CONSTRAINT CONDITIONS AND FUNCTIONAL RELATIONSHIPS BETWEEN VARIABLE REGIME PARAMETERS IN MATHEMATICAL PROGRAMMING FOR THE SELECTION OF DEVICES AND OPERATION CONTROL OF LONG-DISTANCE POWER TRANSMISSION LINES AND ELECTRIC POWER SYSTEMS

R. I. BORISOV, S. G. SLYUSARENKO

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

Mathematical models for selecting the parameters of elements and controlling the operating regimes of interconnections of power systems with long-distance alternating-current transmission lines must reflect, as fully and accurately as possible, the picture of phenomena in the real structure, be surveyable, sufficiently detailed, and amenable to investigation by the available means. At the same time, the level of detail must be sufficient to take into account the diversity of all factors that may prove significant, but not excessive, since that would only complicate the solution of the problems. Thus, definite requirements must be satisfied with respect to the connectedness of regime conditions, requirements on equipment parameters, the stability of parallel operation, heating of individual parts, etc. Failure to take these factors into account may prove significant for the viability of the entire system in certain states, and in this sense such detail appears absolutely necessary.

To solve the posed problems of control and parameter selection, objective and boundary conditions are introduced. The optimality criteria in solving one and the same problem may differ; in that case, some boundary conditions may act as objective functionals. Without further developing the question of rational methods for selecting objective functionals and their relationships [1] as applied to the problems under consideration, we shall dwell on the justification of the adopted forms of writing the equations of boundary conditions and functional relationships between the principal and additional variable regime parameters.

1. Constraints on regime parameters imposed by voltage distribution conditions

The distribution of voltages along the length of lines for different regimes has a very substantial influence on the techno-economic performance indicators of long-distance alternating-current power transmission lines. A large number of studies [2, 3, 4] have been carried out on this question, and important conclusions have been drawn, including the need for deeper voltage regulation in order to improve the economic performance indicators of power-transmission operation [4]. The conditions for reliable operation of overvoltage protection devices establish the permissible upper values of volt—

voltages within \((1.1 \div 1.15)\) of the nominal line voltage [5]. The lower limits of voltage values will be determined by the control ranges of the transformers and by the available reactive power. For this reason, the lower limits of voltage values are not fixed in this work, but are assumed to be specified.

To solve the posed problems, it is necessary to determine within what limits of variation of the transmitted active and reactive powers and the electrical parameters of power transmission lines the adopted voltage conditions prove to be ensured.

Fig. 1. On the distribution of voltage along the length of a long-distance power transmission line

If a lossless line section (Fig. 1) has electrical length \(\Lambda\), then the distribution of voltage along the length \(\lambda\), for specified voltage values at the ends \((U_1\) and \(U_2)\), is written as follows:

\[ |U_{\lambda}| = \left( \left( U_2 \cos \lambda + U_1 \sin \lambda \csc \Lambda \cos \delta - U_2 \sin \lambda \operatorname{ctg} \Lambda \right)^2 + \left( U_1 \sin \lambda \csc \Lambda \sin \delta \right)^2 \right)^{1/2}, \tag{1} \]

where \(\delta\) is the displacement angle of the voltage vectors at the ends.

To determine the location and magnitude of the greatest value \(U_{\lambda m}\), the derivative \(U(\lambda)\) is set equal to zero:

\[ \frac{dU_{\lambda}}{d\lambda}(\lambda, U_1, U_2, \Lambda)=0 \]

or

\[ \begin{aligned} &\left( U_2 \cos \lambda + U_1 \sin \lambda \csc \Lambda \cos \delta - U_2 \sin \lambda \operatorname{ctg} \Lambda \right) \left( - U_2 \sin \lambda \right.\\ &\left. \quad + U_1 \csc \Lambda \cos \delta \cos \lambda - U_2 \cos \lambda \operatorname{ctg} \Lambda \right) + \left( U_1 \sin \lambda \csc \Lambda \sin \delta \right)^2 \cos \lambda =0, \end{aligned} \tag{2} \]

whence

\[ \operatorname{tg} 2\lambda_m = \frac{ -2\left(U_1 U_2 \csc \Lambda \cos \delta - U_2^2 \operatorname{ctg} \Lambda\right) }{ - U_2^2 + U_1^2 \csc^2 \Lambda - 2 U_1 U_2 \csc \Lambda \operatorname{ctg} \Lambda \cos \delta + U_2^2 \operatorname{ctg}^2 \Lambda }. \tag{3} \]

