# 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
Considering the time-varying nature of the PPDs, directly measuring it poses a challenge. Therefore, an optimization objective is constructed based on input-output data, with the objective function expressed as follows:  
\[Q\Big{(}\hat{N}_{i}(\tilde{\kappa},\tilde{\rho})\Big{)}=\Big{(}\Delta\tilde{ \varepsilon}_{si}(\tilde{\kappa}+1,\tilde{\rho}-1)-\hat{N}_{i}(\tilde{\kappa },\tilde{\rho})\Delta\tilde{o}_{i}(\tilde{\kappa},\tilde{\rho}-1)\Big{)}^{2}+ \delta\Big{  
perturbation error measurements \(\delta_{hi}(\tilde{\kappa},\tilde{\rho})=\tilde{z}_{0}(\tilde{\kappa})-\tilde{z}_{hi }(\tilde{\kappa},\tilde{\rho})\) are given. Subsequently, we can derive \(\tilde{\varrho}_{zi}(\tilde{\kappa},\tilde{\rho})=\tilde{\varrho}_{hi}(\tilde{ \kappa},\tilde{\rho})h_{i}(\tilde{\kappa},\tilde{\rho})\), and it is noted that \(0<h_{i}(\tilde{\kappa},\tilde{\rho})\leq 1\).  
Lemma 2 ([37]): _Given the saturation function as defined in (4), it follows that for sensor \(i\), the input gain to output gain correlation can be expressed by \(\Delta\tilde{z}_{zi}(\tilde{\kappa},\tilde{\rho})=g_{i}(\tilde{\kappa},\tilde{ \rho})\Delta\tilde{z}_{hi}(\tilde{\kappa},\tilde{\rho})\), with the constraint \(0\leq g_{i}(\tilde{\kappa},\tilde{\rho})\leq 1\)._  
Lemma 3 ([38; 39]): _Consider the matrix \(\Theta(P)\), defined as an irreducible sub-stochastic iteration varying matrix with all positive diagonal entries. Let \(\Theta\) denote the collection of such matrices \(\Theta(P)\). It is established that the following inequality is satisfied:_  
\[\|\Theta(P)\Theta(P-1)\ldots\Theta(1)\|\leq\Im, \tag{12}\]  
_where \(0<\Im<1\) and \(\Theta(P)\) is arbitrarily selected from \(\Theta\)._  
Theorem 3.1: _Under the premise that the microgrid satisfies Assumptions 1 and 2, if the control parameters \(\beta_{1}\) and \(\delta_{1}\) meet the conditions \(\beta_{1}<1/\max_{i=1,\ldots,N}(\sum_{j=1}^{N}a_{ij}+o_{i})\) and \(\delta_{1}>\delta_{1 min}>0\), respectively, then the DAILC algorithm can achieve the output voltage tracking of the preset reference voltage \(\tilde{\varepsilon}_{0}(\tilde{\kappa})\) for DREGUs with bounded tracking errors._  
Proof: The proof comprises two parts as follows. _Part (i) (Boundedness of the Estimation Value \(\hat{\aleph}_{i}(\tilde{\kappa},\tilde{\rho})\)):_  
Define \(\hat{\aleph}_{i}(\tilde{\kappa},\tilde{\rho})=\aleph_{i}(\tilde{\kappa},\tilde {\rho})-\hat{\aleph}_{i}(\tilde{\kappa},\tilde{\rho})\). From Lemma 2 and the PPD estimation algorithm (11), we have  
\[ \begin{split}\hat{\aleph}_{i}(\tilde{\kappa},\tilde{\rho})=& \aleph_{i}(\tilde{\kappa},\tilde{\rho})-\hat{\aleph}_{i}(\tilde{ \kappa},\tilde{\rho}-1)-\frac{\beta_{2}\Delta\tilde{\varrho}_{i}(\tilde{ \kappa},\tilde{\rho}-1)}{\delta_{2}+\left|\Delta\tilde{\varrho}_{i}(\tilde{ \kappa},\tilde{\rho}-1)\right|^{2}}\\ &\times\left(\Delta\tilde{z}_{zi}(\tilde{\kappa}+1,\tilde{\rho}-1 )-\hat{\aleph}_{i}(\tilde{\kappa},\tilde{\rho}-1)\Delta\tilde{\varrho}_{i}( \tilde{\kappa},\tilde{\rho}-1)\right)\\ =&\tilde{\aleph}_{i}(\tilde{\kappa},\tilde{\rho}-1 )+\Delta\aleph_{i}(\tilde{\kappa},\tilde{\rho})-\frac{\beta_{2}\Delta\tilde{ \varrho}_{i}(\tilde{\kappa},\tilde{\rho}-1)}{\delta_{2}+\left|\Delta\tilde{ \varrho}_{i}(\tilde{\kappa},\tilde{\rho}-1)\right|^{2}}\\ &\times\left(\Delta\tilde{z}_{zi}(\tilde{\kappa}+1,\tilde{\rho}-1 )-\hat{\aleph}_{i}(\tilde{\kappa},\tilde{\rho}-1)\Delta\tilde{\varrho}_{i}( \tilde{\kappa},\tilde{\rho}-1)\right)\\ =&\Delta\aleph_{i}(\tilde{\kappa},\tilde{\rho})+ \left(1-\frac{\beta_{2}(\Delta\tilde{\varrho}_{i}(\tilde{\kappa},\tilde{\rho}-1 ))^{2}}{\delta_{2}+(\tilde{\varrho}_{i}(\tilde{\kappa},\tilde{\rho}-1))^{2}} \right)\hat{\aleph}_{i}(\tilde{\kappa},\tilde{\rho}-1)\\ &+\frac{\beta_{2}(\Delta\tilde{\varrho}_{i}(\tilde{\kappa},\tilde {\rho}-1))^{2}}{\delta_{2}+(\tilde{\varrho}_{i}(\tilde{\kappa},\tilde{\rho}-1 ))^{2}}(1-g_{i}(\tilde{\kappa},\tilde{\rho}-1))\aleph_{i}(\tilde{\kappa},\tilde{ \rho})\end{split}