ETEP z Energy functions analysis in voltage collapse F. Jurado, J. Carpio Abstract zyxwvutsrqp Time-domain approach examines the behaviour of the system, one derermines whether stability has been main- tained or lost. In contrast to the time-domain approach, direct methods determine system stability based on energy functions. The basis of direct methods for the stability assessment of a system is knowledge of the sta- bility region. During the last decade, many researches have thoroughly analysed the use zyx of energy functions for the direct stability assessment of networks. Energy function analysis offers a different geometric view of voltage collapse. The Transient Energy Function, a technique based on Lyapunov stability theory and origi- nally developed for direct stability analysis of power systems, has been successfully used as a voltage stabili- ty index for collapse studies. In this paper rhe simulation results are on the IEEE I73-bus test system. 1 Introduction zyxwvuts Use of energy function methods has seen a long his- tory of development in the power systems literature. With several refinements in its application over the last decade, “transient energy function” (kTEF) methods are gaining acceptance as a very useful supplement to time domain simulation of individual fault scenarios. Energy function analysis offers a different geomet- ric view of voltage collapse. In this approach, a power system operating stable is like a ball which lies at the bot- tom of a valley. The stability can be understood as the ball rolling back to the bottom of the valley when there is a disturbance. As the power system changes, the land- scape of mountains and mountain passes surrounding the valley changes. A voltage collapse corresponds to a mountain pass being lowered so much that with a small perturbation the ball can roll from the bottom of the val- ley over the mountain pass and down the other side of the pass. The height of the lowest mountain pass can be com- puted and used as an index to monitor the proximity to voltage collapse. 2 Direct methods Traditionally, direct methods have been based on the network-reduction model where all the load representa- tions are expressed in constant impedance and the entire network representation is reduced to the generator inter- nal buses. In the classical model, we consider the network re- duction with the classical generator model for transient stability analysis and the loads are modelled as constant impedances. In the one-axis generator model with ex- citers, the model includes one circuit for the field wind- ing of the rotor, i.e.; this model considers the effects of field flux decay. As a result, the voltage behind the di- rect transient reactance is no longer a constant. Sasaki first included such models for direct stability analysis zyxwvu [ zyxwvutsr 11. In the one-axis generator plus automatic voltage regulator model, the exciter control action is included in the generator model. There are two advantages of using the network-pre- serving power system models for direct stability analy- sis [2]. From a modelling viewpoint, it allows more re- alistic representations of power system components, es- pecially load behaviours. From a computational view- point, it allows the use of the sparse matrix technique for the development of faster solution methods for solving non-linear algebraic equations involved in direct meth- ods [3]. In this section we discuss direct methods for net- work-preserving power systems models. The first network-preserving model was developed by Bergen and Hill zyxw [4], who assumed frequency depen- dent real power demands and constant reactive power demands. Narasimhamurthi and Musavi zy [5] moved a step further by considering constant real power and volt- age dependent reactive power loads. Padiyar [6] has in- cluded non-linear voltage dependent loads for both real and reactive powers. Tsolas, Araposthasis and Varaiya [7] developed a network-preserving model with the consideration of flux decay and constant real and reactive power loads. An energy function for a network-preserving model ac- counting for static var compensators and their operating limits was developed by Hiskens and Hill [8]. 3 Energy function 3.1 Models for the power system The energy function to be described in this paper [9] will start from a dynamic model for the power system. It is therefore appropriate to begin with a description of the assumed scenario for voltage collapse, and how dynam- ics come into play. Let us begin with the obvious obser- vation that the physical power system is a dynamic sys- tem; its full range of possible behaviour can not be pre- dicted with a strictly static description. However, in nor- mal operation, the state of the power system is expected to be at or near an operating point. Here we will use the terminology of “operating point” in a physical sense, separate from any assumptions on the nature of the math- ematical model employed to predict system behaviour. ETEP Vol. 1 I, No. 4, July/August 2001 235