The resulting equation (3) is solved with respect to \(\lambda_m\) numerically for power transmission lines of different electrical length \((\Lambda = 15, 30, 45, 60^\circ)\) and over a certain range of variation of the angle \((\delta = 10, 20, 30, 40, 50^\circ)\). The conditions of static stability were not taken into account here, since they, like all boundary conditions, are determined independently. The values \(U_{\lambda m}\) are obtained if the values found from formula (3) are substituted into equation (1). The calculation results make it possible to judge that the voltage values at the extremal point are subject to considerable fluctuations from load changes and lie within \((0.1—1.4)\) when the voltages at the boundaries vary within \((0.9—1.05)\). Thus, for sections with electrical length \(15 \le \Lambda \le 30^\circ\), the extrema are located outside the limits of the sections and the value \(U_{\lambda m}\) does not exceed 1.411.

\[ (\Lambda = 30^\circ;\quad \delta = 20^\circ;\quad \lambda_m = -38.48^\circ;\quad U_{m\lambda}=1.333;\quad U_1=0.95; \]

\[ U_2=1.05). \]

For sections \(\Lambda = 45^\circ\), the extrema are located within the boundaries of the section, with the greatest value \(U_{\lambda m}=1.132\) \((U_1=1.05;\ U_2=1.05;\ \delta=10^\circ;\ \lambda_m=24.136)\). For sections \(\Lambda = 60^\circ\), the extremal points are again located outside the boundaries of the section, which guarantees the voltage distribution in the region of boundary values. Fixing different voltage values at the boundaries of sections of different

length makes it possible to control the distribution of voltages in power transmission systems for different operating conditions.

2. Constraint Conditions and Functional Relationships Between the Variable Parameters of Synchronous Turbogenerators with Respect to Heating of the Stator and Rotor for Different Operating Conditions

For organizing calculations for the control of operating conditions and the selection of device parameters for long-distance AC power transmissions with intermediate systems, it is necessary to express the internal parameters of synchronous turbogenerators (i.e., the values of the magnitude and argument of the EMF behind the synchronous reactances relative to the voltage at the terminals) in terms of the magnitude of the terminal voltage and the delivered powers. This makes it possible to eliminate these variables from the equations of small oscillations when forming constraints on static stability and from the equations of the functional. In another form of constructing the solutions to these problems, the operation of elimination is not supposed to be performed, but the values of the EMF and the internal angles are expressed in the form of two-sided inequalities. Therefore, upper and lower limiting values of the EMF and the internal angles must be established.

As is known, the values of the active and reactive powers delivered by generators are expressed in an elementary way in terms of the EMF value, the internal angle, the reactance, and the terminal voltage. The internal parameters in the form of the EMF value and the actual angle \(\delta\) cannot vary independently. They are limited by heating of the stator and rotor. When generators operate with an inductive load, the action of automatic excitation regulators under a steady-state change in load ensures maintenance of the voltage at the generator terminals with a certain value of the power droop coefficient \((B_p)\). The value of the coefficient \((B_p)\) varies within the range from 1 to 5% and depends on the setting and design of the regulator [6].

\[ U = U_0(1 - B_p P), \tag{4} \]

where \(U_0\) is the no-load voltage at the generator terminals.

This relation makes it possible to determine the voltage at the generator terminals for different values of the coefficient \((B_p)\) and variations of the delivered active power (when operating into an inductive load), taken within the range from \(1.0\) to \(0.3P_{\text{nom}}\). The corresponding limiting values of reactive powers when operating with an inductive load can be determined from overload capability curves with allowance for heating [6]. Then the greatest values of the EMF magnitude for different values of the limiting load and different values of the coefficient \(B_p\) are found from the known formula:

\[ E = \left(\left(U + \frac{QX}{U}\right)^2 + \left(\frac{PX}{U}\right)^2\right)^{1/2}, \]

where \(X\) is the synchronous reactance.

The dependences \(E_{rp} = E_{rp}(P)\) and \(\delta_{rp} = \delta_{rp}(P)\) for different values of the coefficient \((B_p)\) are presented in Fig. 2. The operating values of powers and EMF in service may be smaller than the limiting ones, since the reactive load of generators is determined by the operating condition of consumers and other sources of reactive power. The EMF values of the regulated generator are approximated by a linear three-term function.

on the delivered powers and voltages at the terminals in the region of nine adjacent values of \(U, P\), and \(Q\):

\[ E = v_E U + \tau_E P + q_E P . \tag{5} \]

Fig. 2. Dependences of the internal e.m.f. and generator angles on \(P\) for various power factors

