On the Form of Writing the Initial Equations for Dynamic Stability Calculations by the Lyapunov Function Method
Abstract Generated abstract
The paper addresses how to formulate initial equations for dynamic stability calculations of electric power systems using Lyapunov functions, noting limitations of positional idealizations during electromechanical transients. It proposes combining numerical integration with Lyapunov qualitative criteria by writing state equations through network power balances and reducing each integration interval to equivalent generator, constant bus schemes with parameters fixed from the preceding interval. The resulting intervalwise Lyapunov function includes corrections for weighted average system motion and is shown to coincide with the conventional post fault expression for the same system description. This formulation permits stability assessment for the whole system, individual generators, or subsystems while accounting for changes in operating conditions and network elements during the transient process.
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ON THE FORM OF WRITING THE INITIAL EQUATIONS FOR CALCULATIONS OF DYNAMIC STABILITY BY THE LYAPUNOV FUNCTION METHOD
R. I. BORISOV, I. A. BEZLER, N. E. CHERNYI
(Presented by the Department of Electrical Systems and Networks)
The use of the Lyapunov function method for calculations of the dynamic stability of electric-power systems is hindered by their construction and by the establishment of necessary and sufficient stability criteria. At the same time, the fewer operating-mode parameters and system elements the method of constructing the function takes into account, the weaker the sufficient stability conditions prove to be. Parameters are understood to mean certain variables of the modes and variables of various elements of the circuits, certain combinations of which make it possible to describe the electrical state of the entire system for the time instants of interest.
A whole series of difficulties arises in creating a mathematical model of an electric-power system suitable for the purposes of investigating dynamic stability by the Lyapunov method. Thus, in [4] a general characterization is given of an algorithm for calculating the stability of multimachine systems by Lyapunov criteria, in which the mathematical description of the system corresponds to its positional idealization. Such a representation of the system is acceptable in the normal and established post-fault modes; in a transient process, however, the electrical system should, as far as possible, be described as dissipative [7]. The consequences of this restriction, imposed on the accounting for practically important variables of the operating-mode and system parameters, by narrowing the sufficient stability conditions, do not make it possible to judge whether the transient-process results obtained by this method correspond to those occurring under real conditions. The solution of this problem is possible if the construction of the Lyapunov function and the obtaining of sufficient stability conditions are carried out according to Zubov’s method [5]. However, as follows from a number of works, the limitations imposed by the capabilities of the digital computer do not allow this method to become practically acceptable [6].
However, the posed problem can be solved if the following justifications are made.
Determination of the Values of Criterion Constants in an Established Post-Fault Mode
As noted above, according to [7], in the normal and established post-fault mode the mathematical description of a large electric-power system may be simplified and correspond to its positional idealization. In this case, to determine the coordinates of the net-
saddle point and the critical constant, the method of the \(n\)-dimensional cube is used [4].
The approximate value of the critical constant is determined from the relation:
\[ l \approx \min \omega_i(\varepsilon_i)=\min(2A_i-B_i\varepsilon_i), \tag{1} \]
where
\[ \varepsilon_i=2\operatorname{arctg}\frac{A_i}{B_i}; \tag{2} \]
\[ \left. \begin{aligned} A_i&=\left|\sum_{\substack{j=1\\ j\ne i}}^{n} E_iE_j y_{ij}\cos\alpha_{ij}\cos\delta_{*ij}\right|,\\[4pt] B_i&=\left|\sum_{\substack{j=1\\ j\ne i}}^{n} E_iE_j y_{ij}\cos\alpha_{ij}\cos\delta_{*ij}\right| \end{aligned} \right\}. \tag{3} \]
To find the exact coordinates of the saddle point, the method of steepest descent is used.
The value of the Lyapunov function in the emergency state of the system is calculated by the formula:
\[ V_0=\frac{1}{2}\sum_{i<j}^{n}J_i s_{i0}^{2} -\frac{1}{2} \frac{\left(\sum_{i=1}^{n}J_i s_{i0}\right)^2}{\sum_{i=1}^{n}J_i} +\sum_{i<j}^{n}E_iE_j y_{ij}\cos\alpha_{ij}\times \]
\[ \times\left[\cos\delta_{*ij}\left(1-\cos\Delta\delta_{ij0}\right) -\sin\delta_{*ij}\left(\Delta\delta_{ij0}-\sin\Delta\delta_{ij0}\right)\right]. \tag{4} \]
A judgment about dynamic stability is obtained either by the Lyapunov criterion or by the instability criterion,
\[ V_0\leq l; \tag{5} \]
\[ \max|\delta_{ij}|>\varepsilon. \tag{6} \]
The Lyapunov function (4) as written gives stability conditions close to sufficient ones in a positional electric power system. However, as has already been noted many times, such a mathematical representation of a real system under emergency conditions does not make it possible to describe the electromechanical transient process. It is expedient to describe the electrical state of the system by means of equations of network powers, which make it possible to take into account the characteristics of each element of the network [9].
