformed into the form givenin Table 5
Now by formula (42) we obtain values 119860119901(119873 1199090) 119901 =0 1 4 given in Table 6Using formula (43) we obtain values 119861119901(119873 1199090) for 119901 =0 1 4 given in Table 7Finally by (44) we obtain values 119862119901(119873 1199090) for 119901 =0 1 4 given in Table 8Now by (27)we obtain the elements ofGramsmatrix119867119879 sdot119867 Namely for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119886119894119895 are given by
11988611 = 1198732 (45)
11988612 = 11988621 = 11987911987324 (46)
11988613 = 11988622 = 11988631 = 1198792 sdot (16 (1198733 minus 119873) minus 14119873sdot ctg2119898120596119879)
(47)
11988623 = 11988632 = 1198793 sdot (18 (1198734 minus 21198732) minus 381198732 sdot ctg2119898120596119879) (48)
11988633 = 1198794 sdot ( 130 (31198735 minus 101198733 + 7119873) + 34119873sdot ctg4119898120596119879 minus 12 (1198733 minus 2119873) sdot ctg2119898120596119879)
(49)
8 Mathematical Problems in Engineering
Table 5 The values of functions 119868119901(119873 1199090)1198680(119873 1199090) 0
1198681(119873 1199090) minus1198732 sdot ctg11990901198682(119873 1199090) minus11987322 sdot ctg11990901198683(119873 1199090) 14 sdot (3119873 sdot ctg3 1199090 minus (21198733 minus 3119873) sdot ctg1199090)1198684(119873 1199090) 12 sdot (31198732 sdot ctg3 1199090 minus (1198734 minus 31198732) sdot ctg1199090)
Further for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119887119894119895 are given by
sdot (3119873 sdot ctg3119898120596119879 minus (21198733 minus 3119873) sdot ctg119898120596119879) (58)
11988833 = 11987944sdot (31198732 sdot ctg3119898120596119879 minus (1198734 minus 31198732) sdot ctg119898120596119879)
(59)
The presented representation of Gramrsquos matrix shows thatthe computation of pseudoinverse (119867119879 sdot 119867)minus1 sdot 119867119879 can beperformed more rapidly and with more precision
4 Simulation Results
The proposed phasor measurement scheme is employedto estimate coefficients of the power signal under nonsta-tionary scenarios during which we compared the resultsobtained in the proposed algorithm with some other well-known procedures Practically through the simulation testwe estimated the ability of the algorithms to track harmonicchanges Tests have been performed to show the performanceof the proposed technique with reference to the conditionsof the standard [28] for synchronized phasor measurementsystems in power systems The first is a proposed estimationprocedure verified on the basis of the following test signalwhich is characteristic for oscillating conditions
where 119886(119905) = 1198860 + 1198861 sin 2120587119891119886119905 120601(119905) = 1206010 + 1206011 sin 2120587119891120601119905 1198860 =1206010 = 1 1198861 = 1206011 = 01 119891119886 = 119891120601 = 5 and119873 = 24As can be seen from the above relations the test signal
is sampled with a frequency of 1200 Hz thus forming 24samples within a 20 ms wide window which is the period ofthe signal with a fundamental frequency of 50Hz Figures 1(a)and 1(b) show the size of the estimated amplitude and phaseof the fundamental dynamic phasor using the proposed esti-mation method It is clear that the main difference betweenthe proposed method and some other methods based on theapplication of the least square methods [15 21 27] and themethods based on the application of Kalman filters [16 1723 31] is in the occurrence of the delay in the estimation dueto the use of the data window
One of the established criteria for assessing the qualityof the estimation is based on the Total Vector Error (TVE)where we can estimate the error size in the estimation of thephasor magnitude and angle defining it as
119879119881119864 = 1003816100381610038161003816119883119903 minus 119883119890100381610038161003816100381610038161003816100381610038161198831199031003816100381610038161003816 (61)
where 119883119903 and 119883119890 are real and estimated values Figure 2shows the total vector error of the proposed method in thefirst ten cycles one cycle time delay of estimated phasor iscompensated as in [27] It can be concluded that the erroroccurring is above all the results of the introduced delaysimilar to that of [15 21] Owing to the modification madein calculating the parameters of the phasors the magnitudesof this calculated error are smaller than in other methodsbased on the least squares method with their size becomingcomparable to [16 17 23 38] therefore being much smallerthan [27 39ndash41] The proposed approach can calculate thederivatives of the phasor-phasor speed and accelerationwhich reduces the estimation error In addition it is possibleto calculate the frequency of the processed signal and detectfaults and power swings
Mathematical Problems in Engineering 9
Table 6 The values of 119860119901(119873 1199090)1198600(119873 1199090) 11987321198601(119873 1199090) 119873241198602(119873 1199090) 16 (1198733 minus 119873) minus 14119873 sdot ctg211990901198603(119873 1199090) 18 (1198734 minus 21198732) minus 381198732 sdot ctg211990901198604(119873 1199090) 130 (31198735 minus 101198733 + 7119873) + 34119873 sdot ctg41199090 minus 12 (1198733 minus 2119873) sdot ctg21199090
Table 8 The values 119862119901(119873 1199090)1198620(119873 1199090) 0
1198621(119873 1199090) minus1198734 sdot ctg11990901198622(119873 1199090) minus11987324 sdot ctg11990901198623(119873 1199090) 18 sdot (3119873 sdot ctg31199090 minus (21198733 minus 3119873) sdot ctg1199090)1198624(119873 1199090) 14 sdot (31198732 sdot ctg31199090 minus (1198734 minus 31198732) sdot ctg1199090)
41 Amplitude Oscillation Case The ability of the proposedapproach to track an arbitrary oscillating amplitude is givenin the following example where
119909 (119905) = 119886 (119905) infinsumℎ=0
119888ℎ cos 2120587ℎ1198911119905 (62)
is the input signal to the estimator with 1198911 = 50 Hz andnonzero Fourier coefficients (1198881 = 1 1198883 = 04 and 1198885 = 02)and where 119886(119905) represents an amplitude oscillation given bythe following second-order polynomial
with coefficients 119886 = minus4 119887 = 4 and 119888 = 0 Thecorresponding digital signal is sampled at 24 samplescycle(119891 = 1200Hz) The oscillation of main-original signal andits corresponding estimated signal components are shown in
Figure 3 Based on this representation it can be clearly seenthat the proposed method for estimation is fully conduciveto following the dynamics of all components of the processedinput signal Also we can see that the quality of estimationdecreases with increasing the number of harmonic which arethe results of relatively small and fixed-constant number ofsamples For higher harmonics components this number ofsamples cannot provide precise reconstruction If the numberof samples adaptively changed as the function of the orderof harmonics the accuracy of the estimation would be thesame as on the first-fundamental harmonic component Anadditional problem is certainly the nonideality of the filterfunction
42 Magnitude-Phase Step Test In this section the amplitudeand the phase step tests are performed using the proposedestimation procedureThe test signal is defined as [28] (in thestandard changes in this step are happening separately)
119909 (119905) = 119883119898 (1 + 119896119886119906 (119905 minus 119905119904)) cos (120596119905 + 119896120593119906 (119905 minus 119905119904)) (64)
where 119906(119905) is a unit step function 119896119886 is the amplitude stepsize 119896120593 is the phase step size and 119905119904 is the step instant In thistest a signal with the 10∘ phase angle step and the amplitudestep changes from 09 to 1 pu in the fundamental componentwhich is fed to the algorithm The parameters are chosen as119883119898 = radic2120596 = 2120587 sdot 50119867119911 119896119886 = +01119901119906 119896120601 = 12058718 whilethe measurement noise is set to be negligible The results forthe amplitude and phase step tests are shown in Figure 4 inwhich the upper subplot shows the transient response of theamplitude or the phase while in the lower plot the TVE areillustrated
It is observed that the proposed approach achievesrelatively long transient periods but with small overshootsBased on the results presented in Figure 4 the proposedmethod is capable of performing a precise estimation ofthe amplitude and phase after the expiry of the transitionperiod (the model is more appropriate for smooth amplitudeand phase changes) since it represents the modification ofTailorrsquos-Fourier expansion The duration of this transition inthe proposed approach is shorter than in some othermethodsthat have the same starting point [17 27] which is the resultof the position of the poles of the transfer function withrespect to the unit circle in the 119911 plane (as the poles aremore spaced) In addition there is no large overshoot in theresponse as opposed to [16 17 23] due to the amplitude andphase discontinuities at the step change since no previousassessment is performed concerning a sample window
In Table 9 the performance indices defined in the stan-dard [28] including response time delay time and overshootare listed for both amplitude and phase step change tests Theperformances achieved by the proposed algorithm fulfill theM-Class measurement requirements in the standard
43 Frequency Step Test During this simulation test a signalhaving a 5 Hz frequency step is used to evaluate the possi-bility of a transient response in a frequency step condition(although 5 Hz frequency step is not likely to happen in apower grid this test is designed based on IEEE standard [28])
119909 (119899) = cos 21205871198911119899Δ119905 0 lt 119899 lt 101198731119909 (119899) = cos 2120587 (1198911 + 5) 119899Δ119905 101198731 le 119899 (65)
Table 9 Performance indices for phasor estimation under stepchanges in amplitude and phase using the proposed estimationalgorithm
Response time Delay time OvershootAmplitude step 197ms 886ms 0008 [pu]Phase step 967ms 002 [rad]
The frequency was calculated as a ratio of roots of char-acteristic equation based on extracted unknown coefficients1198601198980 1198601198981 1198601198982 1198611198980 1198611198981 1198611198982 from (12) and 2120587 [15 27]The proposed method shows characteristics as well as theevaluation process similar to the previous type of signal(magnitude-phase step condition) Figure 5 shows that theproposed processing method can be successfully tracked as+5 Hz frequency step versus the traditional approach toprocessing the phasors
44 Harmonic Infiltration Test Input signal definedwith (66)was used to validate the proposed methods in harmoniccondition119909 (119905) = 119886 (119905) cos (1205961119905 + 120601 (119905)) + 005 cos 51205961119905 + 003 cos 71205961119905119886 (119905) = 1198860 + 1198861 sin 2120587119891119886119905120601 (119905) = 1206010 + 1206011 sin 21205871198911206011199051198860 = 1 1206010 = 05 1198861 = 01 1206011 = 005 119891119886 = 119891120601 = 5 119873 = 24
(66)
Figure 6 represents the output of the proposed method inthe harmonic condition According to Figure 6 fundamentalphasor estimation based on proposed modification is freefrom harmonic which shows the superiority of harmonicmodification of the proposed method in this paper Since119873 = 24 in this test it is possible to estimate the dynamicphasor of the first 119873 minus 1 = 23 harmonics To increase therange of harmonics the sampling number per cycle shouldbe increased
45 Frequency Response In order to evaluate the frequencycharacteristic of the proposed estimator its transfer functionis defined in the form
Figure 3 Measured and estimated waveforms in the case of a time-varying signal (62) (a) tracking the first phasors (b) tracking the 3rdphasors (c) tracking the 5th phasors and (d) measured and estimated waveforms of main input signal
12 Mathematical Problems in Engineering
0minus20 40 60 8020Time [ms]
realestim
095
1
105
11
115
12
Am
plitu
de [p
u]
(a)
0minus20 40 60 8020Time [ms]
realestim
minus01
0
01
02
03
Phas
e [ra
d]
(b)
0minus20 40 60 8020Time [ms]
0
05
1
15
2
25
3
TVE
[]
(c)
Figure 4 Estimation of (a) amplitude (b) phase and (c) TVE in percentage
minus10
0
10
Freq
uenc
y (k
Hz)
1510 205Normalized time (cycles)
Figure 5 Frequency estimation during frequency step test
where119883(119911) and 119901(119911) are 119911 transforms of the input (main sig-nal) and output (estimated dynamic phasor) The frequencyresponse of the proposed method is given in Figure 7 Themagnitude responses are unsymmetrical which is completelyunderstandable since it belongs to complex filters In thesurroundings of the fundamental frequency complete elimi-nation of the negative one and complete pass of the positiveone occur The flatter transfer characteristic causes a lowerdistortion in the estimation of the phasor and provides zero-gain in the nonfundamental component In this way the har-monics at the output are removed as opposed to [16 21 23]
46 Error Bounds In order to determine the boundary of thepossible error in the estimation of the phasor signals withvariable envelopes were used
119886 (119905) = 1 + 05 3sum119894=1
119890119905120591119894 cos 21205871198911119905 (68)
where the time constants (120591i) were generated by a uniformrandom process in the interval of [20 40] cycles In a similarway the three frequencies were randomly generated in theintervals of [1 3] [3 5] and [5 7]Hz The error is calculatedby
119903119898119904 minus 119890119903119903119900119903 = radicsum(119886 (119899) minus 119886 (119899)) (69)
where 119886(119899) and 119886(119899) are real and estimated amplitudesrespectively Figure 8 shows the histograms of the errorsattained by proposed method
The RMS error belongs to the range of 3times10minus3 and8times10minus3 for proposed estimation method This error is relatedto elimination of the third derivative term of the Taylorexpansion in phasor estimation process and it is much betterthan error estimated in [16 17 22ndash24 27]
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
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Probability and StatisticsHindawiwwwhindawicom Volume 2018
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Engineering Mathematics
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Operations ResearchAdvances in
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Let 119860119898(119905) and 119861119898(119905) be approximated with two Taylorseries within a short period of time near the reference time119905119903119890119891 = 0 ie
In order to perform the necessary and proposed calcu-lations for the estimation of unknown phasors parametersfrom input phasorlet samples of the processed signal arefirst transformed through the Fourier nonrecursive algorithm[29 30] After that they are introduced in the form of inputparameters to the algorithm proposed in this paper Themathematical model of the algorithm for digital filtering isdefined as
where 119909119898119891119894119897(119899) is nth filtered sample of the mth harmonicvoltage or current signal 119909((119896 + 119899)119879) is the processedsignalndashsample in the timemoment (119896+119899)119879119879 is the samplingperiod and 119873 is the number of samples 119873 = 1198911198911 (119891 =1119879) After the extraction we obtain the vector of the filteredsamples the length of 119873 on which to apply the proposedprocedure For the realization of this filter transformation theFIR structure described in [31] can be used but it was chosen(11) for the reason of obtaining a simpler and faster procedurefor estimating the processed phasors
Let us observe that for 119870 = 2 we obtain the followingsystem of equations
The traditional algorithm for estimating the value of thephasor is based on the subsystem of (12) in which onlytwo column vectors are taken into account in this waythe dynamic phasor is approximated by Tailors zero-orderpolynomial over the interval in which observations are madeThis generates a staircase function with a variable step fromone interval to another Such a model is accurate only whenthe input signal is in the stationary state This is certainly notenough in a situation where oscillations in the power systemoccur in which the first and second derivatives are as relevantas the constant term A matrix representation of this discretesystem (12) has the form
The matrix 119867119879 sdot 119867 is called the Gramrsquos matrix Thebest solution (in the least squares sense) exists providedthat the Grams matrix is invertible which is fulfilled whenthe column vectors H are linearly independent [32] TheGrammian inversion depends on the size of the interval Nand the order describing the model of the signal itself thatis the subject of processing In our approach we assume that2119898120596119873119879 = 2119904120587 for some 119904 isin N (in the case that 119904 = 1 thenumber of samples N covers the whole period)
It is clear that phasors are given by the inverse transformof the phasorlets and that the best solution is given by
Thepseudoinversematrix (119867119879 sdot119867)minus1 sdot119867119879 depends only onthe parameters of the adopted signal model The estimationof the phasors in the center of the evaluation interval whereTailors error is zero is correct if the input signal is welldescribed by the adopted model for which the least meansquared error (LMS) is also zero An LMS error would affectthe estimate if the signal was outside the projection subspaceof the LMS algorithm
3 Gramrsquos Matrix of Dynamic Signal Model
In this section we will investigate the Grams matrix of thedynamic signal model (12) and for each element of thatmatrix an explicit formula will be given
The Gramrsquos matrix 119867119879 sdot 119867 has the form
where119860 = [119886119894119895]3119894119895=1 119861 = [119887119894119895]3119894119895=1 and 119862 = [119888119894119895]3119894119895=1 and for all119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements 119886119894119895 119887119894119895 and 119888119894119895 arefinite trigonometric series
where 120577(119911) is Riemann zeta function [33 34] 120577(119911 119886) isHurvitz zeta function [35] and119867119899(119896) is generalized harmonicnumber [36] The Swiss mathematician Jacob Bernoulli
(1654-1705) derived the formula for the finite sum of powersof consecutive positive integers [37] representing 119878119901(119873) aspolynomial in 119873 of degree 119901 + 1 In the case that thepower119901 takes values in the set 0 1 4 the correspondingpolynomials are presented in Table 1
Now let us consider the finite sums involving trigonomet-ric functions
119877119901 (119873 119909) = Re 1(2i)119901 sdot 119892119901 (119890i2119909) 119868119901 (119873 119909) = Im 1(2i)119901 sdot 119892119901 (119890i2119909)
(38)
The series 119860119901(119873 119909) 119861119901(119873 119909) and 119862119901(119873 119909) can berepresented in the terms of 119878119901(119873) and 119892119901(119909) in the followingway
119860119901 (119873 119909) = 12 sdot 119878119901 (119873) minus 12 sdot Re 1(2i)119901 sdot 119892119901 (119890i2119909) (39)
