(IJEECS) International Journal of Electrical, Electronics and Computer Systems. Vol: 13 Issue: 01, 2013
Effect of Noise Covariance Matrices in Kalman Filter Performance of Power System Harmonics Detection Hisham Alrawashdeh #1, Johnson Asumadu*2, Jumana Alshawawreh*3 #1, *2
Electrical and Computer Engineering Department, Western Michigan university, Kalamazoo, MI, USA *3 Electrical Engineering Department, Tafila Technical University, Tafila, Jordan
filter algorithm from three to one. They mentioned in their paper that, the ratio between Q and R matrices is more important than the real value of these matrices. Scala and Bitmead [5] proposed a new extended Kalman filter to estimate a time-varying frequency. they use three control parameters ε , q and r , Q must be positive definite to have bounded positive controllability matrix. They claimed that the Keywords— Kalman filter, power system harmonics, voltage effective of Q is more significant than R. Yu et al. [6] sag and noise covariance matrices. switched between two values of Q matrix; one for the transient and the other for steady state. At the beginning, I. INTRODUCTION they set Q = 0 , the value of Q was good in steady state In many applications, the operation of the protection and portion, but this value is not sufficient to prevent the control devices need an accurate measurement of the divergence of kalman filter for a step change in the signal, electrical signal, if the measurement signal has high value of then they set Q=I , they showed a good response in the harmonics or noise, this may cause a wrong decision in these transient (step change in signal), they had a large oscillation devices. For that reason, the Kalman filter is widely used in in the steady state period. They used student statistical model power system harmonics detection. Several previous with 95% confident to decide which Q to use. researches showed the affectivity of using Kalman and Abdelsalam et al [7,8], in their proposed algorithm, the extended Kalman filter in power harmonic detection. distorted signal passes first through a hybrid model of In the Kalman filter design, the noise covariance matrices wavelet and Kalman filter to define three parameters; (Q and R) are assumed to be well defined, to get optimal Amplitude, Amplitude slope and harmonics indication, then solution, unfortunately, in many applications, it is hard to the fuzzy controller is used to determine the power quality determine these matrices prior the operation of the Kalman disturbances. Q matrix is set to be 1 in this model, R is filter. These matrices could be estimated and adjusted calculated at each iteration, where the value of R is set to be through online or offline calculations based on the measured the coefficient covariance of the first row of the digital wave data, in such case, the Kalman filter is called adaptive let matrix. Sangsuk-Iam and Bullock [9] analysed the Kalman filter. Kennedy et al. [1] were proposed an adaptive divergence and convergence of the Kalman filter for incorrect extended Kalman filter to estimate the harmonics in power noise covariance matrices, they concluded that, if the system system. They used three signals to test the Kalman filter is modelled accurately, then covariance matrices error will capability to estimate the harmonics under several conditions, never cause the divergence of the Kalman filter. Ajathey concluded that the modifying Q/R ratio is not sufficient Fernández et al. [10] proposed a new fuzzy kalman filter to especially when the S/N ratio exceed the 30 dB. They track stereo visual scheme, it is called principle coordinates proposed an algorithm to compute the Q and R matrices at Kalman filter. Two fuzzy models were used; one of them was each iteration. Routray et al. [2] proposed adaptive extended used to adjust the covariance matrices, while the other was kalman filter for power system frequency measurement, their used to reinitiate the Kalman filter when it losses the track, proposed algorithm resets the covariance matrices based on they only adapted Q matrix in their algorithm. two factors; the error and the convergence of the algorithm, In the previous researches, there was a strong argument, they mentioned that, the covariance matrices should be reset which is more important in kalman filter performance, Q, R as long as some of the parameters are changed. Griffo et al. [3] or the ratio between them, for that purpose, this paper will proposed strategy to solve LQR problems to estimate the study the effect of Q and R matrices in kalman filter frequency in power system. They emphasise that, the optimal performance for different models of power system harmonic value of the LQR problem depends on the relative value detection.. between Q and R matrices, not on their absolute values . in their proposed algorithm, the first estimation of Q and R II. KALMAN FILTER matrices are I and 0 respectively, then the value of R is The foundation of Kalman filter was 1960 by R.E. calculated using nonlinear inequality equation, the choice of Kalman for discrete time systems, then he extended it to the Q value is not clear, but they use it to reduce the negative continuous system [11]. Kalman filter is a recursive, linear, sequence in the current. Bittanti and Savaresi [4] suggested a real time and optimal filter to estimate the state of dynamic new way to reduce the controlling parameters of Kalman noisy system. The noise sources in Kalman filter are assumed Abstract— The prior knowledge of the covariance noise matrices is effecting the performance of the Kalman filter for detecting power system harmonics, In this paper the effect of the noise matrices will be investigated under several circumstances, to determine the quality and the convergence of the Kalman filter in power system harmonics detection.
