## 3 Optimization of Controllable Rectification Based on Advanced Control Strategies
### MPSMC Control Strategy
#### 3.3.2 Design of Sliding Mode Controller  
Traditional control systems depend on PI controllers for setting active power references in the outer loop, employing linear control strategies. Nevertheless, when system parameters or operating conditions vary, the control effectiveness under such strategies is subpar. In comparison, sliding mode control is better equipped to handle nonlinear systems. Once the controller attains the sliding surface, its sensitivity to system parameters diminishes. Hence, to significantly enhance control effectiveness, the outer-loop controller is substituted with a sliding mode controller. According to the law of conservation of energy, the input power of a rectifier should equal its output power. For simplicity, this paper ignores any power loss during rectifier operation. \[P=V_{\text{dc}}i_{\text{dc}}=CV_{\text{dc}}\frac{d}{dt}V_{\text{dc}}+\frac{1} {R_{L}}V_{\text{dc}}^{2} \tag{16}\]  
In order to achieve precise tracking of the output voltage to the expected voltage value, a tracking error \(e\) will be defined, which is used to measure the deviation between the output voltage and the expected voltage. \[e=V_{\text{dc}}^{*}-V_{\text{dc}} \tag{17}\]  
To achieve zero steady-state error, a proportional-integral sliding surface for voltage error is designed using the sliding mode variable structure control principle. \[s=c_{1}e+c_{2}\int edt \tag{18}\]  
In the formula, \(c_{1}>0\), \(c_{2}>0\), taking the derivative of the sliding surface, is obtained by substituting the above formula. \[\dot{s}=-c_{1}\bigg{(}\frac{P}{CV_{\text{dc}}}-\frac{V_{\text{dc}}}{R_{L}C} \bigg{)}+c_{2}(V_{\text{dc}}^{*}-V_{\text{dc}}) \tag{19}\]  
To enhance the efficiency and performance of the system in reaching the sliding mode, a fast power reaching law was employed in the design of the voltage outer-loop controller. \[\dot{s}=-k_{1}s-k_{2}|s\,|\,^{a}sgn(s) \tag{20}\]  
In the formula, \(k_{1}>0\), \(k_{2}>0\), \(0<\alpha\), the \(sgn\) is a signed function passed through \(\alpha\). It is ensured that when the system state moves away from the sliding mode surface, it can swiftly approach the sliding mode to accelerate the convergence speed by using a fast power-law approach law. As the system state gradually approaches the sliding mode surface, reducing the control gain can effectively reduce the chattering phenomenon and achieve a smoother transition. By combining the above equations, we can obtain:  
\[ \begin{array}{c}k_{1}s+k_{2}|s\,|\,^{a}sgn(s)-c_{1}\frac{P}{CV_{\text{dc}}} +c_{1}\frac{V_{\text{dc}}}{R_{L}C}+\\ c_{2}\big{(}V_{\text{dc}}^{*}-V_{\text{dc}}\big{)}=0\end{array}  \tag{21}\]  
Further conversion:  
\[P_{\text{ref}}=\frac{CV_{\text{dc}}}{c_{1}}\bigg{(}k_{1}s+k_{2}|s\,|\,^{a}sgn (s)+\frac{c_{1}V_{\text{dc}}}{R_{L}C}+c_{2}e\bigg{)} \tag{22}\]  
The control law and reference power Pref of the sliding mode controller were obtained using the above formula. Given the complex operating conditions that three-phase rectifiersmay face in actual operation, in order to reduce chattering, the saturation function \(sat\)\((s)\) was adopted to replace the discontinuous sign function \(sgn\)\((s)\). Replacing the sign function \(sgn\)\((s)\) with a saturation function \(sat\)\((s)\) can significantly reduce chattering in sliding mode control, thereby enhancing system stability. This substitution introduces a boundary layer to smooth the control input, avoiding the abrupt changes that occur when the sign function switches. \[sat(s)=\left\{ \begin{array}{ll}1&s>\Delta\\ \frac{s}{\Delta}&|s|\geq\Delta\\ -1&s<-\Delta\end{array} \right. \tag{23}\]  
In the formula, \(\Delta\) is the thickness of the boundary layer and \(\Delta>0\). After replacing the switch function, the above equation can be rewritten as:  
\[P_{\text{ref}}=\frac{CV_{\text{dc}}}{c_{1}}\bigg{(}k_{1}s+k_{2}|s\,|\,^{\alpha }sat(s)+\frac{c_{1}V_{\text{dc}}}{R_{L}C}+c_{2}e\bigg{)} \tag{24}\]  
To enhance system responsiveness and ensure an adjustment time within 0.2, appropriate sliding--mode controller parameters must be calculated.