## 3 Maximum Power Point Estimator
### Modeling of Power Generation Unit
\begin{table}
\begin{tabular}{c c c c} \hline
**Item** & **Symbol** & **Unit** & **Value** \\ \hline DC capacitor & \(C_{Rec}\) & (\(\mu\)F) & 4700 \\ DC filter inductor & \(L_{1}\) & (mH) & 1 \\ DC output capacitor & \(C_{A}\) & (\(\mu\)F) & 470 \\ DC resistor & \(R_{DC}\) & (\(\Omega\)) & 30 \\ Boost inductor & \(L_{2}\), \(L_{3}\) & (mH) & 1 \\ DC capacitor & \(C_{B}\) & (\(\mu\)F) & 10 \\ DC capacitor & \(C_{C}\) & (\(\mu\)F) & 1000 \\ AC filter inductor & \(L_{f}\) & (mH) & 1 \\ AC filter capacitor & \(C_{f}\) & (\(\mu\)F) & 10 \\ AC resistor & \(R_{AC}\) & (\(\Omega\)) & 384 \\ \hline \end{tabular}
\end{table} 
Table 1: Circuit component specifications. DC: direct current; AC: alternating current. Figure 7: Circuit diagram in switch ON/OFF. \[A_{1}=\left[ \begin{array}{ccc}-\frac{1}{R_{M}C_{1}}&-\frac{1}{C_{1}}&0\\ \frac{1}{L}&-\frac{(r_{1}+r_{C})}{L}&0\\ 0&0&-\frac{1}{R_{0}C_{0}}\\ \end{array} \right]B_{1}=\left[ \begin{array}{c}\frac{1}{R_{M}C_{1}}\\ 0\\ 0\end{array} \right] \tag{10}\]  
When the switch G is turned off, G is opened and diode D is conducting, and the energy is charged through the smoothing capacitor \(C_{0}\) from the generator and the reactor L and it is channeled to the load \(R_{0}\). Based on the equivalent circuit, the following equations as in the above case, Equations (11)-(16), are derived by using Kirchhoff's law. \[C_{1}\frac{dV_{T}}{dt}+i_{L}=\frac{V_{M}-V_{T}}{R_{M}} \tag{11}\]  
\[L\frac{di_{L}}{dt}+(r_{L}+r_{D})i_{L}=V_{T}-(V_{D}+V_{o}) \tag{12}\]  
\[C_{0}\frac{dV_{o}}{dt}=-\frac{V_{0}}{R_{0}}+i_{L} \tag{13}\]  
Similarly to the case of the switch being on, the following formula is derived by organizing equations as in Equations (14)-(16). \[u=[V_{M}-V_{D}]^{T} \tag{14}\]  
\[\frac{d\dot{x}}{dt}=A_{2}x+B_{2}u \tag{15}\]  
\[A_{2}=\left[ \begin{array}{ccc}-\frac{1}{R_{M}C_{1}}&-\frac{1}{C_{1}}&0\\ \frac{1}{L}&-\frac{(r_{1}+r_{D})}{L}&-\frac{1}{L}\\ 0&\frac{1}{C_{o}}&-\frac{1}{R_{0}C_{1}}\\ \end{array} \right]B_{2}=\left[ \begin{array}{ccc}\frac{1}{R_{M}C_{1}}&0\\ 0&\frac{1}{L}\\ 0&0\\ \end{array} \right] \tag{16}\]  
The average value of the state variable is defined as in Equation (17):  
\[\overline{x}=\left[\overline{v_{T}i_{L}}\overline{v_{0}}\right]^{T} \tag{17}\]  
The duty cycle when the switch G is turn on is defined as \(D\), and when the switch G is turn off it is defined as in Equation (18). \[D^{\prime}\times\left(D+D^{\prime}=1\right) \tag{18}\]  
The state-space averaging equation can be expressed as in the following Equations (19)-(21):  
\[\frac{d\dot{x}}{dt}=A\dot{x}+Bu \tag{19}\]  
where  
\[A=DA_{1}+\dot{D}A_{2}; \tag{20}\]  
\[B=DB_{1}+\dot{D}B_{2}\]  
Therefore,  
\[A_{1}=\left[ \begin{array}{ccc}-\frac{1}{R_{M}C_{1}}&-\frac{1}{C_{1}}&0\\ \frac{1}{L}&-\frac{(r_{1}+Dr_{C}+\dot{D}r_{C})}{L}&-\frac{\dot{D}}{L}\\ 0&\frac{\dot{D}}{C_{o}}&-\frac{1}{R_{0}C_{0}}\\ \end{array} \right]B_{1}=\left[ \begin{array}{ccc}\frac{1}{R_{M}C_{1}}&0\\ 0&\frac{\dot{D}}{L}\\ 0&0\\ \end{array} \right] \tag{21}\]  
The steady-state value \(=\ \left[V_{T}I_{L}V_{0}\right]^{T}\) can be expressed as in the following equation by \(\frac{d\overline{x(t)}}{dt}=0\). \[x=\ -A^{-1}Bu \tag{22}\]Using the above, the Equation (23) matrix is derived:  
\[\left[ \begin{array}{c}V_{T}\\ I_{L}\\ V_{0}\end{array} \right]=\frac{1}{r+R_{M}}\left[ \begin{array}{c}rV_{M}+\dot{D}V_{D }R_{M}\\ V_{M}-\dot{D}V_{D}\\ \dot{D}V_{M}R_{o}-\dot{D}^{2}V_{D}R_{o}\end{array} \right] \tag{23}\]  
where  
\[r=r_{L}+Dr_{G}+D^{\prime}r_{D}+{D^{\prime}}^{2}R_{o} \tag{24}\]  
The output voltage \(V_{T}\) of the double-stator generator is given in Equation (25). \[V_{T}=\frac{rV_{M}+\dot{D}V_{D}R_{M}}{r_{L}+Dr_{G}+D^{\prime}r_{D}+{D^{\prime }}^{2}R_{o}+R_{M}} \tag{25}\]  
The internal voltage \(V_{M}\) of the double-stator generator is given by Equation (26). \[V_{M}=\frac{(r+R_{M})\,V_{T}-\dot{D}V_{D}R_{M}}{r_{L}+Dr_{G}+D^{\prime}r_{D}+{D ^{\prime}}^{2}R_{o}} \tag{26}\]  
The average value \(I_{L}\) can be expressed as in Equation (27).