Modal damping factor estimation for power systems in normal operation
STEVEN MURETIC2,ED PALMER1, PETER O’SHEA1, GERARD LEDWICH1 & MAJID AL DABBAGH2
1School of Electrical & Electronic Systems Engineering,
Queensland University of Technology,
GPO Box 2434, Brisbane. 4001.
2School of Electrical & Computer Engineering,
RMIT,
GPO Box 2476V, Melbourne, 3001.
AUSTRALIA.
www.eese.bee.qut.edu.au/people/osheap/
Abstract: This paper discusses the estimation of damping factors for post-disturbance modal oscillations in large interconnected power systems. The proposed estimation method is based on variance measurements at a suitable measurement point within the power system under normal operation. The method relies on the fact that a power system is quasi-continuously excited by “disturbances” caused by the switching in and out of electrical appliances/motors. These disturbances, though random, can be reliably characterized statistically (as long as the number of separate customers is large). It will be seen in the paper that this statistical characterization can be used to enable reliable damping factor estimation to occur under normal power system operation. The estimation is based on measurements of power or angle at appropriate positions within the power system. The measured variance can thus be used to deduce estimates of modal damping factors. The proposed method is very computationally efficient and is particularly effective for registering sudden deteriorations in damping.
Keywords: Parameter estimation, power system modelling, power system monitoring.
1. Introduction
Thispaper deals with the problem of estimating the damping factors of inter-area modes in large interconnected power systems.These power systems are assumed to be in “normal operation” and as such, are assumed to be excited by multiple quasi-continuous random disturbances [1].Such disturbances arise naturally as a result of load changes in the form of connections and disconnections. Once excited, each of the disturbances is then damped according to the modal resonances of the power system. This type of scenario can be well modeled with a single continuous random background noise exciting a filterwhose resonances are characteristic of the power system.
The excitation noise for real power systems has been found from empirical observations to have a “1/f” characteristic; that is, the excitation noise can be modeled well as whitenoise that has passed through an integrator. This model is illustrated in Fig. 1. (Note that because the measurements are made in the discrete-time domain the model and subsequent analysis are all presented in sampled form). As indicated in Fig. 1, if the output of the power system is differentiated, the resulting signal, x(n), can be considered to have been obtained from a white noise based excitation of the power system filter, h(n). It is recommended that the measured output, y(n), be taken as the angle of the generator cluster at the measurement point [1], with the steady state (i.e. 50Hz/60Hz) angle component removed.
2. Damping factor estimation in a power system using the signal variance
As per the model in Figure 2, it is assumed that the differentiated output of a quasi-continuously excited power system signal can be modelled by:
, (2.1)
where is stationary white Gaussian noise, * denotes convolution, and is the impulse response of the power system filter. Assume for the present that the power system contains only one real mode.
Then h(n) is the impulse response of a second order filter and is given by:
.(2.2)
The poles of the filter occur at where , , and are the resonant angular frequency, initial phase and damping factor respectively for the mode. The average value for the variance of is [2]:
var{x(n)}=Av{|x(n)|2}=Av{|w(n)*h(n)|2}
=k.Av{|h(n)|}2,(2.3)
where Av{.} denotes the (time) average of {.}, and k=Av{|w(n)|2}. The variance is also proportional to , where is the squared magnitude discrete-time Fourier Transform (DTFT) of , and is the squared magnitude DTFT of . From (2.2) and (2.3) the average power of x(n) is:
The second term in the last line of (2.4a) is small compared with the first term, provided that there are many oscillations of between and . i.e. provided that is not very close to 0. (This second term may actually be interpreted as “interference” between the positive and negative frequency components. The further is from 0, the less the interference is [3], [4]). If this second term is neglected then (2.4a) becomes:
(2.4b)
(2.4b) indicates a simple relationship between , k, N and the variance of the data record. In a practical power system measurement scenario, N (the number of samples in the measured data record) will be known. Also, if the power system is in stable operation for some time with a constant level of input there are various techniques one can perform to get a reliable estimate of k ,(the variance of the white noise input). One could, for example, obtain this estimate by applying Prony’s method to the autocorrelation of x(n) [1]. If a reliable estimate of k is available, then (2.4b) can be used to solve for . In this way, the damping factor can be estimated from variance measurements at the output of a power system.
