In the 1970s and 1980s around the world, it led to blackout that voltage magnitudes at electric power system load buses had rapidly decreased on heavy load. Such a decrease in voltage magnitudes is called voltage collapse. It has been numerically studied in [1-3] that a saddle node bifurcation or a chaotic blue sky bifurcation causes voltage collapse. The system presented in these references consists of an infinite bus, a generator, a load, and transmission lines. The model of the load includes an induction motor, and is described by two first-order differential equations for the magnitude and angle of the load voltage. Although the dynamics of the rotor angle of the generator is governed by a second-order differential equation or a swing equation, the voltage magnitude of the generator bus terminal is constant. The total system is described by four first-order differential equations. The reactive power of the load is chosen as a parameter, and computer simulations had been carried out.

Constant voltage magnitude of the generator bus terminal seems to be an appropriate assumption as long as a power system operates normally because the voltage magnitude of the generator bus terminal is kept constant by an AVR (automatic voltage regulator). The dynamics of an AVR, however, do not always keep the voltage magnitude of the generator bus terminal constant because an AVR has time constants. The voltage magnitude of the generator bus terminal may fluctuate largely when a limit cycle or chaotic attractor appears in the system. Even if a generator operates normally, that is, the system stays in equilibrium, the voltage magnitude of the generator bus terminal will vary with the change of load condition due to the offset of an AVR. Instantaneous voltage of the generator bus terminal will not change sinusoidally when a limit cycle or chaotic attractor appears, but constant voltage magnitude of the generator bus terminal means that instantaneous terminal voltage changes sinusoidally. Under the assumption that the voltage magnitude of the generator bus terminal does not change, discussions of the boundary of stability and the mechanism of voltage collapse are doubtful.

The model of a generator based on internal flux linkage had been studied [4, 5]. This model allows the voltage magnitude of the generator bus terminal to change, and allows the instantaneous terminal voltage to be free wave form. The generator under consideration has three phase armature (or stator) windings and three rotor windings (one filed winding and two damper windings). The instantaneous terminal voltage of any winding is represented by the derivative of the flux linkage, the current flowing in the winding, and the winding resistance. These six windings are magnetically coupled, and the flux linkage of each winding is determined by all the currents. Since the flux linking each winding is a function of the rotor position, the expression for the terminal voltage of each winding is complicated. In order to avoid this complication, armature variables are usually transformed by Park's transformation [4-7]. The new variables are obtained from the projection of the variables on two axes; the direct axis of the rotor and the quadrature axis. Although they are functions of time, they become constant when a generator operates normally. We replace the model of a generator in the references with this model, which is described by seven first-order differential equations including the swing equation.

A generator is usually controlled by an AVR and a GOV (a speed governor). The terminal voltage of the field winding is controlled by an AVR, and the driving torque in a swing equation is controlled by a GOV. If a generator is not connected to an infinite bus and is driven without a GOV, the angular velocity of the rotor (the frequency of the terminal voltage of the armature winding) changes as the driving torque. When a generator is connected to an infinite bus, the angular velocity of the rotor will synchronize with that of the infinite bus, even if the driving torque in the swing equation is constant as in the above references. That is, a GOV is not always required in this case. In order to make the system as simple as possible, we employ an AVR only, which is described by two first-order differential equations.

It is reported in [1] that the equilibrium point which corresponds to a normal operating point of the system is stable before voltage collapse, and it collides with a saddle point, that is, a saddle node bifurcation causes voltage collapse. On the other hand, it is reported in [2] that the equilibrium point becomes unstable before voltage collapse, a chaotic attractor arises through Hopf bifurcation and cascade of period doubling or flip bifurcations, and the collision between a saddle point and the chaotic attractor causes voltage collapse. The system given in [2] differs from that of [1] only in the values of some constants. The values used in Ref. [2] are more real than Ref. [1] except for the transmission lines. The transmission lines given in [1] have resistance and inductance, but the resistances of [2] are zero. We use the values given in [2], and add resistances to the transmission lines.

In this paper, we numerically survey various phenomena which appear in the above system, which is described by the thirteen simultaneous first-order ordinary differential equations, and report bifurcations of limit cycles and chaotic attractors. It is clarified that a global bifurcation or a blue sky bifurcation of a chaotic attractor by an unstable limit cycle causes a voltage collapse. A stable equilibrium point of the differential equation corresponds to normal operation of the power system, and a limit cycle or chaotic attractor corresponds to instability of the power system. From the practical point of view of a power system, stability of an equilibrium will be the more significant problem, yet limit cycles or chaotic attractors will not be interesting. Because loads in the power system become complicated in recent years, it is hard for the power system to stay in one equilibrium point, and may sometimes operate beyond the limit of its stability. This situation makes us concerned with limit cycles, chaotic attractors and their bifurcations in practice.Go to Abstract, Chapter 2, Chapter 3, Chapter 4, Chapter 5