# Stability Analysis in Multi-VSC (Voltage Source Converter) Systems of Wind Turbines
## 5 Simulation Results  
Time domain simulations were carried out by using MATLAB Simulink R2020b and PLECS Blockset in order to validate the eigenvalue-based stability analysis results from Section 3. They were accompanied by the corresponding Fast Fourier Transformation (FFT) analysis, through which we ensured that the dominant frequency in the instant the system tended to become unstable was the same as the critical frequency of instability in the eigenvalue analysis. A weak-grid-case scenario, with an SCR equal to 1.5, was chosen in order to highlight the dynamic interactions in one of the system's converters. A system, as shown in Figure 5, with two VSCs \((n=2)\) was simulated in the time domain. A small step change was applied to the control gain of the tested controller in order to identify cases where instability occurred, and the FFT analysis was performed at the beginning of the instability in all the test cases. A step change was first applied to the proportional gain of the current controller \(K_{P,CC1}\) in \(VSC_{1}\) in order to show the instability. The system became unstable when this step change resulted in a critical proportional gain value equal to \(K_{P,CC1 cr}=99.9\); the corresponding dominant frequency of the unstable case was equal to 3.34 kHz. Figures 13a and 14a show the inductor current in \(I_{L1}\) in \(VSC_{1}\) and highlight this instability case both in the time and frequency domains. Then, the lower resonances, which were caused by the PLL and the AVC in \(VSC_{1}\), were simulated. The PLL proportional gain \(K_{P,PLL1}\) was increased in order to identify the bandwidth in which the system was leading to instability; the critical value \(K_{P,PLL1 cr}\) was equal to 2.55 where the PLL's bandwidth was equal to 113.37 Hz, and the corresponding dominant frequency that led to instability was equal to 182 Hz. These results are shown in Figures 13b and 14b. The same procedure was applied to the AVC in \(VSC_{1}\), where a ramp change was applied to its integral controller \(K_{L,L1}\); the reason was that a ramp indicates a very large number of steps and would be much more accurate to be used for an integrated value than a single step. The critical value of instability was approximately \(K_{L,4 cr}=750\) with a corresponding frequency equal to 44 Hz. These results are shown in Figures 13c and 14c. Figure 12: Participation factors for critically unstable mode during variations of \(K_{L,41}\). The aforementioned results are very close to the corresponding results in Section 3, where the eigenvalue-based stability analysis of the small-signal model with n VSCs was presented; all the results are gathered in Table 2. Therefore, the developed state-space model of the multi-VSC system can describe the interactions introduced by each control loop with very high accuracy. Figure 13: Time domain simulations of the multi-VSC inductor current \(I_{L}\) with \(n=2\) VSCs. The stability impact of the current controller, the PLL and the AVC in the weak grid (SCR = 1.5) is observed, caused after the step/ramp change in \(K_{P,C0}\), \(K_{P,PLL0}\) and \(K_{I,A0}\) at \(\text{t}=1.5\) s: (**a**) step change in CC’s proportional gain, \(K_{P,CCcr}=99.9\); (**b**) step change in PLL’s proportional gain, \(K_{P,PLLcr}=2.55\); (**c**) ramp change in AVC’s integral gain, \(K_{I,a\sigma}=750\). Figure 14: Frequency analysis (FFT) of the multi-VSC inductor current \(I_{L}\) with \(n=2\) VSCs. The dominant oscillation frequency is shown when the control gains under study obtain their critical value: (**a**) dominant oscillation frequency when high-frequency-range instability is caused by step change in \(K_{P,CC}\); (**b**) dominant oscillation frequency when low-frequency-range instability is caused by step change in \(K_{P,PLL}\); (**c**) dominant oscillation frequency when low-frequency-range instability is caused by ramp change in \(K_{I,\mu}\).