**4. Test Power System**

Without loss of generality, the performance of our proposal is proved in an important benchmark of four machine and two area electric power system [1]. Although this system is not big, it presents an interesting and complex behavior for transient stability studies. The two areas are connected by a weak tie, this is an important case for studying the fundamental nature of inter-area oscillations and the inherent difficulty in tuning and controllers coordination.

## *4.1. Case Base*

A base analysis for different types of excitation control is presented in [1]. This power system has three rotor angle modes of oscillation. There is one inter-area mode of 0.55 Hz with generator 1 and 2 swinging against generators 3 and 4 of area 2. Two more are intermachine oscillation local modes, one of 1.09 Hz corresponds to area 1 and the second of 1.12 Hz is for area 2. These modes are determined only with one type of excitation control that changes if the operating condition varies or more dynamic components are involved.

In this analysis, each synchronous generator of Figure 4 is represented by the model Equation (1). Two loads are connected at bus 7 and 9. The initial conditions are based in data set reported in [1]. The active power injected by each generator: *PG*<sup>1</sup> = *PG*<sup>2</sup> = *PG*<sup>4</sup> = 700 MW and, *PG*<sup>3</sup> = 719 MW; terminal voltages: *VtG*<sup>1</sup> = 1.03∠20.2 pu, *VtG*<sup>2</sup> = 1.01∠10.5 pu, *VtG*<sup>3</sup> = 1.03∠ − 6.8 pu, *VtG*<sup>4</sup> = 1.01∠ − 17.0 pu. The initial conditions for state variables of each generator are obtained considering (1)–(4) using this information. The machine parameters are also available in [1] and used on this paper.

**Figure 4.** Four machine power system with StatCom at bus 8.

With this model representation and initial conditions, the system is exposed to a three-phase fault at different nodes with similar results, selecting node 7 on this section to illustrate the results.

The electrical grid without StatCom neither PSS exhibits an unstable performance when the fault is cleared after eight cycles, Figure 5. If the fault is cleared up to seven cycles a stable evolution is observed, but the oscillations have values far from the prefault condition, and with long duration. After 6 s the oscillations continue with very little damping. Figure 5 shows the angular difference between machines with generator number one as a reference, for eigth cycles (unstable operation) and six cycles (stable but oscillating operation). Additional damping required is evidenced. Similar behavior is observed in other system variables.

**Figure 5.** Angular difference of each machine with respect to number one, electrical grid without StatCom neither PSS.
