Sometimes when planning for future operations in an electric power system, it proves useful to have an estimate of the variance of such items as the production cost and the potential outputs of individual generating units. In the past, decisions have been based on only the mean values of such quantities. However in such contexts as short-term buying and selling of power, short-term fuel scheduling, and short-term financial planning, mean values alone may provide too little insight for effective strategy choices.
While the theory of expected values in power system operations is fairly well established, the theory of power system variances is quite new. Most of the literature has appeared only since 1988. During this time three main approaches to power system variances have emerged.
One approach is a deterministic treatment in which Load Duration Curves, traditionally used for computing expected values, are manipulated to yield variances as well. A shortcoming of this approach is that unit outages are assumed to cover the entire length of the simulation period. This causes variances to be over-estimated when the simulation period is fairly long.
A second approach, based on a probabilistic interpretation of the Load Duration Curve, models variances under the assumption of independent hourly unit outages. But this causes variances to be under-estimated when the simulation period is short. Another problem is that the probabilistic loads model creates a systematic loads variance which inflates the computed values in an unjustified way.
A third approach, intermediate between the first two, provides an explicit model of unit outage durations by modeling generating units as Markov processes with specific rates of failure and repair. A major shortcoming of this approach is that its very detailed computer model is too slow for effective implementation in the field.
The systematic loads variance of the second approach is of special interest. One may assume that the total period load energy (the sum of the hourly loads) is a predetermined constant. Then the probabilistic model leads to the idea that this constant value is estimated as the sum of so many random samples of the period's Load Duration Curve. It is shown that the sum-of-samples estimator for the period energy has a binomial distribution, and that its expected value is always equal to the true period energy. However the variance of the estimator is inversely proportional to the number of samples (i.e., load hours), so that the ratio of the estimator's standard deviation in relation to the mean declines as the square root of the period length.
This observation enables one to formulate criteria based on confidence intervals for the minimum length of a simulation period. It is known that as the number of hours grows, the binomial load-sample estimator distribution approaches a normal distribution. This means that the normal table tells one how many hours the period must contain, in order that the load-sample variance may reach an acceptable minimum. Thus it can be insured that the systematic loads variance of the probabilistic approach reduces to a small enough value (in relation to the period energy) that the variances computed with these methods have true objective validity.
The probabilistic approach also raises some interesting questions concerning the statistical nature of power system loads. Data from sets of weekday load curves for a power system in Australia are converted into plots of the mean, variance, and autocovariance of the loads, each as a function of the time of day. Several individual days are also selected for Fourier analysis. Both sets of plots, which are also interesting in their own right, afford some insight into the nature of load as a stochastic process. Further analysis along these lines could even lead to improved short-term load forecasting methodology.
None of the approaches to variance surveyed above offers a useful tool for power system variances in the short-term. The Markov model gives the best treatment of unit outage durations, but it is too slow for this application. For this reason a simplified, streamlined algorithm is developed in which unit outages are modeled by assigning units to load groups of specific length for computation of individual sub-period variances. The full-period variance becomes the sum of these group variances. Preliminary studies indicate that this new streamlined method can compute power system variances very quickly, and with good accuracy.
Finally there is some exploration of expected values and variances in the face of randomized rather than determinate loads (a more realistic picture of the planning-forecasting situation). Preliminary analysis suggests (perhaps surprisingly) that expected values are much more strongly impacted than variances for this case. Methods to deal with expected values for random loads are then developed.