ON THE CALCULATION OF COMPLEX CLOSED NETWORKS TAKING INTO ACCOUNT THE STATIC CHARACTERISTICS OF LOADS

G. A. Nazarenkova

(Presented by Professor, Doctor of Technical Sciences V. K. Shcherbakov)

In considering the need to take into account the static characteristics of loads in the calculation of power-transmission lines, it was found [6] that the error in a calculation without taking into account the static characteristics of loads reaches about 10%. Moreover, the error increases as the voltage deviation at the consumer from the nominal value increases, and the voltage deviation from the nominal value at the consumer will be the greater, the greater the transmitted power and the length of transmission.

Modern closed networks have individual sections of the network with lengths on the order of 80–100 km and more. The configuration of closed networks may be very complex. Therefore, determining the load flow in such networks is a very laborious problem.

Fig. 1.

Loads at the nodes of a closed network are usually specified by powers, which, when determining the load flow, are considered constant. In reality, the power of loads fed from the network varies depending on the applied voltage.

In the practice of calculating closed networks there are several methods for calculating them (the network-transformation method, the loop-equation method, the node-voltage method, etc.).

The most general of these is the node-voltage method. Using this calculation method, it is easiest to take into account the static characteristics of the loads. According to Kirchhoff’s law, equations of current balance are written for the network nodes. The loads in the network branches can be expressed through the node voltages and the branch constants. As a result, a system of equations is obtained with the node voltages as unknowns [1].

For Fig. 1 the equations of current balance at the nodes have the following form:

\[ \left. \begin{aligned} \frac{\dot U_0-\dot A_{01}\dot U_1}{\dot B_{01}} +\frac{\dot U_2-\dot A_{21}\dot U_1}{\dot B_{21}} &=\dot I_1;\\[6pt] \frac{\dot U_1-\dot A_{12}\dot U_2}{\dot B_{12}} +\frac{\dot U_3-\dot A_{32}\dot U_2}{\dot B_{32}} &=\dot I_2;\\[6pt] \frac{\dot U_0-\dot B_{03}\dot U_3}{\dot B_{03}} +\frac{\dot U_2-\dot A_{23}\dot U_3}{\dot B_{23}} &=\dot I_3. \end{aligned} \right\} \tag{1} \]

The currents at the nodes must be known. In addition, the voltage of one of the nodes must be specified. Thus, \((n-1)\) equations are formed (\(n\) is the number of nodes). Since the node loads are usually specified in powers, the currents of the right-hand sides of the equations may be replaced by ratios of the powers to the conjugate values of the voltage—

nodes \(\left(\dfrac{\dot S}{\dot U}\right)\). Solving the resulting system of equations, one finds the voltages at the nodes, and then the power flows in the branches of the network.

The obtained system of equations (1) can be written in the generally accepted form [2]:

\[ \left. \begin{aligned} \frac{1}{\dot B_{01}}\dot U_0 -\left(\frac{\dot A_{01}}{\dot B_{01}}+\frac{\dot A_{21}}{\dot B_{21}}\right)\dot U_1 +\frac{1}{\dot B_{21}}\dot U_2 &=\frac{\dot S_1}{\hat U_1};\\ \frac{1}{\dot B_{12}}\dot U_1 -\left(\frac{\dot A_{12}}{\dot B_{12}}+\frac{\dot A_{32}}{\dot B_{32}}\right)\dot U_2 +\frac{1}{\dot B_{32}}\dot U_3 &=\frac{\dot S_2}{\hat U_2};\\ \frac{1}{\dot B_{03}}\dot U_0 -\left(\frac{\dot A_{03}}{\dot B_{03}}+\frac{\dot A_{23}}{\dot B_{23}}\right)\dot U_3 +\frac{1}{\dot B_{23}}\dot U_2 &=\frac{\dot S_3}{\hat U_3}, \end{aligned} \right\} \tag{2} \]

or

\[ \dot Y_{10}\dot U_0-\dot Y_{11}\dot U_1+\dot Y_{12}\dot U_2 =\frac{\dot S_1}{\hat U_2}; \]

\[ \dot Y_{21}\dot U_1-\dot Y_{22}\dot U_2+\dot Y_{23}\dot U_3 =\frac{\dot S_2}{\hat U_2}; \]

\[ \dot Y_{30}\dot U_0-\dot Y_{33}\dot U_3+\dot Y_{32}\dot U_2 =\frac{\dot S_3}{\hat U_3}. \]

