CONTROL OF REACTIVE POWER OPERATING MODES IN INTERCONNECTED POWER SYSTEMS WITH LONG-DISTANCE AC TRANSMISSION LINES

R. I. BORISOV

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

The present level of development of electric power engineering is characterized by the introduction of power-transmission lines at voltages of 500 and 750 kV, which mainly serve as intersystem ties. The 500-kV lines in operation within the United Dispatch Administration of Siberia perform the role of lines for reversible intersystem exchange with large variations in power flows and high values of the static-stability reserve factor. Power exchanges are carried out only within the limits of neighboring electric power systems. As the capacities of the interconnection, the length, and the nominal voltage of the lines increase, transit transmission of electric energy will also take place. In this case, depending on different conditions, the ratios between the transit and exchange powers of the intermediate systems connected along the route of a long-distance transmission line may change. Under these conditions, many problems of designing and tuning the operating modes of such schemes will require solution. The distribution of active and reactive powers among systems, the role and influence of intermediate systems, the values of the parameters of regulating systems in terminal and intermediate systems, transient processes, reserve provision, reliability, and the technical and economic conditions of their joint operation, and other questions, will require solution.

Raising the operating voltage of long-distance transmission lines to 750 and 1200 kV, and the greater expected load fluctuations of these lines—judging from operating experience with 500-kV transmissions, especially in the initial period of their operation—all these factors will lead to an acute problem of reactive-power control, which may not be solvable under these conditions by the usual and tested methods. In addition, it must be considered that the development of power interconnections is characterized by a certain “connectedness” of states. This connectedness arises from real conditions, and it concerns either the voltage-control conditions in individual parts of the scheme, or specified values of power flows in sections, or load-coverage conditions based on heat-consumption regimes, etc. For such power-system interconnections linked by conditions, the existing methods for calculating operating modes and solving design problems are substantially limited. Mathematical models for the analysis and synthesis of such formations must correspond to their technical content [1]. Growth and development in purely quantitative respects cannot not

lead to qualitatively new phenomena, the timely recognition of which is a major scientific and technical problem. In solving problems of reactive-power control, it is assumed that the intermediate systems are balanced with respect to active power, while transit power is transmitted over the long-distance transmission line to the receiving system. Then the important problem arises of the necessary control of the reactive power of each source. Each intermediate and terminal system is a source capable of favorably affecting the flow distribution of reactive powers, controlling voltage, and influencing losses. For certain other modes of an ac transmission line, they will have a favorable effect on the technical and economic indicators of the intermediate systems and on the reliability of power supply as a whole.

Optimality Conditions for Operating Modes

A power transmission system with intermediate systems is formed in the form of a chain circuit with nonuniform elements (Fig. 1).

Fig. 1. Chain circuits of interconnections of power systems with ac transmission lines

To describe the state of such circuits, a very convenient form is provided by the network-power equations, which express the powers at the beginning \((i)\) and end \((j)\) of each section through the voltages \((U_i\) and \(U_j)\), the angles \((\delta_i\) and \(\delta_j)\), and the parameters of the section \((A_{ij}\angle \varphi_{Aij};\ B_{ij}\angle \varphi_{Bij};\ C_{ij}\angle \varphi_{Cij};\ D_{ij}\angle \varphi_{Dij})\) [2]. The conditions for the flow distribution of active and reactive powers for each node must be satisfied for the corresponding values of the variable operating parameters. The loads of each node may in this case be represented, according to their static characteristics, by constant resistances or constant powers. For the aggregate of all nodes, systems of equations in the variable operating parameters may be formed in the form of the following relations:

\[ \sum^{n} P_{нi}(U_i;\ U_j;\ \delta_i;\ \delta_j;\ (A_{ij})) = 0, \tag{1} \]

\[ \sum^{n} Q_{нi}(U_i;\ U_j;\ \delta_i;\ \delta_j;\ (A_{ij})) = 0, \tag{2} \]

where \(n\) is the number of nodes in the circuit.

