Probabilistic Calculation of Average and Effective Currents of Traction Substations and Sections of Electric Railways

Engineer P. P. Galinskii

The loads of traction substations of electric railways, the loads of transmission lines, and the loads of power systems with a significant proportion of transport consumers are subject to substantial random fluctuations. The influence of these random fluctuations on the operating conditions of elements of the power-supply system can be studied by methods of probability theory. The principal probabilistic characteristics of the currents of substations and railway sections are the average and root-mean-square (effective) value of the current.

Various calculation formulas for the average and effective currents of traction substations, obtained on the basis of probabilistic methods, were proposed by V. E. Rozenfeld [1] in 1937, by K. G. Markvardt [2, 3] in 1948 and 1958, by Yu. P. Porlits [4] in 1959, by A. M. Baranov et al. [5] in 1959, and by S. E. Kuzin [6] in 1962.

The initial quantities in the methods of the authors named are the currents of trains and their number in the supply zone under consideration.

Below a method is presented for calculating the average and effective currents of substations and railway sections, which makes it possible to take into account fluctuations in train currents, the number of trains, the correlation relationship of random fluctuations in train currents, and fluctuations in the occupation by trains of the carrying capacity of railway sections. The method of taking the last factor into account is proposed for the first time.

Let us consider schemes of separate and double-sided supply of trains through the contact network.

The instantaneous value of the substation current \(i_c\) is determined by the number of trains \(n\), their location along the route, and the random fluctuations of train currents.

\[ i_c = \sum_{j=1}^{n} i_j(t, x)\,\gamma_j(x), \tag{1} \]

where \(\gamma(x)\) is the fraction of the train current received from the given substation. With separate supply of trains through the contact network from one substation, \(\gamma = 1\). With double-sided supply,

\[ \gamma(x) = a + bx = \frac{\lambda}{2\lambda + 1} + \frac{1}{2\lambda + 1}\cdot \frac{1}{L}\cdot x, \tag{2} \]

$L$ — distance between substations;

$\lambda=\dfrac{z_{\text{ps}}}{z_0 L}$ — ratio of the substation resistance, taking into account the feeder and return wires, to the resistance of the contact system, taking into account the return conductor (rail, earth);

$x$ — distance from the train to the adjacent substation.

For d.c. substations, their internal resistance may be neglected in comparison with the resistance of the contact system [1]; then $\gamma(x)=\dfrac{x}{L}$. For a.c. substations, design practice [7] indicates the commensurability of the internal resistance of traction substations and the resistance of the contact system. The fact that the correlation coefficient of the currents of identical phases of transformers of a.c. traction substations is close to unity [8] indicates the small influence of the resistance of the contact system in comparison with the resistance of the substations.

Therefore, in engineering calculations for a.c. substations it is permissible to neglect the resistance of the contact system and to take $\gamma=\dfrac{1}{2}$.

If we assume that the element of the probability that a train is located on the track element $dx$ is equal to $\dfrac{dx}{l}$, where $l$ is the length of the feeding zone (with separate feeding $l=L$, with double-sided feeding — $l=2L$), then the mean value of the current $I_c$ and the square of the effective current of the substation $j_c^2$ can be determined from the expressions

\[ I_c = M\left\{ \frac{1}{l^n} \int_{x_1=0}^{l} \int_{x_2=0}^{l} \cdots \int_{x_n=0}^{l} \left[ \sum_{i=1}^{n} i_i(t,x)\gamma_i(x) \right] dx_1\,dx_2\ldots dx_n \right\}; \tag{3} \]

\[ j_c^2 = M\left\{ \frac{1}{l^n} \int_{x_1=0}^{l} \int_{x_2=0}^{l} \cdots \int_{x_n=0}^{l} \left[ \sum_{i=1}^{n} i_i(t,x)\gamma_i(x) \right]^2 dx_1\,dx_2\ldots dx_n \right\}. \tag{4} \]

Here $M$ is the sign of mathematical expectation.

