POSSIBILITIES OF APPLYING LYAPUNOV FUNCTIONS AS A CRITERION FOR THE EQUIVALENCE OF MATHEMATICAL MODELS OF ELECTRIC POWER SYSTEMS

R. I. BORISOV, N. E. CHERNYI

(Presented by the scientific seminar of the Department of Electrical Systems and Networks)

Studies of the dynamic stability of complex electric power systems (EPS) are associated with performing mass calculations under various initial conditions and types of disturbances. This circumstance must be taken into account when choosing a method of analysis, as well as when developing algorithms and operating programs.

One of the possible approaches to solving this problem is based on the application of Lyapunov’s direct method. In studies of electrical systems (ES), an energy approach is widely used for constructing a Lyapunov function, in which the function is defined as the sum of the kinetic (K) and potential (P) energies of the system in its disturbed motion [1].

With this approach, while making it possible to assess the stability of an ES under the action of various disturbances, the Lyapunov-function method can be used to solve other questions connected with the study of operating modes of ES and their control.

Let us consider an ES for which the system of differential equations of disturbed motion, in the positional idealization, has the form [2]

\[ \left. \begin{aligned} \frac{d\Delta\delta_i}{dt} &= s_i;\\ J_i\frac{ds_i}{dt} &= F(\delta); \end{aligned} \right\} \tag{1} \]

where \(s_i\) is the slip of the rotor of a synchronous generator;
\(J_i\) is the inertia constant of a synchronous generator;
\(\delta\) is the column vector of phase coordinates \(\delta_1, \delta_2, \ldots, \delta_n\);
\(F(\delta)\) is a vector function.

For such systems, a Lyapunov function [2] is known which, for the case when infinite-power buses are absent, is written in the following form:

\[ V=\frac{1}{2}\sum_{i=1}^{n}J_i s_i^2 - \frac{\left(\sum_{i=1}^{n}J_i s_i\right)^2}{2\sum_{i=1}^{n}J_i} + \]

\[ +\sum_{i<j} E_i E_j y_{ij}\cos \alpha_{ij}\left[\cos \delta_{*ij}(1-\cos \Delta\delta_{ij})-\sin \delta_{*ij}(\Delta\delta_{ij}-\sin \Delta\delta_{ij})\right]. \tag{2} \]

The fact of stability of the EPS in its dynamic transition is established by the Lyapunov criterion

\[ V_0 \leq V_{kp}, \tag{3} \]

where \(V_0\) is the initial disturbance of the post-fault condition;
\(V_{kp}\) is the critical value of the Lyapunov function in the steady-state post-fault condition.

The initial disturbance of the post-fault condition is determined by numerical integration of the differential equations of motion of the generator rotors and by finding the value of the Lyapunov function for the obtained independent variables—\(\delta_1, \delta_2, \ldots, \delta_n\).

At each step of the numerical integration, by solving the system of algebraic equations, the steady-state condition is calculated, for which the condition must be satisfied

\[ S(X)=0, \tag{4} \]

where \(X\) is the vector of parameters of the EPS condition—\(x_1, x_2, \ldots, x_m\).

Since the network powers in the system, flowing to each node, depend on the voltages of these nodes and the electrical angles between them, the nodal state equations together with the active- and reactive-power balance equations will determine the condition parameters [3]:

\[ \left. \begin{aligned} \sum_{i\ne j}^{n} P_i\left(|U_i|,|U_j|,\delta_{ij}\right)&=0;\\ \sum_{i\ne j}^{n} Q_i\left(|U_i|,|U_j|,\delta_{ij}\right)&=0; \end{aligned} \right\} \tag{5} \]

where \(|U_i|\), \(|U_j|\) are the magnitudes of the voltages of the selected and adjacent nodes;
\(\delta_{ij}\) is the electrical angle between them.

In [4] it is proposed to reduce the system of differential equations of the form (1) to normal form for each equivalent station separately and for the network section associated with it up to the point of connection to the system. As a result, we obtain a new system of differential equations of disturbed motion of the form:

\[ \left. \begin{aligned} \frac{d\Delta\delta_i}{dt}&=s_i;\\ J_i\frac{ds_i}{dt}&=F_i(E_i,U_i,\theta_i,t); \end{aligned} \right\} \tag{6} \]

where \(U_i\) is the voltage at the node where the equivalent generator is connected to the network;

\(\theta_i=\delta_i-\delta_{ui}\) is the electrical angle between the vectors of the emf and voltage.

