2009 2nd International Conference on Power Electronics and Intelligent Transportation System

Frequency Estimation of Distorted Signals in PowerSystems Using Particle Extended Kalman Filter

E. M. Siavashi, S. Afshamia, *M . Tavakoli Bina, Senior Member, M. Karbalai Zadeh, M. R. Baradar

Faculty of Electrical and Computer Engineering , University of Tehran , Tehran , IranEmail: esiavashi @ut.ac .irEmail : safshar@ut.ac.ir

P. O. Box 14395/515, Tehran 14395-lran,'Faculty of Electrical Engineering, K. N. Toosi University of Technology, Tehran , Iran

Email: tavakoli @kntu .ac.ir

Abstract- In this paper, frequency of distorted signal in powersystem has been estimated with particle extended Kalman filter .Base of particle algorithm, extended Kalman filter and particleextended Kalman filter are mentioned. For selecting statevariables, a nonlinear time-variant sinusoidal signal is developedthen a particle extended Kalman filter is applied to detect thefrequency variations. Several tests are performed to show theperformance of PEKF algorithm. Comparison of PEKF withEKF reveals the PEKF preference. Also, these tests prove the fastspeed, good accuracy and robustness against noise. Theseadvantages illustrate that PEKF is more suitable for powersystem applications.

Keyword\'-frequency, state space model, tracking, particleextended Kalman filter (PEKF)

Kalman filter is a well-known observer for tracking the statevariables of system. Literature [15]-[17] used extendedKalman filter (EKF) to improve the tracking performance.

In this paper a novel method has been proposed for trackingfrequency. Particle extended Kalman filter (PEKF) is used toimprove frequency estimation at the present of noise andsevere variation. Also, as heavier changes , robustness of theproposed method is studied in this paper. Hence, robustnessagainst distribution, accuracy and speed of the proposedmethod could be scale to prove good performance of themethod. In order to illustrate performance of the proposedmethod, first, the sinusoidal signal is developed on space state.Then state variables were selected for applying the particleextended Kalman filter.

The using of frequency in power system protection andpower quality has caused that in the resent years was studiedmore about it. Frequency estimation in power system has facewith several unwanted distortion contamination, such asrandom noise, de component, and higher order harmonics, andalso from different forms of frequency variations. Therefore, afrequency estimation method should have ability of tracing inthe noisy and distributed environment. Moreover abovecondition, accuracy and speed are factors for comparison ofestimation methods.

Most of the researches on power system applications duringin last two decades have put more emphasis on frequencyestimation. Several methods are available on the frequencymeasurement for power. The simplest method is the measuringtimes between two zero crossing that indicate to half period ofsignal. Noise and distribution cause Zero-crossing in powersystem does not have desirable work [1]-[2]. Moreover,Several algorithms have been developed in the past fewdecades based on discrete Fourier transform (DFT) [1],[3]least-square error technique [4],[5], adaptive notch filter(ANF) [6],[7], orthogonal components filtered algorithm[8],[9] and phase lock loop (PLL) [10]-[12] as frequencyestimation methods. A comparative study among thesedifferent trackers has been outlined in [13], [14].

Where Wk and Vk are independent white proces s andobservation noises with probability distribution matrixesNCO, Q) and NCO, R) respectively. Also Xk and zk are thehidden state variables and the measurement respectively. Both

II. PAR TICLE E XTENDED K ALMAN FI LTER

Particle filters perform Monte Carlo based estimation basedon point mass approximations of probability densities. Eachstate probability space is represented by a large number ofweighted points (known as particles) which, taken together,approximate the uncertainty of the current system state. Oneapproach that has been proposed for improving particlefiltering is to combine it with another filter such as the EKF.In this approach, each particle is updated at the measurementtime using the EKF, and then resampling is performed usingthe measurement. This is like running a bank of N Kalmanfilters (one for each particle) and then adding a resamplingstep after each measurement. For clarifying the particleextended filter, the system and measurement equations aregiven as follows:

(2)

(I)

INTROD UCTIONI.

978-1-4244-4543-1/09/$25.00 ©2009 IEEE PElTS 2009

174

(5)

f C·) and h(.) could be non-linear functions that are assumedknown.

A. Particle fi lterThe particle filter was invented to numerically implement

the Bayesian estimator. The particle filter can be summarizedin the algorithm 1:

Algorithm I: Particle Filter

I. Initialization:• For i = 1, 2, ..., N

Assuming that the probability distribution function(PDF) of the initia l state P(xo) is known, N initialparticles from the prior P(xo) are generated random ly.These particles are denoted XO,i

• End For2. For k = 1, 2, ...

