Improved recursive Newton-type algorithm for power system relaying and measurement

V.V. Te rz i j a

Indexing term: Computer reiaying, Forgetting factor, Harmonic and frequency measurement, Recursive parameter estimation

Abstract: A new recursive Newton-type algorithm suitable for measurement applications in power systems is presented. In an improved form it is used for power system frequency, phasors and spectra estimation. The key algorithm improvement is provided through a new forgetting factor timing procedure, by which the speed of convergence and accuracy are improved. The results of coimputer simulation, laboratory testing and offline full-scale real-life data processing are given to show the main algorithm properties.

1 Introduction

In substations the presence of microprocessor-basel devices constantly processing system data makes it pos- sible to enhance computer-based relaying, static-state estimation, harmonic measurement, disturbance moni- toring and control. Fast computers with parallel archi- tectures connected through efficient communication networks allow real-t [me measurement of basic quanti- ties (voltage or current phasors and spectra, local sys- tem frequency, time constant of decaying DC component) as well as the calculation of the quantities derived from basic quantities (active and reactive power, the rate of change of basic quantities, etc).

Numerical algorithms aimed at frequency estimation in power systems can be divided into two groups: non- recursive algorithms, or finite impulse-response filters, and recursive algorithms, or infinite impulse-response filters. Recursive algorithms are computationally more efficient in comparison with the corresponding nonre- cursive algorithms.

One of the first numerical algorithms for frequency estimation is presented in [l]. Its performance can be adversely affected by decaying DC components or low signal-to-noise ratio and it requires long measurement windows when the frequency deviation is small [2]. In [3 ] a recursive least error squares algorithm is described in which the nonstationary filter gains are calculated offline and stored in a ring buffer. By this the costly normal matrix inversion is avoided. With the inclusion of the decaying DC component in the signal model, the

0 IEE, 1998 IEE Proceedings online no. 1 998 1540

Paper received 17th December 1996 The author is with the Faculty of Electrical Engineering, University of Belgrade, Belgrade, Yugoslavia

IEE Proc.-Gener. Transm. Distrib., Vol. 145. No. I , January 1998

gains of the recursive filter do not remain cyclic and have to be calculated online, requiring the normal matrix inversion. In [4] a two-stage algorithm, based on a combination of an adaptive extended Kalman filter and an adaptive linear Kalman filter is used for fre- quency estimation in underfrequency load shedding. Statistical properties of the signal to be processed are a prerequisite for the successful estimation of unknown model parameters. In [5] recursive least squares and least mean squares algorithms for the dynamic estima- tion of voltage phasor and frequency are described. Frequency is estimated by a finite derivative of the phase deviation, followed by a moving-average filter.

This paper describes a new recursive Newton-type (RNTA) numerical algorithm suitable for fast measure- ment applications in power systems. It is derived from the nonrecursive Newton-type (NTA) algorithm pre- sented in [6]. It is suitable for phasors, harmonics and local system frequency measurement. In the signal model the frequency is treated as an unknown parame- ter, so the model becomes nonlinear and the strategies of nonlinear estimation are used.

It is not difficult to change the algorithm form or formally establish a recursive estimator from the nonre- cursive one. The corresponding recursive estimator does not have the same features as the nonrecursive one. It is also questionable whether the new recursive estimator will satisfy the measurement requirements. From this point of view the investigation and testing of the RNTA algorithm is of the great interest. Moreover, there exist some cases (always by nonlinear signal mod- els) in which it is not possible to establish a recursive algorithm with satisfactory convergence properties. From the author’s experience, the self-tuning least error squares algorithm [7] yields unsatisfactory results when it is established in the recursive form!

A new forgetting factor tuning procedure is designed and implemented to improve the main properties of the new RNTA algorithm (speed of convergence and accu- racy). This concept of tuning is similar to [7] where the algorithm tuning is heuristically controlled by means of an empirical formula based on the energy of the resid- ual error (error signal) at each iteration.

