Formation of the Electrical Facilities of a Metallurgical Enterprise as a Large System

B. I. Kudrin

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

Let \(x\) be an element of the set \(X\), \(x \in X\), \(y \in Y\), etc., and let \(\{Z_i\}_n\) be a finite collection of sets \(Z_1, Z_2, \ldots, Z_n\), and \(z_i\) their elements, \(i=1, 2, \ldots, n\). The Cartesian product

\[ \hat{Z}=\Pi(Z_i \in \{Z_i\}_n / i=1,2,\ldots,n) \tag{1} \]

is the set of ordered sequences \(z=(z_1, z_2, \ldots, z_n)\), for example, all ordered pairs \(z=(x,y)\), where \(x \in X\), \(y \in Y\).

Let us introduce a measure \(f\) of the interaction of the elements \(x\) of the set \(X\), and denote by \(x_1\) the number of elements whose measure of interaction has the greatest order \(f(x_1)\). With a number of elements \(x_i\), the measure of interaction \(f(x_i)\) of which is one order lower than \(f(x_{i-1})\), \(i=1,2,\ldots,n\), a system \(S\) may be formed, consisting of a set of elements \(x_1+x_2+\cdots+x_k+\cdots+x_n\), where \(k<n\), which may be characterized by the functional

\[ S[f(x_1), f(x_2), \ldots, f(x_n), x_1, x_2, \ldots, x_n]. \tag{2} \]

By changing the order \(k\), we shall obtain different systems. Elements not included in the system \(S\) shall be called the external environment.

Let the system be characterized by a certain set of numerical parameters \(x_1, x_2, \ldots, x_n\), subject to statistical evaluation, and let a set of efficiency indicators \(y_1, y_2, \ldots, y_m\), depending on these parameters, be given:

\[ \begin{aligned} y_1&=\varphi_1(x_1,x_2,\ldots,x_n),\\ y_2&=\varphi_2(x_1,x_2,\ldots,x_n),\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ y_m&=\varphi_m(x_1,x_2,\ldots,x_n). \end{aligned} \tag{3} \]

We shall regard the system as described if, for \(X=X(t)\), we find \(Y=Y(t)\). The complete characterization of the system as a “black box” consists in specifying \(\{f_x(Y)\}\) for all admissible input values \(X\), where \(\{f_x(Y)\}\) denotes the totality of all finite-dimensional distributions of the input parameters \(Y\) for fixed \(X\).

In describing the electrical facilities as a system, the following electrical indicators, proposed by us [3] and used in design, may be taken as basic: the enterprise half-hour maximum load \(P_M\); the demand factor \(k_c\), the ratio of the maximum load to the installed capacity of the plant’s electrical receivers; the annual number of hours of utilization of the maximum \(T\), the ratio of the enterprise’s annual electricity consumption \(A\) to the maximum load (or electricity consumption); the number of installed electric motors \(n\) and their average capacity \(P_c\) (a notional electric motor); the electrical equipment of labor \(A\), the annual amount of electricity consumed per worker; the productivity of the electrical-engineering personnel \(A_p\), the annual amount of electricity consumed (distributed) per electrician (the total electrical-engineering personnel for the enterprise as a whole, \(N_e\)).

The basic electrical indicators, together with technological and technical-and-economic indicators (production of coke, pig iron, steel, rolled products, volume of capital investments, cost of commissioned fixed assets, number of industrial-production personnel \(M_k, M_{ch}, M_c, M_p, K, K_f, N\)), form a Cartesian product, where the value \(\hat z\) is determined by heuristic methods:

\[ \begin{aligned} & M_{k1},\ M_{k2}, \ldots,\ M_{ki}, \ldots,\ M_{kn};\\ & M_{ch1},\ M_{ch2}, \ldots,\ M_{chi}, \ldots,\ M_{chn};\\ & M_{c1},\ M_{c2}, \ldots,\ M_{ci}, \ldots,\ M_{cn};\\ & M_{p1},\ M_{p2}, \ldots,\ M_{pi}, \ldots,\ M_{pn};\\ & K_1,\ K_2, \ldots,\ K_i, \ldots,\ K_n;\\ & K_{f1},\ K_{f2}, \ldots,\ K_{fi}, \ldots,\ K_{fn};\\ & N_1,\ N_2, \ldots,\ N_i, \ldots,\ N_n;\\ & P_{M1},\ P_{M2}, \ldots,\ P_{Mi}, \ldots,\ P_{Mn};\\ & A_1,\ A_2, \ldots,\ A_i, \ldots,\ A_n;\\ & n_1,\ n_2, \ldots,\ n_i, \ldots,\ n_n;\\ & P_{c1},\ P_{c2}, \ldots,\ P_{ci}, \ldots,\ P_{cn};\\ & A_{e1},\ A_{e2}, \ldots,\ A_{ei}, \ldots,\ A_{en};\\ & A_{p1},\ A_{p2}, \ldots,\ A_{pi}, \ldots,\ A_{pn};\\ & N_{e1},\ N_{e2}, \ldots,\ N_{ei}, \ldots,\ N_{en}. \end{aligned} \tag{4} \]

The quantities entering into expression (4) also form a sequence \(T\) of actual observation results \(x(t_1), x(t_2), \ldots, x(t_T)\) over time. The distribution parameters of the random variable \(x(t)\) depend substantially on the time of observation \(t\); the values \(x(t_i)\) and \(x(t_j)\) at two different time instants \(t=t_i\) and \(t=t_j\) are interdependent.

