Voltage and Reactive Power Control of Compensated Long-Distance Power Transmission Lines with Intermediate Systems in Their Self-Balancing Mode

V. I. Gotman, I. A. Bezler

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

Total losses in the high-voltage elements of long-distance power transmission lines (LDPTL) may reach on the order of 8–12% of the transmitted power, which constitutes a large absolute value. In this connection, one of the important operating tasks is the optimal control of operating parameters that ensure minimum, or as close as possible to minimum, active-power losses. A reduction of active-power losses can be contemplated only when the power transmission line is equipped with controllable elements that make it possible to change the operating parameters in the required direction.

The operating mode of a power transmission line is fully characterized by three quantities: \(P\), \(U\), \(Q\). Taking into account that under real conditions the value of active power is of a forced character, \(U\) and \(Q\) have freedom of control.

Control of the parameters \(U\) and \(Q\) can be carried out by changing the excitation current of generators, switching transformer taps, and regulating the power of compensating devices.

To analyze the regularities in the variation of the reactive powers of sections, the conductances (powers) of compensating devices (CD), and voltages that ensure minimum active-power losses, let us turn to the scheme in Fig. 1, which shows a power transmission line with one intermediate system (IS).

Fig. 1. Power transmission scheme

The function of total losses as applied to the scheme (Fig. 1), after insignificant transformations, may be written in the following form:

\[ \Delta P_{\Sigma} = \Delta P_{1p} + \Delta P_{\ell 12} + \Delta P_{2p} + \Delta P_{\ell 23} + \Delta P_{3p} = \]

\[ = \frac{1}{\gamma_1} \operatorname{Re} \left[ \dot{A}_1 \hat{C}_1 U_1^2 + \dot{B}_1 \hat{D}_1 \frac{P_{21}^{2}+Q_{21}^{2}}{U_2^{2}} + (\dot{A}_1 \hat{D}_1+\dot{B}_1 \hat{C}_1)P_{21} + j(\dot{A}_1 \hat{D}_1-\dot{B}_1 \hat{C}_1)Q_{21} \right] - \]

\[ -\frac{K_{\mathrm{p}}}{\gamma_1}\operatorname{Im}\left[\dot A_1 \hat C_1 U_2^2+\dot B_1\hat D_1\frac{P_{21}^2+Q_{21}^2}{U_2^2}+(\dot A_1\hat D_1+\dot B_1\hat C_1)P_{21}+j(\dot A_1\hat D_1-\dot B_1\hat C_1)Q_{21}\right]+ \]

\[ +\frac{1}{\gamma_2}\operatorname{Re}\left[\dot D_2\hat C_2U_2^2+\dot B_2\hat A_2 \frac{[P_{21}-K_{\mathrm{p}}(Q_{21}-Q_{23})]^2+Q_{23}^2}{U_2^2} -(\dot D_2\hat A_2+\dot B_2\hat C_2)\times \right. \]

\[ \left. \times [P_{21}-K_{\mathrm{p}}(Q_{21}-Q_{23})]-j(\dot D_2\hat A_2-\dot B_2\hat C_2)Q_{23}\right] +\frac{K_{\mathrm{p}}}{\gamma_2}\operatorname{Im}\left[-\dot D_2\hat C_2U_2^2-\right. \]

\[ \left. -\dot B_2\hat A_2 \frac{[P_{21}-K_{\mathrm{p}}(Q_{21}-Q_{23})]^2+Q_{23}^2}{U_2^2} +(\dot D_2\hat A_2+\dot B_2\hat C_2)[P_{21}-K_{\mathrm{p}}(Q_{21}-Q_{23})]+\right. \]

\[ \left. +j(\dot D_2\hat A_2-\dot B_2\hat C_2)Q_{23}\right], \tag{1} \]

where

\[ \gamma_1=1-K_{\mathrm{p}}\tan\varphi_1,\qquad \gamma_2=1-K_{\mathrm{p}}\tan\varphi_2, \]

\[ K_{\mathrm{p}}=\tan\psi_{\mathrm{ku}} \]

is the tangent of the angle of losses of the compensating device (0.01–0.005).

