PROPORTIONAL EXCITATION CONTROL OF GENERATORS IN INTERMEDIATE SYSTEMS BASED ON OPERATING PARAMETERS OF LONG-DISTANCE AC TRANSMISSION LINES

R. I. BORISOV, Yu. V. KHRUSHCHEV

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

Intermediate power systems (IPS), located along the route of long-distance AC transmission lines, may serve as the principal or supplementary controllable sources of reactive power (RPS). An important condition for increasing the effectiveness of these functions is the proper choice of control laws based on deviations of operating parameters, or of the primary control, of IPS generators, and the determination of their proportional participation in voltage control on the line as part of other RPS.

The general regularities and limitations governing the solution of these questions are considered. The method is based on replacing all elements connected at intermediate points of the transmission line by static characteristics of their response to external disturbances.

Existing recommendations for choosing the laws of primary control of intermediate RPS concern synchronous and static compensators [1, 2]. They amount to the conclusion that the voltages at the terminals of the RPS, or at the node where the coupling transformer is connected to the line, should be chosen as the deviation-control parameters. The opinions of these authors differ regarding the effectiveness of control by other operating parameters.

It is known that the action of primary control is directed toward maintaining voltages and increasing the limits of power transmission under the conditions of aperiodic stability. In works [3, 4] it is shown that, by an appropriate selection of RPS control laws at intermediate points of the transmission line, it is possible to achieve an unlimited increase in line loading while preserving aperiodic stability, provided that the RPS have sufficient capacity. In this case, equality of voltages can be achieved at all points of the line with uniformly distributed compensation. The maximum power of lines with a finite number of intermediate RPS is limited by the availability of reactive power, overvoltages, and conditions of oscillatory instability.

It is obvious that, from the set of possible dependences $P(\delta, U)$ and, correspondingly, the set of laws of primary control of IPS generators, those should be implemented which would satisfy the requirements of economical voltage distribution along the line, stability of transmission operation, and a number of conditions and limitations, including within the IPS itself.

The requirements for voltage distribution along compensated long lines under the conditions of maximum efficiency are set forth in [5, 6].

If one introduces the requirement of maximizing the static power limit of the line with limited reactive-power resources of the intermediate power system (IPS), then rigid voltage regulation at the buses of its connection may prove not to be the best solution. By implementing certain laws of primary regulation of IPS generators, it is possible to obtain a greater increase in transit power without self-excitation. However, together with the power, the voltages at the IPS connection buses will increase. Thus, an IPS balanced in active and reactive power and connected to the middle of an idealized line with \(\lambda = 120^\circ\), at constant voltages at the ends, permits the transmission of natural power over the line with an 11% margin under static-stability conditions if the voltage at the connection bus is maintained constant \((U - 1 = \text{const.})\). The magnitude of the regulating flow of reactive power from the IPS that provides this margin is \(0.5\,P_{\text{nat.}}\). The same regulating flow makes it possible to increase the margin to 32.5% if the angle along the line is maintained constant \((\delta = \lambda = \text{const.})\). In this case, the dependence of the voltage at the connection bus on the power transmitted over the line is analogous to the half-wave operating mode of the line up to the complete expenditure of the regulating flow of reactive power (Fig. 1).

Fig. 1. Dependences \(P(\delta)\) and \(P(U)\) (shown dashed) at \(Q_{\text{IPS max.}} = 0.5\,P_{\text{nat.}}\).
a) \(\lambda = 60^\circ\);
1.—\(U - 1 = \text{const.}\), 2.—\(\delta = \lambda = \text{const.}\);
b) the same for \(\lambda = 120^\circ\).

As the rigidity of maintaining the transmission angle is weakened, the static-stability margin decreases. In lines with an electrical length up to \(90^\circ\), the static-stability margin is the same when both conditions are satisfied, provided the magnitude of the regulating reactive-power flow of the intermediate power system is limited. However, when the condition \(\delta-\lambda=\text{const.}\) is satisfied, the regulating flow is used up completely at smaller values of the power transmitted over the line than under the condition \(U=1=\text{const.}\) (points \(a_1\) and \(a_2\) in Fig. 1, a).

A further increase in power transit over the line up to the limit occurs through an increase in the angle at a constant value of the reactive power of the intermediate power system. The table presents the limiting values of the electrical lengths of long-distance ac transmission lines at which complete use of the regulating flow of the intermediate power system corresponds to identical values of the active power of the line. The latter, for the limiting electrical lengths, are at the same time the aperiodic limits of transmitted power.

