### ANALYSIS OF THE LOAD CURRENT OF TRACTION SUBSTATION AS RANDOM PROCESS

O. V. Klyuev, E. D. Khmelnitsky, E. G. Nosakov, D. A. Kolesnik

#### Abstract

In this paper, a general approach to the analysis of non-stationary random process (RP) with hidden periodicities, which are determined by the regime of electrical consumption at industrial enterprises is described. It is proposed, as a mathematical model, to accept the representation of the investigated RP as the sum of a deterministic function with a variable mathematical expectation and a stationary normal process with zero mathematical expectation. Subtracting the variable mathematical expectation (ME) from an unsteady RP, we obtain a centered RP, the average value of which with a fairly representative sampling approaches zero and only a deterministic function remains.

Because of the periodicity studied RP, the mathematical expectation can be represented in the form of a Fourier series, the coefficients of which are determined by the least squares method under the condition of minimizing the sum of the ordinates of the centered random process. All calculations with the aim of simplification are performed using matrix algebra.

The question regarding the number of members of the Fourier series, which must be preserved in the expansion, is solved by checking the significance of the coefficient estimates by the Student test with the previous determination of the variance of the coefficient estimates. Testing the hypothesis of equality of variance at different points in time is calculated based on the Cochran criterion.

Based on the proposed methodology, the processing is performed of the general substation current of the traction substation of the Pridneprovska railway. The recording of readings was carried out for 20 days with an interval of 0.5 hours, the total number of measurements 960.

In calculating the estimates of the coefficients of the Fourier series, the first twenty harmonics were taken and the following results were obtained: significant turned out the constant component and 1; 2; 4; 10; 11 harmonics.

An empirical distribution density function is constructed from a sampling of a centered RP.

The matching between the empirical and theoretical distributions was checked by the Pearson criterion, as a result of which, with a probability of 0.95, there is no reason to reject the hypothesis regarding the normality of the distribution.

The correlation function of the deterministic process is determined through the correlation moments. The approximation is performed by a standard exponential dependence.

The proposed calculation methodology gave the following results:

–      the investigated RP is stationary in dispersion and has no stationarity as mathematical expected;

–      the approximation of the mathematical expectation of the load current showed that are statistically significant the constant component and 1; 2; 4; 10; 11 harmonics; the presence of 1; 2; 4 harmonics can be explained by the joint work of the railway and industrial enterprises; periods of 10 and 11 harmonics can correspond to the travel time of the freight train in the zone that is calculated;

–      the degree of relationship of the ordinates of the RP at different points in time can be described by an exponential correlation function, the approximation accuracy is satisfactory.

#### Keywords

random process; variance; load; modeling, mathematical expectation; correlation function; traction substation of direct current

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#### References

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DOI: https://doi.org/10.31319/2519-2884.36.2020.13

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Copyright (c) 2020 O. V. Klyuev, E. D. Khmelnitsky, E. G. Nosakov, D. A. Kolesnik

ISSN (print) 2519-2884

ISSN (online) 2617-8389