# Data-Driven Voltage Control in Isolated AC Microgrids Subject to Sensor Saturation
## 3 The DAILC Algorithm of Voltage Control Subject to Sensor Saturation
### Dynamic Linearization Model of The DREGU  
The AC microgrid is composed of DREGUs. Following the discretization of the large-signal dynamic model (1) of the microgrid, we concentrate solely on secondary voltage control. This results in a single-input single-output nonlinear AC microgrid model, presented as follows [33]:  
\[\underline{z}_{i}(\bar{\kappa}+1)=f_{i}\Big{(}\underline{z}_{i}(\bar{\kappa}),\ldots,\underline{z}_{i}(\bar{\kappa}-n_{\underline{z}}),\vartheta_{i}(\bar {\kappa}),\ldots,\vartheta_{i}(\bar{\kappa}-n_{\rho})\Big{)}+\hbar_{i}(\bar{ \kappa}) \tag{5}\]  
where \(\bar{z}\) denotes the time interval, \(i=1,2,\ldots,N\underline{z}_{i},n_{\bar{\varepsilon}}\) are two unknown positive integers that vary with the scale of the AC microgrids. The terms \(\vartheta_{i}(\bar{\kappa})\in\mathbb{R}\) and \(\underline{z}_{i}(\bar{\kappa})\in\mathbb{R}\) represent the control input voltage and the measured output voltage of DG \(i\) at time \(\bar{\kappa}\), respectively. \(\hbar_{i}(\bar{\kappa})\in\mathbb{R}\) is a bounded disturbance, with \(|\hbar_{i}(\bar{\kappa})|\leq r_{\hbar}\) where \(r_{\hbar}\) is a small positive constant. This model can be linearized, subject to the following two assumptions: in the above, \(\underline{z}\) signifies the time intervals, and for each \(i=1,2,\ldots,N\), the integers \(n_{\underline{z}}\) and \(n_{\rho}\), which are unknown positive values, depend on the AC microgrids' scale. For each distributed generator \(i\), \(\bar{v}_{i}(\bar{\kappa})\) and \(\underline{z}_{i}(\bar{\kappa})\) represent the control input voltage and the output voltage measured, respectively, both belonging to \(\mathbb{R}\) at any given time \(\bar{\kappa}\). Additionally, \(\hbar_{i}(\bar{\kappa})\), a bounded disturbance, adheres to \(|\hbar_{i}(\bar{\kappa})|\leq r_{\hbar}\), with \(r_{\hbar}\) as a minor positive constant. The model is subject to linearization under the stipulated assumptions:  
**Assumption 1**. : _The function \(f(\cdot)\) has continuous partial derivatives with respect to \(\bar{v}(\bar{\kappa}),\vartheta(\bar{\kappa}-1),\ldots,\vartheta(\bar{\kappa }-n_{\rho})\)._  
**Assumption 2**. : _The control system in (5) is generalized Lipschitz, that is to say \(\forall\kappa\in\mathbb{N}^{+},|\Delta_{\bar{\kappa}}^{2}(\bar{\kappa}+1)|\leq b| \Delta\bar{v}(\bar{\kappa})|\;\forall|\Delta\bar{v}(\bar{\kappa})|\
eq 0\), where \(b\) is a positive constant._  
**Lemma 1** ([34]). : _Given Assumptions 1 and 2, for any agent \(i\) with non-zero \(\Delta\bar{v}_{i}(\bar{\kappa})\) and \(\Delta\bar{h}_{i}(\bar{\kappa})\), Equation (1) can be equivalently recast as a dynamic data model for all \(\kappa=1,2,\ldots\)_  
\[\tilde{z}_{si}(\bar{\kappa}+1)-\tilde{z}_{si}(\bar{\kappa})=\aleph_{i}(\bar{ \kappa})(\bar{v}_{si}(\bar{\kappa})-\bar{v}_{si}(\bar{\kappa}-1))+\Delta\bar{ h}_{i}(\bar{\kappa}). \tag{6}\]  
_let \(\aleph_{i}(\bar{\kappa})\) denote the pseudo-partial derivative (PPD) parameters for each distributed generator \(i\), which are dependent on the iterations and vary over time. These parameters comply with the condition \(\aleph_{i}(\bar{\kappa})\leq b\), where \(b\) represents a constant with a positive value._  
**Remark 3**. : _The advantage of the proposed microgrid dynamic model lies in its ability to function without detailed dynamical models for each DREGU, relying solely on input-output data for modeling. Furthermore, the algorithm for estimating the PPD parameters (7), which is elaborated upon subsequently, operates solely on a data-driven basis. It utilizes just the necessary input data alongside the saturated outputs from each DREGU to configure the parameters. Compared to traditional model-based methods [17; 18] and the deep learning techniques that deal with extensive off-line datasets as mentioned in [26; 27], this algorithm demonstrates greater universality and robustness._  
**Remark 4**.