CONSTRAINTS ON THE OPERATING PARAMETERS OF LONG-DISTANCE POWER TRANSMISSION LINES WITH INTERMEDIATE SYSTEMS BASED ON STATIC STABILITY CONDITIONS

R. I. Borisov

(Presented by the scientific seminar of the departments of electric power stations and electric systems and networks)

The network powers \((P_{il}\) and \(Q_j)\) of elements of circuits and of their aggregates are determined through the operating parameters \((U_i, U_j, \delta_i, \delta_j)\), which in turn characterize the states of systems with respect to static and dynamic stability. Establishing stability criteria through operating parameters by means of network-power equations has its positive aspects, since it makes it possible to take into account the influence of the characteristics of each load or of generalized static characteristics of power-system buses, and to do this in calculations in different ways for different buses. Let us suppose that all variable operating parameters \((U)\) and \((\delta)\) are expressed as unknown functions of time under small and short-term disturbances of the operating mode [1]. Then a judgment about static stability can be made from the character of the change of each variable in time. If the free oscillations of each variable in time decay, then the corresponding established operating mode will be stable; if they do not decay, it will be unstable. Since problems of setting the operating mode with allowance for static-stability requirements are being solved, the operating parameters together with the parameters of devices in another group of problems are the quantities sought. At the same time, it is desirable to formulate the constraints on static stability in terms of the same sought parameters that enter into the objective functional, in order to reduce the number of additional variables in the Lagrange formula.

For the first of the generating stations (buses 1, 6, 8) of the circuit in Fig. 1, the equations of free oscillations of the rotor are written as follows:

\[ I_1 \frac{d^2 \Delta \delta_1}{dt^2} = -\Delta P_1 = -\Delta P_1(\Delta E_1,\Delta U_1,\Delta \delta_{01},E_1,U_1,\delta_{01}), \tag{1} \]

where \(\delta_{01}\) is the angle between the e.m.f. vector and the voltage on the buses of the given station.

If the delays in the action of the e.m.f. control systems are not taken into account, then the increment of the e.m.f. will be related to the increments of the external parameters of the circuit (voltages and angles) and their derivatives in the form of the following relation:

\[ \Delta E = \kappa_{U_1}\Delta U_1 + \kappa_{U_2}\Delta U_2 + \ldots + \kappa_{\delta_{01}}\Delta \delta_{01} + \kappa_{\delta_1}\Delta \delta_1 + \kappa_{\delta_2}\Delta \delta_2 + \ldots \]

\[ + \kappa'_{U_1}p\Delta U_1 + \kappa'_{U_2}p\Delta U_2 + \ldots + \kappa'_{\delta_{01}}p\Delta \delta_{01} + \kappa'_{\delta_1}p\Delta \delta_1 + \kappa'_{\delta_2}p\Delta \delta_2 + \ldots \]

\[ + \kappa''_{U_1} p^2 \Delta U_1 + \kappa''_{U_2} p^2 \Delta U_2 + \ldots + \kappa''_{\delta_{01}} p^2 \Delta \delta_{01} + \kappa''_{\delta_1} p^2 \Delta \delta_1 + \kappa''_{\delta_2} p^2 \Delta \delta_2 = \]
\[ = \sum_i \left(\kappa_{U_i}+\kappa'_{U_i}p+\kappa''_{U_i}p^2\right)\Delta U_i + \sum_i \left(\kappa_{\delta_i}+\kappa'_{\delta_i}p+\kappa''_{\delta_i}p^2\right)\Delta \delta_i , \tag{2} \]

where \(\kappa\) are the gain coefficients of the various elements,
\(p\) is the differentiation symbol.

Fig. 1. Schematic diagram of a long-distance power transmission line with an intermediate system

Taking lag into account, the control equation is written as follows:

\[ \Delta E = \left( \frac{\kappa_{U_1}}{1+pT_{U_1}}\Delta U_1 + \frac{\kappa_{U_2}}{1+pT_{U_2}}\Delta U_2 + \ldots + \frac{\kappa_{\delta_{01}}}{1+pT_{\delta_{01}}}\Delta \delta_{01} + \ldots \right. \]
\[ \left. + \frac{\kappa'_{U_1}}{1+pT'_{U_1}}p\Delta U_1 + \ldots + \frac{\kappa'_{\delta_{01}}}{1+pT'_{\delta_{01}}}p\Delta\delta_{01} + \frac{\kappa'_{\delta_1}}{1+pT'_{\delta_1}}p\Delta\delta_1 + \right. \]
\[ \left. + \frac{\kappa''_{U_1}}{1+pT''_{U_1}}p^2\Delta U_1 + \ldots + \frac{\kappa''_{\delta_{01}}}{1+pT''_{\delta_{01}}}p^2\Delta\delta_{1} + \frac{\kappa''_{\delta_1}}{1+pT''_{\delta_1}}p^2\Delta\delta_1 + \ldots \right) \frac{1}{1+pT_{\mathrm{vv}}}\frac{1}{1+pT'_{\mathrm{v}}} \]
\[ = \frac{1}{1+pT_{\mathrm{vv}}}\frac{1}{1+pT_{\mathrm{v}}} \times \left( \sum_i \left( \frac{\kappa_{U_i}}{1+pT_{U_i}} + \frac{\kappa'_{U_i}p}{1+pT'_{U_i}} + \frac{\kappa''_{U_i}p^2}{1+pT''_{U_i}} \right)\Delta U \right. \]
\[ \left. + \sum_i \left( \frac{\kappa_{\delta_i}}{1+pT_{\delta_i}} + \frac{\kappa'_{\delta_i}p}{1+pT'_{\delta_i}} + \frac{\kappa''_{\delta_i}p^2}{1+pT''_{\delta_i}} \right) \right), \tag{3} \]

where \(T_{\mathrm{v}}\) and \(T_{\mathrm{vv}}, T_i\) are the time constants of the field winding, the exciter, and the corresponding element.

Thus, with proportional control by voltage and intrinsic angle without lag, the control equation is written in the following form:

\[ \Delta E = \left(-\kappa_{U_1}\Delta U_1+\kappa_{\delta_{01}}\Delta\delta_{01}\right) \frac{1}{1+pT_{\mathrm{v}}}. \tag{4} \]

In the first form of the calculations, the number of functional variables and constraints is reduced by using approximating polynomials, whereas in the second form the solution is organized over all variables.