Computational, accuracy and tracking properties of the new algorithm are obtained through the algorithm application in frequency, phasor and harmonics meas- urement. After discussing various tuning details, the RNTA algorithm is tested on simulated and laboratory obtained data. The results of the application of the new algorithm in offline real-life data records processing are presented.

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2 Signal model representation

Assume the following observation model of the input signal (arbitrary voltage or current), digitised at the relay (measurement device) location:

in which v(t) is an instantaneous voltage at time t , [ ( t ) a zero mean random noise, x a suitable parameter vec- tor and A(.) is expressed in the following way:

44 = h(z , t ) + E @ ) (1)

M

h(x, t ) = v0e-k + V, sin(kwt + c p k ) (2)

For the generic model (eqn. 2), a suitable vector of unknown parameters is given by

where V, is the magnitude of the decaying DC component at t = 0, Tu is the time constant of the decaying DC component, A4 is the highest order of harmonic in the signal, o is the fundamental angular velocity (equal to 2$, f being frequency), V, is the magnitude of the kth harmonic and q k is the phase angle of the kth harmonic. All mentioned unknown parameters are time-dependent. The signal model adopted is a highly nonlinear function of the unknown frequency, so the application of nonlinear estimation is required.

Assuming that the input signal waveform has been uniformly sampled with ratef, and sampling period T, = lf,, the value of t at a discrete time index is given by t , = mT, and the following discrete representation of the signal model can be used:

(4)

h(x,,t,) = vome-2k +E v,, sin(kwmt,+pkm)

( 5 )

k=l

x = [Vo, Ta, U, Vl , . ' . > V k > 9 1 > . ' . , PklT ( 3 )

wm = h(x,, tm) + Em m = 1 , 2 , 3 , . . . M

k=l

where <m, Tam, vom, ...) Vkm, ..., qkm and tm are referred to the time dependent values E,, Tu, V,, ..., V,, w, ql, ..., qlc and t at the discrete time index m.

The number of unknowns n = 2A4 + 3 (the model order) can be reduced by using some simplified models. In the simplest case the model containing only the fun- damental harmonic has the order n = 3 and x = [U, V,, cp1lT. The main advantages of using the simplified mod- els are the reduction of the real-time computation and hardware requirements.

3 Recursive Newton-type algorithm

In [6] the nonrecursive Newton-type algorithm (NTA) for voltage phasor and local system frequency estima- tion is presented. The key relation of the NTA algo- rithm is given by

where i is an iteration index, J,# is referred to as a left pseudoinverse of Jacobian J,, v is an (m . 1) measurement vector, h(x,, t ) is an (m . 1) vector of nonlinear functions determined by the assumed mathematical model of the input signal and m is the number of samples belonging to the data window. It is shown that the number of iterations could be reduced to i,,, = 1. The use of the nonrecursive NTA algorithm necessitates the normal-equation matrix N, = JTJ, in version at each iteration. This task can be a serious

X Z t l = XI + J!rv - h(x,, t)l (6)

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problem, particularly when the model order is high. For example, taking into account the first seven harmonics, the model order becomes n = 2.7 + 3 = 17, so a (17 . 17) matrix N should be inverted. The computer burden, i.e. the real-time computation, can be significantly reduced by changing the algorithm form: instead of applying the nonrecursive one has to apply the recursive form of the NTA algorithm.

Taking into account the theoretical background of recursive least error squares algorithm [8] it is not diffi- cult to derive RNTA algorithm from the nonrecursive NTA algorithm (eqn. 6)

5 , = ~ ~ - 1 + Pmjm(um - MG,-I)) (7 )

where

M

33 = - = ktcos(kwt + pic) (11) dw

k=l

33+M+k - = v, C O S ( k U t + c p k ) k = 1,. . . , M d 9 k

(13) Vector jzpresents the mth row of the Jacobian matrix in terms of the estimates estimated in the previous (m - 1)st estimation. Here residual r, expresses the differ- ence between v, and h(xm-J, i.e. rm = v, - h(x,-,). The inversion of the normal-equation matrix is performing recursively through eqn. 8. The new algorithm design procedure requires the appropriate choice of sampling frequency, the initial guess for xo and the value of for- getting factor A. The initial guess can be calculated by means of the nonrecursive self-tuning least error squares algorithm presented in [7].