Also these series can be represented in the terms of119878119901(119873) 119877119901(119873 119909) and 119868119901(119873 119909) in the following simple nota-tion
In the case that 119901 takes values in the set 0 1 4the explicit forms of functions 119892119901(119911) obtained by 119901 consec-utive derivations according to formula (35) are presented inTable 2
In the main section of this article the synchronous modelis observed ie the number 119873 is chosen with respect to therelation1198731199090 = 120587 For 119901 = 0 1 4 and 1199110 = 11989011989421199090 this leadsto the values 119864119901(119873 1199090) given in Table 3
If we represent each 1199110119904 in 119864119901(119873 1199110) in its expanded formas a polynomial in sin 1199090 and cos 1199090 of degree 119901 (the degree ofeach term in this polynomial in two variables is the sumof theexponents in each term) then 119877119901(119873 1199090) can be transformedinto the form given in Table 4
6 Mathematical Problems in Engineering
Table2Th
efun
ctions
119892 119901(119911)
119892 0(119911)
119911 119911minus1sdot(119911119873
minus1)119892 1(119911
)2i119911 (119911minus
1)2sdot(119873
119911119873+1minus(119873
+1)119911119873 +
1)119892 2(119911
)(2i)2
119911(119911minus
1)3sdot(1198732119911119873+2minus(2
1198732 +2119873
minus1)119911119873+1+(1198732+2119873
+1)119911119873 minus
119911minus1)
119892 3(119911)
(2i)3119911
(119911minus1)4
sdot(1198733119911119873+3minus(3
1198733+31198732minus3119873
+1)119911119873+2+(3
1198733+61198732minus4)
119911119873+1minus(119873
+1)3119911119873 +
1199112 +4119911+
1)119892 4(119911
)(2i)4119911
(119911minus1)5
sdot(1198734119911119873+4minus(4
1198734 +41198733
minus61198732+4119873
minus1)119911119873+3+(6
1198734+12
1198733minus61198732minus12
119873+11)119911119873+2minus(4
1198734 +121198733+6119873
2minus12
119873minus11)119911119873+1+(119873
+1)4119911119873 minus
1199113 minus111199112minus11
119911minus1)
Mathematical Problems in Engineering 7
Table 3 The values of functions 119864119901(119873 1199090)1198640(119873 1199090) 0
1198641(119873 1199090) 119902111199110122i sdot Im 119911012 11990211 = 119873
11990241 = 1198734119911119873+411990242 = minus41198734 minus 41198733 + 61198732 minus 411987311990243 = 61198734 + 121198733 minus 61198732 minus 1211987311990244 = minus41198734 minus 121198733 minus 61198732 + 12119873
In a similar way by representing each 1199110119904 in 119864119901(119873 1199110)in its expanded form as a polynomial in sin 1199090 and cos1199090 ofdegree 119901 119868119901(119873 1199090) can be transformed into the form givenin Table 5
Now by formula (42) we obtain values 119860119901(119873 1199090) 119901 =0 1 4 given in Table 6Using formula (43) we obtain values 119861119901(119873 1199090) for 119901 =0 1 4 given in Table 7Finally by (44) we obtain values 119862119901(119873 1199090) for 119901 =0 1 4 given in Table 8Now by (27)we obtain the elements ofGramsmatrix119867119879 sdot119867 Namely for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119886119894119895 are given by
11988611 = 1198732 (45)
11988612 = 11988621 = 11987911987324 (46)
11988613 = 11988622 = 11988631 = 1198792 sdot (16 (1198733 minus 119873) minus 14119873sdot ctg2119898120596119879)
(47)
11988623 = 11988632 = 1198793 sdot (18 (1198734 minus 21198732) minus 381198732 sdot ctg2119898120596119879) (48)
11988633 = 1198794 sdot ( 130 (31198735 minus 101198733 + 7119873) + 34119873sdot ctg4119898120596119879 minus 12 (1198733 minus 2119873) sdot ctg2119898120596119879)
(49)
8 Mathematical Problems in Engineering
Table 5 The values of functions 119868119901(119873 1199090)1198680(119873 1199090) 0
1198681(119873 1199090) minus1198732 sdot ctg11990901198682(119873 1199090) minus11987322 sdot ctg11990901198683(119873 1199090) 14 sdot (3119873 sdot ctg3 1199090 minus (21198733 minus 3119873) sdot ctg1199090)1198684(119873 1199090) 12 sdot (31198732 sdot ctg3 1199090 minus (1198734 minus 31198732) sdot ctg1199090)
Further for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119887119894119895 are given by
sdot (3119873 sdot ctg3119898120596119879 minus (21198733 minus 3119873) sdot ctg119898120596119879) (58)
11988833 = 11987944sdot (31198732 sdot ctg3119898120596119879 minus (1198734 minus 31198732) sdot ctg119898120596119879)
(59)
The presented representation of Gramrsquos matrix shows thatthe computation of pseudoinverse (119867119879 sdot 119867)minus1 sdot 119867119879 can beperformed more rapidly and with more precision
4 Simulation Results
The proposed phasor measurement scheme is employedto estimate coefficients of the power signal under nonsta-tionary scenarios during which we compared the resultsobtained in the proposed algorithm with some other well-known procedures Practically through the simulation testwe estimated the ability of the algorithms to track harmonicchanges Tests have been performed to show the performanceof the proposed technique with reference to the conditionsof the standard [28] for synchronized phasor measurementsystems in power systems The first is a proposed estimationprocedure verified on the basis of the following test signalwhich is characteristic for oscillating conditions
where 119886(119905) = 1198860 + 1198861 sin 2120587119891119886119905 120601(119905) = 1206010 + 1206011 sin 2120587119891120601119905 1198860 =1206010 = 1 1198861 = 1206011 = 01 119891119886 = 119891120601 = 5 and119873 = 24As can be seen from the above relations the test signal
is sampled with a frequency of 1200 Hz thus forming 24samples within a 20 ms wide window which is the period ofthe signal with a fundamental frequency of 50Hz Figures 1(a)and 1(b) show the size of the estimated amplitude and phaseof the fundamental dynamic phasor using the proposed esti-mation method It is clear that the main difference betweenthe proposed method and some other methods based on theapplication of the least square methods [15 21 27] and themethods based on the application of Kalman filters [16 1723 31] is in the occurrence of the delay in the estimation dueto the use of the data window
One of the established criteria for assessing the qualityof the estimation is based on the Total Vector Error (TVE)where we can estimate the error size in the estimation of thephasor magnitude and angle defining it as
119879119881119864 = 1003816100381610038161003816119883119903 minus 119883119890100381610038161003816100381610038161003816100381610038161198831199031003816100381610038161003816 (61)
where 119883119903 and 119883119890 are real and estimated values Figure 2shows the total vector error of the proposed method in thefirst ten cycles one cycle time delay of estimated phasor iscompensated as in [27] It can be concluded that the erroroccurring is above all the results of the introduced delaysimilar to that of [15 21] Owing to the modification madein calculating the parameters of the phasors the magnitudesof this calculated error are smaller than in other methodsbased on the least squares method with their size becomingcomparable to [16 17 23 38] therefore being much smallerthan [27 39ndash41] The proposed approach can calculate thederivatives of the phasor-phasor speed and accelerationwhich reduces the estimation error In addition it is possibleto calculate the frequency of the processed signal and detectfaults and power swings
Mathematical Problems in Engineering 9
Table 6 The values of 119860119901(119873 1199090)1198600(119873 1199090) 11987321198601(119873 1199090) 119873241198602(119873 1199090) 16 (1198733 minus 119873) minus 14119873 sdot ctg211990901198603(119873 1199090) 18 (1198734 minus 21198732) minus 381198732 sdot ctg211990901198604(119873 1199090) 130 (31198735 minus 101198733 + 7119873) + 34119873 sdot ctg41199090 minus 12 (1198733 minus 2119873) sdot ctg21199090
Table 8 The values 119862119901(119873 1199090)1198620(119873 1199090) 0
1198621(119873 1199090) minus1198734 sdot ctg11990901198622(119873 1199090) minus11987324 sdot ctg11990901198623(119873 1199090) 18 sdot (3119873 sdot ctg31199090 minus (21198733 minus 3119873) sdot ctg1199090)1198624(119873 1199090) 14 sdot (31198732 sdot ctg31199090 minus (1198734 minus 31198732) sdot ctg1199090)
41 Amplitude Oscillation Case The ability of the proposedapproach to track an arbitrary oscillating amplitude is givenin the following example where
119909 (119905) = 119886 (119905) infinsumℎ=0
119888ℎ cos 2120587ℎ1198911119905 (62)
is the input signal to the estimator with 1198911 = 50 Hz andnonzero Fourier coefficients (1198881 = 1 1198883 = 04 and 1198885 = 02)and where 119886(119905) represents an amplitude oscillation given bythe following second-order polynomial
with coefficients 119886 = minus4 119887 = 4 and 119888 = 0 Thecorresponding digital signal is sampled at 24 samplescycle(119891 = 1200Hz) The oscillation of main-original signal andits corresponding estimated signal components are shown in
Figure 3 Based on this representation it can be clearly seenthat the proposed method for estimation is fully conduciveto following the dynamics of all components of the processedinput signal Also we can see that the quality of estimationdecreases with increasing the number of harmonic which arethe results of relatively small and fixed-constant number ofsamples For higher harmonics components this number ofsamples cannot provide precise reconstruction If the numberof samples adaptively changed as the function of the orderof harmonics the accuracy of the estimation would be thesame as on the first-fundamental harmonic component Anadditional problem is certainly the nonideality of the filterfunction
42 Magnitude-Phase Step Test In this section the amplitudeand the phase step tests are performed using the proposedestimation procedureThe test signal is defined as [28] (in thestandard changes in this step are happening separately)
119909 (119905) = 119883119898 (1 + 119896119886119906 (119905 minus 119905119904)) cos (120596119905 + 119896120593119906 (119905 minus 119905119904)) (64)
where 119906(119905) is a unit step function 119896119886 is the amplitude stepsize 119896120593 is the phase step size and 119905119904 is the step instant In thistest a signal with the 10∘ phase angle step and the amplitudestep changes from 09 to 1 pu in the fundamental componentwhich is fed to the algorithm The parameters are chosen as119883119898 = radic2120596 = 2120587 sdot 50119867119911 119896119886 = +01119901119906 119896120601 = 12058718 whilethe measurement noise is set to be negligible The results forthe amplitude and phase step tests are shown in Figure 4 inwhich the upper subplot shows the transient response of theamplitude or the phase while in the lower plot the TVE areillustrated
It is observed that the proposed approach achievesrelatively long transient periods but with small overshootsBased on the results presented in Figure 4 the proposedmethod is capable of performing a precise estimation ofthe amplitude and phase after the expiry of the transitionperiod (the model is more appropriate for smooth amplitudeand phase changes) since it represents the modification ofTailorrsquos-Fourier expansion The duration of this transition inthe proposed approach is shorter than in some othermethodsthat have the same starting point [17 27] which is the resultof the position of the poles of the transfer function withrespect to the unit circle in the 119911 plane (as the poles aremore spaced) In addition there is no large overshoot in theresponse as opposed to [16 17 23] due to the amplitude andphase discontinuities at the step change since no previousassessment is performed concerning a sample window
In Table 9 the performance indices defined in the stan-dard [28] including response time delay time and overshootare listed for both amplitude and phase step change tests Theperformances achieved by the proposed algorithm fulfill theM-Class measurement requirements in the standard
43 Frequency Step Test During this simulation test a signalhaving a 5 Hz frequency step is used to evaluate the possi-bility of a transient response in a frequency step condition(although 5 Hz frequency step is not likely to happen in apower grid this test is designed based on IEEE standard [28])
119909 (119899) = cos 21205871198911119899Δ119905 0 lt 119899 lt 101198731119909 (119899) = cos 2120587 (1198911 + 5) 119899Δ119905 101198731 le 119899 (65)
Table 9 Performance indices for phasor estimation under stepchanges in amplitude and phase using the proposed estimationalgorithm
Response time Delay time OvershootAmplitude step 197ms 886ms 0008 [pu]Phase step 967ms 002 [rad]
The frequency was calculated as a ratio of roots of char-acteristic equation based on extracted unknown coefficients1198601198980 1198601198981 1198601198982 1198611198980 1198611198981 1198611198982 from (12) and 2120587 [15 27]The proposed method shows characteristics as well as theevaluation process similar to the previous type of signal(magnitude-phase step condition) Figure 5 shows that theproposed processing method can be successfully tracked as+5 Hz frequency step versus the traditional approach toprocessing the phasors
44 Harmonic Infiltration Test Input signal definedwith (66)was used to validate the proposed methods in harmoniccondition119909 (119905) = 119886 (119905) cos (1205961119905 + 120601 (119905)) + 005 cos 51205961119905 + 003 cos 71205961119905119886 (119905) = 1198860 + 1198861 sin 2120587119891119886119905120601 (119905) = 1206010 + 1206011 sin 21205871198911206011199051198860 = 1 1206010 = 05 1198861 = 01 1206011 = 005 119891119886 = 119891120601 = 5 119873 = 24
(66)
Figure 6 represents the output of the proposed method inthe harmonic condition According to Figure 6 fundamentalphasor estimation based on proposed modification is freefrom harmonic which shows the superiority of harmonicmodification of the proposed method in this paper Since119873 = 24 in this test it is possible to estimate the dynamicphasor of the first 119873 minus 1 = 23 harmonics To increase therange of harmonics the sampling number per cycle shouldbe increased
45 Frequency Response In order to evaluate the frequencycharacteristic of the proposed estimator its transfer functionis defined in the form
Figure 3 Measured and estimated waveforms in the case of a time-varying signal (62) (a) tracking the first phasors (b) tracking the 3rdphasors (c) tracking the 5th phasors and (d) measured and estimated waveforms of main input signal
12 Mathematical Problems in Engineering
0minus20 40 60 8020Time [ms]
realestim
095
1
105
11
115
12
Am
plitu
de [p
u]
(a)
0minus20 40 60 8020Time [ms]
realestim
minus01
0
01
02
03
Phas
e [ra
d]
(b)
0minus20 40 60 8020Time [ms]
0
05
1
15
2
25
3
TVE
[]
(c)
Figure 4 Estimation of (a) amplitude (b) phase and (c) TVE in percentage
minus10
0
10
Freq
uenc
y (k
Hz)
1510 205Normalized time (cycles)
Figure 5 Frequency estimation during frequency step test
where119883(119911) and 119901(119911) are 119911 transforms of the input (main sig-nal) and output (estimated dynamic phasor) The frequencyresponse of the proposed method is given in Figure 7 Themagnitude responses are unsymmetrical which is completelyunderstandable since it belongs to complex filters In thesurroundings of the fundamental frequency complete elimi-nation of the negative one and complete pass of the positiveone occur The flatter transfer characteristic causes a lowerdistortion in the estimation of the phasor and provides zero-gain in the nonfundamental component In this way the har-monics at the output are removed as opposed to [16 21 23]
46 Error Bounds In order to determine the boundary of thepossible error in the estimation of the phasor signals withvariable envelopes were used
119886 (119905) = 1 + 05 3sum119894=1
119890119905120591119894 cos 21205871198911119905 (68)
where the time constants (120591i) were generated by a uniformrandom process in the interval of [20 40] cycles In a similarway the three frequencies were randomly generated in theintervals of [1 3] [3 5] and [5 7]Hz The error is calculatedby
119903119898119904 minus 119890119903119903119900119903 = radicsum(119886 (119899) minus 119886 (119899)) (69)
where 119886(119899) and 119886(119899) are real and estimated amplitudesrespectively Figure 8 shows the histograms of the errorsattained by proposed method
The RMS error belongs to the range of 3times10minus3 and8times10minus3 for proposed estimation method This error is relatedto elimination of the third derivative term of the Taylorexpansion in phasor estimation process and it is much betterthan error estimated in [16 17 22ndash24 27]
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
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Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
The matrix 119867119879 sdot 119867 is called the Gramrsquos matrix Thebest solution (in the least squares sense) exists providedthat the Grams matrix is invertible which is fulfilled whenthe column vectors H are linearly independent [32] TheGrammian inversion depends on the size of the interval Nand the order describing the model of the signal itself thatis the subject of processing In our approach we assume that2119898120596119873119879 = 2119904120587 for some 119904 isin N (in the case that 119904 = 1 thenumber of samples N covers the whole period)
It is clear that phasors are given by the inverse transformof the phasorlets and that the best solution is given by
Thepseudoinversematrix (119867119879 sdot119867)minus1 sdot119867119879 depends only onthe parameters of the adopted signal model The estimationof the phasors in the center of the evaluation interval whereTailors error is zero is correct if the input signal is welldescribed by the adopted model for which the least meansquared error (LMS) is also zero An LMS error would affectthe estimate if the signal was outside the projection subspaceof the LMS algorithm
3 Gramrsquos Matrix of Dynamic Signal Model
In this section we will investigate the Grams matrix of thedynamic signal model (12) and for each element of thatmatrix an explicit formula will be given
The Gramrsquos matrix 119867119879 sdot 119867 has the form
where119860 = [119886119894119895]3119894119895=1 119861 = [119887119894119895]3119894119895=1 and 119862 = [119888119894119895]3119894119895=1 and for all119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements 119886119894119895 119887119894119895 and 119888119894119895 arefinite trigonometric series
where 120577(119911) is Riemann zeta function [33 34] 120577(119911 119886) isHurvitz zeta function [35] and119867119899(119896) is generalized harmonicnumber [36] The Swiss mathematician Jacob Bernoulli
(1654-1705) derived the formula for the finite sum of powersof consecutive positive integers [37] representing 119878119901(119873) aspolynomial in 119873 of degree 119901 + 1 In the case that thepower119901 takes values in the set 0 1 4 the correspondingpolynomials are presented in Table 1
Now let us consider the finite sums involving trigonomet-ric functions
119877119901 (119873 119909) = Re 1(2i)119901 sdot 119892119901 (119890i2119909) 119868119901 (119873 119909) = Im 1(2i)119901 sdot 119892119901 (119890i2119909)