If the first form of notation is followed, then:

\[ I_1 p^2 \Delta \delta_1 = - \left( \frac{\partial p}{\partial E_1}(E_1,U_1,\delta_{01})\Delta E_1 + \frac{\partial P_1}{\partial U_1}(E_1,U_1,\delta_{01})\Delta U_1 + \right. \]
\[ \left. + \frac{\partial P_1}{\partial \delta_{01}}(E_1,U_1,\delta_{01})\Delta\delta_{01} \right) = - \left( P_{1E_1}(E_1,U_1,\delta_{01})(-\kappa_{U_1}\Delta U_1+\kappa_{\delta_{01}}\Delta\delta_{01}) \times \right. \]
\[ \left. \times \frac{1}{1+pT_{\mathrm{v}}} + P_{1U_1}(E_1,U_1,\delta_{01})\Delta U_1 + P_{1\delta_1}(E_1,U_1,\delta_{01})\Delta\delta_{01} \right). \]

Substitute, instead of \(E_1\) and \(\delta_{01}\), the approximating polynomials of the delivered powers, which are expressed through the parameters of the external part of the circuit.

\[ \begin{aligned} I_1 p^2 \Delta \delta ={}&-\Bigl(\bar P_{1E_1}(v_{E_1}U_1+\pi_{E_1}P_{01}+q_{E_1}Q_{01};\,U_1;\, v_{\delta_{01}}U_1+\pi_{\delta_{01}}P_{01}+q_{\delta_{01}}Q_{01}) \\ &\times(-\kappa_{U_1}\Delta U_1+\kappa_{\delta_{01}}\Delta\delta_{01}) \frac{1}{1+pT_{\mathrm v}} +\bar P_{1U_1}(v_{E_1}U_1 \\ &+\pi_{E_1}P_{01}+q_{E_1}Q_{01};\,U_1;\, v_{\delta_{01}}U_1+\pi_{\delta_{01}}P_{01}+q_{\delta_{01}}Q_{01})\Delta U_1 \\ &+\bar P_{1\delta_{01}}(v_{E_1}U_1+\pi_{E_1}P_{01}+q_{E_1}Q_{01};\,U_1;\, v_{\delta_{01}}U_1+\pi_{\delta_{01}}P_{01}+q_{\delta_{01}}Q_{01})\Delta\delta_{01}\Bigr); \end{aligned} \]

since \(P_{01}=P_{12}^{*}\) and \(Q_{01}=-Q_{21}\), then

\[ \begin{aligned} &\bar P_{1E_1}\bigl(v_{E_1}U_1+\pi_{E_1}(P_{12}^{*}(U_1,U_2,\delta_1,\delta_2)) +q_{E_1}(-Q_{21}(U_2,U_1,\delta_2,\delta_1));\\ &\qquad U_1;\, v_{\delta_{01}}U_1+\pi_{\delta_{01}}(P_{12}^{*}(U_1,U_2,\delta_1,\delta_2)) +q_{\delta_{01}}(-Q_{21}(U_2,U_1,\delta_2,\delta_1))\bigr) \\ &\qquad\qquad =\bar P_{1E_1}(U_1,U_2,\delta_1,\delta_2); \end{aligned} \]

\[ \bar P_{1U_1}=\bar P_{1U_1}(U_1,U_2,\delta_1,\delta_2); \qquad \bar P_{1\delta_{01}}=\bar P_{1\delta_{01}}(U_1,U_2,\delta_1,\delta_2); \]

\[ I_1p^2\Delta\delta_{01} =-\bigl(\bar P_{1E_1}(U_1,U_2,\delta_1,\delta_2) (-\kappa_{U_1}\Delta U_1+\kappa_{\delta_{01}}\Delta\delta_{01}) \frac{1}{1+pT_{\mathrm v}} \]

\[ +\bar P_{1U_1}(U_1,U_2,\delta_1,\delta_2)\Delta U_1 +\bar P_{1\delta_{01}}(U_1,U_2,\delta_1,\delta_2)\Delta\delta_{01}\bigr); \]

\[ \begin{aligned} I_1p^2\Delta\delta_{01}(1+pT_{\mathrm v}) ={}&-\bigl(\bar P_{1E_1}(U_1,U_2,\delta_1,\delta_2) (-\kappa_{U_1}\Delta U_1+\kappa_{\delta_{01}}\Delta\delta_{01})\\ &+\bar P_{1U_1}(U_1,U_2,\delta_1,\delta_2)\Delta U_1 +\bar P_{1\delta_{01}}(U_1,U_2,\delta_1,\delta_2)\Delta\delta_{01}\bigr); \end{aligned} \]

\[ \begin{aligned} (I_1T_{\mathrm v}p^3+I_1p^2)\Delta\delta_{01} ={}&-\bigl(-\kappa_{U_1}\bar P_{1E_1}(U_1,U_2,\delta_1,\delta_2)\Delta U_1\\ &+\kappa_{\delta_{01}}\bar P_{1E_1}(U_1,U_2,\delta_1,\delta_2)\Delta\delta_{01} +\bar P_{1U_1}(U_1,U_2,\delta_1,\delta_2)\Delta U_1\\ &+T_{\mathrm v}\bar P_{1U_1}(U_1,U_2,\delta_1,\delta_2)p\Delta U_1 +\bar P_{1\delta_{01}}(U_1,U_2,\delta_1,\delta_2)\Delta\delta_{01}\\ &+T_{\mathrm v}\bar P_{1\delta_{01}}(U_1,U_2,\delta_1,\delta_2)p\Delta\delta_{01}\bigr); \end{aligned} \]

\[ \begin{aligned} &\bigl(I_1T_{\mathrm v}p^3+I_1p^2 +T_{\mathrm v}\bar P_{1\delta_{01}}(U_1,U_2,\delta_1,\delta_2)p +\bar P_{1\delta_{01}}(U_1,U_2,\delta_1,\delta_2)\\ &\qquad +\kappa_{\delta_{01}}\bar P_{1E_1}(U_1,U_2,\delta_1,\delta_2)\bigr)\Delta\delta_{01} +\bigl(T_{\mathrm v}\bar P_{1U_1}(U_1,U_2,\delta_1,\delta_2)p\\ &\qquad +\bar P_{1U_1}(U_1,U_2,\delta_1,\delta_2) -\kappa_{U_1}\bar P_{1E_1}(U_1,U_2,\delta_1,\delta_2)\bigr)\Delta U_1=0. \end{aligned} \]

Denote:

\[ \begin{gathered} IT_{\mathrm v}=\alpha_{11};\qquad I=\alpha_{12};\qquad T_{\mathrm v}\bar P_{1\delta_{01}}(U_1,U_2,\delta_1,\delta_2)=\alpha_{13};\\ \bar P_{1\delta_{01}}(U_1,U_2,\delta_1,\delta_2) +\kappa_{\delta_{01}}\bar P_{1E_1}(U_1,U_2,\delta_1,\delta_2)=\alpha_{14};\\ T_{\mathrm v}\bar P_{1U_1}(U_1,U_2,\delta_1,\delta_2)=\alpha_{15};\\ \bar P_{1U_1}(U_1,U_2,\delta_1,\delta_2) -\kappa_{U_1}\bar P_{1E_1}(U_1,U_2,\delta_1,\delta_2)=\alpha_{16}. \end{gathered} \]

