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This paper presents the methods and results of a simulation study of voltage stability of a power system. Important load and reactive power components are included. A nine-machine electric power system is chosen as the working example so that the dynamic interactions among machines in a system are adequately represented. Since it is a very high-order system with many nonlinearities and variable constraints and including both fast transients and slow dynamics, a variable step size is chosen for the integration and a new method is developed to solve the Jacobian matrix equations in each integration step without taking the inverse of a very large matrix directly. Case study results show that all voltage-dependent load and reactive power components are important and must be adequately modeled in voltage stability studies. >

It is important to develop new laboratories for teaching and research in order to keep up with the advancement of modern power engineering1,2. This paper presents salient features of a "micro-machine" dynamic power system developed for teaching Power System Dynamics. It also has been used to develop new stabilizers to improve both dynamic and transient stabilities of power systems.

In this paper the dynamic interactions between mechanical and reactive modes of machines in a multi-machine power system are investigated. The dynamic interactions are expressed in terms of machine angles, transient voltages, and coefficients which are functions of system parameters and operation conditions. The effect of the dynamic interaction can be examined by the electric torque loci on the A6-Aw phase plane. It is found that the interaction between the mechanical modes of machines has significant effect on stability characterized by the natural mechanical oscillation frequency and damping of individual machines.

The paper presents an output feedback excitation control to stabilize the torsional oscillations due to subsynchronous Resonance of a capacitor-compensated power system. The control design is based on an equivalent mass-spring model, modern control laws, and eigenvalue sensitivity technique. With a multiple signal feedback synthesized from a minimum number of system outputs, the control provides proper damping to all torsional modes. Computer simulation tests show that all oscillating modes can be stabilized simultanously for a wide range of capacitor compensation.

Dynamic equivalents of external system are estimated from measurements within the study system. An equivalent model with damping and inertia explicitly expressed is chosen. An intentional disturbance of pulsed excitation is applied and the dynamic interaction between the external system and the study system is intensified for the estimation. The effect of intentional disturbance is restricted to the safe volatge regulation limit, yet much larger than that due to normal load fluctuations. To reduce the computational requirement, a least-squares algorithm with an adapted step-size scheme is used. The technique is applied to athirteen-machine electric power system, and the dynamic equivalents is given several short-circuit tests. The estimated equivalnets are unique for the model chosen, and the test results of the original system and the equivalent system are in good agreements.

Instead of approximating an unknown large power system in the neighboring area by an infinite bus of constant voltage and constant frequency, this paper models the large power system DYNAMICS as one large electric machine so that both the voltage and the frequency of the unknown power system bus may vary. For the parameter estimation, an unknown dynamic power system model is developed, and a local power system model is adapted to accommodate these changes. An estimation technique is applied and an alogrithm is presented. Computer results show that the unknown large power system parameters can be uniquely determined regardless of the power factor of the local system, the local load, a strong or weak intertie, the unknown system inertia and damping, and the initial guess for the estimation.

A physically realizable linear optimal controller in terms of easily measurable state variables for the stabilization of power system is developed in this paper. The controller is based on a fairly accurate fifth order synchronous machine and a first order voltage regulator. The transformation of the system equations into a cannonical form enables us to determine the weighting matrices Q and R of the quadratic cost function with simple procedure and to design an optimal controller which can stabilize a power system under widel. power range operating conditions. Transient response test on the nonlinear model proves the effectiveness of the linear control design.

For stability studies of large power systems, accurate representation of the synchronous machine is required. As pointed out by Canayl, the conventional equivalent circuits for synchronous machines do not give accurate computed field voltage and current values. He suggested several circuits and showed good agreement between his test and calculated results. His circuit parameters were calculated from design.

TWENTY years ago, Kron published the first part of his pioneer work on the application of tensors to the analysis of rotating electric machinery.1 Since that time, many valuable studies2?6 have been made which firmly establish tensor methods as an extraordinarily powerful method for analyzing the performance of electric machines and other complicated physical networks. With the aid of this tool, the engineer can solve intricate network problems in routine manner, thereby obtaining solutions directly.