Note that because of Parseval’s theorem, the variance measurements can be made in either the time domain or the frequency domain. The advantage to doing the measurements in the frequency domain is that the technique can then be applied to individual modes within a multi-modal system (provided all the modes are spectrally well separated). The computation of damping factors from variance measurements has the advantage of being very computationally efficient. Some practical issues in computing the damping factor are explored in the ensuing paragraphs.
Using (2.4b) to determine requires one to determine the variance of x(n). One obvious way to estimate the variance is to simply average the squared magnitude of for all available samples. That is, it can be estimated with the formula:
(2.5)
Determination of the variance of x(n) according to equation (2.5) and the damping factor according to (2.4) is now investigated via simulations. Modal damping factor estimation simulations have been performed using equation (2.4b) in conjunction with a Newton algorithm to determine . The value of k was assumed known and equal to 1. Table 1 summarizes the results of the simulations for the modal damping factor estimation with = 0.01 s-1 for different values of . The impulse response of the filter was given by equation (2.2) with being set to /4 and being set to 0.0 rad/s. The sampling rate was assumed (without loss of generality) to be unity, as per the model in (2.1). 200 independent trials were performed. The results show that the number of samples required to give a damping estimate with less than 10% error is somewhere between 1,024 and 2,048 samples. Tables 2 and 3 show the results of simulations corresponding respectively to damping values of = 0.1 s-1and = 0.001 s-1.
Two trends can be observed in the tables as the damping factor varies. Firstly, lower standard deviation estimates are obtained for a given value of N when there is heavy damping. Secondly, there is a bias in the damping factor estimate due to the interference between positive and negative frequency components, with the bias being greatest for heavy damping. The increase of bias with damping is to be expected because heavily damped modes have a broader spectrum which creates greater “interference” between positive and negative frequency components. For the kinds of modal frequencies and sampling rates employed in practice, the bias tends to be relatively small in practice. The lower standard deviation estimates are obtained for heavy damping scenarios because the signal variance estimates obtained via (2.5) are increasingly unreliable as the damping factor decreases. That is, although finding the signal variance according to equation (2.5) is straightforward and simple, it is inaccurate if there is significant correlation between samples. As explained in [5], it is necessary to explicitly account for inter-sample correlation to obtain accurate variance estimators.
A discussion on how to account for this correlation is provided in [6]. Note that the emphasis in this section and throughout the paper is on simulated rather than real data. This is to provide a check on the accuracy of the estimation process, something which is not possible with measured data where the exact parameters are not known.
No. ofSamples
in Data
Record / Damping Factor of Impulse
Response
(s-1) / Mean
Damping Estimate
over 200
Trials (s-1) / % Error of
Mean
Damping
Estimate / Standard Deviation
128 / 0.01 / 0.0234 / 134 / 0.0164
256 / 0.01 / 0.0161 / 61 / 0.0091
512 / 0.01 / 0.0133 / 33 / 0.0057
1,024 / 0.01 / 0.0112 / 12 / 0.0035
2,048 / 0.01 / 0.0106 / 6 / 0.0025
4,096 / 0.01 / 0.0102 / 2 / 0.0016
8,192 / 0.01 / 0.0102 / 2 / 0.0011
Table 1: Modal damping estimates obtained with the Signal Variance method. = 0.01 s-1.
No. ofSamples
in Data
Record / Damping
Factor of
Impulse
Response (s-1) / Mean
Damping
Estimate
(s-1) / Error of Mean Damping
Estimate (%) / Standard Deviation
128 / 0.1 / 0.0968 / 4 / 0.0292
256 / 0.1 / 0.0954 / 5 / 0.0199
512 / 0.1 / 0.0930 / 7 / 0.0139
1,024 / 0.1 / 0.0909 / 9 / 0.0100
2,048 / 0.1 / 0.0904 / 10 / 0.0074
4,096 / 0.1 / 0.0893 / 11 / 0.0049
8,192 / 0.1 / 0.0899 / 10 / 0.0038
Table 2: Modal damping factor estimates obtained with the Signal Variance technique. = 0.1 s-1.