We shall consider the variation of the currents of individual trains to be independent of the mutual position of the trains on the line. Then the multiple integration reduces to single integration,

\[ I_c = M\left\{ \sum_{i=1}^{n} \frac{1}{l} \int_{0}^{l} i_i(t,x)\gamma_i(x)\,dx \right\} \tag{5} \]

Squaring the polynomial under the integral sign in (4), we obtain

\[ j_c^2 = M\left\{ \sum_{i=1}^{n} \frac{1}{l} \int_{0}^{l} (i_i\gamma_i)^2\,dx + \right. \]

\[ \left. + \sum_{1}^{n(n-1)} \frac{1}{l^2} \int_{x_i=0}^{l} \int_{x_j=0}^{l} i_i\,i_j\,\gamma_i\,\gamma_j\,dx_i\,dx_j \right\} \tag{6} \]

Let trains of the same type operate in the zone under consideration; let the track profile be homogeneous.

Owing to the homogeneity of the profile, the train current may be regarded as independent of its position on the line and may be taken outside the integral sign.

la. Then the integrals prove to be independent of the train number and can be taken outside the summation sign. Integration taking (2) into account transforms the integrals into constant coefficients—the coefficients of the scheme \(C, C_1, C_2\).

\[ I_c=M\left\{C\sum_{i=1}^{n} i_i(t)\right\}=CM(\dot n)\cdot M(i)=C\bar n I, \tag{7} \]

\(n\) is the average number of trains in the feeding zone, \(I\) is the average train current.

\[ j_c^2=M\left\{C_1\sum_{i=1}^{n} i_i^2(t)+C_2\sum_{1}^{n(n-1)} i_i i_j\right\} =C_1M(n)\cdot M(i^2)+ \]

\[ +C_2M[n(n-1)]\cdot M(i_i i_j); \tag{8} \]

\[ j_c^2=C_1\bar n j^2+C_2(n_{\mathrm e}^2-\bar n)I^2\rho. \]

With separate feeding, the scheme coefficients are \(C=C_1=C_2=1\); with two-sided feeding,

\[ C=\frac{1}{2},\qquad C_2=\frac{1}{4},\qquad C_1=\frac{1}{3}\cdot\frac{1+3\lambda+3\lambda^2}{1+4\lambda+4\lambda^2}. \]

\(j\) is the effective train current; \(\rho=1+\sigma^{*2}R\),

\[ \sigma^*=\frac{\sigma}{I} \]

is the coefficient of variation of the train current;

\(\sigma\) is the root-mean-square deviation of the train current;

\(R\) is the correlation coefficient of train currents;

\(n_{\mathrm e}\) is the effective (root-mean-square) number of trains in the feeding zone. It may be determined for particular conditions either experimentally or by theoretical calculations in accordance with an adopted probabilistic scheme (distribution law). Most often, the hypergeometric, binomial, and normal laws of distribution of the number of trains in the feeding zone, or a parallel timetable, are used.

For any law the relation holds

\[ n_{\mathrm e}^2=\bar n^2+\sigma_n^2, \tag{9} \]

where \(\sigma_n\) is the root-mean-square deviation of the number of trains in the feeding zone.

For a parallel timetable, with a sufficiently large average number of trains in the feeding zone, the variance of the number of trains may be neglected and the effective number of trains may be taken as equal to the average,

\[ n_{\mathrm e}\approx \bar n. \]

The probability that a train appears in the feeding zone is equal to the filling of the carrying capacity of the track section by trains,

\[ \varphi=\frac{N}{N_0}, \]

where \(N_0\) is the maximum possible number of trains per day, and \(N\) is the average daily number of trains. Let \(n_0\) be the maximum number of trains that can simultaneously be in the zone under consideration; then the average number of trains in the zone is

\[ \bar n=\varphi n_0. \]

With the binomial distribution law, the variance of the number of trains in the zone, according to [9], is equal to

\[ \sigma_n^2=n_0\varphi(1-\varphi). \tag{10} \]