(38)
The series 119860119901(119873 119909) 119861119901(119873 119909) and 119862119901(119873 119909) can berepresented in the terms of 119878119901(119873) and 119892119901(119909) in the followingway
119860119901 (119873 119909) = 12 sdot 119878119901 (119873) minus 12 sdot Re 1(2i)119901 sdot 119892119901 (119890i2119909) (39)
Also these series can be represented in the terms of119878119901(119873) 119877119901(119873 119909) and 119868119901(119873 119909) in the following simple nota-tion
In the case that 119901 takes values in the set 0 1 4the explicit forms of functions 119892119901(119911) obtained by 119901 consec-utive derivations according to formula (35) are presented inTable 2
In the main section of this article the synchronous modelis observed ie the number 119873 is chosen with respect to therelation1198731199090 = 120587 For 119901 = 0 1 4 and 1199110 = 11989011989421199090 this leadsto the values 119864119901(119873 1199090) given in Table 3
If we represent each 1199110119904 in 119864119901(119873 1199110) in its expanded formas a polynomial in sin 1199090 and cos 1199090 of degree 119901 (the degree ofeach term in this polynomial in two variables is the sumof theexponents in each term) then 119877119901(119873 1199090) can be transformedinto the form given in Table 4
6 Mathematical Problems in Engineering
Table2Th
efun
ctions
119892 119901(119911)
119892 0(119911)
119911 119911minus1sdot(119911119873
minus1)119892 1(119911
)2i119911 (119911minus
1)2sdot(119873
119911119873+1minus(119873
+1)119911119873 +
1)119892 2(119911
)(2i)2
119911(119911minus
1)3sdot(1198732119911119873+2minus(2
1198732 +2119873
minus1)119911119873+1+(1198732+2119873
+1)119911119873 minus
119911minus1)
119892 3(119911)
(2i)3119911
(119911minus1)4
sdot(1198733119911119873+3minus(3
1198733+31198732minus3119873
+1)119911119873+2+(3
1198733+61198732minus4)
119911119873+1minus(119873
+1)3119911119873 +
1199112 +4119911+
1)119892 4(119911
)(2i)4119911
(119911minus1)5
sdot(1198734119911119873+4minus(4
1198734 +41198733
minus61198732+4119873
minus1)119911119873+3+(6
1198734+12
1198733minus61198732minus12
119873+11)119911119873+2minus(4
1198734 +121198733+6119873
2minus12
119873minus11)119911119873+1+(119873
+1)4119911119873 minus
1199113 minus111199112minus11
119911minus1)
Mathematical Problems in Engineering 7
Table 3 The values of functions 119864119901(119873 1199090)1198640(119873 1199090) 0
1198641(119873 1199090) 119902111199110122i sdot Im 119911012 11990211 = 119873
11990241 = 1198734119911119873+411990242 = minus41198734 minus 41198733 + 61198732 minus 411987311990243 = 61198734 + 121198733 minus 61198732 minus 1211987311990244 = minus41198734 minus 121198733 minus 61198732 + 12119873
In a similar way by representing each 1199110119904 in 119864119901(119873 1199110)in its expanded form as a polynomial in sin 1199090 and cos1199090 ofdegree 119901 119868119901(119873 1199090) can be transformed into the form givenin Table 5
Now by formula (42) we obtain values 119860119901(119873 1199090) 119901 =0 1 4 given in Table 6Using formula (43) we obtain values 119861119901(119873 1199090) for 119901 =0 1 4 given in Table 7Finally by (44) we obtain values 119862119901(119873 1199090) for 119901 =0 1 4 given in Table 8Now by (27)we obtain the elements ofGramsmatrix119867119879 sdot119867 Namely for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119886119894119895 are given by
11988611 = 1198732 (45)
11988612 = 11988621 = 11987911987324 (46)
11988613 = 11988622 = 11988631 = 1198792 sdot (16 (1198733 minus 119873) minus 14119873sdot ctg2119898120596119879)
(47)
11988623 = 11988632 = 1198793 sdot (18 (1198734 minus 21198732) minus 381198732 sdot ctg2119898120596119879) (48)
11988633 = 1198794 sdot ( 130 (31198735 minus 101198733 + 7119873) + 34119873sdot ctg4119898120596119879 minus 12 (1198733 minus 2119873) sdot ctg2119898120596119879)
(49)
8 Mathematical Problems in Engineering
Table 5 The values of functions 119868119901(119873 1199090)1198680(119873 1199090) 0
1198681(119873 1199090) minus1198732 sdot ctg11990901198682(119873 1199090) minus11987322 sdot ctg11990901198683(119873 1199090) 14 sdot (3119873 sdot ctg3 1199090 minus (21198733 minus 3119873) sdot ctg1199090)1198684(119873 1199090) 12 sdot (31198732 sdot ctg3 1199090 minus (1198734 minus 31198732) sdot ctg1199090)
Further for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119887119894119895 are given by
sdot (3119873 sdot ctg3119898120596119879 minus (21198733 minus 3119873) sdot ctg119898120596119879) (58)
11988833 = 11987944sdot (31198732 sdot ctg3119898120596119879 minus (1198734 minus 31198732) sdot ctg119898120596119879)
(59)
The presented representation of Gramrsquos matrix shows thatthe computation of pseudoinverse (119867119879 sdot 119867)minus1 sdot 119867119879 can beperformed more rapidly and with more precision
4 Simulation Results
The proposed phasor measurement scheme is employedto estimate coefficients of the power signal under nonsta-tionary scenarios during which we compared the resultsobtained in the proposed algorithm with some other well-known procedures Practically through the simulation testwe estimated the ability of the algorithms to track harmonicchanges Tests have been performed to show the performanceof the proposed technique with reference to the conditionsof the standard [28] for synchronized phasor measurementsystems in power systems The first is a proposed estimationprocedure verified on the basis of the following test signalwhich is characteristic for oscillating conditions
where 119886(119905) = 1198860 + 1198861 sin 2120587119891119886119905 120601(119905) = 1206010 + 1206011 sin 2120587119891120601119905 1198860 =1206010 = 1 1198861 = 1206011 = 01 119891119886 = 119891120601 = 5 and119873 = 24As can be seen from the above relations the test signal
is sampled with a frequency of 1200 Hz thus forming 24samples within a 20 ms wide window which is the period ofthe signal with a fundamental frequency of 50Hz Figures 1(a)and 1(b) show the size of the estimated amplitude and phaseof the fundamental dynamic phasor using the proposed esti-mation method It is clear that the main difference betweenthe proposed method and some other methods based on theapplication of the least square methods [15 21 27] and themethods based on the application of Kalman filters [16 1723 31] is in the occurrence of the delay in the estimation dueto the use of the data window
One of the established criteria for assessing the qualityof the estimation is based on the Total Vector Error (TVE)where we can estimate the error size in the estimation of thephasor magnitude and angle defining it as
119879119881119864 = 1003816100381610038161003816119883119903 minus 119883119890100381610038161003816100381610038161003816100381610038161198831199031003816100381610038161003816 (61)
where 119883119903 and 119883119890 are real and estimated values Figure 2shows the total vector error of the proposed method in thefirst ten cycles one cycle time delay of estimated phasor iscompensated as in [27] It can be concluded that the erroroccurring is above all the results of the introduced delaysimilar to that of [15 21] Owing to the modification madein calculating the parameters of the phasors the magnitudesof this calculated error are smaller than in other methodsbased on the least squares method with their size becomingcomparable to [16 17 23 38] therefore being much smallerthan [27 39ndash41] The proposed approach can calculate thederivatives of the phasor-phasor speed and accelerationwhich reduces the estimation error In addition it is possibleto calculate the frequency of the processed signal and detectfaults and power swings
Mathematical Problems in Engineering 9
Table 6 The values of 119860119901(119873 1199090)1198600(119873 1199090) 11987321198601(119873 1199090) 119873241198602(119873 1199090) 16 (1198733 minus 119873) minus 14119873 sdot ctg211990901198603(119873 1199090) 18 (1198734 minus 21198732) minus 381198732 sdot ctg211990901198604(119873 1199090) 130 (31198735 minus 101198733 + 7119873) + 34119873 sdot ctg41199090 minus 12 (1198733 minus 2119873) sdot ctg21199090
Table 8 The values 119862119901(119873 1199090)1198620(119873 1199090) 0
1198621(119873 1199090) minus1198734 sdot ctg11990901198622(119873 1199090) minus11987324 sdot ctg11990901198623(119873 1199090) 18 sdot (3119873 sdot ctg31199090 minus (21198733 minus 3119873) sdot ctg1199090)1198624(119873 1199090) 14 sdot (31198732 sdot ctg31199090 minus (1198734 minus 31198732) sdot ctg1199090)
41 Amplitude Oscillation Case The ability of the proposedapproach to track an arbitrary oscillating amplitude is givenin the following example where
119909 (119905) = 119886 (119905) infinsumℎ=0
119888ℎ cos 2120587ℎ1198911119905 (62)
is the input signal to the estimator with 1198911 = 50 Hz andnonzero Fourier coefficients (1198881 = 1 1198883 = 04 and 1198885 = 02)and where 119886(119905) represents an amplitude oscillation given bythe following second-order polynomial
with coefficients 119886 = minus4 119887 = 4 and 119888 = 0 Thecorresponding digital signal is sampled at 24 samplescycle(119891 = 1200Hz) The oscillation of main-original signal andits corresponding estimated signal components are shown in
Figure 3 Based on this representation it can be clearly seenthat the proposed method for estimation is fully conduciveto following the dynamics of all components of the processedinput signal Also we can see that the quality of estimationdecreases with increasing the number of harmonic which arethe results of relatively small and fixed-constant number ofsamples For higher harmonics components this number ofsamples cannot provide precise reconstruction If the numberof samples adaptively changed as the function of the orderof harmonics the accuracy of the estimation would be thesame as on the first-fundamental harmonic component Anadditional problem is certainly the nonideality of the filterfunction
42 Magnitude-Phase Step Test In this section the amplitudeand the phase step tests are performed using the proposedestimation procedureThe test signal is defined as [28] (in thestandard changes in this step are happening separately)
119909 (119905) = 119883119898 (1 + 119896119886119906 (119905 minus 119905119904)) cos (120596119905 + 119896120593119906 (119905 minus 119905119904)) (64)
where 119906(119905) is a unit step function 119896119886 is the amplitude stepsize 119896120593 is the phase step size and 119905119904 is the step instant In thistest a signal with the 10∘ phase angle step and the amplitudestep changes from 09 to 1 pu in the fundamental componentwhich is fed to the algorithm The parameters are chosen as119883119898 = radic2120596 = 2120587 sdot 50119867119911 119896119886 = +01119901119906 119896120601 = 12058718 whilethe measurement noise is set to be negligible The results forthe amplitude and phase step tests are shown in Figure 4 inwhich the upper subplot shows the transient response of theamplitude or the phase while in the lower plot the TVE areillustrated
It is observed that the proposed approach achievesrelatively long transient periods but with small overshootsBased on the results presented in Figure 4 the proposedmethod is capable of performing a precise estimation ofthe amplitude and phase after the expiry of the transitionperiod (the model is more appropriate for smooth amplitudeand phase changes) since it represents the modification ofTailorrsquos-Fourier expansion The duration of this transition inthe proposed approach is shorter than in some othermethodsthat have the same starting point [17 27] which is the resultof the position of the poles of the transfer function withrespect to the unit circle in the 119911 plane (as the poles aremore spaced) In addition there is no large overshoot in theresponse as opposed to [16 17 23] due to the amplitude andphase discontinuities at the step change since no previousassessment is performed concerning a sample window
In Table 9 the performance indices defined in the stan-dard [28] including response time delay time and overshootare listed for both amplitude and phase step change tests Theperformances achieved by the proposed algorithm fulfill theM-Class measurement requirements in the standard
43 Frequency Step Test During this simulation test a signalhaving a 5 Hz frequency step is used to evaluate the possi-bility of a transient response in a frequency step condition(although 5 Hz frequency step is not likely to happen in apower grid this test is designed based on IEEE standard [28])
119909 (119899) = cos 21205871198911119899Δ119905 0 lt 119899 lt 101198731119909 (119899) = cos 2120587 (1198911 + 5) 119899Δ119905 101198731 le 119899 (65)
Table 9 Performance indices for phasor estimation under stepchanges in amplitude and phase using the proposed estimationalgorithm
Response time Delay time OvershootAmplitude step 197ms 886ms 0008 [pu]Phase step 967ms 002 [rad]
The frequency was calculated as a ratio of roots of char-acteristic equation based on extracted unknown coefficients1198601198980 1198601198981 1198601198982 1198611198980 1198611198981 1198611198982 from (12) and 2120587 [15 27]The proposed method shows characteristics as well as theevaluation process similar to the previous type of signal(magnitude-phase step condition) Figure 5 shows that theproposed processing method can be successfully tracked as+5 Hz frequency step versus the traditional approach toprocessing the phasors
44 Harmonic Infiltration Test Input signal definedwith (66)was used to validate the proposed methods in harmoniccondition119909 (119905) = 119886 (119905) cos (1205961119905 + 120601 (119905)) + 005 cos 51205961119905 + 003 cos 71205961119905119886 (119905) = 1198860 + 1198861 sin 2120587119891119886119905120601 (119905) = 1206010 + 1206011 sin 21205871198911206011199051198860 = 1 1206010 = 05 1198861 = 01 1206011 = 005 119891119886 = 119891120601 = 5 119873 = 24
(66)
Figure 6 represents the output of the proposed method inthe harmonic condition According to Figure 6 fundamentalphasor estimation based on proposed modification is freefrom harmonic which shows the superiority of harmonicmodification of the proposed method in this paper Since119873 = 24 in this test it is possible to estimate the dynamicphasor of the first 119873 minus 1 = 23 harmonics To increase therange of harmonics the sampling number per cycle shouldbe increased
45 Frequency Response In order to evaluate the frequencycharacteristic of the proposed estimator its transfer functionis defined in the form
Figure 3 Measured and estimated waveforms in the case of a time-varying signal (62) (a) tracking the first phasors (b) tracking the 3rdphasors (c) tracking the 5th phasors and (d) measured and estimated waveforms of main input signal
12 Mathematical Problems in Engineering
0minus20 40 60 8020Time [ms]
realestim
095
1
105
11
115
12
Am
plitu
de [p
u]
(a)
0minus20 40 60 8020Time [ms]
realestim
minus01
0
01
02
03
Phas
e [ra
d]
(b)
0minus20 40 60 8020Time [ms]
0
05
1
15
2
25
3
TVE
[]
(c)
Figure 4 Estimation of (a) amplitude (b) phase and (c) TVE in percentage
minus10
0
10
Freq
uenc
y (k
Hz)
1510 205Normalized time (cycles)
Figure 5 Frequency estimation during frequency step test
where119883(119911) and 119901(119911) are 119911 transforms of the input (main sig-nal) and output (estimated dynamic phasor) The frequencyresponse of the proposed method is given in Figure 7 Themagnitude responses are unsymmetrical which is completelyunderstandable since it belongs to complex filters In thesurroundings of the fundamental frequency complete elimi-nation of the negative one and complete pass of the positiveone occur The flatter transfer characteristic causes a lowerdistortion in the estimation of the phasor and provides zero-gain in the nonfundamental component In this way the har-monics at the output are removed as opposed to [16 21 23]
46 Error Bounds In order to determine the boundary of thepossible error in the estimation of the phasor signals withvariable envelopes were used
119886 (119905) = 1 + 05 3sum119894=1
119890119905120591119894 cos 21205871198911119905 (68)
where the time constants (120591i) were generated by a uniformrandom process in the interval of [20 40] cycles In a similarway the three frequencies were randomly generated in theintervals of [1 3] [3 5] and [5 7]Hz The error is calculatedby
119903119898119904 minus 119890119903119903119900119903 = radicsum(119886 (119899) minus 119886 (119899)) (69)
where 119886(119899) and 119886(119899) are real and estimated amplitudesrespectively Figure 8 shows the histograms of the errorsattained by proposed method
The RMS error belongs to the range of 3times10minus3 and8times10minus3 for proposed estimation method This error is relatedto elimination of the third derivative term of the Taylorexpansion in phasor estimation process and it is much betterthan error estimated in [16 17 22ndash24 27]
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
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Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
(1654-1705) derived the formula for the finite sum of powersof consecutive positive integers [37] representing 119878119901(119873) aspolynomial in 119873 of degree 119901 + 1 In the case that thepower119901 takes values in the set 0 1 4 the correspondingpolynomials are presented in Table 1
Now let us consider the finite sums involving trigonomet-ric functions
119877119901 (119873 119909) = Re 1(2i)119901 sdot 119892119901 (119890i2119909) 119868119901 (119873 119909) = Im 1(2i)119901 sdot 119892119901 (119890i2119909)
(38)
The series 119860119901(119873 119909) 119861119901(119873 119909) and 119862119901(119873 119909) can berepresented in the terms of 119878119901(119873) and 119892119901(119909) in the followingway
119860119901 (119873 119909) = 12 sdot 119878119901 (119873) minus 12 sdot Re 1(2i)119901 sdot 119892119901 (119890i2119909) (39)
Also these series can be represented in the terms of119878119901(119873) 119877119901(119873 119909) and 119868119901(119873 119909) in the following simple nota-tion
In the case that 119901 takes values in the set 0 1 4the explicit forms of functions 119892119901(119911) obtained by 119901 consec-utive derivations according to formula (35) are presented inTable 2
In the main section of this article the synchronous modelis observed ie the number 119873 is chosen with respect to therelation1198731199090 = 120587 For 119901 = 0 1 4 and 1199110 = 11989011989421199090 this leadsto the values 119864119901(119873 1199090) given in Table 3
If we represent each 1199110119904 in 119864119901(119873 1199110) in its expanded formas a polynomial in sin 1199090 and cos 1199090 of degree 119901 (the degree ofeach term in this polynomial in two variables is the sumof theexponents in each term) then 119877119901(119873 1199090) can be transformedinto the form given in Table 4
6 Mathematical Problems in Engineering
Table2Th
efun
ctions
119892 119901(119911)
119892 0(119911)
119911 119911minus1sdot(119911119873
minus1)119892 1(119911
)2i119911 (119911minus
1)2sdot(119873
119911119873+1minus(119873
