Influence of the Generalized Static Characteristics of Intermediate Systems on the Power Transfer Limit of Long-Distance AC Transmission Lines

(Part 1)

R. I. Borisov

(Submitted by the Department of Electrical Systems and Networks)

Introduction

The problem of increasing the distance over which electric power is transmitted by three-phase alternating current continues to remain relevant under the present conditions of electrification of the country. The accumulated experience in the operation and design of high-voltage three-phase-current installations preserves for them certain advantages for their use for this purpose. Long-distance transmission lines have ceased to be regarded and used only as trunk lines intended to solve problems of transit power transmission; they are acquiring the functions of tie lines between neighboring electric systems and of a reference source, distributed in space, for the continuous electrification of the regions through which they pass. At the same time, the process of interconnecting power systems, the growth of their power, the number of power stations and load nodes, and the extent of electric networks is not merely quantitative in character. This process is accompanied by such transformations that qualitatively new phenomena arise, not characteristic of it at earlier stages of development. Such phenomena have included overvoltages of various kinds, stability problems, and many others. There is no doubt that, at the existing and subsequent stages of development of electric-power systems, this law will remain in force. One may expect the consequences of its action in the emergence of new phenomena that will substantially affect the reliability and quality indices of power supply. Knowledge of the objective laws governing the development of electric systems is of great practical importance, since it will make it possible to ensure this development according to the required indices of power supply.

For this reason, the operating problems of the joint operation of long-distance transmission lines and intermediate electric systems (IES) continue to retain important significance. First and foremost, the questions of choosing the connection scheme of an IES to long-distance transmission lines, determining the criteria for the expediency of such connections, the conditions and parameters of regulation in intermediate and terminal systems, reserve conditions, compatibility of the operating modes of long-distance transmission lines and IES, and others, require well-founded solutions.

Problem Statement

This article presents materials from studies on the influence of an intermediate electric system (IES) on the limiting value of transmitted power as determined by static stability. For this purpose, generalized static voltage characteristics of the IES are used.

The voltage at the nodes where the IES is connected will change when the power transmitted over the line changes, which may cause changes in the powers of the loads and generators in the intermediate system. If the IES power is sufficient, the static characteristics of the generators and loads are favorable, or the control parameters have been chosen appropriately and their settings implemented, then the voltage at the IES connection node will be stabilized by the system itself, which should affect the transmitted-power limit in normal and post-emergency operating conditions. The choice of control parameters and their settings must be made in such a way as to satisfy the requirements of joint operation of the long-distance transmission line and the intermediate system.

For the scheme in Fig. 1, a change in voltage at node \(a\) will lead to corresponding changes in the voltage and power of node \(b\). To identify more clearly the influence of the IES characteristics, let us assume that the connection of the transmission line with the IES is carried out through autotransformers without OLTC. These changes in the active and reactive components of power occur in accordance with their static characteristics, and the increments of power in the system as a whole are not the same. The unbalance power will reduce the deviation of the angle between the e.m.f.’s of the sending and receiving systems when the operating condition of the main transmission line changes and the voltage decreases. Consequently, it is possible to formulate the problem of finding the required parameters according to which regulation should be carried out in order to ensure the greatest value of transmitted power. Calculation of the power transmitted under static-stability conditions with allowance for the IES through generalized static characteristics is preferable, since it makes it possible to answer the posed problem in an explicit and simple form.

Research Methodology

The powers of the IES generators (Fig. 1) are functions of three variables—the e.m.f., the voltage at the terminals \((U)\), and the angle between them—while the load powers depend only on the voltage:

\[ P_{\mathrm{g}}=P_{\mathrm{g}}(E,U,\delta), \]

\[ Q_{\mathrm{g}}=Q_{\mathrm{g}}(E,U,\delta), \]

\[ P_{\mathrm{l}}=P_{\mathrm{l}}(U), \]

\[ Q_{\mathrm{l}}=Q_{\mathrm{l}}(U). \]

The power increments are related to the increments of the variables by the known relations:

\[ \Delta P_{\mathrm{g}} = \frac{\partial P_{\mathrm{g}}}{\partial E}\Delta E + \frac{\partial P_{\mathrm{g}}}{\partial U}\Delta U + \frac{\partial P_{\mathrm{g}}}{\partial \delta}\Delta \delta, \]

\[ \Delta Q_{\mathrm{g}} = \frac{\partial Q_{\mathrm{g}}}{\partial E}\Delta E + \frac{\partial Q_{\mathrm{g}}}{\partial U}\Delta U + \frac{\partial Q_{\mathrm{g}}}{\partial \delta}\Delta \delta, \]

\[ \Delta Q_{\mathrm{l}} = \frac{\partial Q_{\mathrm{l}}}{\partial U}\Delta U, \]

\[ \Delta P_{\mathrm{l}} = \frac{\partial P_{\mathrm{l}}}{\partial U}\Delta U. \]

Fig. 1. Connection diagram of the intermediate system.

