CALCULATION OF STEADY-STATE MODES OF COMPENSATED LONG-DISTANCE POWER TRANSMISSION WITH INTERMEDIATE SYSTEMS

R. I. Borisov, I. A. Bezler

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

Newly introduced and existing long-distance AC power transmissions serve as intersystem links between transmitting and receiving systems. But over time such power transmissions are transformed from transit lines into power transmissions with intermediate power takeoffs or with connected intermediate systems (IS). The IS connected to a long-distance power transmission and the adjustable reactive-power compensation installations (RPCI) convert a long-distance power line from passive to active. This leads to a quantitative and qualitative change in the operating parameters of the long-distance power line; the static and dynamic stability of the system as a whole, the limit of transmitted power over sections of the long-distance power line, the reliability of power supply to consumers, and the frequency and voltage levels in the system change. But the change in parameters must occur in such a way that in each specific case an operating mode close to the optimal one is obtained [1—4].

In the present work an algorithm is considered for calculating the steady-state mode of a long-distance power transmission having a longitudinal capacitive compensation installation (LCCI), to the terminals of which IS and RPCI are connected (by RPCI are meant synchronous and static compensators, shunt reactors, and static capacitors).

The algorithm considered is based on [5, 6], taking into account the features imposed by IS and RPCI on the modes of a long-distance power transmission both under normal and post-emergency conditions.

When carrying out a calculation to determine the mode of the long-distance power line proper, and in order to reduce computation time, it is necessary to represent the IS and RPCI by static characteristics. But in an internal study of the IS it is necessary to use the method of chain connection of multi-terminal networks, since it makes it possible simply to take into account any changes in the IS: changes in the impedance of the connection between the IS and the long-distance power line, changes in load, changes in generator control laws, changes in the transformation ratios of coupling transformers, etc.

The problem of calculating the steady-state modes of the power system under consideration (Fig. 1) includes determining the flows of active and reactive power in all branches for a known network diagram and known moduli (vectors) of the voltages of the controlled nodes of the network. In addition, the determina-

the active and reactive powers of stations, loads, and transformation ratios. In this case, various constraints may be imposed on the operating-condition parameters:

a) on the mutual angle of the voltages at the ends of the sections of the long-distance transmission line;

b) on the upper and lower limits of the voltage magnitudes at the controlled nodes of the system;

c) on the transfer capacity of the coupling transformers between the intermediate system (IS) and the long-distance transmission line;

Fig. 1. Diagram of a long-distance power transmission line with an intermediate system, series-compensation installations, and various types of shunt compensation.

and a number of other constraints, which depend on the specific conditions of the problem being solved.

Therefore, at the first stage, for prescribed values of active-power flows and voltage magnitudes at the nodes where the intermediate system (IS) and system elements are connected, the initial operating condition is calculated; as a result, the demand for reactive power at all system nodes, the exchange flow of active power with the balancing power system, and all the remaining operating parameters of the power system are determined. The reactive-power imbalance at the nodes where the IS and system elements are connected is partially or completely covered by shunt compensating devices required for voltage regulation at the nodes where the IS and system elements are connected. Then the prescribed values of active-power flows and voltage magnitudes at the system nodes, together with the calculated quantities of reactive-power imbalance, are fixed operating parameters for calculating the initial operating condition of the resulting power system.

The obtained values of the initial operating condition must satisfy the operating-condition parameters with allowance for the constraints. Bringing the operating variables into the admissible region is ensured by the available degrees of freedom of the system parameters. Thus, in order to limit the upper or lower bound of the voltage magnitude at a node, it is necessary to decrease or increase the reactive power of the reactive-power sources located closest to the node, or to decrease or increase the magnitude of the coeffici-

transformation ratio in the branch connected with the given node. Calculation of the initial operating mode and determination of the final parameters (which are obtained with allowance for the imposed constraints, and the mode is brought into the admissible region) are carried out by any of the iteration methods currently used [3, 7].

For the scheme (Fig. 1), the coefficients of the four-terminal networks have the form:

a) line sections, taking account of the distributed nature of the parameters,

\[ [\dot A_i]= \begin{bmatrix} \operatorname{ch}\dot\gamma l_i & \dfrac{\dot Z}{n}\operatorname{sh}\dot\gamma l_i\\[6pt] \dfrac{n}{\dot Z}\operatorname{sh}\dot\gamma l_i & \operatorname{sh}\dot\gamma l_i \end{bmatrix} \tag{1} \]

where \(i=1,2,\ldots,m\) is the index of the line section;
\(n\) is the number of parallel circuits of the line section;
\(\dot Z,\dot\gamma\) are the surge impedance and propagation coefficient;

b) compensating devices representing series compensation devices (UPK),

\[ [\dot A_k]= \begin{bmatrix} 1 & -jX_k\\ 0 & 1 \end{bmatrix}; \tag{2} \]

c) compensating devices representing shunt compensating devices (UPnK),

\[ [\dot A_k]= \begin{bmatrix} 1 & 0\\ \pm j\dfrac{1}{X_p} & 1 \end{bmatrix}, \tag{3} \]

where

\[ X_p=\frac{U_i^2}{Q_p}, \]

\(i=1,2,\ldots,m\) is the index of the node at which the UPnK is connected to the compensated long-distance power transmission line (before the UPK or after the UPK);
\(U_i\) is the voltage at the node where the UPnK is connected;
\(Q_p\) is the capacity of the UPnK (of any type).

