New Adaptive Linear Combination Structure for Tracking/Estimating Phasor and Frequency of Power System

Transcription

1 28 Journal of Electrical Engineering & echnology Vol. 5, No., pp. 28~35, 2 New Adaptive Linear Cobination Structure for racking/estiating Phasor and Frequency of Power Syste Choowong-Wattanasakpubal and eratu-bunyagul* Abstract his paper presents new Adaptive Linear Cobination Structure (ADALINE) for tracking/estiating voltage-current phasor and frequency of power syste. o estiate the phasors and frequency fro sapled data, the algorith assues that orthogonal coefficients and speed of angular frequency of power syste are unknown paraeters. With adequate sapled data, the estiation proble can be considered as a linear weighted least squares (LMS) proble. In addition to deterining the phasors (orthogonal coefficients), the procedure estiates the power syste frequency. he ain algorith is verified through a coputer siulation and data fro field. he proposed algorith is tested with transient and dynaic behaviors during power swing, a step change of frequency upon islanding of sall generators and disconnection of load. he algorith shows a very high accuracy, robustness, fast response tie and adaptive perforance over a wide range of frequency, fro to 2 Hz. Keywords: Phasor, Frequency, Adaptive, ADALINE, Generator protection. Introduction Voltage phasors and frequency of power syste have always been of interest to power syste engineers. he odern era of phasor and frequency easureent technology is driven by coputer relaying of transission lines and generator protection. In the past, phasors are applied in power syste application with a steady state concept. However in reality, a power syste never reains in a steady state. Frequency of voltage and current change consistently as a result of load and generation variations including fault events. Most power systes operate in a relatively narrow band of frequency, i.e. within.5 Hz fro a noinal value. Nonetheless, under soe circustances, islanding of a sall generators and a disconnection of load can cause frequency variations as large as ± Hz, for exaple islanding condition with hydroelectric generator []-[2]. Such an extree condition is usually resolved by available control actions. Phasor representation is only possible for a pure sinusoid. In practice, a wavefor is often equipped with other signals of different frequencies. As a result, it becoes necessary to extract a fundaental frequency coponent of the signal before representing it by a phasor. he phasor definition also iplies that the signal is constant over tie. Nonetheless, this assuption is only valid for a portion of tie. his tie window is known as data window and is a very iportant paraeter in phasor and frequency estiation. Corresponding Author: Research Division of Provincial Electricity Authority (PEA), hailand. (d_power_syste@hotail.co) * Departent of Electrical Engineering, King Mongkut s University of echnology North Bangkok, hailand. (terata@kutnb.ac.th) Received : Septeber 7, 29; Accepted : Noveber 5, 29 A nuber of nuerical algoriths for easuring phasor and power syste frequency have been published in any literatures [3]-[]. In general, the high speed easureent ade within one or two cycles by applying short data window tends to have greater errors than those techniques using a long data window. A well-known application of frequency easureent is for under-frequency load shedding. Noral operating tie of an under-frequency relay is approxiately 5-6 cycles. Nonetheless, excessively long data window is not a good way for iproving the accuracy of the easureent. During transient, the frequency of power syste ay change rapidly. Consequently with a long data window length N, the process ay cobine significantly different frequency signals and result in significant errors. Considering the process of selecting data window length N with sapling interval Δ, a variety of nuerical algoriths for easuring phasor and power syste frequency in the literature are exained by using two ethods. Method : his ethod estiates the frequency value and at the sae tie adjusts the sapling interval Δ such that the nuber of saples reains an integer value. Accordingly, N reains constant and Δ is varied [6]. his ethod is difficult to be ipleented. Method 2: his ethod estiates the frequency value and adjusts the data window length N. Hence, Δ reains constant and N are varied [7]-[]. However, Method 2 cannot obtain an integer nuber of N and will result in errors. With, Δ =.625 s (32 saples per cycle at 5 Hz), the data window N for corresponding frequencies is shown below. While adjusting data window length N is able to cover a large frequency range fro 2.78 Hz to 57.4 Hz, the data window length N can be very large for real-tie coputations (77. saples at 2.78 Hz).