Formula (1) is valid for loads below the natural power. Under these loads \(Q_{21}\) and \(Q_{32}\) are inductive powers, while \(Q_{12}\) and \(Q_{23}\) are capacitive.

Assuming that \(\tan\varphi_1,\tan\varphi_2\) (\(\cos\varphi_1,\cos\varphi_2\)) are specified by the conditions of normal operation of the generators of the sending station and of the consumers of the receiving system, the partial conditions for minimum losses at fixed voltage \(U_2\) follow from the equations

\[ \frac{\partial \Delta P_{\Sigma}}{\partial Q_{21}}=0,\qquad \frac{\partial \Delta P_{\Sigma}}{\partial Q_{23}}=0. \tag{2} \]

The simultaneous solution of these equations makes it possible to find the values of the reactive powers \(Q_{21}\) and \(Q_{23}\) that provide the optimal voltage drop on the sections

\[ Q_{21\mathrm{m}}=-U_2^2b_{21\mathrm{m}}+P_{21}\frac{K_{\mathrm{p}}M_2}{L},\qquad Q_{23\mathrm{m}}=U_2^2b_{23\mathrm{m}}-P_{21}\frac{K_{\mathrm{p}}M_1}{L}, \tag{3} \]

where

\[ b_{21\mathrm{m}}=\frac{K_{\mathrm{p}}^2(F_1-F_2)+F_1+N_2}{2L},\qquad b_{23\mathrm{m}}=\frac{K_{\mathrm{p}}^2(F_2-F_1)+(M_1)M_2(F_2+N_2)}{2L}, \tag{4} \]

\[ L=M_1+K_{\mathrm{p}}^2(M_1+M_1),\qquad F_1=\operatorname{Im}(\dot B_1\hat C_1)+K_{\mathrm{p}}\operatorname{Re}(\dot B_1\hat C_1)-K_{\mathrm{p}}A_1^2, \]

\[ F_2=\operatorname{Im}(\dot B_2\hat C_2)+K_{\mathrm{p}}\operatorname{Re}(\dot B_2\hat C_2)-K_{\mathrm{p}}A_2^2,\qquad N_2=-K_{\mathrm{p}}^2\operatorname{Im}(\dot B_2\hat C_2)+ \]

\[ +K_{\mathrm{p}}\operatorname{Re}(\dot B_2\hat C_2)+K_{\mathrm{p}}A_2^2, \]

\[ M_1=\operatorname{Re}(\dot B_1\hat D_1)-K_{\mathrm{p}}\operatorname{Im}(\dot B_1\hat D_1),\qquad M_2=\operatorname{Re}(\dot B_2\hat A_2)-K_{\mathrm{p}}\operatorname{Im}(\dot B_2\hat A_2). \]

It should be noted that in equations (3) the second terms are two to three orders of magnitude smaller than the first, so that in practical calculations the influence of \(P_{21}\) on the values \(Q_{\mathrm{m}}\) may be disregarded. Substituting the found relations \(Q_{21\mathrm{m}}\) and \(Q_{23\mathrm{m}}\) into the function of total losses (1) and differentiating it with respect to \(U_2\), we obtain the equation

\[ \frac{d\Delta P_{\Sigma}}{dU_2}=0, \tag{5} \]

which is satisfied by the solution

\[ U_{2\eta}=\sqrt{\frac{P_{21}}{g_{2\eta}}}, \tag{6} \]

where

\[ g_{2\eta}=\sqrt{\frac{H_1+H_2}{W}+\frac{G}{W}}, \tag{7} \]

\[ G=\left[b_{21\mathrm{m}}^{2}M_1+b_{23\mathrm{m}}^{2}M_2+K_{\mathrm{n}}(b_{21\mathrm{m}}+b_{23\mathrm{m}})^2M_2-b_{21\mathrm{m}}(F_1+N_2)-b_{23\mathrm{m}}(F_2+N_2)\right], \]

\[ H_1=\operatorname{Re}(\dot A_1 \hat C_1)-K_{\mathrm{n}}\operatorname{Im}(\dot A_1 \hat C_1),\quad H_2=\operatorname{Re}(\dot D_2 \hat C_2)-K_{\mathrm{n}}\operatorname{Im}(\dot D_2 \hat C_2), \]

\[ W=\frac{M_1(M_1+L)}{L}. \]

The essence of coordinated voltage control ensuring minimum losses is that the overall voltage level of the power transmission must be regulated in strict accordance with the active-power value of the transit schedule; in particular, for the node at which the intermediate system is connected, this relationship is determined by equation (6).

