## 3 Stability Analysis and Improvement in the Multi-Converter Grid-Tied System
### Parameter Optimization  
The \(g(f)\) function derived in Section 3.1 is used to evaluate the stability of the multi-converter grid-tied system. When the system is unstable, researching the influence of VSCs control parameters on system stability is vital. Parameter optimization based on the identification of the stability-weakest parameters is essential for improving system stability, which is performed based on the \(g(f)\) function in this section.  
Let \(g(f)=g(f,\boldsymbol{x})\), where \(\boldsymbol{x}=\{x_{1},x_{2},\ldots,x_{n}\}\) is the set of any adjustable parameter in the multi-converter grid-tied system. As shown in Equation (22), \(g(f,\boldsymbol{x})\) is a piecewise function about \(\boldsymbol{x}\). When the adjustable parameter changes slightly, the \(g(f)\) function can be linearized around the initial values \(\boldsymbol{x}_{0}=\{x_{10},x_{20},\ldots,x_{n0}\}\), according to the Taylor expansion:  
\[\left\{\begin{array}{l}g(f,\boldsymbol{x})\approx g(f,\boldsymbol{x}_{0})+ \mu_{x10}(f)(x_{1}-x_{10})+\mu_{x12}(f)(x_{2}-x_{20})+\ldots+\mu_{xm0}(f)(x_{ n}-x_{n0})\\ \mu_{xm0}(f)=\frac{g_{\mathrm{c}}(f,\boldsymbol{x})}{\partial x_{n}}| \boldsymbol{x}=\boldsymbol{x}_{0}\end{array}\right. \tag{25}\]  
Define \(\mu_{xi0}(f)\) as the sensitivity function of the adjustable parameter \(x_{i}\) at the value of \(x_{i0}\).  
According to Equation (22) and Figure 7, it can be observed that when the function value of the lowest point \(M\) of \(g(f)\) is less than 0, the multi-converter grid-tied system is unstable. The objective of parameter optimization under this criterion is to modify the parameter \(\boldsymbol{x}\) to ensure that the function value of \(M\) is greater than 0, thereby achieving system stability.  
Thus, the following parameter optimization criteria are applied to any adjustable parameter \(\boldsymbol{x}\):  
1. At a specific frequency, if the sensitivity function value \(\mu_{xi0}(f)>0\) for parameter \(x_{i}\), the parameter \(x_{i}\) needs to be increased to increase the value of function \(g(f)\) according to Equation (25). Conversely, if \(\mu_{xi0}(f)<0\), the parameter \(x_{i}\) needs to be decreased.  
Figure 7: Stability evaluation function for the multi-converter grid-tied system.  
If \(|\,\mu_{x_{i0}}(f)|>|\,\mu_{x_{i0}}(f)\,|\), the variation of parameter \(x_{i}\) has a greater impact on system stability at this frequency. So the parameters with large absolute sensitivity values are defined as stability-weakest parameters. When system stability cannot be satisfied by optimizing a single parameter, parameter optimization can be performed by combining stability-weakest parameters.
3. Perform parameter optimization for \(m\) (\(m\geq 1\)) parameters. Assume that the sensitivity values of these parameters have the same sign and equal optimization magnitude. Then, according to Equation (25), the modified values for parameter \(x_{i\,|\,i=1,2,\,...,m}\), to ensure that the function value of \(M\) is greater than 0, are given as: \[x_{i}=x_{i0}+\frac{-g(f,x_{0})}{\sum\limits_{i=1}^{m}\mu_{x_{i0}}(f)}\] (26)  
The process of system parameter optimization based on parameter sensitivity analysis in Figure 8 is as follows.  
Step1: Firstly, the return ratio matrix \(\mathbf{L}(s)\) of the multi-converter grid-tied system is obtained.  
Step2: Then, the stability evaluation function \(g(f)\) of the system is constructed according to the Gerschgorin disc theorem and GNC to determine the system stability.  
Step3: If the system is unstable, get the function and the frequency value of the \(M\), and calculate the parameter sensitivity of each parameter.  
Step4: Finally, the stability-weakest parameters are identified. The parameter optimization values are calculated according to the parameter optimization criteria.  
The parameter optimization value is calculated by several iterations until the function value of \(M\) is above zero, which means the system satisfies the requirement of stability.