The application of correlation and regression analyses, in particular the method of multifactor correlation forecasting, and of professional-logical analysis of prognostic functions [4, 5] using a computer made it possible to conclude that the sequences (4) can be extrapolated and the basic electrical indicators \(Y=Y(t)\) can be forecast over time and from the specified technological indicators \(X=X(t)\).

The quantitative determination of the indicators (3) makes it possible to proceed to the formation of subsystems which, on the one hand, are determined by the principal electrical indicators and, on the other, by the set of objects forming the system—the enterprise as a whole. These subsystems (power supply, maintenance, etc.) have a progressively greater degree of interaction among their elements (power supply on the 220 and 110 kV side, district power supply, electrical equipment of shops, organization and control of the power-supply system, organization of repair of electrical machines, etc.), and their formation is determined by known optimality criteria [6, 7]: by the minimum of reduced costs

\[ F=\min \sum_i \sum_r (C_i^r + E K_i^r), \tag{5} \]

and, for a specified limit \(K_0\) of capital investments, by the maximum provision of needs \(\max \alpha\)

\[ E=\max \alpha \quad \text{for} \quad \sum_i \sum_r K_i^r \leq K_0, \tag{6} \]

where \(C_i\) are current costs; \(K_i\) are capital investments; \(E\) is the normative coefficient of capital investments; \(r\) is the index of a technological variant; \(i\) is a variant of system formation.

The methods of specific indicators and the methods of direct calculation used in forming individual subsystems of an electrical facility should be checked by probabilistic-statistical methods, taking into account that the probability of event \(A_i\), computed under the condition that another event \(B\) has occurred,

\[ P(A_i \mid B)=\frac{P_j \prod P_{ij}}{\sum_j P_j \prod P_{ij}} \tag{7} \]

decreases with an increase in the hierarchy levels. The absence of verification and of the corresponding control contributes to a tendency toward a decrease in the enterprise demand coefficient, which may be characterized by the number of installed transformers per 1 MVA of the enterprise’s half-hour maximum, and is equal to 3.0 for the West Siberian Metallurgical Plant; 2.8 for the Novosibirsk plant; 2.5 for the Cherepovets plant; and 2.0 for the Novolipetsk plant.

Individual subsystems of the electrical facility cannot be described by expressions (1) and (3). In this case the problem, in general form, can be formulated as follows [8].

Let a set of elementary events \(X\) be given, as well as \(\sigma\)—the algebra of its measurable subsets \(U_x\). A random flow \(h(A)\) with phase space \((X, U_x)\) is defined as a system of random variables \(h(A)\), defined on the elements \(A \subset U_x\), possessing the following two properties:

1) \(h(A)\) is an absolutely additive function of the set \(A\);

2) \(h(A)\) assumes only nonnegative integer values.

In this case, queueing theory is applicable [9, 10]. A queueing system, as applied to individual service systems of the electrical facility of a metallurgical enterprise, should be characterized as a multiphase, multichannel system.

a mixed-type system consisting of heterogeneous devices which begin service in the order in which they become free, and to which an unlimited stream of requests arrives, served in the order of arrival with priority for some of the requests.

The electrical facilities that have taken shape at a metallurgical enterprise, like any large system, are characterized by the stability of a number of indicators and by survivability as a whole. Let each element \(x \in X\), where \(U \subset \ldots \subset Y \subset X \subset Z \subset \ldots \subset T\), operate under certain conditions specified for \(X\) by the limits \([A_0, A_m]\), which can be divided into \(m\) intervals—“boxes” \([A_0, A_1], [A_1, A_2], \ldots, [A_{m-1}, A_m]\). Let the number of elements placed in the \(k\)-th box be denoted by \(n_k\); to each row of possible values of \(n_k\) there corresponds a certain number of ways of filling the boxes, each of which is characterized by a “volume” \(g_k\).

Then the number of ways of distributing the elements among \(m\) boxes has the form [11]:

\[ W(n_1,n_2,\ldots,n_m)=\prod_{k=1}^{m}\frac{(n_k+g_k)!}{n_k!\,g_k!}. \tag{8} \]

System stability conditions

\[ \sum_{k=1}^{m} n_k=N=\mathrm{const};\quad \sum_{k=1}^{m} n_k A_k=A=\mathrm{const};\quad f(n_k)=\max. \tag{9} \]

The last condition means that, of all possible rows, the row to which the maximum number of ways corresponds must be realized. Formally, conditions (8) are identical to the conditions formulated for Bose–Einstein statistics.