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(IJEECS) International Journal of Electrical, Electronics and Computer Systems. Vol: 13 Issue: 01, 2013 to be independent Gaussian white additive noise [11-13]. The Kalman state vector X k at any instant can be calculated as follows:
X k = Ak X k −1 + B k U k + W k Z k = C k X k + Vk Where,
electrical signal such as voltage and current will appear as a distorted sinusoidal signal, instead of a pure sinusoidal signal. Based on Fourier series, the electrical signal can be written as follows:
(1)
g (t ) =
∞
∑A
m
cos(2πmf 0 t + φ m )
(4)
m =0
X k : is the state vector,
The harmonic signals can be classified to low and high order harmonics respects to their frequency magnitude, where the amplitude of the harmonic signal is usually inverse Bk : is the input control vector. proportional to the harmonic order, which means that the low Wk : is the process noise, and it is assumed to be white order harmonics have larger values compared to the high order harmonics. For that reason the low order harmonics are Gaussian noise with zero mean and covariance matrix Qk , more effective in signal distortion, and they have the most priority in harmonics detection compared to higher order i.e N (0, Qk ) . harmonics. Z k : is observation of the state X k . The main causes of the power system harmonics are the nonlinear devices, such as; converters, static VAR C k : is the observation matrix. compensator, transformers, rotating machines, arc furnaces, Vk : is the observation noise, and it is also assumed to be fluorescent lightings, power electronics load and unbalanced Gaussian white noise with zero mean and Rk covariance system conditions [14-16]. In transformer, rotating machines and florescent lightning the source of harmonics is the matrix, i.e N (0, Rk ) . saturation of the magnetization curve in these devices, where A recursive means that, it is suitable for real time usually the rotating machines are designed to operate near the application, where there is no need to store much data, it will knee portion for economic reason [14,15]. Due to the load needs only the current measurement and the previous state. variations this may caused the operating point to be in Linear filter means, it will be suitable for linear model or at saturated area. The harmonics affect the performance of the power least linearized model around certain operating conditions. The optimality of the Kalman filter is due to minimizing the system, their effects can be divided to short term and long statistical variance matrix of the state error, the optimality of term [16]. For the short term, the harmonics cause a high the Kalman filter can be achieved under the following distortion in the electrical signal, while losses and voltage stress are increased in the long term [16]. The effects of the conditions [11]: harmonic in electrical apparatus can be summarized as - The system is linear and well modeled. follows [14,15]: - All noise sources are white Gaussian noise. 1. Harmonic losses: there are several types of harmonic - The covariance matrices of the noise are well losses; defined. a. harmonics resonance: the resonance may The Kalman filter calculation can be divided into two occur with one of the harmonic frequencies, stages; prediction and updating stages. In the prediction stage, and this will cause over voltage and high the state is predicted based on the previous instant until a losses in the power system, which may recent data is measured, then the calculation enters the cause a damage of the power system updating stage to modify the predicted state [11-13] as components, or at least shortage the life follows: time of these devices. Predicted stage b. Copper losses of the rotating machines, X k k −1 = Ak X k −1 k −1 + B k −1U k −1 especially due to the low order harmonics. (2) Pk k −1 = Ak Pk −1 k −1 + AkT + Q k The core and the eddy current losses are also increased in transformer and rotating machines. Updating stage c. Corona losses: the harmonics will cause Yk = Z k − C k X k k −1 over voltage in the overhead transmission K k = Pk k −1C kT [C k Pk k −1C kT + R k ] −1 line, which increase the corona discharge, (3) leads to extra losses in power system. X k k = X k k −1 + K k Yk 2. Increasing the dielectric stress: this will shorten the Pk k = [ I − K k C k ]Pk k −1 life time of cables and increases the corona phenomena for overhead transmission lines, which will increase the chance for having breakdown problem. III. HARMONICS IN POWER SYSTEM AND VOLTAGE SAG 3. Changing the shape and the characteristics of the PROBLEM voltage and the current wave form: this will affect The power system harmonics are the components of the performance of the protection devices, the electrical signal, that have a multiple frequencies of the consumer devices quality, the power factor and the fundamental (system) frequency f 0 , which is either 50 or 60 power measurement. Hz [14-16]. Because of the harmonic components, the
Ak : is the transition matrix
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(IJEECS) International Journal of Electrical, Electronics and Computer Systems. Vol: 13 Issue: 01, 2013 4.