In the case where the constants of the sections are equal \([A_1]=[A_2]\), expressions (3), (4), (7) are simplified:

\[ Q_{21\mathrm{m}}=-Q_{23\mathrm{m}}=-U_2^2 b_{2\mathrm{m}}+P_{21}\frac{K_{\mathrm{n}}}{1+2K_{\mathrm{n}}^2}, \tag{8} \]

\[ b_{2\mathrm{m}}= \frac{\operatorname{Im}(\dot B\hat C)-K_{\mathrm{n}}^2\operatorname{Im}(\dot B\hat C)+2K_{\mathrm{n}}\operatorname{Re}(\dot B\hat C)} {2[\operatorname{Re}(\dot B\hat D)-K_{\mathrm{n}}\operatorname{Im}(\dot B\hat D)](1+2K_{\mathrm{n}}^2)}, \tag{9} \]

\[ g_{2\eta}= \sqrt{ \frac{[\operatorname{Re}(\dot D\hat C)-K_{\mathrm{n}}\operatorname{Im}(\dot D\hat C)](1+2K_{\mathrm{n}}^2)} {[\operatorname{Re}(\dot B\hat D)-K_{\mathrm{n}}\operatorname{Im}(\dot B\hat D)](1+K_{\mathrm{n}}^2)} -b_{2\mathrm{m}}^2\frac{(1+2K_{\mathrm{n}}^2)^2}{1+K_{\mathrm{n}}^2} }. \tag{10} \]

The active conductance \(g_\eta\) depends to a considerable degree on losses in the compensating device and increases as the coefficient \(K_{\mathrm{n}}\) increases. In the mode of coordinated voltage control, \(U_2=U_{2\eta}\) according to (6), and the voltages \(U_{1\eta}\) and \(U_{3\eta}\) corresponding to this voltage law are written as:

\[ U_{1\eta} = B\sqrt{ \frac{P_{21}}{g_{2\eta}} \left\{ \left[ \frac{K_{\mathrm{n}}g_{2\eta}}{1+2K_{\mathrm{n}}^2} -b_{2\mathrm{m}} -\frac{\operatorname{Im}(\dot A\hat B)}{B^2} \right]^2 + \left[ g_{2\eta} +\frac{\operatorname{Re}(\dot A\hat B)}{B^2} \right]^2 \right\} }, \tag{11} \]

\[ U_{3\eta} = B\sqrt{ \frac{P_{21}}{g_{2\eta}} \left\{ \left[ b_{2\mathrm{m}}\frac{1+2K_{\mathrm{n}}^2}{1+K_{\mathrm{n}}^2}+T \right]^2 + \left[ \frac{g_{2\eta}}{1+2K_{\mathrm{n}}^2} + \frac{K_{\mathrm{n}}b_{2\mathrm{m}}}{1+K_{\mathrm{n}}^2} -V \right]^2 \right\}(1+K_{\mathrm{n}}^2) }. \tag{12} \]

