# Stability Analysis in Multi-VSC (Voltage Source Converter) Systems of Wind Turbines
## 1 Introduction  
In recent decades, the wind energy sector has experienced significant growth. According to the latest Renewables Global Status Report by REN21, global wind power capacity increased by more than 77 GW in 2022. This includes 68.4 GW of onshore and approximately 8.8 GW of offshore installations, marking a 9% rise in the total operational capacity to an estimate of 906 GW [1]. The Global Wind Energy Council (GWEC) forecasts an addition of 680 GW to the global wind capacity from 2023 to 2027, including 130 GW offshore. By the end of 2030, wind energy is anticipated to reach a landmark 2 TW of installed capacity [2]. The high increase in wind-power penetration has made the power system more efficient and flexible, mainly because it is coming to be dominated by high-power electronic Voltage Source Converters (VSC) [3, 4]. On the other hand, many challenges for the power system's stability and synchronization to the grid have been arising, as the number of components connected to the system is being increased [5]. Therefore, extensive research has been conducted in the area of dynamic interactions between the grid-connected converters of wind turbines on wind farms. The small-signal stability of a converter-based system is the system's ability to overcome a small disturbance and return to a steady state [6]. The main small-signal stability analysis methods for converter-based systems are impedance-based analysis--which is mainly preferred in \"black-box\" systems--and eigenvalue analysis [7]. In impedance-based analysis, the system's equivalent impedances seen from the Point of Common Coupling (PCC) are used in order to represent the small-signal dynamics between the converter control system and the grid [8]. The Nyquist stability criterion evaluates the system's stabilitybased on the system's open-loop gain [7]. In eigenvalue analysis, the state-space model of the whole system is formulated, providing a deep input into the system and control dynamics. Linearization of the nonlinear model is applied at the obtained equilibrium points that correspond to the model's state variables, and then the system's eigenvalues are obtained by implementing a modal analysis. The formulated eigenvalue plots depict the system's stability [9]. State-space modeling is chosen for conducting sensitivity analysis and gaining insights into the internal states of the system, which can subsequently be assessed for stability using eigenvalue analysis. Research on various state-space modeling methods for converter-based systems is well-documented in [10; 11; 12; 13]. These studies emphasize enhanced modularity and offer detailed descriptions of the dynamic states prevalent in wind turbine systems. However, they lack a methodology for computing the equilibrium points around which the model is linearized, as they either set them directly equal to chosen operating points or they rely on ElectroMagnetic Transients (EMTs) simulations tools for obtaining them through power flow analysis. In [14; 15], a Component Connection Method (CCM) was utilized, where the system was decomposed into multiple components and a linear algebra matrix was formed based on the interconnections of the components; the state-space model of the system was obtained by combining the state-space submodels of the components using the linear algebra matrix. However, it was difficult to represent the internal connections caused by the control couplings of the model [16]. A methodology for computing the equilibrium points was also not provided, and chosen operating points were instead used for the model's linearization. In [17], a detailed modular methodology was used to derive the linearized state-space model after computing the equilibrium points; however, it was limited to only using an L filter, without a capacitor in the filter, and this constitutes the dynamic situation of the model, which is much simpler than is the case with LC or LCL filters.