CALCULATION OF THE PARAMETERS OF COMPENSATING DEVICES FOR LONG-DISTANCE POWER TRANSMISSION LINES WITH INTERMEDIATE SYSTEMS FOR A SPECIFIED TRANSFER CAPACITY UNDER STATIC STABILITY CONDITIONS

V. I. GOTMAN

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

Power transmission lines with voltages of 500–1000 kV are intended for transmitting a significant quantity of electric power over distances of tens to hundreds of kilometers. However, the limiting value of the transmitted power and the transmission distance are restricted by the need to satisfy certain requirements. First of all, these requirements consist in ensuring static and dynamic stability, and also in limiting all possible kinds of overvoltages under the conditions of normal operation of the equipment.

The transfer capacity of a power transmission line is a techno-economic category, since ensuring a specified value of transmitted power over a specified distance while observing the above-listed conditions requires certain technical measures that are directly associated with costs. Thus, the problem of ensuring a specified transfer capacity is a combination of a number of technical and economic problems, which at certain stages of their study may be solved independently.

One of the technically acceptable and economically effective means of increasing the transfer capacity of long-distance power transmission lines is series capacitive compensation of the inductive resistance of the line [1].

The problem of selecting the parameters of compensating devices and studying the influence of their location on transfer capacity and voltage distribution for long-distance power transmission lines (LPTL) without intermediate systems (IS) has been addressed at various times by O. B. Tollyto [2, 3, 4], T. V. Zaslavskaya [5], R. I. Borisov [6, 7], I. P. Suda [8], G. E. Pospelov [1], and others.

When selecting the parameters of a series-compensation installation (SCI) for power transmission lines without IS, it is assumed that the voltages at the beginning and end of the power transmission line are constant and that the power of the receiving system is infinite; under these conditions the criterion of aperiodic stability of the transmitting station is the equation

\[ \delta_{12\mathrm{pr}} = 90^\circ \quad (R = 0) \]

and, when the active resistances of the conductors are taken into account, \(\delta_{12\mathrm{pr}} = \Psi_{\mathrm{B}}\), where \(\delta_{12\mathrm{pr}}\) is the limiting angle of divergence of the voltage vectors at the beginning and end of the transmission line. The stability margin in power is taken into account by the corresponding margin in angle.

The problem of choosing the parameters of series compensating devices (SCD) for a specified transfer capacity under the conditions of ensuring static stability and voltage distribution for long-distance power transmission lines without intermediate systems should be regarded as solved [9].

The development of power systems predetermines the gradual “growth” of power transmission lines by means of intermediate connections. Under these conditions, the importance of long-distance power transmission lines increases, since, along with their traditional functions of transmitting power in transit, they will perform the functions of intersystem ties, thus constituting the backbone of large power pools. Intermediate systems are a complex structural element, whose nominal voltage may be equal to, or one or two steps below, the voltage of the long-distance line. In terms of their capacity, intermediate systems will be commensurate with the capacity of the sending station or the receiving system. In the most general case it should be assumed that intermediate systems are unbalanced in active power and, consequently, may operate both in the mode of consuming active power and in the mode of supplying it.

The problem of calculating the parameters of compensating devices for long-distance transmission lines with intermediate systems under static-stability conditions presents considerable difficulties.

Taking into account generally known assumptions, the limiting operating conditions of arbitrarily complex systems, for given initial conditions, can be assessed by practical criteria. In the literature, expressions of practical criteria for complex systems can be found only in matrix form, which are practically unsuitable for an analytical solution of the problem posed because of the high order of the matrix expression. Thus, for a long-distance transmission line with one intermediate system consisting of one load node and a generating station, the order of the matrix is 10! The indicated difficulty was overcome in the following way.

1. Long-distance transmission lines together with intermediate systems can be represented by chain circuits. In calculating limiting operating conditions (and also, in a number of cases, in calculating normal operating conditions) of such circuits, intermediate systems can most simply and at the same time rigorously be taken into account by generalized static characteristics of active and reactive power with respect to the voltage at the node of their connection to the long-distance line. The static characteristics make it possible, at a certain stage in solving the problem posed, to abstract from the specific nature of the system being represented equivalently and to take into account, in generalized form, the reaction of the intermediate system to a change in the operating condition of the long-distance line. The static characteristics are characterized by the slope coefficients \(dQ/dU\), \(dP/dU\), which are determined from calculated or specified operating parameters. A method for calculating the slope coefficients for systems has been developed.

