RECURSIVE TOTAL LEAST-SQUARES ESTIMATION OF FREQUENCY IN THREE-PHASE POWER SYSTEMS

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1 RECURSIVE TOTAL LEAST-SQUARES ESTIMATION OF FREQUENCY IN THREE-PHASE POWER SYSTEMS Reza Arablouei, Kutluyıl Doğançay 2, Stefan Werner 3 2 Institute for Telecommunications Research, University of South Australia, Mawson Lakes SA, Australia 3 Dept. of Signal Processing Acoustics, School of Electrical Engineering, Aalto University, Finl ABSTRACT We propose an adaptive algorithm for estimating the frequency of a three-phase power system from its noisy voltage readings. We consider a second-order autoregressive linear predictive model for the noiseless complex-valued αβ signal of the system to relate the system frequency to the phase voltages. We use this model the noisy voltage data to calculate a total least-square (TLS) estimate of the system frequency by employing the inverse power method in a recursive manner. Simulation results show that the proposed algorithm, called recursive TLS (), outperforms the recursive least-squares () the biascompensated (BC) algorithms in estimating the frequency of both balanced unbalanced three-phase power systems. Unlike BC, does not require the prior knowledge of the noise variance. Index Terms Adaptive signal processing, frequency estimation, inverse power method, linear predictive modeling, total least-squares.. INTRODUCTION In electric power grids, deviation of the system frequency from its nominal value represents an imbalance between load generation. Therefore, variations of the system frequency should be closely watched. Most protection-control applications in electric power systems require accurate fast estimation of the system frequency. An erroneous estimate of the frequency may eventually result in a catastrophic grid failure due to inadequate or delayed load shedding []-[5]. The Clarke s transformation applied to the voltages of a three-phase power system produces a complex-valued signal (known as the signal) that incorporates the information of the three phases. In many applications, the signal can be considered as a faithful representative for a three-phase system [6]. The phase voltages are digitized at the measurement points by sampling at a fixed interval This work was supported in part by the Academy of Finl the Centre of Excellence in Smart Radios Wireless Research (SMARAD) at the Aalto University. quantizing the samples. Therefore, in practice, the observed voltage data consequently the signal are contaminated with noise/error. From a signal-processing point of view, the noisy samples of the signal the sampling rate comprise the available data while the amplitudes of the three phase voltages, the initial phase angle, the system frequency are the unknown parameters. A plethora of techniques have been developed to extract these parameters from the observable data. When the main objective is to estimate the system frequency, a second-order autoregressive (AR2) linear predictive model for time evolution of the noiseless signal can be employed [7]-[]. This model, which we will simply call AR2, linearly relates three consecutive noiseless samples of the signal via a single real-valued parameter that is equal to the cosine of the multiplication of the system angular frequency the sampling interval. Thus, the system frequency can be estimated by identifying the parameter of the AR2 model from the noisy observations of the signal while being safely oblivious to the values of the phase voltage amplitudes the initial phase angle. In other words, since the parameter of the AR2 model depends only on the system frequency the sampling interval, any frequency estimator built on this model is virtually insensitive to the balance state of the three-phase system. Relative values of the voltage amplitudes of three phases determine the balance state. Since the signal is observed with noise, a reliable frequency estimation technique based on the AR2 model should minimize the effect of noise. In [9], [], the leastsquares (LS) method has been used for this purpose. A recursive LS () frequency estimator has also been proposed in []. The LS-based approaches are best suited to counter the effect of noise at the output of a linear model. However, as in the AR2 model, the input of the model is also subject to observational noise, the LS-based frequency estimators are biased. One way to eliminate the estimation bias is to evaluate the bias separately subtract it from the biased estimate [2]. However, evaluation of the bias usually requires prior knowledge of the noise variance or an extra procedure for estimating the noise variance. Alternatively, the total least-squares (TLS) estimation technique can be utilized to compensate for the noise at both

