**3. Results**

Considering a 20 bus bar grid model where current is injected by four generators, the Ad matrix having the damping coefficient difference will be a 24 × 24 cross-block matrix, as shown in Figure 5.

**Figure 5.** 20-bus bar power system.

It is possible to find the master data of the system matrix via the following program. In addition, the system's response to any change in parameters can be plotted and the output current of all generators can be monitored after each failure. In addition, the range of change of parameters can be determined upon system stability and the most appropriate mode can be selected shown as Table 1. The objective is to estimate (<sup>Δ</sup>δ)1<sup>=</sup> 5o,(<sup>Δ</sup>δ)3<sup>=</sup> 3o,(<sup>Δ</sup>δ)2 = (<sup>Δ</sup>δ)4<sup>=</sup> 0 and 5◦, 3◦ deviations in the first and third synchronous generators, respectively. The location of the 24th master data on a complex surface is shown in Figure 6.

The real part of the master data is completely negative, so the system is stable, and each minor fault returns to a stable state after a short time. The following Figure 7 shows how the condition variables of each generator, variation diagrams and other variables are changed by the deviation of the rotor angle by 5 degrees in the first generator and by 3 degrees in the third generator.

In general, the total active electrical power supplied by the generators should always be equal to the active power consumed by the loads, which also includes the losses in the system. A failure in the system can disrupt this balance, causing the rotors of the generators to accelerate or decelerate. If one generator temporarily runs faster than the other does, the angular position of the rotor will increase in connection to that of the slower machine shown as Figure 8. Δω Variation diagram of four generators when (<sup>Δ</sup>δ)1<sup>=</sup> 5o,(<sup>Δ</sup>δ)3<sup>=</sup> 3o,(<sup>Δ</sup>δ)2 = (<sup>Δ</sup>δ)4<sup>=</sup> 0.


**Table 1.** The data of special matrix values.

**Figure 6.** Location of the 24th master data on a complex surface.

**Figure 7.** Shows the damping momentum Δω stability diagram of the generators in the power system.

**Figure 8.** Δv2 Variations diagram of generators in a power system.

As can be seen in Δvs variation diagram of four generators when (Δ <sup>ω</sup>)1<sup>=</sup> 5o,(Δ <sup>ω</sup>)3<sup>=</sup> 3o, (Δ <sup>ω</sup>)2 = (Δ <sup>ω</sup>)4<sup>=</sup> 0, the system has remained stable as the real part of the master data is negative. The minimum changes of rotor angle Δδ and speed of some generators Δ ω did not a ffect the stability of the system and the system regained its stability after these minimum failures.

### *Evaluation of the Proposed Solution Method*

The analysis of linear systems is much simpler than that of nonlinear systems. As can be seen in this article, the position of the matrix state on the complex surface for the main quantities was found because of the analysis of linear systems. Furthermore, the sign of the real portions of the matrix state of the main quantities (positive or negative) showed the stability of the system. If these real parts

are negative, the system is stable and small deviations will not disrupt this stability, but after a short time, the system will regain its equilibrium. Any changes in the parameters of the generator and network will affect the elements of the matrix state; therefore, the location of the main quantities on the complex surface will vary, because of which the stability of the system will be affected. In this thesis, we attempted to examine how changes in the damping coefficient affect the main quantities of the state matrix, and the diagrams of the changes in the real parts of each main quantity were drawn against the changes in the [−200,200] range of the damping coefficient (KD). The system is unstable in KDs where the real part of at least one main quantity is negative. Therefore, the above diagrams show the effect of the damping coefficient on system stability. In addition, these diagrams can be drawn based on changes of other parameters and system stability can be examined. As a result, the stability of large systems in the small signal depends only on finding the main quantities of a very large matrix. In fact, finding the main quantities is possible by finding the roots of a polynomial equation derived from the determinant, det(A-λI). In other words, only n-1 steps are necessary for the conversion of the given matrix to an upper-triangular matrix. In the first stage, the elements below the element (1,1) are reset; in the second stage, the ones under element (2, 2) are reset; and in the final stage, only the ones under element (n, n − 1) are reset.
