Optimization of Parameters and Devices of Long-Distance AC Power Transmission Lines with Intermediate Systems

R. I. Borisov

(Presented by the scientific seminar of the Departments of Electric Power Stations and Electrical Systems and Networks)

Aggregates of sections of long-distance AC power transmission lines and their elements, terminal and intermediate connections, form a complex multiparameter system. The parameters of such a system are certain variables of operating conditions and variables of different elements of the circuits, known combinations of which make it possible to describe the electrical and techno-economic state of the entire system in operational and design calculations.

Establishing the operating parameters and the parameters of circuits in the form of the composition and type of operating equipment is a complex problem, owing to the mutual causal dependence of the action of various factors and the need to satisfy conditions for the required economy and reliability of operation. Attempts at an isolated solution of various problems—for example, the selection of means of longitudinal and transverse compensation [1], the regulation of reactive powers [2], and others—in isolation from mutually conditioning factors inevitably led to uneconomical solutions, since the sum of the local minima of partial solutions in a restricted formulation of the problems could turn out to exceed the minimum value for the solution in its general form. The use of digital computers makes it possible to approach the selection of parameters precisely as multifactor problems of nonlinear programming. In this respect, one should note a whole series of works in which problems of power engineering are posed and solved in a very general form [3, 4, 5]. In the work of I. A. Fedorova [5], the transmission capacity of a long-distance power transmission line under conditions of static stability and conditions of limiting dynamic transitions is formulated in the form of multifactor problems involving the parameters of circuits, the composition of its elements, and the operating parameters. Problems of designing and constructing long-distance power transmission lines are solved by minimizing the cost of transmitting electrical energy. Further generalizations in the formulation and solution of these problems appear entirely necessary and far from exhausted.

Design problems involving the selection of parameters of elements and sections must be solved with allowance for the state, purpose, and development of the entire interconnection scheme and of its individual elements. The correct determination of the functions of long-distance power transmissions at different stages of electrification makes it possible to approach more soundly the problems of their design, such as the selection of the corresponding parameters according to certain objective criteria with allowance for the statistical characteristics of forecast-

…of the operating modes of power transmissions and intermediate systems, and of technical constraints under different conditions. The probabilistic character of a number of the initial data must be taken into account when constructing mathematical models that reflect the process of designing and operating such schemes [6].

For design calculations, data on the composition of the operating equipment and its parameters, and on the design parameters of the power transmission by sections, may serve as variables to be determined. Conditions on static stability, voltage distribution, limitation of short-circuit currents, heating of generators with respect to stator and rotor current, etc., appear as reliability requirements in the form of constraints.

The problems of operational calculations for setting operating modes are, in essence, simpler than the problems of designing the composition and parameters of the elements of schemes. This applies to any form of their formulation, both deterministic and stochastic. Design problems are characterized by the appearance of additional variables to be determined in the form of parameters of scheme elements and equipment composition. In this case the objective function loses its smoothness conditions [7], which substantially complicates the solution of such problems. Moreover, since the scale of conductor cross-sections for long-distance power transmissions and the list of equipment by parameters and composition are discrete quantities, this requires the use of nonlinear integer-programming methods. However, the installed capacity of the terminal and intermediate devices of the systems must correspond to the limiting transit power transmitted over the line. Therefore there is no occasion to speak of variations in the equipment composition over wide limits. The equipment of such a series is unique, and its capacities are limiting. Consequently, the problem of composition can be solved by a simple enumeration of variants, since their number does not exceed two or three. Then, for each variant, the objective function, composed from the equations of network powers, proves to be a smooth function, and the inclusion of constraints cannot violate these properties. As for the elements of the power-transmission line proper, for them the design problem is formulated in full measure.

For a deterministic calculated value of the transmitted power, or for a specified distribution of values of the transmitted powers, one may speak first of all of the parameters of the longitudinal and transverse compensation devices required for line operation, of the values of the surge impedance and the electrical length of the line proper on different sections, etc. In this case the dependences of costs on these parameters may be regarded as smooth functions, since the values of the electrical length and the surge impedance depend not so much on the conductor cross-section of the transmission line as on the line design, while the parameters of the longitudinal compensation devices are assembled from elements whose capacities are much smaller than the capacity of the entire installation or are smoothly regulated. The purpose of shunt-compensation reactors is to regulate voltage and to raise the level of dynamic stability. The latter condition for ensuring dynamic-stability status in the form of a relation to the parameters of the scheme and the parameters of the mode has not yet been formulated and is at the development stage. The problem may be formulated as determining the parameters of unregulated or regulated transverse compensation devices, for a specified distribution of calculated loads, in order to ensure the required effect of their action on the voltages at the nodal points of the scheme.

As is known, the criterion for optimization calculations in electric power engineering is the calculated costs for facilities stabilized in their development over time. The problem under consideration belongs to facilities of this class. Therefore, calculated costs are used as the optimization criterion. Reliability requirements and the technical operating conditions of the scheme and its elements are used as constraints. The mathematical formulation of certain control problems was presented earlier in articles in this collection. However, for design problems their use has certain specific features. Let us first consider the problem of selecting the parameters of power-transmission elements for deterministic forms of initial data on the value of the calculated load.