The operating parameters \(U_1, U_2, \delta_1, \delta_2\) are unknowns to be determined:

\[ (\alpha_{11}p^3+\alpha_{12}p^2+\alpha_{13}p+\alpha_{14})\Delta\delta_{01} +(\alpha_{15}p+\alpha_{16})\Delta U_1=0, \qquad \alpha_1=\alpha_1(U_1,U_2,\delta_1,\delta_2). \tag{5-1} \]

For generating nodes 6 and 8 one can write analogously

\[ (\alpha_{21}p^3+\alpha_{22}p^2+\alpha_{23}p+\alpha_{24})\Delta\delta_{06} +(\alpha_{25}p+\alpha_{26})\Delta U_6=0, \qquad \alpha_2=\alpha_2(U_6,U_5,\delta_6,\delta_5), \tag{5-2} \]

\[ (a_{31}p^3+a_{32}p^2+a_{33}p+a_{34})\Delta\dot{\delta}_{0s}+ (a_{35}p+a_{36})\Delta U_s=0, \tag{5-3} \]

\[ a_3=a_3(U_s,U_7,\dot{\delta}_s,\dot{\delta}_7). \]

The inertia constants and time constants in relative units are written as follows:

\[ I_{*i}=I_{\mathrm{sec}i}\frac{S_{Hi}}{S_b}\frac{1}{\omega_0}; \tag{6} \]

\[ T_{\mathrm{B}i}=T_{pi}\frac{1}{\omega_0}. \tag{7} \]

For the nodal points of the circuit \((1,2,3,4,5,6,7,8)\) in Fig. 1, the equations of free oscillations in the variable operating parameters can be written from the known active- and reactive-power balance conditions

\[ \Delta P_{01}(\Delta E_1;\Delta U_1;\Delta\dot{\delta}_{01};E_1;U_1;\dot{\delta}_{01}) =\Delta P^*_{12}(\Delta U_1;\Delta U_2;\Delta\dot{\delta}_1;\Delta\dot{\delta}_2;U_1;U_2;\dot{\delta}_1;\dot{\delta}_2); \]

\[ \begin{aligned} &\overline{P}_{1F_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) (-\kappa_{U_1}\Delta U_1+\kappa_{\dot{\delta}_{01}}\Delta\dot{\delta}_{01}) \frac{1}{1+pT_{\mathrm{B}}} \\ &\quad+\overline{P}_{1U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta U_1 +\overline{P}_{1\dot{\delta}_0}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta\dot{\delta}_{01} \\ &= \overline{P}^{*}_{12U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta U_1 +\overline{P}^{*}_{12U_2}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta U_2 \\ &\quad+\overline{P}^{*}_{12\dot{\delta}_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta\dot{\delta}_1 +\overline{P}^{*}_{12\dot{\delta}_2}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta\dot{\delta}_2; \end{aligned} \]

\[ \begin{aligned} &\overline{P}_{1E_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) (-\kappa_{U_1}\Delta U_1+\kappa_{\dot{\delta}_{01}}\Delta\dot{\delta}_{01}) +\overline{P}_{1U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)(1+ \\ &\quad+pT_{\mathrm{B}})\Delta U_1 +\overline{P}_{1\dot{\delta}_{01}}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)(1+pT_{\mathrm{B}})\Delta\dot{\delta}_{01} \\ &= \bigl(\overline{P}^{*}_{12U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta U_1 +\overline{P}^{*}_{12U_2}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta U_2 \\ &\quad+\overline{P}^{*}_{12\dot{\delta}_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta\dot{\delta}_1 +\overline{P}^{*}_{12\dot{\delta}_2}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta\dot{\delta}_2\bigr)(1+pT_{\mathrm{B}}); \end{aligned} \]

\[ \begin{aligned} &(1+pT_{\mathrm{B}}) \bigl(\overline{P}_{1U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) -\overline{P}^{*}_{12U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) \\ &\quad-\kappa_{U_1}\overline{P}_{1E_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\bigr)\Delta U_1 +(1+pT_{\mathrm{B}}) \bigl(\overline{P}_{1\dot{\delta}_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) \\ &\quad+\kappa_{\dot{\delta}_{01}}\overline{P}_{1E_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\bigr)\Delta\dot{\delta}_{01} -(1+pT_{\mathrm{B}})\overline{P}^{*}_{12U_2}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta U_2 \\ &\quad-(1+pT_{\mathrm{B}})\overline{P}^{*}_{12\dot{\delta}_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta\dot{\delta}_1 -(1+pT_{\mathrm{B}})\overline{P}^{*}_{12\dot{\delta}_2}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\Delta\dot{\delta}_2=0; \end{aligned} \]

\[ \begin{aligned} &pT_{\mathrm{B}}\bigl(\overline{P}_{1U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) -\overline{P}^{*}_{12U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) +\overline{P}_{1U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) \\ &\quad-\overline{P}^{*}_{12U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) -\kappa_{U_1}\overline{P}_{1E_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\bigr)\Delta U_1 \\ &\quad+ \bigl(pT_{\mathrm{B}}\overline{P}_{1\dot{\delta}_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) +\overline{P}_{1\dot{\delta}_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) \\ &\quad+\kappa_{\dot{\delta}_{01}}\overline{P}_{1E_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\bigr)\Delta\dot{\delta}_{01} -\bigl(pT_{\mathrm{B}}\overline{P}^{*}_{12U_2}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) \\ &\quad+\overline{P}^{*}_{12U_2}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\bigr)\Delta U_2 -\bigl(pT_{\mathrm{B}}\overline{P}^{*}_{12\dot{\delta}_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) \\ &\quad+\overline{P}^{*}_{12\dot{\delta}_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\bigr)\Delta\dot{\delta}_1 -\bigl(pT_{\mathrm{B}}\overline{P}^{*}_{12\dot{\delta}_2}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) \\ &\quad+\overline{P}^{*}_{12\dot{\delta}_2}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)\bigr)\Delta\dot{\delta}_2=0. \end{aligned} \]