The changes in power at the generators and at the load turn out to be different. For a favorable effect on static stability, the choice of regulation parameters and their adjustment should be carried out so that, when the voltage decreases, the reductions in the power components of the generators are smaller than those of the load. Then the necessary excess imbalance arises. It will be still greater if, when the voltage decreases, the power components of the intermediate-system generators increase.

\[ \Delta P_c = \pm \Delta P_g + \Delta P_n, \]

\[ \Delta Q_c = \pm \Delta Q_g + \Delta Q_n. \]

From the increments of power and voltage, the required changes in e.m.f. and angle can be found.

\[ \frac{\partial P_g}{\partial E}\Delta E + \frac{\partial P_g}{\partial U}\Delta U + \frac{\partial P_g}{\partial \delta}\Delta \delta + \frac{\partial P_n}{\partial U}\Delta U = \Delta P_c, \]

\[ \frac{\partial Q_g}{\partial E}\Delta E + \frac{\partial Q_g}{\partial U}\Delta U + \frac{\partial Q_g}{\partial \delta}\Delta \delta + \frac{\partial Q_n}{\partial U}\Delta U = \Delta Q_c, \]

\[ \frac{\partial P_g}{\partial E}\Delta E + \frac{\partial P_g}{\partial \delta}\Delta \delta = F_1, \]

\[ \frac{\partial Q_g}{\partial E}\Delta E + \frac{\partial Q_g}{\partial \delta}\Delta \delta = F_2, \]

where

\[ F_1 = \Delta P_c - \frac{\partial P_g}{\partial U}\Delta U - \frac{\partial P_n}{\partial U}\Delta U, \]

\[ F_2 = \Delta Q_c - \frac{\partial Q_g}{\partial U}\Delta U - \frac{\partial Q_n}{\partial U}\Delta U, \]

then

\[ \Delta E = \frac{\Delta_1}{\Delta} \quad \text{and} \quad \Delta \delta = \frac{\Delta_2}{\Delta}, \]

where

\[ \Delta_1= \begin{vmatrix} F_1 & \dfrac{\partial P_{\mathrm{g}}}{\partial \delta}\\[4pt] F_2 & \dfrac{\partial Q_{\mathrm{g}}}{\partial \delta} \end{vmatrix}; \qquad \Delta_2= \begin{vmatrix} \dfrac{\partial P_{\mathrm{g}}}{\partial E} & F_1\\[4pt] \dfrac{\partial Q_{\mathrm{g}}}{\partial E} & F_2 \end{vmatrix}; \qquad \Delta= \begin{vmatrix} \dfrac{\partial P_{\mathrm{g}}}{\partial E} & \dfrac{\partial P_{\mathrm{g}}}{\partial \delta}\\[4pt] \dfrac{\partial Q_{\mathrm{g}}}{\partial E} & \dfrac{\partial Q_{\mathrm{g}}}{\partial \delta} \end{vmatrix}. \]

The results of determining increments of the operating parameters of the intermediate system under different regulation conditions are presented in Table 1.

The increments of the operating parameters of the IPS, found from the operating conditions of the main transmission line, must not contradict the requirements for maintaining the voltage at consumers, economy, and other technical conditions within the IPS; i.e., the possibility of realizing the required regulation conditions must be compatible with the operation of the consumers and generators of the IPS.

Next, the dependences \(\Delta P_{\mathrm{c}}(\Delta U_{\mathrm{b}})\) and \(\Delta Q_{\mathrm{c}}(\Delta U_{\mathrm{b}})\) must be recalculated directly to the IPS connection node, taking into account the longitudinal and transverse components of the voltage losses.

Calculations were carried out for conditions 1, 2, and 7 to determine the limit of the transmitted power of a lossless line 1000 km long, \(U_{\mathrm{n}}=500\) kV, with an intermediate connection at the center, operating without a voltage drop (Fig. 2).

At the first stage of the calculations the IPS was assumed to be balanced; the ideal power limit, the equalizing-power flow, and the voltage at the middle of the line were determined. At the second stage, using the corresponding static characteristic, the unbalance powers and parameters of the intermediate connection, the self and mutual impedances of the circuit, the receiving-end powers, and the voltage at the IPS connection node were found. The calculations were then repeated until convergence of the results for the voltage at the IPS connection node.

Results and conclusions

The results of calculations by the first method of regulation are presented in Fig. 3 (No. 1); the power \(1.15P_{\mathrm{nat}}\) is the limit for the line without taking the IPS into account. Calculations were carried out for the power of the intermediate system equal to 0.2, 0.4, and 0.6 of the natural power of the line. The capacity of the coupling autotransformer was taken to be equal to the required reserve capacity of the IPS (\(0.1S_{\mathrm{IPS}}\)).

For the second method of regulation the results are presented in Fig. 3 (No. 2). Together with Fig. 3 (No. 1), they make it possible to judge that the influence of the natural static characteristics of the IPS loads on the transmitted-power limit of the line proves to be insignificant. The influence becomes more noticeable at a higher IPS power and the corresponding action on the IPS generators and synchronous compensators, and with compensation of the reactance of the coupling autotransformer.

Thus, regulation according to method No. 7 of Table 1 gives:

7a With \(\Delta U=-0.1\), \(\Delta P_{\mathrm{g}}=0.1\), and \(\Delta Q_{\mathrm{g}}=0.1\)

\[ S_{\mathrm{IPS}}=0.6 \qquad S_{\mathrm{IPS}}=0.6 \qquad S_{\mathrm{IPS}}=0.6 \]

\[ S_{\mathrm{T}}=0.06 \qquad S_{\mathrm{T}}=0.18 \qquad x_{\mathrm{T}}=0 \]

\[ P_{\mathrm{lim}}=1.22 \qquad P_{\mathrm{lim}}=1.25 \qquad P_{\mathrm{lim}}=1.27 . \]

7b With \(\Delta U=-0.1\), \(\Delta P=0.2\), and \(\Delta Q=0.2\)

\[ S_{\mathrm{IPS}}=0.4; \qquad x_{\mathrm{T}}=0; \qquad P_{\mathrm{lim}}=1.415 . \]

The increase in the transmitted-power limit for condition 7b is 23 percent.

Table 1

Fig. 2. Conditions of the example.

Fig. 3. Results of calculations by the first, second, and seventh control methods.