2 input output of the AR2 model. A TLS estimator can eliminate the estimation bias induced by the input noise without performing any explicit bias calculation [3]-[4]. The TLS technique has recently been utilized to estimate the frequency of three-phase power systems based on a firstorder autoregressive widely-linear predictive model for the noiseless signal [5], [6]. In this paper, we derive a recursive TLS () algorithm for adaptive frequency estimation of three-phase power systems. We use the AR2 model for the signal compute a TLS estimate of the model parameter by implementing a single iteration of the inverse power method [7] at each time instant. We verify the effectiveness of the proposed algorithm in estimating the frequency of three-phase power systems through simulated experiments. 2. ALGORITHM DERIVATION The phase voltages of a three-phase power system, sampled at the interval of, are expressed as,, is the system frequency,,, are the peak values, is a constant initial phase angle, is the time index. In practice, the phase voltages are observed with noise. The noisy observations are expressed as,,,, denote the additive noise at the respective phases. We assume that the noises are i.i.d. with zero mean fixed variance. Application of the Clarke s transformation to the observed noisy voltages of the three phases is formulated as [ ] [ ] [ ] ([ ] [ ]). This transformation yields a complex-valued voltage signal (known as the signal) that can represent the three-phase power system [6]. The noiseless signal is calculated as The noisy, signal is given by,. the additive complex noise is related to the noise at the individual phases via We denote the variance of the absolute value of by, i.e., [ ]. It is known that the time evolution of the noiseless signal,, can be described via a second-order autoregressive (AR2) linear predictive model as ( ) () is the model parameter expressed as. This model relates the system frequency to three consecutive noiseless voltage samples of the three phases in a linear manner. However, the measured samples are noisy. Hence, in order to identify the model parameter subsequently the system frequency, a reliable linear estimation technique need be employed that can minimize the effect of noise. Since both input output in the AR2 model, (), are observed with noise, we utilize the TLS estimation technique to identify. For this purpose, we define the input output data vectors as respectively, [ ] [ ] { } is an exponential weighting matrix is the forgetting factor. We compute the TLS estimate of at time instant, denoted by, such that it fits the input data vector,, to the output data vector,, by incurring minimum perturbation in the data, i.e., it holds that

3 are the minimum input output perturbations, respectively. According to the analysis of [4], the TLS estimate is given by or equivalently [ ] [ ] [ ] is the eigenvector corresponding to the smallest eigenvalue of the augmented weighted data covariance matrix [ ] [ ]. The weight accounts for the disparity in the variance of the noise at input output, denoted by, respectively, is calculated as Substituting (4) into (3) solving it for recursive TLS () estimate of as (3) (4) yield a (5) Having calculated via, we can estimate the system frequency The matrix can be written as [ ] 3. SIMULATIONS We compare the frequency estimation performance of the algorithm, i.e., the bias-compensated (BC) algorithm, i.e., The vector is normally found using the eigenvalue decomposition of. A computationally more efficient alternative is to update adaptively by executing a single iteration of the inverse power method [7] at each time instant through the following recursion: Multiplying both sides of (2) by give. (2) dividing by the algorithm, (5), for a three-phase power system Hz, ms, We also set. In Fig., we depict the estimated frequency using different algorithms when the system undergoes several voltage sags making it progressively more unbalanced as shown in Fig. 2. The system is balanced during the initial seconds. Then, the voltage of phase c drops by. After seconds, the voltage of phase a, after further seconds, the voltage of phase b reduce to zero. Zero or almost zero phase voltages can occur in the aftermath of short-circuit to ground or combined asymmetric sags. It is observed from Fig. that all the considered algorithms converge almost equally fast. In Fig. 3, we plot the estimated frequency using different algorithms when a sinusoidal oscillation occurs in the frequency of a balanced system after seconds. The peak the maximum change rate for the oscillation are Hz Hz/s, respectively. In Figs. 4-7, we plot the steady-state bias root-meansquare error, defined as