+1)119911119873 +
1)119892 2(119911
)(2i)2
119911(119911minus
1)3sdot(1198732119911119873+2minus(2
1198732 +2119873
minus1)119911119873+1+(1198732+2119873
+1)119911119873 minus
119911minus1)
119892 3(119911)
(2i)3119911
(119911minus1)4
sdot(1198733119911119873+3minus(3
1198733+31198732minus3119873
+1)119911119873+2+(3
1198733+61198732minus4)
119911119873+1minus(119873
+1)3119911119873 +
1199112 +4119911+
1)119892 4(119911
)(2i)4119911
(119911minus1)5
sdot(1198734119911119873+4minus(4
1198734 +41198733
minus61198732+4119873
minus1)119911119873+3+(6
1198734+12
1198733minus61198732minus12
119873+11)119911119873+2minus(4
1198734 +121198733+6119873
2minus12
119873minus11)119911119873+1+(119873
+1)4119911119873 minus
1199113 minus111199112minus11
119911minus1)
Mathematical Problems in Engineering 7
Table 3 The values of functions 119864119901(119873 1199090)1198640(119873 1199090) 0
1198641(119873 1199090) 119902111199110122i sdot Im 119911012 11990211 = 119873
11990241 = 1198734119911119873+411990242 = minus41198734 minus 41198733 + 61198732 minus 411987311990243 = 61198734 + 121198733 minus 61198732 minus 1211987311990244 = minus41198734 minus 121198733 minus 61198732 + 12119873
In a similar way by representing each 1199110119904 in 119864119901(119873 1199110)in its expanded form as a polynomial in sin 1199090 and cos1199090 ofdegree 119901 119868119901(119873 1199090) can be transformed into the form givenin Table 5
Now by formula (42) we obtain values 119860119901(119873 1199090) 119901 =0 1 4 given in Table 6Using formula (43) we obtain values 119861119901(119873 1199090) for 119901 =0 1 4 given in Table 7Finally by (44) we obtain values 119862119901(119873 1199090) for 119901 =0 1 4 given in Table 8Now by (27)we obtain the elements ofGramsmatrix119867119879 sdot119867 Namely for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119886119894119895 are given by
11988611 = 1198732 (45)
11988612 = 11988621 = 11987911987324 (46)
11988613 = 11988622 = 11988631 = 1198792 sdot (16 (1198733 minus 119873) minus 14119873sdot ctg2119898120596119879)
(47)
11988623 = 11988632 = 1198793 sdot (18 (1198734 minus 21198732) minus 381198732 sdot ctg2119898120596119879) (48)
11988633 = 1198794 sdot ( 130 (31198735 minus 101198733 + 7119873) + 34119873sdot ctg4119898120596119879 minus 12 (1198733 minus 2119873) sdot ctg2119898120596119879)
(49)
8 Mathematical Problems in Engineering
Table 5 The values of functions 119868119901(119873 1199090)1198680(119873 1199090) 0
1198681(119873 1199090) minus1198732 sdot ctg11990901198682(119873 1199090) minus11987322 sdot ctg11990901198683(119873 1199090) 14 sdot (3119873 sdot ctg3 1199090 minus (21198733 minus 3119873) sdot ctg1199090)1198684(119873 1199090) 12 sdot (31198732 sdot ctg3 1199090 minus (1198734 minus 31198732) sdot ctg1199090)
Further for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119887119894119895 are given by
sdot (3119873 sdot ctg3119898120596119879 minus (21198733 minus 3119873) sdot ctg119898120596119879) (58)
11988833 = 11987944sdot (31198732 sdot ctg3119898120596119879 minus (1198734 minus 31198732) sdot ctg119898120596119879)
(59)
The presented representation of Gramrsquos matrix shows thatthe computation of pseudoinverse (119867119879 sdot 119867)minus1 sdot 119867119879 can beperformed more rapidly and with more precision
4 Simulation Results
The proposed phasor measurement scheme is employedto estimate coefficients of the power signal under nonsta-tionary scenarios during which we compared the resultsobtained in the proposed algorithm with some other well-known procedures Practically through the simulation testwe estimated the ability of the algorithms to track harmonicchanges Tests have been performed to show the performanceof the proposed technique with reference to the conditionsof the standard [28] for synchronized phasor measurementsystems in power systems The first is a proposed estimationprocedure verified on the basis of the following test signalwhich is characteristic for oscillating conditions
where 119886(119905) = 1198860 + 1198861 sin 2120587119891119886119905 120601(119905) = 1206010 + 1206011 sin 2120587119891120601119905 1198860 =1206010 = 1 1198861 = 1206011 = 01 119891119886 = 119891120601 = 5 and119873 = 24As can be seen from the above relations the test signal
is sampled with a frequency of 1200 Hz thus forming 24samples within a 20 ms wide window which is the period ofthe signal with a fundamental frequency of 50Hz Figures 1(a)and 1(b) show the size of the estimated amplitude and phaseof the fundamental dynamic phasor using the proposed esti-mation method It is clear that the main difference betweenthe proposed method and some other methods based on theapplication of the least square methods [15 21 27] and themethods based on the application of Kalman filters [16 1723 31] is in the occurrence of the delay in the estimation dueto the use of the data window
One of the established criteria for assessing the qualityof the estimation is based on the Total Vector Error (TVE)where we can estimate the error size in the estimation of thephasor magnitude and angle defining it as
119879119881119864 = 1003816100381610038161003816119883119903 minus 119883119890100381610038161003816100381610038161003816100381610038161198831199031003816100381610038161003816 (61)
where 119883119903 and 119883119890 are real and estimated values Figure 2shows the total vector error of the proposed method in thefirst ten cycles one cycle time delay of estimated phasor iscompensated as in [27] It can be concluded that the erroroccurring is above all the results of the introduced delaysimilar to that of [15 21] Owing to the modification madein calculating the parameters of the phasors the magnitudesof this calculated error are smaller than in other methodsbased on the least squares method with their size becomingcomparable to [16 17 23 38] therefore being much smallerthan [27 39ndash41] The proposed approach can calculate thederivatives of the phasor-phasor speed and accelerationwhich reduces the estimation error In addition it is possibleto calculate the frequency of the processed signal and detectfaults and power swings
Mathematical Problems in Engineering 9
Table 6 The values of 119860119901(119873 1199090)1198600(119873 1199090) 11987321198601(119873 1199090) 119873241198602(119873 1199090) 16 (1198733 minus 119873) minus 14119873 sdot ctg211990901198603(119873 1199090) 18 (1198734 minus 21198732) minus 381198732 sdot ctg211990901198604(119873 1199090) 130 (31198735 minus 101198733 + 7119873) + 34119873 sdot ctg41199090 minus 12 (1198733 minus 2119873) sdot ctg21199090
Table 8 The values 119862119901(119873 1199090)1198620(119873 1199090) 0
1198621(119873 1199090) minus1198734 sdot ctg11990901198622(119873 1199090) minus11987324 sdot ctg11990901198623(119873 1199090) 18 sdot (3119873 sdot ctg31199090 minus (21198733 minus 3119873) sdot ctg1199090)1198624(119873 1199090) 14 sdot (31198732 sdot ctg31199090 minus (1198734 minus 31198732) sdot ctg1199090)
41 Amplitude Oscillation Case The ability of the proposedapproach to track an arbitrary oscillating amplitude is givenin the following example where
119909 (119905) = 119886 (119905) infinsumℎ=0
119888ℎ cos 2120587ℎ1198911119905 (62)
is the input signal to the estimator with 1198911 = 50 Hz andnonzero Fourier coefficients (1198881 = 1 1198883 = 04 and 1198885 = 02)and where 119886(119905) represents an amplitude oscillation given bythe following second-order polynomial
with coefficients 119886 = minus4 119887 = 4 and 119888 = 0 Thecorresponding digital signal is sampled at 24 samplescycle(119891 = 1200Hz) The oscillation of main-original signal andits corresponding estimated signal components are shown in
Figure 3 Based on this representation it can be clearly seenthat the proposed method for estimation is fully conduciveto following the dynamics of all components of the processedinput signal Also we can see that the quality of estimationdecreases with increasing the number of harmonic which arethe results of relatively small and fixed-constant number ofsamples For higher harmonics components this number ofsamples cannot provide precise reconstruction If the numberof samples adaptively changed as the function of the orderof harmonics the accuracy of the estimation would be thesame as on the first-fundamental harmonic component Anadditional problem is certainly the nonideality of the filterfunction
42 Magnitude-Phase Step Test In this section the amplitudeand the phase step tests are performed using the proposedestimation procedureThe test signal is defined as [28] (in thestandard changes in this step are happening separately)
119909 (119905) = 119883119898 (1 + 119896119886119906 (119905 minus 119905119904)) cos (120596119905 + 119896120593119906 (119905 minus 119905119904)) (64)
where 119906(119905) is a unit step function 119896119886 is the amplitude stepsize 119896120593 is the phase step size and 119905119904 is the step instant In thistest a signal with the 10∘ phase angle step and the amplitudestep changes from 09 to 1 pu in the fundamental componentwhich is fed to the algorithm The parameters are chosen as119883119898 = radic2120596 = 2120587 sdot 50119867119911 119896119886 = +01119901119906 119896120601 = 12058718 whilethe measurement noise is set to be negligible The results forthe amplitude and phase step tests are shown in Figure 4 inwhich the upper subplot shows the transient response of theamplitude or the phase while in the lower plot the TVE areillustrated
It is observed that the proposed approach achievesrelatively long transient periods but with small overshootsBased on the results presented in Figure 4 the proposedmethod is capable of performing a precise estimation ofthe amplitude and phase after the expiry of the transitionperiod (the model is more appropriate for smooth amplitudeand phase changes) since it represents the modification ofTailorrsquos-Fourier expansion The duration of this transition inthe proposed approach is shorter than in some othermethodsthat have the same starting point [17 27] which is the resultof the position of the poles of the transfer function withrespect to the unit circle in the 119911 plane (as the poles aremore spaced) In addition there is no large overshoot in theresponse as opposed to [16 17 23] due to the amplitude andphase discontinuities at the step change since no previousassessment is performed concerning a sample window
In Table 9 the performance indices defined in the stan-dard [28] including response time delay time and overshootare listed for both amplitude and phase step change tests Theperformances achieved by the proposed algorithm fulfill theM-Class measurement requirements in the standard
43 Frequency Step Test During this simulation test a signalhaving a 5 Hz frequency step is used to evaluate the possi-bility of a transient response in a frequency step condition(although 5 Hz frequency step is not likely to happen in apower grid this test is designed based on IEEE standard [28])
119909 (119899) = cos 21205871198911119899Δ119905 0 lt 119899 lt 101198731119909 (119899) = cos 2120587 (1198911 + 5) 119899Δ119905 101198731 le 119899 (65)
Table 9 Performance indices for phasor estimation under stepchanges in amplitude and phase using the proposed estimationalgorithm
Response time Delay time OvershootAmplitude step 197ms 886ms 0008 [pu]Phase step 967ms 002 [rad]
The frequency was calculated as a ratio of roots of char-acteristic equation based on extracted unknown coefficients1198601198980 1198601198981 1198601198982 1198611198980 1198611198981 1198611198982 from (12) and 2120587 [15 27]The proposed method shows characteristics as well as theevaluation process similar to the previous type of signal(magnitude-phase step condition) Figure 5 shows that theproposed processing method can be successfully tracked as+5 Hz frequency step versus the traditional approach toprocessing the phasors
44 Harmonic Infiltration Test Input signal definedwith (66)was used to validate the proposed methods in harmoniccondition119909 (119905) = 119886 (119905) cos (1205961119905 + 120601 (119905)) + 005 cos 51205961119905 + 003 cos 71205961119905119886 (119905) = 1198860 + 1198861 sin 2120587119891119886119905120601 (119905) = 1206010 + 1206011 sin 21205871198911206011199051198860 = 1 1206010 = 05 1198861 = 01 1206011 = 005 119891119886 = 119891120601 = 5 119873 = 24
(66)
Figure 6 represents the output of the proposed method inthe harmonic condition According to Figure 6 fundamentalphasor estimation based on proposed modification is freefrom harmonic which shows the superiority of harmonicmodification of the proposed method in this paper Since119873 = 24 in this test it is possible to estimate the dynamicphasor of the first 119873 minus 1 = 23 harmonics To increase therange of harmonics the sampling number per cycle shouldbe increased
45 Frequency Response In order to evaluate the frequencycharacteristic of the proposed estimator its transfer functionis defined in the form
Figure 3 Measured and estimated waveforms in the case of a time-varying signal (62) (a) tracking the first phasors (b) tracking the 3rdphasors (c) tracking the 5th phasors and (d) measured and estimated waveforms of main input signal
12 Mathematical Problems in Engineering
0minus20 40 60 8020Time [ms]
realestim
095
1
105
11
115
12
Am
plitu
de [p
u]
(a)
0minus20 40 60 8020Time [ms]
realestim
minus01
0
01
02
03
Phas
e [ra
d]
(b)
0minus20 40 60 8020Time [ms]
0
05
1
15
2
25
3
TVE
[]
(c)
Figure 4 Estimation of (a) amplitude (b) phase and (c) TVE in percentage
minus10
0
10
Freq
uenc
y (k
Hz)
1510 205Normalized time (cycles)
Figure 5 Frequency estimation during frequency step test
where119883(119911) and 119901(119911) are 119911 transforms of the input (main sig-nal) and output (estimated dynamic phasor) The frequencyresponse of the proposed method is given in Figure 7 Themagnitude responses are unsymmetrical which is completelyunderstandable since it belongs to complex filters In thesurroundings of the fundamental frequency complete elimi-nation of the negative one and complete pass of the positiveone occur The flatter transfer characteristic causes a lowerdistortion in the estimation of the phasor and provides zero-gain in the nonfundamental component In this way the har-monics at the output are removed as opposed to [16 21 23]
46 Error Bounds In order to determine the boundary of thepossible error in the estimation of the phasor signals withvariable envelopes were used
119886 (119905) = 1 + 05 3sum119894=1
119890119905120591119894 cos 21205871198911119905 (68)
where the time constants (120591i) were generated by a uniformrandom process in the interval of [20 40] cycles In a similarway the three frequencies were randomly generated in theintervals of [1 3] [3 5] and [5 7]Hz The error is calculatedby
119903119898119904 minus 119890119903119903119900119903 = radicsum(119886 (119899) minus 119886 (119899)) (69)
where 119886(119899) and 119886(119899) are real and estimated amplitudesrespectively Figure 8 shows the histograms of the errorsattained by proposed method
The RMS error belongs to the range of 3times10minus3 and8times10minus3 for proposed estimation method This error is relatedto elimination of the third derivative term of the Taylorexpansion in phasor estimation process and it is much betterthan error estimated in [16 17 22ndash24 27]
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
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Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
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Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
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Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
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Operations ResearchAdvances in
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
11990241 = 1198734119911119873+411990242 = minus41198734 minus 41198733 + 61198732 minus 411987311990243 = 61198734 + 121198733 minus 61198732 minus 1211987311990244 = minus41198734 minus 121198733 minus 61198732 + 12119873
In a similar way by representing each 1199110119904 in 119864119901(119873 1199110)in its expanded form as a polynomial in sin 1199090 and cos1199090 ofdegree 119901 119868119901(119873 1199090) can be transformed into the form givenin Table 5
Now by formula (42) we obtain values 119860119901(119873 1199090) 119901 =0 1 4 given in Table 6Using formula (43) we obtain values 119861119901(119873 1199090) for 119901 =0 1 4 given in Table 7Finally by (44) we obtain values 119862119901(119873 1199090) for 119901 =0 1 4 given in Table 8Now by (27)we obtain the elements ofGramsmatrix119867119879 sdot119867 Namely for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119886119894119895 are given by
11988611 = 1198732 (45)
11988612 = 11988621 = 11987911987324 (46)
11988613 = 11988622 = 11988631 = 1198792 sdot (16 (1198733 minus 119873) minus 14119873sdot ctg2119898120596119879)
(47)
11988623 = 11988632 = 1198793 sdot (18 (1198734 minus 21198732) minus 381198732 sdot ctg2119898120596119879) (48)
11988633 = 1198794 sdot ( 130 (31198735 minus 101198733 + 7119873) + 34119873sdot ctg4119898120596119879 minus 12 (1198733 minus 2119873) sdot ctg2119898120596119879)
(49)
8 Mathematical Problems in Engineering
Table 5 The values of functions 119868119901(119873 1199090)1198680(119873 1199090) 0
1198681(119873 1199090) minus1198732 sdot ctg11990901198682(119873 1199090) minus11987322 sdot ctg11990901198683(119873 1199090) 14 sdot (3119873 sdot ctg3 1199090 minus (21198733 minus 3119873) sdot ctg1199090)1198684(119873 1199090) 12 sdot (31198732 sdot ctg3 1199090 minus (1198734 minus 31198732) sdot ctg1199090)
Further for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119887119894119895 are given by
sdot (3119873 sdot ctg3119898120596119879 minus (21198733 minus 3119873) sdot ctg119898120596119879) (58)
11988833 = 11987944sdot (31198732 sdot ctg3119898120596119879 minus (1198734 minus 31198732) sdot ctg119898120596119879)
(59)
The presented representation of Gramrsquos matrix shows thatthe computation of pseudoinverse (119867119879 sdot 119867)minus1 sdot 119867119879 can beperformed more rapidly and with more precision
4 Simulation Results
The proposed phasor measurement scheme is employedto estimate coefficients of the power signal under nonsta-tionary scenarios during which we compared the resultsobtained in the proposed algorithm with some other well-known procedures Practically through the simulation testwe estimated the ability of the algorithms to track harmonicchanges Tests have been performed to show the performanceof the proposed technique with reference to the conditionsof the standard [28] for synchronized phasor measurementsystems in power systems The first is a proposed estimationprocedure verified on the basis of the following test signalwhich is characteristic for oscillating conditions
where 119886(119905) = 1198860 + 1198861 sin 2120587119891119886119905 120601(119905) = 1206010 + 1206011 sin 2120587119891120601119905 1198860 =1206010 = 1 1198861 = 1206011 = 01 119891119886 = 119891120601 = 5 and119873 = 24As can be seen from the above relations the test signal
