Electrical Power and Energy Systems 43 (2012) 351–357

Contents lists available at SciVerse ScienceDirect

Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Parameter estimation of low-voltage distribution systems in transientprocesses: Algorithm and simulation for RL circuits

Wu Jiekang a,⇑, Wu Fan b

a School of Automation, Guangdong University of Technology, Chinab College of Science and Technology, China Three Gorges University, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 July 2007Received in revised form 24 April 2012Accepted 29 April 2012Available online 30 June 2012

Keywords:Low-voltage distribution systemsParameter estimationTransient processNumerical differentiation

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.04.030

⇑ Corresponding author. Address: School of AutomatTechnology, No.100, Waihuan Xi Road, Guangzhou HiPanyu District, Guangzhou 510006, China. Tel.: +8618802001098.

E-mail address: wujiekang@163.com (W. Jiekang).

A numerical differentiation based algorithm for parameter measurement of low-voltage distributionsystems in transient processes is presented in this paper. In a transient process of RL circuit, there isnon-periodic component with exponential decay and periodic component of sinusoidal waveform. Inany transient processes of RL circuits, the proposed algorithm may not only accurately track the fre-quency of distribution systems, and estimate the amplitude and phase angle of sinusoidal component,but also estimate the amplitude and time constant of the non-periodic component. The simulation resultsof a studying example carried out for non-noise cases in Matlab show high accuracy, simple computationand less time, the simulation results carried out for noisy cases and harmonic cases may see acceptableerrors, simple computation and less time.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In static processes, the parameter measurement of sinusoidalcurrent or voltage signals of distribution systems is easier to imple-ment than that in transient processes. However, there are not onlyperiodic components of sinusoidal waveform, as like in static pro-cesses, but also non-periodic components with exponential decay.It is more difficult to measure the parameter of the periodic com-ponent, such as the amplitude and phase angle, and the parameterof the non-periodic component, such as the instantaneous valueand time constant. In some application, for example, digital powermetering and digital protection relaying, it is need to measure notonly the amplitude and phase angle of the periodic component ofsinusoidal waveform, but also the instantaneous value and timeconstant of the non-periodic component. At the same time, it isalso more difficult to track the frequency of distribution systemsin a transient processes than in a static processes because of addi-tional non-periodic signals in a transient processes.

Generally, the periodic components are sinusoidal, and the non-periodic components are exponential. The existing techniques, suchas zero crossing technique [1,2], level crossing technique [3], leastsquares error technique [4–6], Newton method [7], Kalman filter[8–12], Fourier transform [13–19], wavelet transform [18], genetic

ll rights reserved.

ion, Guangdong University ofgher Education Mega Center,

20 39322552, mobile: +86

algorithm [20], show good characteristics and better applicationresults in processing the pure sinusoidal signals and multiplesinusoidal signals, but these techniques are not adaptive to beapplied to processing exponential signals. In [21–23], Jun-Zhe Yang,Chih-Wen Liu and Ying-Hong Lin present two new algorithms,namely smart DFT (SDFT) and extended DFT (EDFT), for cases withwaveforms consisting of integer and non-integer harmonics as wellas exponential DC-offset terms. The advantage of these two newmethods lies in three points: (1) higher accuracy than othermethods, (2) they are not affected by time constant of exponentialdelaying DC offset, (3) the estimated signals is permitted to includeharmonics. The shortcoming of these two new methods is that theyneed much more computation and spend much more time thansome algorithms, such as numerical differentiation methods pre-sented by the Wu et al. [24,26] and Wu [25]. It is necessary to devel-op an algorithm with the aim of processing the sinusoidal signalsfor tracking the frequency of distribution systems and measuringthe amplitude and phase angle of periodic component in transientprocesses and in static processes, and of processing the exponentialsignals for measuring the instantaneous value and time constant intransient processes. Non-linear loads bring up harmonics in distri-bution systems [27], and Filter technique helps to decrease param-eter measurement errors [28].

This paper presents a technique, being different from the existingtechniques processing sinusoidal signals and multi-signals. Theproposed technique aims at tracking the frequency of distributionsystems, and measuring the amplitude and phase angle of periodiccomponent in transient processes and in static processes, andmeasuring the instantaneous value and time constant in transient

352 W. Jiekang, W. Fan / Electrical Power and Energy Systems 43 (2012) 351–357

processes. The proposed technique has higher accuracy, more sim-ple computation, and spends lesser time than the other existingmethods.