• For i = 1: N• Using the known process equation and the known

pdf of the process noise, a prio ri particles Xk,i are

achieved as fo llows:• Assign the relative like lihood of each particle as

follows:

(4)

• End For• For i = 1: N

• Norma lizing step: Normalize the likelihoods that areachieved in the prev ious step as follows :

• End For

qiqi = - N- -

L qii= \

• Resamp ling step: a posteriori part icles Xti have beengenerated on the basis of the relative likelihoods q, .

B. Extended kalmanfilterConsider a nonlinear system with state and measurement

equations as (1), (2). Algorithm 2 illustrates The ExtendedKalman filter.

175

Algorithm 2: Extended Kalman Filter

Step I: Predict the state with init ials:

xi = /(xk-I,0)

Step 2: Compute the error covariance:

Step 3: Compute the Kalman gain:

Step 4: Update the state estimate :

Step 5: Update the error covariance

Step 6: Return to step I with updated measurements

Where z is the measurement vector , xl; and Pl; are theapproximate state and covariance, F and H are the Jacobianmatrices of partial derivatives of f and h with respectiveto xl;. At each frame, the filter predicts the current state of thesystem and correct this estimated state using measurement ofthe system. A Kalman gain (Kk ) is computed to find theoptimum feedback gain that minimizes the error covariancebetween the priory and the posteriori estimation. The updatedestimate will then be used to predict the state for the nextframe. Manner achievement of F, H, W and V is shown asbelow:

8/ ~(6)F = 8x (xk-b ub 0)

8/ ~(7)W =-(xk_\,uk,O)

8w

H = ~~ (xk 'O) (8)

V = ~~ (xbO) (9)

III. SIGNAL MODEL

In power system, an observed signal is contaminated withharmonics and noise. Therefore the signal is modeled by:

C. Particle extended kalmanfilterIn a system that is nonlinear, the extended Kalman filter can

be used for state estimation, but the particle filter may givebetter results at the price of additional computational effort.Combining EKF with particle filter results a method thatinvolves better performance. Particle extended Kalmanalgorithm is summarized in algorithm 3:

Algorithm 3: Particl e Extended Kalman Filter

N

V(t)= IAn sin(nmt+~n)+Ekn=l

Where,

(17)

I. Initial ization : k =°• For i = 1,2, ... , N

• Randomly genera te particles Xti aro und the prior pdf

P(xo)• End For

2. For k = 1,2, ... do the follow ing.• For i = 1: N

• Compute the Jaco bian matrices Fk - 1•i and Q k - l .i

of the process model to obtai n priori part icles andcovariance Pk,i using the known processequation:

An amplitude of the n' th harmonic

<Pn phase of the n' th harmonic

n harmonic orderN highest harmonic order

t = n; ~ Sampling time and k sampling instant

co radian frequencyck additive noise (-N(O, R))

Since the amplitude of harmonic components are lowcompared to fundamental, the signal with no harmonics isconsidered. Hence the signal is simplified as below:

( 10) v(t) = Asin(mkTs +~)+ E (18)

• Compute the Jacobian matrix H k,i of themeasurement mode l and then update the a prioriparticles and covar iance to obta in a posterioriparticles and covariance :

Of lFk- 1i =- +, ax X= Xk- l,i

OhlH k i = - -, ax x = xk ,i

Kk . = Rk- ·HkT .(Hk .Rk-· Hk

T + Rk )-l,1 , / , [ , / , 1 , 1

( I I)

( 12)

(13)

(14)

Since in this study main objective is tracking of thefundamental frequency, for applying PEKF in estimationproblem, states variables are formed as below:

X\ (k) =mkTs (19)

x2(k) = A (20)

x3(k) = sin(rokTs +~) (21)

x4(k) = costrokTs + ~)(22)

By consideration state variables above, {(Xk) andh(Xk+1) in (1), (2) are written as follows:

Xk+ . = s; .+ «; .[Zk - h(xk- .)J,1 , I , I , 1

Rk+ · = (I - Kk .n, ·)P.k- ·, I , I , 1 ,1

( 15)

( 16)(23)

Jacobian matrices for linearizing (23), (24) are obtainedfrom (6) and (8) as follows:

• Compute the relative likelihood of eachparticle xt by (4) .

