## 3 Proposed Distributed Secondary Control Technique
### Controller Stability Analysis  
In this section, we will examine the stability analysis of the proposed FDSC. The stability analysis conducted utilizes Lyapunov stability criteria, which enable the evaluation of stability in non-linear systems by making use of a Lyapunov function (LF). Utilizing a quadratic LF \(\mathcal{V}(\varphi)\), the stability requirements satisfying the Lyapunov conditions in a domain around the symmetric point (\(\varphi=0\), \(\varphi\) denoting a state variable) can be determined using the following formula [35], considering a non-linear system \(\dot{\varphi}=f(\varphi)\). If a LF \(\mathcal{V}(\varphi)\) exists such that \(\mathcal{V}(\varphi)=\varphi^{T}\mathrm{Q}\varphi\), and it satisfies the stability requirements below [35]:  
\[\left\{ \begin{array}{ll}&\mathcal{V}(\varphi)>0\qquad\forall\varphi\
eq 0, \\ &\varphi(0)=0\end{array} \right. \tag{15}\]  
\[\dot{\mathcal{V}}(\varphi)\leq 0\qquad\forall\varphi \tag{16}\]  
Then, the non-linear system is considered to be asymptotically stable (\(\lim_{t\to\infty}\varphi(t)=0\), \(\forall\varphi_{o}\
eq 0\)). Calculating \(\dot{\mathcal{V}}(\varphi)\) is essential for verifying system stability, and it is determined by:  
\[\dot{\mathcal{V}}(\varphi)=\dot{\varphi}^{T}(A^{T}M+MA)\varphi \tag{17}\]  
Therefore, in order to establish the stability of a system, \(\dot{\mathcal{V}}(\varphi)\) is required to be negative (less than zero). This will occur if the condition \(\dot{\varphi}^{T}(A^{T}M+MA)\varphi<0\). It is important to emphasize that when the DC microgrid operates in islanded mode, all the converters of the DERs operate in voltage control mode. As a result, the change in the DC voltage implicitly influences the change in the output current. By considering (2) \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & \multicolumn{3}{c}{\(\mathbf{\Delta U_{dc}}\)} \\ \hline  & & PB & PS & Z & NS & NB \\ \cline{2-7}  & PB & S & S & M & M & M \\ \cline{2-7}  & PS & S & S & L & M & M \\ \cline{2-7}  & Z & S & L & L & L & S \\ \cline{2-7}  & NS & M & M & S & S & S \\ \cline{2-7}  & NB & M & M & M & S & S \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fuzzy rules. Figure 6: Outputs’ membership functions. and (12), we can express the derivative of the output current after activating the distributed secondary controller as:  
\[ \begin{split}\dot{l}_{o_{i}}(t)&=\frac{1}{R_{line,i}}( \dot{U}_{i}(t)-\dot{U}_{dc}(t))\\ &=\frac{1}{R_{line,i}}\big{(}\dot{\mathcal{H}}_{i}(t)-(R_{line_{i }}+k_{d_{i}})\cdot\dot{l}_{o_{i}}(t)-\dot{U}_{dc}(t)\big{)}\\ &=\mathcal{P}_{i}\big{(}\dot{\mathcal{H}}_{i}(t)-\dot{U}_{dc}(t) \big{)}\end{split}  \tag{18}\]  
where \(\mathcal{P}_{i}=\frac{1}{2R_{line_{i}}+k_{d_{i}}}\). Furthermore, from (1), it can be deduced that \(\sum\limits_{i\in I_{a}}\dot{l}_{o_{i}}=\dot{U}_{dc}\sum_{j\in I_{L}}1/R_{ load}\); therefore, the derivatives of the output voltage and current of \(DER_{i}\) with respect to the control input from the distributed secondary controller can be obtained from (18) as:  
\[\dot{U}_{dc}(t)=\left(\frac{\sum\limits_{i\in I_{a}}\mathcal{P}_{i}}{\mathcal{ F}}\right)\dot{\mathcal{H}}_{i}(t) \tag{19}\]  
\[\dot{l}_{o_{i}}(t)=\mathcal{P}_{i}\left(\dot{\mathcal{H}}_{i}(t)-\frac{\sum \limits_{i\in I_{a}}\mathcal{P}_{i}}{\mathcal{F}}\dot{\mathcal{H}}_{i}(t)\right) \tag{20}\]  
where \(\mathcal{F}=\sum\limits_{i\in I_{a}}\mathcal{P}_{i}+\sum\limits_{i\in I_{L}}1 /R_{load_{i}}\).