Design calculations for selecting the parameters of individual circuit elements and the parameters of devices \((A_{ij})\) can be formulated by formulas (1) and (2) together with the variable operating parameters. The number of independent equations will then be insufficient for determining all the unknowns. Additional conditions will have to be introduced. However, in such a formulation and solution of problems both for design and for operation, there are certain shortcomings and omissions. The determination of active and reactive powers

of the balancing node and other generating points does not make it possible to judge by what control means the required operating condition is being established, or to judge the heating of the generators. Requirements for maintaining stability, voltage distribution, limiting short-circuit currents, and other technical and economic conditions may prove unsatisfied. A simple enumeration of possible variants makes the problem inconceivably more complicated. To solve these problems, it appears possible to use the apparatus of multifactor analysis in the form of the mathematical theory of experimental design.

Methods for calculating the operating conditions of power systems must ensure the adjustment of states for all elements of a specified set, taking into account the individual features of each element and the constraints imposed by various technical conditions. In constructing a mathematical model of the problem of controlling the reactive-power conditions of power systems, both a deterministic and a stochastic approach to determining the values of transmitted powers and load conditions in the receiving, transmitting, and intermediate systems are possible and have practical significance.

Fig. 2. AC power transmission with one intermediate system

The information in deterministic form relates more to the field of dispatch-control problems, while in stochastic form it is used for solving problems of design and long-term planning. Therefore these problems may be considered and solved at different levels. However, this does not affect the mathematical model of the problem as a certain optimal objective form with technical constraints of various kinds, which act as reliability requirements. The objective form adopted is the calculated costs, while the technical constraints in this problem are conditions on static stability, voltage distribution, limitation of short-circuit currents, heating of generators with respect to stator and rotor current, and the regulating range of transformers. The conditions of connectivity of the operating condition (1), (2) according to Kirchhoff’s first law are taken into account.

The circuit in Fig. 2 is considered for the connection of the transmitting system \(P_1\) and the receiving system \(P_2\) by a long-distance AC transmission line with one intermediate system (IS). Further generalization of the conditions and requirements to circuits that are more complex in composition and configuration does not cause fundamental difficulties.

The directions of active-power flows are indicated in the diagram. The value \(P_{45}\) is known and nonzero, while the flow \(P_{73}\) is taken to be zero. If the composition of the equipment is assumed to be known, which for a specified value of transmitted active power is resolved by a simple enumeration of variants, then the adjustment of the operating condition must be performed by distributing reactive powers among the sources. It must satisfy the requirement of a minimum of operating costs, which consist of the cost expression of losses and

costs for the additionally installed capacity at power plants to cover losses [3]

\[ \min f(m\Pi)=\min\left(\sum P_{\text{loss}} c_m c_p k_y+\sum P_{\text{loss}}c_p\right), \]

where

\(c_m\)—coefficient of coincidence of the losses of the corresponding load with the maximum;

\(c_p\)—unit cost of energy losses;

\(c_p\)—active-power reserve coefficient;

\(k_y\)—unit cost of active power of power plants.

The total losses \(\sum P_{\text{loss}}\) depend on the variable operating parameters and may be written as follows:

\[ \sum_{ij} P_{\text{loss}} = \sum_{ij}\frac{D_{ij}}{B_{ij}}\,U_i^2\cos(\Psi_{Dij}-\Psi_{Bij}) + \sum_{ij}\frac{A_{ij}}{B_{ij}}\cos(\Psi_{Aij}-\Psi_{Bij}) - \sum_{ij}2\frac{U_iU_j}{B_{ij}}\cos(\delta_i-\delta_j)\cos\Psi_{Bij} \to \min . \]

For the conditions under consideration, the number of variable operating parameters is 12 \((m\Pi \in U_1;\ U_2;\ U_3;\ U_4;\ U_5;\ U_6;\ U_7;\ U_8;\ \delta_1;\ \delta_2;\ \delta_3;\ \delta_4;\ \delta_5;\ \delta_6;\ \delta_7;\ \delta_8)\). Finding the values of the variables corresponding to the minimum of expression (4) by methods of differential calculus is inadmissible, since the values of the variable operating parameters are very strictly limited by the technical operating conditions of power systems.

