ON THE PROBLEM OF CALCULATING LOCAL ELECTRIC NETWORKS

E. A. ANTONOV

Introduction

Local electric networks constitute one of the most important links in the power-supply system of cities and industrial districts. The operating conditions of networks and their economic indicators are determined by the layout and the perfection of the calculation. At present, the works of Soviet scientists [1; 2; 3] on the theory of calculation of complex circuits have shown that it is most convenient to replace an electric network by combinations of connections of multi-terminal networks, and to relate the parameters and electrical quantities by means of matrix algebra. In this case, in the course of the calculation, at any stage of it, it becomes possible easily to establish the interrelation between quantities and to obtain equations of great generality.

From the literature known to us on the problem of calculating complex electric circuits, the most general principles are embodied in the method for calculating composite electric circuits developed in a number of works by G. E. Pukhov. The advantages of this method, obvious even in the calculation of symmetrical operating conditions of open networks, are especially manifested in networks of complex configuration.

In the practice of calculating symmetrical networks, the method of the active three-terminal network [3; 4; 5] has been sufficiently developed and is used. However, this method cannot be applied in all cases, for example, in the presence of complex sections and non-through circuits.

In connection with this, we set ourselves the task, using a concrete example of calculating a dispatcher station, of showing the possibility and expediency of applying a more general method for calculating composite electric circuits, based on an active non-through multi-terminal network.

In solving the stated problem, we restrict ourselves to the consideration of local networks, which, however, imposes no limitations on the application of the method to the calculation of networks of electric power systems.

Basic Principles of the Method

The equivalent circuit of an electric network is divided into separate subcircuits, which in the general case are active multi-terminal networks with unequal numbers of terminals. The physical state of each subcircuit is then described by a system of linear equations, which is subsequently expressed by a single generalized equation in matrix form.

For all practically important cases of subcircuits, summary tables of matrices have been compiled in advance. After equalizing the number of terminals, the original circuit, according to the corresponding connection formulas, is replaced by a single calculated multi-terminal network.

Calculation of Radial Lines of Local Electric Networks

When there are several radial lines whose operating conditions are interdependent, it is convenient to represent the network section (Fig. 1) in the form of a multi-terminal network; in this case the return conductors of the network are combined into one common clamp-pole.

Fig. 1

We assume that the voltage on the buses of substation 1 is maintained constant under all operating conditions. For this case we determine the voltages on the buses of substations 2 and 4 as functions of the loads \(\dot W_2\), \(\dot W_4\), and \(\dot W_n\).

The equivalent circuit of the network section (Fig. 1a) is represented in the form of a four-terminal network (Fig. 2), where \(Z_1\), \(Z_2\), and \(Z_4\) are the total impedances of the lines.

Fig. 2

If the currents and voltages of the four-terminal network are divided into input, output, summing, and common quantities, then the relation between them is established by the matrix \(b\) according to equation (1).

\[ \begin{bmatrix} \dot U_1 \\[4pt] \dfrac{\dot W_1}{\hat U_n} \\[6pt] \dfrac{\dot W_4}{\hat U_n} \end{bmatrix} = [b] \begin{bmatrix} \dot U_2 \\[4pt] \dfrac{\dot W_2}{\hat U_n} \\[6pt] \dot U_4 \end{bmatrix} \tag{1} \]

where \(\dot U_n\) is the nominal voltage;

\(\dot U_1, \dot U_2\) and \(U_4\) are the voltages between terminals 1, 2, 4 and the neutral, multiplied by \(\sqrt{3}\); \(-\dfrac{\dot W_1}{\hat U_n}\), \(-\dfrac{\dot W_2}{\hat U_n}\), \(-\dfrac{\dot W_4}{\hat U_n}\) are the terminal currents.