The coefficients of the linear form (5) were determined by the least-squares method [7] and computed from the formulas:

\[ v_E \sum U_i^2 + \tau_E \sum P_i U_i + q_E \sum Q_i U_i = \sum E_i U_i, \]

\[ v_E \sum Q_i P_i + \tau_E \sum P_i^2 + q_E \sum Q_i P_i = \sum E_i P_i, \tag{6} \]

\[ v_E \sum U_i Q_i + \tau_E \sum P_i Q_i + q_E \sum Q_i^2 = \sum E_i Q_i . \]

The results of calculating the coefficients \(v_E\), \(\tau_E\), and \(q_E\) (for \(B_p = 0.03\)) are presented in Table 1. The mean value of the approximation error is less than 3%. The values of the e.m.f. lie within the limits:

\[ 1.66 = \underline{E} \leq E \leq \overline{E} = 3.25 . \tag{7} \]

In a known manner, the set of values of the intrinsic angles of generating stations can be found from the voltage and delivered powers:

\[ \delta = \operatorname{arctg} \frac{\dfrac{PX}{U}} {U + \dfrac{QX}{U}} . \tag{8} \]

The limiting values of the intrinsic angles do not depend on the droop coefficient of the regulators and may vary within the following limits:

\[ 0.18=\underline{\delta}\leq \delta \leq \overline{\delta}=0.92. \tag{9} \]

For sources of infinite power, the heating constraints are removed, and the values of the internal angles and emf may be greater than the limiting values indicated in formulas (7) and (9).

Table

When long-distance power transmission lines operate under low-load conditions, a reactive-power imbalance may arise in individual parts or in an interconnection of power systems, which will lead to an increase in voltage above the maximum permissible values. To reduce or eliminate this impermissible phenomenon, along with the installation and regulation of shunt compensating devices, use may be made of the capability of large turbogenerators to operate in the underexcitation region [8]. As is known, this entails the danger of local overheating of the end packets of the frontal parts of the stator and deterioration of the conditions of static stability [9]. However, as indicated in [8], generator operation in the underexcitation zone may be practiced over a prolonged period. In such regimes, heating of the end packets of the frontal parts must remain within permissible limits. The operating conditions with respect to static stability in the underexcitation region prove to be more severe, since this corresponds to larger values of the internal angles than when operating with an inductive load. It becomes necessary to reduce the active load. This, however, corresponds to the conditions under which generators find themselves in such regimes. In addition, the conditions with respect to static stability for solving the posed problems are formed independently for both the oscillatory and aperiodic forms of instability of the regimes sought. Therefore, the limiting load of generators with respect to heating when operating in the underexcitation region, without taking into account the limit with respect to static stability and self-excitation, may be determined from the apparent power [9] and the rated voltage. This makes it possible to determine the limiting values of the internal emf by magnitude and argument and to approximate the emf, but naturally with other

with the values of the coefficients \((\pi)\) and \((q)\). The values of the coefficients \((\pi)\) and \((q)\) are also calculated by the method of least squares.

The limiting values of the magnitude and angle of the e.m.f. determine the region of their admissible operating values when operating in the underexcited region:

\[ 0.832 = \underline{E}' \leq E \leq \overline{E} = 2.091 \tag{10} \]

\[ 0.024 = \underline{\delta} \leq \delta \leq \overline{\delta} = 1.34 \tag{11} \]

3. Constraint conditions on the level of short-circuit currents with respect to the parameters of the reactances of the connections of long-distance power transmissions with intermediate and receiving systems

The equations of network powers can be written for the state of a network of practically any known configuration and complexity, including for elements and their aggregates in intermediate and receiving systems. To solve the problems posed, the intermediate and terminal power systems must be represented not by partial circuits, but in a sufficiently generalized form. Such a form can be obtained from the constraint conditions on three-phase short-circuit currents in the system under consideration. The requirements for limiting short-circuit currents in terminal and intermediate systems are one of the requirements for the reliability of electric power supply [10]. They may be regarded as an additional component of the reduced costs, since their implementation in systems means the introduction of a whole series of measures associated with expenditures of material values. These requirements for constraints on short-circuit currents can be implemented by various means and methods. To compare their effectiveness, the reduced costs are determined. On the other hand, if the problems of choosing means for limiting short-circuit currents are not posed in the calculations, then one need not carry out their economic equivalencing, and the constraint requirements may be regarded as technical provisions in the form of definite normative conditions. In this form, the requirements for constraints on short-circuit currents reduce, for a given source capacity, to establishing definite values of the separating reactances between the generators of the system and the short-circuit point. Thus, for the scheme in Fig. 2 from paper [11], included in this collection, in the case of a three-phase short circuit at nodes 6 and 7, the value of the supertransient current and the calculated equivalent reactance of the external part of the scheme with respect to the system under consideration and its own supply sources can be established as

\[ I_*'' = \frac{S_{\text{kz}}^{\mathrm{II}}}{S_{\text{ne}}} \;\to\; X_{p*(\text{n})} \quad \text{(from the curves of variation of short-circuit current [12])}, \]

\[ X_{p*}(\text{b}) = X_{p*(\text{n})}^{\mathrm{II}} \frac{S_{\text{b}}}{S_{\text{ne}}}, \]

where \(S_{\text{ne}}\) is the total rated power of all sources,
\(S_{\text{b}}\) is the base power.