And, consequently, the voltage ratios on sections 1—2 and 2—3 in the maximum-efficiency mode are determined as

\[ K_\eta U_{12} = \frac{U_{1\eta}}{U_{2\eta}} = B\sqrt{ \left[ \frac{K_{\mathrm{n}}g_{2\eta}}{1+2K_{\mathrm{n}}^2} -b_{2\mathrm{m}} -\frac{\operatorname{Im}(\dot A\hat B)}{B^2} \right]^2 + \left[ g_{2\eta} +\frac{\operatorname{Re}(\dot A\hat B)}{B^2} \right]^2 }, \tag{13} \]

\[ K_\eta U_{23} = \frac{U_{2\eta}}{U_{3\eta}} = \frac{1}{ B\sqrt{ \left\{ \left[ b_{2\mathrm{m}}\frac{1+2K_{\mathrm{n}}^2}{1+K_{\mathrm{n}}^2}+T \right]^2 + \left[ \frac{g_{2\eta}}{1+2K_{\mathrm{n}}^2} + \frac{K_{\mathrm{n}}b_{2\mathrm{m}}}{1+K_{\mathrm{n}}^2} -V \right]^2 \right\}(1+K_{\mathrm{n}}^2) } }, \tag{14} \]

where

\[ T=\frac{\operatorname{Im}(\dot D\hat B)-K_{\mathrm{n}}\operatorname{Re}(\dot D\hat B)} {B^2(1+K_{\mathrm{n}}^2)}, \quad V=\frac{\operatorname{Re}(\dot D\hat B)+K_{\mathrm{n}}\operatorname{Im}(\dot D\hat B)} {B^2(1+K_{\mathrm{n}}^2)}. \]

The voltage drops \(K_{\eta}U\) of sections in the \(\eta_{\max}\) mode do not depend on operating parameters, but are determined only by the constants of the transmission line and the losses in the compensating devices, and therefore remain constant over the entire interval of coordinated voltage regulation. For long-distance transmission lines with balanced intermediate systems, the voltage drops of the sections are always positive (greater than unity) and increase with increasing section length and losses in the compensating devices. Numerical values of the voltage drops for sections of various lengths and different voltage classes are given in Table 1.

Table 1

Voltage drops in the \(\eta_{\max}\) mode for the case \([A_1]=[A_2]\)

Let us note that, for unchanged values of \(K_{\eta}U\) in the coordinated regulation mode, the voltage difference between the ends of the sections \(\Delta U\) \((\Delta U_{12}=U_{1\eta}-U_{2\eta},\ \Delta U_{23}=U_{2\eta}-U_{3\eta})\) does not remain constant and increases as the active power decreases.

Modern high-power transformers are produced with a regulation range of \(\pm(10\text{–}15)\%\,U_{\text{nom}}\). As applied to high-voltage transmission lines \((U_{\text{nom}}=500\div1000\ \text{kV})\), in which the upper limit for insulation conditions is restricted to \(1.05U_{\text{nom}}\), it is more expedient to manufacture transformers with an asymmetric regulation range: \((+5\text{–}15)\%\,U_{\text{nom}}\), \((+5\text{–}25)\%\,U_{\text{nom}}\). Thanks to this, there are real possibilities, over a certain range of variation of the active transit power, to operate in the minimum-loss mode, or in a mode close to it.

The insulation of the ends of the sections, especially the insulation of equipment located at compensating stations (CS) and, still more, at substations, corresponds more exactly to the rated voltage than does the insulation of the line; and a strict limitation of the voltage at the level \(1.05\times U_{\text{nom}}\) should be applied only to CS and substations of the transmission line. At intermediate points of the sections, some excess over \(U_{\text{nom}}\) may also be permitted. We shall assume that voltage regulation at the ends of the sections is carried out within a certain range: the upper level is limited by \(U_{\text{nom}}\), and the lower level \((U_{\text{low}})\) by the capabilities of the regulating devices.

As the flow of active power increases, the voltages at the ends of the sections \(U_{1\eta}, U_{2\eta}, U_{3\eta}\) (as well as the voltage at any point of the system) increase according to equations (11), (6), (12). The first to reach the upper limit is the voltage at the sending end \((U_{1\eta})\).