Voltage-distribution conditions

The voltage-distribution conditions reduce to two-sided inequalities in the form of requirements for maintaining voltages at the nodal points of the network to which sources or consumers are connected [4].

\[ 0.9=\underline{U_i}\le U_i\le \overline{U_i}=1.05. \tag{5} \]

Conditions for heating of the stator and rotor of generators

For a typical turbogenerator, the internal electrical parameters \((E_i\ \text{and}\ \delta_{0i})\) are approximated by piecewise-linear characteristics of powers and voltages, both when operating with an inductive load and when operating in the underexcitation region [4].

\[ E_{0i}=v_EU_i+\pi_EP_{0i}+q_EQ_{0i}, \tag{6} \]

\[ \delta_{0i}=v_\delta U_i+\pi_\delta P_{0i}+q_\delta Q_{0i}. \tag{7} \]

In the second form, the boundary conditions for the selected parameters are expressed in the form of two-sided inequalities

\[ \underline{E_{0i}}\le E_{0i}\le \overline{E_{0i}}, \tag{8} \]

\[ \underline{\delta_{0i}}\le \delta_{0i}\le \overline{\delta_{0i}}. \tag{9} \]

Coupling conditions based on active- and reactive-power balances at the nodal points of the network

Since the direction of active-power flows in the calculation network (Fig. 2) is considered known, this makes it possible to formulate the equations of active-power balances in the form of formula (1). To formulate

equations of reactive-power balances should be specified by their arbitrary power-flow distribution.

Then

\[ P_{01}(E_1;\ U_1;\ \delta_{01})=P_{12}^{*}(U_1;\ U_2;\ \delta_1;\ \delta_2;\ k_{12}), \tag{10-1} \]

\[ P_{12}(U_1;\ U_2;\ \delta_1;\ \delta_2;\ k_{12})=P_2+P_{23}^{*}(U_2;\ U_3;\ \delta_2;\ \delta_3), \tag{10-2} \]

\[ P_{23}(U_2;\ U_3;\ \delta_2;\ \delta_3)=P_{34}^{*}(U_3;\ U_4;\ \delta_3;\ \delta_4), \tag{10-3} \]

\[ P_{34}(U_3;\ U_4;\ \delta_3;\ \delta_4)=P_{45}^{*}(U_4;\ U_5;\ \delta_4;\ \delta_5), \tag{10-4} \]

\[ P_{45}(U_4;\ U_5;\ \delta_4;\ \delta_5;\ k_{45})+P_{65}(U_6;\ U_5;\ \delta_6;\ \delta_5)=P_5, \tag{10-5} \]

\[ P_{06}(E_6;\ U_6;\ \delta_{06})=P_{65}(U_6;\ U_5;\ \delta_6;\ \delta_5), \tag{10-6} \]

\[ P_{73}(U_7;\ U_3;\ \delta_7;\ \delta_3;\ k_{73})=0, \tag{10-7} \]

\[ P_{45}(U_4;\ U_5;\ \delta_4;\ \delta_5;\ k_{45})=P_{45}, \tag{10-8} \]

\[ P_{08}(E_{08};\ \delta_{08};\ U_8)=P_7+P_{73}^{*}(U_7;\ U_3;\ \delta_7;\ \delta_3;\ k_{73}), \tag{10-9} \]

\[ Q_{01}(E_1;\ \delta_{01};\ U_1)+Q_{21}(U_2;\ U_1;\ \delta_2;\ \delta_1;\ k_{12})=0, \tag{11-1} \]

\[ Q_{12}(U_1;\ U_2;\ \delta_1;\ \delta_2;\ k_{12}) +Q_{32}(U_3;\ U_2;\ \delta_3;\ \delta_2) +Q_{73}(U_7;\ U_3;\ \delta_7;\ \delta_3;\ k_{73})=0, \tag{11-2} \]

\[ Q_{08}(E_8;\ \delta_{08};\ U_7)+Q_{37}(U_3;\ U_7;\ \delta_3;\ \delta_7;\ k_{73})=0, \tag{11-3} \]

\[ Q_{34}(U_3;\ U_4;\ \delta_3;\ \delta_4)+Q_{54}(U_5;\ U_4;\ \delta_5;\ \delta_4;\ k_{45})=0, \tag{11-4} \]

\[ Q_{45}(U_4;\ U_5;\ \delta_4;\ \delta_5;\ k_{45})+Q_{06}(E_6;\ \delta_{06};\ U_6)=0. \tag{11-5} \]