2. From the matrix expression of the practical criterion, compiled with account for the static characteristics of the elements of power systems, we obtained the equation

\[ \sum^{n} \beta_i - \eta_j \sum^{n} \alpha_i = 0, \tag{1} \]

where

\[ \beta_i = \frac{dQ_i}{dU_j}, \qquad \alpha_i = \frac{\partial P_i}{\partial U_j}, \qquad \eta_j = \left. \frac{\partial Q_i}{\partial P_i} \right|_{U_j = \mathrm{const}} . \]

Equation (1) is the algebraic form of the practical criterion, written through the slope coefficients of the static characteristics of the parts of the system adjacent to node \(j\) (an arbitrary node of the system). The directions of the flows of active and reactive power in the sections are taken toward node \(j\). With this direction, the stable state of the system corresponds to the condition

\[ \sum \beta_i - \eta_j \sum \alpha_i < 0 . \]

The derivative $\eta_1$ is determined for that part of the system in which the balancing station or node is located. Owing to the natural positive regulating effect of the static characteristics of the reactive power of the intermediate system (IS), stabilizing the voltage at the junction node contributes to increasing the limiting value of the power transmitted over the long-distance transmission line (LDTL). Any law can be artificially imparted to the natural static characteristics of the reactive power of the IS by connecting regulated sources of reactive power. In this case the slope of the static characteristics can be brought to any desired value.

Proceeding from these premises, the transfer capacity of an LDTL can be increased in two ways: 1) either by increasing the slope coefficients of the static characteristics of the IS by connecting reactive sources at the junction node of the IS; 2) or by acting on the slope coefficients of the static characteristics of sections of the LDTL by introducing longitudinal capacitive compensation; 3) or by a combination of both methods.

The effectiveness of one method or another for a particular scheme can be clarified only by an economic comparison of alternatives. The problem of increasing the transfer capacity by the first method reduces to finding the required values of the slope coefficients of the IS ($\beta_{\mathrm{p}}$) from condition (1). These values of $\beta_{\mathrm{p}}$ must then be compared with the natural ones, and the necessary correction of the values $\beta_{\mathrm{p}}$ must be provided. The values of the reactive powers of additional sources are determined from the balance condition for the corresponding nodes.

Let us consider in greater detail the second method of increasing the transfer capacity. The slope coefficients $\alpha_i$ are due to the regulating effects of the static characteristics of the active power of the load nodes and to the active resistances of the LDTL sections. Their values are 10–15 times smaller than the corresponding values $\beta_i$. For selecting the parameters of compensating devices it is quite acceptable to use lossless equations [9]. Taking $\alpha_i=0$, equation (1) is written as

$$ \sum \beta_i = 0 . $$

Fig. 1. Compensated LDTL with an intermediate system

We shall consider the procedure for calculating the parameters of compensating devices as applied to the scheme of Fig. 1, which shows an LDTL with one IS. For generality, let us assume that the lengths of the LDTL sections are unequal, the compensating devices are installed on both sections and asymmetrically, the active powers of the sections are specified, and $P_1 \ne P_2$. In addition, the operating parameters corresponding to the limiting mode of the system must be specified, i.e., $U_1—U_7$. Depending on the degree of perfection of the voltage regulators installed on the generators of the transmitting station, either the voltage on the high-voltage side of the step-up transformers, or $U_{\mathrm{g}}$ or $E'_q$, can be maintained at an unchanged level. In the latter cases the constants of the first four-terminal network must take into account the reactances of the terminal devices. In what follows we shall assume that $U_1, U_2=\mathrm{const}$,

the former by virtue of “strong-action” regulators, the latter by assuming that the receiving system has infinite power. The value \(\beta_{\mathrm{p}}\) is determined for a specific transmission-system scheme and, consequently, is taken as a known quantity (for the adopted direction, \(\beta_{\mathrm{p}}\) is negative).

