IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 4, APRIL 2011 1089
Robust Frequency-Estimation Method for Distortedand Imbalanced Three-Phase Systems
Using Discrete FiltersPedro Roncero-Sanchez, Member, IEEE, Xavier del Toro Garcıa, Alfonso Parreno Torres,
and Vicente Feliu, Senior Member, IEEE
Abstract—The continuous monitoring of voltage characteristicsin electric power systems, such as in microgrids, is required forpower quality assessment, grid control, and protection purposes.Due to the presence of disturbances in the grid voltage, such asharmonics, imbalances, noise, and offsets introduced by the instru-mentation, among others, the frequency-estimation process has tobe robust against all these disturbances to obtain an accurate esti-mation of the frequency value. This paper presents a fast and accu-rate method to estimate the fundamental frequency of an electricpower system. The estimation method works properly in balancedand imbalanced three-phase systems, and even in single-phase sys-tems. The proposed solution is based on an algebraic method, whichis able to calculate the frequency of a pure sinusoidal signal usingthree samples. A filtering stage is used to increase the robustnessof the algorithm against disturbances in a wide frequency range.Simulation and experimental results show the good performanceof the method for single- and three-phase systems with a high levelof harmonic distortion, even in the presence of amplitude, phase,and frequency changes.
Index Terms—Discrete-time filters, distributed generation (DG),filtering, frequency estimation, harmonics, microgrid, monitoring,power quality.
I. INTRODUCTION
FUTURE electric power systems are expected to raise incomplexity as a result of the integration of distributed gen-
eration (DG) systems, such as renewable energies [1], internalcombustion engines, gas turbines, or microturbines. The in-creasing presence of microgrids with the capability of operatingalone or connected to distribution grids requires a system per-spective to capture the benefits of integrating distributed energyresources into an energy system [2]. The majority of small-scaleDG systems require power electronic converters as interfaceswith electrical distribution grids [3], [4]. On the other hand,power quality solutions to problems, such as voltage sags, har-
Manuscript received July 1, 2010; revised September 14, 2010 and December6, 2010; accepted January 5, 2011. Date of current version June 10, 2011. Thiswork was supported by the European Social Fund and the Council of Castilla-La Mancha under the Research Project PII2I09-0088-1366. Recommended forpublication by Associate Editor P.-T. Cheng.
P. Roncero-Sanchez, X. del Toro Garcıa, and V. Feliu are with the De-partment of Electrical, Electronic, Control Engineering and Communications,Escuela Tecnica Superior de Ingenieros Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain (e-mail: [email protected];[email protected]; [email protected]).
A. Parreno Torres is with Albacete Science and Technology Park, Albacete02006, Spain (e-mail: [email protected]).
Digital Object Identifier 10.1109/TPEL.2011.2107580
monic pollution, or voltage imbalances, among others, have be-come a major concern in order to ensure the reliability of powersystems and maintain the voltage characteristics within certainlimits [5], [6]. Microgrids can improve the service reliability,being responsible for local power quality, voltage regulation,power factor correction, etc., due to the possibilities offered bypower electronic interfaces [7].
The estimation of the frequency in electric power systems,as well as the amplitude and the phase of voltages or currentsis a very important issue to any application related to micro-grids, DG, Flexible Alternating Current Transmission Systems(FACTS), and custom devices, such as dynamic voltage restor-ers, active filters, unified power flow controllers, or Static Syn-chronous Compensators (STATCOM) [8]–[11]. In the case ofmicrogrids, frequency measurement is necessary to implementreal power versus frequency droop controllers in order to main-tain the active power balance [12]. Frequency control is par-ticularly important during transitions between grid-connectedand islanding modes, where high deviations of frequency canoccur. In transitions from islanding to grid-connected modes,the frequency of the microgrid cannot differ from the utility fre-quency more than 0.1% according to standard IEEE 1547 [13].Frequency estimation is also needed for load shedding, fre-quency restoration, and grid code compliance in DG systems.Several methods employ the measurement of frequency drifts forislanding detection [14]–[17]. In all these cases, a frequency-estimation method is implemented in the corresponding DGunit [18]. The adopted method has to be fast, accurate, androbust even in the presence of harmonic distortion, voltage im-balances, magnitude fluctuations, and noise.
There are several standards that have been defined to as-sure the characteristics of the voltage at the customers supplyterminals. The European standard EN-50160 defines the maincharacteristics of the voltage in normal operating conditions inlow- and medium-voltage networks [19]. Considering the volt-age frequency, also referred to as power frequency, deviationsfrom the nominal value might arise due to the mismatch betweengeneration and demand. According to the standard EN-50160,in normal operating conditions, the frequency value must beinside the 50 Hz ± 1% range during 95% of the week and insidethe 50 Hz −6%/+4% range during 100% of the week. On theother hand, the standard IEC 61727, which defines the charac-teristics of the electrical grid for the connection of photovoltaicsystems, establishes that a photovoltaic system has to be discon-nected from the grid when the utility frequency is outside the
1090 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 4, APRIL 2011
rated frequency ±1 Hz range [20]. Regarding standards in test-ing and measurement techniques, the standard IEC 61000-4-30describes the measurement methodologies and accuracy levelsin order to establish the common requirements for measurementdevices [21]. According to this standard, frequency should bemeasured counting the integer number of cycles within a 10-swindow. The required accuracy for A-class equipment is set to±0.01 Hz.
