Equivalent Modeling of Intermediate Systems Along the Route of Long AC Power Transmission Lines Based on Operating Parameters
Abstract Generated abstract
This paper develops an approach for representing intermediate power systems connected along long-distance AC transmission lines by generalized static characteristics derived from operating parameters. It formulates these characteristics as dependencies of active and reactive exchange power on voltage at the connection bus, under assumptions of unchanged regulator settings or constant generator active power, and gives analytical procedures using generator, compensator, and load models. Calculations for a representative intermediate system indicate that the active-power characteristic is governed mainly by load active power and its voltage regulation effect, while the reactive-power characteristic is influenced chiefly by generator-bus voltage stability and coupling impedances. The study also suggests that series compensation in synchronous compensator circuits may improve the regulating capability of intermediate systems, subject to technical and economic evaluation.
Full Text
EQUIVALENCING OF INTERMEDIATE SYSTEMS ALONG LONG-DISTANCE AC POWER TRANSMISSION LINES BY OPERATING PARAMETERS
R. I. BORISOV, V. I. GOTMAN
(Presented by the Department of Electrical Networks and Systems)
When solving operating-regime problems for long-distance AC power transmission lines with intermediate systems, the latter can most simply and rigorously be represented by static characteristics.
By the static characteristics of a bounded part of a power system, or of any of its elements, with respect to a certain calculation point, we shall understand dependencies of the parameters describing the response of this part of the system to a stationary disturbance applied at the calculation point, the source of which is outside the part of the system under consideration. In this, the composition of the equipment, regulator settings, transformer transformation ratios, and also the network diagram in the part of the system under consideration must remain unchanged.
In particular, by the generalized static characteristics of an intermediate system we shall understand the dependence of the parameters \((\Pi_i^l)\) of the regime of branch \(l\) (Fig. 1, a), determined at the point where this branch of the intermediate system is connected to the long-distance transmission line, on the regime parameters at the nodal calculation point \(C\) itself \((\Pi_j^C, \Pi_k^C)\), under the general conditions specified above, i.e., functional dependencies of the form
\[ \Pi_i^l = F(\Pi_j^C, \Pi_k^C). \]
In this case, one of the independent arguments is usually fixed as a parameter.
The active and reactive powers of section \(l\) (Fig. 1, a) are functions of two parameters: the voltage magnitude and the mismatch angle, if \(U_i(E_q, E_q)=\mathrm{const}\) is assumed,
\[ \begin{aligned} P_{\text{eq}} &= P_l = \varphi(U_C,\delta),\\ Q_l &= \Psi(U_C,\delta). \end{aligned} \tag{1} \]
If, in the part of the system for which the static characteristics are determined, the generating sources are equipped only with static frequency regulators, then the condition of unchanged regulator settings of the sources may be replaced by the condition that the active powers of these sources, or the frequency, remain unchanged. In this case the angle \(\delta\) is no longer an independent argument, and expressions (1) are written as*
* Here and below, the active resistances of the elements, as well as active losses, are not taken into account.
\[ \begin{aligned} P_l &= \varphi'(U_c,\delta)\big|_{P_{\mathrm{g}}=\mathrm{const}}=\mathrm{const},\\ Q_l &= \Psi'(U_c,\delta)\big|_{P_{\mathrm{g}}=\mathrm{const}}. \end{aligned} \tag{2} \]
If the generating sources have static frequency regulators, the frequency, along with the power \(P_{\mathrm{g}}\), may be regarded as an additional fixed parameter of the static characteristics.
The power of the load bus of the intermediate system, at constant frequency, is only a function of voltage
\[ P_{\mathrm{load}}(U),\quad Q_{\mathrm{load}}(U). \]
In this case the dependences (2) are written as:
\[ \begin{aligned} P_l &= \varphi(U_c),\\ Q_l &= \Psi(U_c). \end{aligned} \qquad (P_{\mathrm{g}}=\mathrm{const}) \tag{3} \]
The expressions given are valid for an intermediate system of arbitrary structure and configuration.
Thus, under the adopted assumptions concerning the invariability of regulator settings, or, equivalently, the invariability of the active power of the supply sources, the functional dependences (3), while uniquely describing the operating condition of the intermediate system (IS), determine the values of the exchange power flows through which the IS manifests itself when the voltage changes at the connection bus to the long-distance transmission line; i.e., they are the response of the intermediate system when the operating condition in the long-distance transmission line changes.