For the adopted scheme it is necessary to have 6 independent equations (each controlled series capacitor contains three unknowns). Naturally, only five can be written:

\[ Q_{51}=-\frac{U_5^2 A_{1\mathrm{e}}}{B_{1\mathrm{e}}} +\sqrt{\left(\frac{U_1U_5}{B_{1\mathrm{e}}}\right)^2-P_1^2} \tag{2} \]

\[ Q_{15}=\frac{U_1^2D_{1\mathrm{e}}}{B_{1\mathrm{e}}} -\sqrt{\left(\frac{U_1U_5}{B_{1\mathrm{e}}}\right)^2-P_1^2}, \tag{3} \]

\[ Q_{52}=-\frac{U_5^2D_{2\mathrm{e}}}{B_{2\mathrm{e}}} +\sqrt{\left(\frac{U_5U_2}{B_{2\mathrm{e}}}\right)^2-P_2^2} \tag{9} \]

\[ Q_{25}=\frac{U_2^2A_{2\mathrm{e}}}{B_{2\mathrm{e}}} -\sqrt{\left(\frac{U_5U_2}{B_{2\mathrm{e}}}\right)^2-P_2^2}, \tag{5} \]

\[ \beta_{51}+\beta_{52}+\beta_{\mathrm{p}}=0. \tag{6} \]

Matrix and analytical expressions for \(\beta_{51}\) and \(\beta_{52}\) are given in the Appendix.

From the condition limiting the voltages across the capacitor plates of the controlled series capacitors, we determine the balancing reactive powers at the ends of the sections from the expressions:

\[ Q_{51}=-\frac{U_5^2A_2}{B_2} +\sqrt{\left(\frac{U_4U_5}{B_2}\right)^2-P_1^2}, \tag{7} \]

\[ Q_{15}=\frac{U_1^2D_1}{B_1} -\sqrt{\left(\frac{U_1U_3}{B_1}\right)^2-P_1^2}, \tag{8} \]

\[ Q_{52}=-\frac{U_5^2D_3}{B_3} +\sqrt{\left(\frac{U_5U_6}{B_3}\right)^2-P_2^2} \tag{9} \]

\[ Q_{25}=\frac{U_2^2A_4}{B_4} -\sqrt{\left(\frac{U_7U_2}{B_4}\right)^2-P_2^2}, \tag{10} \]

where

\[ [A]_i= \begin{bmatrix} \cos\lambda_i & j\sin\lambda_i\\ j\sin\lambda_i & \cos\lambda_i \end{bmatrix} \quad \text{— for a lossless line, in relative units;} \]

\[ \lambda_i=\alpha_0l_i \quad \text{— the wave length of the sections.} \]

The best use of controlled series capacitors is achieved with equal voltages on both sides of the capacitor bank. This makes it possible to obtain the greatest phase-shifting effect \((\delta_{\mathrm{sc}})\) from the controlled series capacitor; at the same time, the voltages on the plates \((U_3, U_4, U_6, U_7)\) must not exceed a certain value limited by their insulation condition. In calculations by formulas (7)--(10), they may be taken equal to \(U_{\mathrm{nom}}\).

The five equations (2)--(6) presented above contain 6 unknowns \((A_{1\mathrm{e}}, D_{1\mathrm{e}}, B_{1\mathrm{e}}, A_{2\mathrm{e}}, D_{2\mathrm{e}}, B_{2\mathrm{e}})\). The problem, as is evident, is indeterminate, which is quite natural, since for each specified combination of parameters of the first controlled series capacitor, under the conditions of the limiting mode (6), there will unambiguously correspond a quite definite combination for the second controlled series capacitor, i.e., a set of solutions exists. The problem becomes determinate after introducing a coupling equation between the parameters of the first and second controlled series capacitors. As such an equation we have adopted \(\delta_{15}=\delta_{52}\), which can be represented in the following form

\[ \frac{P_1B_{1\mathrm{e}}}{U_1}=\frac{P_2B_{2\mathrm{e}}}{U_2}. \tag{11} \]

Owing to the adopted coupling, a scheme with arms deliberately asymmetric relative to the point of connection of the intermediate system becomes symmetric after compensation; moreover, not only the circuit asymmetry is taken into account, but also the operating-condition asymmetry, which is accounted for through the active powers of the sections and the voltages \(U_1\) and \(U_2\).

As is known [1, 9], symmetric schemes have substantial advantages over asymmetric ones, which are manifested in the fact that, first, there is a better voltage distribution to ensure

for one and the same transfer capacity a smaller expenditure of compensating devices is required; such schemes are simpler in operation and have lower losses. Secondly, as we established earlier, the location of the intermediate system at the electrical center creates conditions for it under which its favorable effect on the operating mode of the power transmission as a whole is manifested with the greatest effectiveness.