2. The proposed method

It is assumed that a sinusoidal voltage is added to a RL circuitwith R being resistance and L being inductance, as shown inFig. 1. The sinusoidal voltage may be formulated:

u ¼ffiffiffi2p

V sinð2pft þuÞ ð1Þ

where u is the bus voltage of low-voltage distribution systems, V isthe amplitude of bus voltage, f is the frequency of low-voltagedistribution systems.

In a RL circuit, the current signal of the transient process is writ-ten as:

iðtÞ ¼ I sinð2pft þuÞ þ De�pt ð2Þ

where i(t) is transient current signal in continuous time, I is theamplitude of sinusoidal component (periodic component), D is theamplitude of non-periodic component, p ¼ 1

s, s is time constant,s ¼ L

R.The transient current signal may be written as a discrete time

form:

iðnÞ ¼ I sinð2pfts þuÞ þ De�pts ð3Þ

where i(n) is the non-sinusoidal current signal in discrete time, n issampling index, ts ¼ nT

N , T is sampling period, N is sampling number.Using numerical differentiation algorithm, the 2nd and 4th dif-

ferential coefficient of i(n), De�pts and p may be calculated with thesampled value i(n) (i = 1,2, . . . ,N), and then the frequency of low-voltage distribution systems is estimated by:

f ¼ 12p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiið4ÞðnÞ � p4De�pts

i00ðnÞ � p2De�pts

sð4Þ

where the 1st, 2nd, 3rd and 4th differential of i(n) with respective tots is seen in Appendix A.

Given central point g in central numerical differentiation, ts isformulated as:

ts ¼ ðg � 1Þ TN

ð5Þ

And estimate the amplitude of non-periodic component by:

D ¼ 1e�pts

ið4ÞðnÞiðnÞ � i00ðnÞi00ðnÞið4ÞðnÞ þ iðnÞp4 � 2p2i00ðnÞ

ð6Þ

Letting the unknown parameters place in the left side and theknown parameters place in the right side, Eq. (19) in Appendix Abecomes the following form:

ð2pf ÞI cosð2pfts þuÞ ¼ i0ðnÞ þ pDe�pts ð7Þ

In the same way, Eq. (20) is rewritten as:

�ð2pf Þ2I sinð2pfts þuÞ ¼ i00ðnÞ � p2De�pts ð8Þ

u

Switch R

L

Fig. 1. RL transient circuit in low-voltage distribution systems.

From (7) and (8), the amplitude of periodic component is esti-mated by:

I ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2pf Þ2½i0ðnÞ þ pDe�pts �2 þ ½i00ðnÞ � p2De�pts �2

ð2pf Þ4

sð9Þ

In order for distinguishing the unknown and known parame-ters, formula (21) and (22) in Appendix A is rewritten as followsrespectively:

� ð2pf Þ3I cosð2pfts þuÞ ¼ ið3ÞðnÞ þ p3De�pts ð10Þð2pf Þ4I sinð2pfts þuÞ ¼ ið4ÞðnÞ � p4De�pts ð11Þ

Dividing (11) by (10), the tangent function of 2pfts + u isobtained:

tgð2pfts þuÞ ¼ � ið4ÞðnÞ � p4De�pts

ð2pf Þ½ið3ÞðnÞ þ p3De�pts �ð12Þ

The phase angle of periodic component is estimated by:

u ¼ atg � ið4ÞðnÞ � p4Depts

2pf ½ið3ÞðnÞ þ p3Depts �

!� 2pfpT

Nð13Þ

3. Implementation steps

The steps for algorithm implementation are written asfollowing:

Step 1: Sampling the signal with a sampling frequency of N � f(N = 512), f is rated frequency of low-voltage distribution sys-tems, and a sampling data sequence is obtained.Step 2: Based on central numerical differentiation of sevenpoints, compute the 1st–4th order differentiation of the signalusing Eqs. (14)–(17):

i0ðnÞ ¼ 160h

45½xðg þ 1Þ � xðg � 1Þ� � 9½xðg þ 2Þ � xðg � 2Þ�f

þ½xðg þ 3Þ � xðg � 3Þ�g ð14Þi00ðnÞ ¼ 1

180h2 �490xðgÞ þ 270½xðg þ 1Þ þ xðg � 1Þ�f

�27½xðg þ 2Þ þ xðg � 2Þ� þ 2½xðg þ 3Þ þ xðg � 3Þ�g ð15Þið3ÞðnÞ ¼ 1

8h3 �13½xðg þ 1Þ � xðg � 1Þ� þ 8½xðg þ 2Þ � xðg � 2Þ�f

�½xðg þ 3Þ � xðg � 3Þ�g ð16Þið4ÞðnÞ ¼ 1

12h4 112xðgÞ � 78½xðg þ 1Þ þ xðg � 1Þ�f

þ24½xðg þ 2Þ þ xðg � 2Þ� � 2½xðg þ 3Þ þ xðg � 3Þ�g ð17Þ

where g is central point in numerical differentiation, h is samplingspace between two sample points.