The power flows at the beginning and at the end of a section are written in the following form:

\[ P_{ij}^{*} = U_i^2\frac{D_{ij}}{B_{ij}}\cos(\Psi_{Dij}-\Psi_{Bij}) - \frac{U_iU_j}{B_{ij}}\cos(\delta_i-\delta_j+\Psi_{Bij}) = P_{ij}^{*}(U_i;\ U_j;\ \delta_i;\ \delta_j), \tag{12-1} \]

\[ P_{ij} = - U_j^2\frac{A_{ij}}{B_{ij}}\cos(\Psi_{Aij}-\Psi_{Bij}) + \frac{U_iU_j}{B_{ij}}\cos(\delta_i-\delta_j-\Psi_{Bij}) = P_{ij}(U_i;\ U_j;\ \delta_i;\ \delta_j), \tag{12-2} \]

\[ Q_{ij}^{*} = U_i^2\frac{D_{ij}}{B_{ij}}\sin(\Psi_{Dij}-\Psi_{Bij}) + \frac{U_iU_j}{B_{ij}}\sin(\delta_i-\delta_j+\Psi_{Bij}) = Q_{ij}^{*}(U_i;\ U_j;\ \delta_i;\ \delta_j), \tag{13-1} \]

\[ Q_{ij} = - U_j^2\frac{A_{ij}}{B_{ij}}\sin(\Psi_{Aij}-\Psi_{Bij}) + \frac{U_iU_j}{B_{ij}}\sin(\delta_i-\delta_j-\Psi_{Bij}) = Q_{ij}(U_i;\ U_j;\ \delta_i;\ \delta_j). \tag{13-2} \]

Active losses are determined as follows:

\[ P_{\mathrm{loss}} = U_i^2\frac{D_{ij}}{B_{ij}}\cos(\Psi_{Dij}-\Psi_{Bij}) + U_j^2\frac{A_{ij}}{B_{ij}}\cos(\Psi_{Aij}-\Psi_{Bij}) - 2\frac{U_iU_j}{B_{ij}}\cos(\delta_i-\delta_j)\cos\Psi_{Bij}. \tag{14-1} \]

The generalized parameters \(A\) of the corresponding section may contain complex current and voltage transformation coefficients, which should be taken into account as variables when solving power-flow problems.

Taking the values of transformation ratios into account as variables in the expressions of the functional (3) and in the constraints (10-13; 10-2; 10-4; 10-5; 10-7; 10-8; 11-1; 11-2; 11-3), in principle, does not complicate the computations. In this case the parameters of each element, in named units, are calculated at their own voltage level. Transformers are introduced into the equivalent circuit not only by parameters referred to any level, but also by elements of ideal transformation with a complex matrix \((\dot A)\), whose parameters are determined from the formulas:

\[ \dot A = \dot k;\quad \dot B = 0;\quad \dot C = 0;\quad \dot D = \frac{1}{\dot k}. \tag{15} \]

The values of the moduli and arguments of the quantities \(\dot k\) form two-sided inequalities:

\[ \underline{\kappa}_{ij} \leq \kappa_{ij} \leq \overline{\kappa}_{ij}, \tag{16} \]

\[ \underline{\Psi}_{\kappa ij} \leq \Psi_{\kappa ij} \leq \overline{\Psi}_{\kappa ij}. \tag{17} \]

In relations (10-1; 10-6; 10-9; 11-1; 11-3; 11-5), following the first form of calculating the EMFs of generating stations, \(E_i\) can be eliminated from the balance equations and from the static-stability constraints by expressing them through the parameters of the external part of the network.