is sampled with a frequency of 1200 Hz thus forming 24samples within a 20 ms wide window which is the period ofthe signal with a fundamental frequency of 50Hz Figures 1(a)and 1(b) show the size of the estimated amplitude and phaseof the fundamental dynamic phasor using the proposed esti-mation method It is clear that the main difference betweenthe proposed method and some other methods based on theapplication of the least square methods [15 21 27] and themethods based on the application of Kalman filters [16 1723 31] is in the occurrence of the delay in the estimation dueto the use of the data window
One of the established criteria for assessing the qualityof the estimation is based on the Total Vector Error (TVE)where we can estimate the error size in the estimation of thephasor magnitude and angle defining it as
119879119881119864 = 1003816100381610038161003816119883119903 minus 119883119890100381610038161003816100381610038161003816100381610038161198831199031003816100381610038161003816 (61)
where 119883119903 and 119883119890 are real and estimated values Figure 2shows the total vector error of the proposed method in thefirst ten cycles one cycle time delay of estimated phasor iscompensated as in [27] It can be concluded that the erroroccurring is above all the results of the introduced delaysimilar to that of [15 21] Owing to the modification madein calculating the parameters of the phasors the magnitudesof this calculated error are smaller than in other methodsbased on the least squares method with their size becomingcomparable to [16 17 23 38] therefore being much smallerthan [27 39ndash41] The proposed approach can calculate thederivatives of the phasor-phasor speed and accelerationwhich reduces the estimation error In addition it is possibleto calculate the frequency of the processed signal and detectfaults and power swings
Mathematical Problems in Engineering 9
Table 6 The values of 119860119901(119873 1199090)1198600(119873 1199090) 11987321198601(119873 1199090) 119873241198602(119873 1199090) 16 (1198733 minus 119873) minus 14119873 sdot ctg211990901198603(119873 1199090) 18 (1198734 minus 21198732) minus 381198732 sdot ctg211990901198604(119873 1199090) 130 (31198735 minus 101198733 + 7119873) + 34119873 sdot ctg41199090 minus 12 (1198733 minus 2119873) sdot ctg21199090
Table 8 The values 119862119901(119873 1199090)1198620(119873 1199090) 0
1198621(119873 1199090) minus1198734 sdot ctg11990901198622(119873 1199090) minus11987324 sdot ctg11990901198623(119873 1199090) 18 sdot (3119873 sdot ctg31199090 minus (21198733 minus 3119873) sdot ctg1199090)1198624(119873 1199090) 14 sdot (31198732 sdot ctg31199090 minus (1198734 minus 31198732) sdot ctg1199090)
41 Amplitude Oscillation Case The ability of the proposedapproach to track an arbitrary oscillating amplitude is givenin the following example where
119909 (119905) = 119886 (119905) infinsumℎ=0
119888ℎ cos 2120587ℎ1198911119905 (62)
is the input signal to the estimator with 1198911 = 50 Hz andnonzero Fourier coefficients (1198881 = 1 1198883 = 04 and 1198885 = 02)and where 119886(119905) represents an amplitude oscillation given bythe following second-order polynomial
with coefficients 119886 = minus4 119887 = 4 and 119888 = 0 Thecorresponding digital signal is sampled at 24 samplescycle(119891 = 1200Hz) The oscillation of main-original signal andits corresponding estimated signal components are shown in
Figure 3 Based on this representation it can be clearly seenthat the proposed method for estimation is fully conduciveto following the dynamics of all components of the processedinput signal Also we can see that the quality of estimationdecreases with increasing the number of harmonic which arethe results of relatively small and fixed-constant number ofsamples For higher harmonics components this number ofsamples cannot provide precise reconstruction If the numberof samples adaptively changed as the function of the orderof harmonics the accuracy of the estimation would be thesame as on the first-fundamental harmonic component Anadditional problem is certainly the nonideality of the filterfunction
42 Magnitude-Phase Step Test In this section the amplitudeand the phase step tests are performed using the proposedestimation procedureThe test signal is defined as [28] (in thestandard changes in this step are happening separately)
119909 (119905) = 119883119898 (1 + 119896119886119906 (119905 minus 119905119904)) cos (120596119905 + 119896120593119906 (119905 minus 119905119904)) (64)
where 119906(119905) is a unit step function 119896119886 is the amplitude stepsize 119896120593 is the phase step size and 119905119904 is the step instant In thistest a signal with the 10∘ phase angle step and the amplitudestep changes from 09 to 1 pu in the fundamental componentwhich is fed to the algorithm The parameters are chosen as119883119898 = radic2120596 = 2120587 sdot 50119867119911 119896119886 = +01119901119906 119896120601 = 12058718 whilethe measurement noise is set to be negligible The results forthe amplitude and phase step tests are shown in Figure 4 inwhich the upper subplot shows the transient response of theamplitude or the phase while in the lower plot the TVE areillustrated
It is observed that the proposed approach achievesrelatively long transient periods but with small overshootsBased on the results presented in Figure 4 the proposedmethod is capable of performing a precise estimation ofthe amplitude and phase after the expiry of the transitionperiod (the model is more appropriate for smooth amplitudeand phase changes) since it represents the modification ofTailorrsquos-Fourier expansion The duration of this transition inthe proposed approach is shorter than in some othermethodsthat have the same starting point [17 27] which is the resultof the position of the poles of the transfer function withrespect to the unit circle in the 119911 plane (as the poles aremore spaced) In addition there is no large overshoot in theresponse as opposed to [16 17 23] due to the amplitude andphase discontinuities at the step change since no previousassessment is performed concerning a sample window
In Table 9 the performance indices defined in the stan-dard [28] including response time delay time and overshootare listed for both amplitude and phase step change tests Theperformances achieved by the proposed algorithm fulfill theM-Class measurement requirements in the standard
43 Frequency Step Test During this simulation test a signalhaving a 5 Hz frequency step is used to evaluate the possi-bility of a transient response in a frequency step condition(although 5 Hz frequency step is not likely to happen in apower grid this test is designed based on IEEE standard [28])
119909 (119899) = cos 21205871198911119899Δ119905 0 lt 119899 lt 101198731119909 (119899) = cos 2120587 (1198911 + 5) 119899Δ119905 101198731 le 119899 (65)
Table 9 Performance indices for phasor estimation under stepchanges in amplitude and phase using the proposed estimationalgorithm
Response time Delay time OvershootAmplitude step 197ms 886ms 0008 [pu]Phase step 967ms 002 [rad]
The frequency was calculated as a ratio of roots of char-acteristic equation based on extracted unknown coefficients1198601198980 1198601198981 1198601198982 1198611198980 1198611198981 1198611198982 from (12) and 2120587 [15 27]The proposed method shows characteristics as well as theevaluation process similar to the previous type of signal(magnitude-phase step condition) Figure 5 shows that theproposed processing method can be successfully tracked as+5 Hz frequency step versus the traditional approach toprocessing the phasors
44 Harmonic Infiltration Test Input signal definedwith (66)was used to validate the proposed methods in harmoniccondition119909 (119905) = 119886 (119905) cos (1205961119905 + 120601 (119905)) + 005 cos 51205961119905 + 003 cos 71205961119905119886 (119905) = 1198860 + 1198861 sin 2120587119891119886119905120601 (119905) = 1206010 + 1206011 sin 21205871198911206011199051198860 = 1 1206010 = 05 1198861 = 01 1206011 = 005 119891119886 = 119891120601 = 5 119873 = 24
(66)
Figure 6 represents the output of the proposed method inthe harmonic condition According to Figure 6 fundamentalphasor estimation based on proposed modification is freefrom harmonic which shows the superiority of harmonicmodification of the proposed method in this paper Since119873 = 24 in this test it is possible to estimate the dynamicphasor of the first 119873 minus 1 = 23 harmonics To increase therange of harmonics the sampling number per cycle shouldbe increased
45 Frequency Response In order to evaluate the frequencycharacteristic of the proposed estimator its transfer functionis defined in the form
Figure 3 Measured and estimated waveforms in the case of a time-varying signal (62) (a) tracking the first phasors (b) tracking the 3rdphasors (c) tracking the 5th phasors and (d) measured and estimated waveforms of main input signal
12 Mathematical Problems in Engineering
0minus20 40 60 8020Time [ms]
realestim
095
1
105
11
115
12
Am
plitu
de [p
u]
(a)
0minus20 40 60 8020Time [ms]
realestim
minus01
0
01
02
03
Phas
e [ra
d]
(b)
0minus20 40 60 8020Time [ms]
0
05
1
15
2
25
3
TVE
[]
(c)
Figure 4 Estimation of (a) amplitude (b) phase and (c) TVE in percentage
minus10
0
10
Freq
uenc
y (k
Hz)
1510 205Normalized time (cycles)
Figure 5 Frequency estimation during frequency step test
where119883(119911) and 119901(119911) are 119911 transforms of the input (main sig-nal) and output (estimated dynamic phasor) The frequencyresponse of the proposed method is given in Figure 7 Themagnitude responses are unsymmetrical which is completelyunderstandable since it belongs to complex filters In thesurroundings of the fundamental frequency complete elimi-nation of the negative one and complete pass of the positiveone occur The flatter transfer characteristic causes a lowerdistortion in the estimation of the phasor and provides zero-gain in the nonfundamental component In this way the har-monics at the output are removed as opposed to [16 21 23]
46 Error Bounds In order to determine the boundary of thepossible error in the estimation of the phasor signals withvariable envelopes were used
119886 (119905) = 1 + 05 3sum119894=1
119890119905120591119894 cos 21205871198911119905 (68)
where the time constants (120591i) were generated by a uniformrandom process in the interval of [20 40] cycles In a similarway the three frequencies were randomly generated in theintervals of [1 3] [3 5] and [5 7]Hz The error is calculatedby
119903119898119904 minus 119890119903119903119900119903 = radicsum(119886 (119899) minus 119886 (119899)) (69)
where 119886(119899) and 119886(119899) are real and estimated amplitudesrespectively Figure 8 shows the histograms of the errorsattained by proposed method
The RMS error belongs to the range of 3times10minus3 and8times10minus3 for proposed estimation method This error is relatedto elimination of the third derivative term of the Taylorexpansion in phasor estimation process and it is much betterthan error estimated in [16 17 22ndash24 27]
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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Engineering Mathematics
International Journal of
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Operations ResearchAdvances in
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
11990241 = 1198734119911119873+411990242 = minus41198734 minus 41198733 + 61198732 minus 411987311990243 = 61198734 + 121198733 minus 61198732 minus 1211987311990244 = minus41198734 minus 121198733 minus 61198732 + 12119873
In a similar way by representing each 1199110119904 in 119864119901(119873 1199110)in its expanded form as a polynomial in sin 1199090 and cos1199090 ofdegree 119901 119868119901(119873 1199090) can be transformed into the form givenin Table 5
Now by formula (42) we obtain values 119860119901(119873 1199090) 119901 =0 1 4 given in Table 6Using formula (43) we obtain values 119861119901(119873 1199090) for 119901 =0 1 4 given in Table 7Finally by (44) we obtain values 119862119901(119873 1199090) for 119901 =0 1 4 given in Table 8Now by (27)we obtain the elements ofGramsmatrix119867119879 sdot119867 Namely for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119886119894119895 are given by
11988611 = 1198732 (45)
11988612 = 11988621 = 11987911987324 (46)
11988613 = 11988622 = 11988631 = 1198792 sdot (16 (1198733 minus 119873) minus 14119873sdot ctg2119898120596119879)
(47)
11988623 = 11988632 = 1198793 sdot (18 (1198734 minus 21198732) minus 381198732 sdot ctg2119898120596119879) (48)
11988633 = 1198794 sdot ( 130 (31198735 minus 101198733 + 7119873) + 34119873sdot ctg4119898120596119879 minus 12 (1198733 minus 2119873) sdot ctg2119898120596119879)
(49)
8 Mathematical Problems in Engineering
Table 5 The values of functions 119868119901(119873 1199090)1198680(119873 1199090) 0
1198681(119873 1199090) minus1198732 sdot ctg11990901198682(119873 1199090) minus11987322 sdot ctg11990901198683(119873 1199090) 14 sdot (3119873 sdot ctg3 1199090 minus (21198733 minus 3119873) sdot ctg1199090)1198684(119873 1199090) 12 sdot (31198732 sdot ctg3 1199090 minus (1198734 minus 31198732) sdot ctg1199090)
Further for all 119894 119895 = 1 2 3 and 119901 = 119894 + 119895 minus 2 the elements119887119894119895 are given by
sdot (3119873 sdot ctg3119898120596119879 minus (21198733 minus 3119873) sdot ctg119898120596119879) (58)
11988833 = 11987944sdot (31198732 sdot ctg3119898120596119879 minus (1198734 minus 31198732) sdot ctg119898120596119879)
(59)
The presented representation of Gramrsquos matrix shows thatthe computation of pseudoinverse (119867119879 sdot 119867)minus1 sdot 119867119879 can beperformed more rapidly and with more precision
4 Simulation Results
The proposed phasor measurement scheme is employedto estimate coefficients of the power signal under nonsta-tionary scenarios during which we compared the resultsobtained in the proposed algorithm with some other well-known procedures Practically through the simulation testwe estimated the ability of the algorithms to track harmonicchanges Tests have been performed to show the performanceof the proposed technique with reference to the conditionsof the standard [28] for synchronized phasor measurementsystems in power systems The first is a proposed estimationprocedure verified on the basis of the following test signalwhich is characteristic for oscillating conditions
where 119886(119905) = 1198860 + 1198861 sin 2120587119891119886119905 120601(119905) = 1206010 + 1206011 sin 2120587119891120601119905 1198860 =1206010 = 1 1198861 = 1206011 = 01 119891119886 = 119891120601 = 5 and119873 = 24As can be seen from the above relations the test signal
is sampled with a frequency of 1200 Hz thus forming 24samples within a 20 ms wide window which is the period ofthe signal with a fundamental frequency of 50Hz Figures 1(a)and 1(b) show the size of the estimated amplitude and phaseof the fundamental dynamic phasor using the proposed esti-mation method It is clear that the main difference betweenthe proposed method and some other methods based on theapplication of the least square methods [15 21 27] and themethods based on the application of Kalman filters [16 1723 31] is in the occurrence of the delay in the estimation dueto the use of the data window
One of the established criteria for assessing the qualityof the estimation is based on the Total Vector Error (TVE)where we can estimate the error size in the estimation of thephasor magnitude and angle defining it as
119879119881119864 = 1003816100381610038161003816119883119903 minus 119883119890100381610038161003816100381610038161003816100381610038161198831199031003816100381610038161003816 (61)
where 119883119903 and 119883119890 are real and estimated values Figure 2shows the total vector error of the proposed method in thefirst ten cycles one cycle time delay of estimated phasor iscompensated as in [27] It can be concluded that the erroroccurring is above all the results of the introduced delaysimilar to that of [15 21] Owing to the modification madein calculating the parameters of the phasors the magnitudesof this calculated error are smaller than in other methodsbased on the least squares method with their size becomingcomparable to [16 17 23 38] therefore being much smallerthan [27 39ndash41] The proposed approach can calculate thederivatives of the phasor-phasor speed and accelerationwhich reduces the estimation error In addition it is possibleto calculate the frequency of the processed signal and detectfaults and power swings
Mathematical Problems in Engineering 9
Table 6 The values of 119860119901(119873 1199090)1198600(119873 1199090) 11987321198601(119873 1199090) 119873241198602(119873 1199090) 16 (1198733 minus 119873) minus 14119873 sdot ctg211990901198603(119873 1199090) 18 (1198734 minus 21198732) minus 381198732 sdot ctg211990901198604(119873 1199090) 130 (31198735 minus 101198733 + 7119873) + 34119873 sdot ctg41199090 minus 12 (1198733 minus 2119873) sdot ctg21199090
Table 8 The values 119862119901(119873 1199090)1198620(119873 1199090) 0
1198621(119873 1199090) minus1198734 sdot ctg11990901198622(119873 1199090) minus11987324 sdot ctg11990901198623(119873 1199090) 18 sdot (3119873 sdot ctg31199090 minus (21198733 minus 3119873) sdot ctg1199090)1198624(119873 1199090) 14 sdot (31198732 sdot ctg31199090 minus (1198734 minus 31198732) sdot ctg1199090)
41 Amplitude Oscillation Case The ability of the proposedapproach to track an arbitrary oscillating amplitude is givenin the following example where
119909 (119905) = 119886 (119905) infinsumℎ=0
119888ℎ cos 2120587ℎ1198911119905 (62)
is the input signal to the estimator with 1198911 = 50 Hz andnonzero Fourier coefficients (1198881 = 1 1198883 = 04 and 1198885 = 02)and where 119886(119905) represents an amplitude oscillation given bythe following second-order polynomial
with coefficients 119886 = minus4 119887 = 4 and 119888 = 0 Thecorresponding digital signal is sampled at 24 samplescycle(119891 = 1200Hz) The oscillation of main-original signal andits corresponding estimated signal components are shown in
Figure 3 Based on this representation it can be clearly seenthat the proposed method for estimation is fully conduciveto following the dynamics of all components of the processedinput signal Also we can see that the quality of estimationdecreases with increasing the number of harmonic which arethe results of relatively small and fixed-constant number ofsamples For higher harmonics components this number ofsamples cannot provide precise reconstruction If the numberof samples adaptively changed as the function of the orderof harmonics the accuracy of the estimation would be thesame as on the first-fundamental harmonic component Anadditional problem is certainly the nonideality of the filterfunction
42 Magnitude-Phase Step Test In this section the amplitudeand the phase step tests are performed using the proposedestimation procedureThe test signal is defined as [28] (in thestandard changes in this step are happening separately)