Fig. 1. Diagram of an intermediate system
In practice, in a number of cases the conditions for which the given static characteristic remains valid need be observed only in the “zone of influence” of the operating point, which may sometimes be limited to the section of the system immediately adjacent to this point. The invariability of the composition of consumers’ equipment should be understood in the sense of invariability of their total averaged characteristics in the operating condition for which the given static characteristic was determined.
Static characteristics may be obtained experimentally or calculated analytically. In an analytical calculation the static characteristics of generators and load, as well as some initial operating condition, must be known. The point of connection of the IS to the long-distance transmission line should be taken as the balancing bus.
The synchronous longitudinal emf of the generators of the IS, equipped with proportional-action automatic excitation regulators, may be determined through the voltage of the \(i\)-th bus by the expression [2]
\[ E_q= \frac{ E_{q0}+K_{0U}\dfrac{x_d}{x_{d\Sigma}}(F-\cos\delta) }{ 1+K_{0U}\dfrac{x_{\mathrm{vn}}}{x_{d\Sigma}} }, \tag{4} \]
where
\[ F=\sqrt{\left(\frac{x_{d\Sigma}}{x_d}\cdot\frac{U_{r0}}{U_i}\right)^2-\left(\frac{x_q x_{d\Sigma}}{x_d x_{q\Sigma}}\right)^2\sin^2\delta}; \]
\(E_{q0}\) is the value of the synchronous emf in the initial operating condition;
\(K_{0U}\) is the voltage gain factor;
\(x_{\mathrm{ex}}\) is the coupling resistance between the generator terminals and the \(i\)-th node,
\(U_{r0}\) is the AVR voltage setting;
\[ x_{d\Sigma}=x_d+x_{\mathrm{ex}}, \qquad x_{q\Sigma}=x_q+x_{\mathrm{ex}}. \]
An additional relation between the quantities \(E_q\) and \(\delta\) is given by the electromagnetic-power equation
\[ P_t=\frac{E_q U_i}{x_{d\Sigma}}\sin\delta+ \frac{U_i^2}{2}\frac{x_d-x_q}{x_{d\Sigma}x_{q\Sigma}}\sin 2\delta . \tag{5} \]
The emf of synchronous compensators equipped with proportional-action AVR can also be calculated from expressions (4) and (5). Neglecting the active power of the compensators, these expressions reduce to
\[ E_q= \frac{ E_{q0}+K_{0U}\left(\dfrac{U_0}{U_j}-\dfrac{x_d}{x_{d\Sigma}}\right) }{ 1+K_{0U}\dfrac{x_{\mathrm{ex}}}{x_{d\Sigma}} }. \tag{6} \]
The static characteristics of the load in the intermediate system (Fig. 1, a) are approximated, with sufficient accuracy for calculations, by linear dependences [3]
\[ P_{\mathrm{load}}=P_{\mathrm{nom}}\left[1+(U_i-U_{i\mathrm{nom}})\frac{dP}{dU}\right], \]
\[ Q_{\mathrm{load}}=Q_{\mathrm{nom}}\left[1+(U_i-U_{i\mathrm{nom}})\frac{dQ}{dU}\right], \tag{7} \]
where \(U_{i\mathrm{nom}}, U_i\) are, respectively, the nominal and current voltages of the \(i\)-th node.
Let us consider the procedure for calculating the static characteristics as applied to the diagram of the intermediate system shown in Fig. 1, b.
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Specify a voltage value \(U_i\) different from the value in the initial operating condition. Using (7), calculate the new values of the load powers. Solve equations (4) and (5) jointly with respect to \(E_q\) and \(\delta\).
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From the found value of \(E_q\), corresponding to \(U_i\) and \(P_t\), determine the balancing reactive power
\[ Q=-\frac{U_i^2}{x_{d\Sigma}}+ \sqrt{\left(\frac{E_q U_i}{x_{d\Sigma}}\right)^2-P_t^2}. \tag{8} \]
- From the known parameters of the \(i\)-th node, the voltage magnitude of the adjacent \(j\)-th node can be determined as
\[ U_j=\frac{1}{U_i}\sqrt{U_i^2\left(U_i^2-2Q_i x_{ij}\right)+\left(P_i^2+Q_i^2\right)x_{ij}^2}. \tag{9} \]
The reactive power \(Q_j\) is determined from an expression analogous to (8), with the corresponding quantities substituted into it. The active power of the sections is determined from the level of balance in the nodes. Similar
calculations are carried out until the required quantities have been determined and are repeated many times.