Thus, for the first station

\[ P_{01}(E_1; U_1; \delta_{01}) = P_{12}^{*}(U_1; U_2; \delta_1; \delta_2), \]

since

\[ E_1 = v_{E_1}(U_1) + x_{E_1} P_{01} + q_{E_1} Q_{01} = v_{E_1}U_1 + P_{12}^{*}(U_1; U_2; \delta_1; \delta_2) + \]

\[ + q_{E_1}\bigl(-Q_{21}(U_2; U_1; \delta_2; \delta_1)\bigr) = E_1(U_1; U_2; \delta_1; \delta_2). \]

Then

\[ P_{01}\bigl(E_1(U_1; U_2; \delta_1; \delta_2); U_1; \delta_{01}\bigr) - P_{12}^{*}(U_1; U_2; \delta_1; \delta_2) = 0 \]

or

\[ P_{\text{n}6,1}(U_1; U_2; \delta_1; \delta_2; \delta_{01}) = 0. \]

The number of variables in the first calculation form (with \(K\) and \(E\) excluded from the number of independent variables) for the circuit in Fig. 2 will be 19:

\[ (\delta_{01}; \delta_{06}; \delta_{08}; U_1; U_2; U_3; U_4; U_5; U_6; U_7; U_8; \delta_1; \delta_2; \delta_3; \delta_4; \delta_5; \delta_6; \delta_7; \delta_8) \]

Under the second form, the number of independent variables will be equal to 25 (3 EMFs and 3 transformation ratios are added).

Constraints imposed by static-stability conditions

The conditions are expressed in the form of a one-sided inequality in all variable operating parameters [5]

\[ a_n(m\Pi) > 0. \tag{18} \]

Or in the form of inequalities in the mutual angles of generating stations:

\[ 0 < \delta_i - \delta_{0j} \leq \overline{\delta}_{ij}. \tag{19} \]

Conditions for constraints on short-circuit currents in power systems

The requirements of the short-circuit-current constraints, as was shown in [4], reduce to determining, in a special subroutine, the values of the separating

reactances \((x_{65}\) and \(x_{87})\) under known conditions concerning the power of a three-phase short circuit at a specified point and the power of the supply sources. These reactance values are specified quantities and, with their parameters \((A_{ij})\), enter into the equivalent circuit.

Justification for the Choice of the Method for Solving the Problem

As noted earlier, the number of variables of the objective functional is twelve; the number of independent variables in the relations (10-1—10-9a, 11-1—11-5) specified in the form of equalities is 19 or 25. In addition, there are additional inequalities (5), (8), (9), (17), (18), (19) with respect to operating-mode variable parameters of various kinds and numbers.

From the relations in the form of equalities or inequalities, the additional variables \(M - m\Pi\) cannot be expressed explicitly in terms of the principal variables of the functional \((m\Pi)\). This causes additional difficulties when using the gradient method or its modifications for solving the problem. Therefore, the method of undetermined Lagrange multipliers [6] is used; for this purpose all inequalities, by introducing additional variables, are written as equalities in the following form:

\[ \begin{aligned} U_1+\bar{x}_1 &= 1.05, && (20\text{-}1)\\ U_2+\bar{x}_2 &= 1.05, && (20\text{-}2)\\ U_3+\bar{x}_3 &= 1.05, && (20\text{-}3)\\ U_4+\bar{x}_4 &= 1.05, && (20\text{-}4)\\ U_5+\bar{x}_5 &= 1.05, && (20\text{-}5)\\ U_6+\bar{x}_6 &= 1.05, && (20\text{-}6)\\ U_7+\bar{x}_7 &= 1.05, && (20\text{-}7)\\ U_8+\bar{x}_8 &= 1.05. && (20\text{-}8) \end{aligned} \qquad \begin{aligned} U_1-\underline{x}_1 &= 0.9, && (21\text{-}1)\\ U_2-\underline{x}_2 &= 0.9, && (21\text{-}2)\\ U_3-\underline{x}_3 &= 0.9, && (21\text{-}3)\\ U_4-\underline{x}_4 &= 0.9, && (21\text{-}4)\\ U_5-\underline{x}_5 &= 0.9, && (21\text{-}5)\\ U_6-\underline{x}_6 &= 0.9, && (21\text{-}6)\\ U_7-\underline{x}_7 &= 0.9, && (21\text{-}7)\\ U_8-\underline{x}_8 &= 0.9. && (21\text{-}8) \end{aligned} \]

\[ \delta_1-\delta_6+\bar{y}_{16}=\bar{\delta}_{16}, \qquad (22\text{-}1) \]

\[ \delta_1-\delta_8+\bar{y}_{18}=\bar{\delta}_{18}, \qquad (22\text{-}2) \]

\[ \delta_6-\delta_8+\bar{y}_{68}=\bar{\delta}_{68}, \qquad (22\text{-}3) \]