In view of the fact that the voltages \(U_2\) and \(U_4\) are the unknowns, we transform the generalized equation with respect to the unknowns, using the rules for transforming matrix equations, and obtain the relation of interest to us (2)

\[ \begin{bmatrix} \dot U_2\\[4pt] \dot U_4\\[4pt] -\dfrac{\dot W_1}{\hat U_n} \end{bmatrix} = [b'] \begin{bmatrix} \dot U_1\\[4pt] -\dfrac{\dot W_2}{\hat U_n}\\[4pt] -\dfrac{\dot W_1}{\hat U_n} \end{bmatrix} \tag{2} \]

In this case, for the four-terminal network of Fig. 2 we obtain the matrix \([b']\)

\[ [b'] = \begin{bmatrix} 1 & -(Z_2+Z_1) & -Z_1\\ 1 & -Z_1 & -(Z_1+Z_4)\\ 0 & 1 & 1 \end{bmatrix}, \tag{3} \]

where \(Z_1, Z_2\) and \(Z_4\) are the total impedances of the corresponding lines.

The elements of the matrix \([b']\) are complex numbers constant for the given section of the network.

In the presence of power take-off at the buses of the intermediate substation (Fig. 1b), the equivalent circuit is represented in the form of an active four-terminal network (Fig. 3).

The transformed generalized equation (2) for the four-terminal network (Fig. 3) has the form (4):

\[ \begin{bmatrix} \dot U_2\\[4pt] \dot U_4\\[4pt] \dfrac{\dot W_1}{\hat U_n} \end{bmatrix} = \begin{bmatrix} 1, & -(Z_2+Z_1), & -Z_1\\ 1, & -Z_1, & -(Z_1+Z_4)\\ 0, & 1, & 1 \end{bmatrix} \begin{bmatrix} \dot U_1\\[4pt] -\dfrac{\dot W_2}{\hat U_n}\\[4pt] \dfrac{\dot W_1}{\hat U_n} \end{bmatrix} + \begin{bmatrix} -\dfrac{\dot W_n}{\hat U_n} Z_1\\[6pt] -\dfrac{\dot W_n}{\hat U_n} Z_1\\[6pt] -\dfrac{\dot W_n}{\hat U_n} \end{bmatrix}. \tag{4} \]

Equation (4) differs from equation (2) by an additional term—a vector parameter that takes into account the active properties of the circuit.

In the presence of transformation elements in the circuit (Fig. 1c), it becomes necessary to determine the voltages on the low-voltage buses of terminal substations \(2'\), \(4'\). Transformers are taken into account by an equivalent circuit in the form of a five-terminal network (Fig. 4).

Fig. 3

The matrix \(b\) of the five-terminal network (Fig. 4) has the form (5):

\[ |b| = \begin{pmatrix} \dot K_{2}, & \dfrac{Z'_{2}}{\hat K_{2}}, & 0, & 0 \\[6pt] 0, & \dfrac{1}{\hat K_{2}}, & 0, & 0 \\[6pt] 0, & 0, & \dfrac{1}{Z'_{4}}, & -\dfrac{\dot K_{1}}{Z'_{4}} \\[6pt] 0, & 0, & \dfrac{\hat K_{4}}{Z'_{4}}, & -\dfrac{\dot K_{1}\hat K_{4}}{Z'_{4}} \end{pmatrix}, \tag{5} \]

where \(Z'_{2}\) and \(Z'_{4}\) are the total impedances of the transformers, and \(\dot K_{2}\) and \(\dot K_{4}\) are complex transformation coefficients.

Fig. 4

To obtain the matrix \(b\) of the equivalent circuit of the entire electric-power transmission system (Fig. 1c), it is necessary to connect the four-terminal network (Fig. 3) with the five-terminal network (Fig. 4).

Equating the number of terminals of the multiports (Figs. 3—4) to 5, we obtain the connection circuit (Fig. 5).

The equivalent five-terminal network for the connection circuit (Fig. 5) is shown in Fig. 6.

Terminal “4” of the equivalent five-terminal network is not connected to the external circuit; consequently, the current \(i_4\) is equal to 0.

