## 2 The Structure of Typical DC Microgrids and Traditional Control Strategies
### Active Disturbance Rejection Control of Outer Voltage Control Loop  
Based on the bidirectional Buck-Boost converter in Figure 4, supercapacitors have the same topology as batteries. The active disturbance rejection control of the outer voltage control loop is shown in Figure 7. According to power balance, the coefficient \(k=v_{bat}/v_{dc}\). \(i_{L}\) is the current of the inductor, \(i_{dc}\) is the current of the converter, and \(i_{load}\) is the load current. Figure 6: Bode diagrams and step response diagrams of voltage control loop: (**a**) Bode diagrams of voltage control loop when \(k_{p}\) changes, (**b**) Bode diagrams of voltage control loop when \(k_{i}\) changes. (**c**) Step response diagrams of voltage control loop when \(k_{p}\) changes, (**d**) Step response diagrams of voltage control loop when \(k_{i}\) changes. Taking the DC bus voltage and inductor current as state variables, it is derived and shown as  
\[C_{bus}\frac{d\vartheta_{dc}}{dt}=\frac{v_{bat}}{v_{dc}}i_{L}-i_{load} \tag{11}\]  
Equation (11) is transformed as  
\[C_{\text{bus}}\cdot v_{\text{dc}}\frac{d\vartheta_{\text{dc}}}{dt}=v_{\text{ bat}}\cdot i_{\text{L}}-i_{\text{load}}\cdot v_{\text{dc}} \tag{12}\]  
Linearizing (12) around the equilibrium point, it is obtained and shown as  
\[C_{\text{bus}}\cdot v_{\text{dc}}\cdot\hat{\vartheta}_{\text{dc}}=v_{\text{ bat}}\cdot\hat{i}_{\text{L}}-I_{\text{load}}\cdot\hat{v}_{\text{dc}}-V_{ \text{dc}}\cdot\hat{i}_{\text{load}} \tag{13}\]  
According to (13), it is derived and shown as  
\[\hat{\vartheta}_{\text{dc}}=\frac{1}{C_{\text{bus}}}D\hat{i}_{\text{L}}- \frac{1}{C_{\text{bus}}}\cdot\frac{I_{\text{load}}}{V_{\text{dc}}}\hat{ \vartheta}_{\text{dc}}-\frac{1}{C_{\text{bus}}}\hat{i}_{\text{load}} \tag{14}\]  
In the outer loop, let input \(u=\hat{i}_{L}\), output \(y=\hat{\vartheta}_{dc}\), and \(h\) is a perturbation, then there is a first-order controlled object and shown as  
\[\dot{y}=-\alpha y+h+b_{0}u \tag{15}\]  
In (15), \(b_{0}=\frac{D}{C_{\text{bus}}}\), \(\alpha=(-\frac{1}{C_{\text{bus}}}\cdot\frac{I_{\text{load}}}{V_{\text{dc}}})\), \(h=-\frac{1}{C_{\text{bus}}}\hat{i}_{load}\). \(h\) is loading current disturbance. According to \(D=V_{bat}/V_{dc}\), it is obtained and shown as  
\[b_{0}=\frac{D}{C_{\text{bus}}}=\frac{1}{C_{\text{bus}}}\cdot\frac{V_{\text{ bat}}}{V_{\text{dc}}} \tag{16}\]  
Then the total perturbations \(f\) is expressed as  
\[f=-\alpha y+h=-\frac{1}{C_{\text{bus}}}\cdot\frac{I_{\text{load}}}{V_{\text{ dc}}}\phi_{\text{dc}}-\frac{1}{C_{\text{bus}}}\hat{i}_{\text{load}} \tag{17}\]  
Both internal and external disturbances are involved in (17), such as disturbances of load current and disturbances of DC bus voltage. Introducing the state variable \(x_{1}=y\), and the expanded state variable \(x_{2}=f\), the state space of the controlled object is obtained as follows:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}+b_{0}u\\ \dot{x}_{2}=\dot{f}\\ y=x_{1}\end{array} \right. \tag{18}\]  
According to (18), the continuous linear expansion state observer is obtained as  
\[\left\{ \begin{array}{l}\dot{\hat{x}}_{1}=\beta_{1}(y-\hat{x}_{1})+\hat{x}_{2 }+b_{0}u\\ \dot{\hat{x}}_{2}=\beta_{2}(y-\hat{x}_{1})\\ \dot{y}=\hat{x}_{1}\end{array} \right. \tag{19}\]  
Figure 7: Active disturbance rejection control of outer voltage loop. In (19), \(\beta_{1}\) and \(\beta_{2}\) are the state observer gains. The linearly expanding state observer allows tracking various variables of the controlled object in real time. For the second-order linearly expanding state observer, this ADRC control is simplified by a proportional controller for the system and shown as  
\[u_{0}=k_{\text{P}}(r-\hat{x}_{1}) \tag{20}\]  
Designing state feedback controllers it is  
\[u=\frac{1}{b_{0}}[k_{\text{P}}(r-\ \hat{x}_{1})-\ \hat{x}_{2}] \tag{21}\]  
In (21), \(r\) is the bus voltage setpoint and \(k_{p}\) is the feedback controller gain. The Laplace transform for the state observer expression of (18) yields  
\[\left\{