It is assumed that over each small, computed time interval the voltage vector at the point of the station branches to the remaining network remains unchanged and independent of changes in the operating condition of the power stations.

Let us determine the value of the voltage vector at the beginning of the \(K\)-th interval for the \(i\)-th connection node:

\[ \dot U_i^{(k)} = \frac{\dot E_i \dot Y_{ii}+\displaystyle\sum_{\substack{j=1\\ i\ne j}}^{n}\dot E_j \dot Y_{ij}} {\dot Y_{ii}+\displaystyle\sum_{\substack{j=1\\ i\ne j}}^{n}\dot Y_{ij}}, \tag{7} \]

where \(\dot Y_{ii}\) is the self-admittance of the generator branch;
\(\dot Y_{ij}\) is the mutual admittance between the connection node and the e.m.f. of the other generators.

Owing to the invariance of the calculated e.m.f.’s in the adopted mathematical model of the EPS, the value of the voltage vector can be written as a function of the phase coordinates \(\delta\) for any instant of time:

\[ \dot U_i = F_1 \cdot (\delta_1,\delta_2,\ldots,\delta_i,\ldots,\delta_n). \tag{8} \]

Thus, in the right-hand side of system (6) there are written vector-functions corresponding to the vector-functions in the right-hand side of system (1):

\[ F(\delta)\equiv F_1(\delta). \tag{9} \]

On the basis of the above reasoning, at each calculation time interval of the transient process the equivalent circuit of a complex EPS can be conditionally reduced to \(n\) simplest circuits—“generator–constant-voltage bus”—with the parameters of the end of the preceding interval. For each of the isolated circuits one can construct a Lyapunov function according to [5]:

\[ V_i^{(k)}=\frac{1}{2}J_i s_1^2 -\frac{E_i U_i}{x_i}\left[\sin\theta_{*i0}(\Delta\theta_i-\sin\Delta\theta_i)+ \right. \]

\[ \left. +\cos\theta_{*i0}(1-\cos\Delta\theta_i)\right], \tag{10} \]

where \(\Theta_{*i0}\) is the electrical angle between the e.m.f. and voltage vectors at the terminals of the \(i\)-th generator in the steady-state, post-fault condition;

\[ \Delta\theta_i=\theta_i^{(k)}\theta_{*i0} \]

is the deviation of the angle \(\theta\) by the end of the calculation interval.

The Lyapunov function for the entire system can be written as the sum of the functions of the individual generators:

\[ V_{\Sigma}^{(k)}=\sum_{i=1}^{n} V_i^{(k)}. \tag{11} \]

In the case where there are no infinite-power buses in the EPS, from expression (11) it is necessary to subtract the term that accounts for the kinetic energy of the weighted-average motion:

\[ V^{(k)}=V_{\Sigma}^{(k)}-K_{\mathrm{cp}}^{(k)}. \tag{12} \]

The proposed algorithm for calculating initial disturbances has been implemented as an autonomous block, which is interfaced with the program for calculating dynamic stability on the M-220 computer of the Siberian Power Engineering Institute, Siberian Branch of the Academy of Sciences of the USSR. The results of calculating a test example for an EPS including 5 synchronous generators are given in Fig. 1; the curves of variation of the relative

tive angles are shown in Fig. 2. The computation time for the control example is 1.5 min. As the number of active nodes increases, the computational time of the problem on the computer increases and, no less importantly, so do the labor costs involved in analyzing the results. Thus, for example, the computation time for the electromechanical process in an EPS consisting of 44 generators is 40–50 min per 1 sec of the process. In addition, the investigator is often

Fig. 1. Curves of the change in the Lyapunov function over time for different types of disturbance.

Fig. 2. Curves of the change in the relative rotor angles of characteristic generators over time.

interested not in the nature of the transient process in all elements, but in the fact of stability or instability of the EPS under the action of a particular disturbance. Attempts aimed at simplifying mathematical models of complex EPSs are therefore natural.

