## 1 Introduction
To better explore and utilize the advantages of PID control, there are currently two main approaches to improving PID control. The first approach is to improve PID control by integrating intelligent methods. References [17; 19] combine dynamic programming and fuzzy control, two control methods without precise models, with PID control [20; 21; 22; 23; 24]. The advantage of these methods is that they can adjust the PID controller parameters in real-time based on the uncertainties of load pulses. However, the performance of the fuzzy control method is highly dependent on the design of fuzzy rules and the selection of membership functions. This typically requires specialized knowledge and experience and may involve extensive experimentation and tuning, resulting in high design complexity and poor robustness. Consequently, it can reduce the real-time performance and accuracy of bus voltage tracking. The second approach to improving PID control is to enhance its modules. By improving the integral module of the PID controller, Active Disturbance-Rejection Control (ADRC) has been proposed. ADRC offers a faster response, smaller overshoot and better robustness, and it has been widely applied to various controlled objects, gaining industry recognition [25; 26]. However, ADRC's performance depends on parameter selection and tuning, and the parameters are relatively sensitive [27; 28; 29]. The primary objective of this study is to address the transient disturbance control in a hybrid energy storage microgrid comprising lithium batteries and supercapacitors. Initially, a hybrid energy storage microgrid was constructed based on its unique characteristics. Subsequently, a traditional second-order linear Active Disturbance-Rejection Controller (ADRC) was enhanced by integrating a BP neural network. This integration facilitates the quantification of uncertainties and the adjustment of parameters \(k_{p}\) and \(k_{d}\). The uncertainty quantification method employed is based on system identification using MATLAB R2023b, which offers the advantage of rapidly quantifying linear disturbances. The main work of this paper is as follows:  
1. The main work of the Section 2 of this paper is to build an overall base model of the lithium battery-supercapacitor hybrid energy storage system, of which the details are, respectively, built lithium battery model, supercapacitor model, DC/DC bidirectional converter model, DC bus model, load-side perturbation model, as well as uncertainty perturbation model of the lithium battery-supercapacitor hybrid energy storage microgrid. 2. In the Section 3 of this paper, uncertainty is introduced on the basis of the built lithium battery-supercapacitor hybrid energy storage microgrid, and the quantization uncertainty model of system identification based on MATLAB is built. After quantization, the Back Propagation Neural Network is introduced to optimize the \(k_{p}\) and \(k_{d}\) parameters of the second-order linear Active Disturbance Rejection Controller. 3. Section 4 of this paper then presents the quantized uncertainty-based BP-ADRC controller in lithium battery-supercapacitor hybrid energy storage microgrids to test the suppression effect of load disturbances and uncertainty disturbances, using the PID controller, the LPV robust controller and the second-order linear ADRC controller as the substrate, to test the bus voltage tracking under the four controllers, and introduce three evaluation indexes, \(IAE\), \(rip\), \(\eta_{a}\), to judge the bus voltage-tracking effect of individual controllers, and then introduce a BP network to optimize the parameters of \(k_{p}\) and \(k_{d}\). Section 4 of this paper tests the proposed uncertainty quantification-based BP-ADRC controller, which is used as the base for the load and uncertainty disturbances. 4. Section 5 of this paper mainly analyzes of Section 4 and the simulation curve data of each group. It compares the curve tracking and oscillation of the BP-ADRC controller in busbar voltage tracking with other controllers in detail.