From (9) and (10) follows the equality

\[ n^2_{\mathrm{e}}-\bar n=\varphi^2 n_0(n_0-1) \tag{11} \]

and the expression for the square of the effective current (8) takes the form

\[ j_{\mathrm{c}}^2=C_1 n_0 \varphi j^2+C_2 \varphi^2 n_0(n_0-1)I^2\rho \tag{12} \]

or

\[ j_{\mathrm{c}}^2=A\varphi+B\varphi^2, \tag{13} \]

where it is denoted that

\[ A=C_1 n_0 j^2,\qquad B=C_2 n_0(n_0-1)I^2\rho . \]

The formulas for the effective current based on the hypergeometric and normal distribution laws, as well as the formula corresponding to the parallel graph, can also be reduced to the form (13).

Formula (12) was derived under the assumption of constant daily traffic volumes, i.e., under the assumption of a constant filling of the capacity of railway sections, \(\varphi=\dfrac{N}{N_0}\), since usually \(N_0\) is a constant quantity.

Let us take into account the fluctuations of daily traffic volumes by finding the mean value of the square of the effective current of a substation (section) over \(k\) days:

\[ j_{\mathrm{c}}^2=\sum_{i=1}^{k} j_{\mathrm{c}i}^2 =\sum_{i=1}^{k}\left(A\varphi_i+B\varphi_i^2\right)= \]

\[ = A\sum_{i=1}^{k}\varphi_i+B\sum_{i=1}^{k}\varphi_i^2 =A\bar\varphi+B\varphi_{\mathrm{e}}^2, \]

where \(\bar\varphi\) is the average filling of the capacity of the railway section by trains over \(k\) days, and \(\varphi_{\mathrm{e}}\) is the effective value of \(\varphi\).

Taking \(N_0\) to be constant, we shall have

\[ \bar\varphi=\sum_{i=1}^{k}\varphi_i =\sum_{i=1}^{k}\frac{N_i}{N_0} =\frac{\bar N}{N_0}, \]

\[ \varphi_{\mathrm{e}}^2=\sum_{i=1}^{k}\varphi_i^2 =\sum_{i=1}^{k}\frac{N_i^2}{N_0^2} =\frac{N_{\mathrm{e}}^2}{N_0^2} =\frac{\bar N^2(1+\sigma_N^{*2})}{\left(\dfrac{\bar N}{\varphi}\right)^2}; \]

\[ \varphi_{\mathrm{e}}^2=\bar\varphi^2(1+\sigma_N^{*2}). \tag{14} \]

Here \(\sigma_N^*=\dfrac{\sigma_N}{N}\) is the coefficient of variation of the daily number of trains.

Thus, we obtain the expression

\[ j_{\mathrm{c}}^2=A\bar\varphi+B\bar\varphi^2(1+\sigma_N^{*2}), \tag{15} \]

which, for the binomial law, takes the form

\[ j_{\mathrm{c}}^2=C_1 n_0\bar\varphi j^2 +C_2\bar\varphi^2(1+\sigma_N^{*2})\,n_0(n_0-1)I^2\rho . \tag{16} \]

The intramonthly fluctuations of daily traffic volumes on the sections of level railway that we investigated, when the capacity was filled with...

...capacity utilization degree $\varphi \approx 0.5$, are characterized by a coefficient of variation of the number of trains equal to $0.12$–$0.14$.

The results of surveys of fluctuations in traffic volume on most sections of the country’s railways are given in [10, Tables 3 and 5]. The coefficients of variation of the daily number of trains, calculated from the data in these tables, proved to be equal to $0.118$ for double-track sections and $0.119$ for single-track sections. These values of $\sigma_N^*$ may be taken as average values for intra-monthly daily fluctuations in the number of trains. (The values of $\sigma_N^*$ for capacity-utilization degrees of railway sections close to zero and close to unity require refinement.)

The necessity of taking into account one or another factor affecting the magnitude of the effective current of a substation follows from its relative significance.

At present there is no information in the literature on the direct determination of the correlation coefficients of train currents $R$, although corresponding methods in which it is proposed to take $R$ into account do exist [6].