One of good features of the RNTA algorithm is that it estimates frequency directly, not as the first deriva- tive of phase angle. By this the algorithm sensitivity to random noise is not so critical as it is known that numerical differentiation is very sensitive to random noise and requires additional postfiltering (e.g. moving average postfiltering).

4

The quality of estimation depends on the preselected forgetting factor A. Through the computer simulation and step parameter changes, the following is con- cluded: the faster convergence can be achieved by set- ting A much lower than 1.0 (e.g. il = O S ) , and if the actual parameter values do not differ considerably from the estimates, better accuracy can be achieved by setting A near 1.0 (e.g. A = 0.99). In practice, il near 1.0 (e.g. A = 0.95) is used as a compromise, resulting in a satisfactory accuracy and relatively slow convergence.

Let rm and R, = C$&-L irpl be the residual error and the sum of the residual error absolute values, belonging to the moving data window with L previous samples,

Iterative tuning of forgetting factor

IEE Proc -Gener Transm Distrrb , Vol 145, No I, January I998

respectively. So the past L residuals are memorised and iteratively used for the R calculation. After numerous computer simulations a new heuristic strategy for itera- tive tuning of forgetting factor A is defined by letting

A, = g(Xo,Ro, RnL) = Xo + (1 - X o ) C R m ’ R o (14) where A, and R, are the forgetting factor and the sum of the residual error absolute values at the mth itera- tion and & and Ro are the tuning parameters defining the tuning function g(&, R,, R,). If the error signal R is small, A will be near 1.0, allowing RNTA algorithm to use more the previous information. If, on the other hand, R is large, A will be near &, allowing the param- eters to be estimated using the most recent data and this will improve the quality of estimation. Tuning parameters & and AIO determine the tuning function (eqn. 14); they control the speed of convergence and hence generally improve the quality of estimation. Tun- ing parameters must be chosen properly in advance and in accordance with the measurement application.

Thus the new RNTA algorithm (eqns. 7-13) is extended with the additional tuning function (eqn. 14). In eqn. 8 instead of A (the constant value), & now appears (the residual error dependent value). The length of the residual error data window L can be determined by the duration of the fundamental period t = 20ms for 50Hz power systems or t = 16.67ms for 60Hz power systems.

The procedure presented is general and can be also used in other applications and problems of recursive algorithms. This means that the ordinary RLS algo- rithm can be extended with the tuning of the forgetting factor too and implemented in the problems of param- eter estimation or system identification. Better results are expected at the beginning of the estimationhdentifi- cation procedure where one usually selects xo = 0 (zero vector) when the expected error is maximal and when A should be much smaller than 1.0. Generally, the better estimates are expected for processes in which a step change of parameter occurs (faults on the elements of power systems, changes in the network topology etc.).

5 Testing RNTA algorithm

The author is currently involved into the project of designing a hardware architecture for real-time fre- quency measurement, so further algorithm testing is viewed from the frequency estimation perspective. Examples of harmonic measurement are given, too.

Modern frequency meters are based on voltage wave processing. Under the assumption that the voltage is strongly prefiltered, 1.he signal model (eqn. 2) can be reduced to the one sinusoid model and in this much simpler form used in the RNTA algorithm for frequency estimation. By this the model order becomes n = 3 and the real-time computation reduces to the minimum.