Interference between the power system and the communication system: which may cause a signal losses in the communication system. To ensure the power system stability, the voltage at each bus must be maintain within certain limits, otherwise the voltage may collapse and causing the voltage sag problem, in such case the protection devices need to detect the voltage deviation and operate before the voltage sag occurs. In case the voltage signal has high harmonics and noise, then the protection devices may be operate at wrong value of the voltage, using kalman filter in such case will reduce the measurement noise and help he protection devices to operate precisely.
IV. HARMONICS MODELLING FOR KALMAN FILTER In power system the order of harmonics is odd, there are no existence of even harmonics. Generally the state variables can be chosen as follows:
X (k ) = [x1 (k ) x 2 (k ) x 3 (k ) K x 2 n (k )]'T
x1 (k ) = A cos( wkT + θ 1 )
collapse problem. The noise covariance matrices will be varied in different experiments to investigate the kalman performance as follows: A. Experiment 1: The signal will kept constant through the simulation, while the value of the covariance matrices will be changed to investigate the kalman filter performance. Suppose the voltage signal has first and third order of harmonics as follows:
y (t ) = 1.414 cos(100πt + π 6) + 0.3 cos(300πt + π ) + ... 5
0.1 cos(500πt + π ) (5) 8 The kalman filter model in this experiment will assume the existence of the fundamental component and treat the harmonic components beside the measurement noise as a disturbance in the signal. The number of state variables in this case is two, and the plant matrices will be as follows: cos( wT ) − sin( wT ) 0 A= , B = sin( wT ) cos(wT ) 0 C = [1 0] ,, D = [0]
x 2 (k ) = A sin( wkT + θ 1 )
Fig. 1 shows the plot of the signal with harmonics and the signal with noise. The value of the noise standard deviation of the measurement noise is set to be 0.1. In this experiment, the first and the third order of harmonic will be treated as a noise, which means that, even though if the R in kalman filter is set to be 0.1, it is still not the true value of the noise covariance matrix. Fig. 2 shows the plot of the fundamental signal and the estimated fundamental signal using kalman filter, there is slight difference between them at the beginning of estimation, then the difference between them goes to zero. Fig. 3 shows the fundamental signal amplitude and the estimated amplitude using kalman filter, and it is also shows the error signal (i.e difference between the estimated value and the real value) using kalman filter, the value of the noise covariance matrices are Q = 0, R = 0.1. as it is shown in Fig.3 the kalman filter converges to the fundamental amplitude so fast.
x 3 (k ) = A cos(3wkT + θ 3 ) x 4 (k ) = A sin(3wkT + θ 3 ) M x 2 n −1 (k ) = A cos(nwkT + θ n )
x 2 n (k ) = A sin( nwkT + θ n ) Where n is the order of harmonic. The state equations will be as follows: X (k + 1) = AX (k ) + B U (k ) + wk Y (k + 1) = C X (k ) + D U (k ) + v k
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V. SIMULATION RESULTS The kalman filter in this paper will be used to estimate the amplitude of the fundamental signal to avoid the voltage ©IJEECS
(IJEECS) International Journal of Electrical, Electronics and Computer Systems. Vol: 13 Issue: 01, 2013
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The Q matrix is kept constant and the value of R is changed from 0.01 to 100. Fig. 4 shows the amplitude of the fundamental signal using kalman filter for different values of R less than the true value and starts from 0.01 up to 0.1. The kalman filter converges for all the values of R less than 0.1, for more carefully investigate, Fig. 5 and Fig. 6 show a zooming areas; one at the beginning of the simulation and the other one at the end of simulation. The variations of the amplitude due to the R value mostly occur at the beginning of kalman filter operation (transient period), then the performance of the kalman filter is not affected by changing R value. Fig. 7 shows the amplitude of the fundamental signal using kalman filter for different values of R greater than 0.1 up to 100. Fig. 8 and Fig. 9 show zooming areas; one at the beginning and the other at the end of the simulation area. The variation of the kalman filter output in case of R greater than 0.1 is less than the variation of the output that estimated for R less than 0.1. .
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Fig. 3 fundamental signal using kalman filter for R=0.1.