Denote:

\[ \overline{P}_{1U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) -\overline{P}^{*}_{12U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)=\beta_{11}; \]

\[ \overline{P}_{1U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) -\overline{P}^{*}_{12U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) -\kappa_{U_1}\overline{P}_{1F_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)=\beta_{12}; \]

\[ T_{\mathrm{B}}\overline{P}^{*}_{12U_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)=\beta_{15}; \qquad \overline{P}^{*}_{12U_2}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)=\beta_{16}; \]

\[ T_{\mathrm{B}}\overline{P}_{1\dot{\delta}_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)=\beta_{13}; \qquad P_{1\dot{\delta}_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2) +\kappa_{\dot{\delta}_{01}}\overline{P}_{1E_1}(U_1,U_2,\dot{\delta}_1,\dot{\delta}_2)=\beta_{14}; \]

\[ \begin{gathered} T_{\mathrm{в}}P_{12q_1}^{*}(U_1,U_2,\delta_1,\delta_2)=\beta_{17};\quad P_{12q_1}^{*}(U_1,U_2,\delta_1,\delta_2)=\beta_{18};\\ T_{\mathrm{в}}\dot P_{12q_2}^{*}(U_1,U_2,\delta_1,\delta_2)=\beta_{19};\quad \overline{P}_{12q_2}^{*}(U_1,U_2,\delta_1,\delta_2)=\beta_{1-10};\\ (p\beta_{11}+\beta_{12})\Delta U_1+(p\beta_{13}+\beta_{14})\Delta\delta_{01} -(p\beta_{15}+\beta_{16})\Delta U_2-(p\beta_{17}+\beta_{18})\Delta\delta_1\\ -(p\beta_{19}+\beta_{1-10})\Delta\delta_2=0; \tag{5-4}\\ \beta_1=\beta_1(U_1,U_2,\delta_1,\delta_2). \end{gathered} \]

Analogously, equations may be written for the 6th and 8th nodes.

\[ \begin{gathered} (p\beta_{21}+\beta_{22})\Delta U_6+(p\beta_{23}+\beta_{24})\Delta\delta_{06} -(p\beta_{25}+\beta_{26})\Delta U_5-(p\beta_{27}+\beta_{28})\Delta\delta_6\\ -(p\beta_{29}+\beta_{2-10})=0; \tag{5-5}\\ \beta_2=\beta_2(U_6,U_5,\delta_6,\delta_5); \end{gathered} \]

\[ \begin{gathered} (p\beta_{31}+\beta_{32})\Delta U_8+(p\beta_{33}+\beta_{34})\Delta\delta_{08} -(p\beta_{35}+\beta_{36})\Delta U_7-(p\beta_{37}+\beta_{38})\Delta\delta_8\\ -(p\beta_{39}+\beta_{3-10})=0; \tag{5-6}\\ \beta_3=\beta_3(U_8,U_7,\delta_8,\delta_7). \end{gathered} \]

Analogously to (5-4; 5-5; and 5-6), equations from the reactive-power balances may be written for the same nodes:

\[ \begin{gathered} (p\gamma_{11}+\gamma_{12})\Delta U_1+(p\gamma_{13}+\gamma_{14})\Delta\delta_{01} -(p\gamma_{15}+\gamma_{16})\Delta U_2-(p\gamma_{17}+\gamma_{18})\Delta\delta_1\\ -(p\gamma_{19}+\gamma_{1-10})\Delta\delta_2=0; \tag{5-7}\\ \gamma_1=\gamma_1(U_1,U_2,\delta_1,\delta_2); \end{gathered} \]

\[ \begin{gathered} p\gamma_{21}+\gamma_{22})\Delta U_6+(p\gamma_{23}+\gamma_{24})\Delta\delta_{06} -(p\gamma_{25}+\gamma_{26})\Delta U_5-(p\gamma_{27}+\gamma_{28})\Delta\delta_6\\ -(p\gamma_{29}+\gamma_{2-10})\Delta\delta_5=0; \tag{5-8}\\ \gamma_2=\gamma_2(U_6,U_5,\delta_6,\delta_5); \end{gathered} \]

\[ \begin{gathered} p\gamma_{31}+\gamma_{32})\Delta U_8+(p\gamma_{33}+\gamma_{34})\Delta\delta_{08} -(p\gamma_{35}+\gamma_{36})\Delta U_7-(p\gamma_{37}+\gamma_{38})\Delta\delta_8\\ -(p\gamma_{39}+\gamma_{3-10})\Delta\delta_7=0; \tag{5-9}\\ \gamma_3=\gamma_3(U_8,U_7,\delta_8,\delta_7). \end{gathered} \]

The equations of the 2nd, 3rd, 4th, 5th, and 7th nodes will have a different structure. Let us first write the active-power equations:

\[ \begin{gathered} P_{12}(U_1,U_2,\delta_1,\delta_2)=P_2(U_2)+P_{23}^{*}(U_2,U_3,\delta_2,\delta_3);\\ P_{\mathrm{nb}10}(U_1,U_2,U_3,\delta_1,\delta_2,\delta_3)=0; \tag{5-10} \end{gathered} \]

\[ P_{23}(U_2,U_3,\delta_2,\delta_3)=P_{34}^{*}(U_3,U_4,\delta_3,\delta_4);\quad P_{\mathrm{nb}11}(U_2,U_3,U_4,\delta_2,\delta_3,\delta_4)=0; \tag{5-11} \]

\[ P_{73}(U_7,U_3,\delta_7,\delta_3)=0;\quad P_{\mathrm{ogr}_{12}}(U_7,U_3,\delta_7,\delta_3)=0; \tag{5-12} \]

\[ \begin{gathered} P_{34}(U_3,U_4,\delta_3,\delta_4)=P_{45}^{*}(U_4,U_5,\delta_4,\delta_5);\\ P_{\mathrm{nb}13}(U_3,U_4,U_5,\delta_3,\delta_4,\delta_5)=0; \end{gathered} \tag{5-13} \]

\[ P_{45}(U_4,U_5,\delta_4,\delta_5)=P_{45};\quad P_{\mathrm{ogr}14}(U_4,U_5,\delta_4,\delta_5)=0; \tag{5-14} \]

\[ \begin{gathered} P_{45}(U_4,U_5,\delta_4,\delta_5)+P_{65}(U_6,U_5,\delta_6,\delta_5)=P_5(U_5);\\ P_{\mathrm{nb}15}(U_4,U_5,U_6,\delta_4,\delta_5,\delta_6)=0; \end{gathered} \tag{5-15} \]

\[ \begin{gathered} P_{87}(U_8,U_7,\delta_8,\delta_7)=P_{73}^{*}(U_7,U_3,\delta_7,\delta_3)+P_7(U_7);\\ P_{\mathrm{nb}16}(U_3,U_7,U_8,\delta_3,\delta_7,\delta_8)=0. \end{gathered} \tag{5-16} \]

