STATISTICAL ANALYSIS OF LOAD CONDITIONS IN THE SIBERIAN POWER SYSTEMS

R. I. BORISOV, A. G. MILYUSHKIN

(Presented by the scientific seminar of the Department of Electric Systems and Networks)

A significant group of problems in power engineering is based on the magnitude and character of electrical loads, which, owing to the multifactor nature of the influences, have a random, stochastic character. All power installations, being intended for repeated use under similar conditions, are never, in the course of operation, under identical conditions, since consumers of electric power are switched on randomly, depending on a number of causes and factors, including the technological features of production, the organization of the population’s domestic and work schedules, natural illumination, ambient temperature, etc. Therefore, in order to avoid unjustified decisions in calculation practice, it is necessary to apply scientific probabilistic-statistical methods of solution. “However, probabilistic calculation methods are effective and yield reliable data when they are based not on inferences and all kinds of assumptions, but on reliable statistical data and regularities revealed by the method of mathematical statistics” [1]. The operating conditions of existing power facilities, as a rule, can be determined sufficiently accurately and simply on the basis of operational measurements, which are, in essence, discrete hourly records of the values of active and reactive loads in the daily reports of the dispatching services of unified power systems.

The accumulated data are generalized indicators of the functioning of power systems and contain a great deal of information, the subsequent use of which, after appropriate processing of the statistical material, can lead to a noticeable improvement in the quality of the results obtained within the limits of the attainable accuracy of calculation determined by the methods applied. At the same time, the questions of collecting, storing, and processing information in order to obtain estimates of parameters of the required quality with an optimal amount of initial statistical material require their own solution.

In the present work, some of these questions are investigated using the available statistical data on the loads of the Siberian power systems, with the use of the methodology set forth in the literature [2]. The source material consisted of data on the hourly active and reactive loads of power systems A and B. The statistics were processed for the period 1968–1972.

1. Distribution laws of the quantities under study

The regularities in the grouping of the frequencies with which the quantities \(P\) and \(Q\) fall into individual intervals of a distribution series make it possible to determine probabilistic characteristics in the form of the mathematical expectation \(m_1\), the variance \(m_2\), and the standard deviation \(\sigma\) of a random variable.

\[ m_1=a=\sum_{1}^{N} P_i \rho_i, \tag{1} \]

\[ m_2=\sum_{1}^{N} (P_i-a)^2 \rho_i, \tag{2} \]

\[ \sigma=\sqrt{m_2}. \tag{3} \]

However, these characteristics are point characteristics and do not determine, as a whole, the possible changes in random variables. The most universal characteristic of a random variable is the distribution law, i.e., a mathematical expression establishing the relationship between the possible values of the random variable and their corresponding probabilities. In the general case, the distribution law of a sum does not coincide with the distribution law of its summands [3], and deviations may occur from the normal distribution law axiomatically accepted by some authors, which is described by the function

\[ F(x)=\frac{1}{\sigma_x\sqrt{2\pi}}\int e^{-\frac{(x_i-a)^2}{2\sigma_x}}, \tag{4} \]

where \(x_i\) is the value of the variable;
\(a\) is the mathematical expectation;
\(\sigma_x\) is the standard deviation.

Therefore, in the study, in addition to the basic characteristics sufficient for the normal law, it is necessary to determine the skewness \(A\) and the excess \(\mathcal{E}\):

\[ A=\frac{m_3}{\sigma^3}, \tag{5} \]

where

\[ m_3=\sum_{1}^{N}(x_i-a)^3\cdot\rho_i; \]

\[ \mathcal{E}=\frac{m_4}{\sigma^4}-3, \tag{6} \]

where

\[ m_4=\sum_{1}^{N}(x_i-a)^4\cdot\rho. \]

From the values of the last two quantities one can draw preliminary conclusions about the degree of approximation of the distribution under study to the normal distribution. In the course of the calculation, the question is resolved of determining the optimal interval step, i.e., the number of classes in the distribution series,

since, in order to obtain reliable estimates of the numerical characteristics of operating parameters, a large amount of statistical material is required, which makes it necessary to group it by classes with subsequent averaging for the convenience of further processing. The question of which class should include values equal to boundary values is not fundamental in this case, since the discretization of a continuous process, such as electric-power consumption, is, as a rule, associated with errors.