119909 (119905) = 119883119898 (1 + 119896119886119906 (119905 minus 119905119904)) cos (120596119905 + 119896120593119906 (119905 minus 119905119904)) (64)
where 119906(119905) is a unit step function 119896119886 is the amplitude stepsize 119896120593 is the phase step size and 119905119904 is the step instant In thistest a signal with the 10∘ phase angle step and the amplitudestep changes from 09 to 1 pu in the fundamental componentwhich is fed to the algorithm The parameters are chosen as119883119898 = radic2120596 = 2120587 sdot 50119867119911 119896119886 = +01119901119906 119896120601 = 12058718 whilethe measurement noise is set to be negligible The results forthe amplitude and phase step tests are shown in Figure 4 inwhich the upper subplot shows the transient response of theamplitude or the phase while in the lower plot the TVE areillustrated
It is observed that the proposed approach achievesrelatively long transient periods but with small overshootsBased on the results presented in Figure 4 the proposedmethod is capable of performing a precise estimation ofthe amplitude and phase after the expiry of the transitionperiod (the model is more appropriate for smooth amplitudeand phase changes) since it represents the modification ofTailorrsquos-Fourier expansion The duration of this transition inthe proposed approach is shorter than in some othermethodsthat have the same starting point [17 27] which is the resultof the position of the poles of the transfer function withrespect to the unit circle in the 119911 plane (as the poles aremore spaced) In addition there is no large overshoot in theresponse as opposed to [16 17 23] due to the amplitude andphase discontinuities at the step change since no previousassessment is performed concerning a sample window
In Table 9 the performance indices defined in the stan-dard [28] including response time delay time and overshootare listed for both amplitude and phase step change tests Theperformances achieved by the proposed algorithm fulfill theM-Class measurement requirements in the standard
43 Frequency Step Test During this simulation test a signalhaving a 5 Hz frequency step is used to evaluate the possi-bility of a transient response in a frequency step condition(although 5 Hz frequency step is not likely to happen in apower grid this test is designed based on IEEE standard [28])
119909 (119899) = cos 21205871198911119899Δ119905 0 lt 119899 lt 101198731119909 (119899) = cos 2120587 (1198911 + 5) 119899Δ119905 101198731 le 119899 (65)
Table 9 Performance indices for phasor estimation under stepchanges in amplitude and phase using the proposed estimationalgorithm
Response time Delay time OvershootAmplitude step 197ms 886ms 0008 [pu]Phase step 967ms 002 [rad]
The frequency was calculated as a ratio of roots of char-acteristic equation based on extracted unknown coefficients1198601198980 1198601198981 1198601198982 1198611198980 1198611198981 1198611198982 from (12) and 2120587 [15 27]The proposed method shows characteristics as well as theevaluation process similar to the previous type of signal(magnitude-phase step condition) Figure 5 shows that theproposed processing method can be successfully tracked as+5 Hz frequency step versus the traditional approach toprocessing the phasors
44 Harmonic Infiltration Test Input signal definedwith (66)was used to validate the proposed methods in harmoniccondition119909 (119905) = 119886 (119905) cos (1205961119905 + 120601 (119905)) + 005 cos 51205961119905 + 003 cos 71205961119905119886 (119905) = 1198860 + 1198861 sin 2120587119891119886119905120601 (119905) = 1206010 + 1206011 sin 21205871198911206011199051198860 = 1 1206010 = 05 1198861 = 01 1206011 = 005 119891119886 = 119891120601 = 5 119873 = 24
(66)
Figure 6 represents the output of the proposed method inthe harmonic condition According to Figure 6 fundamentalphasor estimation based on proposed modification is freefrom harmonic which shows the superiority of harmonicmodification of the proposed method in this paper Since119873 = 24 in this test it is possible to estimate the dynamicphasor of the first 119873 minus 1 = 23 harmonics To increase therange of harmonics the sampling number per cycle shouldbe increased
45 Frequency Response In order to evaluate the frequencycharacteristic of the proposed estimator its transfer functionis defined in the form
Figure 3 Measured and estimated waveforms in the case of a time-varying signal (62) (a) tracking the first phasors (b) tracking the 3rdphasors (c) tracking the 5th phasors and (d) measured and estimated waveforms of main input signal
12 Mathematical Problems in Engineering
0minus20 40 60 8020Time [ms]
realestim
095
1
105
11
115
12
Am
plitu
de [p
u]
(a)
0minus20 40 60 8020Time [ms]
realestim
minus01
0
01
02
03
Phas
e [ra
d]
(b)
0minus20 40 60 8020Time [ms]
0
05
1
15
2
25
3
TVE
[]
(c)
Figure 4 Estimation of (a) amplitude (b) phase and (c) TVE in percentage
minus10
0
10
Freq
uenc
y (k
Hz)
1510 205Normalized time (cycles)
Figure 5 Frequency estimation during frequency step test
where119883(119911) and 119901(119911) are 119911 transforms of the input (main sig-nal) and output (estimated dynamic phasor) The frequencyresponse of the proposed method is given in Figure 7 Themagnitude responses are unsymmetrical which is completelyunderstandable since it belongs to complex filters In thesurroundings of the fundamental frequency complete elimi-nation of the negative one and complete pass of the positiveone occur The flatter transfer characteristic causes a lowerdistortion in the estimation of the phasor and provides zero-gain in the nonfundamental component In this way the har-monics at the output are removed as opposed to [16 21 23]
46 Error Bounds In order to determine the boundary of thepossible error in the estimation of the phasor signals withvariable envelopes were used
119886 (119905) = 1 + 05 3sum119894=1
119890119905120591119894 cos 21205871198911119905 (68)
where the time constants (120591i) were generated by a uniformrandom process in the interval of [20 40] cycles In a similarway the three frequencies were randomly generated in theintervals of [1 3] [3 5] and [5 7]Hz The error is calculatedby
119903119898119904 minus 119890119903119903119900119903 = radicsum(119886 (119899) minus 119886 (119899)) (69)
where 119886(119899) and 119886(119899) are real and estimated amplitudesrespectively Figure 8 shows the histograms of the errorsattained by proposed method
The RMS error belongs to the range of 3times10minus3 and8times10minus3 for proposed estimation method This error is relatedto elimination of the third derivative term of the Taylorexpansion in phasor estimation process and it is much betterthan error estimated in [16 17 22ndash24 27]
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
sdot (3119873 sdot ctg3119898120596119879 minus (21198733 minus 3119873) sdot ctg119898120596119879) (58)
11988833 = 11987944sdot (31198732 sdot ctg3119898120596119879 minus (1198734 minus 31198732) sdot ctg119898120596119879)
(59)
The presented representation of Gramrsquos matrix shows thatthe computation of pseudoinverse (119867119879 sdot 119867)minus1 sdot 119867119879 can beperformed more rapidly and with more precision
4 Simulation Results
The proposed phasor measurement scheme is employedto estimate coefficients of the power signal under nonsta-tionary scenarios during which we compared the resultsobtained in the proposed algorithm with some other well-known procedures Practically through the simulation testwe estimated the ability of the algorithms to track harmonicchanges Tests have been performed to show the performanceof the proposed technique with reference to the conditionsof the standard [28] for synchronized phasor measurementsystems in power systems The first is a proposed estimationprocedure verified on the basis of the following test signalwhich is characteristic for oscillating conditions
where 119886(119905) = 1198860 + 1198861 sin 2120587119891119886119905 120601(119905) = 1206010 + 1206011 sin 2120587119891120601119905 1198860 =1206010 = 1 1198861 = 1206011 = 01 119891119886 = 119891120601 = 5 and119873 = 24As can be seen from the above relations the test signal
is sampled with a frequency of 1200 Hz thus forming 24samples within a 20 ms wide window which is the period ofthe signal with a fundamental frequency of 50Hz Figures 1(a)and 1(b) show the size of the estimated amplitude and phaseof the fundamental dynamic phasor using the proposed esti-mation method It is clear that the main difference betweenthe proposed method and some other methods based on theapplication of the least square methods [15 21 27] and themethods based on the application of Kalman filters [16 1723 31] is in the occurrence of the delay in the estimation dueto the use of the data window
One of the established criteria for assessing the qualityof the estimation is based on the Total Vector Error (TVE)where we can estimate the error size in the estimation of thephasor magnitude and angle defining it as
119879119881119864 = 1003816100381610038161003816119883119903 minus 119883119890100381610038161003816100381610038161003816100381610038161198831199031003816100381610038161003816 (61)
where 119883119903 and 119883119890 are real and estimated values Figure 2shows the total vector error of the proposed method in thefirst ten cycles one cycle time delay of estimated phasor iscompensated as in [27] It can be concluded that the erroroccurring is above all the results of the introduced delaysimilar to that of [15 21] Owing to the modification madein calculating the parameters of the phasors the magnitudesof this calculated error are smaller than in other methodsbased on the least squares method with their size becomingcomparable to [16 17 23 38] therefore being much smallerthan [27 39ndash41] The proposed approach can calculate thederivatives of the phasor-phasor speed and accelerationwhich reduces the estimation error In addition it is possibleto calculate the frequency of the processed signal and detectfaults and power swings
Mathematical Problems in Engineering 9
Table 6 The values of 119860119901(119873 1199090)1198600(119873 1199090) 11987321198601(119873 1199090) 119873241198602(119873 1199090) 16 (1198733 minus 119873) minus 14119873 sdot ctg211990901198603(119873 1199090) 18 (1198734 minus 21198732) minus 381198732 sdot ctg211990901198604(119873 1199090) 130 (31198735 minus 101198733 + 7119873) + 34119873 sdot ctg41199090 minus 12 (1198733 minus 2119873) sdot ctg21199090
Table 8 The values 119862119901(119873 1199090)1198620(119873 1199090) 0
1198621(119873 1199090) minus1198734 sdot ctg11990901198622(119873 1199090) minus11987324 sdot ctg11990901198623(119873 1199090) 18 sdot (3119873 sdot ctg31199090 minus (21198733 minus 3119873) sdot ctg1199090)1198624(119873 1199090) 14 sdot (31198732 sdot ctg31199090 minus (1198734 minus 31198732) sdot ctg1199090)
41 Amplitude Oscillation Case The ability of the proposedapproach to track an arbitrary oscillating amplitude is givenin the following example where
119909 (119905) = 119886 (119905) infinsumℎ=0
119888ℎ cos 2120587ℎ1198911119905 (62)
is the input signal to the estimator with 1198911 = 50 Hz andnonzero Fourier coefficients (1198881 = 1 1198883 = 04 and 1198885 = 02)and where 119886(119905) represents an amplitude oscillation given bythe following second-order polynomial
with coefficients 119886 = minus4 119887 = 4 and 119888 = 0 Thecorresponding digital signal is sampled at 24 samplescycle(119891 = 1200Hz) The oscillation of main-original signal andits corresponding estimated signal components are shown in
Figure 3 Based on this representation it can be clearly seenthat the proposed method for estimation is fully conduciveto following the dynamics of all components of the processedinput signal Also we can see that the quality of estimationdecreases with increasing the number of harmonic which arethe results of relatively small and fixed-constant number ofsamples For higher harmonics components this number ofsamples cannot provide precise reconstruction If the numberof samples adaptively changed as the function of the orderof harmonics the accuracy of the estimation would be thesame as on the first-fundamental harmonic component Anadditional problem is certainly the nonideality of the filterfunction
42 Magnitude-Phase Step Test In this section the amplitudeand the phase step tests are performed using the proposedestimation procedureThe test signal is defined as [28] (in thestandard changes in this step are happening separately)
119909 (119905) = 119883119898 (1 + 119896119886119906 (119905 minus 119905119904)) cos (120596119905 + 119896120593119906 (119905 minus 119905119904)) (64)
where 119906(119905) is a unit step function 119896119886 is the amplitude stepsize 119896120593 is the phase step size and 119905119904 is the step instant In thistest a signal with the 10∘ phase angle step and the amplitudestep changes from 09 to 1 pu in the fundamental componentwhich is fed to the algorithm The parameters are chosen as119883119898 = radic2120596 = 2120587 sdot 50119867119911 119896119886 = +01119901119906 119896120601 = 12058718 whilethe measurement noise is set to be negligible The results forthe amplitude and phase step tests are shown in Figure 4 inwhich the upper subplot shows the transient response of theamplitude or the phase while in the lower plot the TVE areillustrated
It is observed that the proposed approach achievesrelatively long transient periods but with small overshootsBased on the results presented in Figure 4 the proposedmethod is capable of performing a precise estimation ofthe amplitude and phase after the expiry of the transitionperiod (the model is more appropriate for smooth amplitudeand phase changes) since it represents the modification ofTailorrsquos-Fourier expansion The duration of this transition inthe proposed approach is shorter than in some othermethodsthat have the same starting point [17 27] which is the resultof the position of the poles of the transfer function withrespect to the unit circle in the 119911 plane (as the poles aremore spaced) In addition there is no large overshoot in theresponse as opposed to [16 17 23] due to the amplitude andphase discontinuities at the step change since no previousassessment is performed concerning a sample window
In Table 9 the performance indices defined in the stan-dard [28] including response time delay time and overshootare listed for both amplitude and phase step change tests Theperformances achieved by the proposed algorithm fulfill theM-Class measurement requirements in the standard
43 Frequency Step Test During this simulation test a signalhaving a 5 Hz frequency step is used to evaluate the possi-bility of a transient response in a frequency step condition(although 5 Hz frequency step is not likely to happen in apower grid this test is designed based on IEEE standard [28])
119909 (119899) = cos 21205871198911119899Δ119905 0 lt 119899 lt 101198731119909 (119899) = cos 2120587 (1198911 + 5) 119899Δ119905 101198731 le 119899 (65)
Table 9 Performance indices for phasor estimation under stepchanges in amplitude and phase using the proposed estimationalgorithm
Response time Delay time OvershootAmplitude step 197ms 886ms 0008 [pu]Phase step 967ms 002 [rad]
The frequency was calculated as a ratio of roots of char-acteristic equation based on extracted unknown coefficients1198601198980 1198601198981 1198601198982 1198611198980 1198611198981 1198611198982 from (12) and 2120587 [15 27]The proposed method shows characteristics as well as theevaluation process similar to the previous type of signal(magnitude-phase step condition) Figure 5 shows that theproposed processing method can be successfully tracked as+5 Hz frequency step versus the traditional approach toprocessing the phasors
44 Harmonic Infiltration Test Input signal definedwith (66)was used to validate the proposed methods in harmoniccondition119909 (119905) = 119886 (119905) cos (1205961119905 + 120601 (119905)) + 005 cos 51205961119905 + 003 cos 71205961119905119886 (119905) = 1198860 + 1198861 sin 2120587119891119886119905120601 (119905) = 1206010 + 1206011 sin 21205871198911206011199051198860 = 1 1206010 = 05 1198861 = 01 1206011 = 005 119891119886 = 119891120601 = 5 119873 = 24
(66)
Figure 6 represents the output of the proposed method inthe harmonic condition According to Figure 6 fundamentalphasor estimation based on proposed modification is freefrom harmonic which shows the superiority of harmonicmodification of the proposed method in this paper Since119873 = 24 in this test it is possible to estimate the dynamicphasor of the first 119873 minus 1 = 23 harmonics To increase therange of harmonics the sampling number per cycle shouldbe increased
45 Frequency Response In order to evaluate the frequencycharacteristic of the proposed estimator its transfer functionis defined in the form
Figure 3 Measured and estimated waveforms in the case of a time-varying signal (62) (a) tracking the first phasors (b) tracking the 3rdphasors (c) tracking the 5th phasors and (d) measured and estimated waveforms of main input signal
12 Mathematical Problems in Engineering
0minus20 40 60 8020Time [ms]
realestim
095
1
105
11
115
12
Am
plitu
de [p
u]
(a)
0minus20 40 60 8020Time [ms]
realestim
minus01
0
01
02
03
Phas
e [ra
d]
(b)
0minus20 40 60 8020Time [ms]
0
05
1
15
2
25
3
TVE
[]
(c)
Figure 4 Estimation of (a) amplitude (b) phase and (c) TVE in percentage
minus10
0
10
Freq
uenc
y (k
Hz)
1510 205Normalized time (cycles)
Figure 5 Frequency estimation during frequency step test
where119883(119911) and 119901(119911) are 119911 transforms of the input (main sig-nal) and output (estimated dynamic phasor) The frequencyresponse of the proposed method is given in Figure 7 Themagnitude responses are unsymmetrical which is completelyunderstandable since it belongs to complex filters In thesurroundings of the fundamental frequency complete elimi-nation of the negative one and complete pass of the positiveone occur The flatter transfer characteristic causes a lowerdistortion in the estimation of the phasor and provides zero-gain in the nonfundamental component In this way the har-monics at the output are removed as opposed to [16 21 23]
46 Error Bounds In order to determine the boundary of thepossible error in the estimation of the phasor signals withvariable envelopes were used
119886 (119905) = 1 + 05 3sum119894=1
119890119905120591119894 cos 21205871198911119905 (68)
where the time constants (120591i) were generated by a uniformrandom process in the interval of [20 40] cycles In a similarway the three frequencies were randomly generated in theintervals of [1 3] [3 5] and [5 7]Hz The error is calculatedby
119903119898119904 minus 119890119903119903119900119903 = radicsum(119886 (119899) minus 119886 (119899)) (69)
where 119886(119899) and 119886(119899) are real and estimated amplitudesrespectively Figure 8 shows the histograms of the errorsattained by proposed method