The value of active power \(P_{21}^{10}\), starting from which it is necessary to pass over to the fixed-voltage mode \(U_1\), will be found from equation (11) under the condition \(U_{1\eta}=U_{\text{nom}}\).

\[ P_{21}^{10}= \frac{U_{\text{nom}}^{2}g_{2\eta}} {B^{2}\left\{ \left[ \frac{-K_{\text{p}}g_{2\eta}}{1+2K_{\text{p}}^{2}} -b_{2m} -\frac{\operatorname{Im}(\hat{A}\hat{B})}{B^{4}} \right]^{2} + \left[ g_{2\eta} +\frac{\operatorname{Re}(\hat{A}\hat{B})}{B^{2}} \right]^{2} \right\}} \tag{15} \]

As the load subsequently increases, at certain values of \(P_{21}\) it is necessary to fix the voltage \(U_2\) and then \(U_3\). The power \(P_{21,\text{upper}}^{23}\),

at which the equal-voltage mode begins on section 2--3 and, consequently, the transition to the mode of equal and fixed voltages \(U_{\mathrm{nom}}\) throughout the entire power transmission line is completed, is determined from the expression

\[ P^{23}_{21\mathrm{upper}}= \frac{U_{\mathrm{nom}}^{2}(1+K_{\mathrm{n}}^{2})(1+2K_{\mathrm{n}}^{2})} {(1+2K_{\mathrm{n}}^{2})^{2}+K_{\mathrm{n}}^{2}} \left[ (1+2K_{\mathrm{n}}^{2})V-K_{\mathrm{n}}(g_{2\mathrm{m}}+T)+ \sqrt{ \frac{(1+2K_{\mathrm{n}}^{2})^{2}+K_{\mathrm{n}}^{2}} {B^{2}(1+K_{\mathrm{n}}^{2})} -\left[(1+2K_{\mathrm{n}}^{2})(g_{2\mathrm{m}}+T)+K_{\mathrm{n}}V\right]^{2} } \right]. \tag{16} \]

In an analogous manner, as the transmitted power decreases, the general voltage level on the sections is reduced; in this case, because of the positivity of the voltage differences, the first to reach the lower limit, restricted by the regulating capabilities of the step-down transformer, is the voltage at the receiving end of the power transmission line \((U_{3})\).

The value \(P^{30}_{21}\), corresponding to the fixing of the voltage \(U_{3}\), is determined from equation (12) under the condition \(U_{3\eta}=U_{\mathrm{lower}}\):

\[ P^{30}_{21}= \frac{U_{\mathrm{lower}}^{2}g_{2\eta}} {B^{2} \left\{ \left[ g_{2\mathrm{m}}\frac{1+2K_{\mathrm{n}}^{2}}{1+K_{\mathrm{n}}^{2}}+T \right]^{2} + \left[ \frac{g_{2\eta}}{1+2K_{\mathrm{n}}^{2}} + \frac{K_{\mathrm{n}}g_{2\mathrm{m}}}{1+K_{\mathrm{n}}^{2}} - V \right]^{2} \right\} }. \tag{17} \]

With a subsequent decrease of \(P_{21}\), the voltages \(U_{2}\) and \(U_{1}\) reach the lower limit.

The equal-voltage mode at the nodal points of the entire power transmission line begins when \(U_{1}=U_{\mathrm{lower}}\) is reached (it is assumed that all transformers of the high-voltage section have one and the same regulating range), to which \(P^{12}_{21\mathrm{lower}}\) corresponds:

\[ P^{12}_{21\mathrm{lower}}= \frac{U_{\mathrm{lower}}^{2}(1+2K_{\mathrm{n}}^{2})} {(1+2K_{\mathrm{n}}^{2})^{2}+K_{\mathrm{n}}^{2}} \left[ I+K_{\mathrm{n}}b_{2\mathrm{m}} + \sqrt{ \frac{(1+2K_{\mathrm{n}}^{2})^{2}+K_{\mathrm{n}}^{2}}{B^{2}} - \left[R+(1+2K_{\mathrm{n}}^{2})g_{2\mathrm{m}}\right]^{2} } \right] \tag{18} \]

where

\[ I= \frac{ K_{\mathrm{n}}\operatorname{Im}(\dot A\dot B) - (1+2K_{\mathrm{n}}^{2})\operatorname{Re}(\dot A\dot B) }{B^{2}}, \qquad R= \frac{ K_{\mathrm{n}}\operatorname{Re}(\dot A\dot B) + (1+2K_{\mathrm{n}}^{2})\operatorname{Im}(\dot A\dot B) }{B^{2}}. \]

Expressions (15)--(18) are equations describing the boundaries of the transition zones of the regularities of optimal control of voltages, reactive-power flows in the sections, and the conductances (powers) of the compensating devices, taking account of operating voltage constraints.