Table 1

In connection with these features, it may prove expedient to introduce combined (stepwise and continuous) regulation of the reactive-power flow from the intermediate power system. Within the operating constraints, stepwise regulation will provide an economical distribution of voltage along the line, while continuous regulation will act to increase static-stability margins within the technical constraints.

All operating parameters by which the basic regulation of the generators of the intermediate power system may be carried out are conventionally divided into two groups: internal and external. The internal parameters (magnitudes of voltages and currents, power flows, internal relative angles, and also the voltage magnitude at the connection node of the intermediate power system) constitute a set relating only to the elements of the intermediate power system. The remaining operating parameters are external. These also include the phases of voltages, currents, and emf’s within the intermediate power system.

It is convenient to characterize the reaction of the intermediate power system to a steady-state increment of a given operating parameter of the transmission by deviations of the active and reactive powers of the intermediate power system at the connection node and by generalized influence, or slope, coefficients [8].

For the scheme (Fig. 2), the regulation equations of the intermediate power system can be written as follows:

\[ \Delta P_4=\sum_{\text{п}}\rho_{\text{п}}\Delta \Pi; \tag{1} \]

\[ \Delta Q_4=\sum_{\text{п}}q_{\text{п}}\Delta \Pi, \]

where

\[ \Delta P_4,\ \Delta Q_4 \]
are the increments of the powers of the intermediate power system;

\[ \Pi \]
is an operating parameter of the transmission;

\[ \rho_{\text{п}},\ q_{\text{п}} \]
are the slope coefficients of the active and reactive powers of the intermediate power system with respect to the corresponding parameter \(\Pi\).

The coefficients \(\rho_{\text{п}}, q_{\text{п}}\) \((\Pi \ne U_4)\) are due to the basic regulation of the generators of the intermediate power system by external parameters and, if the latter is absent,

are equal to zero. The coefficients \(\rho_{i4}, q_{i4}\) for any control laws differ from zero unless a special condition is imposed that reduces them to zero.

Constructing the functions \(P(\delta_1)\) and \(P(U_4)\) by successive weighting of the transmission operating mode under different laws of excitation control of the generators of the intermediate power station requires a large amount of computation even for the simplest circuit shown in Fig. 2. It is possible to limit oneself to determining the directions and values of the linear increments of these functions, which are determined by the sign and magnitude of the total derivatives of the initial operating mode \(\dfrac{dP}{d\delta_1}\) and \(\dfrac{dP}{dU_4}\). At the same time, the aperiodic stability of the initial operating mode is checked by the sign of the free term of the characteristic equation.

Fig. 2

To determine the derivatives \(\dfrac{dP}{d\delta_1}\) and \(\dfrac{dP}{dU_4}\), let us apply a small steady-state disturbance \(\Delta P\) to the rotor shaft of the remote generator. Owing to the equality of the mechanical and electromagnetic torques on the generator shaft in steady-state operating modes, the increment \(\Delta P\) can be expressed through the operating parameters of the transmission as follows:

\[ \frac{\partial P_{14}}{\partial \delta_1}\Delta\delta_1 + \frac{\partial P_{14}}{\partial \delta_4}\Delta\delta_4 + \frac{\partial P}{\partial U_4}\Delta U_4 = \Delta P . \tag{2} \]

The equations of the active- and reactive-power balances at the bus of connection of the intermediate power station, after linearization, take the following form:

\[ \Delta P_{41}+\Delta P_{42}+\Delta P_4=0, \tag{3} \]

\[ \Delta Q_{41}+\Delta Q_{42}+\Delta Q_4=0. \tag{4} \]

Let us express the power increments in equations (3), (4) through the same parameters as in (2). As a result we obtain:

\[ \left( \frac{\partial P_{41}}{\partial \delta_1} + \sum_{\mathrm{п}}\rho_n\frac{\partial \Pi}{\partial \delta_1} \right)\Delta\delta_1 + \left( \frac{\partial P_{41}}{\partial \delta_4} + \frac{\partial P_{42}}{\partial \delta_4} + \sum_{\mathrm{п}}\rho_n\frac{\partial \Pi}{\partial \delta_4} \right)\Delta\delta_4 + \]

\[ + \left( \frac{\partial P_{41}}{\partial U_4} + \frac{\partial P_{42}}{\partial U_4} + \sum_{\mathrm{п}}\rho_n\frac{\partial \Pi}{\partial U_4} \right)\Delta U_4 = 0, \tag{5} \]