\[ \begin{aligned} \delta_1+\bar{y}_1 &= \bar{\delta}_1, && (23\text{-}1)\\ \delta_2+\bar{y}_2 &= \bar{\delta}_2, && (23\text{-}2)\\ \delta_3+\bar{y}_3 &= \bar{\delta}_3, && (23\text{-}3)\\ \delta_4+\bar{y}_4 &= \bar{\delta}_4, && (23\text{-}4)\\ \delta_5+\bar{y}_5 &= \bar{\delta}_5, && (23\text{-}5)\\ \delta_6+\bar{y}_6 &= \bar{\delta}_6, && (23\text{-}6)\\ \delta_7+\bar{y}_7 &= \bar{\delta}_7, && (23\text{-}7)\\ \delta_8+\bar{y}_8 &= \bar{\delta}_8, && (23\text{-}8) \end{aligned} \qquad \begin{aligned} \delta_1-\underline{y}_1 &= \underline{\delta}_1, && (24\text{-}1)\\ \delta_2-\underline{y}_2 &= \underline{\delta}_2, && (24\text{-}2)\\ \delta_3-\underline{y}_3 &= \underline{\delta}_3, && (24\text{-}3)\\ \delta_4-\underline{y}_4 &= \underline{\delta}_4, && (24\text{-}4)\\ \delta_5-\underline{y}_5 &= \underline{\delta}_5, && (24\text{-}5)\\ \delta_6-\underline{y}_6 &= \underline{\delta}_6, && (24\text{-}6)\\ \delta_7-\underline{y}_7 &= \underline{\delta}_7, && (24\text{-}7)\\ \delta_8-\underline{y}_8 &= \underline{\delta}_8. && (24\text{-}8) \end{aligned} \]

\[ \begin{gathered} \kappa_{45}+\bar z_{45}=\bar\kappa_{45}, \qquad (25\text{-}1)\\ E_1+\bar S_1=\bar E_1, \qquad (27\text{-}1)\\ \kappa_{73}+\bar z_{73}=\bar\kappa_{73}, \qquad (25\text{-}2)\\ E_6+\bar S_6=\bar E_6, \qquad (27\text{-}2)\\ \kappa_{12}+\bar z_{12}=\bar\kappa_{12}, \qquad (25\text{-}3)\\ E_8+\bar S_8=\bar E_8, \qquad (27\text{-}2) \end{gathered} \qquad \begin{gathered} \kappa_{12}-\underline z_{12}=\underline\kappa_{12}, \qquad (26\text{-}1)\\ E_1-\underline S_1=\underline E_1, \qquad (28\text{-}3)\\ \kappa_{45}-\underline z_{45}=\underline\kappa_{45}, \qquad (26\text{-}2)\\ E_6-\underline S_6=\underline E_6, \qquad (28\text{-}2)\\ \kappa_{73}-\underline z_{73}=\underline\kappa_{73}, \qquad (26\text{-}3)\\ E_8-\underline S_8=\underline E_8. \qquad (28\text{-}3) \end{gathered} \tag{2} \]

Since all constraints have been reduced to equalities, the method of undetermined Lagrange multipliers can be used for the solution. The Lagrange function may be written as follows:

\[ \begin{aligned} L={}& f(m\Pi)+\sum_i \lambda_i P_{\text{unb}i}(M\Pi) +\sum_i \psi_i Q_{\text{unb}i}(M\Pi) +\sum_i \nu_i(U_i+\bar x_i-\bar U_i)+ \\ &+\sum_i \chi_i(U_i-\underline x_i-\underline U_i) +\sum_i \vartheta_i(\delta_i+\bar y_i-\bar\delta_i) +\sum_i \Theta_i(\delta_i-\underline y_i-\underline\delta_i)+ \\ &+\sum_{ij} t_{ij}(\delta_i-\delta_j+\bar y_{ij}-\bar\delta_{ij}) +\sum_i \eta_i(\kappa_{ij}+\bar z_{ij}-\bar\kappa_{ij}) +\sum_i \rho_i\left(\kappa_{ij}-\underline z_{ij}^{\,ij}-\underline\kappa_{ij}\right)+ \\ &+\sum_i \omega_i(E_i+\bar S_i-\bar E_i) +\sum_i \Omega(E-\underline S_i-\underline E_i), \end{aligned} \tag{29} \]

where \(\lambda;\ \mu;\ \nu;\ \chi;\ \vartheta;\ \Theta;\ t;\ \eta;\ \rho;\ \omega;\ \Omega\) are undetermined Lagrange multipliers.