Fig. 5

The matrix parameter of the equivalent five-terminal network is calculated by formula (6), and the vector parameter by formula (6a).

\[ \zeta = \begin{bmatrix} \zeta'_{11}\zeta''_{11} & \zeta'_{12}+\zeta'_{11}\zeta''_{12} \\[4pt] \zeta'_{21}\zeta''_{11}+\zeta''_{21} & \zeta'_{22}+\zeta''_{22}+\zeta'_{21}\zeta''_{12} \end{bmatrix} \tag{6} \]

\[ \zeta_{0} = \begin{bmatrix} \zeta'_{10}+\zeta'_{11}\zeta''_{10} \\[4pt] \zeta'_{20}+\zeta''_{20}+\zeta'_{21}\zeta''_{10} \end{bmatrix}, \tag{6a} \]

where \(\zeta'_{11}, \zeta'_{12}, \zeta'_{21}, \zeta'_{22}\) are the components of the matrix parameter of the first subcircuit,

\(\zeta''_{11}, \zeta''_{12}, \zeta''_{21}, \zeta''_{22}\) are those of the second,

\(\zeta'_{10}, \zeta'_{20}\) are the vector parameters of the first subcircuit,

\(\zeta''_{10}, \zeta''_{20}\) are those of the second.

Fig. 6

After the corresponding calculations, we obtain the equation of the equivalent circuit in matrix form (7).

The current \(\dot I_{4}\), as indicated above, is equal to zero; consequently, there is no need to compute the elements of the third row of the matrix and of the vector parameter.

\[ \begin{bmatrix} \dot U_{1}\\[0.8em] \dfrac{\dot W_{1}}{\hat U_{n}}\\[0.8em] 0\\[0.8em] \dfrac{\dot W_{4}}{\hat U_{n}} \end{bmatrix} = \begin{bmatrix} \dfrac{Z_{1}+Z_{4}}{Z_{4}}\,\dot K_{2} & \dfrac{Z_{1}+Z_{4}}{Z_{4}}\,Z'_{2} + \dfrac{Z_{1}Z_{2}+Z_{1}Z_{4}+Z_{2}Z_{4}}{Z_{4}\hat K_{2}} & -\dfrac{Z_{1}}{Z_{4}} & 0 \\[1.0em] \dfrac{\dot K_{2}}{Z_{4}} & \dfrac{Z'_{2}}{Z_{4}\hat K_{2}} + \dfrac{Z_{1}+Z_{2}}{Z_{4}\hat K_{2}} & -\dfrac{1}{Z_{4}} & 0 \\[1.0em] b_{31} & b_{32} & b_{33} & b_{34} \\[1.0em] 0 & 0 & \dfrac{\hat K_{1}}{Z'_{1}} & -\dfrac{\dot K_{4}\hat K_{4}}{Z'_{1}} \end{bmatrix} \times \begin{bmatrix} \dot U'_{2}\\[0.8em] \dfrac{\dot W_{2}}{\hat U_{n}}\\[0.8em] \dot U_{n}\\[0.8em] \dot U_{4} \end{bmatrix} + \begin{bmatrix} \dfrac{\dot W_{n}}{\hat U_{n}}Z_{1}\\[0.8em] \dfrac{\dot W_{n}}{\hat U_{n}}\\[0.8em] b_{30}\\[0.8em] 0 \end{bmatrix} \tag{7} \]

Equation (7) can be transformed into an equation of form (8).

\[ \begin{bmatrix} \dot U''_{2}\\[0.8em] \dfrac{\dot W_{1}}{\hat U_{n}}\\[0.8em] 0\\[0.8em] \dot U''_{4} \end{bmatrix} = \begin{bmatrix} \dfrac{Z_{4}}{(Z_{1}+Z_{4})\hat K_{2}} & -\dfrac{1}{\dot K_{2}\hat K_{2}} \left( Z'_{2} + \dfrac{Z_{1}Z_{2}+Z_{1}Z_{4}+Z_{2}Z_{4}}{Z_{1}+Z_{4}} \right) & \dfrac{Z_{1}}{(Z_{1}+Z_{4})\hat K_{2}} & 0 \\[1.0em] \dfrac{1}{Z_{1}+Z_{4}} & -\dfrac{Z_{4}}{(Z_{1}+Z_{4})\hat K_{2}} & -\dfrac{1}{Z_{1}+Z_{4}} & 0 \\[1.0em] b'_{31} & b'_{32} & b'_{33} & b'_{34} \\[1.0em] 0 & 0 & \dfrac{1}{\hat K_{1}} & \dfrac{Z'_{1}}{\dot K_{4}\hat K_{1}} \end{bmatrix} \times \]