Step 3: Compute the solution z of quadratic Eq. (47) using Eq.(48).Step 4: Compute a and b using a ¼

ffiffiffiz3p

and b ¼ � r3a respectively.

Step 5: Compute the value of variable s using Eq. (49).Step 6: Compute the solution y of quadratic Eq. (51) using Eq.(52).Step 7: Calculate p using Eq. (37).Step 8: Calculate the time constant of non-periodic component:s ¼ 1

p.Step 9: Calculate the estimated value of non-periodic compo-nent at time ts using Eq. (53).Step 10: Estimate the amplitude of non-periodic componentusing Eq. (6).Step 11: Estimate the frequency f of distribution systems usingEq. (4).

Fig. 3. Sensitivity analysis of frequency estimation for fluctuation of its value.

Fig. 4. Sensitivity analysis of frequency estimation for fluctuation of the amplitudeof periodic component.

W. Jiekang, W. Fan / Electrical Power and Energy Systems 43 (2012) 351–357 353

Step 12: Estimate the amplitude of periodic signals usingEq. (9).Step 13: Estimate the phase angle of periodic signals usingEq. (13).

4. Simulation results

4.1. Simulation in non-noise cases

In Matlab, the following current signal is used to simulate thetransient current of low-voltage distribution systems with RLcircuits:

iðtÞ ¼ffiffiffi2p

I sinð2pft þuÞ þ De�pt ð18Þ

where p ¼ 1s, s is time constant of non-periodic component.

In the simulation, relative error is used for illustrating the accu-racy of the proposed method and is defined to be equal to(xe � x0) � 100/x0, where xe is the estimated value, x0 is the realvalue.

Fig. 2 shows frequency estimation of the periodic signals intransient process for the cases, in which the time constant is0.073 s and the instantaneous value is 43.928 A. In this simulatedcase, the time constant is set to a small value and the instanta-neous value is set to a great value. The simulated results show that:(1) because the sampling frequency is set to larger integer multi-plier of the rated frequency of distribution systems, for example50 Hz � 512, the estimated errors of frequency estimation of peri-odic component is smallest when the estimated frequency goesnear the rated frequency of distribution systems, and (2) the esti-mated errors become greater when the estimated frequency variesbeyond the rated frequency, and (3) the proposed method has verywide dynamic characteristic in frequency, for example the esti-mated frequency is permitted to vary from 40 Hz to 60 Hz (in factthe estimated frequency range is far wider than that), (4) the esti-mated errors is very small and it’s absolute value of the smallestone is not lees than 0.0002.

Fig. 3 shows the sensitivity analysis of frequency estimation ofdistribution systems in transient processes for fluctuation of fre-quency. When the frequency of distribution systems varies from10 Hz to 10000 Hz, the estimation errors almost converge to 0.

Fig. 4 shows the sensitivity analysis of frequency estimation ofdistribution systems in transient processes for fluctuation of theamplitude of periodic component. When the amplitude of periodiccomponent varies from 0 to 20 A, the estimation errors fluctuatebetween �0.00004% and 0.00004%. When the amplitude of peri-odic component varies from 0 to 6000 A, the estimation errors fluc-tuate between �0.000006% and 0.00001%. When the amplitude of

Fig. 2. Frequency estimation of transient process for the case, in which the timeconstant is 0.073 s and the instantaneous value is 43.928 A.

Fig. 5. Sensitivity analysis of frequency estimation for fluctuation o of the phaseangle of periodic component.

periodic component varies from 20 A to 200 A, the estimationerrors are greater than that when the amplitude of periodic com-ponent varies from 200 A to 6000 A.

Fig. 5 shows the sensitivity analysis of frequency estimation ofdistribution systems in transient processes for fluctuation of phaseangle of periodic component. In this simulated case, the phase an-gle is set to vary from 0� to 360�. Except the estimation errors offrequency arrive at �6% and 8% respectively when the phase angle

Fig. 6. Sensitivity analysis of frequency estimation for fluctuation of the amplitudeof non-periodic component.

Fig. 7. Sensitivity analysis of the amplitude estimation of periodic component forfluctuation of the amplitude of periodic component.