5. I Computer simulated tests First, a set of static tests are performed. The sampling frequency is selected to bef , = 600Hz and the forget- ting factor A = 0.95 := const. With the initial guess for xo correct selectedicalculated, true values are obtained in the frequency range limited by the frequency f = f J 2 (60012 = 300Hz).

Secondly, the influmce of the forgetting factor on the algorithm convergence is investigated. An input sinusoidal test signal only frequency modulated (step

IEE Proc-Genev. Transm. Distrib., Vol. 145, No. 1, January 1998

frequency change from 50 to 45Hz at t = 0.02s) is processed with AI = 0.50, 4 = 0.99 and A, = g(0.5, 0.06, R). For the A, tuning R is calculated from L = 12 samples. The true and estimated frequencies are depicted in Fig. 1. For AI = 0.50 the fastest convergence and limited accuracy are obtained. For ;12 = 0.99 the convergence is very slow. With A3 a compromise between fast convergence and a good accuracy is achieved. The changes of il and R are depicted in Fig. 2. At the beginning the values of R increase controlling the values of A and the speed of convergence. When the new value (45Hz) is reached, R decreases and A becomes approximately A = 0.95 improving the accuracy in the new state. The preceding test confirms the fact that better estimates will be provided through the improved RNTA algorithm than through the basic RNTA algorithm (RNTA algorithm without the tuning of forgetting factor).

I I I I

0.00 0.05 0.10 0 .I5 0.20 time.s

Fig. 1 Frequency estimates for step change of signal frequency 1 a = 0.50 2 a = 0.99 3 a = g(.) 4 exact frequency

0.0 0.00 0.05 0.10 0.15 0.20

time,s

Fig.2

-R

Changes of il and R a _ _ _

201 I I I

0.00 0.30 1.00 1.50 2.00 time,s

Fig.3 Frequency estimates in dynamic test

In the following dynamic test two sinusoidal test sig- nals with frequencies fi = 50 ~ 5t - 5t2 and f 2 = 50 + 5t + 5t2 are processed with the algorithm settings from the previous tests. The estimated and true frequencies are depicted in Fig. 3, whereas the corresponding errors

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e(t) for thefi case are plotted in Fig. 4. Note that at the end of simulation the rate of frequency changes reaches 25Hz/s! For the f i case the change of A is depicted in Fig. 5 .

n a-

2 1.0- U

.- - a

0.5 5

0.lOr

7th 5th \ \ 3 rd

-

0.08

2 0.05

0.03

0.00

N

- a

0.0 0.5 1.0 1.5 2.0 time,s

Fig. 4 Calculated errors infrequency estimation

0.90

0.0 0.5 1.0 1.5 2.0 time,s

Fig.5 Changes of /t in dynamic test (casefi)

N

w =.

37 L7 57 67 77

Maximum estimation errors in terms of ksNR K s N R ~ ~ B

Fig. 6 (i) 1 = 0.5

(iii) I = 0.9 (iv) 1 = 1.0

a = 0.8

Computer testing with the signals corrupted by ran- dom noise are provided. A sinusoidal 50Hz input test signal with the superimposed additive white zero-mean gaussian random noise is used as an input for the tests. The random noise is selected to obtain a prescribed value of signal-to-noise ratio ( ICsNR), defined as k , = 20 log (A/2o), where A is the magnitude of the signal fundamental harmonics and CJ is the noise standard deviation. The maximum frequency estimation errors against k , for A E (0.5; 0.8; 0.9; 1.0) andf, = 600Hz are shown in Fig. 6. Accuracy increases as the value of A increases. If the forgetting factor is iteratively calcu- lated the sensitivity to random noise depends on the selected tuning function g(.).