The Routh-Hurwitz criterion and the D-partition method are used in the steady-state stability study of a synchronous machine connected to an infinite bus bar. The machine has a continuously acting voltage regulator and a speed governor. The effects of the saliency and the short-circuit ratio of the synchronous machine, and that of the tie-line resistance and reactance on the stability of the system, are investigated.

The stabilization of power systems with excitation control using speed or power signals has been very well developed,1-6 while the application of optimal control to power system stability study is relatively new.7 Both are based on linear models. In this paper, a comparison is made of these two techniques. Signals derived from a lower order linear model are tested on a high order non-linear system modelled as a single machine-infinite bus as well as multi-machine systems.

Neither the analog study of electric machines[1]-[4] nor the modeling technique in feedback control systems is new. However, modeling a multimachine system with parameters directly determined from tests and checking the steady-state and transient performances from analog studies with those from actual machine tests are very complicated but indispensable for complete modeling.

In recent years important research has been done in the area of system optimization by control engineers. Many theoretical results have been published but application examples have mainly been on low-order systems. An attempt is made to apply a certain class of optimal control theory, known as the state regulator problem, to obtain an optimal controller to improve the dynamic response of a power system. The system differential equations are written in the first-order state variable form. A cost functional is then chosen, and the matrix Riccati equation is solved. Puri's and Gruver's method is applied for the numerical computation, and the system is made initially stable by shifting the system eigenvalues.

For an optimal linear regulator design a performance function of the quadratic form must be chosen. The question arises of how to decide the weighting matrix Q of the performance function. A new method is developed in this paper to determine Q in conjunction with a left shift of the dominant eigenvalues as far as the practical controllers permit. The method is then applied to the optimal control design of a typical power system. Three cases are investigated, the first with an optimal excitation control uE, the second with optimal governor controls uG and uG, with and without the dash-pot, and the third with uE plus uG control. The stabilizing signals thus obtained are given nonlinear tests on the same power system. It is found from the results that the optimal controls are more effective than conventional excitation control, that the optimal governor control without dash-pot is just as good as the optimal excitation control, and that the optimal uE plus uG control is the best way to stabilize a power system.

Although there are many publications on power system stability, the application of Liapunov's direct method to the study is relatively new. Gless[l] used an energy integral as a Liapunov function to study 1-, 2-, and 3-machine problems. El-Abiad and Nagappan,[2] also based their studies of the transient stability of a multi-machine system with losses on energy concepts.

The general machine described in this paper, and also in a previous one by the author,1 includes all conventional electric machines and models of analysis as special cases. In this paper, the typical elements of the inductance matrix and torque tensor are derived. From these, the impedance matrix and torque tensor of any electric machine can be readily found. Examples of application are shown. No Christoffel transformations are necessary. The paper provides unified and systematic ways of analyzing electric machines.

The optimal linear regulator design technique of determining the weighting matrix Q with dominant eigenvalue shift, developed in a companion paper[6], is applied to the optimal stabilization of a multi- machine system. Two cases are studied, the first is with one optimal controller uE1 on one machine of the m-machine system and the other with m optimal controllers uEM for the m-machine system. For comparison arison a one optimal controller uE on one machine with the rest approximated as an infinite system and m optimal controllers of individual idual design are also investigated. Nonlinear tests are given. It is found that uEI control is better than uE control, uEM control is better still than UEI control, and individual uE controllers are not good for the overall system.

It has been known for many years that organic solutes can serve as osmotic regulators in algae. Craigie & McLachlan (1964) reported that the glycerol content of Dunaliella tertiolecta in a 2.5 M NaCl solution is 133 times greater than in 0.025 M. Gimmler (1981) observed that in Dunaliella parva, the content of glycerol, as well as those of starch and endogenous phosphates, is directly related to the salinity of the medium. Kauss (1967) discovered a direct correlation between the amount of floridoside in Ochromonas malhamensis and the osmotic potential of the external medium. The same relationship was also observed in the macroalgae Porphyra perforata and Iridophycus flaccidum (Kauss 1968). Similarly, cyclohexanetetrol was found to be involved in resistance to osmotic stress in Pavlova lutheri (Craigie 1969 as Monochrysis lutheri). Hiller & Greenway (1968) reported that Chlorella accumulates sucrose when stressed with external osmoticants.