354 W. Jiekang, W. Fan / Electrical Power and Energy Systems 43 (2012) 351–357

goes near 180� and 360�, the estimation errors of frequency almostbecomes 0.

Fig. 6 shows the sensitivity analysis of frequency estimation ofdistribution systems in transient processes for fluctuation of theamplitude of non-periodic component. The estimated errors ofdistribution system frequency approximate 0 when the amplitudeof non-periodic component varies from 0 to 2000 A, while the esti-mated errors fluctuate between �0.006% and 0.004% when theamplitude of non-periodic component varies from 2000 A to10000 A.

The errors of amplitude estimation of periodic component fluc-tuate between �0.03% and 0.03%, when time constant of non-peri-odic component is lower than 0.2 s, while the errors of amplitudeestimation of periodic component is close to 0 when time constantof non- periodic component is greater than 0.2 s.

When the sampling frequency is lower than about 100000 Hz,the estimation errors of distribution system frequency convergeto 0. When the sampling frequency is greater than about100000 Hz, the estimation errors increase and fluctuate between�0.15% and 0.15%.

Table 1 shows the simulation results of the amplitude andphase angle estimation of the periodic component. It is seen fromTable 1 that the estimated accuracy of the amplitude and phase an-gle estimation of the periodic component using the proposedmethod is very high.

The estimated errors of the amplitude estimation of periodiccomponent approximate 0 when the frequency of periodic compo-nent varies from 30 Hz to 100 Hz, while the estimated errors fluctu-ate between �0.3% and 0.3% when the frequency of periodiccomponent varies from 10 Hz to 30 Hz.

Table 1Amplitude and phase angle estimation in transient process.

Amplitude (A) Phase angle (�)

Real value Estimated value Real value Estimated value

2 2.000005 13 12.99999015 15.000007 35 34.99999546 46.000006 76 75.99999968 67.999964 100 99.99954682 82.000050 160 159.994234

100 99.999958 250 250.052100130 130.000038 280 280.338674160 160.000183 350 351.035603190 190.000220 0 0.000073220 219.999277 45 43.869656

10 10.0007281 130 130.269853

Fig. 7 shows the sensitivity analysis of amplitude estimation ofperiodic component of distribution systems in transient processesfor fluctuation of the amplitude of periodic component. In this sim-ulation case, the estimated errors of the amplitude estimation ofperiodic component fluctuate between �0.001% and 0.0006%.

When the amplitude of non-periodic component varies from 0to 1000 A, the estimated errors of amplitude estimation of periodiccomponent approximate to 0. With increase of the amplitude ofnon-periodic component, the estimated errors of amplitude esti-mation of periodic component also increase, but the errors fluctu-ate between �0. 01% and 0.015%.

The estimated errors of amplitude estimation of periodic com-ponent fluctuate between �0. 01% and 0.02%, when time constantof non-periodic component is lower than 0.2 s, while the estimatederrors of amplitude estimation of periodic component is close to 0when time constant of non-periodic component is greater than0.2 s.

When the sampling frequency varies from 0 to 100000 Hz, theestimated errors of amplitude estimation of periodic componentapproximate to 0. With increase of the sampling frequency ofnon-periodic component, the estimated errors of amplitude esti-mation of periodic component also increase, but the errors fluctu-ate between �0.3% and 0.3%.

Fig. 8 shows the sensitivity analysis of phase angle estimation ofperiodic component of distribution systems in transient processesfor fluctuation of frequency of period component. The estimated er-rors of the phase angle estimation of periodic component fluctuate

Fig. 8. Sensitivity analysis of the phase angle estimation of periodic component forfluctuation of the frequency of distribution systems.

Table 2Instantaneous value and time constant estimation in transient process.

Instantaneous value (A) Time constant (s)

Real value Estimated value Real value Estimated value

2 1.999999 0.001 0.00148615 14.999996 0.05 0.05009346 45.999992 0.3 0.30000868 68.000049 5 5.00002982 81.999990 16 15.999874

100 99.999945 25 25.000050130 130.000052 45 45.000000160 159.999908 83 83.000273190 189.999981 10 10.000244220 220.000903 90 89.999350

68 68.0008117 100 99.894552

Table 4Parameter estimation in transient process in harmonic cases.