\[ \times \begin{bmatrix} \dot U_1\\[0.6em] \dfrac{\dot W_2}{\hat U_n}\\[0.8em] \dot U_4\\[0.6em] \dfrac{\dot W_1}{\hat U_n} \end{bmatrix} + \begin{bmatrix} \dfrac{\dot W_n Z_1 Z_4}{\hat U_n \dot K_2 (Z_1+Z_4)}\\[1em] \dfrac{\dot W_n}{\hat U_n}\left(1-\dfrac{Z_1}{Z_1+Z_4}\right)\\[1em] 0\\[0.6em] 0 \end{bmatrix} \tag{8} \]

In the practice of calculating local networks there is no need to determine the phase of the voltage, which makes it possible to introduce certain additional simplifications. We introduce into the calculations only the absolute value of the voltage and of the transformation ratio. We regard the product of an element of a row of the matrix by the current vector as the value of the longitudinal component of the voltage drop.

The equivalent five-terminal network (Fig. 6) of the network section (Fig. 1в) has an idle terminal “4,” which may be brought inside; as a result, the five-terminal network will be a four-terminal network.

In the latter case the matrix-vector parameters of the generalized equation of the four-terminal network will have the form (9)

\[ \begin{bmatrix} \dot U'_2\\[0.8em] \dot U'_1\\[0.8em] \dfrac{\dot W_1}{\hat U_n} \end{bmatrix} = \begin{bmatrix} \dfrac{1}{\dot K_2} & -\dfrac{Z_2+Z'_2+Z_1}{\dot K_2 \dot K_2} & -\dfrac{Z_1}{\dot K_2 \dot K_1} \\[1.1em] \dfrac{1}{\dot K_2} & -\dfrac{Z_1}{\dot K_1 \dot K_2} & -\dfrac{Z_1+Z_4+Z'_1}{\dot K_2 \dot K_1} \\[1.1em] 0 & -\dfrac{1}{\hat K_2} & \dfrac{1}{\hat K_1} \end{bmatrix} \times \begin{bmatrix} \dot U_1\\[0.8em] \dfrac{\dot W_2}{\hat U_n}\\[0.8em] \dfrac{\dot W_1}{\hat U_n} \end{bmatrix} + \begin{bmatrix} -\dfrac{1}{\dot K_2}\dfrac{W_n}{\hat U_n}Z_1\\[1.1em] -\dfrac{1}{\dot K_n}\dfrac{W_n}{\hat U_n}Z_1\\[1.1em] \dfrac{W_n}{\hat U_n} \end{bmatrix} \tag{9} \]

For illustration of the application of the expressions obtained, a numerical example is given. A section of an electrical network is given, for which the current loads and the constants of the lines and transformers are known.

Constants of the lines

\[ L_1: l=10\ \text{km},\quad \mathrm{MG}-70,\quad Z_1=5.04\mid 57^\circ 30' \ \text{ohm}; \]

\[ L_2: l=6\ \text{km},\quad \mathrm{MG}-50,\quad Z_2=3.46\mid 49^\circ 20' \ \text{ohm}; \]

\[ L_4: l=12\ \text{km},\quad \mathrm{MG}-50,\quad Z_4=6.93\mid 49^\circ 20' \ \text{ohm}. \]