In general form, the system of node-voltage equations may be written as:

\[ \left. \begin{aligned} -\dot Y_{11}\dot U_1-\dot Y_{12}\dot U_2+\ldots+\dot Y_{1n}\dot U_n &=\frac{\dot S_1}{\hat U_1};\\ \dot Y_{21}\dot U_1-\dot Y_{22}\dot U_2+\ldots+\dot Y_{2n}\dot U_n &=\frac{\dot S_2}{\hat U_2};\\ &\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\\ \dot Y_{n1}\dot U_1+\dot Y_{n2}\dot U_2+\ldots-\dot Y_{nn}\dot U_n &=\frac{\dot S_n}{\hat U_n}. \end{aligned} \right\} \tag{3} \]

Thus, a system of nonlinear equations is obtained. To solve this system of equations directly is a very difficult task. Most simply, a system of nonlinear equations is solved by the iteration method [2 and 4]. Before using the iteration method, it is necessary to bring the system of equations to a form convenient for iteration. This can be done in two ways. In one of the methods it is assumed that the network has one voltage level, and the node with specified voltage coincides with the node adjacent to the regulating station (for example, \(\dot U_2\)). From the corresponding node equations (3), one node voltage is isolated from each. For this purpose all equations of the system are divided by their diagonal coefficients.

\[ \left. \begin{aligned} \dot U_1&=\dot a_{12}\dot U_2+\dot a_{13}\dot U_3+\ldots+\dot a_{1n}\dot U_n -\dot b_{11}\frac{\dot S_1}{\hat U_1};\\ \dot U_2&=\dot a_{21}\dot U_1+\dot a_{23}\dot U_3+\ldots+\dot a_{2n}\dot U_n -\dot b_{22}\frac{\dot S_2}{\hat U_2};\\ \dot U_3&=\dot a_{31}\dot U_1+\dot a_{32}\dot U_2+\ldots+\dot a_{3n}\dot U_n -\dot b_{33}\frac{\dot S_3}{\hat U_3};\\ &\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\\ \dot U_n&=\dot a_{n1}\dot U_1+\dot a_{n2}\dot U_2+\ldots+\dot a_{n(n-1)}\dot U_{n-1} -\dot b_{nn}\frac{\dot S_n}{\hat U_n}; \end{aligned} \right. \]

\[ \left. \begin{gathered} \left( \dot a_{12}=\frac{Y_{12}}{Y_{11}};\quad \dot a_{21}=\frac{Y_{21}}{Y_{22}}\ \text{and so on};\right.\\ \left. \dot b_{11}=\frac{1}{Y_{11}};\quad \dot b_{22}=\frac{1}{Y_{22}}\ \text{and so on}. \right) \end{gathered} \right\} \tag{4} \]

The system of equations (3) can also be reduced to a form convenient for iteration in another way.

In the equations of system (3), the known terms are transferred to the right-hand side, and the resulting system of equations is solved for the unknown node voltages (for this it is best to use matrices). As a result one obtains:

\[ \left. \begin{aligned} \dot U_1 &= \dot a_{12}\dot U_2+\dot b_{11}\frac{\dot S_1}{\dot U_1} +\dot b_{13}\frac{\dot S_3}{\dot U_3}+\ldots+\dot b_{1n}\frac{\dot S_n}{\dot U_1};\\ &\qquad\qquad\qquad\qquad\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ \dot U_n &= \dot a_{n2}\dot U_2+\dot b_{n1}\frac{\dot S_1}{\dot U_1} +\dot b_{n3}\frac{\dot S_3}{\dot U_3}+\ldots+\dot b_{nn}\frac{\dot S_n}{\dot U_n}. \end{aligned} \right\} \tag{5} \]

Having assigned initial values of the node voltages, from the system of equations (4) we find their first refined value. Then, from the obtained node voltages, we find the second approximations, and so on. Similarly for equations (5). To obtain the result more rapidly, accelerated iteration may be used (which consists in the immediate use of the obtained approximations of the unknowns for determining the subsequent unknowns). Consequently, when solving the system of equations for the node voltages by the iteration method, allowance for the static characteristics of loads does not complicate the calculation. After the first approximations of the node voltages have been determined, the power consumed by the current receivers at the given voltages is found from the static characteristics of the loads. Then the second approximations of the node voltages are found. From the second approximations of the voltages, the power consumed by the current receivers is again determined from the static characteristics, and so on. Accounting for the static characteristics of the loads does not slow down the iteration process. In setting up the system of equations for the node voltages, the load is usually considered as specified on the high-voltage side of the step-down transformers, and the voltages are determined there as well. If it is necessary to find the voltage on the low-voltage side of the transformers, then in the case where the calculation is carried out without taking into account the static characteristics of the loads, this causes no difficulty. It is more difficult to solve this question in a calculation that takes into account the static characteristics of the loads. Usually, however, the permissible deviations of the voltage from nominal in the various operating conditions of the network are given with respect to the nominal network voltage on the low-voltage side of the transformers. It is therefore more convenient that the load be specified on the low-voltage side of the step-down transformers, and that the system of equations for the node voltages be compiled taking into account the constant step-down transformers.