Amplitude (A) Phase angle (�) Instantaneousvalue (A)

Time constant (s)

Realvalue

Estimatedvalue

Realvalue

Estimatedvalue

Realvalue

Estimatedvalue

Realvalue

Estimatedvalue

2 2.3526 13 13.2635 2 1.6733 0.001 0.0764315 15.4563 35 35.6542 15 15.3566 0.05 0.1284646 45.3847 76 76.3645 46 46.4634 0.3 0.483968 67.2460 100 100.5274 68 67.5924 5 5.467782 82.4733 160 160.7621 82 81.7891 16 15.6345

100 100.4897 250 250.6235 100 100.6430 25 25.3652130 129.1562 280 280.5978 130 129.7543 45 45.3561160 160.3874 350 350.3976 160 160.5264 83 82.6245190 190.4826 0 0.3242 190 190.2645 10 10.4260220 219.3841 45 44.7638 220 220.4652 90 90.5736

10 9.6834 130 129.8354 68 68.3546 100 100.3872

Table 3Parameter estimation in transient process with white Guassian noises.

Amplitude (A) Phase angle (�) Instantaneousvalue (A)

Time constant (s)

Realvalue

Estimatedvalue

Realvalue

Estimatedvalue

Realvalue

Estimatedvalue

Realvalue

Estimatedvalue

2 2.2957 13 12.8301 2 2.3758 0.001 0.00386215 14.5820 35 34.9023 15 15.4272 0.05 0.08234646 46.6120 76 76.1422 46 45.8253 0.3 0.52645268 67.1959 100 99.9639 68 67.7326 5 5.73853382 82.2084 160 160.4812 82 82.6384 16 16.826436

100 100.3538 250 249.9595 100 99.7254 25 25.934354130 130.4284 280 279.8800 130 129.4823 45 44.238983160 160.5898 350 350.0122 160 160.2341 83 82.346772190 190.2226 0 0.0240 190 191.0492 10 10.928364220 220.6885 45 45.3059 220 219.7528 90 90.478710

10 9.9258 130 129.7422 68 68.5371 100 99.346782

W. Jiekang, W. Fan / Electrical Power and Energy Systems 43 (2012) 351–357 355

between �0.00005% and 0.00015% when the frequency of periodiccomponent varies from 40 Hz to 60 Hz.

Table 2 shows the simulation results of the amplitude and timeconstant estimation of the non-period component. In any transientprocesses, the amplitude and time constant of the non-period com-ponent may be estimated at a high accuracy using the proposedalgorithm.

4.2. Simulation in noising cases

A current signal with non-periodic components in white Guas-sian noising cases is considered, and a method based on the FIR fil-ter technique proposed in [24,25] is used to filter white Guassian

noise from the current signal. After white Guassian noise is filteredfrom the current signal, the method proposed in Section 2 is usedto estimate the parameters of low-voltage distribution systems intransient processes with non-periodic components in white Guas-sian noising cases, and the simulation result is shown in Table 3. Itis seen from Table 3 that the proposed measurement method maydo well in noising cases, although its measurement error is greaterthan that in non-noise cases.

4.3. Simulation in harmonic cases

In distribution systems, there are many non-linear loads such asadjustable speed drives and variable frequency drives, batterychargers, UPS, and any other equipment powered by switched-mode power supply equipment. The non-linear loads make har-monics rich in distribution systems. In harmonic cases, the pro-posed method still does well after the FIR filter techniqueproposed in [24,25] is used to filter the 2nd and higher order har-monics from the signal, although its estimation errors are greaterthan that in non-noise cases, as shown in Table 4.

5. Conclusions

Based on central numerical differentiation and Lagrange inter-polation, an algorithm is presented in this paper. The proposedalgorithm may not only accurately estimate the frequency, ampli-tude and phase angle of periodic components of distribution sys-tems in any transient processes at any time, but also accuratelyestimate the amplitude and the time constant of non-periodiccomponents. Comparing with other existing methods, the methodproposed in this paper has the following advantages:

(1) Higher accuracy than other existing methods: except someextreme cases, the estimation errors for the frequency,amplitude and phase angle of periodic components as wellas the amplitude and the time constant of non-periodic com-ponents are smaller than 0.3% in absolute value.

(2) More simple in computation: using numerical differentiation,the proposed method shows more simple in formulation.

(3) Lesser time for computation: under a CPU with a main speedof 2.0 GHz, the time spent on computation is at most 40 ms(two periods of the estimated signal of the periodic compo-nent). If a faster CPU, for example 3.0 GHz, the computationtime may decreases at a great range, for example 10 ms (halfperiod of the estimated signal of the periodic component).

(4) The proposed method is not affected by time constant ofexponential delaying DC offset.

