PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE NAMED AFTER S. M. KIROV,
ORDER OF THE OCTOBER REVOLUTION AND ORDER OF THE RED BANNER OF LABOR

Volume 295 1976

LIMITING OPERATING CONDITIONS OF COMPENSATED LONG-DISTANCE AC POWER TRANSMISSION WITH INTERMEDIATE SYSTEMS

R. I. BORISOV, I. A. BEZLER, V. I. GOTMAN

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

This paper presents the results of studies on the influence of the principal parameters of intermediate systems (IS) and shunt-compensation installations (ShCI), their connection points and placement along the route of a long-distance power transmission line, the type of ShCI, the initial operating condition, and a number of other factors on the limiting operating conditions of compensated long-distance power transmission lines (CLDPT), as well as on the limiting values of the relative angles between stations corresponding to the limit of aperiodic stability.

In determining the limiting values of the relative rotor angles of the synchronous machines of the IS and ShCI, they must be represented in the form of generalized constant four-terminal networks with equivalent values of the e.m.f. of the power sources (Fig. 1). The parameters of the generalized four-terminal networks are determined by the chain-connection method. In the system considered, the loads and reactive-power sources are represented by admittances. The IS generators and the synchronous compensators of the ShCI are replaced by constant values of e.m.f. applied behind the corresponding resistance [1]. When comparing results for a long-distance power transmission line with an IS and with different types of ShCI, all other conditions being equal, the method used to account for the synchronous machines of the receiving

Fig. 1. Schematic diagram of a long-distance power transmission line with an intermediate system and series- and shunt-compensation installations.

No. 2 and the transmitting No. 1 systems is not of primary importance and does not affect the qualitative picture of the results obtained.

After replacing the controlled shunt reactors, intermediate systems, and controlled longitudinal-compensation devices by quadripoles (Fig. 1), the electromagnetic-power equation for each equivalent station is functionally dependent on the relative angles between their rotors, with the emf’s of the synchronous machines constant, and takes the form

\[ P_i=P_{ii}+\sum_{\substack{j=1\\ j\ne i}}^{n} P_{ij}\cdot \sin(\delta_{ij}-\alpha_{ij}), \tag{1} \]

where

\[ P_{ii}=\frac{E_i^2\cdot F_i}{B_i\cdot M_0}\sin\alpha_{ii},\qquad \alpha_{ii}=90^0+\varphi_i-\psi_{B_i}-\psi_0; \]

\[ P_{12}=\frac{B_3\cdot B_4}{H_1}E_1E_2;\qquad P_{14}=\frac{B_2\cdot B_3}{H_1}\cdot\frac{B_4\cdot N_1}{H_0}E_1E_4; \]

\[ P_{13}=\frac{B_2\cdot B_4}{H_1}E_1E_3;\qquad P_{23}=\frac{H_0}{H_1}E_2E_3; \]

\[ P_{24}=\frac{B_3\cdot N_1}{H_1}E_2E_4;\qquad P_{34}=\frac{B_2N_1}{H_1}E_3E_4; \]

\[ \delta_{ij}=\delta_i-\delta_j; \]

\[ \dot F_1=F_1\angle\varphi_1=\dot D_1\dot M_0-\dot B_4\dot N_0;\qquad \dot F_2=F_2\angle\varphi_2=\dot A_2\dot M_0-\dot B_3\dot H_0; \]

\[ \dot F_3=F_3\angle\varphi_3=\dot A_3\dot M_0-\dot B_2\dot H_0;\qquad \dot F_4=F_4\angle\varphi_4=\dot A_4\dot M_0-\dot B_1\dot N_0; \]

\[ \dot M_0=M_0\angle\psi_0=(\dot B_4\dot A_1+\dot B_1)\dot N_0+\dot B_1\cdot\dot B_4\cdot\dot B_{23}; \]

\[ \dot H_0=H_0\angle\eta_0=\dot B_k(\dot B_4\dot A_1+\dot B_1)+\dot B_1\dot B_4; \]

\[ \dot H_1=H_1\angle\eta_1=\dot B_2\dot B_3(\dot B_4\dot A_1+\dot B_1)+\dot B_{23}\dot H_0; \]

\[ \dot N_0=\dot B_2\dot B_3+\dot B_k\dot B_{23}; \]

\[ \dot N_1=N_1\angle\varepsilon_1=\dot B_4\dot A_1+\dot B_1-\dot B_4; \]

\[ \alpha_{12}=90^0+\psi_{B3}+\psi_{B4}-\eta_1;\qquad \alpha_{23}=90^0+\eta_0-\eta_1; \]

\[ \alpha_{13}=90^0+\psi_{B2}+\psi_{B4}-\eta_1;\qquad \alpha_{24}=90^0+\psi_{B3}+\varepsilon_1-\eta_1; \]

\[ \alpha_{14}=90^0+\psi_{B2}+\psi_{B3}+\psi_{B4}+\varepsilon_1-\eta_0-\eta_1;\qquad \alpha_{34}=90^0+\psi_{B2}+\varepsilon_1-\eta_1. \]

In equation (1), the self and mutual conductances are expressed through the generalized constants of the quadripoles of the corresponding parts of the system elements.

