ON THE MEASUREMENT OF MEAN VALUES OF ELECTRICAL QUANTITIES

V. A. ZAKHARENKO, A. V. SHMOILOV

(Presented by the scientific seminar of the Department of Electric Power Stations)

In evaluating the operation of complex systems, in particular power systems, in constructing and analyzing the functioning of automatic control devices, and in studying mathematical models of real physical processes, it is necessary to use the characteristics of random processes.

In such cases, the probabilistic approach to the problems posed supplements or replaces the deterministic approach, which is associated with overcoming serious mathematical difficulties. Therefore, the development and investigation of methods and devices for measuring the parameters of random processes is an important task, to which much work has been devoted in recent years [1–9].

The probabilistic characteristic most often used for practical purposes is the mathematical expectation. The mathematical expectation is used both as an independent parameter of a random process and as a necessary mathematical operation in determining the mean power, variance, and correlation function, which are used mainly for the analysis of nondeterministic quantities.

In estimating the mathematical expectation of stationary ergodic processes, two methods are used. Either one and the same process is considered at different instants of time in order to accumulate a set of observed quantities, or a set of random quantities is observed at one and the same instant of time. The ergodic theorem of mathematical statistics states that for stationary processes the two methods are equivalent. That is, for a stationary ergodic process, the probabilistic characteristics can be obtained with probability close to unity as the result of some time averaging of one realization \(x(t)\) of sufficiently long (theoretically infinite) duration [6].

The mathematical expectation characterizes the mean values of a random process. Thus, for a stationary ergodic random process \(X(t)\), the mathematical expectation \(M[X(t)]\) is determined as

\[ M[X(t)] \simeq \frac{1}{T}\int_{0}^{T} x(t)\,dt, \tag{1} \]

where

\(T\) is the averaging period, \(x(t)\) is a realization of the process.

Algorithm (1) can be implemented in hardware by devices of analog or discrete type \([1 \div 3]\). In discrete analysis, part of the information is lost. In addition, this leads to statistical errors that are difficult to estimate [2]. In this connection, the analog method of measurement is the most preferable.

The structure of an analog device (Fig. 1) that determines the mean value of a random process follows directly from expression (1). For this purpose it is necessary to feed the random process \(X(t)\), at a certain scale, to an integrator, integrate it, divide by the averaging time, and record the result.

Fig. 1. Block diagram of the averaging device: 1 — input block; 2 — integrator; 3 — division device; 4 — indicating device

The specifics of the analysis of statistical data for electric-power-engineering problems require averaging over long periods of time, and this leads to the problem of constructing a high-quality integrator for long time intervals. Integrators constructed on the basis of operational amplifiers of modern analog computers do not satisfy these requirements. A simple hardware implementation of the division operation is also associated with technical difficulties.

Taking the foregoing into account, we propose the following algorithm for computing the mathematical expectation:

\[ M[X_0(t+\Delta t)] = M[X_0(t)] + \frac{M[X_t(\Delta t)] - M[X_0(t)]}{n}, \tag{2} \]

where

\(M[X_0(t+\Delta t)]\) — the mathematical expectation of a stationary ergodic random process over an averaging period equal to \(t+\Delta t\);

\(M[X_0(t)]\) — the same over an averaging period equal to \(t\);

\(M[X_t(\Delta t)]\) — the mathematical expectation over the averaging period \(\Delta t\), beginning at time \(t\);

\[ n=\frac{t+\Delta t}{\Delta t} \]

— the number of intervals.

Algorithm (2), in implementation, makes it possible either to use existing integrators or to substantially increase the integration time. Division in measuring \(M[X_t(\Delta t)]\) is replaced by a scale factor determined by the quantity \(\Delta t\). Division by \(n\), in comparison with division by \(T\) in algorithm (1), is performed quite simply.

At present, using this algorithm, we are working out the structure and individual elements of the instrument. In particular, a circuit has been developed for high-quality integration over half an hour, as well as a device for reading the measurement result.

Conclusions

1. The problems of analyzing the operation of complex power systems raise the tasks of constructing instruments for statistical analysis.

2. One method is proposed for constructing an instrument for determining the mean of a random process.

References

1. G. A. Ball. Instrumental Correlation Analysis of Random Processes. “Energiya,” 1968.

2. J. Bendat, A. Piersol. Measurement and Analysis of Random Processes. “Mir,” 1971.

3. G. Ya. Mirskii. Instrumental Determination of the Characteristics of Random Processes. “Energiya,” 1967.

4. I. M. Markovich. Modes of Electric Power Systems. “Energiya,” 1969.

5. F. Lange. Correlation Electronics. Sudpromgiz, 1963.

6. V. S. Pugachev. Theory of Random Functions and Its Application to Problems of Automatic Control. Fizmatgiz, 1962.

7. V. F. Nesteruk, A. N. Porfir’eva. On one method for determining the correlation function of normal random processes. Izvestiya VUZov, “Priborostroenie,” 1962, No. 6.

8. V. I. Chaikovskii. Methods for the experimental determination of correlation functions. Izvestiya VUZov, “Radiotekhnika,” 1960, No. 5.

9. S. Karlin. Foundations of the Theory of Random Processes. “Mir,” 1971.