Many frequency measurement techniques have been pre-sented during the last decades. The zero-crossing detectionmethod is one of the earliest techniques [22], [23], and is widelyemployed and easy to implement by means of digital electronics.Nevertheless, it is very sensitive to noise, notches, and harmonicdistortion, which cause multiple zero crossing [24]. Appropriatefiltering and large time windows are therefore required to obtainaccurate measurements [21]. From this perspective, it providesvery poor performance in terms of frequency tracking. A differ-ent alternative is the use of the discrete Fourier transform (DFT)as a filtering technique to obtain the fundamental frequency [25].This methodology, however, is sensitive to frequency variations,since the time window employed by the DFT algorithm has tobe a multiple of the fundamental period in order to obtain ac-curate results. Recursive DFT algorithms have been proposedto overcome this problem [25]–[27]. Nevertheless, several pe-riods of the fundamental frequency are required to obtain thefrequency value accurately and with the appropriate level of res-olution. Different filtering techniques have also been developed,such as Kalman filters [28] and recursive weighted least-squareestimation [29].
Other methods are based on phase-locked loops (PLLs),which have become the state-of-the-art technique for grid syn-chronization, providing the phase angle of the fundamental fre-quency component, and additionally, frequency and amplitudeinformation with fast dynamic response [30]. In three-phasesystems, the synchronous reference frame (SRF) version of thePLL is widely employed. SRF–PLLs are very sensitive to thepresence of low-frequency harmonics and voltage imbalances,and the PLL bandwidth needs to be reduced to obtain accurateresults in the phase-angle estimation. The frequency estimation(i.e., the derivative of the phase angle), however, will containconsiderable oscillations, and therefore, unacceptable errors.Some PLL-based techniques have been proposed to improvethe frequency-estimation accuracy and enhance the dynamic re-sponse in case of distorted voltages. In [31], a new PLL methodis presented, which is able to estimate frequency in approxi-mately five cycles of the fundamental frequency with a steady-state error limited to 0.02 Hz in the presence of several types ofdisturbances. In [32], a frequency-locked-loop method is pre-sented for imbalanced and distorted voltages. This method isable to estimate the frequency in approximately two cycles ofthe fundamental frequency with a maximum steady-state errorof approximately 0.35 Hz. In all cases, several cycles of the fun-damental frequency are necessary to obtain a proper accuracylevel.
In recent years, different studies that deal with the frequencyestimation or synchronization problems have been developedin order to improve the performance that the aforementioned
methods offer. In general, these studies try to speed up the pro-cess while maintaining the accuracy levels. Enhanced robust-ness against disturbances and wide frequency-estimation rangeare also sought. In [33] and [34], a synchronization methodbased on adaptive notch filters is presented; Yazdani et al. usean adaptive notch filter estructure with adders, multipliers, andintegrators to estimate the frequency, amplitude, and phase ofthe fundamental component of a distorted input signal. A sta-bility analysis is also presented. The simulation results show agood transient response for the estimated frequency (approxi-mately 50 ms for an input signal of 60 Hz as rated frequency(i.e., three cycles), with a frequency deviation of +3 Hz) and thefrequency-estimation error is around 0.02 Hz. Nevertheless, theperformance for different signal-to-noise ratios (SNRs) is notstudied. Freijedo et al. [35] show a synchronization method forbalanced/imbalanced three-phase systems based on the com-bination of different moving-average filters, implemented inthe SRF, in order to eliminate all the possible even harmonicspresent in the input signal. A more robust version of this solutionis also developed by increasing the number of filters to handlefrequency variations. An inconvenience of this method is thefact that the frequency is calculated from the time derivative ofthe phase, which can amplify undesired noises. Experimentalresults show that the phase estimation is carried out in one pe-riod of the fundamental frequency component of a very distortedinput signal for the robust version of the method. Nevertheless,the study is not focused on frequency estimation and there is nodetailed information about the precision of the method.
In [36], Fedele et al. propose a method based on modulatingfunctions together with a second-order orthogonal generatorsystem, which is used to obtain two signals in quadrature froma single-phase voltage. The synchronization method is split intothe frequency tracking problem, and the phase and amplitudeestimations. The proposed algorithm can estimate the frequencywith imbalances, voltage sags, and harmonic components, withan estimation time around 50 ms. Although the algorithm worksproperly with high harmonic and noise content within the fre-quency range defined by the standard EN-50160 [19], no de-tailed information about the frequency-estimation error or sen-sitivity to noise is provided.
A different approach to estimate the frequency of a signal ispresented in [37]. The proposed method can estimate simulta-neously the magnitude and the frequency in a wide frequencyrange. Different adaptive FIR filters are designed in order tocancel out harmonic components together with a recursive al-gorithm to estimate the frequency. The author uses a forgettingfactor to improve the accuracy of the method. However, thebetter the accuracy is, the slower the time response becomes.The results obtained show an excellent accuracy (errors around4 mHz) for an SNR between 50 and 70 dB with settling timesbetween 27 and 180 ms (the fastest and slowest responses, re-spectively) plus the time necessary to buffer input data for theFIR filters.
A simple method to obtain the fundamental component of thegrid voltage is presented in [38]. Mathematical transformationsallow the extraction of the voltage fundamental harmonic bymeans of delayed signals, which introduces the desired phase
RONCERO-SANCHEZ et al.: ROBUST FREQUENCY-ESTIMATION METHOD FOR DISTORTED AND IMBALANCED THREE-PHASE SYSTEMS 1091
shifting. The method shows good results when there are no er-rors between the actual voltage frequency and the value used inthe algorithm as fundamental frequency. Nonetheless, this solu-tion is not robust to grid frequency variations; in this case, thephase and amplitude estimations of the fundamental componentcontain errors in steady state.
