Proceedings
of the Tomsk Polytechnic Institute named after S. M. Kirov,
Holder of the Order of the October Revolution and the Order of the Red Banner of Labor

Vol. 244 1972

CONDITIONS FOR MAXIMUM EFFICIENCY OF LONG-DISTANCE POWER TRANSMISSION LINES WITH INTERMEDIATE SYSTEMS

V. I. Gotman

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

The most general cases for long-distance power transmission lines (LDPTL) with intermediate systems (IS) should be considered to be operating conditions under which the values of the active-power flows on the sections of the line are not identical. In this case, the laws governing the variation of these values may, in the general case, be quite arbitrary. Economically correct regulation of the reactive powers and voltages of the sections, in the case of unbalanced IS, can be conceived only as coordinated regulation of all sections of the line subject to the condition of a minimum of the total transmission losses.

As a basis for the investigation we shall take the power transmission considered in [1], using the same notation. As independent variables we take \(P_{21}, P_{23}, Q_{21}, Q_{23}, U_2, \operatorname{tg}\varphi_1, \operatorname{tg}\varphi_2\). Summing the losses in the individual elements after minor transformations, we obtain the expression

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

\[ = \frac{1}{\gamma_1}\operatorname{Re}G_1 - \frac{K_p}{\gamma_1}J_mG_1 + \frac{1}{\gamma_2}\operatorname{Re}G_2 - \frac{K_p}{\gamma_2}J_mG_2 + P_{23}-P_{21}+K_p(Q_{21}-Q_{23}), \tag{1} \]

where

\[ G_1 = \dot{A}_1\hat{C}_1U_2^2 + \dot{B}_1\hat{D}_1 \frac{P_{21}^{2}+Q_{21}^{2}}{U_2^{2}} + (\dot{A}_1\hat{D}_1+\dot{B}_1\hat{C}_1)P_{21} + j(\dot{A}_1\hat{D}_1-\dot{B}_1\hat{C}_1)Q_{21}, \]

\[ G_2 = \dot{D}_2\hat{C}_2U_2^2 + \dot{B}_2\hat{A}_2 \frac{P_{23}^{2}+Q_{23}^{2}}{U_2^{2}} - (\hat{A}_2\dot{D}_2+\dot{B}_2\hat{C}_2)P_{22} - j(\dot{D}_2\hat{A}_2-\dot{B}_2\hat{C}_2)Q_{23}, \]

\[ \gamma_1=1-K_p\operatorname{tg}\varphi_1,\qquad \gamma_2=1-K_p\operatorname{tg}\varphi_2,\qquad K_p=\operatorname{tg}\psi_{ku} \]
is the tangent of the loss angle of the compensating devices.

The values of the reactive powers \(Q_{21m}\) and \(Q_{23m}\), which ensure optimal voltage drops on sections 1--2 and 2--3, respectively, for given values of \(P_{21}, P_{23}, \cos\varphi_1, \cos\varphi_2\) and fixed voltage \(U_2\), are determined from the conditions:

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

Equations (2) are independent and have the solutions

\[ Q_{21m}=-U_2^{2}b_{21m},\qquad Q_{23m}=U_2^{2}b_{23m}, \tag{3} \]

where

\[ \varepsilon_{21\mathrm{m}}= \frac{ \operatorname{Im}(\dot B_1 \hat C_1)+K_{\mathrm{p}}\operatorname{Re}(\dot B_1 \hat C_1)-K_{\mathrm{p}}A_1^2+K_{\mathrm{p}}\gamma_1 }{ 2\left[\operatorname{Re}(\dot B_1 \hat D_1)-K_{\mathrm{p}}\operatorname{Im}(\dot B_1 \hat D_1)\right] }, \tag{4} \]

\[ \varepsilon_{23\mathrm{m}}= \frac{ \operatorname{Im}(\dot B_2 \hat C_2)+K_{\mathrm{p}}\operatorname{Re}(\dot B_2 \hat C_2)-K_{\mathrm{p}}A_2^2+K_{\mathrm{p}}\gamma_2 }{ 2\left[\operatorname{Re}(\dot B_2 \hat A_2)-K_{\mathrm{p}}\operatorname{Im}(\dot B_2 \hat A_2)\right] }. \tag{5} \]