\[ \left( \frac{\partial Q_{41}}{\partial \delta_1} + \sum_{\mathrm{п}}q_n\frac{\partial \Pi}{\partial \delta_1} \right)\Delta\delta_1 + \left( \frac{\partial Q_{41}}{\partial \delta_4} + \frac{\partial Q_{42}}{\partial \delta_4} + \sum_{\mathrm{п}}q_n\frac{\partial \Pi}{\partial \delta_4} \right)\Delta\delta_4 + \]

\[ + \left( \frac{\partial Q_{41}}{\partial U_4} + \frac{\partial Q_{42}}{\partial U_4} + \sum_{\mathrm{п}}q_n\frac{\partial \Pi}{\partial U_4} \right)\Delta U_4 = 0. \tag{6} \]

The right-hand side of the system of equations (2, 5, 6) is not equal to zero, since the disturbance \(\Delta P\) has been applied. Solving this system with respect to \(\Delta \delta_1\) and \(\Delta U_4\), we obtain:

\[ \Delta \delta_1=\frac{D'}{D}\Delta P, \tag{7} \]

\[ \Delta U_4=\frac{D''}{D}\Delta P. \tag{8} \]

Passing to infinitesimal increments, we transform (7) and (8) to the following form:

\[ \frac{dP}{d\delta_1}=\frac{D}{D'}, \tag{9} \]

\[ \frac{dP}{dU_4}=\frac{D}{D''}. \tag{10} \]

It is obvious that the determinant \(D\) in (7–10) is the free term of the characteristic equation of the power transmission system when it operates on the infinite bus, if the AVR of the remote generator is absent or is taken into account in simplified form. The determinants \(D'\) and \(D''\) are algebraic complements of \(D\).

It should be noted that the slope coefficients of the active power of the intermediate power system \(\rho_n\) are determined by the static characteristics of the active power of the loads and, as a rule, are considerably smaller than the corresponding coefficients \(q_n\).

\[ \rho_n \ll q_n. \]

Let us consider the influence of the laws of the basic control of the generators of the intermediate power system with respect to internal and external \((P_{41}, \delta_1, \delta_4)\) operating parameters on the aperiodic stability of the power transmission and on the relationship between the increments of the power transmitted over the line \(\Delta P\), the angle \(\Delta \delta_1\), and the voltage \(\Delta U_4\) (Fig. 2).

Taking \(\Delta P_4=\rho_n=0\), we transform the derivatives (9, 10) to the following form:

\[ \frac{dP}{d\delta_1}=\frac{D}{D'}= \frac{D_0+q_{u4}D_1+q_{\delta1}D_2+q_{\delta4}D_3+q_{p41}D_4} {D'_0+q_{u4}D'_1+q_{\delta1}D'_2+q_{\delta4}D'_3+q_{p41}D'_4}, \tag{11} \]

\[ \frac{dP}{dU_4}=\frac{D}{D''}= \frac{D_0+q_{u4}D_1+q_{\delta1}D_2+q_{\delta4}D_3+q_{p41}D_4} {D''_0+q_{u4}D''_1+q_{\delta1}D''_2+q_{\delta4}D''_3+q_{p41}D''_4}, \tag{12} \]

where

\[ D_i,\; D'_i,\; D''_i \;(i=0,1,2,3,4) \]

are the coefficients of the initial operating condition.

In the absence of basic control of the generators of the intermediate power system with respect to external parameters, expressions (11, 12) acquire the simpler form:

\[ \frac{dP}{d\delta_1}=\frac{D}{D'}= \frac{D_0+q_{u4}D_1} {D'_0+q_{u4}D'_1}, \tag{13} \]

\[ \frac{dP}{dU_4}=\frac{D}{D''}= \frac{D_0+q_{u4}D_1} {D''_0+q_{u4}D''_1}, \tag{14} \]

Since \(D''_1=0\), the derivative \(\dfrac{dP}{dU_4}\) depends linearly on \(q_{u4}\):

\[ \frac{dP}{dU_4}=\frac{1}{D''_0}\left(D_0+q_{u4}D_1\right). \tag{15} \]

If the voltage slope coefficient of the reactive power of the intermediate system tends to limiting values \((\pm\infty)\), then we shall have

\[ \left.\frac{dP}{d\delta_1}\right|_{qU_4\to\mp\infty} = \frac{D_1}{D'_1}. \tag{16} \]