The frequency-estimation method presented in this paper isbased on an algebraic law to estimate the frequency of a sinu-soidal signal by computing only three samples. A filtering stagehas been designed to provide the method with the required ro-bustness against harmonic distortion and frequency deviations,while keeping a good tradeoff between speed and accuracy inthe estimation. The algorithm is valid for single- and three-phasesystems, even in case of imbalances.
The paper is organized as follows. Section II introduces thetheoretical background of the frequency-estimation problem aswell as the discrete filters used in this paper. The proposedmethod is described in Section III, focusing on the robustnessimprovement of the method for changes in the frequency of theinput signal. The algorithm sensitivity to noise is also analyzed.Comprehensive experimental results are presented in Section IVto validate the proposed method and evaluate its performance.Finally, the main conclusions are given in Section V.
II. THEORETICAL BACKGROUND
A. Problem Formulation
The frequency-estimation algorithm proposed in this paper isbased on the well-known discrete-oscillator law
x(k) = 2 cos (ω0Ts)x(k − 1) − x(k − 2) (1)
which is able to generate a sinusoid with the designated fre-quency ω0 for a given sampling period Ts and providing someinitial conditions [39].
Let x(k) be a sinusoidal signal of the form
x(k) = A sin(ω0Tsk + φ) (2)
where A and φ are the amplitude and the phase, respectively.Equation (1) can be rewritten as follows:
x(k) − 2 cos (ω0Ts)x(k − 1) + x(k − 2) = 0. (3)
Therefore, the angular frequency ω0 can be calculated from(3) as follows:
ω0 =1Ts
arccos[x(k) + x(k − 2)
2x(k − 1)
]. (4)
Notice that (4) can be written in a continuous-time version asfollows:
ω0 =1Ts
arccos[X(s) + X(s)e−2Ts s
2X(s)e−Ts s
]. (5)
The algebraic law (4) needs only three samples to computethe angular frequency ω0 . Therefore, if the sampling period ischosen small enough, the estimation process of ω0 will be veryfast and can be used to calculate the frequency of an electri-cal system. Nevertheless, the method has two issues that mustbe addressed in order to successfully calculate the frequency.
First, the method is very sensitive to perturbed sinusoids (e.g., ameasured voltage that contains harmonic components or noise).This problem can be solved by means of appropriate filtering, asexplained in the next section. Second, the method is numericalill-conditioned when the term x(k − 1) is close to zero. This is-sue can be solved by taking the previous estimated value whenthis condition occurs.
B. Generation of Even Harmonic Componentsin Electrical Systems and Filtering
Let us consider an electrical three-phase system, in whichthe line-to-neutral voltages (va , vb , and vc ) typically containthe fundamental harmonic and odd harmonic components (h =1, 3, 5, 7, 9, . . . ,m; h being the odd harmonic order and m anodd positive integer). These voltages can be written as follows:
va =m∑
h=1
Ah sin (hω1t + ϕah) (6)
vb =m∑
h=1
Bh sin(
hω1t −2π
3h + ϕbh
)
=m∑
h=1
Bh sin (hω1t + φbh) (7)
vc =m∑
h=1
Ch sin(
hω1t −4π
3h + ϕch
)
=m∑
h=1
Ch sin (hω1t + φch) (8)
where Ah , Bh , and Ch are the respective amplitudes of theharmonic components of the line-to-neutral voltages, ω1 standsfor the fundamental frequency, and ϕah
, ϕbh, and ϕch
are thephases of the voltages va , vb , and vc , respectively. The phasesφbh
and φchwill be used in the rest of the section for simplicity.
Notice that if the three-phase system is imbalanced, the am-plitudes Ah,Bh, and Ch and/or the phases ϕah
, ϕbh, and ϕch
will be different.If only one of the line-to-neutral voltages is taken into ac-
count, a new signal vsI can be obtained by making the simplecalculation vsI = v2
a as follows:
vsI =
[m∑
h=1
Ah sin (hω1t + ϕah)
]2
=m∑
h=1
A2h
2−
m∑h=1
A2h
2cos [2 (hω1t + ϕah
)]
+ A1
m∑h=3
Ah cos[(1 − h) ω1t + (ϕa1 − ϕah)]
− A1
m∑h=3
Ah cos [(1 + h) ω1t + (ϕa1 + ϕah)]
1092 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 4, APRIL 2011
Signal vsI contains a dc value plus even harmonic compo-nents considering that h only takes odd integer values. Hence,all its components are multiples of two times the fundamentalfrequency.
The same procedure can be applied using two or more line-to-neutral voltages as, for example, vsI I = (va − vb)
2
vsI I = (va − vb)2 = v2
a + v2b − 2va · vb . (10)
The harmonic components of signal vsI I will be a multipleof two if all the addends in (10) have even harmonics. Equation(9) demonstrates that the addends v2
a and v2b have only even
harmonics. Furthermore, 2va · vb can be written as follows:
2va · vb = 2
[m∑
i=1
Ai sin (iω1t + ϕai)
]
×[
m∑h=1
Bh sin (hω1t + φbh)
]
= 2m∑
i=1
[Ai sin (iω1t + ϕai
)m∑
h=1
Bh sin (hω1t + φbh)
]
=m∑
i=1
[Ai
m∑h=1
Bh cos [(i − h) ω1t + (ϕai− φbh
)]
]
−m∑
i=1
[Ai
m∑h=1
Bh cos [(i + h)ω1t + (ϕai+ φbh
)]
]
(11)
with i = 1, 3, 5, 7, 9, . . . ,m.Equation (11) also shows that the addend 2va · vb only con-
tains even harmonic components. Therefore, it can be concludedthat the signal vsI I will only contain even harmonics.