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5.2 Distorted signal processing A series of computer tests are provided with distorted input signals. In Fig. 7 a signal including the funda- mental, 3rd, 5th and 7th harmonics is depicted. Until t = 0.02s it is a pure sinusoidal 50Hz signal. At t = 0.02s it is distorted with the higher harmonics and its fre- quency is step changed from 50 to 49Hz. Fig. 8 shows the estimated amplitudes of all harmonic components (all estimates are exact). Fig. 9 depicts true and esti- mated frequencies when the extended model is used, whereas Fig. 10 shows the frequency estimates when the simple one sinusoid voltage model is used. As expected, the unacceptable errors occur in the second case.

time,s Fig.7 Input signal

1st

I I I I

0.00 0.01 0.02 0.03 0.OL 0.05 time,s

Fig. 9 Frequency estimates: extended model

L71 I I I I

0.00 0.01 0.02 0.03 0.OL 0.05 time,s

Fig. 10 Frequency estimates: one sinusoid model

IEE Proc.-Gener. Transm. Distrib., Vol. 145, No. 1, January 1998

In some measurement applications it is important to have an algorithm insensitive to decaying DC compo- nents, which is typical in processing currents during faults for the purpose of distance or overcurrent pro- tection. Fig. 11 presents the frequency estimates of the signal containing decaying DC component. Three methods are used: DFT, RLS (without a DC compo- nent assumed in the signal model) and RNTA (with a DC component assumed in the signal model). The last case yields the best estimates.

537 51.31 0

I I '.

L7.0 L I I J I I I I

time,s 0.02 0.05 0.07 0.10 0.12 0.15 0.17 0.20

Fig. 11 ~ RNTA

Frequency estimates with DC component

DFT RLS

_ _ _ _

Fig. 12 One-line diagram #of the nine-bus test system

5.3 Dynamic simulation of multimachine 60- Hz power system This Section investigates the RNTA algorithm proper- ties in a multimachine 60-Hz power system immediately following a severe generation-load imbalance. Of inter- est here is the generator disconnection which may lead to automatic load shedding. Consider a three-machine 60-Hz power system ['9] depicted in Fig. 12 and suppose that on the transformer between nodes 3 and 9 a short circuit occurred lasting 50ms, so that it is disconnected from the network. Thereby, generator 3 is disconnected as well, and the frequencies of generators 1 and 2 decrease in a quasioscillatory manner. The magnitude of frequency of these oscillations, often referred to as synchronising oscillations, is dependent on the rotor inertia constant H and the electrical location of the generator. After a period of time the synchronising oscillations decay and. frequencies of all remaining gen- erators approach a new system frequency.

Using generator voltage samples v1 and v2, the RNTA algorithm estimated the frequencies of genera- tors l and 2. The rotor inertia constants of the genera- tors are H I = 8.4s and HZ = 4.01s. The dynamic simulation integration step is selected to give voltages with sampling frequency f, = 600Hz. Fig. 13 depicts estimated frequencies of generators 1 and 2. During the fault on the transformer severe voltage drops occur in the whole network (see Fig. 14 which presents esti- mated amplitude of the generator 1 voltage), so the

IEE Proc-Gener. Transm. Disrrib., Vol. 145, No. I , Junuary 1998

estimates are not exact in this period. Later, the estima- tor reaches the true values. The relative errors in esti- mating the frequency and voltage amplitude, excluding the period during the fault, are less than 0.16 and 0.2%, respectively. Fig. 15 presents a typical change of il dur- ing estimation procedure.

59

58 -

57

56

-

-

-

551 I I I , I I I

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 time,s

Fig. 13 Estimatedfrequency of generators 1 and 2

2 ' ' '3hic--- 1.0

0.51 I I I I I I I

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 time,s

Fig. 14 Estimated generator 1 voltage amplitude

0.80 Y 0.75 I I I I I I

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 time,s

Fig. 15 Changes of duringfrequency estimation

The tests presented proved that the new algorithm is suitable for the USA 60-Hz power systems.