The RMS error belongs to the range of 3times10minus3 and8times10minus3 for proposed estimation method This error is relatedto elimination of the third derivative term of the Taylorexpansion in phasor estimation process and it is much betterthan error estimated in [16 17 22ndash24 27]
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Table 8 The values 119862119901(119873 1199090)1198620(119873 1199090) 0
1198621(119873 1199090) minus1198734 sdot ctg11990901198622(119873 1199090) minus11987324 sdot ctg11990901198623(119873 1199090) 18 sdot (3119873 sdot ctg31199090 minus (21198733 minus 3119873) sdot ctg1199090)1198624(119873 1199090) 14 sdot (31198732 sdot ctg31199090 minus (1198734 minus 31198732) sdot ctg1199090)
41 Amplitude Oscillation Case The ability of the proposedapproach to track an arbitrary oscillating amplitude is givenin the following example where
119909 (119905) = 119886 (119905) infinsumℎ=0
119888ℎ cos 2120587ℎ1198911119905 (62)
is the input signal to the estimator with 1198911 = 50 Hz andnonzero Fourier coefficients (1198881 = 1 1198883 = 04 and 1198885 = 02)and where 119886(119905) represents an amplitude oscillation given bythe following second-order polynomial
with coefficients 119886 = minus4 119887 = 4 and 119888 = 0 Thecorresponding digital signal is sampled at 24 samplescycle(119891 = 1200Hz) The oscillation of main-original signal andits corresponding estimated signal components are shown in
Figure 3 Based on this representation it can be clearly seenthat the proposed method for estimation is fully conduciveto following the dynamics of all components of the processedinput signal Also we can see that the quality of estimationdecreases with increasing the number of harmonic which arethe results of relatively small and fixed-constant number ofsamples For higher harmonics components this number ofsamples cannot provide precise reconstruction If the numberof samples adaptively changed as the function of the orderof harmonics the accuracy of the estimation would be thesame as on the first-fundamental harmonic component Anadditional problem is certainly the nonideality of the filterfunction
42 Magnitude-Phase Step Test In this section the amplitudeand the phase step tests are performed using the proposedestimation procedureThe test signal is defined as [28] (in thestandard changes in this step are happening separately)
119909 (119905) = 119883119898 (1 + 119896119886119906 (119905 minus 119905119904)) cos (120596119905 + 119896120593119906 (119905 minus 119905119904)) (64)
where 119906(119905) is a unit step function 119896119886 is the amplitude stepsize 119896120593 is the phase step size and 119905119904 is the step instant In thistest a signal with the 10∘ phase angle step and the amplitudestep changes from 09 to 1 pu in the fundamental componentwhich is fed to the algorithm The parameters are chosen as119883119898 = radic2120596 = 2120587 sdot 50119867119911 119896119886 = +01119901119906 119896120601 = 12058718 whilethe measurement noise is set to be negligible The results forthe amplitude and phase step tests are shown in Figure 4 inwhich the upper subplot shows the transient response of theamplitude or the phase while in the lower plot the TVE areillustrated
It is observed that the proposed approach achievesrelatively long transient periods but with small overshootsBased on the results presented in Figure 4 the proposedmethod is capable of performing a precise estimation ofthe amplitude and phase after the expiry of the transitionperiod (the model is more appropriate for smooth amplitudeand phase changes) since it represents the modification ofTailorrsquos-Fourier expansion The duration of this transition inthe proposed approach is shorter than in some othermethodsthat have the same starting point [17 27] which is the resultof the position of the poles of the transfer function withrespect to the unit circle in the 119911 plane (as the poles aremore spaced) In addition there is no large overshoot in theresponse as opposed to [16 17 23] due to the amplitude andphase discontinuities at the step change since no previousassessment is performed concerning a sample window
In Table 9 the performance indices defined in the stan-dard [28] including response time delay time and overshootare listed for both amplitude and phase step change tests Theperformances achieved by the proposed algorithm fulfill theM-Class measurement requirements in the standard
43 Frequency Step Test During this simulation test a signalhaving a 5 Hz frequency step is used to evaluate the possi-bility of a transient response in a frequency step condition(although 5 Hz frequency step is not likely to happen in apower grid this test is designed based on IEEE standard [28])
119909 (119899) = cos 21205871198911119899Δ119905 0 lt 119899 lt 101198731119909 (119899) = cos 2120587 (1198911 + 5) 119899Δ119905 101198731 le 119899 (65)
Table 9 Performance indices for phasor estimation under stepchanges in amplitude and phase using the proposed estimationalgorithm
Response time Delay time OvershootAmplitude step 197ms 886ms 0008 [pu]Phase step 967ms 002 [rad]
The frequency was calculated as a ratio of roots of char-acteristic equation based on extracted unknown coefficients1198601198980 1198601198981 1198601198982 1198611198980 1198611198981 1198611198982 from (12) and 2120587 [15 27]The proposed method shows characteristics as well as theevaluation process similar to the previous type of signal(magnitude-phase step condition) Figure 5 shows that theproposed processing method can be successfully tracked as+5 Hz frequency step versus the traditional approach toprocessing the phasors
44 Harmonic Infiltration Test Input signal definedwith (66)was used to validate the proposed methods in harmoniccondition119909 (119905) = 119886 (119905) cos (1205961119905 + 120601 (119905)) + 005 cos 51205961119905 + 003 cos 71205961119905119886 (119905) = 1198860 + 1198861 sin 2120587119891119886119905120601 (119905) = 1206010 + 1206011 sin 21205871198911206011199051198860 = 1 1206010 = 05 1198861 = 01 1206011 = 005 119891119886 = 119891120601 = 5 119873 = 24
(66)
Figure 6 represents the output of the proposed method inthe harmonic condition According to Figure 6 fundamentalphasor estimation based on proposed modification is freefrom harmonic which shows the superiority of harmonicmodification of the proposed method in this paper Since119873 = 24 in this test it is possible to estimate the dynamicphasor of the first 119873 minus 1 = 23 harmonics To increase therange of harmonics the sampling number per cycle shouldbe increased
45 Frequency Response In order to evaluate the frequencycharacteristic of the proposed estimator its transfer functionis defined in the form
Figure 3 Measured and estimated waveforms in the case of a time-varying signal (62) (a) tracking the first phasors (b) tracking the 3rdphasors (c) tracking the 5th phasors and (d) measured and estimated waveforms of main input signal
12 Mathematical Problems in Engineering
0minus20 40 60 8020Time [ms]
realestim
095
1
105
11
115
12
Am
plitu
de [p
u]
(a)
0minus20 40 60 8020Time [ms]
realestim
minus01
0
01
02
03
Phas
e [ra
d]
(b)
0minus20 40 60 8020Time [ms]
0
05
1
15
2
25
3
TVE
[]
(c)
Figure 4 Estimation of (a) amplitude (b) phase and (c) TVE in percentage
minus10
0
10
Freq
uenc
y (k
Hz)
1510 205Normalized time (cycles)
Figure 5 Frequency estimation during frequency step test
where119883(119911) and 119901(119911) are 119911 transforms of the input (main sig-nal) and output (estimated dynamic phasor) The frequencyresponse of the proposed method is given in Figure 7 Themagnitude responses are unsymmetrical which is completelyunderstandable since it belongs to complex filters In thesurroundings of the fundamental frequency complete elimi-nation of the negative one and complete pass of the positiveone occur The flatter transfer characteristic causes a lowerdistortion in the estimation of the phasor and provides zero-gain in the nonfundamental component In this way the har-monics at the output are removed as opposed to [16 21 23]
46 Error Bounds In order to determine the boundary of thepossible error in the estimation of the phasor signals withvariable envelopes were used
119886 (119905) = 1 + 05 3sum119894=1
119890119905120591119894 cos 21205871198911119905 (68)
where the time constants (120591i) were generated by a uniformrandom process in the interval of [20 40] cycles In a similarway the three frequencies were randomly generated in theintervals of [1 3] [3 5] and [5 7]Hz The error is calculatedby
119903119898119904 minus 119890119903119903119900119903 = radicsum(119886 (119899) minus 119886 (119899)) (69)
where 119886(119899) and 119886(119899) are real and estimated amplitudesrespectively Figure 8 shows the histograms of the errorsattained by proposed method
The RMS error belongs to the range of 3times10minus3 and8times10minus3 for proposed estimation method This error is relatedto elimination of the third derivative term of the Taylorexpansion in phasor estimation process and it is much betterthan error estimated in [16 17 22ndash24 27]
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
It is observed that the proposed approach achievesrelatively long transient periods but with small overshootsBased on the results presented in Figure 4 the proposedmethod is capable of performing a precise estimation ofthe amplitude and phase after the expiry of the transitionperiod (the model is more appropriate for smooth amplitudeand phase changes) since it represents the modification ofTailorrsquos-Fourier expansion The duration of this transition inthe proposed approach is shorter than in some othermethodsthat have the same starting point [17 27] which is the resultof the position of the poles of the transfer function withrespect to the unit circle in the 119911 plane (as the poles aremore spaced) In addition there is no large overshoot in theresponse as opposed to [16 17 23] due to the amplitude andphase discontinuities at the step change since no previousassessment is performed concerning a sample window
In Table 9 the performance indices defined in the stan-dard [28] including response time delay time and overshootare listed for both amplitude and phase step change tests Theperformances achieved by the proposed algorithm fulfill theM-Class measurement requirements in the standard
43 Frequency Step Test During this simulation test a signalhaving a 5 Hz frequency step is used to evaluate the possi-bility of a transient response in a frequency step condition(although 5 Hz frequency step is not likely to happen in apower grid this test is designed based on IEEE standard [28])
119909 (119899) = cos 21205871198911119899Δ119905 0 lt 119899 lt 101198731119909 (119899) = cos 2120587 (1198911 + 5) 119899Δ119905 101198731 le 119899 (65)
Table 9 Performance indices for phasor estimation under stepchanges in amplitude and phase using the proposed estimationalgorithm
Response time Delay time OvershootAmplitude step 197ms 886ms 0008 [pu]Phase step 967ms 002 [rad]
The frequency was calculated as a ratio of roots of char-acteristic equation based on extracted unknown coefficients1198601198980 1198601198981 1198601198982 1198611198980 1198611198981 1198611198982 from (12) and 2120587 [15 27]The proposed method shows characteristics as well as theevaluation process similar to the previous type of signal(magnitude-phase step condition) Figure 5 shows that theproposed processing method can be successfully tracked as+5 Hz frequency step versus the traditional approach toprocessing the phasors
44 Harmonic Infiltration Test Input signal definedwith (66)was used to validate the proposed methods in harmoniccondition119909 (119905) = 119886 (119905) cos (1205961119905 + 120601 (119905)) + 005 cos 51205961119905 + 003 cos 71205961119905119886 (119905) = 1198860 + 1198861 sin 2120587119891119886119905120601 (119905) = 1206010 + 1206011 sin 21205871198911206011199051198860 = 1 1206010 = 05 1198861 = 01 1206011 = 005 119891119886 = 119891120601 = 5 119873 = 24
(66)
Figure 6 represents the output of the proposed method inthe harmonic condition According to Figure 6 fundamentalphasor estimation based on proposed modification is freefrom harmonic which shows the superiority of harmonicmodification of the proposed method in this paper Since119873 = 24 in this test it is possible to estimate the dynamicphasor of the first 119873 minus 1 = 23 harmonics To increase therange of harmonics the sampling number per cycle shouldbe increased
45 Frequency Response In order to evaluate the frequencycharacteristic of the proposed estimator its transfer functionis defined in the form
Figure 3 Measured and estimated waveforms in the case of a time-varying signal (62) (a) tracking the first phasors (b) tracking the 3rdphasors (c) tracking the 5th phasors and (d) measured and estimated waveforms of main input signal
12 Mathematical Problems in Engineering
0minus20 40 60 8020Time [ms]
realestim
095
1
105
11
115
12
Am
plitu
de [p
u]
(a)
0minus20 40 60 8020Time [ms]
realestim
minus01
0
01
02
03
Phas
e [ra
d]
(b)
0minus20 40 60 8020Time [ms]
0
05
1
15
2
25
3
TVE
[]
(c)
Figure 4 Estimation of (a) amplitude (b) phase and (c) TVE in percentage
minus10
0
10
Freq
uenc
y (k
Hz)
1510 205Normalized time (cycles)
Figure 5 Frequency estimation during frequency step test
where119883(119911) and 119901(119911) are 119911 transforms of the input (main sig-nal) and output (estimated dynamic phasor) The frequencyresponse of the proposed method is given in Figure 7 Themagnitude responses are unsymmetrical which is completelyunderstandable since it belongs to complex filters In thesurroundings of the fundamental frequency complete elimi-nation of the negative one and complete pass of the positiveone occur The flatter transfer characteristic causes a lowerdistortion in the estimation of the phasor and provides zero-gain in the nonfundamental component In this way the har-monics at the output are removed as opposed to [16 21 23]
46 Error Bounds In order to determine the boundary of thepossible error in the estimation of the phasor signals withvariable envelopes were used
119886 (119905) = 1 + 05 3sum119894=1
119890119905120591119894 cos 21205871198911119905 (68)
where the time constants (120591i) were generated by a uniformrandom process in the interval of [20 40] cycles In a similarway the three frequencies were randomly generated in theintervals of [1 3] [3 5] and [5 7]Hz The error is calculatedby
119903119898119904 minus 119890119903119903119900119903 = radicsum(119886 (119899) minus 119886 (119899)) (69)
where 119886(119899) and 119886(119899) are real and estimated amplitudesrespectively Figure 8 shows the histograms of the errorsattained by proposed method
The RMS error belongs to the range of 3times10minus3 and8times10minus3 for proposed estimation method This error is relatedto elimination of the third derivative term of the Taylorexpansion in phasor estimation process and it is much betterthan error estimated in [16 17 22ndash24 27]
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 11
0 001 002 003 004 005 006Time [s]
First harmonic
originalestimated
minus02minus015
minus01minus005
0005
01015
02025
Am
plitu
de [p
u]
(a)
0 001 002 003 004 005 006Time [s]
Third harmonic
originalestimated
minus01minus008minus006minus004minus002
0002004006008
01
Am
plitu
de[p
u]
(b)
0 001 002 003 004 005 006Time [s]
Fifth harmonic
originalestimated
minus005minus004minus003minus002minus001
0001002003004005
Am
plitu
de [p
u]
(c)
0 001 002 003 004 005 006Time [s]
Signals x(t) and x1(t)
x(t)x1(t)
minus04
minus03
minus02
minus01
0
01
02
03
04
Am
plitu
de[p
u]
(d)
Figure 3 Measured and estimated waveforms in the case of a time-varying signal (62) (a) tracking the first phasors (b) tracking the 3rdphasors (c) tracking the 5th phasors and (d) measured and estimated waveforms of main input signal
12 Mathematical Problems in Engineering
0minus20 40 60 8020Time [ms]
realestim
095
1
105
11
115
12
Am
plitu
de [p
u]
(a)
0minus20 40 60 8020Time [ms]
realestim
minus01
0
01
02
03
Phas
e [ra
d]
(b)
0minus20 40 60 8020Time [ms]
0
05
1
15
2
25
3
TVE
[]
(c)
Figure 4 Estimation of (a) amplitude (b) phase and (c) TVE in percentage
minus10
0
10
Freq
uenc
y (k
Hz)
1510 205Normalized time (cycles)
Figure 5 Frequency estimation during frequency step test
where119883(119911) and 119901(119911) are 119911 transforms of the input (main sig-nal) and output (estimated dynamic phasor) The frequencyresponse of the proposed method is given in Figure 7 Themagnitude responses are unsymmetrical which is completelyunderstandable since it belongs to complex filters In thesurroundings of the fundamental frequency complete elimi-nation of the negative one and complete pass of the positiveone occur The flatter transfer characteristic causes a lowerdistortion in the estimation of the phasor and provides zero-gain in the nonfundamental component In this way the har-monics at the output are removed as opposed to [16 21 23]
46 Error Bounds In order to determine the boundary of thepossible error in the estimation of the phasor signals withvariable envelopes were used
119886 (119905) = 1 + 05 3sum119894=1
119890119905120591119894 cos 21205871198911119905 (68)
where the time constants (120591i) were generated by a uniformrandom process in the interval of [20 40] cycles In a similarway the three frequencies were randomly generated in theintervals of [1 3] [3 5] and [5 7]Hz The error is calculatedby
119903119898119904 minus 119890119903119903119900119903 = radicsum(119886 (119899) minus 119886 (119899)) (69)
where 119886(119899) and 119886(119899) are real and estimated amplitudesrespectively Figure 8 shows the histograms of the errorsattained by proposed method
The RMS error belongs to the range of 3times10minus3 and8times10minus3 for proposed estimation method This error is relatedto elimination of the third derivative term of the Taylorexpansion in phasor estimation process and it is much betterthan error estimated in [16 17 22ndash24 27]
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
12 Mathematical Problems in Engineering
0minus20 40 60 8020Time [ms]
realestim
095
1
105
11
115
12
Am
plitu
de [p
u]
(a)
0minus20 40 60 8020Time [ms]
realestim
minus01
0
01
02
03
Phas
e [ra
d]
(b)
0minus20 40 60 8020Time [ms]
0
05
1
15
2
25
3
TVE
[]
(c)
Figure 4 Estimation of (a) amplitude (b) phase and (c) TVE in percentage
minus10
0
10
Freq
uenc
y (k
Hz)
1510 205Normalized time (cycles)
Figure 5 Frequency estimation during frequency step test
where119883(119911) and 119901(119911) are 119911 transforms of the input (main sig-nal) and output (estimated dynamic phasor) The frequencyresponse of the proposed method is given in Figure 7 Themagnitude responses are unsymmetrical which is completelyunderstandable since it belongs to complex filters In thesurroundings of the fundamental frequency complete elimi-nation of the negative one and complete pass of the positiveone occur The flatter transfer characteristic causes a lowerdistortion in the estimation of the phasor and provides zero-gain in the nonfundamental component In this way the har-monics at the output are removed as opposed to [16 21 23]
46 Error Bounds In order to determine the boundary of thepossible error in the estimation of the phasor signals withvariable envelopes were used
