- Email: [email protected]
Recursive estimation of power system harmonics
Electric Power Systems Research 47 (1998) 143 – 152
Recursive estimation of power system harmonics Maamar Bettayeb *, Uvais Qidwai Electrical Engineering Department, King Fahd Uni6ersity of Petroleum and Minerals, PO Box 1730, Dhahran 31261, Saudi Arabia Received 23 February 1998; accepted 30 March 1998
Abstract This paper presents the application of well known recursive estimation techniques to the important problem of power system harmonics in a noisy environment. on-line estimation of harmonic amplitudes and phases is performed using several variants of recursive least square (RLS) algorithms, known for their simplicity of computation and good convergence properties. The estimates are updated recursively as samples of the harmonic signals are received. A noisy harmonic signal from the AC bus of a six pulse rectifier is used as a test signal in the simulation. The various RLS algorithms are evaluated under different signal to noise ratios (SNR) and are shown to produce good harmonic magnitude and phase estimates even for a 0 dB SNR. Due to their simplicity, these algorithms are appropriate for on-line implementation in polluted power systems. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Harmonics; Estimation; Least square estimator; Recursive estimation
1. Introduction In recent decades, the increasing use of non-linear loads has led to elevated harmonic levels in the power system network, which produces several problems related to power system operation, protection and control. This has raised worldwide interest in harmonic studies, including harmonic modeling, estimation, mitigation and a variety of related subjects. This paper addresses the problem of harmonic estimation with specific emphasis on the use of estimation techniques for on-line application. Accurate harmonic estimation is a very important task in industrial power system. These estimates are useful both for a better characterization of electrical machinery under non sinusoidal conditions and for efficient design of compensatory filters and other equipment. Hence, there is a need to continuously monitor various signals in a power system for harmonics. The difficulty in measurement of harmonics stems from the fact that most of the harmonic producing sources are dynamic in nature and produce time varying amplitudes in voltage and current waveforms. Therefore, fast
* Corresponding author. Tel.: +966 3 8602277; fax: + 966 3 8603535; e-mail: [email protected] 0378-7796/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0378-7796(98)00063-7
and accurate estimates for amplitude and phase of these frequency components are needed. This requires estimation algorithms that have less computational burden and can therefore be used on-line. This paper presents estimation algorithms that can be used in on-line recursive estimation of amplitudes and phases of different frequency components present in a distorted power system voltage signal. The most commonly used classical estimation technique is based on the fast fourier transform (FFT) of the signal. Apart from the classical algorithm, many extensions and improvements have been published [1– 5]. These methods utilized the original FFT with different windowing and interpolation techniques to produce better estimates. The FFT based techniques are good for noise rejection but have the inherent problems of spectral leakage, aliasing and picket fence effect. Kalman filtering has been used for a number of power systems in order to estimate different states or parameters [6,7]. Kalman filtering technique utilizes simple, linear and robust algorithm to estimate the magnitude of the known harmonics embedded in the signal along with stochastic noise. It gives a better noise rejection and estimation compared to FFT algorithms. Among the classical methods of estimation of frequencies and their respective amplitudes and phases,
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
144
Table 1 Harmonic content of the test voltage signal at the load bus for the six pulse rectifier Harmonic order (Hz)
Amplitude (p.u.)
Phase (degrees)
Fundamental (60) 5th (300) 7th (420) 11th (660) 13th (780)
0.95 0.09 0.043 0.03 0.033
−2.02 82.1 7.9 −147.1 162.6
maximum likelihood estimation algorithms have been used by many researchers in the area [8,9]. One reason being a very established theoretical structure of this algorithm and the near ideal properties that this algorithm possessed. However, the estimator suffers the formulation complexity problems which makes it unsuitable for on-line implementation. The estimates also tend to be sensitive to small changes in the fundamental frequency. Another popular method is least absolute value estimation. This method is based upon the minimization of sum of the absolute values of residuals. The LAV norm [10] can often be a potential alternative to the classical least square (LS) norm in estimation problems. The algorithm can be used to optimally track the harmonics in the power system network. This algorithm can also handle time varying parameters, rejects bad measurements and at the same time works well in the moderately high noise levels. However, LAV is extremely sensitive to the deviations in the fundamental frequency and requires a prior measurement for the fundamental frequency to correct for this discrepancy. A very popular method for parameter estimation is least square (LS) algorithm. The algorithm is based upon the quadratic error objective function and is very powerful in estimating system parameters under specific conditions [11,12]. It has been used by many researchers to estimate the frequencies and their deviations in a signal. Many other algorithms are also found in the literature.