Assuming that the relative parameters of the elements of the external and internal parts are identical, the resulting reactance \(X_{p*(\text{b})}\) can be divided inversely proportionally to the powers of the external and own supply sources:

\[ X_{(\text{b})}^{\mathrm{II}} = \frac{ X_{\text{c}}^{\mathrm{II}} X_{\text{vn}}^{\mathrm{II}} }{ X_{\text{c}}^{\mathrm{II}} + X_{\text{vn}}^{\mathrm{II}} }, \tag{12} \]

\[ X_c^{\prime\prime}=X_{c(n)}^{\prime\prime}\frac{S_b}{S_{pc}}, \tag{13} \]

\[ X_{vn}^{\prime\prime}=X_{vn(n)}^{\prime\prime}\frac{S_b}{S_{vn(n)}}. \tag{14} \]

Assuming \(X_{vn(n)}^{\prime\prime}=X_{c(n)}^{\prime\prime}\), we obtain

\[ S_c X_c^{\prime\prime}=X_{vn}^{\prime\prime}S_{vn}. \tag{15} \]

Solving equations (12) and (15) with respect to \(X_c^{\prime\prime}\), we find the reactance that must be assigned to the given system:

\[ X_c^{\prime\prime}=X_b^{\prime\prime}\left(1+\frac{S_{vn}}{S_c}\right). \tag{16} \]

The system reactance \(X_c^{\prime\prime}\), in turn, is divided into two series-connected parts, \(X_{g(n)}^{\prime\prime}\) and \(X_c(\Delta)\). Since the number of generators in the receiving system is assumed known for each calculated regime, it becomes possible to determine the resulting reactance attributable to the generators,

\[ X_{g(n)}^{\prime\prime}=\frac{X_g^{\prime\prime}}{n}. \]

The remaining reactance \(X_c(\Delta)\) is found from the formula for series addition,

\[ X_c(\Delta)=X_c^{\prime\prime}-X_g^{\prime\prime}(n). \]

For calculations of normal regimes, the value \(X_{g(n)}^{\prime\prime}\) is replaced by \(X_{g(n)}\), according to the synchronous reactance of the generators. Calculations of the parameters \(X_{(\Delta)}\) for intermediate and receiving systems are performed in a special subprogram.

References

1. D. A. Arzamastsev, A. L. Myzin, Yu. S. Sklyarov. Problems of Optimal Development of a Power System. Textbook. Sverdlovsk, 1969.

2. V. A. Venikov, D. V. Kholmskii. On the operation of long-distance power transmission lines in regimes of optimal voltages. Scientific and Technical Bulletin of Higher Educational Institutions, “Energetika,” 1959, No. 3.

3. V. A. Venikov, I. P. Silda. Calculations of Regimes of Long-Distance AC Power Transmission Lines. “Vysshaya shkola,” 1966.

4. V. A. Venikov. On the influence of deep voltage regulation of long-distance extra-high-voltage power transmission lines on the techno-economic indicators of their operation. Scientific and Technical Bulletin of Higher Educational Institutions, “Energetika,” 1958, No. 3.

5. High-Voltage Engineering. Under the general editorship of D. V. Razevig. Moscow–Leningrad, “Energiya,” 1964.

6. I. D. Markovich. Regimes of Power Systems. Moscow, “Energiya,” 1969.

7. B. M. Shchigulev. Mathematical Processing of Observations. Moscow, “Nauka,” 1969.

8. W. Y. Balet and R. L. Webb. Reactive Power and Its Control in a Large Metropolitan Electric Supply System. PAS-87, pp. 49–53.

9. N. A. Polyakov. Modern Large Two-Shaft Turbogenerators. Moscow, “Energiya,” 1972.

10. R. I. Borisov, V. D. Kozyrev. On estimating the appropriate value of short-circuit currents. “Elektrichestvo,” 1968, No. 8.

11. R. I. Borisov. Control of operating regimes with respect to reactive power in interconnected power systems with long-distance AC power transmission lines. Article in this collection.

12. S. A. Ulyanov. Electromagnetic Transient Processes in Electrical Systems. Moscow, “Energiya,” 1964.