APPLICATION OF THE CONTROL DISTURBANCE METHOD FOR DETERMINING CHARACTERISTIC NODES OF COMPLEX LOAD CONNECTION IN DYNAMIC STABILITY CALCULATIONS

R. I. BORISOV, N. E. CHERNYI

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

A modern electric-power system (EPS) belongs to the class of large systems and, as a consequence, is characterized by a great variety of elements included in it. To ensure the reliable operation of such a system, constant analysis of its operating conditions is necessary. In doing so, one must constantly address questions concerning the choice of a mathematical model of the system that describes the processes occurring in it.

In analyzing the dynamic stability of electric systems (ES), the question of choosing a method for representing complex loads is very important. The principal methods of replacing load elements are described in the monographs of P. S. Zhdanov and D. I. Azar’ev [1, 2]. In the course of further work, methods for replacing load elements were refined [3, 4, 5], and methods for accounting for them in dynamic-stability calculations were proposed [4, 6, 7, 8, 9]. From a number of works [4, 6, 7, 10] it follows that the method of representing the load can substantially affect the results of the analysis.

Less studied remains the question of identifying characteristic nodes that exert a substantial influence on the nature of electromechanical transient processes. In [4] it is noted that replacing the local load of a generating station, comparable with it in power, by dynamic characteristics gives good agreement between experimental and field tests. In [6] it is recommended that the loads located “electrically” closer than others to the place of disturbance be taken into account by dynamic characteristics. In a number of calculation cases, the recommendations given yield good results.

However, in complex, multi-loop ES, the choice of a method for representing one or another load depends on combinations of the parameters of the system elements and its operating parameters. We shall show the complexity of these dependencies by the example of calculating the synchronous dynamic stability of a simple two-machine ES, the equivalent circuit of which is given in Fig. 1a.

The influence of the location of the connection nodes of an identical complex load \((z_i)\), identical in composition and magnitude, the method of its representation, and also the points of occurrence \((z_j)\) of a single type of short circuit on the nature of the dynamic transition was considered. The load was represented by mathematical models that are limiting in their essence—by dynamic characteristics or by a constant resistance.

By varying the combination of nodes \(i\) and \(j\), we obtain the dependence of the error of the results, caused by neglecting transients in the load, on the “electrical” location of the load-connection node and the point of application of the disturbance:

a)

b)

Fig. 1. a) Schematic diagram of the electrical system under study; b) dependence of the error of the results on the method of specifying the load, the node of its connection, and the point of application of the disturbance.

\[ \Delta \%=\varphi(z_i,\ z_j), \tag{1} \]

where \(\Delta \%\) is the error of the results, in %;

\(z_i\) is the “electrical” distance of the node at which the complex load is connected;

\(z_j\) is the “electrical” distance of the point of application of the disturbance.

The obtained dependence, constructed in the space of the corresponding coordinates, is shown in Fig. 1b. If, in addition, one takes into account

to (1), such factors as the load power \((S_{\mathrm{n}}\ne \mathrm{const})\), the composition and parameters of the load elements \((K_{\mathrm{n}})\), the ratio between the generated powers \((S_1\) and \(S_2)\), the parameters of the generators and their control systems \((B)\), and the type of disturbance \((K^{(i)})\), then the error function may be written as

\[ \Delta \%=\varphi_1[z_i,z_j,S_{\mathrm{n}},S_1,S_2,B,K^{(i)},K_{\mathrm{n}}]. \tag{2} \]

Analyzing (2), one may conclude that even for the simple case of a two-machine power system it is difficult to obtain any general recommendations concerning the method of load replacement. This problem becomes still more complicated when it is necessary to study the dynamic stability of a complex power system.

In this case the models of load nodes considered above acquire a number of substantial shortcomings. It should be acknowledged that specifying all load elements by the corresponding exact characteristics is the standard from the point of view of their influence on the nature of the transient process in the power system. In Fig. 2, curves 1 correspond to the results of calculating the relative rotor angles of characteristic generators of a complex power system when all load elements are specified by dynamic characteristics. For comparison, the same figure also gives (curves 2) the results of a dynamic-stability calculation when all load elements are specified by constant resistances. It is not difficult to see that specifying the entire load by constant resistance leads to significant distortions of the pattern of the dynamic transient. In the first case, however, the dimensionality of the problem increases sharply, making it practically unrealizable for real systems. Moreover, the objective presence of errors in the initial information reduces the reference value of exact mathematical models.

Let us consider expressions describing the behavior of a composite load when it is specified by its dynamic characteristics. The load characteristics in dynamic-stability calculation algorithms are accounted for by complex admittances, which are taken as constant over the calculation interval. In the general case, the admittances are determined by the sum of various components:

\[ \dot y_{\mathrm{n}}=\dot y_{\mathrm{osv}}+\dot y_{\mathrm{pr}}+\dot y_{\mathrm{as}}, \tag{3} \]

where \(\dot y_{\mathrm{osv}}\) is the admittance of the lighting-load component;
\(\dot y_{\mathrm{pr}}\) is the admittance of the other-load component;
\(\dot y_{\mathrm{as}}\) is the admittance of the asynchronous load.

Lighting and other loads are accounted for by static characteristics. For them, if the change in frequency at the connection node is not taken into account, one may write

\[ \dot y_{\mathrm{osv}}+\dot y_{\mathrm{pr}}=f(U), \tag{4} \]

where \(U\) is the voltage at the connection node.

The admittance of the asynchronous load is determined according to the equations of motion of an asynchronous motor [1, 2]. Therefore

\[ \dot y_{\mathrm{as}}=f_1(Us,), \tag{5} \]

where \(s\) is the rotor slip of the asynchronous motor.