For any power transmission line having control means, five characteristic zones can be distinguished. The extent of each zone depends on the parameters, the length and the number of sections, the loss coefficient of the compensating devices, the voltage class of the power transmission line, the regulating range of the transformers, and the voltage-limit levels.

The first zone, at \(P_{21}\leq P^{12}_{21\mathrm{lower}}\), is characterized by the fact that the lowest possible voltage level is maintained at the ends of all sections.

In the second \(P^{12}_{21\mathrm{lower}}\leq P_{21}\leq P^{30}_{21}\) and fourth \(P^{10}_{21}\leq P_{21}\leq P^{23}_{21\mathrm{upper}}\) zones, there is a successive transition to the mode of fixed voltages on the sections: in the second, to \(U_{\mathrm{lower}}\); in the fourth, to \(U_{\mathrm{nom}}\). The mode of the fifth zone, \(P_{21}\geq P^{23}_{21\mathrm{upper}}\), is analogous to the mode of the first zone, except that the voltages are maintained at the highest permissible level.

In the third zone, \(P^{30}_{21}\leq P_{21}\leq P^{10}_{21}\), a coordinated voltage-control mode is maintained.

In the zone of coordinated voltage regulation, the reactive powers \(Q_{21\mathrm{m}}\) and \(Q_{23\mathrm{m}}\) must be regulated according to the following laws

\[ Q_{21\mathrm{m}}=-P_{21}\frac{b_{21\mathrm{m}}}{g_{2\eta}},\quad Q_{23\mathrm{m}}=P_{21}\frac{b_{23\mathrm{m}}}{g_{2\eta}}, \tag{19} \]

and the voltage \(U_2\) according to equation (6).

It is not difficult to show that the operating parameters of the sending and receiving ends of the power transmission line obey analogous regularities:

\[ U_{1\eta}=\sqrt{\frac{P_{12}}{g_{1\eta}}},\qquad U_{3\eta}=\sqrt{\frac{P_{32}}{g_{3\eta}}}, \tag{20} \]

\[ Q_{12\mathrm{m}}=U_{1\eta}^{2}b_{12\mathrm{m}} =P_{12}\frac{b_{12\mathrm{m}}}{g_{1\eta}},\quad Q_{23\mathrm{m}}=-U_{3\eta}^{2}b_{32\mathrm{m}} =-P_{32}\frac{b_{32\mathrm{m}}}{g_{3\eta}}, \tag{21} \]

where

\[ b_{12\mathrm{m}}= \frac{ \operatorname{Im}(\dot A_1\hat C_1) +\operatorname{Im}(\dot B_1\hat D_1)(g_{2\eta}^{2}+b_{21\mathrm{m}}^{2}) +\operatorname{Im}(\dot B_1\hat C_1)g_{2\eta} +\operatorname{Re}(\dot B_1\hat C_1-\dot A_1\hat D_1)b_{21\mathrm{m}} }{ K_{\eta}^{2}U_{12} } \]

\[ b_{32\mathrm{m}}= \{\operatorname{Im}(\dot D_2\hat C_2)+\operatorname{Im}(\dot B_2\hat A_2) [(g_{2\eta}+2K_{\eta}b_{23\mathrm{m}})^2+b_{23\mathrm{m}}^2] -\operatorname{Im}(\dot B_2\hat C_2)(g_{2\eta} \]

\[ +2K_{\eta}b_{23\mathrm{m}}) +\operatorname{Re}(\dot B_2\hat C_2-\dot D_2\hat A_2)b_{23\mathrm{m}}\} K_{\eta}^{2}U_{23}. \]