Substituting the laws found for \(Q_{21\mathrm{m}}\) and \(Q_{23\mathrm{m}}\) into the expression for the total losses and differentiating it with respect to \(U_2\), we obtain the equation \(d\Delta P_\Sigma/dU_2=0\), which is satisfied by the solution

\[ U_{2\eta}=\sqrt[4]{ \frac{P_{21}^2 g_{23}^2+P_{23}^2 g_{21}^2} {2g_{21}^2g_{23}^2} }, \tag{6} \]

where

\[ g_{21}=\sqrt{ \frac{H_1+H_2}{2M_1} -0.5\left(\varepsilon_{21\mathrm{m}}^2+\frac{M_2}{M_1}\varepsilon_{23\mathrm{m}}^2\right) }, \]

\[ g_{23}=\sqrt{ \frac{H_1+H_2}{2M_2} -0.5\left(\frac{M_1}{M_2}\varepsilon_{21\mathrm{m}}^2+\varepsilon_{23\mathrm{m}}^2\right) }, \tag{7} \]

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

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

For given values of \(\cos\varphi_1\), \(\cos\varphi_2\), \(P_{21}\), and \(P_{23}\), the laws \(Q_{21\mathrm{m}}\), \(Q_{23\mathrm{m}}\), and \(U_{2\eta}\) correspond to the conditions for the maximum efficiency of the entire electric-power transmission \((\eta_{\Sigma\max})\); in the general case, the maximum values of the efficiencies of the sections \((\eta_{1\max}, \eta_{2\max})\) occur at different times and are different for each pair of combinations of \(P_{21}\) and \(P_{23}\).

If it is assumed that the values \(\varphi_1\) and \(\varphi_2\) are not constrained by the operating conditions and have a degree of freedom, then, by a rational choice of the control laws for \(\varphi_1\) and \(\varphi_2\), an additional reduction of losses can be achieved. To find the laws of optimal regulation of the phase-shift angles, cumbersome investigations for the minimum of the total losses are not required; rather, they follow directly from the conditions that the losses in the first and third reactors be equal to zero, i.e.,

\[ \Delta P_{1\mathrm{p}}=0,\qquad \Delta P_{3\mathrm{p}}=0. \tag{8} \]

and consequently,

\[ \varphi_{1\mathrm{opt}}=\arctan\frac{Q_{12}}{P_{12}},\qquad \varphi_{2\mathrm{opt}}=\arctan\frac{Q_{32}}{P_{32}}. \tag{9} \]

The meaning of these laws is simple: the reactive power of the generators, in magnitude and direction, must be regulated so that it fully corresponds to the reactive power of the sending end of the first section, i.e., \(Q_{01}=Q_{12}\); in this case the first reactor is in no-load operation and may be disconnected. The condition \(\varphi_{2\mathrm{opt}}\) will be satisfied when the reactive power \(Q_{32}\) is completely absorbed by the receiving system. For loads below the natural load, in the maximum-efficiency regime \(\cos\varphi_{2\mathrm{opt}}\) is lagging, and reactive power enters the receiving system together with active power. The reactive power \(Q_{12}\) in these regimes is capacitive, and to ensure \(\varphi_{1\mathrm{opt}}\) the generators of the sending station must operate in underexcited mode. The implementation of such regimes makes it possible, in a number of cas-

cases not only to reduce losses, but also to reduce the installed capacity of compensating devices in the power transmission. Limitations on the underexcited mode for synchronous generators may arise from self-excitation conditions, and at significant loads additionally from the condition of ensuring static stability. At present, so-called synchronous-asynchronous generators [2] are at the stage of research and development; for these, modes with a leading $\cos\varphi$, along with lagging ones, are operating modes. The use of such generators for operation in combination with high-voltage lines will make it possible to realize fully the advantages of optimal regulation of the phase-shift angle at the sending end.

Analyzing the function of total losses (1) for the case when optimal regulation of the phase-shift angles at the sending and receiving ends of the power transmission is carried out according to equation (9), we obtain the laws of variation of $Q_{21m}$, $Q_{23m}$, and $U_{2\eta}$, completely analogous to the previous ones (3), (6), in which the reactive and equivalent active conductances of the sections have the following form:

\[ b_{21m}=\frac{\operatorname{Im}(\dot B_1 \hat C_1)+K_{\mathrm{p}}}{2\operatorname{Re}(\dot B_1 \hat D_1)},\qquad b_{23m}=\frac{\operatorname{Im}(\dot B_2 \hat C_2)+K_{\mathrm{p}}}{2\operatorname{Re}(\dot B_2 \hat A_2)}, \tag{10} \]