Thus,

\[ \dot y_{\mathrm{н}} = F(U,s). \tag{6} \]

According to the work of the Institute of Electrodynamics of the Academy of Sciences of the Ukrainian SSR, in particular [6], as a first approximation we shall consider it possible to assign all load elements typical dynamic characteristics. This assumption makes it possible to reduce as much as possible the labor required to compile the initial information when carrying out the control calculation. In what follows we shall use the method of control disturbance [11]. In its implementation, the type of disturbance and the place of its occurrence are specified; then the transient process is calculated. In the transient process, deviations of the variables in expression (6) will be observed; during the process it is written as

Fig. 2. Curves of the change in the relative rotor angles of the characteristic generators of a complex power system over time for different methods of substituting the complex load.

\[ \dot y_{\mathrm{н}} \pm \Delta \dot y_{\mathrm{н}} = F_1(U \pm \Delta U;\ s \pm \Delta s). \tag{7} \]

Let us introduce the concept of specified accuracies over the interval for the variables:

\(\varepsilon\) — maximum voltage deviation,
\(\xi\) — maximum deviation of the slip value of the asynchronous-motor rotor.

When the specified accuracies are exceeded at the “K” instant of time, the powers and conductances of the load branches on which these deviations are observed are recalculated. The new values of the branch parameters are adopted as the calculated ones for the subsequent intervals until the accuracies are exceeded again. The remaining elements remain unchanged. Upon completion of the checking calculation, we obtain the characteristic nodes of complex load, for which refinement of the initial information is desirable.

Fig. 3. Block diagram of the algorithm for identifying characteristic nodes of complex-load connection.

Flowchart labels:

• Calculation of the electrical operating condition over the interval
• Identification of load nodes
• Check \(\Delta U\)
• \(\Delta U > \varepsilon\)
• \(\Delta U < \varepsilon\)
• Check \(\Delta S\)
• \(\Delta S < \xi\) — to the next node
• \(\Delta S > \xi\)
• Determination of the conductances of the asynchronous motor
• Determination of the conductances of lighting and other load
• Determination of \(\dot{y}_i\)
• Counter of the number of changes
• All nodes?
• no
• yes
• End of the checking calculation?
• no — to continue the calculation
• yes
• Printout of the number of changes for each load node

The block diagram of the logical algorithm described is shown in Fig. 3.

The method considered was verified using the dynamic-stability calculation program IV-4 for the M-220 computer of the Institute of Electrodynamics of the Academy of Sciences of the Ukrainian SSR, as applied to the example considered above (Fig. 2). The following were adopted: \(\varepsilon = 0.15 U_{\mathrm{n}}\), \(\xi = 0.5 \dfrac{\text{r.u.}}{\text{s}}\). The equivalent circuit of the investigated EPS contains 17 nodes for connecting complex loads. After analyzing the results of the control calculation, it was possible to identify 6 characteristic nodes for connecting complex loads.

The results of calculating the relative rotor angles of the generators when specifying only the loads of the characteristic nodes by dynamic characteristics are shown in Fig. 2, curves 3. The results presented indicate their good agreement with the results of the reference calculation (curves 1).

Conclusions

1. In studies of electromechanical transients in an EPS, it is constantly necessary to address the question of selecting a mathematical model of the system that describes these processes. The selected model, on the one hand, must correctly reflect the nature of the processes occurring in the real object and, on the other hand, minimize the labor required in preparing the initial information and analyzing the results obtained.

2. The method of representing complex loads has a noticeable influence on the results of dynamic-stability calculations for an EPS. It is difficult to establish general recommendations for choosing the method of their representation for real EPSs.

3. Using the control-disturbance method, it is possible to obtain characteristic nodes for connecting complex loads that have a substantial influence on the nature of electromechanical transients in an EPS. This makes it possible to refine the computational mathematical models used in specific dynamic-stability studies.

4. When compiling libraries of computational circuits for the computer complex of an automated dispatch-control system for an EPS, preliminary optimization of the computational models is necessary in order to ensure maximum computational speed with sufficient accuracy of the results. The method proposed in this article for determining characteristic nodes for connecting complex loads can be used in the general optimization algorithm.

5. The magnitudes of the specified accuracies of the deviations of variables may substantially affect the results of control calculations. Work should be continued to determine their permissible values.

References

1. P. S. Zhdanov. Stability of Electric Systems. Moscow, Gosenergoizdat, 1948.

2. D. I. Azar’ev. Mathematical Modeling of Electric Systems. Moscow, Gosenergoizdat, 1962.

3. I. M. Sirota. Dynamic characteristics of asynchronous load. In: Collection of Proceedings of the Institute of Energy, Academy of Sciences of the Ukrainian SSR, issue 4, 1949.

1. Yu. E. Gurevich. The influence of load parameters on the dynamic stability of generators. Elektrichestvo, 1969, No. 1.

2. Yu. E. Gurevich, L. E. Libova. Problems of accounting for load in studies of complex power systems. Reports at the Third All-Union Conference on the Stability and Reliability of Power Systems of the USSR. Leningrad, “Energiya,” 1973.

3. V. N. Avramenko. The influence of the characteristics of composite load on the dynamic stability of power systems. Automation and Relay Protection of Electrical Systems. Kiev, “Naukova Dumka,” 1966.

4. V. A. Venikov. Transient electromechanical processes in electrical systems. “Vysshaya Shkola,” 1970.

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7. R. I. Borisov, N. E. Chernyi. The influence of the method of replacing composite load on the results of dynamic-stability studies. In: Electrification of Metallurgical Enterprises of Siberia, Tomsk, 1971.

8. N. N. Shchedrin. Simplification of electrical systems in modeling. Moscow, “Energiya,” 1966.