\[ g_{1\eta}=[\operatorname{Re}(\dot A_1\hat C_1) +\operatorname{Re}(\dot B_1\hat D_1)(g_{2\eta}^{2}+b_{21\mathrm{m}}^{2}) -\operatorname{Im}(\dot B_1\hat C_1)b_{2\eta} \]

\[ +\operatorname{Re}(\dot B_1\hat C_1+\dot A_1\hat D_1)g_{2\eta}]K_{\eta}^{-2}U_{12}, \]

\[ g_{3\eta}=\{-\operatorname{Re}(\dot D_2\hat C_2) -\operatorname{Re}(\dot B_2\hat A_2)[(g_{2\eta}+2K_{\eta}b_{23\mathrm{m}})^2+b_{23\mathrm{m}}^2] +\operatorname{Im}(\dot B_2\hat C_2)b_{23\mathrm{m}} \]

\[ +\operatorname{Re}(\dot B_2\hat C_2+\dot D_2\hat A_2)(g_{2\eta}+2K_{\eta}b_{23\mathrm{m}})\} K_{\eta}^{2}U_{23}, \]

The laws of regulation of reactor powers are determined by the sum of the reactive-power flows at the corresponding ends of the sections. With unchanged values of the phase-shift angles (\(\varphi_1\) and \(\varphi_2\)), the conductances of reactors 1, 2, and 3 in the \(\eta_{\max}\) mode remain unchanged, and their power in this case is regulated automatically by changing the general voltage level applied to them.

A characteristic feature of the coordinated regulation mode is that the regularities of change in the reactive powers of the ends of the sections and of the reactors are linear functions of the corresponding values of active powers (or of one and the same power of either end); the active powers along the entire power transmission line are also related to one another by a linear dependence, which predetermines the linear character of the losses.

On the basis of formulas (6), (20), one can establish a simple relationship between the parameters in the maximum-efficiency mode:

\[ \frac{U_{1\eta}}{U_{2\eta}} = \sqrt{\frac{g_{2\eta}}{g_{1\eta}\eta_{12\max}}}, \qquad \frac{U_{2\eta}}{U_{3\eta}} = \sqrt{\frac{g_{3\eta}}{g_{2\eta}\eta_{23\max}}}. \tag{22} \]

For a power transmission line with several self-balancing intermediate systems in the coordinated voltage-regulation mode, the process of power transfer consists of a number of completely similar transfer processes over the individual sections, so that, under equal conditions, the voltage drops on equal sections will have one and the same value:

\[ K_{\eta u}= \frac{U_{\mathrm{н}1}}{U_{\mathrm{k}1}} = \frac{U_{\mathrm{н}2}}{U_{\mathrm{k}2}} =\cdots= \frac{U_{\mathrm{н}n}}{U_{\mathrm{k}n}} = \sqrt{\frac{g_{\mathrm{k}}}{g_{\mathrm{н}}\eta_{\max}}}. \tag{23} \]

Thus, in the zone of coordinated control the efficiency is determined only by the constants of the sections and by the loss coefficient of the compensating device, and consequently remains unchanged.

Corona on the conductors can be taken into account approximately in operating-condition calculations in the form of a per-unit active conductance \(g_{\mathrm{c}}\), uniformly distributed along the line or along its individual section and dependent on weather conditions and on the average voltage. For an extended power transmission line, the per-unit conductance for the various sections can be allowed for differentially. The calculation procedure and the expressions given do not change when corona is present. However, the generalized constants entering into the calculation expressions will depend on the magnitude of the corona conductance and hence also on the voltage distribution. Therefore the calculation has to be carried out by successive approximations.

In the second and fourth zones, taking into account the imposed voltage constraints, operating conditions are realized that differ from the optimum, but with minimum deviations from it.

Fig. 2. a) transition zones for a 500-kV transmission line; (b) for a 1000-kV transmission line

Using the expressions given above, the transition zones were calculated for different voltage classes as a function of the length of the power transmission line, for a 15% voltage-control range \((\Delta U_{\mathrm{reg}} = U_{\mathrm{nom}} - 0.85 \cdot U_{\mathrm{nom}})\), as applied to the scheme of Fig. 1 with \([A_1]=[A_2]\).