\[ g_{21\mathrm{e}}= \sqrt{ \frac{\operatorname{Re}(\dot A_1 \hat C_1)+\operatorname{Re}(\dot D_2 \hat C_2)} {2\operatorname{Re}(\dot B_1 \hat D_1)} -0.5\left( b_{21m}^{2} +\frac{\operatorname{Re}(\dot B_2 \hat A_2)} {\operatorname{Re}(\dot B_1 \hat D_1)} b_{23m}^{2} \right) }, \tag{11} \]

\[ g_{23\mathrm{e}}= \sqrt{ \frac{\operatorname{Re}(\dot A_1 \hat C_1)+\operatorname{Re}(\dot D_2 \hat C_2)} {2\operatorname{Re}(\dot B_2 \hat A_2)} -0.5\left( \frac{\operatorname{Re}(\dot B_1 \hat D_1)} {\operatorname{Re}(\dot B_2 \hat A_2)} b_{21m}^{2} +b_{23m}^{2} \right) }. \tag{12} \]

Since the general form of expressions (3), (6) does not depend on the operating conditions of reactors 1 and 3, the equations obtained below are equally valid both under optimal regulation of the angles $\varphi_1$, $\varphi_2$ and under the condition of their limitation.

The voltage ratios in the first and second sections in the coordinated regulation mode are determined by the expressions:

\[ K_{\eta U12}=\frac{U_{1\eta}}{U_{2\eta}} = B_1 \sqrt{ \left[ b_{21m}+\frac{\operatorname{Im}(\dot A_1 \hat B_1)}{B_1^{2}} \right]^{2} + \left[ \frac{\sqrt{2}\,g_{21\mathrm{e}}g_{23\mathrm{e}}} {g_{23\mathrm{e}}^{2}+K_{P}^{2}g_{21\mathrm{e}}^{2}} + \frac{\operatorname{Re}(\dot A_1 \hat B_1)}{B_1^{2}} \right]^{2} }, \tag{13} \]

\[ K_{\eta U23}=\frac{U_{2\eta}}{U_{3\eta}} = \frac{1}{ B_2 \sqrt{ \left[ b_{23m}+\frac{\operatorname{Im}(\dot D_2 \hat B_2)}{B_2^{2}} \right]^{2} + \left[ \frac{\sqrt{2}\,g_{21\mathrm{e}}g_{23\mathrm{e}}K_{P}} {g_{23\mathrm{e}}^{2}+K_{P}^{2}g_{21\mathrm{e}}^{2}} - \frac{\operatorname{Re}(\dot D_2 \hat B_2)}{B_2^{2}} \right]^{2} } }, \tag{14} \]

where

\[ K_{P}=P_{23}/P_{21}. \tag{15} \]

The voltage ratios in the sections in the zone of coordinated regulation depend on the ratio of the active powers of the sections ($K_{P}$) and, consequently, in the general case are variable. In contrast to the case of a balanced intermediate system, when the optimal ratios were always positive, here, along with positive ratios at $P_{21}\geq P_{23}$, negative ones are also possible ($K_U<1$) under the condition $P_{21}<P_{23}$, i.e., only in the case when the intermediate system operates in the active-power output mode.

In order to introduce, in the proper sequence, voltage constraints at the controlled nodes, it is necessary to know the distribution

voltage drops on the sections. Since, for given section lengths and a given value of \(K_{\mathrm{p}}\), the optimal drops depend only on \(K_{\mathrm{p}}\), the values of \(K_{\mathrm{p}}\) corresponding to equality of the voltages at the ends of the sections may serve as a basis for judging the voltage distribution over the sections. They are determined from equations (13), (14) under the condition \(K_{\eta U12}=1\) and \(K_{\eta U23}=1\):

\[ K_{12\mathrm{p}}=g_{339}\sqrt{ 4n_1\left(n_1-\sqrt{n_1^2-\frac{1}{m_1}}\right) -\frac{2}{m_1}-\frac{1}{g_{219}^2} }, \tag{16} \]

\[ K_{23\mathrm{p}}= \left[ g_{219}\sqrt{ 4n_2\left(n_2+\sqrt{n_2^2-\frac{1}{m_2}}\right) -\frac{2}{m_2}-\frac{1}{g_{239}^2} } \right]^{-1}, \tag{17} \]