Equivalent results can be obtained by using other alternatives,such as (va + vb)
2 or (va − vb)2 + (vb − vc)
2 .Finally, notice that the signals vsI and vsI I can be obtained by
means of a single voltage sensor, either in single- or three-phasesystems, since vsI I can be calculated measuring the line-to-linevoltage vab (vsI I = (va − vb)2 = v2
ab ).As it is mentioned in the previous section, (5) cannot estimate
the frequency of a signal containing harmonics. For this reason,a filtering stage is required. If the estimation process is carriedout by using the signals vsI or vsI I , only even harmonics haveto be filtered.
In order to solve this issue, different filters can be applied.In this paper, the filtering stage has been designed by using the
Fig. 1. Block diagram of the frequency-estimation method.
following constant time-delay filter:
Hh(s) =1 + e−
πω 1 h s
2. (12)
Filter (12) effectively cancels out the positive and nega-tive sequences of the h-order harmonic since Hh(jω1h) =Hh(−jω1h) = 0. This filter has been adjusted with unity staticgain (Hh(0) = 1), which implies that the denominator of (12)is equal to 2. Moreover, this filter has a repetitive property,since the frequency response is repeated for every odd integermultiple of the hth harmonic (i.e., hω1 , 3hω1 , and 5hω1). Thisfeature is very useful to reduce the number of filters to be de-signed. Finally, the constant time delay of the filter is given bythe following expression:
Lh =π
ω1h. (13)
III. PROPOSED METHOD
As explained in Section II, the frequency estimator proposedin this paper is based on the algebraic expression defined in(5), which is only valid for a pure sinusoidal signal with nodistortion. In order to successfully estimate the fundamentalfrequency of the grid voltage, appropriate filtering based on filter(12) is required to attenuate the distortion that it may contain.Furthermore, the proposed method is expected to provide a fastestimation and to be robust against frequency deviations andharmonic distortion in balanced and imbalanced three-phasesystems. Additionally, it is desirable to obtain a method that canbe applied to single-phase systems.
The block diagram of the proposed method is depicted inFig. 1. The method can be divided into three different blocks:the generation of even harmonic components, the filtering stage,and the frequency calculation. The filtering process and thefrequency calculation are shown in detail in Fig. 2.
The generation of even harmonic components is used to cal-culate the signal vs (i.e., vsI or vsI I ); in the case of single-phasesystems, vs will be v2
a and, if a three-phase system is considered,then the signal vs will be calculated using one of the line-to-linevoltages, e.g., vab (vs = v2
ab ). The measurement instrumenta-tion is, therefore, reduced to only one voltage sensor for anyelectrical system.
A fast estimation of the fundamental frequency can only beachieved by reducing the delay introduced by the filtering stageas much as possible, while obtaining a pure sinusoidal signalsuitable for (5). According to (13), the delay introduced by thefilters described in Section II-B is inversely proportional to thefiltered frequency. The total delay of the filtering stage can be,
RONCERO-SANCHEZ et al.: ROBUST FREQUENCY-ESTIMATION METHOD FOR DISTORTED AND IMBALANCED THREE-PHASE SYSTEMS 1093
Fig. 2. Detail of the filtering process and the frequency calculation.
Fig. 3. (a) Bode diagram of the magnitude of filters H4 (s) (- -) and (H4 (s))4 (–), with 4f1 = 200 Hz, and (b) detail of the magnitude diagram around 200 Hz.
therefore, reduced if the frequencies eliminated by the filtersare higher, while using the lowest order harmonic componentto calculate the frequency. Several ideas are combined in theproposed method to reduce the delay introduced by the filteringstage.
1) The calculation of the signal vs , as explained inSection II-B, generates a dc value [see (9)]. This dc com-ponent can be eliminated by adding a filter based on theso-called first-difference filter [40], which only introducesa small delay Ld that can be adjusted to eliminate the dccomponent while keeping the gain at high frequencies low
Hdc(s) = 1 − e−Ld s . (14)
2) According to (9), the lowest harmonic in vs , excluding thedc component, is twice the fundamental frequency (2ω1).
The proposed method combines both ideas and performs thefrequency estimation using the second harmonic, which is thelowest frequency component, while the rest of harmonic contentthat might typically appear is filtered. It is important to noticethat vs will always contain the second harmonic either in single-or three-phase systems. Furthermore, in both balanced and im-balanced three-phase systems, the second harmonic is alwayspresent. The generated second harmonic will have a consid-erable amplitude, resulting in a higher SNR, and hence, moreaccurate estimation results can be obtained.
Filter (12) is useful if the frequency of the harmonic to becanceled is a priori known. Nevertheless, as the frequency of
an electrical grid can undergo variations, this filter could failto attenuate the desired harmonic components; consequently, amore robust filter is needed for frequency changes.