• End For• For i = 1: N

• Sca le the likelihoods is obta ined as shown in (5) .• End For• Resampling step: on the basis of the relative likelih oods

qi refine Xti and Pt i'

176

h(Xk) = x2(k)x3 (k) (24)

o 01 0o cos(x](k))o -sin(x] (k))

Sin(~, (k)) Jcos(x](k))

(25) ..c'" 0Cii

M] = X4 (k) cos(x](k)) - X3 (k) sin(x] (k))

M 2 =-x4(k)sin(x] (k))-x3(k)cos(x] (k))

..i:I~ -0.5

Measurement equation is shown as follows:

(26)

-1.50 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.06 0.09 0.1

Time (sec)

(a)

(b)

460 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.06 0.09 0.1

Time (sec)

Fig. I. a) Actual Signal for Test Ib) Estimated Frequency with PEKF and EKF and Real Frequency

1

- - - Real Frequency I-'- ' EKF Method- PEKF Method

II ...I \ •

I """"L__ ._...._.

51

47

_ 50 ...........-QN

::.'"'g 49..:Ie-eIL 46

Tracking time Steady state error (Hz)

Test Signals PEKF EKF PEKF EKF

Test Signal I - 0.5cycle - 3cycle .00001 0.0003

Test Signal 2 - Icycle - 2cycle 0 0

Test Signal 3 - L5cycle NotTracking 0.001 00

V. CONC LUSION

In this paper is used particle extended Kalman filter fortracking the frequency of power system. . State space modelof the nonlinear time-varying system is developed using exactanalytical equations. For showing the performance of particleextended Kalman filter, several tests as different level noiseand variation of the frequency have been applied, thencompared with extended Kalman filter. Simulation resultsshow that PEKF has better accuracy and speed than EKF.Also, robustness against noise and severe variation of state arethe other special features of PEKF method.

TABLE ITracking time and steady state error for three test signals

By using (18)-(26), PEKF is applied for tracking frequencyas it is shown simulation section.

IV. SIMULATION RESULTS

In simulation, different noise level and step change in thefundamental frequency is considered to test the performanceof particle extended Kalman filter for tracking frequency. Forall tests, fundamental frequency and amplitude are considered50 Hz and I Pu, respectively. Hence state variables Xl andX 2 are initialized 50 Hz and I Pu. Also, step change infrequency for all tests is applying at 0.2 second. For PEKFalgorithm, 100 particles in each sample time are considered.

a) Test Signal I: 2 Hz step-down changes in frequency withlow noise: The frequency of the signal is suddenly decreasedfrom 50 Hz to 48 Hz. Signal-to-noise ratio (SNR) for this caseis 50 dB. Fig. 1 illustrates actual signal and the tracking offrequency with both methods EKF and PEKF. As it is revealsthat PEKF has estimated frequency fast and accurately thanEKF. Very fast estimation of frequency with PEKF isconsiderable in Fig (1)-(b). Hence, this test confirms the fastspeed of PEKF method.

b) Test Signal 2: I Hz step-up change in frequency withhigh noise: The frequency of the signal change from 50 Hz to51 Hz with 10 dB measurement noise. The contaminatedsignal with high level noise and estimated frequency withPEKF and EKF have been shown in Fig. 2. In this test, PEKFhas been better performance than EKF, too. However, severenoise effect on both performance. Therefore, ability oftracking in present of strong noise is confirmed by this test.

c) Test Signal 3: Severe step-down (8Hz) in frequency withhigh noise: At 0.2 sec, in present of high level noise (10 dBSNR), frequency of actual signal varied from 50 Hz to 42 Hz.Results of this test are shown Fig. 3. It is shown that PEKFsucceeds to estimate new frequency. However EKF methodhas lacking ability to track the frequency. This test reveals thatPEKF robust against severe change.

These tests confirm the high robustness of particle extendedKalman filter against noise and variation of state. Trackingtime and steady state error are mentioned in Table I for bothmethods PEKF and EKF.

177

1 .5 r-~-~-~-~~-~-~-~~-,

_ 1 i r\1 t' l'~~ (II:\! l{i",

0 5

1l t\ '1'1 j \ 1\ I \

I \ f I,l I~\ I[

i" \I \ I \ \ I I.~ W ~ VVV

- 1 . 5~---:-'::c--:-'::,______:=____:_::-:----:-':::--:-'::,______:=____:_';;;:----:-':::--;;'o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Time (sec)

(a)

-1.5.0L-O~.0""1----:-o .':::o2:--o~.0""3 ----:-0.":-04:--0~.0::-5----:-0.':::06,------:0:-':.0=-7 ----:-0.'.::08'------:0:-':.0=-9 ----:-0.1

Time (sec)

(a)

PEKF Method

El ~~ : : jo 0.01 0.02 0.03 0.04 0.05 0 06 007 0.08 0 09 0.1

~ :1~ 46e~ 420'e 36LL

34 o

PEKF Method

~ : : J0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

EKF Method

Fl•~----- ::::jo 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Time (sec)

(b)

Fig. 2. a) Actual Signal for Test 2b) Estimated Frequency with PEKF and EKF and Real Frequency

REFERENCES

[I] M. M. Begovic, P. M. Djuric, S. Dunlap, and A. G. Phadke, "Frequencytracking in power networks in the presence of harmonics ," IEEE Trans.Power Del., vol. 8, no. 2, pp. 480--486, Apr. 1993.