The simultaneous solution of equations (2)--(6), (11) makes it possible to find the equivalent constants of both sections:

\[ B_{1e}=\frac{U_1U_5}{\sqrt{2P_1}} \sqrt{\,1+\sqrt{\frac{a_0^2}{a_0^2+2P_1^2}}\,}, \qquad B_{2e}=\frac{U_2}{K_pU_1}B_{1e}, \]

\[ A_{1e}=\left[\sqrt{\left(\frac{U_1U_5}{B_{1e}}\right)^2-P_1^2}-Q_{51}\right]\frac{B_{1e}}{U_5^2}, \qquad A_{2e}=\left[\sqrt{\left(\frac{U_2U_5}{B_{2e}}\right)^2-P_2^2}+Q_{25}\right]\frac{B_{2e}}{U_2^2}, \]

\[ D_{1e}=\left[\sqrt{\left(\frac{U_1U_5}{B_{1e}}\right)^2-P_1^2}+Q_{15}\right]\frac{B_{1e}}{U_1^2}, \qquad D_{2e}=\left[\sqrt{\left(\frac{U_2U_5}{B_{2e}}\right)^2-P_2^2}-Q_{52}\right]\frac{B_{2e}}{U_5^2}, \]

\[ C_{1e}=\frac{1-A_{1e}D_{1e}}{B_{1e}}, \qquad C_{2e}=\frac{1-A_{2e}D_{2e}}{B_{2e}}, \]

where

\[ a_0=\frac{\beta_pU_5+2(Q_{52}-Q_{51})}{1+K_p}, \qquad K_p=\frac{P_2}{P_1}. \]

The next stage consists in finding the constants of the compensating devices. For the first section the following matrix relation is valid

\[ [\dot A_{1e}]=[\dot A_1][\dot A_{1k}][\dot A_2] \]

or, in expanded form,

\[ \dot A_{1e}=(\dot A_1\dot A_{1k}+\dot B_1\dot C_{1k})\dot A_2+ (\dot A_1\dot B_{1k}+\dot B_1\dot D_{1k})\dot C_2, \tag{12} \]

\[ \dot B_{1e}=(\dot A_1\dot A_{1k}+\dot B_1\dot C_{1k})\dot B_2+ (\dot A_1\dot B_{1k}+\dot B_1\dot D_{1k})\dot D_2, \tag{13} \]

\[ \dot C_{1e}=(\dot C_1\dot A_{1k}+\dot D_1\dot C_{1k})\dot A_2+ (\dot C_1\dot B_{1k}+\dot D_1\dot D_{1k})\dot C_2, \tag{14} \]

\[ \dot D_{1e}=(\dot C_1\dot A_{1k}+\dot D_1\dot C_{1k})\dot B_2+ (\dot C_1\dot B_{1k}+\dot D_1\dot D_{1k})\dot D_2. \tag{15} \]

These equations constitute a linear system with four unknowns \((\dot A_{1k}, \dot B_{1k}, \dot C_{1k}, \dot D_{1k})\), of which only three are independent. It is expedient to reduce the system (12)--(15) to a system of three linear equations with three unknowns. For this purpose it is sufficient to solve equation (14) for \(\dot C_{1k}\) and substitute the resulting relation successively into (12), (13), (15). After minor transformations, taking into account that for a lossless line and series-capacitor compensators the constants are purely real and imaginary quantities, we obtain a system of three algebraic equations with real coefficients:

\[ \cos\lambda_2 A_{1k}-\sin\lambda_2\cdot B_{1k}+0\cdot D_{1k} =\cos\lambda_1 A_{1e}+\sin\lambda_1 C_{1e}, \]

\[ \sin\lambda_2\cos\lambda_2 A_{1k} +(\cos^2\lambda_1-\sin^2\lambda_2)B_{1k} +\sin\lambda_1\cos\lambda_1D_{1k} = \cos\lambda_1\cos\lambda_2B_{1e} +\sin\lambda_1\sin\lambda_2C_{1e}, \tag{16} \]

\[ 0\cdot A_{1k}-\sin\lambda_1B_{1k}+\cos\lambda_1D_{1k} = \cos\lambda_2D_{1e}+\sin\lambda_2C_{1e}. \]

From the known constants \(A_{1k}, B_{1k}, D_{1k}\), it is not difficult to calculate the longitudinal capacitive reactance and the transverse conductances

\[ x_{34}=B_{1k},\qquad Y_{30}=\frac{1-D_{1k}}{B_{1k}},\qquad Y_{40}=\frac{1-A_{1k}}{B_{1k}}. \tag{17} \]

The calculation of the parameters of the CPP for the second section is carried out using equations analogous to (16), (17).

