POWER SYSTEM TRANSIENT STABILITY AS AFFECTED BY VOLTAGE REGULATOR AND GOVERNOR THROUGH LIAPUNOV'S DIRECT METHOD

Abstract

The problem of power-system stability has interested the research workers for the last several • decades. The mathematical model to describe this system, taking into account the voltage-regulator and governor actions, is a nonlinear vector differential equation involving product-type trigonometrical nonlinearities. The complexity of the problem has forced workers to divide the problem, somewhat artificially, to the powersystem transient-stability and dynamic-stability problems. While the first study uses a nonlinear mathematical model of the system, the second study considers a linearised mathematical model. To solve the transient -stability problem a number of techniques have been used by several workers. Methods based on numerical integra tion using digital computers and analog simulation of the system can solve this problem but these are laborious methods. Other methods, like equal-area and energyintegral criteria, and the phase-plane technique, cons ider a very approximate system model, always ignoring the voltage-regulator action and often ignoring the governor action as well. Yet another method is the Liapunov's direct method which comments, conservatively, on Ihe system stability without solving the system differential equations and can be used as a supplemen tary method to the other existing methods. This method requires the use of a Liapunov function. Unluckily, the generation of a suitable Liapunov function, inspite of several known techniques to generate it, requires a lot of skill and ingenuity. Among the various techniques to generate a Liapunov function the best ones seem to be the Popov's method and the variable-gradient method. Liapunov functions have been generated by researchers for single and multi-machine power systems considering the governor action but explicit expressions of the Liapunov function have not been generated except for the case of simple speed governing with a single time lag of the prime-mover-governing scheme. No work exists till date, to the best of the author's knowledge, regarding the power-system transient-stability study with a realistic voltage regulator using Liapunov's direct method. The present author has attempted to solve the powersystem transient-stability problem as affected by voltage regulator and governor through Liapunov's direct method. The variable-gradient method is used to systematically generate the Liapunov functions. The inclusion of the voltage-regulator dynamics in the system mathematical model, suitable for the study undertaken, poses a problem. The apparantly simple relationships of the machine terminal voltage , V+ =j(V?d+ V^ ), and the load current, I=JUd+Iq)> in -\ terms of the direct-,and quadrature-axis components of the machine terminal voltage and current are very complicated relationships when expressed in terms of the system state variables. Accordingly, under different approximations, a number of voltage-regulator mathe matical models have been proposed for the various cases studied here and their accuracies are checked by the numerical method. It is later found that the more accurate is the voltage-regulator(avr) representation the less flexible is the proposed Liapunov function from the point of view of the system-parameter coordination. The avr, considered here, is actuated by the system operat ing parameters (i.e. voltage, current and load angle) and their time derivatives. It is assumed that the avr is of the electronic type with negligible time delay and it feeds the machine field directly. The prime-mover governor is actuated by the machine lead-angle, speed and acceleration signals,and is represented by a differential equation involving either a single tim-o lag for simpler studies or two time lags. The author's approach to this problem has been to start with a relatively simple system and then progressively take more and mere system details. The present work considers in the beginning the single-machine transient stability as affected by forced (derivative-type) governing. A Liapunov function is derived for the case of single time lag in the governor control and the effect of parameter variations on the critical fault-clearing time is studied. A nonlinear-model parameter-plane study is introduced for the first time which uses the constraints of the Liapunov function derived.This gives a region in the parameter plane which assures asymptotic stability of the system over a wide range of system operating conditions.Next, Liapunov functions are developed for different descriptions of the governing scheme involving two time lags in its control. For all cases the machine transient saliency and non linear damping are included in the system model. The transient-stability problem of a multimachine system, without voltage regulator, is studied next. The nonuniform type of damping power is considered. A Liapunov function is generated including the fieldflux decay of all machines for a lossless system, ignoring the transient saliency of the machines and any regulator action. This Liapunov function has the flexibility of putting varying weightages on the kinetic-energy terms due to relative speeds among various machines and their absolute speeds. Such a Liapunov function is also given ignoring the flux decays but now including the system driving-point conductances. Next, including the governor control on one machine only and ignoring the flux decays and system transfer conductances, Liapunov functions are developed,* the governor control is of different descriptions involving one or two time lags in its control mechanism. A 4-machine system is used for illustrating two of the Liapunov functions derived here. The present work now includes the voltageregulator action. First the transient stability of a single machine, connected to an infinite busbar and equipped with a 3-term prime-mover governor and a 2-term voltage regulator,is studied. The voltage regulator is actuated by deviation in the machine ter minal voltage and its first time derivative. Synchronous machines of both salient-pole and turbo types are considered. The mathematical model considers the non linear damping phenomenon for the case of salientpole machine and the quadrature-axis rotor-flux decay for the case of turboalternator. A number of Liapunov functions are systematically generated with different assumptions in the avr mathematical model. The effect of variations in the regulator parameters is studied on the critical fault-clearing and circuit-breaker reclosing times for a 3-phase fault. A nonlinear-model parameter-plane study is carried out for a case qonsider. ing only the avr dynamics and it is compared with the conventional linear-model parameter-plane study. Explicit expressions are also given to cover the cases of governing with two time lags. The case of a doully-excited synchronous generator, connected to an infinite busbar is next taken up. The direct-axis avr is actuated by deviation in the machine terminal voltage and its first derivative while the quadrature-axis regulator is actuated by the load-angle deviation and its first two derivatives. Both these regulators, assumed to have negligible time delay, feed the respective machine fields directly. Under different approximations in mathematical modelling of the voltage regulators, two Liapunov functions are generated. The governor is again of the 3-term type with one or two time lags. The effect of parameter variations on the critical fault-clearing time is studied using the more flexible type of Liapunov func tion. A general form of the voltage regulator is now considered. The single-machine transient-stability problem is studied with an avr which senses the machine terminal voltage, load current and their first derivat ives, and the load angle along with its first two derivatives.The avr, having negligible time lag, feeds the machine field directly. The machine is also fitted with a 3-term governor involving one or two time lags. Liapunov functions are generated for two approximate avr mathematical models and the effect of parameter variations on the critical fault-clearing and circuitbreaker reclosing times is studied. A brief study is included here to determine the effect of parameter variations on the system transient response for small disturbances through Liapunov's direct method using linear model of the system and a performance index. Lastly, the multimachine transient-stability problem with regulators is considered. Only one machine is assumed to be fitted with regulators. The avr is actu ated by the machine terminal voltage, load current, and their first time derivatives while the governor is of the 3-term type involving one or two time lags. Under, various simplifying assumptions Liapunov functions are generated for 2-machine and multimachine systems. The effect of parameter variations is studied for 2-machine and 4-machine systems on the critical fault-clearing time.