4 phase c phase b phase a frequency (Hz) frequency (Hz) BC true Fig.. Estimated frequency of the system of Fig BC true Fig. 3. Estimated tracked frequency Fig. 2. Voltages of the three phases in the considered system. [ ] [ ] respectively, against the signal-to-noise ratio (SNR), which is considered to be, for different algorithms. In Figs. 4 5, the system is balanced while, in Figs. 6 7, short circuit between phase a ground has made the system unbalanced. We run the algorithms for seconds to reach the steady state. We evaluate the expectations by taking the ensemble average over independent trials the steady-state values by averaging over the last seconds. 4. CONCLUSION We proposed an adaptive algorithm for estimating the frequency of three-phase power systems. To this end, we utilized an intrinsic second-order autoregressive linear predictive model for the noiseless signal of the system whose parameter is equal to the cosine of the system frequency. Then, we found a TLS estimate for the model s parameter using the noisy observations of the signal employing the inverse power method in a recursive manner. The resultant recursive TLS () algorithm compensates for noise at both input output of the model. We showed through simulation examples that the algorithm achieves considerably lower frequency estimation bias root mean-square error compared with the algorithm for both balanced unbalanced three-phase power systems; is more efficacious than the bias-compensated algorithm in eliminating the estimation bias while bypassing the need for prior knowledge of the noise variance; has a good capability to track the changes in the system frequency. REFERENCES [] A. Ipakchi F. Albuyeh, Grid of the future, IEEE Power Energy Mag., vol. 7, pp , Mar./Apr. 29. [2] B. Subudhi, P. K. Ray, S. R. Mohanty, A. M. Pa, A comparative study on different power system frequency estimation techniques, Int. J. Autom. Control, vol. 3, no. 2/3, pp , May 29. [3] M. H. J. Bollen, Voltage sags in three-phase systems, IEEE Power Eng. Rev., vol. 2, pp. 8 5, Sep. 2. [4] R. C. Dugan, M. F. McGranaghan, S. Santoso, H. W. Beaty, Electrical Power Systems Quality, 2 nd ed., New York: McGraw-Hill, 23. [5] M. H. J. Bollen, I. Y. H. Gu, S. Santoso, M. F. McGranaghan, P. A. Crossley, M. V. Ribeiro, P. F. Ribeiro, Bridging the gap between signal power, IEEE Signal Process. Mag., vol. 26, pp. 3, Jul. 29. [6] E. Clarke, Circuit Analysis of A-C Power Systems, New York: Wiley, 943. [7] M. K. Mahmood, J. E. Allos, M. A. H. Abdul-Karim, Microprocessor implementation of a fast simultaneous amplitude frequency detector for sinusoidal signals, IEEE Trans. Instrum. Meas., vol. IM-34, p.43-47, Sep. 985.

5 steady-state root-mean-square error (db) steady-state root-mean-square error (db) steady-state bias (db) steady-state bias (db) BC Fig. 4. Steady-state frequency estimation bias versus SNR when the system is balanced BC Fig. 6. Steady-state frequency estimation bias versus SNR when the system is unbalanced due to a phase-to-ground fault. 2 BC 2 BC Fig. 5. Steady-state frequency estimation root-mean-square error versus SNR when the system is balanced Fig. 7. Steady-state frequency estimation root-mean-square error versus SNR when the system is unbalanced due to a phase-to-ground fault. [8] P. K. Dash, D. P. Swain, A. Routary, A. C. Liew, An adaptive neural network approach for the estimation of power system frequency, Electric Power Syst. Research, vol. 4, pp. 23-2, 997. [9] A. López, J.-C. Montaño, M. Castilla, J. Gutiérrez, M. D. Borrás, J. C. Bravo, Power system frequency measurement under nonstationary situations, IEEE Trans. Power Del., vol. 23, pp , Apr. 28. [] A. Abdollahi F. Matinfar, Frequency estimation: a leastsquares new approach, IEEE Trans. Power Del., vol. 26, pp , Apr. 2. [] M. D. Kušljević, J. J. Tomić, L. D. Jovanović, Frequency estimation of three-phase power system using weighted-least-square algorithm adaptive FIR filtering, IEEE Trans. Instrum. Meas., vol. 59, pp , Feb. 2. [2] P. Stoica T. Söderström, Bias correction in least-squares identification, Int. J. Control, vol. 35, pp , 982. [3] I. Markovsky S. Van Huffel, Overview of total leastsquares methods, Signal Process., vol. 87, pp , 27. [4] G. H. Golub C. F. Van Loan, An analysis of the total least squares problem, SIAM J. Numer. Anal., vol. 7, no. 6, pp , Dec. 98. [5] R. Arablouei, S. Werner, K. Doğançay, Adaptive frequency estimation of three-phase power systems with noisy measurements, in Proc. Int. Conf. Acoust., Speech Signal Process., Vancouver, Canada, May 23, pp [6] R. Arablouei, S. Werner, K. Doğançay, Estimating frequency of three-phase power systems via widely-linear modeling total least-squares, in Proc. IEEE Int. Workshop Computational Advances Multi-Sensor Adaptive Process., Saint Martin, Dec. 23, pp [7] J. W. Demmel, Applied Numerical Linear Algebra, Philadelphia, PA: SIAM, 997.