Fig. 2.

Let there be the network shown in Fig. 2. The loads are specified by powers on the low-voltage side of the step-down substations.

The phase voltage of the feeding point \(U_0\) is given.

The parameters of the lines and transformers are known. It is necessary to find the voltages on the low-voltage side of the step-down substations \(U_1, U_2, U_3\), referred to the high-voltage side.

The relation between the phase voltages and currents at the beginning and end of any link of power transmission may be expressed as:

\[ \begin{gathered} \dot U_n=\dot A\dot U_k+\dot B\dot I_k;\quad \dot I_n=\dot C\dot U_k+\dot D\dot I_k,\\ \text{or}\\ \dot U_k=\dot D\dot U_n-\dot B\dot I_n;\quad \dot I_k=-\dot C\dot U_n+\dot A\dot I_n, \end{gathered} \left\} \tag{6} \]

where \(A, B, C, D\) are the generalized constants of the given transmission element, whose values depend on the equivalent circuit of the element. It is assumed that line sections are replaced by a \(\Pi\)-type equivalent circuit, and transformers by a T-type equivalent circuit.

The current-balance equations at the nodes may be written as:

\[ I_{\mathrm{I}} = I_{a\mathrm{I}} - I_{b\mathrm{I}}; \]

\[ I_{\mathrm{II}} = I_{b\mathrm{II}} + I_{v\mathrm{II}}; \]

\[ I_{\mathrm{III}} = I_{g\mathrm{III}} - I_{v\mathrm{III}}. \]

As a result of substituting the current values from equations (6) and carrying out certain transformations, a system of three equations with three unknown node voltages is obtained:

\[ \left( \dot C_1+\frac{\dot A_1\dot A_a}{\dot B_a} +\frac{\dot A_1\dot A_b}{\dot B_b} \right)\dot U_1 -\frac{\dot A_2\dot U_2}{\dot B_b} = \frac{\dot U_0}{\dot B_a} - \left( \frac{\dot A_a\dot B_1}{\dot B_a} +\frac{\dot A_b\dot B_1}{\dot B_b} +\dot D_1 \right)\dot I_1 +\frac{\dot B_2\dot I_2}{\dot B_b}; \]

\[ -\frac{\dot A_1\dot U_1}{\dot B_b} + \left( \dot C_2+\frac{\dot A_b\dot A_2}{\dot B_b} +\frac{\dot A_v\dot A_2}{\dot B_v} \right)\dot U_2 -\frac{\dot A_3\dot U_3}{\dot B_v} = \]

\[ = \frac{\dot B_1\dot I_1}{\dot B_b} - \left( \dot D_2+\frac{\dot A_b\dot B_2}{\dot B_b} +\frac{\dot A_v\dot B_2}{\dot B_v} \right)\dot I_2 +\frac{\dot B_3\dot I_3}{\dot B_v}; \]

\[ -\frac{\dot A_2\dot U_2}{\dot B_v} + \left( \dot C_3+\frac{\dot A_2\dot A_3}{\dot B_2} +\frac{\dot A_v\dot A_3}{\dot B_v} \right)\dot U_3 = \]

\[ = \frac{\dot U_0}{\dot B_2} +\frac{\dot B_2\dot I_2}{\dot B_v} - \left( \frac{\dot A_b\dot B_3}{\dot B_2} +\frac{\dot A_v\dot B_3}{\dot B_v} +\dot D_3 \right)\dot I_3. \]

Analogously, one may form a system of nodal equations for any number of nodes in the network. The number of terms in the coefficients of the voltage and current of the node for which the equation is being formed is equal to the number of incoming and outgoing branches. The structure of the coefficients for currents and unknown voltages is simple. Writing all the coefficients for any node of the network is not particularly difficult. The node currents are determined as

\[ \frac{\dot S_m}{\dot U_m}. \]

When the static characteristics of the powers consumed by the loads are taken into account in the calculation, they will be functions of the voltage applied to them:

\[ \dot S_m=f(\dot U_m). \]