Mathematical model for selecting the parameters of series and shunt compensation installations for deterministic values of transmitted powers

Let us consider the scheme in Fig. 1, in which the receiving system \(\Pi_1\) and the transmitting system \(\Pi_2\) are connected by a long-distance power transmission line with one intermediate system balanced in active power. These conditions and schemes may also be different, more complex and special, which does not violate the construction of the mathematical model. The transmitted power \(P_{89}\) is taken as the calculated value. The flow \(P_{11-5}\) is, by condition, taken equal to zero.

Fig. 1. On control of the parameters of series and shunt compensating devices of an AC power transmission line

The variable part of the reduced costs when transmitting the specified power \(P_{89}\) is written as follows:

\[ \sum Z = Z_{y_2}+Z_{\lambda_{23}}+Z_{y_3}+Z_{c_{34}}+Z_{y_4}+Z_{\lambda_{56}}+ Z_{y_6}+Z_{y_7}+Z_{\lambda_{78}}+Z_{y_8} = \sum^{6} Z_y+\sum^{4} Z_\lambda+\sum^{2} Z_c, \]

\[ \begin{aligned} \sum Z={}& Z_{y_2}(y_2,P_{23}^{*})+ Z_{\lambda_{23}}(\lambda_{23},P_{23})+ Z_{y_3}(y_3,P_{23})+ Z_{c_{34}}(x_{c34},P_{34})+\\ &+Z_{y_4}(y_4,P_{34})+ Z_{\lambda_{45}}(\lambda_{45},P_{45})+ Z_{\lambda_{56}}(\lambda_{56},P_{56})+ Z_{y_6}(y_6,P_{60})+\\ &+Z_{c_{67}}(x_{c67},P_{56})+ Z_{y_7}(y_7,P_{67})+ Z_{\lambda_{78}}(\lambda_{78},P_{78})+ Z_{y_8}(y_8,P_{78}). \end{aligned} \tag{2} \]

In the last relation the electrical lengths of the sections are assumed known, although the quantities that determine them may be sought, while the parameters of the compensation-device elements are taken as unknowns \((y_2, y_3, y_4, y_5, y_6, y_7, y_8, x_{c34}, x_{c67})\). Since the powers in each section depend on the operating parameters of the corresponding sections, then

costs prove to be in functional dependence on these operating parameters and circuit parameters.

\[ \sum Z = \sum Z(y_2,\ y_3,\ y_4,\ y_5,\ y_6,\ y_7,\ y_8,\ x_{34},\ x_{67},\ U_2,\ U_3,\ U_4,\ U_5,\ U_6, \]
\[ U_7,\ U_8,\ \delta_2,\ \delta_3,\ \delta_4,\ \delta_5,\ \delta_6,\ \delta_7,\ \delta_8) \to \sum Z(m\Pi)\to \min . \tag{3} \]

The parameters of the elements of the compensating devices (CD) must be chosen in such a way that the reduced costs are the smallest and the required technical operating conditions of the power transmission and of the elements of the terminal and intermediate systems are ensured. These requirements, in the form of constraints on operating parameters, were formulated earlier. It is assumed that the conductances of shunt CD can take any values from \(-\infty\) to \(+\infty\). The parameters of series CD are limited, as is known, by the conditions of static stability and by the values of short-circuit currents; namely: the value of the parameters \(x_{ij}\) must be greater than zero, but less than the modulus of the parameter \(B_{ij}\) of the uncompensated line.

\[ 0 < \pi x_{ij} < \pi |B_{ij}|, \]

where \(\pi\) is an enumeration symbol.

All constraints in the form of two-sided inequalities on circuit parameters and operating parameters can be written in generalized form as

\[ \underline{\pi \Pi_i} \le \pi \Pi_i \le \overline{\pi \Pi_i}; \]

where

\[ \Pi_i \in (E_1;\ E_{10};\ E_{12};\ \delta_{01};\ \delta_{010};\ \delta_{012};\ k_{12};\ k_{8,9};\ k_{11,12};\ U'_1;\ U'_2;\ U'_3; \]
\[ U'_4;\ U'_5;\ U'_6;\ U_7;\ U_8;\ \delta_1;\ \delta_2;\ \delta_3;\ \delta_4;\ \delta_5;\ \delta_6;\ \delta_7;\ \delta_8). \]

Constraints in the form of one-sided inequalities are represented in generalized form as follows:

\[ 0 < \pi \Pi_j \le \overline{\pi \Pi_j}, \tag{6} \]

where

\[ \Pi_j \in (\delta_{E_1}-\delta_{F_{10}};\ \delta_{F_1}-\delta_{F_{12}};\ \delta_{F_{10}}-\delta_{E_{12}};\ x_{34};\ x_{67}). \]