Fig. 7

Constants of the transformers

\[ T_2: 5600\ \text{kVA},\quad K_2=5.45,\quad Z'_2=16.5\mid 82^\circ 18' \ \text{ohm}. \]

\[ T_4: 7500\ \text{kVA},\quad K_4=5.45,\quad Z'_4=12.3\mid 82^\circ 24' \ \text{ohm}. \]

Loads

\[ \dot{i}_2=469\mid 31^\circ 50'\ a, \]

\[ \dot{I}_4=545\mid 29^\circ 30'\ a, \]

\[ \dot{i}_n=40\mid 31^\circ 50'\ a. \]

The matrix-vector parameters of circuit (16) are equal to:

\[ [b]= \begin{array}{|c|c|c|} \hline 1 & -(4.98+j6.87) & -(2.72+j4.25) \\ \hline 1 & -(2.72+j4.25) & -(7.24+j9.53) \\ \hline 0 & 1 & 1 \\ \hline \end{array} ,\qquad (16) \]

\[ [b_0]= \begin{array}{|c|} \hline -\left(2{,}72+j4{,}25\right)\dfrac{\dot W_{\mathrm{н}}}{\hat U_{\mathrm{н}}} \\[1.2em] \hline -\left(2{,}72-j4{,}25\right)\dfrac{W_{\mathrm{н}}}{\hat U_{\mathrm{н}}} \\[1.2em] \hline -\dfrac{W_{\mathrm{н}}}{\hat U_{\mathrm{н}}} \\[1.2em] \hline \end{array} \tag{11} \]

As a result of the calculations we have:

\[ U_2=36{,}7\ \text{kV}. \]

\[ U_1=35{,}3\ \text{kV}. \]

The matrix-vector parameters of circuit \((1\text{в})\) are equal to:

\[ [b]= \begin{array}{|c|c|c|} \hline 0{,}183 & -\left(0{,}245+j0{,}785\right) & -\left(0{,}092+j0{,}143\right) \\ \hline 0{,}183 & -\left(0{,}092+j0{,}143\right) & -\left(0{,}303+j0{,}725\right) \\ \hline b_{31} & b_{32} & b_{33} \\ \hline \end{array}, \]

\[ [b_0]= \begin{array}{|c|} \hline -\left(0{,}504+j0{,}780\right)\dfrac{\dot W_{\mathrm{н}}}{\hat U_{\mathrm{н}}} \\[1.2em] \hline -\left(0{,}504+j0{,}780\right)\dfrac{\dot W_{\mathrm{н}}}{U_{\mathrm{н}}} \\[1.2em] \hline b_{30} \\ \hline \end{array}. \]

The elements of the third row need not be calculated, since the current \(I\) is not determined by us.

As a result of the calculation we have:

\[ U_2=6{,}5\ \text{kV}. \]

\[ U_1=6{,}5\ \text{kV}. \]

Conclusions

The application of the method of composite electrical circuits to the calculation of local networks, as is evident from the example, introduces simplifications. The availability of tables of ready-made matrix and vector parameters of characteristic network sections will make it possible to automate the calculation process and reduce it to the addition and multiplication of matrices of subschemes.

The generalized equation in matrix form for a section of a network—a subcircuit—makes it possible to obtain shorter and more general formulas, and also to establish more clearly the interrelations among electrical quantities.

The circumstances indicated make it possible to recommend the method presented, especially in operational calculations, to a wide circle of persons without special training.

REFERENCES

1. Voronov R. A. General Theory of Four-Terminal and Multi-Terminal Networks. Gosenergoizdat, 1951.

2. Pukhov G. E. Theory of the method of subcircuits. Elektrichestvo, No. 8, 1952.

3. Zelyakh E. V. Fundamentals of the General Theory of Linear Electric Circuits. Publishing House of the Academy of Sciences of the USSR, 1951.

4. Melnikov N. A. Calculations of Operating Conditions of Electrical-System Networks. Gosenergoizdat, 1950.

5. Melnikov N. A., Knyazevsky B. A. Analogy in transformations of electrical systems with e.m.f. and with transformations. VZEI, Collected Works, 1951.