Of the six mutual angles entering equations of the form (1), only three are independent variables; they are related by the relation

\[ \delta_{12}+\delta_{23}+\delta_{34}+\delta_{41}=0. \tag{2} \]

The limiting operating condition of the system is found by successive loading of the initial steady normal condition by redistributing active powers among the generator stations. Then the maximum value of active power that can be transmitted from the remote station No. 1 to the receiving No. 2, the intermediate No. 3 system, and the synchronous compensator performing the functions of a controlled longitudinal-compensation device,

on the basis of equation (1) will occur at \(\delta_{12}=90^\circ+\alpha_{12}\); \(\delta_{13}=90^\circ+\alpha_{13}\); \(\delta_{14}=90^\circ+\alpha_{14}\) and will be

\[ P_1=P_{11}+P_{12}+P_{13}+P_{14}. \]

To determine \(P_{1\mathrm{lim}}\) in the general case, it is necessary to establish the interrelation between \(P_1\) and the three independent mutual angles, i.e., from the limiting values of the angles the corresponding limiting active-power flows can be determined. Then, to obtain the required dependence, it is necessary to use the practical criterion for the limiting operating condition of the following form:

\[ \frac{dP_1}{d\delta_{12}} = \frac{\partial P_1}{\partial P_{12}} + \frac{\partial P_1}{\partial P_{13}} \frac{\partial \delta_{13}}{\partial \delta_{12}} + \frac{\partial P_1}{\partial \delta_{14}} \frac{\partial \delta_{14}}{\partial \delta_{12}} =0, \tag{3} \]

The expanded equation (3), in form and essence, is analogous to the expression obtained in [2], but with a larger number of terms.

Fig. 2. Dependence of the limiting power of the long-distance AC power transmission line on the position of the intermediate system and the type of compensating devices: 1 — a synchronous condenser is connected to the input of the compensating device, and the intermediate system is connected to the output; 2 — the intermediate system is replaced by static characteristics; 3 — a shunt reactor is connected to the input of the compensating device; 4 — the intermediate system is connected to the input of the compensating device, and a synchronous condenser to the output.

Equation (3), taking (2) into account, makes it possible to obtain the dependence between the mutual angles of the synchronous machines of the system at which transmitting system No. 1 delivers maximum power.

In Fig. 2, according to equations (1)–(3), the region of limiting values of the transmitted active power at transmitting station No. 1 is plotted.

depending on the operating mode of the IS and the type of CShC; in this case the CShC is located in the middle of a long-distance transmission line 1000 km long, and the degree of compensation of the inductive resistance of the line is taken as \(K_c=50\%\).

In investigating the operating modes of an LDPTL and reducing computation time, it is possible to use the static characteristics of the IS and CShC as functions of the voltage at the point of their connection [3]. The regulating properties of the static characteristics are characterized by slope coefficients, and their magnitude depends on many factors: the active and reactive powers of the loads, the coupling impedances between the IS and CShC and the node of connection to the LDPTL, as well as on the adopted control laws.

In order to determine the influence of the principal parameters of the static characteristics of the IS and CShC on the limiting operating modes of a CLDPTL, an equation has been obtained for the transmitted power \(P_1\) at the end of the first section of the CLDPTL (Fig. 1) as a function of the voltage of the node of connection of the IS and CShC and of their static characteristics:

\[ P_1= \frac{ a_1\cdot a_2+d_1\cdot d_2^{\,2} +\sqrt{(a_1\cdot a_2+d_1\cdot d_2^{\,2})^2+(a_2^{\,2}+d_2^{\,2})(c_1d_2^{\,2}-a_1^{\,2})} }{ a_2^{\,2}+d_2^{\,2} }, \tag{4} \]

where

\[ d_1= \frac{\operatorname{Re}(\dot D_2\cdot \hat B_2)}{B_2^2}U_6^2 - \frac{\operatorname{Re}(\dot A_1\hat B_1)}{B_1^2}U_5^2 +P_{\mathrm{ps}}; \]