where

\[ n_1=\frac{\operatorname{Re}(\dot A_1\hat B_1)}{B_1^2 m_1},\qquad m_1= \left[ \theta_{21\mathrm{m}}+\frac{\operatorname{Im}(\dot A_1\hat B_1)}{B_1^2} \right]^2 + \left[ \frac{\operatorname{Re}(\dot A_1\hat B_1)}{B_1^2} \right]^2 -\frac{1}{B_1^2}, \]

\[ n_2=\frac{\operatorname{Re}(\dot D_2\hat B_2)}{B_2^2 m_2},\qquad m_2= \left[ \theta_{23\mathrm{m}}+\frac{\operatorname{Im}(\dot D_2\hat B_2)}{B_2^2} \right]^2 + \left[ \frac{\operatorname{Re}(\dot D_2\hat B_2)}{B_2^2} \right]^2 -\frac{1}{B_2^2}. \]

By comparing the actual value \(K_{\mathrm{p}}=P_{23}/P_{21}\) with the calculated values \(K_{12\mathrm{p}}\) and \(K_{23\mathrm{p}}\) for the first and second sections, respectively, one can judge the actual pattern of voltage distribution in the power transmission. Values \(K_{\mathrm{p}}<K_{12\mathrm{p}}, K_{23\mathrm{p}}\) correspond to positive voltage drops on the sections \((K_{U12}>1,\ K_{U23}>1)\), and, in the opposite case, \(K_{\mathrm{p}}>K_{12\mathrm{p}}, K_{23\mathrm{p}}\), to negative drops \((K_{U12}, K_{U23}<1)\).

The quantities \(K_{12\mathrm{p}}\) and \(K_{23\mathrm{p}}\) vary over a sufficiently wide range, from 1 to \(\infty\). For real section lengths they are always greater than unity and decrease as the length increases. Numerical values of \(K_{12\mathrm{p}}\) and \(K_{23\mathrm{p}}\) as functions of the section lengths and the transmission-line voltage for \(\cos\varphi_1=1\) and \(\cos\varphi_2=0.92\) are given in Table 1.

Analyzing the results of the calculations, we note that for any combination of section lengths the inequality \(K_{23\mathrm{p}}>K_{12\mathrm{p}}\) always holds. As the section lengths increase, this difference diminishes. For long-distance transmission lines with an unbalanced intermediate system, three characteristic combinations of voltage drops on the sections may occur:

1. \(K_{\mathrm{p}}<K_{12\mathrm{p}}, K_{23\mathrm{p}}\)—on both sections the drop is positive, increasing as \(K_{\mathrm{p}}\) decreases;
2. \(K_{12\mathrm{p}}<K_{\mathrm{p}}<K_{23\mathrm{p}}\)—in this case the first section has a negative drop, while the second has a positive one; transmission lines with relatively short equal sections have a greater probability of such a combination, and, in the case of unequal sections, when \(l_{12}<l_{23}\);
3. And, finally, when \(K_{\mathrm{p}}>K_{12\mathrm{p}}, K_{23\mathrm{p}}\), both sections have a negative drop, increasing as the inequality strengthens. A negative drop may be more characteristic of transmission lines with relatively long equal sections and, in the case of unequal sections, when \(l_{12}>l_{23}\).

Depending on the magnitude of \(K_{\mathrm{p}}\), i.e., on the character of the optimal drops on the sections, the order of successive voltage limitation is determined.

With positive drops on both sections, at the upper limiting level the voltage \(U_1\) is fixed first, then \(U_2\), and last \(U_3\). The order of voltage limitation at the lower level under the same conditions is the reverse: \(U_3, U_2, U_1\). With negative drops on the sections, the voltage at the receiving end first reaches the upper limit; subsequently \(U_2\) and \(U_1\) reach this level. Obvi-

Table 1

Values of \(k_{12p}\) and \(k_{23p}\), corresponding to \(k_1 U_{12}\), \(k_1 U_{23}=1\) in the \(\gamma_{i\max}\) regime

the order of fixing the voltages at the ends of the sections at the lower level is the reverse: \(U_1, U_2, U_3\). With the same character of the drops in all sections, the order of limiting the voltages is in strict correspondence with the order in which the nodes of the long-distance transmission line follow, i.e., either from the beginning of the transmission line to its end, or from the end to the beginning, independently of the number of sections. This strict sequence is violated in the case when the first section has a negative drop and the second a positive one. The voltage \(U_2\) is the first to reach the upper limit; subsequently the voltages \(U_1\) and \(U_3\) rise to this level. At the lower level, \(U_1\) and \(U_3\) are fixed first, and \(U_2\) last.