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Amplitude of the fundemetal signal using kalman filter
(IJEECS) International Journal of Electrical, Electronics and Computer Systems. Vol: 13 Issue: 01, 2013
0 0 cos( wT ) − sin( wT ) sin( wT ) cos(wT ) 0 0 A= 0 0 cos(3wT ) − sin(3wT ) 0 sin(3wT ) cos(3wT ) 0
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B. Experiment 2: The 1st harmonic will be included in the kalman filter matrices, which means that; there will be four state variables, and the plant matrices will be adjusted as follows:
The value of R will also be changed as it is done in Experiment 1 inorder to investigate the performance of the kalman filter . Fig. 10 and Fig. 11 show the fundamental and the 1st harmonic signal estimated by the kalman filter respectively for R=0.1 ( the true value of R). As it is shown in Fig. 10 and Fig. 11 the variation of the estimated 1st harmonic signal is more than the variation of the estimated fundamental signal. Fig, 12 shows the error signal ( difference between the estimated amplitude of the fundamental signal and the true value) using the model in experiment 1 and the model in experiment 2. The fundamental signal variation using the model in experiment 1 is larger at the beginning of the
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(IJEECS) International Journal of Electrical, Electronics and Computer Systems. Vol: 13 Issue: 01, 2013 simulation compared to that found using the model in experiment 2, then the deviation of both models goes to zero.
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Fig. 12 Error signal ( for the fundamental component) using kalman filter models of experiment1 and experiment 2 for R= 0.1.
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Fig. 13 to Fig. 21 show the error signal using model in experiment 1 and model in experiment 2 for different values of R. Fig. 13 to Fig. 20 show the error signal at the beginning of the simulation, for long run of the error signal goes to zero for both models. As the value of R is increased the variation and the difference between the two models decrease, while both models took longer time to get zero error signal. when the value of R is set to be small in kalman filter, this will means that, the kalman filter can trust the measured signal because the error due to noise will be small, which means that the kalman filter will give large weight for the measured input to estimate the fundamental component. This will illustrate the fact that, when R has large values in kalman model, the deviation on error signal become small and almost the same for most of these values, since the model will not trust the measured values and depend more on the predicted values. In this case the kalman model will be more sensitive to the initial values, since it will trust more the predicted values and the predicted value is strongly affected of the initial values. The initial values in the previous figure
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©IJEECS
(IJEECS) International Journal of Electrical, Electronics and Computer Systems. Vol: 13 Issue: 01, 2013 R =0.08
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(IJEECS) International Journal of Electrical, Electronics and Computer Systems. Vol: 13 Issue: 01, 2013
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Fig. 24 Error signal using kalman filter models of experiment1, experiment 2 and experiment 3 for R= 0.02. R =0.05 2
C. Experiment 3: The 1st and the 3rd harmonic will be consider in this kalman filter model , which makes the number of variables equal s 6 and the model matrices as follows: Z A1 Z cos( jwT ) − sin( jwT ) A = Z A2 Z , A j = sin( jwT ) cos( jwT ) Z Z A3
0 0 T Z= , B = [0 0 0 0 0 0] 0 0 C = [1 0 1 0 1 0] , D = 0 . the R is changed as it is done in previous two experiments from 0.01 to 1000. Fig. 23 to Fig.34 Show the result of the error signal using the models in the three experiments for different values of R. experiment 3 model has the least deviation of the error signal for small value of R, as R is increased the three models of kalman filter have approximately the same error signal. Fig. 35 shows the error signal of the three models using different initial state values, where the initial value in this figure is
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Fig. 26 Error signal using kalman filter models of experiment1, experiment 2 and experiment 3 for R= 0.08.
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Fig. 23 Error signal using kalman filter models of experiment1, experiment 2 and experiment 3 for R= 0.01.
©IJEECS
(IJEECS) International Journal of Electrical, Electronics and Computer Systems. Vol: 13 Issue: 01, 2013 R =0.1
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Fig. 27 Error signal using kalman filter models of experiment1, experiment 2 and experiment 3 for R= 0.1.
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Fig. 30 Error signal using kalman filter models of experiment1, experiment 2 and experiment 3 for R= 1.
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Fig. 28 Error signal using kalman filter models of experiment1, experiment 2 and experiment 3 for R= 0.5.
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Fig. 31 Error signal using kalman filter models of experiment1, experiment 2 and experiment 3 for R= 0.2.
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Fig. 29 Error signal using kalman filter models of experiment1, experiment 2 and experiment 3 for R= 0.8.