In general form, the system of equations for nodal voltages, with transformers taken into account, may be written as:

\[ \left. \begin{aligned} \dot a_{11}\dot U_1-\dot a_{12}\dot U_2-\ldots-\dot a_{1n}\dot U_n &=\dot a_{10}\dot U_0-\dot b_{11}\dot I_1+\ldots+\dot b_{1n}\dot I_n;\\ -\dot a_{21}\dot U_1+\dot a_{22}\dot U_2-\ldots-\dot a_{2n}\dot U_n &=\dot a_{20}\dot U_0+\dot b_{21}\dot I_1-\dot b_{22}\dot I_2+\ldots+\dot b_{2n}\dot I_n;\\ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\\ -\dot a_{n1}\dot U_1-\dot a_{n2}\dot U_2-\ldots+\dot a_{nn}\dot U_n &=\dot a_{n0}\dot U_0+\dot b_{n1}\dot I_1+\ldots-\dot b_{nn}\dot I_n, \end{aligned} \right\} \tag{7} \]

where \(a\), with the corresponding indices, are the coefficients of the node voltages; \(b\), with the corresponding indices, are the coefficients of the currents. If the known right-hand sides of the equations are denoted by \(C\), then the system of equations (7) may also be written in matrix form:

\[ \dot A_{mn}\dot U_n=\dot C. \]

or, otherwise:

$$ \left\{ \begin{array}{cccc} -\dot a_{11}, & -\dot a_{12}, & \ldots, & -\dot a_{1n}\\ -\dot a_{21}, & \dot a_{22}, & \ldots, & -\dot a_{2n}\\ \cdot & \cdot & \cdot & \cdot\\ -\dot a_{n1}, & -\dot a_{n2}, & \ldots, & -\dot a_{nn} \end{array} \right\} \left\{ \begin{array}{c} \dot U_1\\ \dot U_2\\ \cdot\\ \dot U_n \end{array} \right\} = \left\{ \begin{array}{c} \dot C_1\\ \dot C_2\\ \cdot\\ \dot C_n \end{array} \right\}. \tag{8} $$

The resulting system of equations is small, and it is simplest to solve such a system by the iteration method.

The system of equations (7) can be brought to a form convenient for iteration by dividing each equation of the system by its diagonal element (since the diagonal elements of the matrix \(A\) predominate over the remaining elements):

$$ \dot U_1-\frac{\dot a_{12}}{\dot a_{11}}\dot U_2-\ldots-\frac{\dot a_{1n}}{\dot a_{11}}\dot U_n =\frac{\dot C_1}{\dot a_{11}}; $$

$$ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot $$

$$ -\frac{\dot a_{n1}}{\dot a_{nn}}\dot U_1-\frac{\dot a_{n2}}{\dot a_{nn}}\dot U_2-\ldots+\dot U_n =\frac{\dot C_n}{\dot a_{nn}}, $$

or

$$ \left. \begin{array}{l} \dot U_1=\dot a'_{12}\dot U_2+\ldots+\dot a'_{1n}\dot U_n+\dot C'_1;\\[4pt] \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\[4pt] \dot U_n=\dot a'_{n1}\dot U_1+\ldots+\dot a'_{n,n-1}\dot U_{n-1}+\dot C'_n. \end{array} \right\} \tag{9} $$

The obtained system of equations can also be transformed to a form convenient for iteration in another way. To do this, it is necessary to solve the system of equations (8) with respect to the unknown node voltages.

In general form this solution may be written as follows:

$$ (\dot U_1,\dot U_2,\ldots,\dot U_n)=A^{-1}(C_1,C_2,\ldots,C_n), $$

where \(A^{-1}\) is the inverse matrix.

In expanded form:

$$ \left. \begin{array}{l} \dot U_1=\dot a_{10}\dot U_0+\dot b_{11}\dfrac{\dot S_1}{\hat U_1} +\dot b_{12}\dfrac{\dot S_2}{\hat U_2} +\ldots+\dot b_{1n}\dfrac{\dot S_n}{\hat U_n};\\[6pt] \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\[6pt] \dot U_n=\dot a_{n0}\dot U_0+\dot b_{n1}\dfrac{\dot S_1}{\hat U_1} +\dot b_{n2}\dfrac{\dot S_2}{\hat U_2} +\ldots+\dot b_{nn}\dfrac{\dot S_n}{\hat U_n}. \end{array} \right\} \tag{10} $$

Thus, the system of nodal equations may be compiled both in the form of the system of equations (9) and in the form of the system of equations (10). It is necessary to determine under what conditions the stability of the iteration process will be ensured when solving equations of this kind. The rate of convergence of the iteration process depends on how successfully the form of the equations is chosen and how successfully the initial values of the unknowns are chosen, on the basis of which the entire iterative process will be constructed. It is assumed that the calculation is carried out taking into account the static characteristics of the loa-

loads. Consequently, the load powers in the equations are functions of the applied voltage.