The requirement of a robust version of the filter (12) canbe obtained by making the higher derivatives of the frequencyresponse of Hh(s) equal to zero at frequency ω1h, which may beachieved with a filter with transfer function (Hh(s))n (n beinga positive integer greater than 1) [41]. Effectively, the frequencyresponse of this filter and its n − 1 derivatives are equal to zeroat s = jω1h. However, the total delay introduced by the filter isnow multiplied by n
Lhn=
nπ
ω1h. (15)
In order to illustrate how the filter response varies, Fig. 3shows the magnitude of the frequency response of two filtersadjusted to eliminate the fourth harmonic component with thefundamental frequency f1 = 50 Hz. Fig. 3(a) shows that themagnitude of the filter (H4(s))
4 is close to zero in a widerfrequency region around 200 Hz than the one obtained with thefilter H4(s), which can be seen in more detail in Fig. 3(b).
Finally, the parameter n can be designed for a maximum givenfrequency deviation, as specified in the standards to be adopted,such as the standard EN 50160 [19]. A value of 3 provides agood tradeoff between accuracy and delay in the estimation (seeSection IV for more details).
The total Bode of the filtering stage is depicted in Fig. 4,showing the magnitude of the frequency response of the filters
1094 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 4, APRIL 2011
Fig. 4. Magnitude of the frequency response of the complete filtering stage.
combined in cascade to eliminate the dc component and theundesired harmonics 4th, 6th, 8th, 10th, 12th, 14th, 16th, and18th.
Additionally, the offset introduced by the sensors producesa harmonic component of the fundamental frequency after thecalculation of the signal vs . Although the offset amplitude is verysmall, a 50-Hz filter is used to remove this offset and improvethe accuracy of the estimator. Since 50 Hz is a low-frequencycomponent that would add a considerable delay if (12) is used,a faster notch filter is used to eliminate this component by usingthe following expression:
Hω1 (s) = 1 − 2 cos(ω1L)e−Ls + e−2Ls. (16)
The total time delay of this filter is 2L. The design of theparameter L must take into account a tradeoff between thetime delay and the filter gain at high frequencies (i.e., a smallvalue of L would produce an undesired magnification at highfrequencies).
Once all the harmonic components are eliminated, (5) is ap-plied to the resulting signal, as shown in the block diagram ofFig. 2.
Taking into account the time delays introduced by (5) and(16), together with the time delays defined in (14) and (15), thenecessary total time to carry out the frequency estimation canbe calculated as follows:
tset = 2Ts + Ld + 2L +∑h∈M
nπ
ω1h,
M = {4, 6, 8, 10, . . . , 18} . (17)
As mentioned in Section II, the frequency estimator is verysensitive to the presence of noise. The influence of noise on theestimation accuracy has been studied by simulating the com-plete system in the presence of additive white Gaussian noiseat different SNR levels and with different transformations toobtain the signal vs . The frequency of a highly distorted three-phase system with a fundamental frequency of 50 Hz is used toanalyze the influence of noise. The proposed method is testedby using one or two of the line-to-neutral voltages and usingdifferent transformations to calculate vs . In the results shownin Fig. 5, it can be observed that the frequency-estimation rmserror is very low for the typical SNR values of the voltage
Fig. 5. Frequency-estimation rms error for different values of SNR usingvs = v2
a , vs = (va − vb )2 , and vs = (va + vb )2 .
grid (50–70 dB) [37], particularly when the method employsthe transformation (va − vb)2 = v2
ab . Further simulations showthat, in the case of three-phase systems, the best results are alsoobtained for the same transformation: (va − vb)2 = v2
ab . Hence,the optimum solution will be the use of a single voltage sen-sor to measure one of the line-to-line voltages and to calculatevs = v2
ab (vs = v2bc or vs = v2
ca ).Since the method is implemented in a microprocessor-based
system, filters (12), (14), and (16) must be expressed in thediscrete-time domain. Ideally, the sampling period Ts shouldbe chosen so that the delays of the filters are integer multiplesof Ts . Unfortunately, due to the number of filters and otherconsiderations, some of the delays will not be integer multiplesof the sampling period. Different alternatives can be consideredto solve this issue, such as rounding the delay to the nearestlower or higher integer [42]. This solution, however, results inerrors introduced by the difference between the real delay andthe integer approximation.
In this paper, a linear interpolation has been implementedachieving good results in the frequency region of interest ofeach filter. The respective delays of filters (12), (14), and (16)can be written as follows:
where dh , dd , and d are the nearest lower integer multiplesof Ts for filters (12), (14), and (16), respectively, and τh , τd ,and τ are positive real numbers lower than 1. The proposedlinear interpolation for a general delay Lg = (dg + τg ) Ts can bewritten using the z-transform as Z{e−Lg s} ≈ (1 − τg ) z−dg +τg z
−dg −1 . Notice that, if the delay Lg is an integer multipleof Ts , and therefore, τg = 0, this linear interpolation coincideswith the z-transform of an integer delay (z−dg ).
Finally, the z-transforms of filters (12), (14), and (16) yieldthe following:
Z{
1 + e−Lh s
2
}≈ 1 + (1 − τh) z−dh + τhz−dh −1
2(21)
RONCERO-SANCHEZ et al.: ROBUST FREQUENCY-ESTIMATION METHOD FOR DISTORTED AND IMBALANCED THREE-PHASE SYSTEMS 1095
Fig. 6. (a) Bode diagram of the magnitude of filters (H12 (s))3 (–) and(H12 (z))3 (– ·), and with the delay rounded to the nearest integer (- -). (b) Detail of themagnitude around 600 Hz.
with 2L = (da + τa) Ts .The implementation of filters (21)–(23) in a digital processor
is very simple and only requires enough amount of memory tostore the respective samples in each case.