The active-power imbalance equations form a set of 14 equations, and the reactive-power imbalance equations a set of 12 equations:

\[ \pi P_{\mathrm{imb}\, i}(m'\Pi)=0, \]

\[ \pi Q_{\mathrm{imb}\, i}(m'\Pi)=0. \]

where

\[ (m'\Pi)\in (E_1;\ E_{10};\ E_{12};\ \delta_{01};\ \delta_{010};\ \delta_{012};\ k_{12};\ k_{89};\ k_{11,12}; \]
\[ U'_1;\ U'_2;\ U'_3;\ U'_4;\ U'_5;\ U'_6;\ U_7;\ U_8;\ \delta_1;\ \delta_2;\ \delta_3;\ \delta_4;\ \delta_5;\ \delta_6;\ \delta_7;\ \delta_8; \]
\[ y_2;\ y_3;\ y_4;\ y_5;\ y_6;\ y_7;\ y_8;\ x_{34};\ x_{67}). \]

In a known manner, by introducing additional variables, the inequalities are transformed into equalities, after which the Lagrange function can be formed; from it a system of nonlinear algebraic equations with respect to all unknowns is obtained. The total number of all unknowns in determining the CD parameters and the laws of their regulation is 120. For the sets of transmitted active-power operating conditions, the required regulation laws and the powers of shunt and series CD are thus established, us-

proceeding from the principle of a minimum of reduced costs with constraints for different conditions.

Mathematical Model for Selecting the Parameters of KU for Probabilistic Forms of Initial Data on Transmitted Powers

The load of long-distance power transmissions is a random variable with definite probability values. The same may be said of weather conditions, mechanical loads, and states involving disturbances of the operating regime. Other factors participating in the calculation—for example, the standard coefficient of efficiency of capital investments, maintenance costs, and others—are deterministic quantities. Design problems must be solved for combined conditions of specifying initial data in the form of probabilistic and deterministic sets. The objective function is formulated as the mathematical expectation of the reduced costs, i.e., as a probabilistic category of the minimum of costs expected on average [6]. The technical conditions cannot be formulated according to the mathematical expectation, since these conditions must be guaranteed for every operating regime, and not on average and with the probability of these events equal to unity.

The distribution of transmitted powers obeys definite laws. However, the parameters of KU may be determined from the minimum of the objective functional also as constant quantities independent of the load. If the distribution of the values of the thirty-minute maxima over the calculated time interval were the same, then the problem would be solved by the minimum of reduced costs. Since the number of repetitions of the maxima of thirty-minute loads turns out to be different, it is necessary to introduce the mathematical expectation, as the number of repetitions multiplied by the value of each load, and to approximate the parameters of the schemes according to the mathematical expectation. Suppose that, for the scheme in Fig. 1, it is required to determine the parameters of the series and shunt KU, if the probability distribution of the active load \(P_{89}\) for the end of the power transmission is known, \(P_{89}=P_{89}(\rho)\),

\(\rho\) is the probability of the event.

The expected unit costs for all sections and elements of the KU may be determined as follows:

\[ \sum З=\sum_t \sum_m З(m\Pi)\cdot \varphi_t . \tag{9} \]

The constraints in the form of equalities and inequalities (5), (6), (7) may be written in the form of sums, since for each variable they must be satisfied for the whole set of regimes,

\[ \sum_t \pi P_{\delta t}(m\Pi)=0, \tag{10} \]

\[ \sum_t \pi Q_{н\delta t}(m\Pi)=0, \tag{11} \]

\[ \sum_t \pi \underline{\Pi}_i \leq \sum_t \pi \Pi_i \leq \sum_t \pi \overline{\Pi}_i, \tag{12} \]

\[ 0<\sum_t \pi \Pi_j \leq \sum_t \pi \overline{\Pi}_j. \tag{13} \]

The model in stochastic form can be developed for studies of the economic range of voltage regulation, for solving problems of active- and reactive-power distribution in interconnections of this kind. It appears possible to evaluate the efficiency of implementing optimal and other operating modes. Questions concerning the joint operation of 1150-, 750-, and 500-kV power transmissions can be posed and solved, and problems of the development strategy of electric networks can be formulated both with an excess and with a deficiency of initial information.

REFERENCES

1. R. I. Borisov. Selection of optimal parameters of compensating devices for a long bus connected with intermediate systems. Proceedings of the Power Engineering Institute, Siberian Branch, USSR Academy of Sciences, issue 14, Novosibirsk, 1962.

2. R. I. Borisov. Selection of optimal voltage-regulation parameters in intermediate systems connected by a long AC line. Reports of the meeting on problems of electric-power transmission over distances of 2000–3000 km. Novosibirsk, 1960.

3. L. A. Krumm. Gradient method for optimizing operating modes of electric-power systems. Elektrichestvo, 1963, No. 6.

4. G. B. Levental, L. S. Popyrin. Optimization of thermal power installations. Moscow, “Energiya,” 1970.

5. I. A. Fedorova. Elements of analysis and synthesis of electric-power transmission systems using a digital computer. Author’s abstract of a doctoral dissertation. Minsk, 1970.

6. E. V. Tsvetkov. On the methodology for designing hydroelectric power-station power systems. Proceedings of VNIIE, issue XII, 1961.

7. D. Hadley. Nonlinear and Dynamic Programming. Moscow, “Nauka,” 1967.