Equations for reactive power:

\[ \begin{gathered} Q_{12}(U_1,U_2,\delta_1,\delta_2)+Q_{32}(U_3,U_2,\delta_3,\delta_2)=Q_2(U_2);\\ Q_{\mathrm{nb}17}(U_1,U_2,\delta_1,\delta_2,\delta_3)=0; \end{gathered} \tag{5-17} \]

\[ \begin{gathered} Q_{23}(U'_2,\, U_3,\, \delta_2,\, \delta_3)+Q_{73}(U_7,\, U'_3,\, \delta_7,\, \delta_3)+Q_{13}(U_1,\, U_3,\, \delta_1,\, \delta_3)=0;\\ Q_{\mathrm{н}5_{18}}=0; \end{gathered} \tag{5-18} \]

\[ \begin{gathered} Q_{34}(U_3,\, U_4,\, \delta_3,\, \delta_4)+Q_{54}(U_5,\, U'_4,\, \delta_5,\, \delta_4)=0;\\ Q_{\mathrm{н}6_{19}}(U_3,\, U_4,\, U_5,\, \delta_3,\, \delta_4,\, \delta_5)=0; \end{gathered} \tag{5-19} \]

\[ \begin{gathered} Q_{45}(U_4,\, U_5,\, \delta_4,\, \delta_5)+Q_{65}(U_6,\, U'_5,\, \delta_6,\, \delta_5)=Q_5(U'_5);\\ Q_{\mathrm{н}6_{20}}(U_4,\, U_5,\, U_6,\, \delta_4,\, \delta_5,\, \delta_6)=0; \end{gathered} \tag{5-20} \]

\[ \begin{gathered} Q_{37}(U_3,\, U_7,\, \delta_3,\, \delta_7)+Q_{87}(U_8,\, U_7,\, \delta_8,\, \delta_7)=Q_7(U'_7);\\ Q_{\mathrm{н}6_{21}}(U_3,\, U_7,\, U_8,\, \delta_3,\, \delta_7,\, \delta_8)=0. \end{gathered} \tag{5-21} \]

For small voltage deviations from any stationary value, the dependences of the load powers of the given node on voltage are taken into account, since for small voltage deviations the action of voltage-regulating devices at the load-connection nodes should not be taken into account because of their insensitivity. Loads are taken as inertia-free, with regulating effect with respect to active power equal to 1 and 6, and with respect to reactive power equal to two.

The number of variables in equations (5-4—5-21) is 19; the number of variables is also 19 \((U_1 \div U_8;\ \delta_1 \div \delta_8;\ \delta_{01};\ \delta_{06};\ \delta_{08})\). These equations are written for the increments of the variables, with a right-hand side different from zero. From them one can find expressions for the voltage increments \(\Delta U_1\), \(\Delta U_6\), \(\Delta U_8\) and substitute them into 5-1, 5-2, and 5-3, respectively.

\[ \Delta U_1=\frac{f_{11}(p)}{F_{11}(p)}\,\Delta\delta_{01} +\frac{f_{12}(p)}{F_{12}(p)}\,\Delta\delta_{06} +\frac{f_{13}(p)}{F_{13}(p)}\,\Delta\delta_{08}; \]

\[ \Delta U_6=\frac{f_{21}(p)}{F_{21}(p)}\,\Delta\delta_{01} +\frac{f_{22}(p)}{F_{22}(p)}\,\Delta\delta_{06} +\frac{f_{23}(p)}{F_{23}(p)}\,\Delta\delta_{08}; \]

\[ \Delta U_8=\frac{f_{31}(p)}{F_{31}(p)}\,\Delta\delta_{01} +\frac{f_{32}(p)}{F_{32}(p)}\,\Delta\delta_{06} +\frac{f_{33}(p)}{F_{33}(p)}\,\Delta\delta_{08}. \]

Then three equations will be obtained for the changes of the intrinsic angles of each station; each equation will include all 16 unknown variable operating parameters.

\[ \begin{gathered} \left(a_{11}p^3+a_{12}p^2+a_{13}p+a_{14}+(a_{15}p+a_{16})\frac{f_{11}(p)}{F_{11}(p)}\right)\Delta\delta_{01} +(a_{15}p+a_{16})\times\\ \times \frac{f_{12}(p)}{F_{12}(p)}\,\Delta\delta_{06} +(a_{15}p+a_{16})\frac{f_{13}(p)}{F_{13}(p)}\,\Delta\delta_{08}=0; \end{gathered} \]

\[ \begin{gathered} \left(a_{21}p^3+a_{22}p^2+a_{23}p+a_{24}+(a_{25}p+a_{26})\frac{f_{21}(p)}{F_{21}(p)}\right)\Delta\delta_{01} +(a_{25}p+a_{26})\times\\ \times \frac{f_{22}(p)}{F_{22}(p)}\,\Delta\delta_{06} +(a_{25}p+a_{26})\frac{f_{23}(p)}{F_{23}(p)}\,\Delta\delta_{08}=0; \end{gathered} \]

\[ \begin{gathered} \left(a_{31}p^3+a_{32}p^2+a_{33}p+a_{34}+(a_{35}p+a_{36})\frac{f_{31}(p)}{F_{31}(p)}\right)\Delta\delta_{01} +(a_{35}+a_{36})\times\\ \times \frac{f_{22}(p)}{F_{22}(p)}\,\Delta\delta_{06} +(a_{35}p+a_{36})\frac{f_{33}(p)}{F_{33}(p)}\,\Delta\delta_{08}=0. \end{gathered} \]

The criteria of aperiodic stability may be formulated in the form of a one-sided inequality for the sign of the constant term of the characteristic equation, or in the form of Hurwitz inequalities for ra-

operation of underexcited generators, if oscillatory instability is possible. Static-stability conditions in this form are included in the calculations as a subroutine.

In the second form of writing the constraint conditions for static stability, the approximating polynomials for the e.m.f., the internal angles of the generators, and the transformation ratios of the power transformers are not used, and these variables also appear as unknowns. In addition, the parameters of certain circuit elements or sections may also be unknown. The small-oscillation equations of the electromechanical transient process are written in the form of relation (1).