As an example, Fig. 6 shows a comparison of the frequencyresponses of three filters designed to cancel the 12th har-monic with a sampling period Ts = 5000−1 s, where L12 =8.333 × 10−4 s, d12 = 4, and τ12 = 0.16667. The Bode di-agram shows the magnitude of the ideal filter (H12(s))3
compared with those obtained with the linear interpolation(H12(z))3 =
(0.5
(1 + (1 − τ12)z−4 + τ12z
−5))3
and with thedelay rounded to the nearest integer (H ′
12(z))3 = (0.5(1 +z−4))3 . Fig. 6(a) and (b) shows that the frequency responseof the linear interpolation fits very well the ideal response inthe frequency region of 600 Hz, whereas the approximation tothe nearest integer exhibits a higher error at 600 Hz. Hence, thisapproximation is less suitable to cancel a harmonic of order hthan the one obtained with the linear interpolation. Moreover,Fig. 6(a) shows how the amplitude of the linear interpolationdiminishes as frequency increases, which can also be beneficialto attenuate possible high-frequency signals such as noise.
IV. EXPERIMENTAL RESULTS
In order to evaluate the performance of the proposed estima-tion method, several experiments were carried out. The experi-mental setup consists of a programmable power source ELGAR
TABLE IPARAMETERS OF FILTERS FOR EVEN HARMONICS
SW10500 (10.5 kVA) for the generation of different voltagewaveforms, which were measured by means of three Hall-effectsensors (LEM LV-25P). The algorithm was implemented in areal-time platform (dSPACE DS1103), which also acquired theanalog signals provided by the voltage sensors. The samplingfrequency was set to 5 kHz, whereas the maximum limits ofthe frequency-variation range were established in accordancewith the standard EN-50160 [19]. For this frequency range, theparameter n of the robust filters was chosen equal to 3. The timedelays of the dc filter and the 50-Hz filter were set to 1 and 5 ms(Ld = 1 ms and L = 2.5 ms), respectively. With these values,the parameters of the dc filter are dh = 5 and τh = 0, whereasthe parameters of the filter, which suppresses the 50-Hz com-ponent, are d = 12, τ = 0.5, da = 25, and τa = 0, respectively.The parameters of the filters for even harmonics are summarizedin Table I, while Fig. 7 shows the Bode diagram of the filteringstage implemented in discrete time, which is very similar to the
1096 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 4, APRIL 2011
Fig. 7. Magnitude of the frequency response of the filtering stage implementedin discrete time.
TABLE IIHARMONIC COMPONENTS IN THE WAVEFORMS USED FOR DIFFERENT TESTS
one plotted in Fig. 4. Finally, taking into account (17), the totaltime necessary for the estimation to be carried out is 33.8 ms.
Two different waveforms were generated in order to carry outseveral tests. Waveform 1 consists of a distorted set of three-phase voltages; the voltages only contain odd harmonics untilthe 17th component, and the limits of the harmonic componentswere in accordance with the standard EN-50160 [19]. On theother hand, waveform 2 was performed by adding some oddharmonic components with a high amplitude in order to ob-tain a very distorted voltage. In particular, the 5th, 7th, 11th,and 13th components were added. In both cases, the fundamen-tal harmonic amplitude of the line-to-neutral voltage was 311 V(220 Vrms). The total harmonic distortions for waveforms 1 and2 are THDv1 = 10.67% and THDv2 = 31.6%, respectively.Table II summarizes the relative amplitudes of the harmoniccomponents to the fundamental one for both the waveforms,while Fig. 8 shows the measured line-to-neutral voltages inboth cases.
The aim of the first test is to demonstrate that the frequencycan be estimated either by using the measurement of one ortwo line-to-neutral voltages (i.e., v2
a or (va − vb)2). For thispurpose, only waveform 1 has been used with the followingsequence of events: 1) the initial frequency at t = 0 s is 50 Hz;2) at t = 3 s, a frequency step change from 50 to 47 Hz isapplied; and 3) finally, at instant t = 6 s, the frequency steps up
Fig. 8. Two generated sets of three-phase voltages for the experimental results:(a) waveform 1, and (b) waveform 2.
from 47 to 52 Hz. Fig. 9 shows the results obtained with oneand two line-to-neutral voltages: in both the cases, the algorithmtakes approximately 30 ms to estimate the frequency (one anda half cycles of the 50-Hz frequency), and it is observed that,besides the transient response achieved, the algorithm is able toproperly estimate the frequency for the two aforementioned stepchanges [see Fig. 9(a) and (b)]. The results show that there areno significant differences in the use of one or two line-to-neutralvoltages. Therefore, this algorithm is also suitable for frequencyestimation in single-phase systems.