Modern power system utilizes microprocessors for the purpose of control and protection. This require real-time calculations for the basic waveforms in voltage and current waveforms based upon the measured data in the presence of high noise. The existence of a very small time window to take the necessary action, requires the use of recursive techniques to estimate the amplitudes and phases of different frequency components present in the signal. Recursive estimation is also useful because of the fact that there are always changes in the system like changes in loading, changes in topologies of various outgrowths in the system and occurrence of faults. These changes produces different type of non-linearities and hence produces different harmonics with varying amplitudes and phases. This requires on-line adaptation in the estimates so that the necessary actions can be taken at the right moment. The objective of this work is the application of several variants of recursive least square for on-line estimation of power system harmonics. Due to time variation in the power network, it is important to periodically update the harmonic estimates by using the latest available signal samples. It is shown that recursive least square algorithms implemented in this work, are capable of efficient estimation of harmonics buried in noise. This paper presents the estimation of harmonics present in a distorted voltage or current waveform. The distortion of the signal is further enhanced by considering the case of different noise levels. The methods used here are different least square (LS) based algorithms; least square estimator (LS), weighted least square estimator (WLS), recursive least square estimator information form (RLS-I), recursive least square estimator covariance form (RLS-C), extended least square estimator form 1 (ELS-1) and extended least square estimator form 2 (ELS-2) [14–16]. The first two methods use batch processing and are a basis of comparison and evaluation for the recursive least square algorithms. The rest of the paper is arranged as follows. Section 2 briefly describes the various methods of estimation used in this paper. Section 3 states the estimation problem and its formulation for single as well as multiple harmonic estimation cases. Section 4 presents the simulation results and Section 5 gives the conclusion of the work
2. Estimation algorithms
Fig. 1. Test system: a two bus architecture with six pulse full wave ridge rectifier supplying the load.
The algorithms used in this work are based upon the criterion of least square norms. These have been categorized as batch and recursive algorithms and are summarized next.
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
145
Fig. 2. Test signal (a) and its spectrum (b).
2.1. Batch least square algorithms This includes the least square estimator (LS) and the weighted least square estimator (WLS). For the linear model,
found in the literature [14–16]. The enhanced performance gained through these modifications are faster convergence, less deviations, recursive calculations and consistency of estimates. In particular, the feature for recursive calculation is attractive in connection with using the algorithms on-line. This work utilizes the following algorithms.
where, Z(k) is the noisy measurement, f(k) is the system structure matrix, u(k) is the vector for unknown parameters that are to be estimated and n(k) is the additive noise. The estimate for the required parameter vector u, can be obtained by
2.2.1. Recursi6e least square estimator-information form (RL-I) This is one variation to use the WLS recursively. This is comprised of a three step calculation in each iteration;
u. LS(k) =[f(k)f T(k)] − 1f(k)Z(k)
P − 1(k+1)= P − 1(k)+ f T(k+1)w(k+ 1)f(k+1)
The weighted least square estimator (WLS) is given by
KW (k+1)= P(k+1)f T(k+1)w(k+1)
u. (k) =[FTWF] − 1FTWZ The weighing matrix W is used to give more importance to some of the observations as compared to others. Usually, the most recent observations are given more importance and hence, are weighted more heavily. The usual choice of this matrix is
Æu 1 0 … 0 Ç Ã Ã 0 m2 … 0 Ã W(k)= Ã Ã Ã È 0 0 … m kÉ The choice of m lies between 0 and 1; however, it is usually recommended to use, m close to 1.
2.2. Recursi6e least square algorithms To improve the performance in the least square algorithm, many extensions and modifications are
u. (k+1)= u. (k)+ KW (k+1)[z(k+ 1)− f(k+ 1)u. (k)] These equations are initialized by taking some initial values for the estimate at instants k, u. (k) and P. For P, a rule of thumb is usually used which requires P= aI, where a is a large number and I is the identity matrix of order n× n, where n is the number of parameters to be estimated.
2.2.2. Recursi6e least squares estimator— co6ariance form (RLS-C) This is the second variation for recursive usage of WLS. This algorithm corresponds to the following computations: KW (k+1)= P(k)f T(k+1)
f(k+ 1)P(k)f T(k+1)+
1 w(k+ 1)
n
−1
P(k+ 1)=[1− KW (k+1)f(k+1)]P(k) u. (k+ 1)= u. (k)+KW (k+1)[z(k+ 1)− f(k+ 1)u. (k)]
146
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
Fig. 3. Test signal with noise and its spectrum.
The choice of P remains the same as for the information form. The advantage of this algorithm compared to last one is that the matrix inversion is converted to scalar division.
K(k+ 1)=P(k+ 1)f T(k+1) u. (k+1)= u. (k)+ K(k+ 1)[z(k+ 1)−f(k+ 1)u. (k)] 3. Harmonic estimation problem
2.2.3. Extended least square estimator-1 (ELS-1) This algorithm is simpler to calculate and provides a method to eliminate the bias present in the methods based upon the actual least square algorithm. This technique rearranges the observation matrix F and formulates the problem as follows: P − 1(k +1)= P − 1(k) + f T(k)f(k) u. (k +1)=u. (k)+P(k +1)f T(k)[z(k + 1) −f(k)u. (k)]
2.2.4. Extended least square estimator-2 (ELS-2) This algorithm is another modification of the extended least square that is especially suitable for fast convergence. P(k+ 1)= P(k)− P(k)f T(k)f(k)P(k) [w(k)+ f(k)P(k)f T(k)] − 1
In this section, the harmonic estimation problem is formulated for both fundamental harmonic estimation and the multiple harmonic case.
3.1. Fundamental harmonic estimation The primary objectives of power system fundamental harmonic signal estimation and retrieval are: to provide reference signal for active filtering schemes; to analyze the voltage peak value in order to warn against any possible voltage collapse; and to decide about the reactive power requirements of the system at that instant. For the case of a signal with fundamental frequency v0, the harmonics are the multiples of this frequency. Hence, the structure can be written as follows:
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
147
noise or other harmonics—as the additive noise; the new structure will become, Z(k)=A1 sin (v0k+ f1)+ m(k) By simple manipulation, this equation can be modified for the purpose of estimation: Z(k)=[sinv0kcosv0k]
n
a + mk b
or in the standard form, Z(k)= F(k)u+ m(k) where a and b are the parameters to be estimated and are given by a= A1 cos f1 b= A1 sin f1 The actual required parameters are A1 and f1, which can be given in terms of a and b as Fig. 4. Estimation strategy.