We performed calculations of static characteristics for the intermediate system shown in Fig. 1,b*. In the initial operating condition, the natural power flowed along the long-distance transmission line, and the intermediate system was considered self-balanced with respect to both active and reactive power. As a constraint, the overload of generators and synchronous compensators with respect to rotor current was taken as 3.5. The calculations determined the influence of regulation of generators (and synchronous compensators), the load power, the load $\cos \varphi$, the regulating effects of the active and reactive load powers, the magnitudes of the coupling impedances of the nodes, and the initial operating condition on the generalized static characteristics of the intermediate system. The results obtained make it possible to draw certain conclusions that are general for any intermediate system.
Thus, it was found that the slope coefficient of the generalized static characteristics of an intermediate system with respect to active power is determined mainly by the magnitude of the active power of the loads and their regulating effects. It decreases as the regulating effects and the active power of the loads decrease and as their electrical distance from the node connecting the intermediate system to the long-distance transmission line increases.
Of all the variety of factors that were taken into account in this work, the most significant influence on the slope coefficient of the generalized static characteristics with respect to reactive power is exerted by the voltage stability on the generator buses and by the magnitudes of the coupling impedances of the intermediate-system nodes. With an increase in $K_{\omega}$, the coefficient of the static characteristics increases and reaches its limit (other conditions being equal) when the voltage at the generator terminals is maintained unchanged. For the adopted parameters of the circuit (Fig. 1,b), the value
\[ K_{\omega}=50\ \frac{\text{excitation units}}{\text{voltage units}} \]
corresponds to $U_{\mathrm{g}}=\mathrm{const}$. A change in the load $\cos \psi$ does not affect the static characteristics. Two initial operating conditions of the intermediate system with respect to reactive power have identical static characteristics, shifted relative to each other along the ordinate axis. The magnitude of the regulating effect of the load has an insignificant influence (Fig. 2a, $1 — dQ/dU = 2$, $2 — dQ/dU = 4$).
The second factor exerting a significant influence on the coefficient of the static characteristics is the magnitude of the coupling impedances of the intermediate-system nodes. Fig. 2b presents static characteristics calculated for different values of the coupling impedances: in the first case $x_{c1}=0$, in the second $x_{c1}=0.142$; in this case, over a considerable interval of voltage variation, the characteristic is rectilinear. The curvature of the characteristics is associated with the nonlinearity of losses, the share of which decreases significantly as the impedances decrease.
The voltage stability of a long-distance transmission line can be increased to a considerable degree by synchronous compensators installed on the low-voltage side of the coupling transformers. However, realization of the capacity of synchronous compensators, as well as of the reserve capacity of the power-supply sources, is hindered by the considerable value of the synchronous reactance of the compensators and of the coupling transformer. This difficulty can be eliminated by switching capacitors into the circuit of the synchronous compensator. This is illustrated in Fig. 2c (No. 1 $x_{c1}=0$; No. 2 $x_{c1}=0.104$, $x_{c2}=0.886$).
* The calculations were carried out in relative units at $U_b=230\ \text{kV}$, $S_b=S_{\text{nat}}(220)=137\ \text{MVA}$, $Z_b=386\ \Omega$.
Fig. 2a. Generalized static characteristics of the intermediate system
Fig. 2b
Series compensation of the inductance of synchronous compensators may prove to be an effective means of increasing the regulating capabilities of intermediate systems. However, this requires its own technical and economic justification.
[Figure not reproduced]
Fig. 2в
References
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I. D. Anisimova et al. Methodology for Calculating the Stability of Automated Electric Power Systems. Vysshaya shkola, Moscow, 1966.
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G. E. Pospelov, V. T. Fedin. “The Influence of Static Load Characteristics on the Ratings of Compensating Devices for Voltage Regulation in Electric Networks.” In: Proceedings: Voltage Regulation in Electric Networks. Energiya, Moscow, 1968.