To estimate the value of $R$, we shall confine ourselves to the following numerical example. Suppose that over three consecutive equal intervals of time $T$ (days, ten-day periods), the relative specific energy consumption for train traction, due to the influence of weather conditions, is characterized by the numbers $0.9;\ 1.0;\ 1.1$. These ratios of specific energy consumptions were obtained for real changes in air temperature and wind force by calculation according to a known method [11].

Let us estimate the value of $R$, assuming that all trains change their average currents to the same degree in the corresponding periods of time. The mean value of the pairwise product of the currents of different trains is equal to

$$ \overline{i_i i_j}=I^2(1+\sigma^{*2}R)= \frac{1}{3T}\int_0^{3T} i_i(t)i_j(t)\,dt =\frac{1}{3}(I_1^2+I_2^2+I_3^2)= $$

$$ =\frac{1}{3}I^2(0.9^2+1.0^2+1.1^2)=I^2\cdot 1.007. $$

Consequently, $\sigma^{*2}R=0.007$, or

$$ R=\frac{0.007}{\sigma^{*2}}. \tag{*} $$

The variance of the train current will be minimal when, within the consecutive time intervals $T$, the current magnitude is constant and equal to $0.9;\ 1.0$, and $1.1$ of the average magnitude over the whole time.

$$ j_{\min}^2=\frac{1}{3T}\int_0^{3T} i_i^2(t)\,dt =I^2\cdot 1.007=I^2(1+\sigma^{*2}). $$

$$ \sigma_{\min}^{*2}=0.007;\qquad \sigma_{\min}^{*}=0.084. $$

Below are the values of the coefficient of correlation of train currents, calculated from (*) for various $\sigma^*$.

This calculation does not eliminate the need for an experimental determination of $R$, but it makes it possible to judge its approximate magnitude and to assess the significance of the correlation of train currents in the aggregate of all

factors in the formation of the current variance of a substation or a railway section.

Let a railway section be fed as a cantilever from one power system over a single-circuit line through several DC traction substations; the supply of trains through the contact network is sectionalized. In this case the average and effective currents of the feeder, substation, and sections of the transmission line between the substation connection points can be determined by the same formulas (7) and (16) with \(C=C_1=C_2=1\).

Let us assess the significance of the individual factors in the formation of the variance of the currents of railway sections of different length. To do this, from (7) and (16) we shall find the variance of the current of a substation (section) and express it in units of the average current.

\[ \sigma_{ic}^{2}=i_c^{2}-I_c^{2} \]

\[ \sigma_{ic}^{*2}=\frac{\sigma_{ic}^{2}}{I_c^{2}} =\frac{\sigma^{*2}}{n}+\frac{1-\bar{\varphi}}{n} +\sigma_N^{*2}\frac{\bar n-\varphi}{n} +R\sigma^{*2}(1+\sigma_N^{*2})\frac{\bar n-\varphi}{n} \tag{17} \]

The first of the terms on the right-hand side of equation (17) takes into account fluctuations of train currents \((i)\), the second—the fluctuation in the number of trains \((n)\) in the supply zone at the average utilization of the carrying capacity of the railway section, the third—the fluctuation of the daily volume of traffic \((N)\), and the fourth term takes into account the mutual correlation of train currents \((R)\).

Figure 1 gives diagrams, calculated from (17), of the significance of each of the four named factors as a function of the average number of trains in the supply zone. It is evident from the figure that when the number of trains is small \((n<5)\), corresponding to a feeder and substation zone, the predominant influence is exerted by two factors—the fluctuations in the number of trains and in their currents \((n,i)\). In this case the fluctuation of the daily volume of traffic and the correlation of train currents \((N,R)\) may be disregarded. However, for extended railway sections \((n>10)\) the significance of the last two factors is substantial and may even become predominant.

Analogous calculations for schemes with double-sided supply and a double-circuit line become more complicated, but the character of the significance diagrams for the individual factors is preserved.

