## 4 Results
### Experimental Results  
To validate the practical performance of the proposed algorithm in terms of tracking velocity of MPPT and MPPT efficiency, a small scale PV system is built. In the PV system, DSP controlled boost DC-DC converter is implemented for the voltage regulation of the PV system and this boost converter can transfer the PV power to a 26 V DC-link voltage which is connected to two 13 V lead acid batteries. The sunlight is simulated by two generic 50 W electrical light bulbs. Under the test conditions given by  
Figure 16: Second transition caused by the irradiation increasing. Figure 14: Initial tracking time: FLC vs. P and O (0.1 V). Figure 15: First transition caused by the irradiation drop. Table 3, the Boulder 15 W can maximally generate 3.1 W of power. The practical MPPT efficiency [29] can be calculated by the Equation (4)  
\[\eta_{MPPT}=\frac{\textit{average power conducted by algorithm}}{\textit{the potential MPP}} \tag{4}\]  
The objective of the boost DC-DC converter is to force the PV voltage to converge the reference value which is provided by the MPPT algorithm. Perturbing the duty ratio of the PWM signal, which is fed into the converter, can rebuild the steady-state proportion between the input voltage and output voltage. The transient response of the input terminal may not be always satisfied due to the nonlinear characteristics of the switching-mode circuit and to unknown dynamics. Hence, a voltage controller is normally added into the inner control loop of a photovoltaic system in order to fulfill the voltage regulation of PVs and to improve the transient responses. The topology of the photovoltaic boost DC-DC converter is shown in Figure 17. Parameters of the photovoltaic system are listed in Table 3. Characteristics of the Boulder 15 W operating under test conditions are summarized in Table 4.  
By applying [PERSON]'s voltage and current laws, the state space equations of the input terminal can be obtained:  
\[ \begin{split}\frac{d}{dt}\left[\begin{array}{c}\widetilde{I_{L}}\\ \widetilde{V_{pv}}\end{array}\right]&=\left[\begin{array}{cc}-\frac{R_{L}}{L }&\frac{1}{L}\\ -\frac{R_{pv}(1-R_{L}R_{c})}{LC\left(1-R_{pv}\right)}&\frac{CR_{R}R_{c}-1}{LC \left(1-R_{pv}\right)}\end{array}\right]\left[\begin{array}{c}\widetilde{I_{L }}\\ \widetilde{V_{pv}}\end{array}\right]+\left[\begin{array}{c}-\frac{V_{bat}}{L} \\ \frac{R_{c}R_{pv}V_{bat}}{LC\left(1-R_{pv}\right)}\end{array}\right]\widetilde{d} \\ &\widetilde{y}=\left[0\,1\right]\left[\begin{array}{c}\widetilde{I_{L}}\\ \widetilde{V_{pv}}\end{array}\right]\\ G_{vd}(s)=\frac{A}{s^{2}+Ns+M}\end{split}  \tag{5}\]  
where:  
\[ \begin{split}& A=-V_{bat}\left(\frac{1}{L^{2}}+\frac{R_{c}R_{pv} ^{2}-R_{c}R_{pv}}{L^{2}C\left(1-R_{pv}\right)^{2}}-\frac{R_{c}R_{pv}}{L\left( 1-R_{pv}\right)}s\right)\\ & N=\left(\frac{R_{L}}{L}-\frac{R_{c}R_{pv}-1}{LC\left(1-R_{pv}\right)} \right)\\ & M=\frac{R_{L}R_{c}R_{pv}-R_{L}+R_{pv}L-CR_{L}^{2}R_{c}R_{pv}}{L^{2}C \left(1-R_{pv}\right)}\end{split} \]  
Based on the linear approximation of Equation (5), a PI controller is designed as the voltage controller. The digitalized transfer function of the PI controller [37] is given by Equation (6). \[G_{PI}(z)=0.1+\frac{0.05}{1-z^{-1}} \tag{6}\]  
The partial embedded code related to the digital PI controller is generated by using MATLAB/Simulink Embedded Coder Toolbox. The Simulink diagram for the embedded coder is shown in Figure 19. A 0.5 V step response of the inner control loop of the photovoltaic system is illustrated by Figure 20.  
Figure 19: MATLAB/Simulink diagram of the PI controller. Figure 18: The small signal model of the input terminal.