Z(k) = A1sin(v0k + f1) +A2sin(2v0k +f2) +···+ 6(k) where, Ai is the amplitude of the ith harmonic, fi is its phase and n(k) is the additive noise. For the estimation of the fundamental sinusoid, it is necessary to consider the rest of the distortion present in the signal — either
A1 = a 2 + b 2 f1 = tan − 1
b a
3.2. Multiple harmonic estimation The estimation of multiple harmonics in power system is carried out in order to:
Fig. 5. Fundamental harmonic with single frequency estimation technique.
148
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
Fig. 6. Fundamental harmonic with multiple frequency estimation technique.
tune proper filters to improve the system performance; discover the characteristics of the signal present in the system for future planning, control and protection; and use appropriate capacitors and other VAR compensators in order that the combined impedance is not close to the resonance value for some of the harmonics present. The same routines can also be used for multiple frequency estimation, i.e. to estimate the amplitudes and phases of different harmonics in the signal. The modification is in the observation matrix f and parameter matrix u. In this case, the assumption which was made earlier for considering the harmonics with the noise collectively, is now discarded. Each harmonic has its representative term appearing in the matrix f. The new equations for the model, based upon the values from Table 1 are as follows: F(k)=
Æ sin (v 0t1) cos (v 0t1) sin (5v 0t1) cos (5v 0t1) … Ã sin (v 0t2) cos (v 0t2) sin (5v 0t2) cos (5v 0t2) … Èsin (v 0tk ) cos(v 0tk ) sin (5v 0tk ) cos (5v 0tk ) … sin (13v 0t1) sin (13v 0t2) sin (13v 0tk )
cos (13v 0t1) Ç cos (13v 0t2) Ã cos (13v 0tk )É
and u(k)=[a1 b1 a5 b5 … a13 b13]T
4. Simulation results The distorted signal is constructed by using the values used by many authors in this area [6,13]. The signal is a distorted voltage waveform at the terminals of a load bus for a load driven by a six pulse rectifier, in a simple two bus configuration, as shown in Fig. 1. Table 1 shows the harmonics present in the system and Fig. 2 shows the resulting signal with its frequency spectrum. The sample signal thus formulated is further distorted with zero mean additive Gaussian noise. Different noise powers were used however, the two high power noise cases being given here. These correspond to signal to noise ratios (SNR) of 10 and 0 dB. The resulting signal is shown in Fig. 3. Four different forgetting factors were used, l=1, l= 0.98, l= 0.95 and l=0.9. These were used for faster convergence and to study their effects on algorithm robustness. It was found that lB 0.95 gave poor results. For each algorithm of estimation, 20 realizations were established for each forgetting factor and SNR and the averaged estimate was plotted as a bar graph. The simulation strategy is shown in Fig. 4. The selection of sampling time and the harmonic frequencies that are to be estimated, is the initial step
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
149
Fig. 7. Fifth harmonic with multiple frequency estimation technique.
for the algorithm. With this selection, the time vector and the structure matrix F can be formulated beforehand. The next step is to select the method for estimation from the four recursive methods RLS-L RLS-C, ELS-1 and ELS-2. The samples are then obtained recursively and estimates are calculated. When the time vector is exhausted, the algorithm can either restart or terminate. The simulation results presented below are based on estimation for one whole period, with a sampling frequency of 1620 Hz, which is slightly higher than the Nyquist frequency for the test signal.
4.1. Fundamental harmonic case Fig. 5 shows the results for estimation of fundamental frequency. The true values that are being estimated are the amplitude and phase of the fundamental frequency component. The actual value for the amplitude is 0.95 p.u. and that for the phase is − 2.02°. The results show almost exact estimation. In case of 10 dB SNR, the maximum deviation is about 0.4%, while in the case of 0 dB SNR it is 0.95%. However, the estimates of phase show deviations of 4 and 8% for 10 and 0 dB SNR cases, respectively.
4.2. Multiple harmonic case Fig. 6 shows the results of estimates obtained for fundamental harmonic by using the multiple harmonic estimation algorithm. As can be seen from the results in
these figures, the estimation is again fairly close to the exact value for almost all of the algorithms used. Any discrepancy could be a result of numerical approximation and finite word length effects, especially for the calculation of phase. The maximum deviation in amplitude estimate is 0.002 p.u., corresponding to an error of 2%. Similarly for the case of phase, except for one estimate, the results are within 5% error. Although the presented results are for the Gaussian noise case, similar results were obtained for uniform noise distribution. Fig. 7 shows the estimation for the amplitude of fifth harmonic. The actual value is 0.09 p.u. and the phase is 82.1°. In the case of amplitude estimation, the maximum error for 10 dB SNRs is about 1.5%, while for 0 dB the error increases to 6.7%. The reason is the adding of noise components at the same frequency with the fifth harmonic component to produce a higher amplitude harmonic. However, the estimation for phase is much closer to the actual, producing an error of about 1%. Fig. 8 shows the estimates for the seventh harmonic. The true value is 0.043 p.u and 7.9°. The maximum error in the estimation of amplitude is 8% and this is the case when SNR is 0 dB. Similarly, the error in phase is about 7.6%. However, what should be noted here is the larger deviation of values for different algorithms. In fact, the higher frequencies are more difficult to track in the presence of noise by the help of the algorithms used. However, the percentage error is well within the acceptable range.
