## 3 DC Bus Voltage Control Strategy
### Active Disturbance Rejection Controller  
LADRC is a linearized control method. Its core idea is to use a simple integrator chain as the standard form and then generalize the parts of the system dynamics that differ from the standard form as generalized disturbances, compensating for them. Therefore, LADRC can effectively handle all uncertainties, with advantages such as a simple topology, low-parameter tuning difficulty, and wide applicability [25; 26; 27]. The structure of the ADRC primarily consists of three parts: Linear tracking differentiator (LTD), linear extended state observer (LESO), and linear state error feedback (LSEF). For the DC voltage control of the UPQC system, it can be simplified as a first-order control. In the design of the voltage-current dual-loop control system of the inverter, the control bandwidth of the current inner loop is usually much larger than that of the voltage outer loop. When designing the controller for the voltage outer loop, the dynamic process of the current inner loop can be ignored, and the output current \(i_{d}\) is considered to track the given value \(i_{d}^{*}\) of the voltage control outer loop in real time. For a first-order controlled object, the following can be derived:  
\[u_{0}=f[\omega(t),t]+(b-b_{0})i_{d}^{*}+b_{0}i_{d}^{*} \tag{7}\]  
where \(f[\omega(t),t]\) represents the model errors, input power, and other internal and external disturbances and uncertainties; \(b\) is the unknown controller gain; and \(b_{0}\) is the set approximate controller gain.  
Consider \(f[\omega(t),t]\) and \((b-b_{0})i_{d}^{*}\) as the total disturbances, and design a second-order LESO for state variable observation and disturbance estimation:  
\[\left\{\begin{array}{l}e=z_{1}-u_{dc}\\ \dot{z}_{1}=z_{2}-\beta_{1}e+b_{0}i_{d}^{*}\\ \dot{z}_{2}=-\beta_{2}e\end{array}\right. \tag{8}\]  
\[\left\{\begin{array}{l}\beta_{1}=2\omega_{0}\\ \beta_{2}=\omega_{0}^{2}\end{array}\right. \tag{9}\]  
where \(e\) is the state variable observation error; \(\beta_{1}\) and \(\beta_{2}\) are the correction coefficients of the extended state observer, and \(\omega_{0}\) is the observation bandwidth of the observer.  
In ADRC, \(z_{1}\), \(z_{2}\) and \(b_{0}\) are key control variables. \(z_{1}\) represents the system's total disturbance estimation, including both external disturbances and internal system dynamics. \(z_{2}\) is the second-order disturbance estimate obtained via the LESO, which represents the rate of change of the disturbance or its higher-order dynamic features. \(b_{0}\) is the control gain used to adjust the effectiveness of disturbance compensation, ensuring robust system performance. By estimating these disturbances in real time, ADRC allows for dynamic compensation of both external disturbances and internal uncertainties.  
\[i_{d}^{*}=\frac{1}{b_{0}}[k_{\mathrm{P}}(u_{dc}^{*}-z_{1})-z_{2}] \tag{10}\]  
where \(u_{dc}^{*}\) is the expected value of the DC side capacitor voltage, and \(k_{\mathrm{P}}\) is the proportional gain of the controller.  
By substituting Equation (10) into Equation (7), we obtain:  
\[u_{0}=k_{\mathrm{P}}(u_{dc}^{*}-z_{1}) \tag{11}\]  
The simplified first-order LADRC control structure can be obtained, as shown in Figure 4.