2 Choowong-Wattanasakpubal and eratu-bunyagul 29 Data window length N Frequency (Hz) his paper presents new adaptive linear cobination structure for tracking/estiating voltage-current phasor and frequency in power syste. It is suitable for real-tie application of phasor and frequency estiation providing high accuracy and can overcoe the above entioned proble of Method and Method 2. In estiating phasors and frequency of power syste fro sapled data, the algorith assues that orthogonal coefficients and speed of angular frequency of power syste are unknown paraeters. With adequate sapled data, one can consider the estiation proble as a linear weighted least squares (LMS) proble. Hence, coputation and accuracy of the new algorith are not related to the data window length N as in the case of Discrete Fourier ransfor (DF) or other techniques that use the nuber of saples N for estiating the phasor and frequency of power syste. Furtherore, the technique can easure phasors with a wide frequency range fro Hz to 2 Hz with ease of ipleentation. he paper is divided into six sections. Next section presents new adaptive linear cobination structure for tracking/estiating voltage-current phasor and frequency in power syste and shown theoretical derivation of the necessary equations. he following section presents siulations results for transient response during power swing, step change of frequency for islanding of sall generators and disconnecting of load using Matlab progra. After that, ipleentation and results fro field data will be deonstrated in the next section and is followed by conclusion section. Finally, future work is presented in the final section. and frequency tracking. herefore, the structure of ADALINE has to be odified in order to be able to tracking phasor and frequency of power syste. Fig. illustrates the structure of ADALINE for tracking/estiating dynaic voltagecurrent phasor and frequency in power syste. he structure is fored by the linear cobination of tie varying connection vector Xt (). he tie varying connection vector is ultiplied by the weighting vector Wt, () and then sued up to produce the linear output yt (). he weighting vector can be adjusted using Least Mean Square algorith (LMS). his will provide the output yt () that is close to the input signal x() t (voltage or current signal). Unlike other ethods, the proposed new algorith assues that the phasor and frequency are dynaic. he ain procedure is divided into two stages. First, weighting vector is estiated. hen, the angular speed of fundaental frequency of the input vector will be adjusted by to obtain the actual value. Stages : Let assue that the observation odel of an input signal (voltage or current) at the easureent location can be expressed as (). y () t = Acos( θ +Δ ωt+ ω t) () Where A is the agnitude of signal. θ is the phase angle. ω is the angular speed of fundaental frequency. S Δ ω is a sall change in angular speed of fundaental frequency. t is the tie of observation. Applying the trigonoetry relationship of cos( a+ b) = cos( a) cos( b) -sin( a) sin( b) to the above equation will result in y ( t) = Acos( θ +Δωt) cos( ωst) - Asin( θ +Δωt) sin( ω t) w w 2 Δω Δω cos( ω S t) Δω -sin( ω S t) Σ s s yt () et () x() t Σ (2) 2. he Methodology he ADALINE structure was introduced as a powerful estiation tool [2] as present in the Appendix. However, this structure is not yet suitable for the proble of phasor Fig.. ADALINE structure for tracking phasor and frequency.

3 3 New Adaptive Linear Cobination Structure for racking/estiating Phasor and Frequency of Power Syste Rearranging the above equations in atrix fro will result in (3). yt () = X() t Wt () (3) Where yt () represents the output signal fro the observation odel. X () t as (4) represents the tie varying connection vector and Wt () as (5) represents the paraeters to be estiated. X ( t) = [cos( ωst) -sin( ωst)] (4) Wt () = w w (5) [ ] 2 Where w and w 2 are orthogonal coefficients of voltage or current signal which can be represented by (6)-(7). w Acos( = +Δ (6) w Asin( 2 = +Δ (7) Accordingly, w represents real part of voltage or current phasor and w represents iaginary part of voltage 2 or current phasor. For discrete-tie signals, the weighting vector can be updated by using Widrow-Hoff delta rule as following. α ek ( ) X( k) W( k+ ) = W( k) + X ( k) X ( k) Where k is the index of iteration and α is the learning paraeter. Substituting X( k) X ( k) = into (8) will result in (8) W( k + ) = W( k) + α e( k) X( k) (9) he perfect learning in (9) is copleted when the error 2 ( e ) is brought to zero. he weighting vector W( k ) of voltage and current signal at the easureent location will be presented in phasor or coplex for. As a result, the syste can track voltage and current phasor. Stages 2: A study of previous work has revealed that ost of voltage and current phasor estiation are not suitable for extracting the fundaental frequency coponents. his is because the observation odel neglects a sall change in angular speed of fundaental frequency Δ ω. In this paper, the effect of Δ ω is