(5) The proposed method has very wide dynamic characteristicin frequency, amplitude, phase angle and time constant, forexample the estimated frequency is permitted to vary from40 Hz to 60 Hz (in fact the estimated frequency range is farwider than that).

Although harmonics is not included in the proposed algorithm,when some other algorithms, such as digital FIR, is adopted to filterthe noises and to decompose the fundamental component andother high harmonics from the estimated signals, the methodmay be applied in the same way as that in this paper. The contri-bution of this paper lies in providing a basic method for parameterestimation of periodic and non-periodic components in any tran-sient processes of distribution systems. In some extent, the pro-posed method is applied to the practical electric distributionsystems.

356 W. Jiekang, W. Fan / Electrical Power and Energy Systems 43 (2012) 351–357

Acknowledgments

This work is financially supported by the National HighTechnology Research and Development Program of China (863Program) (2007AA04Z197), National Natural Science Foundationof China (50767001), Specialized Research Fund for the DoctoralProgram of Higher Education (20094501110002).

Appendix A

The 1st, 2nd, 3rd and 4th differential of i(n) with respective to ts

is respectively formulated as following:

i0ðnÞ ¼ ð2pf ÞI cosð2pfts þuÞ � pDe�pts ð19Þi00ðnÞ ¼ ð2pf Þ2I sinð2pfts þuÞ þ p2De�pts ð20Þið3ÞðnÞ ¼ �ð2pf Þ3I cosð2pfts þuÞ � p3De�pts ð21Þið4ÞðnÞ ¼ ð2pf Þ4I sinð2pfts þuÞ þ p4De�pts ð22Þ

where i0(n) and i00ðnÞ is 1st and 2nd differential with respective to ts,i(k)(n) (i = 3.4) is ith differential of i(n) with respective to ts, k is dif-ferential index.

Decomposing out I sin(2pts + u) from formula (3), and pluggingit into (20) and (22), the following two formulas are obtained:

i00ðnÞ � Dp2e�pts ¼ �ð2pf Þ2 iðnÞ � De�pts� �

ð23Þið4ÞðnÞ � Dp4e�pts ¼ ð2pf Þ4 iðnÞ � De�pts

� �ð24Þ

Dividing (24) by (23), the expression of (2pf)2 is expressed bythe following formula:

ð2pf Þ2 ¼ � ið4ÞðnÞ � Dp4e�pts

i00ðnÞ � Dp2e�ptsð25Þ

After plugged the item (2pf)2, formula (23) is converted as thefollowing form:

ið4ÞðnÞDþ iðnÞDp4 � 2Dp2i00ðnÞh i

e�pts ¼ ið4ÞðnÞiðnÞ � i00ðnÞi00ðnÞ ð26Þ

After simple transformation, the formula (19) and (21) isrespectively expressed as follows:

i0ðnÞ þ Dpe�pts ¼ ð2pf ÞI cosð2pfts þuÞ ð27Þið3ÞðnÞ þ Dp3e�pts ¼ �ð2pf Þ3I cosð2pfts þuÞ ð28Þ

Dividing (28) by (27), the expression of (2pf)2 is also obtained:

ð2pf Þ2 ¼ � ið3ÞðnÞ þ Dp3e�pts

i0ðnÞ þ Dpe�ptsð29Þ

Making the right side of formula (25) and (29) to be equal andmoving the same items together, the following expression iswritten:

Dpið4ÞðnÞ � Dp4i0ðnÞ þ Dp2ið3ÞðnÞ � Dp3i00ðnÞh i

e�pts

¼ ið3ÞðnÞi00ðnÞ � ið4ÞðnÞi0ðnÞ ð30Þ

The following two coefficients are defined:

K1 ¼ ið4ÞðnÞiðnÞ � i00ðnÞi00ðnÞ ð31ÞK2 ¼ ið3ÞðnÞi00ðnÞ � ið4ÞðnÞi0ðnÞ ð32Þ

Dividing (26) by (30) and moving the same items together, weobtain a quartic equation of p:

½K1i0ðnÞ þ K2iðnÞ�p4 þ K1i00ðnÞp3 þ �K1ið3ÞðnÞ � 2K2i00ðnÞh i

p2

� K1ið4ÞðnÞpþ K2ið4ÞðnÞ ¼ 0 ð33Þ

Eq. (33) may be simplified as the following form:

K3p4 þ K4p3 þ K5p2 þ K6pþ K7 ¼ 0 ð34Þ

where K3 = K1i0(n) + K2i(n).