Fig. 1. 1 — empirical distribution of the weekly sample statistic for “\(P\)”;
2 — empirical distribution of the general population.

An illustration of the results obtained may be Figs. 1 and 2, which contain the distributions of \(P\) and \(Q\), whose parameters characterize the electric-power consumption regime of system A for 1971.

Testing the hypothesis of normality of the distribution of the quantities under study for distributions characterized by the largest values of skewness and kurtosis, using Pearson’s criterion, showed that the hypothesis of a normal distribution can be accepted. Consequently, variants with the best indicators of skewness and kurtosis are clearly consistent with the law of normal distribution. It should be noted that the maximum values of skewness and kurtosis characterize the weekly random samples. This indicates a greater influence of individual errors in the preparation of information, or of isolated surges that occurred, on the integral characteristics of small samples [4]. Analysis of statistical samples of different sizes showed that the minimum statistical random sample with a representativeness error \(\Delta\) of no more than three percent may be taken to be the weekly sample. The actual values of the sample means and root-mean-square deviations are determined within the limits

\[ \bar{x} \pm O_{\bar{x}} \quad \text{and} \quad \sigma_x \pm O_\sigma . \tag{7} \]

In this case the errors in determining them are found from the expressions

\[ O_{\bar{x}} = t_a \frac{\sigma_x}{\sqrt{m}} \quad \text{and} \quad O_\sigma = t_a \frac{\sigma_x}{\sqrt{2m}}, \tag{8} \]

where \(m\) is the sample size;

\(t_a\) is the normalized deviation of the normal distribution. For \(t_a\) equal to two, the probability of error is no more than 5%; for \(t_a\) equal to three, no more than 1%.

The sufficient convergence of the distribution laws of the twenty-four-hour sample and the annual population can be judged from the results in Figs. 1 and 2.

Fig. 2. 1 — empirical distribution of the weekly sample statistic for “\(Q\)”; 2 — empirical distribution of the general population.

As a first approximation in determining the interval step, one may use the value found from the formula

\[ K \simeq \frac{x_{\max} - x_{\min}}{1 - 3.2 \lg m}, \tag{9} \]

where \(m\) is the sample size;

\(x_{\max}, x_{\min}\) are the largest and smallest values in the distribution series of the quantity under study.

2. Correlation Analysis

Mathematically, the process of load variation of any power-system facility is most fully reflected in the concept of a nonstationary random-

of a random process, all properties of which are characterized by an \(R\)-dimensional distribution function. The process of electricity consumption is characterized by the value of apparent power, which is two-dimensional,

\[ S = P + jQ . \tag{10} \]

However, in the general case it is impossible to characterize the distribution law of “\(S\)” by the distribution laws of “\(P\)” and “\(Q\)”; it is necessary to know the dependence between the quantities “\(P\)” and “\(Q\).” Owing to the random nature of the process, the dependence cannot be functional, and in some cases it may not exist at all; therefore the problem is posed of identifying the presence of dependence, its character, and its closeness, i.e., of establishing the correlation between two concomitant processes of consumption of active and reactive energy in the system. This dependence can be characterized by means of conditional distribution laws [4], i.e., empirical regressions of \(P\) on \(Q\) and \(Q\) on \(P\), from which the form of the dependence of the quantities under study can be established.

