## 5 Results and Discussion
### Simulation Results  
#### 5.2.1 Results under Different Controller Configurations  
This section describes the obtained statistical results related to the stability of critical eigenvalues (also termed as modes) for four different controller configurations: case study with no controllers (base case), case study with only PSSs, case study with only BESS controllers, and case study with the proposed strategy of using PSS and BESS controllers. The base case was run by disconnecting all PSSs and BESSs. The obtained result using the PSSs assessment method (described in Section 3) was first compared with MCS (10,000 simulations with 95% confidence interval) to verify its accuracy. Figure 12 shows that the PDF of the most critical eigenvalue (nearest eigenvalue to origin), obtained using the analytical method, very closely resembles MCS.Furthermore, the average root mean square [26] and computation time of the analytical method were found to be 0.1731% and 72.6 s, respectively, highlighting the accuracy and computational efficiency of the analytical method. Figure 11: Modified test system with RES. The probabilistic stability indices obtained for all critical electromechanical modes (15 in total) under the no-controller case study are shown under the base case heading in Table 2. The probability of the damping constant being greater than \(-0.1\) was very high for Modes 1 to 4 under the base case. Furthermore, the damping factor of all modes had a 100% probability of being lower than 10% for the case without controllers. Both of these values suggest that the given test system under the base case has much potential to be small-signal unstable. This result led us to investigate potential solutions for this problem. Note in Table 2 and here afterwards in our paper, the symbol (\") refers to the ditto mark. The second controller configuration only considered PSS operation and assumed all BESSs to be disconnected. The PSSs located at Generators 1 to 12 were then simultaneously tuned using the proposed method. A population size of 50 fireflies was chosen, as this size is commonly selected by most swarm-based intelligence algorithms that are used to solve optimization problems related to low-frequency stability [11; 19]. The values of \(\beta\), \(\alpha_{t}\), and \(\gamma\) were calculated from the average length of controller parameters and are shown in Table 3. The optimization algorithm was solved using Intel (R) Core (TM) i5-2400 CPU, 3.1 GHz processing speed, with 8 GB RAM, and it took 3.15 h to compute. However, as the optimization algorithm provides the stochastic output, 30 repeated evaluations were performed and the average was used for analysis in this paper [24]. All programs were written using DIGSILENT programming language [25]. Table 2, under the PSS category, shows that, except for Modes 2 and 3, there was a low likelihood that the damping constant of other modes was greater than their margin. From a damping=factor perspective, Modes 1 to 3 had high probability of being less than the critical damping value, which shows that only employing PSSs is insufficient to provide required damping under stochastic PVG and WTG penetration. However, probabilistic statistics related to SSS were much improved in