\[ d_2= \frac{\operatorname{Im}(\dot D_2\cdot \hat B_2)}{B_2^2}U_6^2 - \frac{\operatorname{Im}(\dot A_k\hat B_k)}{B_k^2}U_6^2 +Q_{\mathrm{ps}}; \]

\[ a_2= \frac{\operatorname{Re}(\dot D_2\cdot \hat B_2)}{B_2^2}U_6^2 +P_{\mathrm{ps}}; \]

\[ \begin{aligned} a_1=\frac{1}{2}\Bigg[ &\frac{U_2^2U_6^2}{B_2^2} -\frac{U_5^2U_6^2}{B_k^2} +\left(\frac{\operatorname{Im}(\dot A_1\hat B_1)}{B_1^2}U_5^2+Q_{\mathrm{ps}}\right)^2 \\ &+\left(\frac{\operatorname{Im}(\dot D_2\hat B_2)}{B_2^2}U_6^2+Q_{\mathrm{ps}}\right)^2 +\left(\frac{\operatorname{Im}(\dot D_2\hat B_2)}{B_2^2}U_6^2+Q_p\right)^2 \\ &-\left(\frac{\operatorname{Im}(\dot A_k\hat B_k)}{B_k^2}U_6^2+Q_p\right)^2 -\left(\frac{\operatorname{Im}(\dot A_1\hat B_1)}{B_1^2}U_5^2 -\frac{\operatorname{Im}(\dot D_2\cdot \hat B_2)}{B_2^2}U_6^2\right)^2 \\ &-2\frac{\operatorname{Im}(\dot A_1\hat B_1)}{B_1^2}\cdot \frac{\operatorname{Im}(\dot A_k\hat B_k)}{B_k^2}U_5^2U_6^2 +2Q_{\mathrm{ps}}Q_p-Q_{\mathrm{ps}}^2-2d_2^2+a_2^2 \Bigg]; \end{aligned} \]

\[ \begin{aligned} c_1={}& \frac{U_1^2U_5^2}{B_1^2} -\frac{U_5^2U_6^2}{B_k^2} +\frac{U_2^2U_6^2}{B_2^2} -\left(\frac{\operatorname{Im}(\dot A_1\hat B_1)}{B_1^2}U_5^2+Q_{\mathrm{ps}}\right)^2 \\ &-\left(\frac{\operatorname{Im}(\dot D_2\hat B_2)}{B_2^2}U_6^2+Q_{\mathrm{ps}}\right)^2 +\left(\frac{\operatorname{Im}(\dot A_1\hat B_1)}{B_1^2}U_5^2 +\frac{\operatorname{Im}(\dot A_k\hat B_k)}{B_k^2}U_6^2\right)^2-\; \end{aligned} \]

\[ -2\frac{\operatorname{Im}\left(A_{1}\hat B_{1}\right)}{B_{1}^{2}} \frac{\operatorname{Im}\left(D_{2}\hat B_{2}\right)}{B_{2}^{2}}U_{5}^{2}U_{6}^{2} -\left(\frac{\operatorname{Re}\left(A_{1}\hat B_{1}\right)}{B_{1}^{2}}\right)^{2}U_{5}^{4} +2Q_{p}d_{2}+Q_{\text{pc}}^{2}-d_{2}^{2}-a_{2}^{2}. \]

When the voltages at the connection nodes of the IS and the controlled series-compensation device are varied, expression (4) makes it possible to determine \(P_{1}\) directly; the limiting condition of transmitting system No. 1 corresponds to the equality \(\dfrac{dP_{1}}{dU_{6}}=0\). If active losses are neglected, expression (4) can be simplified.

It follows from the analysis of Fig. 2 that the limiting value of the transmitted power from the transmitting system depends to a considerable degree on the type of controlled series-compensation device, the point of connection of the IS and the controlled series-compensation device, and also on the static characteristics of the IS and the controlled series-compensation device.

Fig. 3. Load-angle characteristics of a CLDPT with an IS: 1 — the IS is connected to the output of the controlled series-compensation device, and a synchronous compensator is connected to the input; 2 — the IS is connected to the input of the controlled series-compensation device, and a synchronous compensator is connected to the output; 3 — the IS is connected to the output of the controlled series-compensation device, and a shunting reactor (static compensator) is connected to the input.