The formulas for the average and effective current of a substation (7), (8), and (15) were derived for trains of the same type and for a uniform profile. In the general case, trains of different types operate in the supply zone, the track profile is nonuniform, and there are points of settlement and formation of freight traffic.

Let us divide the supply zone into \(P\) sections differing either in the character of the profile or in traffic intensity. For each section of uniform profile and specified traffic intensity, we shall replace trains of different types by trains of an equivalent type.

The average current of an equivalent-type train on section \(\pi\) is

\[ I_{\pi}=\sum_{i=1}^{q}\alpha_{\pi i}I_{\pi i}, \]

the variance of the current of an equivalent-type train is

\[ \sigma_{\pi}^{2}=\sum_{i=1}^{q}\alpha_{\pi i}\sigma_{\pi i}^{2}, \]

square of the effective current of a train of the equivalent type

\[ j_{\pi}^{2}=I_{\pi}^{2}+s_{\pi}^{2}, \]

\(q\) — the number of different types of trains on section \(\pi\),
\(a_{\pi i}\) — the share of trains of type \(i\) on section \(\pi\).

Such equivalencing in the presence of correlation of train currents is not logically absolutely rigorous. However, as calculations show, the error due to the logical inconsistency is practically imperceptible.

Fig. 1. Diagram of the relative significances in the formation of the variance of the current of a railway section: fluctuations in the number of trains \(n\), their currents \(i\), mutual correlation of train currents \(R\), and fluctuations in the daily traffic volume \(N\).

The calculation was performed by formula (17) for \(\sigma^{*}=0.4\), \(R=0.04\), \(\sigma^{*}_{N}=0.12\).

Proceeding further, for each of the sections and combinations of sections, with all the operations that were performed in deriving formulas (7), (8), and (15), we obtain formulas for the average and effective current of a substation for the most general case

\[ I_{c}=\sum_{\pi=1}^{P}\bar n_{\pi} I_{\pi} C_{\pi}, \tag{18} \]

\[ j_c^2 = \sum_{\pi=1}^{P} \overline{n}_{\pi} j_{\pi}^{2} C_{1\pi} + \sum_{\pi=1}^{P} \left(n_{\pi e}^{2}-\overline{n}_{\pi}\right) I_{\pi}^{2} \rho_{\pi} C_{\pi}^{2} + 2 \sum_{\psi<\pi} \overline{n_{\pi} n_{\psi}}\, I_{\pi} I_{\psi}\, \rho_{\pi\psi} C_{\pi} C_{\psi}. \tag{19} \]

Formula (8) is a special case of formula (19). If the binomial law of train distribution is adopted, then, similarly to (15), we obtain

\[ j_c^2 = \sum_{\pi=1}^{P} \overline{n}_{\pi} j_{\pi}^{2} C_{1\pi} + \sum_{\pi=1}^{P} \left(1+\sigma_{N\pi}^{*2}\right) \overline{n}_{\pi}\left(\overline{n}_{\pi}-\varphi_{\pi}\right) I_{\pi}^{2}\rho_{\pi} C_{\pi}^{2} + \]

\[ + 2 \sum_{\psi<\pi} \left(\overline{n}_{\pi}\cdot\overline{n}_{\psi} + \sigma_{N\pi\psi}^{*2}\, \overline{n}_{\pi\mathrm{tot}}\cdot \overline{n}_{\psi\mathrm{tot}} \right) I_{\pi} I_{\psi} C_{\pi} C_{\psi}, \tag{20} \]