Depending on the signs of the coefficients \(D_1\) and \(D''_0\), the derivative \(\frac{dP}{dU_4}\) tends either to \((\mp\infty)\) or to \((\pm\infty)\). In both cases the condition \(qU_4\to\mp\infty\) corresponds to the result \(U_4\to\text{const.}\)

As the calculations performed have shown, the ratio \(\frac{D_1}{D'_1}\) is equal to the partial derivative \(\frac{dP}{d\delta}\) of the weakest section of the transmission line. If the equivalent electrical length of at least one section lies in the second quadrant \((90^\circ<\lambda_{\mathrm{e}}<180^\circ)\), then in all operating conditions

\[ \frac{D_1}{D'_1}<0;\qquad D_1<0;\qquad D'_1>0, \]

which is an indication of instability.

It follows from (16) that a criterion of steady-state stability of the form \(\frac{dP}{d\delta_1}\) may assume positive values both for positive and for negative slope coefficients \(q_{U4}\). Under conditions in which, in each of the sections, \(\lambda_{\mathrm{e}}<90^\circ\), there is only a small segment of the \(q_{U4}\) axis where the free term \(D\) and the algebraic complement \(D'\) do not coincide in sign and \(\frac{dP}{d\delta_1}<0\).

However, as is evident from (13) and (14), the free term of the characteristic equation is positive only when

\[ qU_4\cdot D_1>-D_0. \tag{17} \]

The range of values of \(q_{U4}\) for which \(\frac{dP}{d\delta_1}>0\), but \(D<0\), is located on the negative half-axis \(q_{U4}\) in all operating conditions from \(P=0\) up to the limiting loading of the weakest section, if, by means of stepwise control, the voltage is maintained at the nominal level. A special check of the characteristic equation has shown that, in this range of values of \(q_{U4}\), the power transmission can operate stably only with an inertial intermediate connection. In particular, if instead of an intermediate system a bank of static capacitors is connected, tuning the power transmission to the half-wave mode and possessing, as is known, a negative slope coefficient, then, together with the free term, all coefficients of the characteristic equation acquire a negative sign. Depending on the transmission operating condition and the settings of the regulators of the remote station, the roots of such an equation may have no positive real parts. When an inertial connection is connected, the leading coefficient of the characteristic equation does not depend on the slope coefficients and, if it has a positive sign, then the condition \(D<0\) is an indication of instability even when \(\frac{dP}{d\delta_1}>0\).

It is not difficult to verify that, under the condition \(D<0\), the increments of active power \(\Delta P\) and voltage \(\Delta U'_4\) coincide in sign. Therefore, by means of an inertial connection with primary control only by internal parameters, it is impossible to obtain a stably operating power transmission in which, as the transmitted active power increases, the voltage at the connection node would increase and, consequently,

it is impossible to implement such solutions as tuning to a half-wave line, compensation of the operating mode, and the like. This conclusion is valid for transmissions with one intermediate tap, in which the wave length of each section does not exceed \(90^\circ\), and is consistent with the analogous conclusion in [7].

Thus, from the conditions of static stability under primary control of generators by internal parameters from a single IPP, one can require only an increase in the positive values of the slope coefficient \(q_{U4}\), which is equivalent to the requirement of the most rigid voltage control at the junction node.

It should be noted that the capability of the IPP to increase the slope coefficient \(q_{U4}\) is very limited. If no special measures are taken (for example, compensation of the impedance of the connection between the IPP and the transmission line), then the point of constant voltage inside the IPP can be “brought closer” to the junction node only by means of generator AVR. However, as the point of constant voltage is “moved away” from the terminals of the generators assigned for this purpose, the dependence of generator voltages on the transmission operating mode increases, which in the main also limits the possibilities of rigid control at the junction node. Thus, a change in the reactive-power flow of the tap in both directions within the rated power of the coupling transformer is provided by voltage deviations at the terminals on the IPP side within \(\Delta U_3 = \pm u_k\%\), if in this case \(U_4=\text{const.}=1\), \(P_4=0\) (Fig. 2).

The participation of the IPP in voltage control on the line increases substantially when an on-load tap changer is introduced at the coupling transformer, making it possible to change the reactive-power flow discretely. If one allows the voltages at the terminals of the IPP generators to depend on the transformation ratio of the coupling transformer, then the rigidity of voltage control at the junction node can be increased considerably by choosing it as a parameter of the primary control of the assigned generators. If \(u_k=12\%\), then by changing the transformation ratio within \(k_t=0.86\text{--}1.18\), it is possible to obtain \(U_3=\text{const.}=1\) under the same conditions as above.