considered and the frequency is adjusted to its actual value. A sall change in angular speed of the fundaental frequency is coputed accurately fro the weighting vector according to (). w ( k) w ( k)- w ( k) w ( k) Δ ω = ( w( k) + w2( k)) () Fro (), w ( k) and w 2 ( k) are the derivative of weighting vector with respect to tie. A differentiation ethod can be used to copute w ( k) and w 2 ( k) according to () and (2), where Δ t is the sapling tie interval. w( k)- w( k-) w ( k) = Δt w2( k)- w2( k-) w 2( k) = Δt () (2) For discrete-tie signal, the angular speed of the fundaental frequency according to the observation odel in () can be updated to the actual value by using (3). ω ( k+ ) = ω ( k) + α Δ ω (3) S 3. Siulation Results Coprehensive evaluation of the proposed algorith was carried out by conducting several test cases. he first case focuses on a range of the easureent frequency. he second test deals ainly with exaining the transient response of the algorith in tracking/estiating phasor and frequency for all changes in aplitude, phase angle and frequency. he third test case considers a rate of change of frequency. he fourth and fifth test cases deonstrate a dynaic perforance of the processing unit under power swing condition. he last test case is for testing noise property of the processing unit. All test scenarios applies a sapling rate of f s = 4 saples per second and a learning paraeter α =.2. In this research, it is found that decreasing the learning factor will reduce the prediction error, but at the expense of increasing the convergence tie. In contrast, increasing the learning factor will increase the prediction error draatically while reducing the convergence tie. herefore, there is a trade-off in selecting the value of α for fast convergence and fast response with a iniu prediction error. Much research has been conducted to deterine a proper value of α. Nonetheless, ost coon approach in deterining the value relies on trial and error ethod [2]-[3]. It was found that the value of the learning factor that gives the best phasor and frequency tracking perforance is.2. CASE: he ability in estiating frequency value over a wide range of frequency changes is investigated using a sinusoidal test signal. he algorith is tested by raping up the syste frequency (reference frequency of input signal) fro Hz to 2 Hz with rate of change 5 Hz/s and a signal to noise ratio (SNR) =6 db. he result in Fig. 2 shows that the values of estiated frequency can follow the change of reference frequency. In addition, it shows that the algorith can satisfactorily perfor frequency easureent over a wide range of frequency between Hz to 2 Hz. S

4 Choowong-Wattanasakpubal and eratu-bunyagul 3 Estiated Frequency(Hz) 3 2 Estiate Frequency Reference Frequency Next, a sudden change in frequency, agnitude and phase angle is applied at t = s fro 5 Hz to 45 Hz, p.u. to.5 p.u. and -35 degree to -55 degree respectively. According to Fig. 5, the estiated frequency, agnitude and phase angle can track the actual value of the input signal with total delay tie less than 6 s. In addition, at steadystate, the values of frequency agnitude and phase angle are close to the actual value of the input signal Fig. 2. Frequency easureent between Hz to 2 Hz. CASE2: he transient responses of the algorith are investigated. It is assued that the power syste is initially operating close to noinal frequency (5 Hz), and a sudden change in frequency occurs at t = s. he frequency of input signal is changed fro 5 Hz to 49 Hz with a signal to noise ratio (SNR) = 6 db and 5% third haronic. he tie response of the algorith for a step change in frequency is shown in Fig.3. he third haronic of 5% of fundaental frequency agnitude and zero-ean Gaussian white noise with a signal to noise ratio of 6 db are also included. Obviously, frequency estiation using the proposed algorith with online adaptation results in less estiation errors within the range of.75 Hz at a steady state as show in Fig. 4. Frequency Estiation (Hz) Fig. 3. Frequency estiation for a sudden change in frequency..2 Freq(Hz) Mag(p.u.) Angle (deg) s s s Fig. 5. Frequency, agnitude and phase angle estiation for a sudden change in frequency, agnitude and phase angle. CASE3: Perforance for rate of change of frequency is tested using siulated voltage wavefors with different rates of frequency raping (Hz/s). In Fig.6, the reference frequency starts fro 5 Hz with + Hz/s rate of change. he frequency estiation starts fro 5 Hz and tracks the frequency variation. Fro the figure, the axiu error in the frequency tracking is less than.25 Hz. Error(Hz) Estiated Frequency(Hz) Estiate Frequency Reference Frequency ie(s) Error of Frequency Estiation (Hz) ie(s) Fig. 4. Absolute error Fig. 6. Frequency estiation with rate of change + Hz/s. Subsequently, the rate of change of frequency is increased to +2 Hz/s. he frequency starts with a 5 Hz value and reaches 54 Hz after.2 s. he frequency estiation starts fro 5 Hz and tracks frequency variation. According to Fig.7, the axiu error in the frequency tracking is less than.25 Hz.