where \(n_{\pi e}\) is the effective number of trains on section \(\pi\); \(\overline{n_{\pi} n_{\psi}}\) is the mean value of the product of the numbers of trains simultaneously located on sections \(\pi\) and \(\psi\); \(C_{\pi}=a+b\overline{x}_{\pi}=r(\overline{x}_{\pi})\) is determined from (2), \(\overline{x}_{\pi}\) is the distance from the mean point of section \(\pi\) to the adjacent substation; \(\overline{n}_{\pi}, \overline{n}_{\psi}\) are the mean numbers of trains on sections \(\pi\) and \(\psi\); \(C_{1\pi}=C_{\pi}^{2}+\frac{1}{12}b^{2}l_{\pi}^{2}\simeq C_{\pi}^{2}\); \(l_{\pi}\) is the length of section \(\pi\); \(\sigma_{N\pi}^{*}\), \(\sigma_{N\pi\psi}^{*}\) are the coefficient of variation of the daily number of trains on section \(\pi\) and the coefficient of variation of the daily number of trains passing through both sections \(\pi\) and \(\psi\); \(\overline{n}_{\pi\mathrm{tot}}, \overline{n}_{\psi\mathrm{tot}}\) are the mean numbers of trains on sections \(\pi\) and \(\psi\) formed by trains passing through both sections \(\pi\) and \(\psi\);

\[ \rho_{\pi}=1+\sigma_{\pi}^{*2} R_{\pi}, \qquad \rho_{\pi\psi}=1+\sigma_{\pi}^{*}\sigma_{\psi}^{*}R_{\pi\psi}, \]

\(R_{\pi}\) is the correlation coefficient of the currents of trains simultaneously located on section \(\pi\); \(R_{\pi\psi}\) is the correlation coefficient of the currents of two trains, one of which is located on section \(\pi\), the other on section \(\psi\).

In the particular case when there are no points of settling and formation of freight traffic and, moreover, \(\rho_{\pi}=\rho_{\pi\psi}=\rho\), from (20) we obtain

\[ j_c^2 = \sum_{\pi=1}^{P} \overline{n}_{\pi} j_{\pi}^{2} C_{1\pi} + I_c^{2}\rho\left(1+\sigma_N^{*2}\right) \left( 1-\frac{\overline{\varphi}}{I_c^{2}} \sum_{\pi=1}^{P} \overline{n}_{\pi} I_{\pi}^{2} C_{\pi}^{2} \right) \tag{21} \]

where \(I_c\) is the mean current of the substation, determined by (18).

Conclusion: 1. A method is proposed for calculating the average and effective currents of substations and railway sections, making it possible to take into account fluctuations in the number of trains, their currents, the mutual correlation of train currents, and fluctuations in the daily volumes of traffic.

1. Mutual correlation and fluctuations in daily traffic volumes have a substantial effect on the magnitude of the effective currents of extended railway sections. When calculating the currents of feeders and individual substations, the influence of these factors is insignificant and they need not be taken into account.

References

1. Rosenfeld V. E., Calculation of Traction Networks, Transtemizdat, 1937.

2. Markvardt K. G., Power Supply of Electric Railways, Transzheldorizdat, 1958.

3. Markvardt K. G., Power Supply of Electrified Railways, Transzheldorizdat, 1948.

4. Porplits Yu. P., Application of methods of mathematical statistics in the design of longitudinal power-supply circuits for main-line railways, Proceedings of MIIT, issue 104, TZhI, 1959.

5. Baranov A. M., Miroshnichenko R. I., Segal L. G., Adadurova E. V., Operating requirements for the parameters of power-supply devices of railways electrified with direct current, Proceedings of VNIIZhT, issue 174, TZhI, 1959.

6. Kuzin S. E., Determination of the maximum loads of traction substations of electric railways, Proceedings of LIIZhT, issue 193, Transzheldorizdat, 1962.

7. Davydov V. N., Measures for voltage stabilization and reactive-power compensation provided for in railway electrification designs, Reports at the Conference on Voltage Quality and Regulation, Moscow, 1961.

8. Timofeev D. V., Regularities in the variation of traction-load currents, Elektrichestvo, No. 2, 1963.

9. Smirnov N. V., Dunin-Barkovskii I. V., A Short Course of Mathematical Statistics for Technical Applications, Fizmatgiz, 1959.

10. Karetnikov A. D., Vorob’ev N. A., Improvement of Train Timetables and Better Utilization of the Capacity of Railway Lines, Transzheldorizdat, 1960.

11. Rosenfeld V. E. et al., Fundamentals of Electric Traction, Transzheldorizdat, 1957.