It should be noted that a positive conductance of the CPP corresponds to capacitance, while a negative one corresponds to inductance. This problem has also been solved for a long-distance electric-power transmission line with two intermediate systems when compensating an arbitrary number of sections.

Appendix

Calculation of the Steepness Coefficients of the Static Characteristics of Reactive Power with Respect to Voltage

The steepness coefficients are calculated for a specific circuit from the known operating-condition parameters.

[Figure: a line section with nodes labeled \(U_1,\delta_1\) and \(U_5,\delta_5\); arrows and labels \(P_{15}, Q_{15}\), \(P_{51}, Q_{51}\), and \(\Delta P,\Delta Q\).]

Fig. 2. On the determination of the coefficient
\(\beta_{51}\)

For the first section (Fig. 1), shown in Fig. 2, we have the following conditions: \(U_1=\mathrm{const}\), \(U_5=\mathrm{var}\). The voltage phase is fixed at node 5 \((\delta_5=\mathrm{const})\); this same node is taken as the balancing node for active power. Since \(U_1=\mathrm{const}\), the first node is automatically the balancing node for reactive power. According to the adopted conditions, for this section the matrix equation is valid

\[ \begin{bmatrix} 0 & \dfrac{\partial P_{15}}{\partial U_5} & \dfrac{\partial P_{15}}{\partial \delta_1} & 0 \\ 0 & \dfrac{\partial P_{51}}{\partial U_5} & \dfrac{\partial P_{51}}{\partial \delta_1} & -1 \\ -1 & \dfrac{\partial Q_{15}}{\partial U_5} & \dfrac{\partial Q_{15}}{\partial \delta_1} & 0 \\ 0 & \dfrac{\partial Q_{51}}{\partial U_5} & \dfrac{\partial Q_{51}}{\partial \delta_1} & 0 \end{bmatrix} \begin{bmatrix} \Delta Q_{15}\\ \Delta U_5\\ \Delta \delta_1\\ \Delta P_{51} \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ 0\\ \Delta Q_{51} \end{bmatrix} \tag{P-1} \]

Whence

\[ \beta_{51}=\frac{dQ_{51}}{dU_5} = \frac{ -\left(\partial P_{15}/\partial U_5\right)\left(\partial Q_{51}/\partial \delta_1\right) + \left(\partial P_{15}/\partial \delta_1\right)\left(\partial Q_{51}/\partial U_5\right) }{ \partial P_{15}/\partial \delta_1 }. \tag{P-2} \]

The expressions for the active and reactive powers at the ends of the section, according to the adopted directions, are written as

\[ P_{15}=\frac{U_1U_5}{B}\sin(\delta_1-\delta_5),\qquad Q_{51}=\frac{U_5^2A}{B}-\frac{U_1U_5}{B}\cos(\delta_5-\delta_1). \tag{P-3} \]

Differentiating expressions (П-3) with respect to the independent variables, after minor transformations we obtain:

\[ \frac{\partial P_{15}}{\partial U_5}=\frac{P_{15}}{U_5}, \qquad \frac{\partial P_{15}}{\partial \delta_1} = \sqrt{\left(\frac{U_1U_5}{B}\right)^2-P_{15}^{\,2}}, \tag{П—4} \]

\[ \frac{\partial Q_{51}}{\partial U_5} = \frac{2U_5A}{B} - \frac{1}{U_5} \sqrt{\left(\frac{U_1U_5}{B}\right)^2-P_{15}^{\,2}}, \qquad \frac{\partial Q_{51}}{\partial \delta_1}=P_{15}. \tag{П—5} \]

Substituting the values of the partial derivatives from (П—4), (П—5) into expression (П—2), we obtain the steepness coefficient in the following form:

\[ \beta_{51} = -\frac{2U_5A}{B} + \frac{U_1^{2}U_5}{B^2} \left[ \left(\frac{U_1U_5}{B}\right)^2-P_{15}^{\,2} \right]^{-1/2} \tag{П—6} \]

In expression (П—6), in comparison with (П—2), the opposite sign has been taken. This corresponds to the direction of the reactive power \(Q_{51}\) into node No. 5, which was adopted in the presentation of the material [see Fig. 1 and equation (6)]. The steepness coefficient of the second section has an analogous form:

\[ \beta_{52} = -\frac{2U_5D}{B} + \frac{U_2^{2}U_5}{B^2} \left[ \left(\frac{U_2U_5}{B}\right)^2-P_{52}^{\,2} \right]^{-1/2} \tag{П—7} \]

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