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Fig. 32 Error signal using kalman filter models of experiment1, experiment 2 and experiment 3 for R= 10.
©IJEECS
(IJEECS) International Journal of Electrical, Electronics and Computer Systems. Vol: 13 Issue: 01, 2013 it doesn't go to zero. As the value of Q is increased the error signal in the three models is reduced but the deviation is increased.
R =100 2 error signal using model in experiment1 error signal using model in experiment2 error signal using model in experiment3
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Fig. 33 Error signal using kalman filter models of experiment1, experiment 2 and experiment 3 for R= 100. R =1000 2 error signal using model in experiment1 error signal using model in experiment2 error signal using model in experiment3
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Fig. 34 Error signal using kalman filter models of experiment1, experiment 2 and experiment 3 for R= 1000.
VI. CONCLUSION AND FUTURE WORKS the kalman filter was used to detect the fundamental amplitude of a signal has; first harmonic, third harmonic and noise, for that purpose the kalman filter was modelled in three different ways; the first one treat the harmonics in the signal as a measurement noise, the second model treat the third harmonic as noise and the last one modelled all the harmonics components in the kalman filter matrices, In general the performance of the third model is better than the second one and the second model is better than the first one, but in other hand the complexity and the number of calculation in model three is greater than model 2 and model 2 is greater than model 1, there is several cases, where the performance of the three models is the same. The effect of R value in the kalman filter was examined carefully in this paper, where it is founded that when the value of R is small the kalman filter react more with the measured signal which causes deviation at the beginning of the kalman filter operation, while when R has a large value, the kalman filter will trust more the predicted value which cause the kalman filter to converge to none true value of the output. As the value of Q is increased in the three models, the kalman filter puts more weight in the measured value, which causes a variations in the estimated output because of the noisy part in the measured signal.. For Future work a dynamic signal will be used to test the effect of noise covariance matrices, where the amplitude and the frequency will be changed. Then based on the results an adaptive kalman filter for power system harmonics may propose.
R =1000 2 error signal using model in experiment1 error signal using model in experiment2 error signal using model in experiment3
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Fig. 35 Error signal using kalman filter models of experiment1 and experiment 2 for R= 1000 and state variables initial estimate
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Until now the Q is set to be zero in all the above figures, the worst results for all the three models were found when the value of R has a large value, for this case Q will be changed to figure its effect in kalman filter performance. Fig. 36 shows the error signal using the three models for different values of Q and constant R (i.e R=1000). When the value of Q is small, the three models have approximately the smae error signal, the error signal in this case has low deviation and ©IJEECS
0.1
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(IJEECS) International Journal of Electrical, Electronics and Computer Systems. Vol: 13 Issue: 01, 2013 Q =0.001 2 error signal using model in experiment1
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Fig. 36 Error signal using kalman filter models of experiment1 and experiment 2 and experiment 3 for different values of Q, R= 1000 and state variables initial estimate are
[2
0 0.3 0 0.1 0]T .
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©IJEECS
Hisham O. Alrawashdeh received her M.SC in communication Engineering from Mutah university, Jordan 2007, and he is now pursing his Ph.D. degrees in Electrical and Computer Engineering, WMU, USA. Eng. Alrawashdeh is specialized in electronics and communication engineering His areas of research include power system harmonics, adaptive filters, fiber optics, modern control. Johnson A. Asumadu (S’82, M’94, SM’00) received his B.S. from the University of Science Technology, Ghana, in 1975, his M.S. from Aston University, England, in 1978, his MEE from the Rensselaer Polytechnic Institute, USA, in 1983, and his Ph.D. from the University of MissouriColumbia, USA, in 1987. He is now a faculty member at Western Michigan University, Michigan. He has published several technical papers and books, has a number of patents and has been involved in numerous government and industrial-sponsored projects. His current research interests include power electronics and control engineering including fuzzy logic applications, and microprocessor / microcontroller embedded applications. Jumana A. Alshawawreh received her M.SC in power Engineering and Control from JUST, Jordan 2004, and Ph.D. degrees in Electrical and Computer Engineering from WMU, USA in 2011. Dr. Alshawawreh is specialized in power system, electrical machines and control. She conducted several researches in the field of fuzzy control, optimal flow, genetic algorithms, sub synchronous resonance in power system. She is Faculty member in TTU in Jordan, since 2011. Her areas of research include power system stability, power system simulation, power system dynamics, machines, harmonics, adaptive control, fuzzy control, optimization methods, genetic algorithms and renewable energy.
