Theoretical Analysis of the Interaction between Power System Stability Modes with the Normal Forms of Vector Fields

Because of the strong nonlinear characteristics appearing in the modern power system, the strong interaction does exist between the angle stability and the voltage stability, which were conventionally studied insularly. However, as power system is a complex unified system, the angle instability always happens together with the voltage instability. The authors propose a novel method to analyze such kind of stability problem. In the proposed method, the theory of normal forms of vector fields used to treat a auxiliary dynamic system according to the power system. By use of this method, the interaction between response modes caused by the nonlinearity of power system can be analyzed. Consequently, the eigenvalue analysis method is extended to cope with performance analysis of power system with heavy nonlinearity. The effectiveness of the proposed methodology is verified on a 4-bus power system, and some conclusions are reached.

I. Introduction

One of the most clear and concise classification, provided by the IEEE/CIGRE Joint Task Force on Stability Terms and Definitions [1], is shown in Fig. 1. For the purpose of simplifying the analysis, studies of the angle stability and the voltage stability for power systems are usually performed independently. Reference [2] studied voltage instability decoupled from angle instability. Reference [3] decomposed the power system into separate subsystems and studied the relationship between the angle stability and the voltage stability. Reference [4] studied the instability modes, the angle instability mode and the voltage instability mode, at different operation conditions, and highlighted the factors that distinguished the angle and the voltage stability. Reference [5] used the direct method constituting a general energy function, and examines the relationship between the angle instability and the voltage instability from the viewpoint of this energy function. Reference [6] studied the characterization of power system small disturbance stability with models incorporating voltage variation. the relevant sub-categories and time scales defined with short term covering the first 20 seconds following an event, and long term relating to effects examined over several minutes. It shows that the definitions of power system stability are overlapped in time. As power system is a unified system, all kinds of its instability are inter–connected. Therefore, they are indivisible problems. For example, a possible outcome of voltage instability is loss of load in an area, or tripping of transmission lines and other elements by their protective systems leading to cascading outages. Loss of synchronism of some generators may result from these outages or from operating conditions that violate field current limit. Progressive drop in bus voltages can also be associated with rotor angle instability. The loss of synchronism of machines as rotor angles between two groups of machines approach 180° causes rapid drop in voltages at intermediate points in the network close to the electrical center. Reference [1] proposed such kind of viewpoint and investigated this problem. The focus of this paper is on the characteristics of the angle stability and the voltage stability. According to this point of view in reference [1], power system angle stability and voltage stability are tow aspects of power system stability, which can not be separated from each other clearly in the real power system because they can be associated with each other. Insular investigation of them may result in an inaccuracy or even wrong result. In fact, interaction between the angle and the voltage instability does exist in the power system, especially, when the system operating condition is close to the stability limit, which is resulted from the restriction of the environment and the interconnecting networks. Investigation results show that in these situations, because of the existence of the nonlinearity behaviors, the angle stability and the voltage stability can not be distinguished exactly [7]. Therefore, new methodologies should be developed to study such problems.