150
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
Fig. 8. Seventh harmonic with multiple frequency estimation technique.
Fig. 9 represents the estimates for the 11th harmonic amplitudes and phase, the true values being 0.03 p.u. and − 147.1°. The amplitude estimates show a maximum error of 6.5%. The error in the phase estimate is seen, from Fig. 9 (c) and (d), to be about 4%.
Fig. 10 shows the estimates for the amplitude and phase of the 13th harmonic; the true value for amplitude is 0.033 p.u. The estimation error for most of the algorithms is about 7%, while the worst estimates have deviated by about 12%. The phase estimation error is about 5%.
Fig. 9. Eleventh harmonic with multiple frequency estimation technique.
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
151
Fig. 10 Thirteenth harmonic with multiple frequency estimation technique.
In general therefore, the algorithms have been very successful in estimating the amplitudes and phases of sinusoids in the signal. .
5. Conclusion This paper presented the estimation algorithms based on the well-known least square criterion and their usage for the estimation of different frequency components in a voltage signal. Most of the estimates are within 3–4% error. Maximum deviation is within 9 – 10%. The results obtained from the recursive estimation are fairly close to the estimates obtained by the batch algorithms. This finding enables the use of recursive algorithms to be implemented on-line and provide estimates with timevarying tracking capability. The estimates for fundamental with single frequency estimation algorithm, are closer to the true values, compared to the estimates with multiple frequency estimation algorithm. The reason lies in the calculation of SNR. With multiple frequency estimation, the signal power is the sum of the powers in all harmonics and hence the corresponding noise power is large. However, with single frequency estimation, the signal power is only due to the fundamental frequency. Hence, part of the noise power is shared by harmonics and overall stochastic power in the spectrum decreases. This produces better estimates.
The estimates thus obtained, provide the necessary information about the signal frequency contents. Based on this information, important decisions for system component designs, operation and protection can be taken. Although the results are given for Gaussian noise only, similar results were obtained for uniform noise distribution.
Acknowledgements The authors acknowledge the King Fahd University of Petroleum and Minerals for its support.
References [1] T. Lobos, New recursive method for real time determination of basic wave form of voltage and currents, IEE Proc. C 136 (6) (1989) 347 – 351. [2] K.F. Eichhorn, T. Lobos, Recursive real-time calculation of basic waveforms of signals, IEE Proc. C 138 (6) (1991) 469–470. [3] Z. Staroszczyk, A. Chwaleba, High accuracy harmonics identification and power measurements in power systems, Instrumentation and Measurement Technology Conf., Hamamatsu, Japan, 1994, pp. 1305 – 1308. [4] D.J. Nyarko, K.A. Stromsmoe, A new approach to the estimation of harmonics of digitized periodic waveforms, IEEE Wescanex’95 Proc., 1995, pp. 18 – 23.
152
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
[5] J. Xi, J.F. Chicharo, A new algorithm for improving the accuracy of periodic signal analysis, IEEE Trans. Instrum. Meas. 45 (4) (1996) 827 – 831. [6] P.K. Dash, A.M. Sharaf, A Kalman filtering approach for estimation of power system harmonics, Proc. 3rd Int. Conf. on Harmonics in Power Systems, 1988, Nashville, TN, pp. 34 – 40. [7] H. Ma, A.A. Girgis, Identification and tracking of harmonic sources in a power system using Kalman filter, IEEE Trans. Power Deliv. 11 (1996) 1659–1665. [8] T. Gouraud, F. Auger, M. Guglielmi, M. Machmoum, S. Siala, M.F. Benkhoris, A maximum likelihood approach to harmonics measurements in power systems, IEEE 4th Workshop on Computers in Power Electronics, Trois-Rivieres, Quebec, Canada, 1994, pp. 224–249. [9] D.C. Rife, R.R. Boorstyn, Single tone parameter estimation from discrete time observations, IEEE Trans. Inform. Theory 20 (5) (1974) 591 – 598. [10] S.A. Soliman, A.M. Al-Kandani, K. El-Nagar, M.E. El-
.
[11]
[12]
[13]
[14] [15] [16]
Hawary, New dynamic filter based on LAV algorithm for online tracking of power system harmonics, IEE Proc. C 142 (1) (1995) 37 – 44. M.S. Sachdev, M.M. Giray, A least error square technique for determining power system frequency, IEEE Trans. Power Appar. Syst. 104 (2) (1985) 437 – 443. V. Terzija, M. Djuric, B. Kovacevic, A new self tuning algorithm for the frequency estimation of distorted signals, IEEE Trans. Power Deliv. 10 (4) (1995) 1779 – 1785. E.A. Abu Al-Feilat, I. El-Amin, M. Bettayeb, Power system harmonic estimation a comparative study, Electr. Power Syst. Res. 29 (1994) 91 – 97. J.M. Mendel, Lessons in Digital Estimation Theory, PrenticeHall, Englewood Cliffs, NJ, 1987. T. Soderstrom and P. Stocia, System Identification, PrenticeHall, Englewood Cliffs, NJ, 1989. L. Ljung and T. Soderstrom, Theory and Practice of Recursive Identification, MIT Press, Cambridge, MA, 1983.