It is important which kind of AID converter is imple- mented in the analogue input data digitisation. The number of bits determines the accuracy of data conver- sion, i.e. the quantisation error. To get an insight into these errors the following test is provided. The input sequence of voltage samples is processed in such a way to round off every voltage sample value to the 2nd, 3rd, 5th, 6th and 7th digit, respectively. The new volt- age sequences generated in the described way are than used as an input to the RNTA algorithm. Based on estimated and true values, the maximal relative errors 6 in terms of number of significant digits (Nd) in the fre- quency estimation are calculated. They are shown in Table 1.

Table 1: Maximal relative errors -~ ~ ~

Nd 1 2 3 4 5. 6 7

6( Yo ) 0.67 0.67 0.042 0.025 0.016

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5.4 La bora tory testing In these tests data records are obtained from a function generator using an eight-bit digital oscilloscope with sampling frequency f, = 500Hz. The true frequency is measured using an HP 5234L counter. The recorded data are processed offline on a personal computer. The results of the test in which the input signal frequency is step changed from 50 to 47Hz in steps of 1Hz are depicted in Fig. 16. Better results are obtained by using the improved RNTA algorithm. The values of the iter- atively tuned forgetting factor A = g(0.5, 0.06, R) are depicted in Fig. 17.

N I x c W 2 U

E! L

0.0 0.5 1.0 1.5 2.0 time,s

Fig. 16

-.__ exact frequency

Estimated frequency in laboratory testing ~ a = g o a = 0.975 - - -

1.0r

0.0 0.5 1.0 1.5 2.0 time,s

Fig. 17 Changes of /1 in laboratory testing

5.5 Real-life conditions tests To prepare the studies necessary to determine the con- ditions and possibilities of parallel operation of the Romanian Electric Power System (RENEL) within the European Interconnection UCPTE, preliminary trial parallel operation of the RENEL with the electric power systems of Albania, Greece and the former Yugoslav republics was performed on 28th October, 1993.

50.00-

L8.85! I I I 1 I I

0 20 LO 60 80 100 120 time,s

Fig. 18 Frequency estimated in real-lge test

In the testing voltage samples are acquired using the data acquisition system (12-bit A/D converter, sam- pling frequency 1600Hz) installed at Faculty of Electri- cal Engineering in Belgrade. Data are recorded and

20

then processed offline. The recorded data have been first prefiltered by means of the 4th-order lowpass But- terworth filter and further processed by the RNTA algorithm. The tuning function is selected to be A = g(0.9, 0.1, R). The frequency estimated in the test of 200 MW generating unit disconnection in HPP Bajina Basta (Yugoslavia, Serbia) with the secondary load-fre- quency control switched off is depicted in Fig. 18. In the 17th second the 200MW generating unit is discon- nected, whereas in the 21st second a portion of custom- ers in former Yugoslavia (approximately 60 MW) is disconnected, as shown in Fig. 18 as a temporarily fre- quency increase (time = 21-30 seconds).

The system frequency is also measured from the volt- age signal by using HP 3457A multimeter. The input voltage is prefiltered by the analogue lowpass filter with cut-off frequency 150Hz. The maximum difference between the respective values was less than 10-3Hz. Such an accuracy satisfies the frequency relaying requirements and some other measurement applications in power systems. The frequency estimates obtained by the new algorithm without tuning the forgetting factor are similar to the estimates depicted in Fig. 18, but they are not sharp enough, i.e. they are corrupted by noise error.

6 Conclusion

A new recursive Newton-type algorithm has been pre- sented developed from the nonrecursive Newton-type algorithm [6] and improved by designing a new strategy of choosing the forgetting factor as a function of the sum of the residual error absolute values. This approach is general and could be utilised in some other recursive algorithms which include the forgetting of old data (forgetting factor). Through the extensive algo- rithm testing it is shown that the new algorithm can be successfully applied as a reliable measurement tool in power system, which is proved through computer-simu- lated, laboratory and full-scale real-life tests.

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IEE Proc-Gener. Trunsni. Distrib., Vol. 145, No. I , January 1998