Writing the state equations by generator nodes and calculating the Lyapunov function from known parameters
Since the network powers in the system (Fig. 1), flowing to each node, depend on the voltages of these nodes and on the electrical angles between the voltage vectors, the state equation by nodes, together with the balance equations of active and reactive powers, will determine the parameters of the mode of these states.
\[ \left. \begin{aligned} \sum_{i=1}^{n}P_i(U_i,U_j,\delta_{ij})&=0\\[4pt] \sum_{i=1}^{n}Q_i(U_i,U_j,\delta_{ij})&=0 \end{aligned} \right\}. \tag{7} \]
For the scheme shown in Fig. 1, the electromechanical transient process is described by a system of \(n\) differential equations of the form:
\[ J_i \frac{d x_i}{d t} = F_i, \tag{8} \]
where
\[ F_i = \Phi_i(E_i, U_i, \theta_i, z_i, t), \tag{9} \]
where \(E_i, U_i\) are, respectively, the emf and the voltages at the terminals of the \(i\)-th generator;
\[ \theta_i = \delta_E - \delta_U \]
is the electrical angle between the vectors of the corresponding variables;
\(z_i\) is the resistance of the \(i\)-th generator.
Solving (8) by methods of numerical integration, at each computed interval the values of the variable operating parameters and system elements are taken to be unchanged. On this basis, for the duration of the given interval, the scheme of the complex system shown in Fig. 1 can be reduced to \(n\) simplest schemes of the “generator—bus of constant voltage and frequency” type, with parameters corresponding to the end of the preceding interval (Fig. 2).
Fig. 1
Fig. 2
The values of the powers delivered by the \(i\)-th generator to the system, according to (9), are determined as
\[ P_i = \frac{E_i U_i}{z_i} \sin \theta_i, \tag{10} \]
\[ Q_i = \frac{E_i U_i}{z_i} \cos \theta_i . \]
The increment of the Lyapunov function can be calculated from the expression for the function constructed in [2] for the simplest system
\[ V_i = \frac{1}{2} J_i s^2 - \frac{E_i U_i}{z_i} \sin(\theta_{*i0}-\alpha_i) (\Delta \theta_i - \sin \Delta \theta_i) + \tag{11} \]
\[ + \frac{E_i U_i}{z_i} (\theta_{*i0}-\alpha_i) (1-\cos \Delta \theta_i), \]
where
\[ s = \frac{d \Delta \theta_i}{d t} \]
is the slip of the \(i\)-th machine relative to the rotating voltage vector.
The value of the Lyapunov function for the \(i\)-th generator upon reaching the post-fault characteristic is determined as
\[ V_{i0} = \sum_{k=1}^{m} V_{ik}. \tag{12} \]
The value of the Lyapunov function on the \(k\)-th interval for the entire system, when it moves relative to the synchronously rotating axis, is determined as the sum of the partial values
\[ V'_k = \sum_{i=1}^{n} V_{ik}. \tag{13} \]
It is obvious that in the general case the stability of the system should be considered not with respect to the synchronous axis, but in their mutual motion. This circumstance is easy to take into account if the additional term
\[ \frac{\left(\sum_{i=1}^{n} J_i s_i\right)^2}{2\sum_{i=1}^{n} J_i}, \]
which characterizes the kinetic energy of the weighted-average motion of the system [8], is subtracted from the expression. Here
\[ s_i=\frac{d\Delta\delta_i}{dt} \]
is the slip of the \(i\)-th machine relative to the synchronously rotating axis.
Thus, the final value of the Lyapunov function on the \(k\)-th interval is determined as
\[ V_k=V'_k-\frac{\left(\sum_{i=1}^{n} J_i s_i\right)^2}{2\sum_{i=1}^{n} J_i}. \tag{14} \]
The value upon reaching the post-emergency characteristic is equal to the sum of the values of \(m\) final time intervals
\[ V_0=\sum_{k=1}^{m} V_k. \tag{15} \]
It is not difficult to note that, for one and the same electric power system, with the same mathematical description, expression (15) is identically equal to (4). Therefore, the stability of the system under consideration can be established by Lyapunov’s stability criteria (5) or instability criteria (6), calculated in the established post-emergency regime.
The stability of individual generating stations is established analogously, according to the corresponding criteria.
Conclusions
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The combination of numerical integration methods and qualitative analysis methods that have found application in practice substantially broadens the range of dynamic-stability problems solvable for complex electric power systems.
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The equations of network powers considered make it possible to take into account the features of each operating condition and each element of the network of interest.
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The values of the Lyapunov functions determined with the aid of these equations over the actually considered time of the transient process take into account changes in the network of both operating-condition and operational character.
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Stability assessment can be performed both for individual generators or parts of the system and for the system as a whole.
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