Waveform 2 has been used in order to test the behavior ofthe estimation algorithm when three-phase balanced voltagesare generated with different and simultaneous disturbances. Inthis case, the signal vs is (va − vb)2 = v2
ab . The time sequencecomprises different cases: 1) the initial frequency is 50 Hz and a30% voltage sag occurs at t = 0.1 s; 2) at instant t = 0.2 s, thevoltage sag disappears and the frequency steps down to 47 Hz;3) after this, the frequency increases its value to 52 Hz; and 4) fi-nally, a phase shift of 20◦ takes place. Fig. 10 shows the resultsobtained for the complete sequence: the highly distorted voltagewaveform in phase A is plotted in Fig. 10(a) (see Fig. 8(b) andTable II for more details); and Fig. 10(b) shows the time responseof the estimation process: once again, the estimation is carriedout in approximately 30 ms, and the final value converges to
RONCERO-SANCHEZ et al.: ROBUST FREQUENCY-ESTIMATION METHOD FOR DISTORTED AND IMBALANCED THREE-PHASE SYSTEMS 1097
Fig. 9. Time response of the estimation process for step changes in the fre-quency using v2
a and (va − vb )2 : (a) step from 50 to 47 Hz, and (b) step from47 to 52 Hz.
the real one with independence of the disturbance present in themeasured voltage. Although the transient response of the fre-quency estimation exhibits a considerable overshoot when thephase shift occurs, the performance of the method is good forthe rest of the disturbances and changes of the tested sequence.Finally, Fig. 10(c) shows the filtered signal vsf obtained after thefiltering process; it can be observed that this waveform is closeto a pure sinusoidal waveform without harmonic content andits frequency is twice the estimated frequency, corresponding tothe second harmonic.
A detail of the time response of the frequency-estimationprocess can be seen in Fig. 11 (corresponding with the timeinterval in which the real frequency is 52 Hz). The frequencyestimation is very accurate in steady state with a maximumabsolute error in the range of 15 mHz. The results obtained byusing waveform 2 suggest that the frequency estimation can becarried out with highly distorted waveforms, such as currentmeasurements of nonlinear loads (e.g., uncontrolled rectifiers).
The estimation process has also been tested in an imbalancedsituation. In this case, waveform 2 has been used to generate theline-to-neutral voltages va and vb , where the fundamental har-monic amplitudes are 311 and 218 V, respectively. Furthermore,
Fig. 10. Experimental results with waveform 2 under different disturbances:(a) measured phase-to-neutral voltage, (b) time response of the frequency esti-mation, and (c) filtered signal vsf .
Fig. 11. Detail of the time response of the estimation process for a fundamentalfrequency f = 52 Hz.
1098 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 4, APRIL 2011
Fig. 12. Time response of the estimation process for an imbalanced situa-tion with a step change in the frequency using (va − vb )2 : (a) line-to-neutralvoltages, and (b) frequency estimation.
the phase shift of voltage vb with respect to voltage va is −110◦.The line-to-neutral voltage vc is a sinusoidal waveform with anamplitude of 156 V and a phase shift with respect to va equalto −270◦. The initial frequency of the voltages is 50 Hz, and atinstant t = 0.2 s, the frequency steps up to 52 Hz. The estima-tion process has been carried out using the signal vs = v2
ab . Thethree imbalanced voltages can be seen in Fig. 12(a), whereasFig. 12(b) shows that the frequency estimation converges to theactual value, corroborating that the algorithm is able to obtainthe real frequency value even under imbalanced situations.
Table III summarizes the rms errors for different waveformsand for the limits of the frequency range defined in the standardEN-50160 [19]. The best results are obtained when the realfrequency is 50 Hz, achieving a reduced error even with the hightotal harmonic distortion present in waveform 2 (the maximumrms error is 6.3 mHz), while the highest errors are producedwhen the frequency deviation is maximum (−3 Hz with respectto the 50 Hz value); in this case, waveform 2 shows again theworst results with an rms error of 27 mHz (note that the rmserror obtained with waveform 1 for that frequency value is muchsmaller).
TABLE IIIFREQUENCY-ESTIMATION RMS ERRORS FOR DIFFERENT WAVEFORMS
TABLE IVFREQUENCY-ESTIMATION RMS ERROR FOR DIFFERENT ORDER
OF THE FILTERS (n = 2, n = 3, AND n = 4)
A comprehensive simulation has been carried out beyondthe frequency limits established by the standard EN-50160 inorder to evaluate the performance of the algorithm for differentorders of the filters used in the filtering process. The simulationresults have been obtained using waveform 2 without noise.Table IV shows the rms error of the frequency estimation. Itcan be observed that, even though the fastest time responseis obtained for a filtering stage with order n = 2 (24.7 ms,i.e., 1.24 cycles of the fundamental frequency), the estimationaccuracy is very poor for frequency values lower than 48 Hz andgreater than 52 Hz. Hence, this option should be restricted toapplications operating within a small frequency range near therated frequency (e.g., 50 Hz), which require a fast time responseof the estimation process, or when the harmonic content ofthe measured voltages is low. As an example, the frequency-estimation method has been tested using the waveform specifiedin [34] with a total harmonic distortion of 5% and the samefrequency deviation value (see [34] for more details); in thiscase, the rms error obtained for n = 2 and a frequency jumpfrom 50 to 53 Hz is 0.016 Hz, which is a similar result in termsof accuracy when compared to other methods.
RONCERO-SANCHEZ et al.: ROBUST FREQUENCY-ESTIMATION METHOD FOR DISTORTED AND IMBALANCED THREE-PHASE SYSTEMS 1099
Fig. 13. AC motor drive connected to a 2.2-kW induction motor.