5 32 New Adaptive Linear Cobination Structure for racking/estiating Phasor and Frequency of Power Syste Error(Hz) Estiated Frequency(Hz) Estiate Frequency Reference Frequency ie(s) Fig. 7. Frequency estiation with rate of change + 2 Hz/s. CASE4: he easureent perforances under power swing condition are presented in the fourth to the sixth test case. It is assued that the power syste is initially operating close to the noinal frequency (5 Hz) and a transient stability has been initiated. he agnitude of a corresponding voltage vt () signal is odulated with a frequency f as follow. [ π ] vt ( ) = 2 +.2sin(2 f t) sin( ωt) (4) he frequency f varies between - Hz. Fig.8 shows the result obtained fro the phasor agnitude estiation according to the input in (4) with ω = 2π 5Hz and f = 5 Hz. Under these conditions, it is clear that the calculated phasor agnitude can follow the odulated aplitude perfectly. CASE5: During power swings condition, both agnitudes and frequency of voltage and current in a power syste ay be odulated. he equivalent test for dynaic perforance is provided by using aplitude and frequency odulation function as follow. [ π ] vt () = 2 +.2sin(2 ft) sin( ωt) (5) ω = ω + Asin(2 π ft) (6) he frequency of input signal varies sinusoidally between ( ω + A ) and ( ω - A ) with a frequency ω and the aplitude of input signal varies sinusoidally between (+.2 ) and (-.2 ) with a frequency f. In Case 5, the frequency input is varied between 5 Hz and 49 Hz ( A =, ω = 2π 5), with f = Hz and f = 5 Hz. he aplitude and frequency estiation fro the proposed algorith is shown in Fig. and Fig. respectively. Fro the figures, the proposed algorith shows a high quality dynaic response. he estiated frequency fluctuates around the reference frequency according to Fig...5 Estiated Phasor Magnitude Phasor Magnitude v(t) Fig. 8. Magnitude estiation for aplitude odulation at 5 Hz. he result for Estiated Phasor Magnitude f = Hz is shown in Fig Phasor Magnitude V(t) Fig. 9. Magnitude estiation for aplitude odulation at Hz. Estiated Phasor Magnitude Phasor Magnitude v(t) Fig.. Magnitude estiation for aplitude and frequency odulation. Estiated Frequency(Hz) Estiate Frequency Reference Frequency Fig.. Frequency estiation for aplitude and frequency odulation.

