# Stability Analysis in Multi-VSC (Voltage Source Converter) Systems of Wind Turbines
## 3 Equilibrium Points Computation and Eigenvalue Analysis in a Multi-VSC System
### Eigenvalue Analysis  
The grid strength is defined by the Short Circuit Ratio (SCR), which is the ratio of the short circuit power by all VSCs at PCC \(S_{SC}\) and the rated power of the inverter \(S_{N}\). The SCR is given by  
\[SCR=\frac{S_{SC}}{S_{N}}=\frac{\frac{3}{2}\frac{(V_{S})^{2}}{|Z_{S}|}}{\frac{3 }{2}nV_{PCC}I_{Ldref}} \tag{16}\]  
Hence, the strength of the grid is influenced by its impedance, with an increase in impedance resulting in a reduction in the short circuit power. Also, the more converters that are connected to the system, with the same grid impedance, the higher the total rated inverter power becomes and the grid becomes weaker.  
In this study, a scenario with a weak grid and an SCR of 1.5 was examined. The eigenvalue analysis of the small-signal model was conducted using the control parameters specified in Table 1. The bandwidth of the current controller was set to 1 kHz, targeting a closed-loop current regulation at 1/20 of the switching frequency; the standard bandwidth for the PLL is 7.27 Hz. The grid inductance \(L_{S}\) was obtained by (16), and the grid resistance \(R_{S}\) was assumed to be zero.  
An eigenvalue sensitivity analysis was performed to assess the small-signal stability and the interactions influenced by the controllers across both low- and high-frequency ranges. This analysis adjusted the control parameters of the system's control structures to pinpoint the levels associated with instability. The equilibrium states of the integrated state-space model were recalculated with each adjustment of the tested controller until instability was caused. Instability is detected when the real part of a complex eigenvalue becomes positive, indicating negative damping. Consequently, a new eigenvalue analysis was conducted each time a parameter was adjusted, ultimately producing an eigenvalue trace that illustrated the stability trend of the tested control parameter. This method is illustrated in Figure 6.  
At first, the proportional gain of the current controller for a single VSC K\({}_{PCC1}\) was swept from 0.1 (deep blue) to 10 (deep red) times its standard value K\({}_{PCC0}\). The resulting eigenvalues indicating instability are displayed in Figure 7, highlighting the critical gain and frequency. As the proportional gain of the current controller increased, the system tended toward instability, with the critical frequency approximately equal to 1/6 of the sampling frequency \(f_{S}\).  
Subsequently, low-frequency interactions were analyzed by varying the PLL proportional gain \(K_{P,PLL1}\) of a single VSC in the system from 0.1 (deep blue) to 20 (deep red) times the default value \(K_{P,PLL0}\). In that way, the critical PLL bandwidth for instability was determined. The findings, shown in Figure 8, indicate that instability occurred when the PLL proportional gain reached approximately 2.4759, corresponding to a critical PLL bandwidth of about 110.07 Hz.  
The outer loop of the AVC was also leading to instability in the low-frequency range. This is depicted in Figure 9, where the integral gain \(K_{I,A1}\) is varied from 0.1 (deep blue) to 200 (deep red) times the default value \(K_{I,a0}\). The eigenvalue analysis shows that the critical case of instability was when \(K_{I,acc}=857\) with a dominant frequency close to the 1 st harmonic (\(f_{\sigma}=45\) Hz).  
Figure 6: Methodology for eigenvalue analysis in an n-VSC system when control parameters are varied.  
Figure 7: Eigenvalue trace of current controller’s proportional gain variation \(K_{P,CC1}\), which varied from 0.1 (deep blue) to 10 (deep red) times \(K_{P,CC0}\). Green arrow means instability point.