The product of the matrices of the elements of series-connected four-terminal networks gives the matrix of the equivalent four-terminal network

\[ [\dot A]=\prod_{j=1}^{m}[\dot A_j]. \tag{4} \]

When intermediate power withdrawals, as well as power transfers between the PS and the long-distance transmission line and loads in the system, are represented by static characteristics, the latter are approximated by second-degree polynomials [7] of the form

\[ \left. \begin{aligned} P_i&=a_0+a_1\cdot U_i+a_2\cdot U_i^2,\\ Q_i&=b_0+b_1\cdot U_i+b_2\cdot U_i^2, \end{aligned} \right\} \tag{5} \]

where \(a_i\) and \(b_i\) are the polynomial coefficients, which depend on the composition of consumers.

The steady-state equations of the equivalent four-terminal network can be represented in matrix-vector form

\[ [\dot I_i]=[A]\cdot \begin{bmatrix} \dot U_j\\ \dot I_j \end{bmatrix}; \tag{6} \]

\[ [\dot U_i]=[A]\cdot \begin{bmatrix} \dot U_j\\ \dot I_j \end{bmatrix}; \tag{7} \]

\[ \dot U\,\hat I_i=P_i-jQ_i. \tag{8} \]

Loads and power flows may also be represented by constant quantities.

The calculation of the steady-state mode begins with determining the reactive power accompanying the active power [8], and determining the balancing powers at the system nodes. After this, starting from the busbars of receiving system No. 2, the operating parameters at all nodes are determined successively in accordance with Fig. 1 and taking into account expressions (6–8). In doing so, to determine the steady-state operating parameters of the quadripole that replaces the section of the long-distance transmission line \(l_2\), the operating parameters of the receiving system \(P_2, Q_2, U_2, \varphi_2\) are used; and for the remaining quadripoles, the desired operating parameters found from the calculation of the preceding quadripole are used, taking account of the balancing powers. After completion of the first iteration cycle, a repeated cycle is carried out, having first determined the refined parameters of the loads, power flows, and power balances at the system nodes; the principal condition for terminating the calculation may be the inequality \(\varepsilon > 10^{-2}\div 10^{-4}\) (depending on the calculation accuracy), which takes into account the difference between the results for the operating parameters of the last cycle and those of the preceding cycle at all nodes of the system. Depending on the circuit, there may be 3–9 cycles. To accelerate convergence of the solution of nonlinear equations of the form (6–8), after the third cycle an accelerating step may be applied. The calculation results are presented in Tables 1 and 2.

Analyzing the results obtained, when calculating the steady-state operating parameters it may be concluded that, when maintaining the voltage at the outputs of the series-compensating installation (SCI) at the same level, different magnitudes of transverse compensating devices are required. When an intermediate system (IS) is connected to the inputs of the SCI, less reactive power is required from the IS at the input than from the transverse compensating installation (TCI) at the output of the SCI. Moving the SCI together with the IS from transmitting system No. 1 toward receiving system No. 2 leads to a reduction in the reactive power of the TCI at the output and, in the mode \(P_{\text{is}}>0\), at the input of the SCI, while in modes \(P_{\text{is}}\leqslant 0\) it leads to an increase in the reactive power of the TCI at the input of the SCI. Connecting the IS to the outputs of the SCI leads to the same results as connecting the IS to the inputs of the SCI. In the mode \(P_{\text{is}}=0\), the total reactive power of the TCI does not depend on the point of connection of the IS; in modes \(P_{\text{is}}>0\), less total reactive power of the TCI is required when the IS is connected to the inputs of the SCI, while in modes \(P_{\text{is}}<0\), less installed TCI capacity is required when the IS is connected to the outputs of the SCI (Table 1).

In a compensated long-distance power transmission line, the distribution of voltage and current depends significantly on the point of connection of the IS: in all modes, the indicators \(U\) and \(I\) are better when the IS is connected to the outputs of the SCI, with the exception of the mode \(P_{\text{is}}<0\), in which, when the IS is moved together with the SCI from the transmitting to the receiving systems, the results deteriorate somewhat and the best mode is the one with the IS connected to the inputs of the SCI (Table 2). The calculation results showed that a more favorable distribution mode of \(U\) and \(I\) when the IS is connected to the inputs of the SCI is observed for \(l_1\leqslant 0.5l\).

Table 1

Table 2

Note. The calculations were performed under the following conditions:

\[ P_{\mathrm{base}}=P_{\mathrm{nat}}=891\ \text{MW},\quad U_{\mathrm{b}}=U_{\mathrm{n}}=500\ \text{kV},\quad K_C=50\%,\quad P_2=1,8;\quad U_2=1,\quad \varphi_2=0, \]

\[ Q_2=0,06,\quad l_1=l_2=500\ \text{km}. \]

The distribution mode of \(U\) and \(I\) in the system under consideration is strongly influenced by the types of UPC, as well as by the electrical “remoteness” of the PS supply sources from the point at which they are connected to the long-distance power transmission line. The distribution of \(U\) and \(I\) in the PS can be changed by connecting compensating devices either at the point where the PS is connected to the long-distance power transmission line or to the low-voltage winding of the coupling transformer. In the calculations the UPCs (of any type) were represented by static characteristics obtained from additional subroutines.

Conclusions

1. The algorithm considered for calculating the steady-state mode of a long-distance power-transmission line takes into account the distributed nature of the parameters of the line sections, and the loads as specified by constant resistances—

...or static characteristics, as well as series-compensation installations, to the terminals of which the intermediate systems and various types of controllable series-compensation devices are connected.

1. From the characteristics of the automatic-control systems and the parameters one can judge the necessity of using synchronous or static compensators, or shunt reactors.

2. The optimum mode with respect to the distribution of \(U\) and \(I\) is the mode in which the intermediate systems are connected to the terminals of the controllable series-compensation devices.

REFERENCES

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