From the results obtained it follows that there is a positive correlation dependence, fairly close to linear. This is indicated by the correlation coefficients, the magnitude of which for all statistics varies within the range from 0.5 to 0.9. Therefore, the obtained dependences are approximated by a linear equation, by the method of least squares [6],

\[ \overline{Q} = m_q + r_{pq}\frac{\sigma_q}{\sigma_p}(P - m_p) \tag{11} \]

in the form

\[ \overline{Q} = kP + C, \tag{12} \]

where

\[ k = r_{pq}\frac{\sigma_q}{\sigma_p} = \operatorname{tg}\varphi;\quad C = m_q - \operatorname{tg}\varphi \cdot m_p . \tag{13} \]

The obtained dependences can be used in operating-condition calculations for the approximate estimation of one quantity through another. Estimates of increased accuracy, proceeding from the normality of the distribution of the quantity \(Q\), can be found from the formula

\[ Q = \overline{Q} + t_a \cdot \sigma_q, \tag{14} \]

where \(t_a\) is the normalized deviation of the normal distribution. The value \(t_a\) depends on the nature of the calculations being performed and, accordingly, on the requirements imposed on the reliability of the estimates used in calculating the parameters.

3. Dynamics of the Parameters

The adoption of optimal decisions in power-engineering problems requires, wherever possible, accurate forecasts of electricity-consumption conditions. Forecasting load conditions on the basis of a detailed analysis of the structure of consumers in electric-power systems, using regulated electricity-consumption standards for individual groups and subsequent synthesis, is associated with great difficulties. Therefore it is necessary to resort to an approximate forecast by extrapolating the dynamics of the principal parameters characterizing the electricity-consumption conditions over a number of past years. The forecast, in turn, depends on the stability of the con—

Fig. 3. \(m_p, m_q\)—mathematical expectations of active and reactive loads; \(W_p\)—electric-power consumption by systems.

Fig. 4. \(P_m\)—maximum value of active load.

averaged parameters describing the process. As can be seen from Figs. 3 and 4, the dynamics of the parameters are monotonic and tend toward a steady increase for both systems, although the rates of increase for systems A and B are different, which indicates the need for similar studies for specific power systems and facilities. The coefficients of variation $\gamma$ show that system A has a denser load curve than system B; the dynamics of the value $\sigma$ indicate a tendency toward a further flattening of the load curves of systems A and B.

Fig. 5.

Of interest is the change in intra-annual modes of electric power consumption, as well as the tendencies of individual indicators characterizing electric power consumption in corresponding periods over a number of years.

The increase in electric power consumption in corresponding periods of 1969—1972 for system B is characterized by more stable rates of growth in the spring-summer period. The dynamics of average monthly loads during the year reflect the influence of stable regular seasonal changes in load and are close to a sinusoid (shifted to the right relative to the starting point); however, the intra-annual increase in electric power consumption is characterized by deviations in individual months from the annual average, which is probably connected with the simultaneous introduction of new significant loads or with significant temperature fluctuations in the given region.

Conclusions

1. It has been established that the distribution of loads characterizing electric power consumption in systems A and B, in time intervals from a week to a year, agrees with the normal distribution and can be described by a law with parameters found from experiment.

1. A weekly statistical sample from the monthly population reliably (with a small error of representativeness) characterizes the monthly cycle of electricity consumption, and a twelve-week sample correspondingly characterizes the annual cycle.

2. The dependence between the magnitudes of active and reactive loads has a high degree of correlation and can be described with sufficient accuracy by a linear functional relationship.

The dynamics of the principal parameters describing electricity-consumption cycles are characterized by steady growth and may serve as a basis for forecasts for the near term.

References

1. I. A. Syromyatnikov. Directions of scientific research and tasks for their implementation in the application of probabilistic and statistical methods in power engineering. Proceedings of the conference. Kiev, 1963.

2. B. M. Shchigolev. Mathematical Processing of Observations. Moscow, “Nauka,” 1969.

3. H. Cramer. Random Variables and Probability Distributions. IL, 1947.

4. N. V. Smirnov, I. V. Dunin-Barkovsky. A Course in Probability Theory and Mathematical Statistics. Moscow, “Nauka,” 1969.

5. A. G. Ivakhnenko, V. G. Lapa. Prediction of Random Processes. Kiev, “Naukova dumka,” 1971.

6. E. S. Wentzel. Probability Theory. Moscow, “Nauka,” 1969.