# Stability Analysis in Multi-VSC (Voltage Source Converter) Systems of Wind Turbines
## 2 State-Space Model of a System with n Voltage Source Converters  
The focus of this study was on the grid-side converters of wind turbines in a wind farm and their potential dynamic interactions with the grid. The control architecture of a grid-connected single-converter system is depicted in Figure 1. This system incorporates a grid-following converter that utilizes vector Current Control (CC). The concept is derived from [22]; it employs a PLL to synchronize the converter with the grid, assuming a consistent DC link voltage at the inverter. The regulation of active power adheres to a specified reference value using open-loop control, which is considered adequate under the assumption of an ideal converter. The input feed-forward voltage into the CC loop, which is the voltage at PCC \(V_{PCC}\), integrates a low-pass filter in the d-q domain (FF, LPF in Figure 1). An Alternating-Voltage magnitude Controller (AVC), which includes a low-pass filter (AVC, LPF in Figure 1), is also utilized to regulate the voltage fed at the PLL of each VSC, based on the study in [23]. The AVC is of high importance in order to ensure the system's normal operation, especially in weak-grid cases. The control topologies of the PLL, the current controller and the AVC are shown in Figures 2, 3 and 4, respectively.  
The assumption utilized for deriving the initial state-space model of the multi-converter system is that all VSCs have homogeneous dynamics; meaning, their topology and control philosophy are all the same. In the configuration of the parallel connection of \(n\) such converters the converters share a common point of synchronization as well as a common PCC; therefore, the transformer's leakage inductance of each VSC was not considered in this work, in order to simplify the complexity of the final model, which is shown in Figure 5.  
A state-space model for a single-wind-turbine system has been formulated, as detailed in [23], to characterize small-signal instabilities. This model is outlined in (1) and features nonlinear state equations that capture the system's control dynamics. Each component of the system is represented by these equations, and the complete nonlinear state-space model arises from the combination of the individual state-space models of each system component.  
\[\begin{split}\dot{x}&=Ax+R(x,u)\\ y&=S(x,u)\end{split} \tag{1}\]  
where \(R(x,u)\) describes the non-linear dependencies in the model. The state equations describing the small-signal model of the single-VSC system in Figure 1 are presented in Appendix A.  
Figure 4: Control structure of the Alternating-Voltage Controller (AVC) in Figure 1[23].  
Figure 5: Multi-VSC-based grid where the sources are wind turbines and feed power to \(V_{DC}\).  
The abovementioned nonlinear model of one VSC can be used to compose a system with n parallel VSCs corresponding to n wind turbines on a wind farm. The state equations of each grid-connected converter are combined in order to form the total state-space model of the system with n VSCs. The holistic nonlinear state-space model is given in (2):  
\[\begin{bmatrix}\dot{x}_{VSC1}\\ \dot{x}_{VSC2}\\ \vdots\\ \dot{x}_{VSCn}\end{bmatrix} =\begin{bmatrix}A_{VSC1}&0&\cdots&0\\ 0&A_{VSC2}&\ddots&\vdots\\ \vdots&&\ddots&0\\ 0&\cdots&0&A_{VSCn}\end{bmatrix}\begin{bmatrix}x_{VSC1}\\ x_{VSC2}\\ \vdots\\ x_{VSCn}\end{bmatrix} \tag{2}\] \[+\begin{bmatrix}R_{VSC1}&0&\cdots&0\\ 0&R_{VSC2}&\ddots&\vdots\\ \vdots&&\ddots&0\\ 0&\cdots&0&R_{VSCn}\end{bmatrix}\begin{bmatrix}x_{VSC1},u_{VSC1}\\ x_{VSC2},u_{VSC2}\\ \vdots\\ x_{VSCn},u_{VSCn}\end{bmatrix}\]  
The system parameters of the system in Figure 1 and the control parameters of the corresponding state-space model in Appendix A are listed in Table 1.