PROGRAM FOR CALCULATING THE STEADY-STATE OPERATING MODE OF ELECTRICAL SYSTEMS

R. I. BORISOV, S. G. SLYUSARENKO

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

This paper considers an algorithm and a program for calculating the steady-state operating modes of electrical systems on the basis of nodal-power equations by the Newton–Raphson method [1].

For carrying out research calculations, an experimental program has been developed for calculating the steady-state operating modes of electrical systems by the nodal-power method. The modification of the method consists in using generalized equations of network powers [2]. These equations correspond to representing the elements of an electrical network, including the elements of long-distance power transmission lines, as four-terminal networks.

Fig. 1.

The nodal-power equations describing the steady-state mode of an electrical system, for the calculated node (Fig. 1), are written in general form as

\[ \sum_{j \in \Pi_i} P_{ij} + P_{\mathrm{H}i} - P_{\Gamma i} = 0;\quad \sum_{j \in \Pi_i} Q_{ij} + Q_{\mathrm{H}i} - Q_{\Gamma i} = 0, \tag{1} \]

where \(P_{\Gamma i}, Q_{\Gamma i}\) are generated powers;
\(P_{\mathrm{H}i}, Q_{\mathrm{H}i}\) are load powers;
\(P_{ij}, Q_{ij}\) are network powers;
\(\Pi_i\) is the set of nodes connected to the calculated one.

In the expressions for network powers, the operating-mode parameters enter in natural quantities, since conversion is carried out automatically through explicit accounting for transformation ratios in the expressions for the generalized constants of four-terminal networks.

The topology of the network is revealed by means of a special table consisting of the impedance moduli of the connecting elements. An example of constructing such a table is shown in Fig. 2.

The table is a square matrix. The elements of the rows indicate the presence of connections of the corresponding node with all the others and are used in forming the coefficients \(\dot B\) of the equivalent four-terminal networks. In what follows we shall call them connection rows. Since the matrix is symmetric with respect to the main diagonal, only the triangular form is used. The order in which the connection rows are formed is indicated by arrows.

Fig. 2.

The following operating-mode parameters may be taken as dependent variables: \(P_{\Gamma}, Q_{\Gamma}; P_{\mathrm{H}} Q_{\mathrm{H}}; U; \delta; |\dot B|; K_{\mathrm{T}}\). The program, whose block diagram is given in Fig. 3, is compiled in machine codes for the BESM-4 computer and consists of the following main parts:

1) a forming part (blocks 0, 1);
2) the actual program for calculating the steady-state operating mode (blocks 2–18);
3) a subprogram transforming the second part for calculating power losses (block 17).

Let us consider the functional purposes of some of the blocks in more detail.

Block 0

1) processes the initial data;
2) forms the logical part of the program.

Block 1 restores the instructions.

Block 3

1) selects, from the matrix of triangular form, the connection row of the calculated node;
2) selects, from the arrays of initial data, \(U_i; \delta_i; P_{\Gamma i}, Q_{\Gamma i}; P_{\mathrm{H} i}, Q_{\mathrm{H} i}\);
3) forms the operating-mode parameters that are dependent variables;
4) converts \(\delta_i\) into radians.

Block 6

1) selects \(U_j\) and \(\delta_j\) from the initial-data vectors;
2) checks whether the branch under consideration is a transformer, and if this proves to be the case, then

![Fig. 3 flowchart]

Fig. 3.

Flowchart labels:

1. Input
2. —
3. Start of the cycle over nodes
4. Formation of the row of connections and of the initial data of the calculated node
5. Start of the branch cycle
6. \(z_{ij}=0?\) — Yes / No
7. Formation of the initial data of the branch under consideration
8. Start of the double cycle
9. Formation of the vector of power mismatches \(W(X)\) and of the Jacobi matrix \(\dfrac{\partial W}{\partial X}\)
10. \(\Psi'_B=\Psi_B-\dfrac{\pi}{2}\)
11. Counter of the double cycle
12. Counter of the number of branches
13. Final formation of \(W(X)\) and \(\dfrac{\partial W}{\partial X}\)
14. Counter of the number of nodes
15. — Yes / No
16. \(X := X+\Delta X\)
17. \(W(X)<\varepsilon?\) — Yes / No
18. Forming block
19. \(\Delta X=-\left|\dfrac{\partial W}{\partial X}\right|^{-1}W(X)\)
20. Print
    Stop

a) selects the corresponding value of the transformation coefficient from the array of initial data or forms it, if it is a dependent variable;

b) calculates \(\dot B\) and \(\dot M\), taking into account the character of the connection to the calculated node;

3) takes into account whether \(B_{ij}\), \(U_j\), \(\delta_i\) are dependent variables or not, and converts \(\Psi_B\), \(\Psi_M\), \(\delta_i\) into radians.

Block 8, the first time for the active and the second time for the reactive components of the mismatch functions of the calculated node, performs

1) calculation of the network power and accumulation of the value in the corresponding place of the power-mismatch vector;

2) calculation of the values of the derivatives with respect to the dependent variables from

expressions of the network power, accumulating them in the corresponding positions of the Jacobian matrix.

Block 12 assembles the Jacobian matrix with the values of the derivatives of the imbalance functions of the calculation nodes with respect to \(P_{\Gamma}, Q_{\Gamma}; P_{\mathrm{H}}, Q_{\mathrm{H}}\).

Block 14 performs the check.

If block 17 has operated, it transfers control to block 19.

Block 17

1) replaces block 8 with the block for calculating losses;
2) disables the operation of block 12.

Block 18

1) calculates the correction vector \(\Delta X\) by calling standard programs;
2) converts \(\Delta \delta\) of the correction vector into degrees. The operating-mode parameters that are dependent variables are supplied with special flags. The formation of dependent variables is carried out according to the requirement of [2]. The program is capable of calculating a system whose equivalent circuit contains up to 20 nodes.

REFERENCES

1. L. A. Krumm. Application of the Newton–Raphson method for calculating the steady-state operating mode of complex electrical systems. Izv. AN SSSR, “Energetika i transport,” 1965, No. 5.

2. R. I. Borisov, S. G. Slyusarenko. Toward a unified algorithm for calculating network powers. In the present collection.