A good tradeoff between accuracy and dynamic response isobtained for n = 3; the total time necessary for the estimationis approximately one and a half cycle of the fundamental fre-quency, and the estimation error has been reduced by one orderof magnitude, at least, with respect to the case of n = 2. Thisoption can operate in the whole frequency range specified inTable IV and is suitable for voltage signals with high harmoniccontent (recall that the total harmonic distortion of waveform 2is 31.6%). The option of using higher order filters (e.g., n = 4) issuitable for applications, which require a very accurate value ofthe estimated frequency or which use measurements with highharmonic content; in this case, the required time to estimatethe frequency is approximately two cycles of the fundamentalfrequency, and very small estimation errors are achieved in thewhole frequency range.
Finally, in order to test the proposed method in a real acline, the frequency of the electrical grid available in the lab-oratory has been estimated. Moreover, a commercial 3-kW acmotor drive (ACS350-03E-07A3-4) supplying a 2.2-kW three-phase squirrel-cage motor (3GAA092004-ASE by ABB) hasbeen involved in the test (see Fig. 13). The grid voltage has arated line-to-line voltage equal to 380 V with a fundamentalfrequency of f = 50 Hz, whereas the switching frequency ofthe ac motor drive is 4 kHz.
Fig. 14 shows the three line-to-neutral voltages of the elec-trical grid measured with the oscilloscope when the ac driveis working; the three voltages are not purely sinusoidal, andthe amplitude of phase B is higher than the amplitudes ofother phases. The frequency measured with the oscilloscopeis 50.0046 Hz.
As shown in Fig. 15, different odd harmonics are presentin the line-to-line voltage vab ; the fundamental component hasan amplitude of 556 V, the amplitudes of the 5th, 7th, 11th,and 13th harmonic components are very noticeable, while the49th harmonic is the highest frequency component measured.The estimated frequency using the signal vs = v2
ab is shown inFig. 16. The method works properly even in the presence ofthe ac motor drive and the high-order harmonics of the voltage
Fig. 14. Line-to-neutral voltages in the laboratory.
Fig. 15. Harmonic spectrum detail of the voltage vab .
Fig. 16. Estimated grid frequency in the laboratory (f = 50 Hz).
waveform vab . The frequency estimation has a small steady-stateerror below 20 mHz, and no disturbances are noticeable.
V. CONCLUSION
Frequency estimation plays an important role to ensure thecontrol and reliability of power systems such as microgrids. Thisestimation has to be accurate and fast even in the presence of
1100 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 4, APRIL 2011
disturbances, such as harmonic components, voltage imbal-ances, etc.
According to these requirements, this paper proposes a fastidentification method to estimate the fundamental frequency forbalanced and imbalanced three-phase systems and for single-phase systems in a wide frequency range, even in the presenceof harmonic distortion. The estimation method is based on analgebraic law, which provides a fast convergence to the solution,together with a simple mathematical transformation to generateeven harmonics. Furthermore, a filtering stage has been usedto eliminate the undesired harmonic components. The resultingfrequency-estimation method is robust against the presence ofharmonic distortion and frequency deviations, and exhibits smallestimation errors in steady state. The algorithm can be imple-mented with voltage or even current measurements up to certainlimits. Experimental results show the excellent performance ofthe algorithm in terms of dynamic response and accuracy.
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Pedro Roncero-Sanchez (M’07) received the M.Sc.degree in electrical engineering degree from Univer-sidad Pontificia Comillas, Madrid, Spain, in 1998,and the Ph.D. degree in industrial engineer from Uni-versidad de Castilla-La Mancha, Ciudad Real, Spain,in 2004.
He is a currently Senior Lecturer at the Schoolof Industrial Engineering, Universidad de Castilla-La Mancha. His research interests include control ofpower electronic converters, power quality, renew-able energy systems, and energy storage devices.
Xavier del Toro Garcıa received the B.Sc. and M.Sc.degrees in automatic control and industrial electron-ics from the Universitat Politecnica de Catalunya,Catalunya, Spain, in 1999 and 2002, respectively, andthe Ph.D. degree from the University of Glamorgan,Wales, U.K., in 2008.
From September 2005 until October 2006, he wasa Marie Curie Research Fellow at Politecnico di Bari,Bari, Italy. Since 2008, he has been a Researcher inthe University of Castilla-La Mancha, Ciudad Real,Spain. His research interests include power electron-
ics, renewable energy sources, energy storage systems, and power quality.
Alfonso Parreno Torres received the B.Eng. de-gree in electronic engineering from Universidad deValencia, Valencia, Spain, in 2006. He is currentlyworking toward the Ph.D. degree at the Universidadde Castilla-La Mancha, Ciudad Real, Spain.
He is with the Albacete Science and TechnologyPark, Albacete, Spain. His research interests includecontrol, power electronics, power quality, and signal-processing algorithms for electrical power systems.
Vicente Feliu (SM’08) received the M.S. degree(Hons.) in industrial engineering and the Ph.D. de-gree from the Polytechnical University of Madrid,Madrid, Spain, in 1979 and 1982, respectively.
From 1980 to 1994, he was with the ElectricalEngineering Department, Universidad Nacional deEducacion a Distancia, Madrid, where he was a FullProfessor in 1990 and the Head of the Departmentfrom 1991 to 1994. From 1987 to 1989, he was aFulbright Scholar at the Robotics Institute, CarnegieMellon University, Pittsburgh, PA. He is currently
with the School of Industrial Engineering, Universidad de Castilla-La Mancha,Ciudad Real, Spain. His research interests include multivariable and digital con-trol systems, and kinematic and dynamic control of rigid and flexible robots.