For the CLDPT under consideration, with an IS and a controlled series-compensation device (synchronous compensator), the characteristic equation has the form [3]

\[ p^{6}+A_{2}p^{4}+A_{4}p^{2}+A_{6}=0, \tag{6} \]

where

\[ A_{2}=a_{21}^{12}+a_{31}^{13}+a_{41}^{14}; \]

\[ A_{4}=a_{21}^{12}\cdot a_{31}^{13}-a_{21}^{13}\cdot a_{31}^{13} +a_{21}^{12}\cdot a_{41}^{14}-a_{21}^{14}\cdot a_{41}^{12} +a_{31}^{13}\cdot a_{41}^{14}-a_{31}^{14}\cdot a_{41}^{13}; \]

\[ A_{6}=a_{21}^{12}\cdot a_{31}^{13}\cdot a_{41}^{14} -a_{21}^{12}\cdot a_{31}^{14}\cdot a_{41}^{13} +a_{31}^{13}\cdot a_{41}^{14}\cdot a_{21}^{12} -a_{31}^{12}\cdot a_{41}^{14}\cdot a_{21}^{13} +a_{41}^{12}\cdot a_{21}^{13}\cdot a_{31}^{14} -a_{41}^{12}\cdot a_{21}^{14}\cdot a_{31}^{13}; \]

\[ a_{21}^{12}=\frac{1}{T_{I1}}\frac{\partial P_{1}}{\partial \delta_{12}} -\frac{1}{T_{I2}}\frac{\partial P_{2}}{\partial \delta_{12}}; \qquad a_{21}^{13}=\frac{1}{T_{I1}}\frac{\partial P_{1}}{\partial \delta_{13}} -\frac{1}{T_{I2}}\frac{\partial P_{2}}{\partial \delta_{13}}; \]

\[ a_{31}^{13}=\frac{1}{T_{l1}}\frac{\partial P_1}{\partial \delta_{13}}-\frac{1}{T_{l3}}\frac{\partial P_3}{\partial \delta_{13}};\qquad a_{21}^{14}=\frac{1}{T_{l1}}\frac{\partial P_1}{\partial \delta_{14}}-\frac{1}{T_{l2}}\frac{\partial P_2}{\partial \delta_{14}}; \]

\[ a_{41}^{14}=\frac{1}{T_{l1}}\frac{\partial P_1}{\partial \delta_{14}}-\frac{1}{T_{l4}}\frac{\partial P_4}{\partial \delta_{14}};\qquad a_{31}^{12}=\frac{1}{T_{l1}}\frac{\partial P_1}{\partial \delta_{12}}-\frac{1}{T_{l3}}\frac{\partial P_3}{\partial \delta_{12}}; \]

\[ a_{41}^{12}=\frac{1}{T_{l1}}\frac{\partial P_1}{\partial \delta_{12}}-\frac{1}{T_{l4}}\frac{\partial P_4}{\partial \delta_{12}};\qquad a_{31}^{14}=\frac{1}{T_{l1}}\frac{\partial P_1}{\partial \delta_{14}}-\frac{1}{T_{l3}}\frac{\partial P_3}{\partial \delta_{14}}; \]

\[ a_{41}^{13}=\frac{1}{T_{l1}}\frac{\partial P_1}{\partial \delta_{13}}-\frac{1}{T_{l4}}\frac{\partial P_4}{\partial \delta_{13}}, \]

where \(T_{li}\) are the inertia constants of synchronous machines.

The determination of the limiting values of the relative angles under the conditions of static stability of the system reduces to an analysis of the roots of the characteristic equation (5), in which, according to the assumptions adopted, only an aperiodic loss of stability is possible, and the boundary condition for aperiodic stability is the condition: \(A_6=0\). Since receiving system No. 2 performs the functions of infinite-capacity buses, the static frequency characteristics of the system elements are not taken into account, and \(T_{l2} >>> T_{l1}, T_{l3}, T_{l4}\). Then, when expanding the constant term of the characteristic equation \((A_6)\), the common term will be the expression \(\dfrac{1}{T_{l1}\cdot T_{l3}\cdot T_{l4}}\), which has little effect on the limit of transmitted power; consequently, the power limit will coincide with the limit of aperiodic stability of the system.

Figure 3 presents the load-angle characteristics of transmitting system No. 1 as a function of the location of the PS and the type of UPpK.

Analysis of the characteristics shows that the limit of transmitted power depends to a significant degree on the point at which the PS is connected and on the type of UPpK.

REFERENCES

1. V. A. Venikov et al. Investigation by means of a digital computer of the static stability of complex automatically controlled systems. Elektrichestvo, 1967, No. 11.

2. R. I. Borisov, V. I. Gotman. Limiting operating conditions of long-distance AC power transmission with intermediate systems under static-stability conditions. Izvestiya VUZov SSSR, Energetika, 1971, No. 8.

3. P. S. Zhdanov. Stability of Electric Power Systems. Moscow, Gosenergoizdat, 1948.