Above, conditions have been established that ensure the maximum efficiency of the entire transmission line for arbitrary laws \(P_{21}\) and \(P_{23}\), assuming that their values at any instant are known and specified quantities. In this case, the maximum efficiency does not depend on the absolute values of the power flows of the sections, but is determined only by their ratio, by the constants of the sections, and by the loss coefficient of the compensating device.

The optimum ratio for which the maximum efficiency reaches its greatest value is determined by the equation

\[ \frac{d\eta_{\Sigma\max}}{dK_P}=0, \tag{18} \]

which has the solution

\[ K_{P\mathrm{opt}.}=\frac{g_{23\eta}}{g_{213}g_{21\eta}} \sqrt{2g_{213}^{2}-g_{21\eta}^{2}}. \tag{19} \]

Taking into account the equation relating the conductances of the sections,

\[ g_{21\eta}^{2}g_{23\eta}^{2}+g_{23\eta}^{2}g_{213}^{2} =2g_{213}^{2}g_{213}^{2}, \tag{20} \]

the optimum value \(K_P\) can be represented in another form,

\[ K_{P\mathrm{opt}.}= -\frac{g_{233}g_{23\eta}} {g_{213}\sqrt{2g_{233}^{2}-g_{23\eta}^{2}}}, \tag{21} \]

where

\[ g_{21\eta}=\sqrt{\frac{H_1}{M_1}-g_{21\mathrm{m}}^{2}},\qquad g_{23\eta}=\sqrt{\frac{H_1}{M_1}-g_{23\mathrm{m}}^{2}}. \tag{22} \]

Equation (6), taking into account relation (15), can be represented in the following forms:

\[ U_{2\eta}= \sqrt[4]{\frac{P_{21}^{2}\left(g_{233}^{2}+K_P^{2}g_{213}^{2}\right)} {2g_{213}^{2}g_{233}^{2}}} \quad \text{or} \quad U_{2\eta}= \sqrt[4]{\frac{P_{23}^{2}\left(g_{233}^{2}+K_P^{2}g_{213}^{2}\right)} {2g_{213}^{2}g_{233}^{2}K_P^{2}}}. \tag{23} \]

Substituting \(K_{P\mathrm{opt}}\) from (19) into the first form of the expression for \(U_{2\eta}\), and \(K_{P\mathrm{opt}}\) from (21) into the second form of the expression for \(U_{2\eta}\), according to (23) we obtain

\[ U_{2\eta}=\sqrt{\frac{P_{21}}{g_{21\eta}}}, \qquad U_{2\eta}=\sqrt{\frac{P_{23}}{g_{23\eta}}}. \tag{24} \]

From this we obtain the simplest and most transparent form for writing the optimum ratio of the powers of the sections:

\[ K_{P\mathrm{opt}}=\frac{g_{23\eta}}{g_{21\eta}}=\frac{P_{23}}{P_{21}}. \tag{25} \]

For a transmission line with several intermediate systems, the greatest value of the total maximum efficiency occurs under the condition

\[ \frac{P_{1\mathrm{K}}}{P_{2\mathrm{H}}} =\frac{g_{1\eta}}{g_{2\eta}},\quad \frac{P_{2\mathrm{K}}}{P_{3\mathrm{H}}} =\frac{g_{2\eta}}{g_{3\eta}},\ \ldots,\ \frac{P_{(n-1)\mathrm{K}}}{P_{n\mathrm{H}}} =\frac{g_{(n-1)\eta}}{g_{n\eta}}. \tag{26} \]

The efficiency of the sections under these conditions also reaches its maximum values. From relations (26) it follows that the power flows of the sections must be proportional to their conductances, i.e., extended sections must be more heavily loaded than short ones.

Thus, the smallest losses in the presence of unequal sections occur with unbalanced intermediate systems, and, consequently, with equal sections—in the case of self-balancing intermediate systems.

It should be noted that, when conditions (26) are observed, irrespective of the ratio of the section lengths, in the mode of coordinated voltage regulation only positive drops always occur.

REFERENCES

1. V. I. Gotman, I. A. Bezler. Voltage and reactive-power regulation of compensated long-distance power transmission lines with intermediate systems in their self-balancing mode. “Izv. TPI,” present collection.

2. V. A. Venikov. Transient electromechanical processes in electrical systems. “Vysshaya shkola,” Moscow, 1970.