K4 ¼ K1i00ðnÞK5 ¼ �K1ið3ÞðnÞ � 2K2i00ðnÞK6 ¼ �K1ið4ÞðnÞK7 ¼ K2ið4ÞðnÞ

Eq. (34) may be further simplified as the following form:

p4 þ K8p3 þ K9p2 þ K10pþ K11 ¼ 0 ð35Þ

where K8 ¼ K4K3;K9 ¼ K5

K3;K10 ¼ K6

K3;K11 ¼ K7

K3.

When the following formula is defined:

y ¼ pþ K8

4ð36Þ

Then, the time constant s is expressed by:

p ¼ y� K8

4ð37Þ

Eq. (35) becomes the following quartic equation:

y4 ¼ �K12y2 � K13y� K14 ¼ 0 ð38Þ

where

K12 ¼ K9 �3K2

8

8

K13 ¼ K10 �K8K9

2þ K3

8

2

K14 ¼ K11 �K8K10

4þ K2

8K9

16� K4

8

256

If a variable s is introduced, the following expression isobtained:

ðy2 þ sÞ2 ¼ y4 þ 2y2sþ s2 ð39Þ

Plugging (38) into (39), the expression (39) becomes the follow-ing form:

ðy2 þ sÞ2 ¼ ð2s� K12Þy2 � K13y� K14 þ s2 ð40Þ

Making Eq. (40) to be a square number, the following conditionmust be satisfied:

�K13ð Þ2 � 4ð2s� K12Þ �K14 þ s2� �¼ 0 ð41Þ

For variable s, Eq. (41) becomes the following cubic equation:

s3 þ K15s2 þ K16sþ K17 ¼ 0 ð42Þ

where

K15 ¼ �K12

2K16 ¼ �K14

K17 ¼K12K14

2� K2

13

8

Letting s ¼ x� K153 the Eq. (42) is transformed into the following

form:

x3 ¼ rxþ q ð43Þ

where r ¼ � K16 � K2153

� �.

q ¼ � 2K315

27� K15K16

3� K17

!

W. Jiekang, W. Fan / Electrical Power and Energy Systems 43 (2012) 351–357 357

Defining x = a � b and (a � b)3 = a3 � 3a2b � 3ab2 � b3, the fol-lowing formula is obtained:

a3 � b3 ¼ ða� bÞðr þ 3abÞ þ q ð44Þ

The condition satisfying Eq. (43) is:

r ¼ �3ab ð45Þ

So, the formula (44) becomes the another form:

a3 � b3 ¼ q ð46Þ

Multiply (45) by 27a3, a quadratic equation is obtained:

27z2 � 27qzþ r3 ¼ 0 ð47Þ

where z = a3.In (47), the solution of z is:

z ¼27q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið27qÞ2 � 4 � 27 � r3

q2 � 27

ð48Þ

From z, the value of variable s is computed:

s ¼ a� b� K15

3ð49Þ

where a ¼ffiffiffiz3p; b ¼ � r

3a.From (41), the following equation is obtained:

2s� K12 ¼K2

13

4ð�K14 þ s2Þ ð50Þ

Plugging (50) into (40), a quadratic equation is obtained:

K20y2 þ K21yþ K22 ¼ 0 ð51Þ

where

K15 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ðs2 � K14Þ

qK16 ¼ �K13

K17 ¼ sK20 þ 2ðs2 � K14Þ

The solution of quadratic Eq. (51) is:

y ¼�K21 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2

21 � 4K20K22

q2K20

ð52Þ

Plugging y into (37), p is computed.From (26), we obtain the mathematical formulas of non-peri-

odic component:

De�pts ¼ ið4ÞðnÞiðnÞ � i00ðnÞi00ðnÞið4ÞðnÞ þ iðnÞp4 � 2p2i00ðnÞ

ð53Þ

References

[1] Moore PJ, Carranza RD, Johns AT. Model system tests on a new numericmethod of power system frequency measurement. IEEE Trans Power Deliv1996;11:696–701.

[2] Begovic MM, Djuric PM, Dunlap S, Phadke AG. Frequency tracking in powernetworks in the presence of harmonics. IEEE Trans Power Deliv 1993;8:480–6.

[3] Nguyen CT, Srinivasan KA. A new technique for rapid tracking of frequencydeviations based on level crossings. IEEE Trans Power Apparat Syst1984;103:2230–6.

[4] Kamwa I, Grondin R. Fast adaptive schemes for tracking voltage phasor andlocal frequency in power transmission and distribution systems. IEEE TransPower Deliv 1992;7:789–95.