Further investigation showed that the laws of primary control by the angles \(\delta_1\) and \(\delta_4\) are identical in their effect on static stability and on the voltage at the junction node. Both laws, with different degrees of influence, make it possible to obtain a stable (\(D>0\)) transmission in which an increase in the transmitted active power is accompanied by a rise of voltage at the junction node. At the same time, more complex difficulties arise with regulating the voltages at the terminals of the IPP generators and at the junction node. The magnitude of the reactive-power flow from the IPP and the voltage at the junction node are determined by the transmission angle, and therefore the use of an on-load tap changer of the coupling transformer is ineffective for their regulation. The effect of switching the transformer taps appears as a certain change in the slope coefficients of the IPP reactive power with respect to the angles \(\delta_1\) and \(\delta_4\) and, accordingly, in the rigidity of control of these angles, as well as as a change in the voltage at the terminals of the coupling transformer on the IPP side.

Cascaded connection of uncontrolled compensating devices for the purpose of regulating the voltage at the junction node also does not give the desired effect, since such connection leads to a change in the transmission angle, while the IPP, by restoring the angle, restores the total reactive-power flow of the tap and the voltage at the junction node. Therefore, over wide ranges of variation of the active power of the line, nonlinear dependences of the IPP reactive power on the transmission angles are required. Such dependences can be implemented…

are carried out by switching the settings of the measuring elements of the automatic excitation regulators (AER) of the generators of the intermediate power station, which regulate the transmission angle, as a function of the active power transmitted over the line.

An electric transmission acquires interesting properties when the main control of the generators of the intermediate power station is by the active power of the line. The constant term of the characteristic equation does not depend on the action of proportional AERs under external disturbances, since the latter are not taken into account in its determination. Consequently, the slope coefficient \(q_{p41}\) has no influence on the magnitude of the constant term if active-power losses in the electric transmission are not taken into account, since in this case \(\Delta P_{41}=-\Delta P\). When losses are taken into account, the constant term depends only weakly on \(q_{p41}\).

The total derivatives \(\dfrac{dP}{d\delta_1}\) and \(\dfrac{dP}{dU_4}\) relate the applied disturbance to the increments of the operating parameters of the transmission and therefore depend to a considerable extent on \(q_{p41}\). Main control of the generators of the intermediate power station by the deviation of the active power of the line makes it possible to obtain a transition from one steady-state transmission condition to another at positive values of the derivatives \(\dfrac{dP}{d\delta_1}\) and \(\dfrac{dP}{dU_4}\). It is quite possible that the constant term of the characteristic equation will become negative before the new steady state is reached.

Conclusions

1. In lines with an electrical length up to \(90^\circ \div 80^\circ\), regulation of the transmission angle and regulation of the voltage at the connection node by means of an intermediate power station connected at the midpoint of the line give the same effect in improving the static stability of the transmission, if the resources of the intermediate power station are limited to \(0 \div 2P_{\text{nat}}\). The stiffness of angle regulation for such transmissions should not be large, in order to reduce the dependence of the voltage at the connection node on the power transmitted over the line.

2. In lines with an electrical length from \(90^\circ \div 80^\circ\) to \(180^\circ\), under the same conditions, regulation of the transmission angle by means of the intermediate power station makes it possible to increase the static-stability margin. With equal reactive-power resources of the intermediate power station, the margin increases with an increase in the idealized line length and with strengthening of the stiffness of transmission-angle regulation.

3. Main control of the generators of the intermediate power station by the active power of the line is of little effectiveness for increasing the level of static stability of the transmission, despite the fact that in principle it makes it possible to obtain arbitrary dependences of the angle and voltage at the connection node on the power transmitted over the line.

4. Voltage deviations at the terminals of the coupling transformer on the side of the intermediate power station, caused by a change in the reactive-power flow regulating the voltage at the connection node, can be compensated by means of an on-load tap changer (OLTC). Voltage deviations on both sides of the coupling transformer that occur when regulating the transmission angle can be compensated as a result of the combined use of the OLTC of the coupling transformer and switching of the settings of the measuring elements of the AER of the generators of the intermediate power station, which regulate the angle.

5. In the absence of main control of the generators by external parameters, it is impossible, by means of a single intermediate power station, to obtain a stably operating electric transmission in which the increments of the active power of the line and of the voltage at the connection node have the same sign.

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