Since

\[ \Delta E=\left(-\kappa_{U_1}\Delta U_1+\kappa_{\delta_{01}}\Delta\delta_{01}\right)\frac{1}{1+pT_{\mathrm{B}}}, \]

then

\[ I_1\frac{d^2\Delta\delta_{01}}{dt^2} = -\left(\overline{P}_{1E_1}(E_1,U_1,\delta_{01}) \left(-\kappa_{U_1}\Delta U_1+\kappa_{\delta_{01}}\Delta\delta_{01}\right) \frac{1}{1+pT_{\mathrm{B}}} +\overline{P}_{1U_1}(E_1,U_1,\Delta\delta_{01})\Delta U_1 +P_{1\delta_{01}}(E_1,U_1,\delta_{01})\Delta\delta_{01}\right); \]

\[ I_1(1+pT_{\mathrm{B}})p^2\Delta\delta_{01} = -\left(P_{1E_1}(E_1,U_1,\delta_{01}) \left(-\kappa_{U_1}\Delta U_1+\kappa_{\delta_{01}}\Delta\delta_{01}\right) +(1+pT_{\mathrm{B}}) \left(\overline{P}_{1U_1}(E_1,U_1,\delta_{01})\Delta U_1 +\overline{P}_{1\delta_{01}}(E_1,U_1,\delta_{01})\Delta\delta_{01}\right)\right); \]

\[ I_1(1+pT_{\mathrm{B}})p^2\Delta\delta_{01} = -\left(pT_{\mathrm{B}}\overline{P}_{1U_1}(E_1,U_1,\delta_{01}) +\overline{P}_{1U_1}(E_1,U_1,\delta_{01}) -\kappa_{U_1}\overline{P}_{1E_1}(E_1,U_1,\delta_{01})\right)\Delta U_1 \]

\[ +\left(pT_{\mathrm{B}}\overline{P}_{1\delta_{01}}(E_1,U_1,\delta_{01}) +\overline{P}_{1\delta_{01}}(E_1,U_1,\delta_{01}) +\kappa_{\delta_{01}}\overline{P}_{1E_1}(E_1,U_1,\delta_{01})\right)\Delta\delta_{01}; \]

\[ \left(I_1T_{\mathrm{B}}p^3+I_1p^2 +pT_{\mathrm{B}}P_{1\delta_{01}}(E_1,U_1,\delta_{01}) +\overline{P}_{1\delta_{01}}(E_1,U_1,\delta_{01}) +\kappa_{\delta_{01}}\overline{P}_{1E_1}(E_1,U_1,\delta_{01})\right)\Delta\delta_{01} \]

\[ +\left(pT_{\mathrm{B}}\overline{P}_{1U_1}(E_1,U_1,\delta_{01}) +\overline{P}_{1U_1}(E_1,U_1,\delta_{01}) -\kappa_{U_1}\overline{P}_{1E_1}(E_1,U_1,\delta_{01})\right)\Delta U_1 =0. \]

Denote:

\[ I T_{\mathrm{B}}=z_{11};\qquad I_1=z_{12};\qquad T_{\mathrm{B}}\overline{P}_{1\delta_{01}}(E_1,U_1,\delta_{01})=z_{13}; \]

\[ \overline{P}_{1\delta_{01}}(E_1,U_1,\delta_{01}) +\kappa_{\delta_{01}}\overline{P}_{1E_1}(E_1,U_1,\delta_{01})=z_{14};\qquad T_{\mathrm{B}}\overline{P}_{1U_1}(E_1,U_1,\delta_{01})=z_{15}; \]

\[ \overline{P}_{1U_1}(E_1,U_1,\delta_{01}) -\kappa_{U_1}\overline{P}_{1E_1}(E_1,U_1,\delta_{01})=z_{16}. \]

Thus,

\[ (z_{11}p^3+z_{12}p^2+z_{13}p+z_{14})\Delta\delta_{01} +(z_{15}p+z_{16})\Delta U_1=0, \tag{9-1} \]

where

\[ z_1=z_1(E_1,U_1,\delta_{01}). \]

For the rotors of the stations at nodes 6 and 8, analogous relations may be written:

\[ (z_{21}p^3+z_{22}p^2+z_{23}p+z_{24})\Delta\delta_{06} +(z_{25}p+z_{26})\Delta U_6=0 \tag{9-2} \]

with

\[ z_2=z_2(E_6,U_6,\delta_{06}); \]

\[ (z_{31}p^3+z_{32}p^2+z_{33}p+z_{34})\Delta\delta_{08} +(z_{35}p+z_{36})\Delta U_8=0 \tag{9-3} \]

with

\[ z_3=z_3(E_8,U_8,\delta_{08}). \]

For the nodes of connection of the generating stations 1, 6, 8, the small-oscillation equations of the operating-mode parameters are written from the conditions of the balances of active and reactive powers:

\[ \Delta P_{p1}(E_1,U_1,\delta_{01},\Delta E_1,\Delta U_1,\Delta\delta_{01}) = \Delta P^{*}_{12}(U_1,U_2,\delta_1,\delta_2,\kappa_{12},\Delta U_1,\Delta U_2, \Delta\delta_1,\delta\Delta_2,\Delta\kappa_{12}), \tag{9-4} \]

where \(\kappa_{12}\) is the transformation ratio of the transformer.

\[ \begin{aligned} \overline P_{1E_1}(E_1,U_1,\delta_{01})\Delta E_1 &+\overline P_{1U_1}(E_1,U_1,\delta_{01})\Delta U_1 +\overline P_{1\delta_{01}}(E_1,U_1,\delta_{01})\Delta\delta_{01} \\ &=\overline P^{*}_{12U_1}(U_1,U_2,\delta_1,\delta_2,\kappa_{12})\Delta U_1 +\overline P^{*}_{12U_2}(U_1,U_2,\delta_1,\delta_2,\kappa_{12})\Delta U_2 \\ &\quad+\overline P^{*}_{12\delta_1}(U_1,U_2,\delta_1,\delta_2,\kappa_{12})\Delta\delta_1 +\overline P^{*}_{12\delta_2}(U_1,U_2,\delta_1,\delta_2,\kappa_{12})\Delta\delta_2 \\ &\quad+\overline P^{*}_{12\kappa_{12}}(U_1,U_2,\delta_1,\delta_2,\kappa_{12})\Delta\kappa_{12}. \end{aligned} \]

Substitute \(\Delta E_1\) from equation (4) into (9-4) and perform the necessary transformations:

\[ \begin{aligned} &\overline P_{1E_1}(E_1,U_1,\delta_{01})(-\kappa_{U_1}\Delta U_1+\kappa_{\delta_{01}}\Delta\delta_{01}) \frac{1}{1+pT_{\mathrm{v}1}} +\overline P_{1U_1}(E_1,U_1,\delta_{01})\Delta U_1 \\ &\quad+\overline P_{1\delta_{01}}(E_1,U_1,\delta_{01})\Delta\delta_{01} =\Delta P^{*}_{2}(U_1,U_2,\delta_1,\delta_2,\kappa_{12},\Delta U_1,\Delta U_2,\Delta\delta_1,\Delta\delta_2,\Delta\kappa_{12}); \end{aligned} \]

\[ \begin{aligned} &\overline P_{1E_1}(E_1,U_1,\delta_{01})(-\kappa_{U_1}\Delta U_1+\kappa_{\delta_{01}}\Delta\delta_{01}) +\bigl(\overline P_{1U_1}(E_1,U_1,\delta_{01})\Delta U_1 \\ &\quad+\overline P_{1\delta_{01}}(E_1,U_1,\delta_{01})\Delta\delta_{01}\bigr)(1+pT_{\mathrm{v}1}) \\ &=(1+pT_{\mathrm{v}1})\cdot \Delta P^{*}_{12}(U_1,U_2,\delta_1,\delta_2,\kappa_{12}, \Delta U_1,\Delta U_2,\Delta\delta_1,\Delta\delta_2,\Delta\kappa_{12}); \end{aligned} \]

