# Data-Driven Voltage Control in Isolated AC Microgrids Subject to Sensor Saturation
## 2 Preliminaries and Problem Formulation
### Sensor Saturation  
In AC microgrids, voltage transformers collect \(\tilde{z}_{i}(\tilde{\kappa})\), the voltage output from each DG, using instruments like data acquisition systems and phasor measurement units. However, under conditions such as transient voltage or current spikes, electrical faults, or line overloads, these transformers may become saturated. Adverse weather conditions and harsh working environments can also accelerate the aging of sensors, increasing the risk of saturation. Once saturation occurs, the sensor output will deviate from the actual measured values, leading to inaccuracies in the voltage secondary control design, which relies solely on measured outputs to update control protocol inputs. Moreover, the presence of measurement noise is natural and inevitable.  
Based on the analysis above, we define sensor saturation in the AC microgrids as follows:  
\[\tilde{z}_{si}(\tilde{\kappa})=\sigma_{s}(\tilde{z}_{di}(\tilde{\kappa}))= \left\{\begin{array}{ll}-s,&\text{if }\tilde{z}_{di}(\tilde{\kappa})<-s\\ \tilde{z}_{di}(\tilde{\kappa}),&\text{if }-s\leq\tilde{z}_{di}(\tilde{\kappa}) \leq s\\ s,&\text{if }\tilde{z}_{di}(\tilde{\kappa})>s\end{array}\right. \tag{4}\]  
where \(\tilde{z}_{si}(\tilde{\kappa})\) is the sensor saturation output voltage, \(\tilde{z}_{di}(\tilde{\kappa})=\tilde{z}_{i}(\tilde{\kappa})+\tilde{h}_{i}( \tilde{\kappa})\) corresponds to the sensor's input including noise, and \(\tilde{h}_{i}(\tilde{\kappa})\) signifies noise in the measurements. \(s>0\) is referred to as the saturation threshold, and \([-s,s]\) defines the sensor's actual maximum measurement range.  
**Remark 1**: _With the widespread application of new energy devices, the components of AC microgrids have become increasingly diverse and complex. AC microgrids are inherently nonlinear and heterogeneous. In such environments, sensors are prone to saturation, and the system is often affected by various disturbances, posing challenges to the stable operation of microgrids. Under these conditions, traditional model-based control methods [20; 21] are often inadequate as they usually require accurate system models and parameters, which are difficult to obtain in practice. Therefore, it is particularly important and necessary to develop a control method based on input-output data, which does not rely on system models but directly utilizes data for voltage control of the microgrid._  
**Remark 2**.: _It should be acknowledged that the hard-threshold saturation model used in this paper is an idealized representation. In practice, the saturation behavior of sensors may exhibit more complex forms, such as soft limiting (where the output gradually approaches the saturation limit), dead zones, or non-symmetrical characteristics due to aging. Nevertheless, the choice of the hard-threshold model is justified for several reasons. Firstly, it is a standard model widely used in the control systems literature to represent physical constraints and captures the core nonlinear nature of saturation. Secondly, it presents a challenging, non-smooth scenario for controller design, which serves as a rigorous test for the robustness of our proposed DAILC algorithm. The primary contribution of this work lies in the data-driven adaptive control algorithm itself, rather than in the high-fidelity modeling of sensors. Therefore, using the hard-threshold model as a representative benchmark is appropriate. Investigating the controller's performance under more sophisticated sensor models is a valuable direction for our future work._