[2] D. W. P. Thomas and M. S. Woolfson, "Evaluation of frequency trackingmethods," IEEE Trans. Power Del., vol. 16, no. 3, pp. 367-371, Jul.2001.

[3] 1. Yang and C. Liu, "A precise calculation of power system frequency,"IEEE Trans. Power Del., vol. 16, no. 3, pp, 361-366, Jul. 2001.

[4] M. S. Sachdev and M. M. Giray, "A least square technique fordetermining power system frequency," IEEE Trans. Power App. Syst. ,vol. PAS-104, no. 5, pp, 427-443, May 1985.

[5] A. K. Pradhan, A. Toutray, and A. Basak, "Power system frequencyestimation using least mean square technique," IEEE Trans. Power Del.,vol. 20, no. 3, pp. 1812-1816, Jul. 2005.

[6] M. Mojiri, M. Karimi-Ghart emani, and A. Bakhshai, "Time-DomainSignal Analysis Using Adaptive Notch Filter," IEEE. Trans. SignalProcess., vol. 55, no. 1, pp.85-93, January 2007.

[7] M. Mojiri, M. Karimi-Ghart emani, and A. Bakhshai, "Estimation ofPower System Frequency Using an Adaptive Notch Filter," IEEE. Trans.Instrum. Meas., vol. 56, no. 6, pp.2470-24 77, December 2007.

[8] 1. Szafran and W. Rebizant, "Power system frequency estimation ,"Proc. Inst. Elect. Eng., Gen., Transm. Distrib., vol. 145, no. 5, pp.578-582,Sep.1998.

[9] T. S. Sidhu and M. S. Saehdev, "An iterative technique for fast andaccurate measurement of power system frequency," IEEE Trans. PowerDel., vol. 13, no. 1, pp, 109-115, Jan. 1998.

178

EKF Method

j ~~o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Time (sec)

(b)

Fig. 3. a) Actual Signal for Test 3b) Estimated Frequency with PEKF and EKF and Real Frequency

[10] 1-1. Karimi, M. Karimi-Ghartemani , and M. R. Iravani, "Estimation offrequency and its rate of change for applications in power systems,"IEEE Trans. Power Del., vol. 19, no. 2, pp. 472-480, Apr. 2004.

[11] M. Karimi-Ghart emani, and M. R. 1ravani, "A Method forSynchronization of Power Electronic Converters in Polluted andVariable-Frequency Environments," IEEE Trans. Power syst., vol. 19,no. 3, pp. 1263-1270, August. 2004.

[12] S. Shinnaka, "A Robust Single-Phase PLL System With Stable and FastTracking," IEEE Trans. Ind. Appl., vol. 44, no. 2, pp. 624-633,March/April. 2008.

[13] P. Tiehavsky et al., "Comparative study of four adaptive frequencytrackers," IEEE Trans. Signal Processing, vol. 45, pp. 1473-1484,June 1997.

[14] D. W. P. Thomas and Malcolm S. Woolfson, "Evaluation of FrequencyTracking Methods," IEEE Trans. Power Del., vol. 16, no. 3, pp, 367­371, July. 2001.

[15] P. K. Dash, R. K. Panda, and G. Panda, "An extended complex Kalmanfilter for frequency measurement of distorted signals," IEEE Trans.Instrum. Meas., vol. 49, no. 4, pp. 746-753, Aug. 2000.

[16] C. Huang, C. Lee, K. Shih, and Y. Wang, ., "Frequency Estimation ofDistorted Power System Signals Using a Robust Algorithm," IEEETrans. Power Del., vol. 23, no. 1, pp. 41-50, January . 2008.

[17] A. Routray, A. K. Pradhan, and K. P. Rao, "A novel Kalman filterfor frequency estimation of distorted signal in power systems," IEEETrans. Instrum. Meas., vol. 51, no. 3, pp. 469-479, Jun. 2002.