Recursive estimation of power system harmonics Maamar Bettayeb *, Uvais Qidwai Electrical Engineering Department, King Fahd Uni6ersity of Petroleum and Minerals, PO Box 1730, Dhahran 31261, Saudi Arabia Received 23 February 1998; accepted 30 March 1998
Abstract This paper presents the application of well known recursive estimation techniques to the important problem of power system harmonics in a noisy environment. on-line estimation of harmonic amplitudes and phases is performed using several variants of recursive least square (RLS) algorithms, known for their simplicity of computation and good convergence properties. The estimates are updated recursively as samples of the harmonic signals are received. A noisy harmonic signal from the AC bus of a six pulse rectifier is used as a test signal in the simulation. The various RLS algorithms are evaluated under different signal to noise ratios (SNR) and are shown to produce good harmonic magnitude and phase estimates even for a 0 dB SNR. Due to their simplicity, these algorithms are appropriate for on-line implementation in polluted power systems. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Harmonics; Estimation; Least square estimator; Recursive estimation
1. Introduction In recent decades, the increasing use of non-linear loads has led to elevated harmonic levels in the power system network, which produces several problems related to power system operation, protection and control. This has raised worldwide interest in harmonic studies, including harmonic modeling, estimation, mitigation and a variety of related subjects. This paper addresses the problem of harmonic estimation with specific emphasis on the use of estimation techniques for on-line application. Accurate harmonic estimation is a very important task in industrial power system. These estimates are useful both for a better characterization of electrical machinery under non sinusoidal conditions and for efficient design of compensatory filters and other equipment. Hence, there is a need to continuously monitor various signals in a power system for harmonics. The difficulty in measurement of harmonics stems from the fact that most of the harmonic producing sources are dynamic in nature and produce time varying amplitudes in voltage and current waveforms. Therefore, fast
* Corresponding author. Tel.: +966 3 8602277; fax: + 966 3 8603535; e-mail: [email protected] 0378-7796/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0378-7796(98)00063-7
and accurate estimates for amplitude and phase of these frequency components are needed. This requires estimation algorithms that have less computational burden and can therefore be used on-line. This paper presents estimation algorithms that can be used in on-line recursive estimation of amplitudes and phases of different frequency components present in a distorted power system voltage signal. The most commonly used classical estimation technique is based on the fast fourier transform (FFT) of the signal. Apart from the classical algorithm, many extensions and improvements have been published [1– 5]. These methods utilized the original FFT with different windowing and interpolation techniques to produce better estimates. The FFT based techniques are good for noise rejection but have the inherent problems of spectral leakage, aliasing and picket fence effect. Kalman filtering has been used for a number of power systems in order to estimate different states or parameters [6,7]. Kalman filtering technique utilizes simple, linear and robust algorithm to estimate the magnitude of the known harmonics embedded in the signal along with stochastic noise. It gives a better noise rejection and estimation compared to FFT algorithms. Among the classical methods of estimation of frequencies and their respective amplitudes and phases,
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
144
Table 1 Harmonic content of the test voltage signal at the load bus for the six pulse rectifier Harmonic order (Hz)
Amplitude (p.u.)
Phase (degrees)
Fundamental (60) 5th (300) 7th (420) 11th (660) 13th (780)
0.95 0.09 0.043 0.03 0.033
−2.02 82.1 7.9 −147.1 162.6
maximum likelihood estimation algorithms have been used by many researchers in the area [8,9]. One reason being a very established theoretical structure of this algorithm and the near ideal properties that this algorithm possessed. However, the estimator suffers the formulation complexity problems which makes it unsuitable for on-line implementation. The estimates also tend to be sensitive to small changes in the fundamental frequency. Another popular method is least absolute value estimation. This method is based upon the minimization of sum of the absolute values of residuals. The LAV norm [10] can often be a potential alternative to the classical least square (LS) norm in estimation problems. The algorithm can be used to optimally track the harmonics in the power system network. This algorithm can also handle time varying parameters, rejects bad measurements and at the same time works well in the moderately high noise levels. However, LAV is extremely sensitive to the deviations in the fundamental frequency and requires a prior measurement for the fundamental frequency to correct for this discrepancy. A very popular method for parameter estimation is least square (LS) algorithm. The algorithm is based upon the quadratic error objective function and is very powerful in estimating system parameters under specific conditions [11,12]. It has been used by many researchers to estimate the frequencies and their deviations in a signal. Many other algorithms are also found in the literature.
Modern power system utilizes microprocessors for the purpose of control and protection. This require real-time calculations for the basic waveforms in voltage and current waveforms based upon the measured data in the presence of high noise. The existence of a very small time window to take the necessary action, requires the use of recursive techniques to estimate the amplitudes and phases of different frequency components present in the signal. Recursive estimation is also useful because of the fact that there are always changes in the system like changes in loading, changes in topologies of various outgrowths in the system and occurrence of faults. These changes produces different type of non-linearities and hence produces different harmonics with varying amplitudes and phases. This requires on-line adaptation in the estimates so that the necessary actions can be taken at the right moment. The objective of this work is the application of several variants of recursive least square for on-line estimation of power system harmonics. Due to time variation in the power network, it is important to periodically update the harmonic estimates by using the latest available signal samples. It is shown that recursive least square algorithms implemented in this work, are capable of efficient estimation of harmonics buried in noise. This paper presents the estimation of harmonics present in a distorted voltage or current waveform. The distortion of the signal is further enhanced by considering the case of different noise levels. The methods used here are different least square (LS) based algorithms; least square estimator (LS), weighted least square estimator (WLS), recursive least square estimator information form (RLS-I), recursive least square estimator covariance form (RLS-C), extended least square estimator form 1 (ELS-1) and extended least square estimator form 2 (ELS-2) [14–16]. The first two methods use batch processing and are a basis of comparison and evaluation for the recursive least square algorithms. The rest of the paper is arranged as follows. Section 2 briefly describes the various methods of estimation used in this paper. Section 3 states the estimation problem and its formulation for single as well as multiple harmonic estimation cases. Section 4 presents the simulation results and Section 5 gives the conclusion of the work
2. Estimation algorithms
Fig. 1. Test system: a two bus architecture with six pulse full wave ridge rectifier supplying the load.