Putting \(dL=0\) at the extremum point, one can form the required number of equations for solving the problem:

\[ \begin{aligned} dL={}& \sum_i^M \frac{\partial L}{\partial \Pi_i}\,d\Pi +\sum_i \frac{\partial L}{\partial \lambda_i}\,d\lambda_i +\sum_i \frac{\partial L}{\partial \psi_i}\,d\psi_i +\sum_i \frac{\partial L}{\partial \nu_i}\,d\nu_i+ \\ &+\sum_i \frac{\partial L}{\partial \bar x_i}\,d\bar x_i +\sum_i \frac{\partial L}{\partial \chi_i}\,d\chi_i +\sum_i \frac{\partial L}{\partial \underline x_i}\,d\underline x_i +\sum_i \frac{\partial L}{\partial \vartheta_i}\,d\vartheta_i+ \\ &+\sum_i \frac{\partial L}{\partial \bar y_i}\,d\bar y_i +\sum_i \frac{\partial L}{\partial \Theta_i}\,d\Theta_i +\sum_i \frac{\partial L}{\partial \underline y_i}\,d\underline y_i +\sum_i \frac{\partial L}{\partial t_{ij}}\,dt_{ij}+ \\ &+\sum_{ij} \frac{\partial L}{\partial \bar y_{ij}}\,d\bar y_{ij} +\sum_i \frac{\partial L}{\partial \eta_i}\,d\eta_i +\sum_{ij} \frac{\partial L}{\partial \bar Z_{ij}}\,d\bar Z_{ij} +\sum_i \frac{\partial L}{\partial \rho_i}\,d\rho_i+ \\ &+\sum_{ij} \frac{\partial L}{\partial \underline Z_{ij}}\,d\underline Z_{ij} +\sum_i \frac{\partial L}{\partial \omega_i}\,d\omega_i +\sum_i \frac{\partial L}{\partial \bar S_i}\,d\bar S_i +\sum_i \frac{\partial L}{\partial \Omega_i}\,d\Omega_i +\sum_i \frac{\partial L}{\partial \underline S_i}\,d\underline S_i=0. \end{aligned} \tag{30} \]

Equating the expression of each sum to zero, we obtain the required number of nonlinear algebraic equations with respect to all unknown operating-condition parameters \((E_i;\ \delta_i;\ \underline U_i;\ \kappa_{ij})\) and the additional variables of the Lagrange form \((\lambda_i;\ \mu_i;\ \nu_i;\ \chi_i;\ x_i;\ \text{etc.})\). To solve systems of algebraic nonlinear equations, the method is used

linear extrapolation by expanding the Lagrange function in a Taylor series at the point of the feasible solution and subsequently solving systems of linear equations with respect to the increments by the Gauss–Seidel scheme [7].

REFERENCES

1. R. I. Borisov. Programmed control and construction of interconnections of electric-power systems with long-distance alternating-current transmission lines. Technical report, state registration no. 71073552, Tomsk, 1971.

2. V. A. Venikov. The development of power engineering and social progress in the light of Lenin’s ideas on the significance of electrification. Elektrichestvo, 1970, no. 12.

3. N. A. Melnikov. Electric Networks and Systems. Moscow, “Energiya,” 1969.

4. R. I. Borisov. Conditions of constraints and functional relationships between variable operating parameters in mathematical programming for the selection of devices and control of the operation of long-distance power transmissions and electric systems. Article in this collection.

5. R. I. Borisov. Constraints on operating parameters of long-distance power transmissions with intermediate systems under static-stability conditions. Article in this collection.

6. Ch. Karr, Ch. Howe. Quantitative Methods for Decision-Making in Management and Economics. Moscow, “Mir,” 1966.

7. R. I. Borisov. Optimal Mathematical Programming in Electric Power Engineering. Tomsk, TPI, 1973.