6 Choowong-Wattanasakpubal and eratu-bunyagul 33 Error of Frequency Estiation(Hz) It can be seen fro Fig. 2 that the axiu error of the frequency tracking is less than.28 Hz Fig. 4. Instantaneous phasor and frequency eter Fig. 2. Absolute error. CASE6: he algorith was tested under the presence of zero-ean Gaussian noise in the input signal (additive noise). he rando noise standard deviation was selected to obtain a prescribed Signal-to-Noise-Ratio (SNR) defined as: SNR = 2log A 2 σ (6) Where A is the agnitude of the signal. σ is the noise standard deviation. As shown in Fig. 3, a better sensitivity is obtained when additive noise is added. Accordingly, the axiu error increases considerably. Nevertheless, for SNR>4dB, the axiu error is less than. Hz. 2.5 Maxiu Error (Hz) 2.5 converter (A/D) with a sapling rate at 4 khz. his A/D digit is 2 bits providing a non-anti-aliasing RC filter. he current signal is delivered to the data acquisition syste fro a 6/5 A current transforer. he LabView software package is applied to the syste ipleentation [4]. he LabView environent is very friendly for the progra developent using graphical prograing language. It uses terinology, icons and ideas being failiar to scientists and engineers, and relies on graphical sybols rather than textual for language prograing. In addition, it provides extensive galleries of functions and subroutines for ost prograing tasks, including data acquisition. he ability of aplitude and frequency estiation during a large power disturbance is investigated using a real data fro a 5 kv transission syste shown in Fig.5. he voltage wavefor consists of aplitude variation, frequency deviation and haronics. Under these conditions, it is clear fro Fig.5 that the calculated phasor agnitude can follow the odulated aplitude perfectly. he instantaneous frequency of the fundaental coponent was obtained using the zero-crossing ethod [5] and the ethod proposed by this paper. he results fro both ethods are shown and copared in Fig.6, focusing only on the wavefor distortion in both frequency and aplitude. Fro the figure, the results fro using zero-crossing ethod show a decreasing trend, but the line is non-sooth and coprised of several step changes. his phenoenon is caused by the rapid frequency drop and the haronic interference. Because the actual values are unknown, this SNR (db) Fig. 3. Maxiu error of estiated frequency. 4. Ipleentation Instantaneous phasor and frequency etering was developed by using the techniques described previously and is shown in Fig. 4. he data acquisition syste is based on a personal coputer being coposed of an analog to digital Estiated Phasor Magnitude Fig. 5. Voltage wavefor and agnitude estiation.

7 34 New Adaptive Linear Cobination Structure for racking/estiating Phasor and Frequency of Power Syste paper copares the proposed ethod with the widely used zero-crossing ethod. Fro Fig. 6, it can be seen that the behaviors of the two ethods are very siilar in trend, but the ethod proposed by this paper shows a sooth frequency drop and the results are atched well with the voltage wavefor. Estiated frequency Method proposed by this paper Zero-crossing Fig. 6. Frequency estiation. 5. Conclusion he paper describes and deonstrates the new adaptive linear cobination structure for estiating electrical paraeters of sinusoidal signals. he proposed technique is suitable for tracking phasor and frequency over a wide range of frequency variations which can be caused by changes in load and generation, power swing or fault events. he algorith provides several benefits, which are:. he algorith applies a fixed sapling interval Δ and can estiate phasor and frequency accurately. In addition, it allows ease of ipleentation into a relay or other phasor easureent devices. 2. he algorith possesses high quality dynaic responses. It is able to track phasor and frequency during situations below : ransient events, i.e. a sudden or step changes in phasor agnitude, frequency and phase angle. During power swing events, i.e. both agnitudes and frequency of voltage and current signal ay be odulated. 3. he algorith is able to track frequency change over a wide range fro Hz to 2 Hz with a sapling rate f = 4 khz. s 4. Good noise property, i.e. for SNR > 4 db, the axiu error is less than. Hz. 6. Future Work he authors intend to study in details about the ipleentation of the techniques presented in this paper for utilities. his syste can be a very useful tool for fault locator devices. Acknowledgents he authors would like to thank the Research Division of Provincial Electricity Authority hailand (PEA) for their supporting to this project. In addition, the authors wish to extend their thanks to y advisor Asst. Prof. Dr. eratu Bunyagul. Appendix ADALINE, LMS and Widro-Hoff deta rule [2]. he ADALINE structure is shown in Fig. 7. Its output is constructed fro the linear cobination of its input vector X k = [ x x2... xn] at any given tie. he input vector is ultiplied by the weighting vector Wk = [ w w2... wn], and then these weighted inputs are sued to produce the linear output yk ( ) = X( k) Wk ( ). x x 2 x n w w 2 w n Σ yk () Fig. 7. Basic ADALINE structure. ek ( ) x( k) In order for the ADALINE output yk ( ) to precisely iic the desired value of x( k ), the weighting vector is adjusted utilizing an adaptation rule based ainly on LMS algorith. his rule is also known as Widro-Hoff delta rule, and is given by (7). α ek ( ) X( k) Wk ( + ) = Wk ( ) + X ( k) X ( k) Σ (7) Where ek ( ) = xk ( )- yk ( ) is the prediction error at any tie k, yk ( ) is the estiated signal agnitude at tie k and α is the learning paraeter (reduction factor). Perfect learning is achieved when the error is brought to zero.