The algorithms used in this work are based upon the criterion of least square norms. These have been categorized as batch and recursive algorithms and are summarized next.
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
145
Fig. 2. Test signal (a) and its spectrum (b).
2.1. Batch least square algorithms This includes the least square estimator (LS) and the weighted least square estimator (WLS). For the linear model,
found in the literature [14–16]. The enhanced performance gained through these modifications are faster convergence, less deviations, recursive calculations and consistency of estimates. In particular, the feature for recursive calculation is attractive in connection with using the algorithms on-line. This work utilizes the following algorithms.
where, Z(k) is the noisy measurement, f(k) is the system structure matrix, u(k) is the vector for unknown parameters that are to be estimated and n(k) is the additive noise. The estimate for the required parameter vector u, can be obtained by
2.2.1. Recursi6e least square estimator-information form (RL-I) This is one variation to use the WLS recursively. This is comprised of a three step calculation in each iteration;
u. LS(k) =[f(k)f T(k)] − 1f(k)Z(k)
P − 1(k+1)= P − 1(k)+ f T(k+1)w(k+ 1)f(k+1)
The weighted least square estimator (WLS) is given by
KW (k+1)= P(k+1)f T(k+1)w(k+1)
u. (k) =[FTWF] − 1FTWZ The weighing matrix W is used to give more importance to some of the observations as compared to others. Usually, the most recent observations are given more importance and hence, are weighted more heavily. The usual choice of this matrix is
Æu 1 0 … 0 Ç Ã Ã 0 m2 … 0 Ã W(k)= Ã Ã Ã È 0 0 … m kÉ The choice of m lies between 0 and 1; however, it is usually recommended to use, m close to 1.
2.2. Recursi6e least square algorithms To improve the performance in the least square algorithm, many extensions and modifications are
u. (k+1)= u. (k)+ KW (k+1)[z(k+ 1)− f(k+ 1)u. (k)] These equations are initialized by taking some initial values for the estimate at instants k, u. (k) and P. For P, a rule of thumb is usually used which requires P= aI, where a is a large number and I is the identity matrix of order n× n, where n is the number of parameters to be estimated.
2.2.2. Recursi6e least squares estimator— co6ariance form (RLS-C) This is the second variation for recursive usage of WLS. This algorithm corresponds to the following computations: KW (k+1)= P(k)f T(k+1)
f(k+ 1)P(k)f T(k+1)+
1 w(k+ 1)
n
−1
P(k+ 1)=[1− KW (k+1)f(k+1)]P(k) u. (k+ 1)= u. (k)+KW (k+1)[z(k+ 1)− f(k+ 1)u. (k)]
146
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
Fig. 3. Test signal with noise and its spectrum.
The choice of P remains the same as for the information form. The advantage of this algorithm compared to last one is that the matrix inversion is converted to scalar division.
K(k+ 1)=P(k+ 1)f T(k+1) u. (k+1)= u. (k)+ K(k+ 1)[z(k+ 1)−f(k+ 1)u. (k)] 3. Harmonic estimation problem
2.2.3. Extended least square estimator-1 (ELS-1) This algorithm is simpler to calculate and provides a method to eliminate the bias present in the methods based upon the actual least square algorithm. This technique rearranges the observation matrix F and formulates the problem as follows: P − 1(k +1)= P − 1(k) + f T(k)f(k) u. (k +1)=u. (k)+P(k +1)f T(k)[z(k + 1) −f(k)u. (k)]
2.2.4. Extended least square estimator-2 (ELS-2) This algorithm is another modification of the extended least square that is especially suitable for fast convergence. P(k+ 1)= P(k)− P(k)f T(k)f(k)P(k) [w(k)+ f(k)P(k)f T(k)] − 1
In this section, the harmonic estimation problem is formulated for both fundamental harmonic estimation and the multiple harmonic case.
3.1. Fundamental harmonic estimation The primary objectives of power system fundamental harmonic signal estimation and retrieval are: to provide reference signal for active filtering schemes; to analyze the voltage peak value in order to warn against any possible voltage collapse; and to decide about the reactive power requirements of the system at that instant. For the case of a signal with fundamental frequency v0, the harmonics are the multiples of this frequency. Hence, the structure can be written as follows:
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
147
noise or other harmonics—as the additive noise; the new structure will become, Z(k)=A1 sin (v0k+ f1)+ m(k) By simple manipulation, this equation can be modified for the purpose of estimation: Z(k)=[sinv0kcosv0k]
n
a + mk b
or in the standard form, Z(k)= F(k)u+ m(k) where a and b are the parameters to be estimated and are given by a= A1 cos f1 b= A1 sin f1 The actual required parameters are A1 and f1, which can be given in terms of a and b as Fig. 4. Estimation strategy.