In [6, 7] certain possibilities of applying Lyapunov functions in EPS modeling are considered. It is shown that the shape of the hypersurface representing the Lyapunov function determines the characteristics of the corresponding system.

Let us show the possibilities of applying function (12) for the equivalent representation of individual subsystems. To this end, let us consider a particular case—

case when the subsystem being equivalenced has one node of connection to the nontransformable subsystem, Fig. 3. In design and operating practice, such a problem occurs both in cases of connecting new electric power systems and in cases where a remote power plant operates into a receiving system.

Fig. 3. Structural diagram of the electrical system under study:
1 — nontransformable subsystem,
2 — transformable subsystem.

A similar problem was solved in [8], where invariance of the kinetic energy of rotor motion in the original and equivalent systems was taken as the generalized equivalencing criterion. Function (12) makes it possible to take into account the invariance of the total energy of the entire system. In this case the equivalencing criterion may be formulated as follows:

\[ V^{(k)} = V_{\mathrm{e}}^{(k)}, \tag{13} \]

In addition to (13), the condition

\[ V_i^{(k)} = V_{i\mathrm{e}}^{(k)}, \tag{14} \]

must be satisfied, where \(V_i^{(k)}\) is the value of the function for the original subsystem;
\(V_{i\mathrm{e}}^{(k)}\) is the value of the function for the equivalent subsystem.
In this case the values of the initial disturbances of the post-emergency regime remain identical for the original and equivalent models of the electric-power system.

In addition to (14), the condition of invariance of active power must be fulfilled:

\[ \sum_{i=1}^{n} P_i = P_{\mathrm{e}}. \tag{15} \]

The actual process of equivalencing the selected subsystem is divided into 3 main stages:

1. The steady-state regime of the complete electric-power system is calculated. By known equivalencing methods, the quantities \(\dot{E}_{i\mathrm{e}}, \dot{Y}_{i\mathrm{e}}, J_{i\mathrm{e}}, S_{i\mathrm{e}}\) are determined (network transformation, moment rule, summation).

2. The transient process in the equivalent electric-power system is calculated and the equivalencing error, \(\Delta V_{i\mathrm{e}}^{(k)}\), is estimated.

3. From the value of \(\Delta V_{i\mathrm{e}}^{(k)}\) over the calculation interval, the values \(E_{i\mathrm{e}}^{(k)}\) (according to the magnitude of the potential energy) and \(J_{i\mathrm{e}}^{(k)}\) (according to the magnitude of the ki-

netic energy), the corrected values are taken as the calculated ones on the following interval.

The proposed equivalencing procedure can easily be implemented on a computer, but it requires knowledge of the values of the function \(V_i(t)\) for the power-system model being sought. Its concrete implementation is possible in combination with the method of test disturbances [9]. The use of this method consists in specifying the type of disturbance at the connection node; then the transient process is calculated in the subsystem singled out for equivalencing. It seems permissible to take, as the test disturbance, the region of possible voltage values at the connection node:

\[ 0 \leq U_i \leq U_{i \max} \tag{16} \]

For various values of the test disturbance, in combination with probable active-power flows along the connection branch, the time characteristics of function (14) are constructed; on their basis an archive of standard characteristics can be organized in the computer memory. In the course of calculating the dynamic stability of a power system at the second stage of equivalencing, the corresponding characteristics are selected.

Extending the equivalencing procedure presented here to the general case, when the subsystem being simplified has several connection nodes, is difficult because of the probabilistic dependence of the operating-state parameters at each of the nodes. To identify these dependences, it would be necessary to organize the collection and formation of a large amount of statistical material, which is not available at present.

Conclusions

1. The use of network-power equations in combination with the features of methods for numerical integration of differential equations makes it possible to determine the value of the Lyapunov function for a power system as the sum of the functions of individual synchronous generators.

2. The proposed form of writing the Lyapunov function (12) makes it possible to use it as a criterion of equivalence for mathematical models of power systems.

3. The considered procedure for simplifying power-system models is easily implemented for a special case occurring in complex power systems. Its implementation for the general case is made difficult by the absence of statistical material.

References

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