Typical static characteristics for a mixed load are determined sufficiently well by the following empirical formulas [3]:

\[ \left. \begin{aligned} P &= P_n\left(0.83 - 0.3\frac{U}{U_n} + 0.47\frac{U^2}{U_n^2}\right);\\ Q &= Q_n\left(4.9 - 10.1\frac{U}{U_n} + 6.2\frac{U^2}{U_n^2}\right). \end{aligned} \right\} \tag{11} \]

For voltage deviations of up to 20% from the nominal value, the curves constructed from these formulas coincide with the static characteristics for a mixed load, and for deviations above 20% the divergence of the curves is insignificant.

When studying the convergence of the iteration process, the value of the load power \(S\) computed from formulas (11) is substituted into the system of equations (9)—(10) instead of the load power.

The general form of the system of equations (9) and (10) is:

\[ \dot U_1 = \dot F_1(\dot U_1,\dot U_2,\ldots,\dot U_n); \]

\[ \ldots\ldots\ldots\ldots\ldots \]

\[ \dot U_n = F_n(\dot U_1,\dot U_2,\ldots,\dot U_n). \]

For equations of types (9) and (10), the iteration process converges when the following conditions are satisfied [5]:

\[ \left|\frac{\partial \dot F_1}{\partial \dot U_1}\right| + \left|\frac{\partial \dot F_2}{\partial \dot U_1}\right| +\ldots+ \left|\frac{\partial \dot F_n}{\partial \dot U_1}\right| <1; \]

\[ \left|\frac{\partial \dot F_1}{\partial \dot U_2}\right| + \left|\frac{\partial \dot F_2}{\partial \dot U_2}\right| +\ldots+ \left|\frac{\partial \dot F_n}{\partial \dot U_2}\right| <1; \]

\[ \ldots\ldots\ldots\ldots\ldots \]

\[ \left|\frac{\partial \dot F_1}{\partial \dot U_n}\right| + \left|\frac{\partial \dot F_2}{\partial \dot U_n}\right| +\ldots+ \left|\frac{\partial \dot F_n}{\partial \dot U_n}\right| <1. \]

In order for convergence to be sufficiently rapid and for it to make sense under the given conditions to apply the iteration method, it is necessary that each of these quantities be considerably less than unity. Thus, in order to obtain criteria for convergence of the iteration process for the given system of equations, it is necessary to differentiate all equations of the system with respect to all unknowns. In doing so, one has to deal with complex numbers and with quantities conjugate to the unknown voltages. In the calculation of closed networks, when the lengths of individual line sections are of the order of 80–100 km or less, the transverse component of the voltage drop has little effect on the magnitude of the voltage modulus. Consequently, as a first approximation it may be assumed that all unknown voltages are directed along the real axis. Taking into account that, in obtaining the convergence criterion for the iteration process, the moduli of the corresponding quantities are needed, this assumption will not introduce a large error. As calculation practice has shown, the convergence criterion for equations of the form (10) is almost always considerably less than unity, while for equations of the form (9) it is close to unity. Consequently, the iteration process for equations of the form (10) will converge more rapidly than for equations (9).

Conclusions

1. It is simplest to take into account the static characteristics of loads in the calculation of closed networks when using the nodal-voltage method.

2. In doing so, it is desirable to carry out the calculation with allowance for step-down transformers by the iteration method.

3. The iteration process converges faster for a system of equations of the form (10).

Literature

1. Shcherbakov V. K., Calculation of voltages and power-flow distribution in complex electrical systems under normal operating conditions, Jubilee Collection of the Tomsk Industrial Institute, 1940.

2. Fazylov Kh. F., Theory and Methods for Calculating Electrical Systems, Academy of Sciences of the Uzbek SSR, 1953.

3. Melnikov N. A., Calculations of Operating Conditions of Networks of Electrical Systems, Gosenergoizdat, 1950.

4. Fadeeva V. N., Computational Methods of Linear Algebra, Gostekhizdat, 1950.

5. Scarborough J., Numerical Methods of Mathematical Analysis, Gostekhizdat, 1934.

6. Shcherbakov V. K. and Nazarenkova G. A., Is it permissible in calculations of electrical networks to neglect the change in load power under voltage deviations at consumer substations? Izvestiya TPI, vol. 72, 1952.