119886 (119905) = 1 + 05 3sum119894=1
119890119905120591119894 cos 21205871198911119905 (68)
where the time constants (120591i) were generated by a uniformrandom process in the interval of [20 40] cycles In a similarway the three frequencies were randomly generated in theintervals of [1 3] [3 5] and [5 7]Hz The error is calculatedby
119903119898119904 minus 119890119903119903119900119903 = radicsum(119886 (119899) minus 119886 (119899)) (69)
where 119886(119899) and 119886(119899) are real and estimated amplitudesrespectively Figure 8 shows the histograms of the errorsattained by proposed method
The RMS error belongs to the range of 3times10minus3 and8times10minus3 for proposed estimation method This error is relatedto elimination of the third derivative term of the Taylorexpansion in phasor estimation process and it is much betterthan error estimated in [16 17 22ndash24 27]
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
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Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 13
09
095
1
105
085
08
11
115
12
realestim
Am
plitu
de (p
u)
10 15 205 250Normalized time (cycles)
Figure 6 Amplitude estimation in harmonic conditions
47 Noise Subharmonics and Interharmonics Infiltration It isuseful to analyze the estimation error of the proposedmethodwhen the noise level changes A criterion named transientmonitor (TM) is used for this evaluation [21] TM is an indexto monitor a sudden change of the measured signal and iscalculated by
[119905119899] = [119878119899 minus 119878119899]119879119872 = 119899=119903sum
where 119878119899 is the real (measured) value of sample and 119878119899 isthe recomputed sample obtained by the estimated dynamicphasor Estimation error 119905119899 is the difference between thesetwo quantities TM is calculated by estimation error in everysample and can be used as a quality measure of phasorestimation Figure 9 shows the TM as a function of noisevariance for the proposed and somewell-knownmethods forphasors estimation As the noise variance increases highererrors are obtained so there is an upward trend when noiselevel increases According to Figure 9 there is a stable trendfor least squares-based methods and our proposed methodbefore critical point with variance of about 10minus2 Afterthis point transient monitor increases steeply by increasingerror variance Generally Kalman based methods [16 17 23]show lower values compared to least square-based methodshowever our proposed algorithm has the best performancerelative to other Taylor expansion based methods when noisevariance is low according to Figure 9 Also we can concludethat the proposed method depicts a downward trend byincreasing the sampling frequency because the estimationaccuracy increases with a high sampling rate and contrary to[21] does not increase its noise sensitivity at a high samplingrate
The standard [28] does not explicitly define the behaviourof estimation methods in the presence of interharmonicand subharmonic components which can be found in the
installations in which these measurements are applied Ingeneral interharmonics can be very difficult to detect andisolate if they are located in the band of interest of thephasor dynamic model In order to obtain an even morecomplete picture of the performance of the phasor estimationmethod described here a subharmonic frequency of 20 Hzas well as two interharmonic components frequency of 575Hz and 85 Hz [5] with a 10 amplitude with respect tothe fundamental component amplitude has been added toa sinusoidal signal with the aforementioned contents Anout-of-band interference test is defined to test the capabilityto filter interfering signals that could be aliased into themeasurement and it is considered only for M class and it iseliminated for P class [39]
The precision in estimation of the proposed algorithmsis directly related to their filtering capabilities When theinterharmonic is in the passband of the algorithm (designedto include dynamic phenomena of interest) as for inter-harmonic components on 575 Hz the TVE is about 3whereas for interharmonics in the stopband frequencies TVEdepends on the stopband attenuation In this case for aninterharmonic frequency of 85 Hz TVE is about 05 Thesame situation is for subharmonics components on 20 HzThe obtained results are better than the results obtained in[5 39]
The proposed approach in estimation of phasor valueswas also tested in a situation when processing the input signalcontains the third harmonic with amplitude settled to 20of the fundamental component and an interharmonic fre-quency 75 Hz This level of the harmonics is very commonlypresent in current waveforms in the control applications suchas a current tracking in shunt active power filters (APFs) inmarine networks or smart distributed grids with renewableenergy sources [39] TVE in estimation of third harmonicsin this test was about 1 In all above-mentioned tests thenumber of samples was N=24
48 Computational Complexity and Simulation Time Theinputs of the algorithm are matrix119867 of the type119873times2(119870+1)and matrix 119909119891119894119897119898 of the type 119873 times 1 given by (16) and (13)respectively The output of the algorithm is column matrix119910119898 of the type 2(119870 + 1) times 1 obtained by (19) The proposedalgorithm can be decomposed in the following steps
(A1) Calculation of the elements of Gramrsquos matrix 119867119879 sdot 119867by (45)-(59)
(A2) Inversion (119867119879 sdot 119867)minus1 of Gramrsquos matrix 119867119879 sdot 119867(A3) Multiplication of (119867119879 sdot 119867)minus1 with 119867119879(A4) Multiplication of (119867119879 sdot 119867)minus1 sdot 119867119879 with 119909119891119894119897119898
(A1) If the formulas (45)-(59) are used for calculation of theelements of Grams matrix119867119879 sdot119867 then the task is reduced to4(119870+1)2 consecutive calculations of specific values based onpolynomials If 1198880 denotes the maximum of all computationaltimes needed for executing (45)-(59) then the computationaltime for119867119879 sdot 119867 does not exceed 4(119870 + 1)21198880
(A2) The solution to (19) includes the inversion of theGramrsquosmatrix To find the inverse of 2(119870+1)times2(119870+1)-matrix
14 Mathematical Problems in Engineering
0 2minus20
1
2
mag
nitu
de
Normalized frequency ff1(a)
0 1 2
0
5
minus5
phas
e
Normalized frequency ff1(b)
Figure 7 Frequency response (a) magnitude response (b) phase response
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
(A3) In this step the matrix (119867119879 sdot 119867)minus1 of type 2(119870 +1) times 2(119870 + 1) is to be multiplied by the matrix 119867119879 of the type2(119870 + 1) times 119873 which takes the 4(119870 + 1)2119873 FLOPs(A4) In this step the matrix (119867119879 sdot 119867)minus1 sdot 119867119879 of the type2(119870+ 1) times119873 is to be multiplied by the matrix 119909119891119894119897119898 of the type119873 times 1 which takes the 2(119870 + 1)119873 FLOPsBased on the all above-mentioned procedures-steps the
operations-FLOPs which practically defined its computa-tional costs If the step (A1) is replaced with
(Arsquo1) Multiplication of119867119879 with 119867
then the computational time of this step is increased withrespect to the time needed for performing (A1) Since 119867 is119873times2(119870+1)-matrix it follows that119867119879 sdot119867 is 2(119870+1)times2(119870+1)-matrix and multiplication of matrices gives an algorithmthat takes time 4(119870 + 1)2119873 to multiply the matrix 119867119879 of
type 2(119870 + 1) times 119873 with the matrix 119867 of type 119873 times 2119870Clearly the time needed for such an algorithm depends onthe number of samples 119873 On the other hand the time 1198880 in(A1) is independent of119873 so the number of samples does notinfluence the speed of (A1)This enables us to deal with a largenumber of samples without decreasing the performanceof algorithm Also if the time 1198880 is the maximum of allcomputational times needed for executing (45)-(59) clearlyit is the time needed for realization of (54) If ctg119909 is regardedas ctg119909 asymp 1119909 minus 1199093 then its computational time takes3 FLOPs for basic arithmetic operations Consequently thecomputational time for (54) can be approximated with finitenumber 119888which is the number of basic arithmetic operations(addition subtraction multiplication and division) in (54)
In order to practically (in real conditions) determinethe time needed for the computation of unknown phasorsparameters according to the proposedmethod the frequencyof the processed signal is increased which can clearly definewhether the method can be used for offline or online pro-cessing A comparison was made with well-known methodsin the literature and the resulting comparisons were given inFigure 10 The hardware features of the following character-istics were used Intel(R) Core(TM) i5-4460 32GHz 16 GBRAMThe proposedmodification in the algorithm describedhere showed that the time required for the calculation wassignificantly less than in other methods based on the leastsquares method [24 27] and approaching the speed bymethods [22 23 39ndash41]
5 Conclusions
In this paper we used the dynamic phasor concept to estimatethe variable amplitude and phase in processing of oscillatingsignals in modern power grids The specific modification ofTaylor-Fourier expansion and corresponding Gram matrixgive us the possibility to develop more precise and compu-tational attractive algorithm for estimation of all unknownparameters of processing signal in all conditions definedwith standard Simulation results demonstrate the accuracyand capability of the proposed method in view of TVEharmonic condition frequency tracking and computationtime The proposed method has been investigated under
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 15
Tran
siant
Mon
itor
Variance
Ref [15]Ref [23]Ref [16]
Ref [21]Ref [17]
Ref [22]
Ref [27]Proposed method
10minus3
10minus8 10minus6 10minus4 10minus2 100
10minus2
100
101
102
10minus1
Figure 9 Transient monitor versus noise variance
0 20 40 60 80 100 120sample per cycle
Com
puta
tion
time
Ref [22]
Ref [27]Proposed method
Ref [15]Ref [23]Ref [16]
Ref [17]Ref [21]
10minus3
10minus2
100
101
102
103
10minus1
Figure 10 Computation time versus sampling frequency
different conditions and found to be a valuable and efficienttool for detection of signal components under dynamicconditions The simulations results have shown that theproposed technique provides accurate estimates and offersthe possibility to track the phase frequency and amplitudechanges of nonstationary signals Due to its specific formwhich enables the reduction of both the computational costand the memory requirements of the algorithm the proposedalgorithm can be very useful for real-time digital systems
Data Availability
The numerical data obtained analytical relations and pre-sented graphics data used to support the findings of this studyare included within the article
Conflicts of Interest
The authors declare that they have no conflicts of interest
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
16 Mathematical Problems in Engineering
Acknowledgments
The research is supported by the Ministry of EducationScience and Technological Development Republic of SerbiaGrant Nos 42009 172057 and 174013
References
[1] J A Rohten J R Espinoza J A Munoz et al ldquoEnhancedPredictive Control for a Wide Time-Variant Frequency Envi-ronmentrdquo IEEE Transactions on Industrial Electronics vol 63no 9 pp 5827ndash5837 2016
[2] I Kamwa S R Samantaray and G Joos ldquoWide frequencyrange adaptive phasor and frequency PMU algorithmsrdquo IEEETransactions on Smart Grid vol 5 no 2 pp 569ndash579 2014
[3] S-K Chung ldquoHarmonic power flow determination using thefast Fourier transformrdquo IEEE Transactions on Power Deliveryvol 2 no 2 pp 530ndash535 1991
[4] C A G Marques M V Ribeiro C A Duque P F Ribeiro andE A B Da Silva ldquoA controlled filtering method for estimatingharmonics of off-nominal frequenciesrdquo IEEE Transactions onSmart Grid vol 3 no 1 pp 38ndash49 2012
[5] P Castello M Lixia C Muscas and P A Pegoraro ldquoImpactof the model on the accuracy of synchrophasor measurementrdquoIEEETransactions on Instrumentation andMeasurement vol 61no 8 pp 2179ndash2188 2012
[6] P K Dash K Kr and R K Patnaik ldquoDynamic phasor andfrequency estimation of time-varying power system signalsrdquoInternational Journal of Electrical Power amp Energy Systems vol44 no 1 pp 971ndash980 2013
[7] S-H Kang D-G Lee S-R Nam P A Crossley and Y-CKang ldquoFourier transform-based modified phasor estimationmethod immune to the effect of the DC offsetsrdquo IEEE Trans-actions on Power Delivery vol 24 no 3 pp 1104ndash1111 2009
[8] I Kamwa A K Pradhan and G Joos ldquoAdaptive phasorand frequency-tracking schemes for wide-area protection andcontrolrdquo IEEE Transactions on Power Delivery vol 26 no 2 pp744ndash753 2011
[9] X Jin F Wang and Z Wang ldquoA dynamic phasor estimationalgorithm based on angle-shifted energy operatorrdquo ScienceChina Technological Sciences vol 56 no 6 pp 1322ndash1329 2013
[10] D-G Lee S-H Kang and S-R Nam ldquoPhasor estimationalgorithm based on the least square technique during CTsaturationrdquo Journal of Electrical Engineering amp Technology vol6 no 4 pp 459ndash465 2011
[11] A G Phadke and B Kasztenny ldquoSynchronized phasor and fre-quency measurement under transient conditionsrdquo IEEE Trans-actions on Power Delivery vol 24 no 1 pp 89ndash95 2009
[12] M A Platas-Garza and J A De La O Serna ldquoDynamic phasorand frequency estimates throughmaximally flat differentiatorsrdquoIEEE Transactions on Instrumentation and Measurement vol59 no 7 pp 1803ndash1811 2010
[13] J A De la O Serna ldquoPhasor estimation from phasorletsrdquo IEEETransactions on Instrumentation and Measurement vol 54 no1 pp 134ndash143 2005
[14] J Ren and M Kezunovic ldquoReal-time power system frequencyand phasors estimation using recursive wavelet transformrdquoIEEE Transactions on Power Delivery vol 26 no 3 pp 1392ndash1402 2011
[15] J A de laO Serna ldquoDynamic phasor estimates for power systemoscillationsrdquo IEEE Transactions on Instrumentation and Meas-urement vol 56 no 5 pp 1648ndash1657 2007
[16] J A De La O Serna and J Rodrıguez-Maldonado ldquoInstan-taneous oscillating phasor estimates with TaylorK-Kalmanfiltersrdquo IEEE Transactions on Power Systems vol 26 no 4 pp2336ndash2344 2011
[17] J A De La O Serna and J Rodrıguez-Maldonado ldquoTaylor-Kalman-Fourier filters for instantaneous oscillating phasor andharmonic estimatesrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 4 pp 941ndash951 2012
[18] J A De La O Serna ldquoSynchrophasor estimation using Pronyrsquosmethodrdquo IEEE Transactions on Instrumentation and Measure-ment vol 62 no 8 pp 2119ndash2128 2013
[19] J A De la O Serna and K E Martin ldquoImproving phasor mea-surements under power system oscillationsrdquo IEEE Transactionson Power Systems vol 18 no 1 pp 160ndash166 2003
[20] J A de la O Serna ldquoReducing the error in phasor estimatesfrom phasorlets in fault voltage and current signalsrdquo IEEETransactions on Instrumentation and Measurement vol 56 no3 pp 856ndash866 2007
[21] A Muoz and J A de la O Serna ldquoShanksrsquo Method for DynamicPhasor Estimationrdquo IEEE Transactions on Instrumentation andMeasurement vol 57 no 4 pp 813ndash819 2008
[22] M D Kusljevic and J J Tomic ldquoMultiple-resonator-basedpower system Taylor-Fourier harmonic analysisrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp554ndash563 2015
[23] J Liu F Ni J Tang F Ponci and A Monti ldquoAmodified Taylor-Kalman filter for instantaneous dynamic phasor estimationrdquo inProceedings of the 2012 3rd IEEE PES Innovative Smart GridTechnologies Europe ISGTEurope 2012 GermanyOctober 2012
[24] D Belega D Fontanelli and D Petri ldquoDynamic phasor andfrequency measurements by an improved taylor weighted leastsquares algorithmrdquo IEEE Transactions on Instrumentation andMeasurement vol 64 no 8 pp 2165ndash2178 2015
[25] C Qian and M Kezunovic ldquoDynamic synchrophasor esti-mation with modified hybrid methodrdquo in Proceedings of the2016 IEEE Power and Energy Society Innovative Smart GridTechnologies Conference ISGT 2016 USA September 2016
[26] C Huang X Xie and H Jiang ldquoDynamic Phasor EstimationThrough DSTKF under Transient Conditionsrdquo IEEE Transac-tions on Instrumentation and Measurement vol 66 no 11 pp2929ndash2936 2017
[27] J Khodaparast andM Khederzadeh ldquoDynamic synchrophasorestimation by Taylor-Prony method in harmonic and non-harmonic conditionsrdquo IET Generation Transmission amp Distri-bution vol 11 no 18 pp 4406ndash4413 2017
[28] ldquoIEEE standard for synchrophasor measurements for powersystemsrdquo IEEE Std C371181-2011 (Revision of IEEE Std C37118-2005) pp 1ndash61 2011
[29] S Rechka E Ngandui J Xu and P Sicard ldquoPerformance eval-uation of harmonics detection methods applied to harmonicscompensation in presence of common power quality problemsrdquoMathematics and Computers in Simulation vol 63 no 3-5 pp363ndash375 2003
[30] P B Petrovic and D Rozgic ldquoComputational effective modifiedNewtonndashRaphson algorithm for power harmonics parametersestimationrdquo IET Signal Processing vol 12 no 5 pp 590ndash5982018
[31] P Petrovic ldquoAlgorithm for simultaneous parameters estimationof multi-harmonic signalrdquo Metrology amp Measurement Sistemsvol 19 no 4 pp 693ndash702 2012
[32] A Ben-Israel and T N E GrevilleGeneralized Inverses Theoryand Applications Springer Berlin Germany 2nd edition 2003
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 17
[33] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 1976
[34] A IviC The Riemann zeta-function John Wiley amp Sons IncNew York USA 1985
[35] R Dwilewicz and J Minac ldquoThe Hurwitz zeta function as aconvergent seriesrdquo RockyMountain Journal of Mathematics vol36 no 4 pp 1191ndash1219 2006
[36] D E Knuth Fundamental Algorithms vol 1 Addison-WesleyReading Mass USA 3rd edition 1997
[37] R J Dwilewicz ldquoCauchy-Riemann theory an overviewrdquo inGeometry Seminars 2005ndash2009 (Italian) pp 59ndash95 Univ StudBologna Bologna 2010
[38] J Khodaparast andM Khederzadeh ldquoLeast square and Kalmanbased methods for dynamic phasor estimation a reviewrdquoProtection and Control of Modern Power Systems vol 2 no 12017
[39] M D Kusljevic J J Tomic and P D Poljak ldquoMaximally Flat-Frequency-Response Multiple-Resonator-Based HarmonicAnalysisrdquo IEEE Transactions on Instrumentation and Meas-urement vol 66 no 12 pp 3387ndash3398 2017
[40] J A De La O Serna ldquoSynchrophasor measurement with poly-nomial phase-locked-loop Taylor-Fourier filtersrdquo IEEE Trans-actions on Instrumentation and Measurement vol 64 no 2 pp328ndash337 2015
[41] M Karimi-Ghartemani M Mojiri A Bakhshai and P Jain ldquoAphasormeasurement algorithm based on phase-locked looprdquo inProceedings of the 2012 IEEE PES Transmission and DistributionConference and Exposition T and D 2012 USA May 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