[5] Sachdev MS, Giray MM. A least error squares technique for determining powersystem frequency. IEEE Trans Power Apparat Syst 1985;104:437–43.

[6] Giray MM, Sachdev MS. Off-nominal frequency measurements in electricdistribution systems. IEEE Trans Power Deliv 1989;4:1573–8.

[7] Terzija VV, Djuric MB, Kovacevic BD. Voltage phasor and local systemfrequency estimation using newton type algorithm. IEEE Trans Power Deliv1994;9:368–1374.

[8] Sachdev MS, Wood HC, Johnson NG. Kalman filtering applied to power systemmeasurements for relaying. IEEE Trans Power Apparat Syst 1985;104:3565–73.

[9] Girgis AA, Hwang TLD. Optimal estimation of voltage phasors and frequencydeviation using linear and nonlinear Kalman filter: theory and limitations. IEEETrans Power Apparat Syst 1984;103:2943–9.

[10] Girgis AA, Peterson WL. Adaptive estimation of power system frequencydeviation and its rate of change for calculating sudden power systemoverloads. IEEE Trans Power Deliv 1990;5:85–594.

[11] Lobos T, Rezmer J. Real time determination of power system frequency. IEEETrans Instrum Meas 1997;46:877–81.

[12] Phadke AG, Thorp JS, Adamiak MG. A new measurement technique for trackingvoltage phasors, local system frequency, and rate of change of frequency. IEEETrans Power Apparat Syst 1983;102:1025–38.

[13] Yang JZ, Liu CW. A precise calculation of power system frequency. IEEE TransPower Deliv 2001;16:361–6.

[14] Yang JZ, Liu CW. A precise calculation of power system frequency and phasor.IEEE Trans Power Deliv 2000;15:494–9.

[15] Lu SL, Lin CE, Huang CL. Power frequency harmonic measurement usinginteger periodicic extension method. Electr Distrib Syst Res 1998;44:107–15.

[16] Moore PJ, Johns AT. A new numeric technique for high-speed evaluation ofpower system frequency. IEE Proc-Gener Transm Distrib 1994;141:529–36.

[17] Szafran J, Rebizant W. Power system frequency estimation. IEE Proc-GenerTransm Distrib 1998;145:578–82.

[18] Kuang WT, Morris AS. Using short-time Fourier transform and wavelet packetfilter banks for improved frequency measurement in a Doppler robot trackingsystem. IEEE Trans Instrum Meas 2002;5:440–4.

[19] Girgis AA, Ham FM. A new FFT-based digital frequency relay for load shedding.IEEE Trans Power Apparat Syst 1982;101:433–9.

[20] Arjona MA, Hernandez C, Cisneros-Gonzalez M, Escarela-Perez R. Estimation ofsynchronous generator parameters using the standstill step-voltage test and ahybrid genetic algorithm. Int J Electr Power Energy Syst 2012;35:105–11.

[21] Yang JZ, Liu CW. A new family of measurement technique for tracking voltagephasor, local system frequency, harmonics and DC offset. 2000 IEEE powerengineering society summer meeting, vol. 3; 16–20 July, 2000. p. 1327–32.

[22] Lin YH, Liu CW. A new DFT-based phasor computation algorithm fortransmission line digital protection. In: Transmission and distributionconference and exhibition 2002: Asia Pacific, vol. 3. IEEE/PES; 2002. p. 1733–7.

[23] Yang JZ, Liu CW. Complete elimination of DC offset in current signals forrelaying applications. In: IEEE power engineering society winter meeting, vol.3; 2000. p. 1933–8.

[24] Wu JK, Long J, Wang JX. High-accuracy, wide-range frequency estimationmethods for power system signals under non-sinusoidal conditions. IEEE TransPower Deliv 2005;20:366–74.

[25] Wu JK. Numerical differentiation and digital FIR filter based algorithm forpower system measurement. In: IEEE PES general meeting, San Francisco,California, USA; 12–16 June, 2005. p. 1112–9.

[26] Wu JK, Liang Y, Chen GT. Neuro-fuzzy and digital FIR filter approach for powersystem measurement denoising. In: International conference on innovativecomputing, information and control (ICICIC2006), Beijing, China; August 30–September 1; 2006.

[27] Askarzadeh Alireza, Rezazadeh Alireza. Parameter estimation and comparisonof the load models with considering distribution network directly orindirectly. Int J Electr Power Energy Syst 2010;32:965–8.

[28] Meredith LS, Alireza KZ, Thomas HO. A novel adaptive flicker measurementtechnique. Int J Electr Power Energy Syst 2011;33:1686–94.