# Data-Driven Voltage Control in Isolated AC Microgrids Subject to Sensor Saturation
## 3 The DAILC Algorithm of Voltage Control Subject to Sensor Saturation
### Design of The DAILC Algorithm
\right]^{T}\] \[\boldsymbol{r}(\tilde{\kappa}) =\text{diag}(r_{1}(\tilde{\kappa},\tilde{\rho}),r_{2}(\tilde{ \kappa},\tilde{\rho}),\ldots,r_{N}(\tilde{\kappa},\tilde{\rho}))\]  
and  
\[\tilde{\underline{z}}_{s}(\tilde{\kappa}+1,\tilde{\rho}) =\left[\tilde{\underline{z}}_{s1}(\tilde{\kappa}+1,\tilde{\rho} )&\tilde{\underline{z}}_{s2}(\tilde{\kappa}+1,\tilde{\rho})&\cdots&\tilde{ \underline{z}}_{sN}(\tilde{\kappa}+1,\tilde{\rho})\right]^{T}\] \[\mathbf{G}(\tilde{\kappa},\tilde{\rho}) =\text{diag}\left(g_{1}(\tilde{\kappa},\tilde{\rho}),\,g_{2}( \tilde{\kappa},\tilde{\rho}),\,\ldots,\,g_{n}(\tilde{\kappa},\tilde{\rho})\right)\]  
where \(g_{i}(\tilde{\kappa},\tilde{\rho})\) is defined as \(g_{i}(\tilde{\kappa},\tilde{\rho})=\hat{\aleph}_{i}(\tilde{\kappa},\tilde{ \rho})/(\delta_{1}+|\hat{\aleph}_{i}(\tilde{\kappa},\tilde{\rho})|^{2})\). The disturbance error, denoted as \(\tilde{\varrho}_{hi}(\tilde{\kappa},\tilde{\rho})\), can be defined by the following equation: \(\tilde{\varrho}_{hi}(\tilde{\kappa},\tilde{\rho})=\tilde{\underline{z}}_{0}( \tilde{\kappa})-\tilde{\underline{z}}_{hi}(\tilde{\kappa},\tilde{\rho})\). Consequently, we can derive the following relationship: \(\tilde{\varrho}_{hi}(\tilde{\kappa},\tilde{\rho})=\tilde{\varrho}_{i}(\tilde {\kappa},\tilde{\rho})-\tilde{h}_{i}(\tilde{\kappa},\tilde{\rho})\). After substituting the saturation function, we obtain \(\tilde{\varrho}_{hi}(\tilde{\kappa},\tilde{\rho})r_{i}(\tilde{\kappa},\tilde{ \rho})\). \(r_{i}(\tilde{\kappa},\tilde{\rho})\) is given by  
\[r_{i}(\tilde{\kappa},\tilde{\rho})=\left\{ \begin{array}{ll}\frac{\tilde{ \underline{z}}_{0}-s}{\tilde{\varrho}_{hi}(\tilde{\kappa})},&\text{if }\tilde{ \underline{z}}_{hi}(\tilde{\kappa})>s\\ 1,&\text{if }-s\leq\tilde{\underline{z}}_{hi}(\tilde{\kappa})\leq s\\ \frac{\tilde{\varrho}_{hi}(\tilde{\kappa},\tilde{\rho})}{\tilde{\varrho}_{hi}( \tilde{\kappa})},&\text{if }\tilde{\underline{z}}_{hi}(\tilde{\kappa})<-s\end{array} \right. \tag{17}\]  
where \(0<r_{i}(\tilde{\kappa},\tilde{\rho})\leq 1\). Equation (7) can be transformed into the following form:  
\[\tilde{\underline{z}}_{s}(\tilde{\kappa}+1,\tilde{\rho})=\tilde{\underline{z}}_ {s}(\tilde{\kappa}+1,\tilde{\rho}-1)+\mathbf{G}_{\tilde{\aleph}}(\tilde{\kappa},\tilde{\rho})\Delta\tilde{\varrho}(\tilde{\kappa},\tilde{\rho})+\Delta \boldsymbol{h}(\tilde{\kappa},\tilde{\rho}) \tag{18}\]  
where \(\Delta\tilde{\varrho}(\tilde{\kappa},\tilde{\rho})=\tilde{\varrho}(\tilde{ \kappa},\tilde{\rho})-\tilde{\varrho}(\tilde{\kappa},\tilde{\rho}-1)\). The following vector-form distributed controller is described:  
\[ \begin{split}\tilde{\boldsymbol{v}}(\tilde{\kappa},\tilde{\rho}) =\tilde{\boldsymbol{v}}(\tilde{\kappa},\tilde{\rho}-1)+\beta_{1}\mathbf{G}( \tilde{\kappa},\tilde{\rho})(L+O)\boldsymbol{r}(\tilde{\kappa},\tilde{\rho}) \\ \times\left(\tilde{\varrho}(\tilde{\kappa}+1,\tilde{\rho}-1)- \boldsymbol{h}(\tilde{\kappa},\tilde{\rho})\right)\end{split}  \tag{19}\]  
After combining Equations (18) and (19), the expression is transformed as follows:  
\[ \begin{split}\Delta\underline{\underline{z}}_{s}(\tilde{\kappa}+1, \tilde{\rho})&=\mathbf{G}_{\tilde{\kappa}}(\tilde{\kappa},\tilde{ \rho})\beta_{1}\mathbf{G}(\tilde{\kappa},\tilde{\rho})(L+O)\boldsymbol{r}( \tilde{\kappa},\tilde{\rho})\\ &\times\left(\tilde{\varrho}(\tilde{\kappa}+1,\tilde{\rho}-1)- \boldsymbol{h}(\tilde{\kappa},\tilde{\rho})\right)+\Delta\boldsymbol{h}( \tilde{\kappa},\tilde{\rho})\end{split}