# Grid-Connected PV Systems Controlled by Sliding via Wireless Communication
## 3 The DC/DC Converter and DC/AC Converter Controllers
### DC/DC Converter Sliding Controller  
There are numerous studies up to now about the control algorithm implemented in the DC/DC converter to track the PV maximum power point while maintaining a constant output voltage. The results of the backstepping algorithm are presented in [19], where results show that the algorithm guarantees the tracking of the PV maximum power point with different weather conditions and the answer of the algorithm is good. However, in this work, it is proposed to test another algorithm to improve the controller dynamic behavior: the non-linear sliding control method which is a variable structure controller (VSC). This algorithm has already been used in [13] in islanding systems. In this new case, the sliding control will be applied to the buck-boost converter, connected between the PV array and the DC/AC converter. The wide operating range allowed by the sliding control increases the DC/DC converter dynamic performance. The high-speed switching law used by the control, makes the system go to a switching function (also denominated sliding surface). In addition, the control makes the system stay in the surface. The value of the DC/DC converter elements and the equations corresponding to the switching surface determine the system dynamic behavior, which is robust under changes of the parameter and/or disturbances. The buck-boost equations, using the state averaging method and the \(D\) parameter, working in continuous conduction mode (CCM) are presented in Equations (2)-(5). \[\dot{x}_{1}=\frac{ip_{PV}}{C_{1}}-\frac{x_{2}}{C_{1}}D \tag{2}\]  
\[\dot{x}_{2}=-\frac{x_{3}}{L}+\frac{(x_{1}+x_{3})}{L}D \tag{3}\]  
\[\dot{x}_{3}=\frac{x_{2}}{C}-\frac{x_{3}}{RC}-\frac{x_{2}}{C}D \tag{4}\]  
Figure 5: Topology of the inverter. \[\dot{x}_{4}=x_{4} \tag{5}\]  
where \(x_{1}=v_{PV};x_{2}=i_{L};x_{3}=v_{o};x_{4}=\int(v_{PV}-v^{r}_{PV})dt\) and \(v^{r}_{PV}\) is the reference voltage to be reached. Equations (2)-(5) can be expressed in matrix form, Equation (6). \[\dot{x}=Ax+\delta+D(Bx+\gamma) \tag{6}\]  
Matrices are presented in Equations (7)-(10). \[A=\left( \begin{array}{cccc}0&0&0&0\\ 0&0&-\frac{1}{L}&0\\ 0&\frac{1}{L}&-\frac{1}{RC}&0\\ 0&0&0&1\end{array} \right) \tag{7}\]  
\[\delta=\left( \begin{array}{c}\frac{i_{pc}}{C_{1}}\\ 0\\ 0\\ 0\end{array} \right) \tag{8}\]  
\[B=\left( \begin{array}{cccc}0&-\frac{1}{C_{1}}&0&0\\ \frac{1}{L}&0&\frac{1}{L}&0\\ 0&-\frac{1}{L}&0&0\\ 0&0&0&0\end{array} \right) \tag{9}\]  
\[\gamma=\left( \begin{array}{c}0\\ 0\\ 0\\ 0\end{array} \right) \tag{10}\]  
The surface appropriate for the studied application is proposed in Equation (11). The trajectories around that surface should converge to it and stay on the sliding surface. The proposed switching function guarantees the voltage regulation of the buck-boost converter. \[S(x)=k_{1}(x_{1}-v_{PV}{}^{r})+x_{2}+k_{2}x_{4} \tag{11}\]  
where \(k_{1}\) and \(k_{2}\) are the sliding design parameters. The condition for the sliding surface existence if \(S(x)=\mathrm{d}S/\mathrm{d}t=0\). To define the sliding surface, a control law must be found, which also assures the sliding mode. The control input is composed of two components called corrective control, \(D_{c}\), and equivalent control, \(D_{eq}\), Equation (12). \[D=D_{c}+D_{eq} \tag{12}\]  
Equation (13) express, in a typical way for this controller, the corrective control, which compensates the deviations from the sliding surface. \[D_{c}=-k_{c}sgn(S) \tag{13}\]  
where \(k_{c}\) must be positive. Equation (14) defines the sign of the sliding surface. \[sgn(S)=\left\{ \begin{array}{c}+1,S(x)>0\\ -1,S(x)<0\end{array} \right. \tag{14}\]  
Equation (15) presents the control which makes the sliding surface null to remain on the sliding surface. \[D_{eq}=-\left[\frac{\partial S}{\partial x}B(x)\right]^{-1}\left(\frac{ \partial S}{\partial x}(Ax+\delta)\right) \tag{15}\]Including Equations (5) and (11)-(13) in (15), Equation (16) is obtained.