Z(k) = A1sin(v0k + f1) +A2sin(2v0k +f2) +···+ 6(k) where, Ai is the amplitude of the ith harmonic, fi is its phase and n(k) is the additive noise. For the estimation of the fundamental sinusoid, it is necessary to consider the rest of the distortion present in the signal — either
A1 = a 2 + b 2 f1 = tan − 1
b a
3.2. Multiple harmonic estimation The estimation of multiple harmonics in power system is carried out in order to:
Fig. 5. Fundamental harmonic with single frequency estimation technique.
148
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
Fig. 6. Fundamental harmonic with multiple frequency estimation technique.
tune proper filters to improve the system performance; discover the characteristics of the signal present in the system for future planning, control and protection; and use appropriate capacitors and other VAR compensators in order that the combined impedance is not close to the resonance value for some of the harmonics present. The same routines can also be used for multiple frequency estimation, i.e. to estimate the amplitudes and phases of different harmonics in the signal. The modification is in the observation matrix f and parameter matrix u. In this case, the assumption which was made earlier for considering the harmonics with the noise collectively, is now discarded. Each harmonic has its representative term appearing in the matrix f. The new equations for the model, based upon the values from Table 1 are as follows: F(k)=
Æ sin (v 0t1) cos (v 0t1) sin (5v 0t1) cos (5v 0t1) … Ã sin (v 0t2) cos (v 0t2) sin (5v 0t2) cos (5v 0t2) … Èsin (v 0tk ) cos(v 0tk ) sin (5v 0tk ) cos (5v 0tk ) … sin (13v 0t1) sin (13v 0t2) sin (13v 0tk )
cos (13v 0t1) Ç cos (13v 0t2) Ã cos (13v 0tk )É
and u(k)=[a1 b1 a5 b5 … a13 b13]T
4. Simulation results The distorted signal is constructed by using the values used by many authors in this area [6,13]. The signal is a distorted voltage waveform at the terminals of a load bus for a load driven by a six pulse rectifier, in a simple two bus configuration, as shown in Fig. 1. Table 1 shows the harmonics present in the system and Fig. 2 shows the resulting signal with its frequency spectrum. The sample signal thus formulated is further distorted with zero mean additive Gaussian noise. Different noise powers were used however, the two high power noise cases being given here. These correspond to signal to noise ratios (SNR) of 10 and 0 dB. The resulting signal is shown in Fig. 3. Four different forgetting factors were used, l=1, l= 0.98, l= 0.95 and l=0.9. These were used for faster convergence and to study their effects on algorithm robustness. It was found that lB 0.95 gave poor results. For each algorithm of estimation, 20 realizations were established for each forgetting factor and SNR and the averaged estimate was plotted as a bar graph. The simulation strategy is shown in Fig. 4. The selection of sampling time and the harmonic frequencies that are to be estimated, is the initial step
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
149
Fig. 7. Fifth harmonic with multiple frequency estimation technique.
for the algorithm. With this selection, the time vector and the structure matrix F can be formulated beforehand. The next step is to select the method for estimation from the four recursive methods RLS-L RLS-C, ELS-1 and ELS-2. The samples are then obtained recursively and estimates are calculated. When the time vector is exhausted, the algorithm can either restart or terminate. The simulation results presented below are based on estimation for one whole period, with a sampling frequency of 1620 Hz, which is slightly higher than the Nyquist frequency for the test signal.
4.1. Fundamental harmonic case Fig. 5 shows the results for estimation of fundamental frequency. The true values that are being estimated are the amplitude and phase of the fundamental frequency component. The actual value for the amplitude is 0.95 p.u. and that for the phase is − 2.02°. The results show almost exact estimation. In case of 10 dB SNR, the maximum deviation is about 0.4%, while in the case of 0 dB SNR it is 0.95%. However, the estimates of phase show deviations of 4 and 8% for 10 and 0 dB SNR cases, respectively.
4.2. Multiple harmonic case Fig. 6 shows the results of estimates obtained for fundamental harmonic by using the multiple harmonic estimation algorithm. As can be seen from the results in
these figures, the estimation is again fairly close to the exact value for almost all of the algorithms used. Any discrepancy could be a result of numerical approximation and finite word length effects, especially for the calculation of phase. The maximum deviation in amplitude estimate is 0.002 p.u., corresponding to an error of 2%. Similarly for the case of phase, except for one estimate, the results are within 5% error. Although the presented results are for the Gaussian noise case, similar results were obtained for uniform noise distribution. Fig. 7 shows the estimation for the amplitude of fifth harmonic. The actual value is 0.09 p.u. and the phase is 82.1°. In the case of amplitude estimation, the maximum error for 10 dB SNRs is about 1.5%, while for 0 dB the error increases to 6.7%. The reason is the adding of noise components at the same frequency with the fifth harmonic component to produce a higher amplitude harmonic. However, the estimation for phase is much closer to the actual, producing an error of about 1%. Fig. 8 shows the estimates for the seventh harmonic. The true value is 0.043 p.u and 7.9°. The maximum error in the estimation of amplitude is 8% and this is the case when SNR is 0 dB. Similarly, the error in phase is about 7.6%. However, what should be noted here is the larger deviation of values for different algorithms. In fact, the higher frequencies are more difficult to track in the presence of noise by the help of the algorithms used. However, the percentage error is well within the acceptable range.
