## 2 The Structure of Typical DC Microgrids and Traditional Control Strategies
### PI Double Closed-Loop Control  
PI double closed-loop control is widely utilized in HESS. DC/DC converters of batteries are chosen as a typical example to illustrate the principle of PI control. The bidirectional DC/DC converter of batteries is shown in Figure 4. \(V_{dc}\) is the DC bus voltage, \(C_{bus}\) is the DC bus capacitance, and \(L\) is the inductor of the battery converter. SW\({}_{1}\) and SW\({}_{2}\) represent the power switches of the battery converter. The states of the power switches are 0 (off) or 1 (on). D, \(i_{L}\) and \(i_{load}\) are the duty cycle, inductor current, and load current, respectively. \(V_{bat}\) is the battery voltage. Based on Figure 4, state equations are derived and shown as  
\[\left\{ \begin{array}{l}L\frac{di_{1}}{dt}=V_{\text{bat}}-D\cdot V_{\text{dc }}\\ C_{bus}\frac{dV_{\text{dc}}}{dt}=D\cdot i_{L}-i_{load}\end{array} \right. \tag{9}\]  
PI double closed-loop control adopted in bidirectional Buck-Boost battery converters is shown in Figure 5. The outer voltage control loop is used to maintain stable DC bus voltage, and the inner current control loop determines the charging or discharging currents of batteries. Figure 3: The principle of the low-pass filter (LPF). In the depiction presented in Figure 5, the discrepancy between the reference voltage and the DC bus voltage is calculated and subsequently utilized as the input signal for the exterior voltage control loop. The output derived from this outer PI controller serves as the benchmark current for the interior loop. Leveraging the deviation between the reference current and the inductor current, the interior PI controller's outputs contribute to the generation of PWM signals. The voltage and current controllers' transfer functions are \(G_{v}(s)\) and \(G_{i}(s)\) with \(k_{vi}\) and \(k_{vp}\) representing the voltage controller's integral and proportional coefficients, respectively, while analogous coefficients for the current controller are also specified. \[\left\{ \begin{aligned} G_{i}(s)&=k_{ip}+\frac{k_{ii}} {s}\\ G_{v}(s)&=k_{vp}+\frac{k_{vi}}{s}\end{aligned} \right. \tag{10}\]  
Subsequently, utilizing the voltage control loop's transfer function, bode plots are constructed and presented in Figure 6a,b. These plots offer insights into the system's dynamic behavior. Additionally, step response diagrams illustrating the voltage control loop's performance under varying \(k_{p}\) and \(k_{i}\) values are shown in Figures 6c and 6d, respectively. To ensure robust stability margins, as evident in Figure 6a,b, the optimal PI parameters for the battery converter's outer loop are determined to be \(k_{vp}=1.5\), \(k_{vi}=200\). These parameters facilitate minimal overshoot and swift regulation, as demonstrated in Figure 6c,d. Furthermore, the battery converter's inner current loop PI parameters are set to \(k_{ip}=45\) and \(k_{ii}=0.1\), respectively. Analogously, for the supercapacitor converter, the optimal outer-loop PI parameters are identified as \(k_{pSC}=1\) and \(k_{iSC}=100\), while the inner-loop PI parameters are set to \(k_{pSC1}=0.1\) and \(k_{iSC1}=20\), respectively. These parameter selections aim to optimize the converter's performance characteristics. However, the disturbance immunity of traditional PI control is extremely weak, and due to the essential characteristic of PI control, the smallest overshoot and shortest regulation time are not accomplished at the same time. It is essential to introduce a new control strategy to increase response speed, decrease overshoots, and improve immunity to disturbances. Figure 4: The topology of the bidirectional Buck–Boost converters in HESS. Figure 5: The block diagram of PI double closed-loop control. Active Disturbance Rejection Control Combined with Improved MPC of Large-Capacity HESS in DC Microgrids  
A comprehensive hybrid energy storage system (HESS) incorporates a network of n batteries, each equipped with a dedicated battery converter, alongside a supercapacitor and its corresponding converter.