\[ \begin{aligned} &\bigl((1+pT_{\mathrm{v}1})\overline P_{1\delta_{01}}(E_1,U_1,\delta_{01}) +\kappa_{\delta_{01}}\overline P_{1E_1}(E_1,U_1,\delta_{01})\bigr)\Delta\delta_{01} \\ &\quad+\bigl((1+pT_{\mathrm{v}1})\overline P_{1U_1}(E_1,U_1,\delta_{01}) -\overline P^{*}_{12U_1}(U_1,U_2,\delta_1,\delta_2,\kappa_{12}) \\ &\qquad-\kappa_{U_1}\overline P_{1E_1}(E_1,U_1,\delta_{01})\bigr)\Delta U_1 -(1+pT_{\mathrm{v}1})\overline P^{*}_{12U_2}(U_1,U_2,\delta_1,\delta_2,\kappa_{12})\Delta U_2 \\ &\quad-(1+pT_{\mathrm{v}1})\overline P^{*}_{12\delta_1}(U_1,U_2,\delta_1,\delta_2,\kappa_{12})\Delta\delta_1 -(1+pT_{\mathrm{v}1})\overline P^{*}_{12\delta_2}(U_1,U_2,\delta_1,\delta_2,\kappa_{12})\Delta\delta_2 \\ &\quad-(1+pT_{\mathrm{v}1})\overline P^{*}_{12\kappa_{12}}(U_1,U_2,\delta_1,\delta_2,\kappa_{12})\Delta\kappa_{12}=0. \end{aligned} \]

Denote

\[ T_{\mathrm{v}1}\bigl(\overline P_{1U_1}(E_1,U_1,\delta_{01}) -\overline P^{*}_{12U_1}(U_1,U_2,\delta_1,\delta_2,\kappa_{12})\bigr) =\beta_{11}(E_1,U_1,\delta_{01},U_2,\delta_1,\delta_2,\kappa_{12}); \]

\[ \overline P_{1U_1}(E_1,U_1,\delta_{01}) -\overline P^{*}_{12U_1}(U_1,U_2,\delta_1,\delta_2,\kappa_{12}) -\kappa_{U_1}\overline P_{1E_1}(E_1,U_1,\delta_{01}) = \beta_{12}(E_1,U_1,\delta_{01},U_2,\delta_1,\delta_2,\kappa_{12}); \]

\[ T_{\mathrm{v}1}\cdot \overline P_{1\delta_{01}}(E_1,U_1,\delta_{01})=\beta_{13}; \]

\[ \overline P_{1\delta_{01}}(E_1,U_1,\delta_{01}) +\kappa_{\delta_{01}}\overline P_{1E_1}(E_1,U_1,\delta_{01})=\beta_{14}; \]

\[ T_{\mathrm{v}1}\overline P^{*}_{12U_2}=\beta_{15};\qquad \overline P^{*}_{12U_2}=\beta_{16};\qquad T_{\mathrm{v}1}\overline P^{*}_{12\delta_1}=\beta_{17}; \]

\[ \overline P^{*}_{12\delta_1}=\beta_{18};\qquad T_{\mathrm{v}1}\cdot\overline P^{*}_{12\delta_2}=\beta_{19};\qquad \overline P^{*}_{12\delta_2}=\beta_{110}. \]

Then

\[ (p\beta_{11}+\beta_{12})\Delta U_1 +(p\beta_{13}+\beta_{14})\Delta\delta_{01} -(p\beta_{15}+\beta_{16})\Delta U_2 -(p\beta_{17}+\beta_{18})\Delta\delta_1 \]
\[ -(p\beta_{19}+\beta_{110})\Delta\delta_2 -(p\beta_{111}+\beta_{112})\Delta\kappa_{12}=0, \]

where

\[ \beta_1=\beta(E_1,U_1,\delta_{01},U_2,\delta_1,\delta_2,\kappa_{12}), \]

for the 6th node:

\[ (p\beta_{21}+\beta_{22})\Delta U_6 +(p\beta_{23}+\beta_{24})\Delta\delta_{06} -(p\beta_{25}+\beta_{26})\Delta U_5 -(p\beta_{27}+\beta_{28})\Delta\delta_6 \]
\[ -(p\beta_{29}+\beta_{210})\Delta\delta_5=0, \tag{9-5} \]

where

\[ \beta_2=\beta(E_6;U_6;\delta_{06};U_5;\delta_5;\delta_6). \]

For the eighth node

\[ (p\beta_{31}+\beta_{32})\Delta U_8 +(p\beta_{33}+\beta_{34})\Delta\delta_{08} -(p\beta_{35}+\beta_{36})\Delta U_7 -(p\beta_{37}+\beta_{38})\Delta\delta_{08} \]
\[ -(p\beta_{39}+\beta_{310})\Delta\delta_7=0, \tag{9-6} \]

where

\[ \beta_3=\beta(E_8;\delta_{08};U_8;U_7;\delta_7;\delta_8). \]

Similarly to (5-7; 5-8; 5-9), equations are written for the reactive-power balances for the same nodes, whose coefficients are denoted by \(\gamma\).

For the first node:

\[ (p\gamma_{11}+\gamma_{12})\Delta U_1+(p\gamma_{13}+\gamma_{14})\Delta\delta_{01} -(p\gamma_{15}+\gamma_{16})\Delta U_2-(p\gamma_{17}+\gamma_{18})\Delta\delta_1- \]

\[ -(p\gamma_{19}+\gamma_{110})\Delta\delta_2-(p\gamma_{111}+\gamma_{112})\Delta\kappa_{12}=0, \]

where

\[ \gamma_1=\gamma_1(E_1,U_1,\delta_{01},U_2,\delta_1,\delta_2,\kappa_{12}). \]

Or, more briefly:

\[ \Delta\gamma_{\mathrm{nb}_1}=\Delta\gamma_{\mathrm{nb}_1}(E_1,U_1,\delta_{01},U_2,\delta_1,\delta_2,\kappa_{12}, \Delta U_1,\Delta U_2,\Delta\delta_1,\Delta\delta_2,\Delta\kappa_{12})=0. \tag{9-7} \]

For the sixth node

\[ \Delta\gamma_{\mathrm{nb}_6}=\Delta\gamma_{\mathrm{nb}_6}(E_6,U_6,\delta_{06},U_5,\delta_5,\delta_6, \Delta U_6,\Delta\delta_{06},\Delta U_5,\Delta\delta_5,\Delta\delta_6)=0. \tag{9-8} \]

For the eighth node

\[ \Delta\gamma_{\mathrm{nb}_8}=\Delta\gamma_{\mathrm{nb}_8}(E_8,U_8,\delta_{08},U_7,\delta_8,\delta_7, \Delta U_8,\Delta\delta_{08},\Delta U_7,\Delta\delta_8,\Delta\delta_7)=0. \tag{9-9} \]

As can be seen from a comparison of the systems of equations (5-7; 7-8; 5-9) and (9-7; 9-8; 9-9), they are identical in form (the system of equations (9-7) contains an additional term in the increment of the additional variable from the changes in the transformation coefficient), but have different contents of the variables in the coefficients of these equations.