150
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
Fig. 8. Seventh harmonic with multiple frequency estimation technique.
Fig. 9 represents the estimates for the 11th harmonic amplitudes and phase, the true values being 0.03 p.u. and − 147.1°. The amplitude estimates show a maximum error of 6.5%. The error in the phase estimate is seen, from Fig. 9 (c) and (d), to be about 4%.
Fig. 10 shows the estimates for the amplitude and phase of the 13th harmonic; the true value for amplitude is 0.033 p.u. The estimation error for most of the algorithms is about 7%, while the worst estimates have deviated by about 12%. The phase estimation error is about 5%.
Fig. 9. Eleventh harmonic with multiple frequency estimation technique.
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
151
Fig. 10 Thirteenth harmonic with multiple frequency estimation technique.
In general therefore, the algorithms have been very successful in estimating the amplitudes and phases of sinusoids in the signal. .
5. Conclusion This paper presented the estimation algorithms based on the well-known least square criterion and their usage for the estimation of different frequency components in a voltage signal. Most of the estimates are within 3–4% error. Maximum deviation is within 9 – 10%. The results obtained from the recursive estimation are fairly close to the estimates obtained by the batch algorithms. This finding enables the use of recursive algorithms to be implemented on-line and provide estimates with timevarying tracking capability. The estimates for fundamental with single frequency estimation algorithm, are closer to the true values, compared to the estimates with multiple frequency estimation algorithm. The reason lies in the calculation of SNR. With multiple frequency estimation, the signal power is the sum of the powers in all harmonics and hence the corresponding noise power is large. However, with single frequency estimation, the signal power is only due to the fundamental frequency. Hence, part of the noise power is shared by harmonics and overall stochastic power in the spectrum decreases. This produces better estimates.
The estimates thus obtained, provide the necessary information about the signal frequency contents. Based on this information, important decisions for system component designs, operation and protection can be taken. Although the results are given for Gaussian noise only, similar results were obtained for uniform noise distribution.
Acknowledgements The authors acknowledge the King Fahd University of Petroleum and Minerals for its support.
References [1] T. Lobos, New recursive method for real time determination of basic wave form of voltage and currents, IEE Proc. C 136 (6) (1989) 347 – 351. [2] K.F. Eichhorn, T. Lobos, Recursive real-time calculation of basic waveforms of signals, IEE Proc. C 138 (6) (1991) 469–470. [3] Z. Staroszczyk, A. Chwaleba, High accuracy harmonics identification and power measurements in power systems, Instrumentation and Measurement Technology Conf., Hamamatsu, Japan, 1994, pp. 1305 – 1308. [4] D.J. Nyarko, K.A. Stromsmoe, A new approach to the estimation of harmonics of digitized periodic waveforms, IEEE Wescanex’95 Proc., 1995, pp. 18 – 23.
152
M. Bettayeb, U. Qidwai / Electric Power Systems Research 47 (1998) 143–152
[5] J. Xi, J.F. Chicharo, A new algorithm for improving the accuracy of periodic signal analysis, IEEE Trans. Instrum. Meas. 45 (4) (1996) 827 – 831. [6] P.K. Dash, A.M. Sharaf, A Kalman filtering approach for estimation of power system harmonics, Proc. 3rd Int. Conf. on Harmonics in Power Systems, 1988, Nashville, TN, pp. 34 – 40. [7] H. Ma, A.A. Girgis, Identification and tracking of harmonic sources in a power system using Kalman filter, IEEE Trans. Power Deliv. 11 (1996) 1659–1665. [8] T. Gouraud, F. Auger, M. Guglielmi, M. Machmoum, S. Siala, M.F. Benkhoris, A maximum likelihood approach to harmonics measurements in power systems, IEEE 4th Workshop on Computers in Power Electronics, Trois-Rivieres, Quebec, Canada, 1994, pp. 224–249. [9] D.C. Rife, R.R. Boorstyn, Single tone parameter estimation from discrete time observations, IEEE Trans. Inform. Theory 20 (5) (1974) 591 – 598. [10] S.A. Soliman, A.M. Al-Kandani, K. El-Nagar, M.E. El-
.
[11]
[12]
[13]
[14] [15] [16]
Hawary, New dynamic filter based on LAV algorithm for online tracking of power system harmonics, IEE Proc. C 142 (1) (1995) 37 – 44. M.S. Sachdev, M.M. Giray, A least error square technique for determining power system frequency, IEEE Trans. Power Appar. Syst. 104 (2) (1985) 437 – 443. V. Terzija, M. Djuric, B. Kovacevic, A new self tuning algorithm for the frequency estimation of distorted signals, IEEE Trans. Power Deliv. 10 (4) (1995) 1779 – 1785. E.A. Abu Al-Feilat, I. El-Amin, M. Bettayeb, Power system harmonic estimation a comparative study, Electr. Power Syst. Res. 29 (1994) 91 – 97. J.M. Mendel, Lessons in Digital Estimation Theory, PrenticeHall, Englewood Cliffs, NJ, 1987. T. Soderstrom and P. Stocia, System Identification, PrenticeHall, Englewood Cliffs, NJ, 1989. L. Ljung and T. Soderstrom, Theory and Practice of Recursive Identification, MIT Press, Cambridge, MA, 1983.