8 Choowong-Wattanasakpubal and eratu-bunyagul 35 References [] P. Kundur, Power Syste Stability and Control: McGraw-Hill, 994. [2] A.G. Phadke and J.S. horp, Synchronized Phasor Measureents and heir Applications: Springer, 28. [3] K.E. Martin, G. Benouyal, M.G. Adaiak, M. Begovic, R.O. Burnett, Jr., K.R. Carr, A. Cobb, J.A. Kusters, S.H. Horowitz, G.R. Jensen, G.L. Michel, R.J. Murphy, A.G. Phadke, M.S. Sachdev, J.S. horp, IEEE Standard for Synchrophasors for Power Systes, IEEE rans Power Delivery, Vol.3, Issue, pp.73-77, Jan., 998. [4] J.A. de la O Serna and K.E. Martin, Iproving phasor easureents under power systes oscillations, IEEE rans. Power Syst., Vol.8, No., pp.6-66, Feb., 23. [5] A.G. Phadke, J.S. horp and M.G. Adaiak, A new easureent technique for tracking voltage phasors, local syste frequency, and rate of change of frequency, IEEE rans. Power App. Syst., Vol.PAS-2, No.5, pp.25-38, May, 983. [6] G. Benouyal, An adaptive sapling-interval generator for digital relaying, IEEE rans. Power Delivery, Vol.4, pp , July, 989. [7] D. Hart, D. Novosel, Yi Hu, B. Sith and M. Egolf, A new frequency tracking and phasor estiation algorith for generator protection, IEEE rans. Power Deliver, Vol.2, pp , July, 997. [8]. Lobos, Nonrecursive ethods for real-tie deterination of basic wavefors of voltages and currents, IEE Proc., Vol. 36, pp , Nov., 989. [9] K.-F. Eichhorn and. Lobos, Recursive real-tie calculation of basic wavefors of signals, IEE Proc., Vol.38, pp , Nov., 99. [] B. Boashash, Estiating and interpreting the instantaneous frequency of a signal: Part I: Fundaentals, Part II: Algoriths and applications, Proc. IEEE, Vol.8, pp , Apr., 992. [] A. Cichocki and. Lobos, Artificial neural networks for real-tie estiation of basic wavefors of voltages and currents, IEEE rans. Power Syst., Vol.9, pp.62-68, May, 994. [2] B. Widrow and M. A. Lehr, 3 years of adaptive neural networks: Perceptrons, adaline and backpropagation, Proc. IEEE., Vol.78, pp , Sept. 99. [3] P.K. Dash, D.P. Swain, A.C. Liew and S. Rahan, An adaptive linear cobiner for on-line tracking of power syste haronics, IEEE rans. Power Syst., Vol., pp , Nov [4] LabView Analysis VI Reference Manual. Austin, X, National Instruents Corp, 996. [5] V. Friedan, A Zero crossing Algorith for Estiation of the frequency of a Single Sinusoid in White Noise, IEEE ransaction on signal processing, Vol.42, No.6, pp , June, 994. Choowong-Wattanasakpubal He was born in Songkhla, hailand, on March 5, 976. He received bachelor and aster degree in Electrical Engineering fro King Mongkut's Institute of echnology North Bangkok, Bangkok, hailand in 999 and 23, respectively. He is studying Ph.D. degree at Electrical Engineering at King Mongkut's University of echnology North Bangkok. Currently, he is an Assistant Chief of Power Analysis Section at Research Division of Provincial Electricity Authority hailand (PEA). His ain fields of research are power syste protection, fault location, grounding and digital signal processing (DSP) application to power syste. erata Bunyagul He was born in Phattalung, hailand, on 974. He received bachelor degree in Electrical Engineering fro King Mongkut's Institute of echnology North Bangkok. He received aster degree in Electrical Engineering fro University of Manchester Institute of Science and echnology, UK. He received the Ph.D. in Electrical Engineering fro University of Manchester Institute of Science and echnology, UK. Currently, he is an Assistant Professor in the Departent of Electrical Engineering at King Mongkut s University of echnology North Bangkok. He has consulted widely with governent agencies and the electrical industry. His ain fields of research are Power systes protection, autoation syste and digital signal processing (DSP) application to power syste.