Let us show the presence of stability for the electrical facilities that are taking shape. Let \(x \in X\) be defined by quantitative and qualitative characteristics [3]. Then the element \(x\), on the one hand, represents a unit (individual)—an electric motor, transformer, etc.; on the other hand, it is a type size (kind). For example, type size 28A includes motors A71-2, A72-4, A81-6, A82-8; transformer type size TM1000. As the type size for cables, a cable connecting two substations is taken, differing in cross section, make, and length (discreteness 100 m), for example \((3 \times 95)\) AAB, length 1600–1700 m.

Consider all elements or their representative samples forming the electrical facilities: \(x \in X\) electric motors, \(y \in Y\) transformers, \(z \in Z\) cables, etc. Form the subsets \(a_i, b_i, c_i\), \(X_i \subset X;\ Y_i \subset Y;\ Z_i \subset Z\), in which the numbers of elements \(x_i, y_i, z_i\) of each kind are numerically successively equal to \(i=1,2,\ldots,k\). If \(n_i\) is the total number of kinds (type sizes) in the subset (class) \(a_i\), then the total number of type sizes (kinds) in the system (sample) is

\[ S=\sum_{i=1}^{k} n_i; \tag{10} \]

the number of elements in the class

\[ u_i=a_i n_i; \tag{11} \]

the number of elements in the system (sample)

\[ U=\sum_{i=1}^{k} a_i n_i; \tag{12} \]

the relative frequency of occurrence of a class

\[ \omega_i=\frac{n_i}{S}. \tag{13} \]

The studies carried out by the author covered \(U=27632\) electrical machines, \(S=5618\) type-sizes; \(a_i=1\) occurred once for \(n_1=2868\) type-sizes (12 annual samples for metallurgical enterprises of Siberia). The results were checked at the West Siberian Metallurgical Plant for installed transformers, \(U=541,\ S=48\), and high-voltage cables, \(U=1285,\ S=263\).

The studies showed that the formation of the electrical facilities and of individual subsystems can be characterized by a regularity that has an objective character: a countable set of elements (individuals), all of which can be assigned to some system, the number of species of one class (family), and the number of species itself are distributed in such a way that each of the majority of species is represented by a small number of elements; and as the number of elements of one species increases, the number of these species decreases.

The decreasing number of species, with an increasing number of elements in each species, is each time based successively on an increasing number of species, each of which is represented by a number of elements decreasing to unity.

This regularity can be represented in the form

\[ \omega_i = k a_i^{-b}, \tag{14} \]

where \(k,\ b\) are constants, and has analogues [12] indicating the generality of the laws governing the formation of technological and biological ecosystems. R. Fisher, instead of the hyperbola (14), used a logarithmic series as a convergent series with a finite sum, representing the distribution frequencies of species containing different numbers of elements in the form (the zero term is not included) [12]

\[ n_1,\frac{n_1x}{2},\frac{n_1x^2}{3},\ldots,\frac{n_1x^{n-1}}{n}. \tag{15} \]

The sum of all species and elements is related to the variable \(x<1\) by the following expressions:

\[ S=\frac{n_1}{x}(-\ln\overline{1-x});\ U=n_1(1-x). \tag{16} \]

The discrepancy between the regularity proposed by us and the logarithmic series for all investigated samples of electrical machines did not exceed 1.4% by species.

REFERENCES

1. Kuratovskii K., Mostovskii A. Set Theory. Mir, 1966.
2. Buslenko N. P., Kalashnikov V. V., Kovalenko I. N. Lectures on the Theory of Complex Systems. Sovetskoe radio, 1973.

1. Electrification of Metallurgical Enterprises of Siberia. Tomsk, Tomsk State University Press, 1971.

2. Aivazyan S. A., Frenkel A. A. Application of statistical methods in economics.—“Electrical Engineering Industry. Series: General Industry Questions,” 1973, No. 5 (408), No. 6 (409).

3. Hauenstein G. Methods of Forecasting in a Socialist Economy. Moscow, Progress, 1971.

4. Methodological Provisions for Optimal Sectoral Planning in Industry. Novosibirsk, Nauka, 1972.

5. Sherman E. B., Shirinskaya N. Kh. Economic-mathematical modeling of the problem of the development and specialization of transformer production.—“Electrical Engineering Industry. Series: General Industry Questions,” 1972, No. 10 (391).

6. Khinchin A. Ya. Works on the Mathematical Theory of Mass Service. Moscow, Fizmatgiz, 1963.

7. Kudrin B. I., Borisov R. I. Application of queuing theory in the design of electrical-repair shops at metallurgical plants.—Collection of the NVII, No. 11, physical-mathematical issue. Krasnoyarsk, 1971.

8. Novikov O. A., Petukhov S. I. Applied Questions of Queuing Theory. Moscow, Sovetskoe Radio, 1969.

9. Andreev N. B., Bakirov T. S., Zavalishin N. N., Efimov V. M. Distribution of the organisms of a stable biocenosis according to the magnitudes of the energy flux.—Proceedings of the Fourth Conference of Zoologists of Siberia. Novosibirsk, 1972.

10. Williams C. B. Patterns in the Balanse of Nature (and Related Problems in Quantitative Ecology). Academic Press London and New York, 1964.