Number of Samples inData Record / Damping Factor of Impulse Response (s-1) / Mean
Damping
Estimate (s-1) / %Error of
Mean
Damping Estimate / Standard Deviation
128 / 0.001 / 0.0153 / 1430 / 0.0127
256 / 0.001 / 0.0090 / 800 / 0.0075
512 / 0.001 / 0.0044 / 340 / 0.0034
1,024 / 0.001 / 0.0026 / 160 / 0.0020
2,048 / 0.001 / 0.0017 / 70 / 0.0011
4,096 / 0.001 / 0.0014 / 40 / 0.0006
8,192 / 0.001 / 0.0012 / 20 / 0.0004
Table 3: Modal damping factor estimates obtained with the signal variance technique. = 0.001 s-1.
3. Issues associated with the signal variance estimation method
There are a number of issues with the signal variance method for the damping factor estimation method discussed in the previous section. The first concern deals with the fact that the method assumes knowledge of the variance of in equation (2.4b). i.e. it assumes a knowledge of . Fortunately, in most power systems quasi-stationary operation occurs over a protracted period of time. One can use the output from this type of “normal operation” of the power system operation to derive estimates of the modal damping factors, frequencies, amplitudes, phases and input “noise” level. The process is described in [1].
The second concern with the signal variance method is that equation (2.4b) is a non-linear equation which requires a numerical technique for the computation of . In the simulations Newton’s Method was used to estimate the damping factor once the variance had been determined. This, however, requires making an initial guess/estimate of the damping factor. The concern is how this would work in a real power system scenario where the damping factor is unknown. Fortunately, the right hand side of (2.4b) is a monotonic function and so convergence tends to occur fairly readily in practice. For example, with N=1,000, =/4 and 0.1 s-1, initial estimates of the damping factor for the Newton’s Method technique between the range of 0.0035 s-1 to 4.5 s-1 resulted in convergence.
The third concern with the techniques in the previous section is as already stated, the failure to account for the correlation between samples in the variance estimation. One way to improve these results is to “whiten” (de-correlate) the “colored” signal. This is done because the best estimators for variance are obtained by “whitening” the observation before trying to estimate the variance [5]. The equation in (2.4b) is then adjusted to take account of the whitening process. Details, including supporting simulations, are provided in [6].
4. Conclusion
As discussed previously, assuming quasi-stationary behaviour in a power system, one can use the techniques in [1] to derive estimates of the modal damping factors, frequencies, amplitudes, phases and input disturbance or “noise” levels. The techniques described in [1], however, require long observation times to yield accurate estimates, particularly in the case of damping factor estimates. Typically, 3 hours of data would be required to obtain damping factor estimates which are accurate to within 0.02s-1. One thing which is critically required in a power system, however, is the rapid detection of any sudden and dangerous deterioration of modal damping. For this type of problem the exact estimation of the damping factor is not crucial; what is important is to detect whether or not a dangerous deterioration has occurred. For this purpose the variance based measures described in this paper have significant potential. That is, if the damping of one of the modes in a power system suddenly reduces (or even changes to negative damping) the variance of the power system output would increase significantly. Moreover, the more serious this damping deterioration, the more quickly it could be detected from the variance record. One could typically detect that a dangerous damping (i.e. < 0.01 s-1 or less) existed within about 25 seconds, assuming there was 5Hz sampling. Remedial actions could then be put into place quickly. The variance based measures of damping described in this paper therefore have significant potential for disturbance monitoring.
References
[1] Ledwich, G., and Palmer, E., “Modal Estimates from normal operation of Power Systems”, Power Engineering Society Winter Meeting,, IEEE, Volume 2, pp.1527-1531 Vol. 2, 2000.
[2] P. Z. Peebles, Probability, random variables, and random signal principles, 4th ed. ed. New York, NY : McGraw Hill, 2000.
[3] P. O'Shea, "The use of sliding spectral windows for parameter estimation in power system disturbance monitoring," IEEE Transactions on Power Systems, vol. 15, 2000, pp. 1261-1267.
[4] P. O'Shea, "A high resolution spectral analysis algorithm for power system disturbance monitoring," IEEE Transactions on Power Systems, vol. 17, 2002, pp.676-680.
[5] S. Kay, “Modern Spectrum Analysis”, Prentice-Hall, 1988.
[6] S. Muretic, “Modal parameter estimation for large interconnected power systems”, Master of Engineering Dissertation, RMIT, 2003.