For nodes 2, 3, 4, 5, and 7 and the operating conditions for transmitted power, equations in small increments of the operating parameters may be written in the form of the system of equations (5-7—5-2), into which the transformation coefficients on the sections 1-2; 4-5; 7-3 will additionally enter.

\[ \Delta P_{\mathrm{nb}_2}(U_1,U_2,U_3,\delta_1,\delta_2,\delta_3,\kappa_{12}, \Delta U_1,\Delta U_2,\Delta U_3,\Delta\delta_1,\Delta\delta_2,\Delta\delta_3,\Delta\kappa_{12})=0. \tag{9-10} \]

\[ \Delta P_{\mathrm{nb}_3}(U_2,U_3,U_4,\delta_2,\delta_3,\delta_4, \Delta U_2,\Delta U_3,\Delta U_4,\Delta\delta_2,\Delta\delta_3,\Delta\delta_4)=0; \tag{9-11} \]

\[ \Delta P_{\mathrm{nb}_4}(U_3,U_4,U_5,\delta_3,\delta_4,\delta_5,\kappa_{45}, \Delta U_3,\Delta U_4,\Delta U_5,\Delta\delta_3,\Delta\delta_4,\Delta\delta_5,\Delta\kappa_{45})=0; \tag{9-12} \]

\[ \Delta P_{\mathrm{nb}_5}(U_4,U_5,U_6,\delta_4,\delta_5,\delta_6,\kappa_{45}, \Delta U_4,\Delta U_5,\Delta U_6,\Delta\delta_4,\Delta\delta_5,\Delta\delta_6,\Delta\kappa_{45})=0; \tag{9-13} \]

\[ \Delta P_{\mathrm{ogr}_1}(U_4,U_5,\delta_4,\delta_5,\kappa_{45}, \Delta U_4,\Delta U_5,\Delta\delta_4,\Delta\delta_5,\Delta\kappa_{45})=0; \tag{9-14} \]

\[ \Delta P_{\mathrm{ogr}_2}(U_3,U_7,\delta_3,\delta_7,\kappa_{37}, \Delta U_3,\Delta U_7,\Delta\delta_3,\Delta\delta_7,\Delta\kappa_{37})=0; \tag{9-15} \]

\[ \Delta P_{\mathrm{nb}_7}(U_3,U_7,U_8,\delta_3,\delta_7,\delta_8,\kappa_{37}, \Delta U_3,\Delta U_7,\Delta U_8,\Delta\delta_3,\Delta\delta_7,\Delta\delta_8,\Delta\kappa_{37})=0; \tag{9-16} \]

\[ \Delta Q_{\mathrm{nb}_2}(U_1,U_2,U_3,\delta_1,\delta_2,\delta_3,\kappa_{12}, \Delta U_1,\Delta U_2,\Delta U_3,\Delta\delta_1,\Delta\delta_2,\Delta\delta_3,\Delta\kappa_{12})=0; \tag{9-17} \]

\[ \Delta Q_{\mathrm{nb}_3}(U_2,U_3,U_4,\delta_2,\delta_3,\delta_4, \Delta U_2,\Delta U_3,\Delta U_4,\Delta\delta_2,\Delta\delta_3,\Delta\delta_4)=0; \tag{9-18} \]

\[ \Delta Q_{\mathrm{nb}_4}(U_3,\Delta U_4,U_5,\delta_3,\delta_4,\delta_5,\kappa_{45}, \Delta U_3,\Delta U_4,\Delta U_5,\Delta\delta_3,\Delta\delta_4,\Delta\delta_5,\Delta\kappa_{45})=0; \tag{9-19} \]

\[ \Delta Q_{\mathrm{nb}_5}(U_4,U_5,U_6,\delta_4,\delta_5,\delta_6,\kappa_{45}, \Delta U_4,\Delta U_5,\Delta U_6,\Delta\delta_4,\Delta\delta_5,\Delta\delta_6,\Delta\kappa_{45})=0; \tag{9-20} \]

\[ \Delta Q_{\mathrm{nb}_7}(U_3,U_7,U_8,\delta_3,\delta_7,\delta_8,\kappa_{37}, \Delta U_3,\Delta U_7,\Delta U_8,\Delta\delta_3,\Delta\delta_7,\Delta\delta_8,\Delta\kappa_{37})=0; \tag{9-21} \]

The number of equations of power imbalances is 21, and the number of unknown variables in these equations is 25

\[ (E_1;\ E_6;\ E_8;\ \delta_{01};\ \delta_{06};\ \delta_{08}; \]

\[ \kappa_{12};\ \kappa_{45};\ \kappa_{73};\ U_1;\ U_2;\ U_3;\ U_4;\ U_5;\ U_6;\ U_7;\ U_8;\ \delta_1;\ \delta_2;\ \delta_3;\ \delta_4;\ \delta_5;\ \delta_6;\ \delta_7;\ \delta_8). \]

To estimate the influence and eliminate some of the variables from the mismatch equations (8-4—8-21), multifactor analysis may be applied in the form of the mathematical apparatus of experimental design. Let \(k_{12}\), \(k_{45}\), and \(k_{73}\) and one of the voltages be known. Then the static-stability conditions reduce, in accordance with (7), to a certain relation in all variable operating parameters (25 P) in the form of the inequality

\[ a_n(25\text{ P}) > 0. \tag{10} \]

More simply, for approximate calculations the static-stability conditions may be written in the form of one-sided inequalities for the relative angles of all terminal devices:

\[ 0 < \delta_{01} - \delta_{06} \le \tilde{\delta}_{06}; \tag{11-1} \]

\[ 0 < \delta_{01} - \delta_{08} \le \tilde{\delta}; \tag{11-2} \]

\[ 0 < \delta_{06} - \delta_{08} \le \tilde{\delta}_{08}; \tag{11-3} \]

REFERENCES

1. I. Pechon and other. Sensitivity in Power Systems IEEE, Transations of PAS, vol. 87, pp. 1367—1374.