## 2 System Description
### Clarke and Park Transformations
#### 2.2.2 Park Transformation  
The two orthogonal variables \(v_{a}\) and \(v_{\beta}\) obtained with the Clarke transformation are provided to a system of rotating vectors, \(v_{d}\) and \(v_{q}\), which rotate at an angle \(\theta\). The Park transformation is defined by the following equation considering \(v_{a}\) and \(v_{\beta}\) the orthogonal variables resulting from the application of the Clarke transformation [15]:  
\[\left[ \begin{array}{c}v_{d}\\ v_{q}\end{array} \right]=\left[ \begin{array}{cc}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{array} \right]\cdot\left[ \begin{array}{c}v_{a}\\ v_{\beta}\end{array} \right] \tag{10}\]  
and the inverse transformation is:  
\[\left[ \begin{array}{c}v_{a}\\ v_{\beta}\end{array} \right]=\left[ \begin{array}{cc}\cos\theta&-\sin\theta \\ \sin\theta&\cos\theta\end{array} \right]\cdot\left[ \begin{array}{c}v_{d}\\ v_{q}\end{array} \right] \tag{11}\]  
Considering the variables \(v_{a}\) and \(v_{\beta}\) concerning expressions obtained from (8) and (9), we obtain the variables \(v_{d}\) and \(v_{q}\)[15]:  
\[v_{d}=V_{a}\cdot\sin\omega t\cdot\cos\theta-V_{a}\cdot\cos\omega t\cdot\sin\theta \tag{12}\]  
\[v_{q}=-V_{a}\cdot\sin\omega t\cdot\sin\theta-V_{a}\cdot\cos\omega t\cdot\cos\theta \tag{13}\]  
If the angle \(\theta\) varies in time with the same velocity as the argument of the sinusoidal component of the phase from the three-phase system, namely \(\omega t\), and if we consider the substitution \(\theta=\omega t\), we obtain [15]:  
\[v_{d}=V_{a}\cdot(\sin\omega t\cdot\cos\omega t-\cos\omega t\cdot\sin\omega t)=0 \tag{14}\]  
\[v_{q}=-V_{a}\cdot(\sin^{2}\omega t+\cos^{2}\omega t)=-V_{a} \tag{15}\]  
From Equations (14) and (15), resulting from the previously assumed conditions, the variables \(v_{d}\) and \(v_{q}\) obtained by the Park transformation are constant values in time for a perfectly balanced voltage. This method uses the PV array's incremental conductance (dI/dV) to calculate the sign change in the power variation versus voltage variation (dP/dV). Knowing that the power is expressed concerning current, when (dI/dV) is equal to \(-\)I/V or both dV and dI are null, the algorithm knows that the maximum power point is reached. Thus, it ends and returns the corresponding value of the operating voltage for MPP [16]. Alternatively, when the variations are different (dI/dV) from \(-\)I/V, respectively, dV is null, and dI is positive or negative but not zero, the approach is the following: before returning to the voltage value of the MPP, the operating voltage is either increased or decreased. Figure 3 represents the flow chart of the incremental conductance method, and it offers a better understanding of the process [16]. In Figure 4, we have the circuit in the Matlab/Simulink implementation for the incremental conductance algorithm. Both solar panel current and voltage are measured and fitted as input signals for the algorithm. The difference between each signal is obtained by delaying the corresponding input signals (I and V) and subtracting them from the values of the signals sampled and obtained from _Vdelta_ and _Idelta_ signals [17]. The memory block was set to zero for the initial calculation. The block Switch 3 compares the value of _Vdelta_ with zero and divides the algorithm into the top part and the bottom part. The top corresponds to the case when the value of _Vdelta_ is zero. The following comparison is for current values. If the _Idelta_ variation is zero, the adjustment signal passes without any change, and the maximum power point is found. If _Idelta_ is positive, the adjustment signal is incremented by 0.01, and the negative adjustment signal is incremented. This calculation is a linear approximation of the current-voltage slope characteristic near the MPP. The bottom part corresponds to the case when there is a variation of the voltage signal, which means that _Vdelta_ can be any value minus the zero value that has already been calculated. When these conditions are satisfied, the MPPT has been found, and the system stops oscillating [17].