Figure 2a gives the calculation results for a 500-kV transmission line with \(3 \times \mathrm{ACSR}\text{-}500\) conductors \((P_{\mathrm{nat}} = 900\ \mathrm{MW})\) at a loss coefficient \(K_{\mathrm{p}}=0.01\); Fig. 2b, for a 1000-kV transmission line with conductor type \(4 \times \mathrm{ACSR}\text{-}1000\) \((P_{\mathrm{nat}} = 3920\ \mathrm{MW})\) and \(K_{\mathrm{p}}=0.005\), in the absence of corona.

Table 2

Notes: 1. $\Delta P_{1\Sigma}$ (MW) corresponds to optimal operating conditions; 2. $\Delta P_{2\Sigma}$ (MW) corresponds to $U_1=U_2=U_3=U_{\mathrm{nom}}=\mathrm{const}$; 3. $(\Delta P_{2\Sigma}-\Delta P_{1\Sigma})\%=$

\[ =\frac{\Delta P_{2\Sigma}-\Delta P_{1\Sigma}}{\Delta P_{1\Sigma}}\,100\% \]

Table 3

Notes: 1. $\Delta P_{1\Sigma}$ (MW) corresponds to optimal operating conditions; 2. $\Delta P_{2\Sigma}$ (MW) corresponds to $U_1=U_2=U_3=U_{\mathrm{nom}}=\mathrm{const}$; 3. $(\Delta P_{2\Sigma}-\Delta P_{1\Sigma})\%=$

\[ =\frac{\Delta P_{2\Sigma}-\Delta P_{1\Sigma}}{\Delta P_{1\Sigma}}\,100\% \]

The total range of load variation \((\Delta P_{21}=P^{23}_{21\,\text{upper}}-P^{12}_{21\,\text{lower}},\ 2\text{--}4\ \text{zones})\), within which optimal or partially optimal voltage control is possible, gradually narrows as the length of the transmission line increases: from \(0.78\,P_{\text{nat}}\) for a 500 kV transmission line and \(0.475\,P_{\text{nat}}\) for a 1000 kV transmission line at \(l_{\text{sec}}=200\ \text{km}\), to \(0.315\,P_{\text{nat}}\) for both transmission lines at \(l_{\text{sec}}=1000\ \text{km}\). Although the total range of zones 2–4 narrows as the length of the sections increases, the overall power level nevertheless rises. The loss coefficient \(K_{\text{p}}\) has a significant influence on the level at which the zones are located, and as it increases the zones shift into the region of higher loads. Thus, at \(K_{\text{p}}=0.01\), the limiting load for a 1000 kV transmission line is \(1\text{--}1.15\,P_{\text{nat}}\). The presence of corona losses shifts the zone boundaries substantially into the region of higher loads.

Tables 2 and 3 give the results of calculating total losses for various values of transmitted power for two cases: 1 corresponds to the mode of coordinated voltage control, taking into account the constraints \(1\text{--}0.85\,U_{\text{nom}}\); 2 is when in all modes \(U_1=U_2=U_3=U_{\text{nom}}=\text{const}\). The circuit parameters are taken to be the same as in the calculation of the transition zones. As follows from the calculations, as the load decreases, the difference in losses for the first and second cases increases and, in general, is considerable.

Conclusions

Taking into account that the operating load of modern 500 kV transmission lines is on the order of \(P_{\text{nat}}\), coordinated voltage control for 500 kV transmission lines, and especially for 750–1000 kV transmission lines, is quite feasible and has practical significance.

References

1. V. A. Venikov, I. P. Suda. Calculations of Operating Conditions of Long-Distance AC Transmission Lines. Moscow, “Vysshaya shkola,” 1966.

2. D. V. Kholmskii. Voltage control of long-distance transmission lines by means of controllable transformers. Izv. vuzov, “Energetika,” 1959, No. 7.