## 4 Interleaved Inverter
### Control Strategy  
Once the reference power \(P_{R}^{*}\) is available by means of the already discussed MPPT algorithm, the control strategy of the inverter is responsible for assuring that this power is drained only by the fundamental direct component of the grid voltages.  
Decomposing the grid voltage space vector into its harmonics is shown as follows:  
\[\mathbf{v}_{\mathbf{s}}=\sum_{h=-\infty}^{h=+\infty}\mathbf{v}_{\mathbf{s}}^ {h}\mathrm{e}^{ih\omega t}=\sum_{h=-\infty}^{h=+\infty}V_{\mathbf{s}}^{h} \mathrm{e}^{i\psi^{h}}, \tag{26}\]  
The reference space vector current has to satisfy the following:  
\[\mathbf{i}_{\mathbf{s}}^{*}=\frac{2}{3}\frac{P_{R}^{*}}{V_{s}^{1}}\mathrm{e}^{ i\psi^{1}}, \tag{27}\]  
The quantities \(V_{s}^{1}\) and \(\psi^{1}\) are estimated by means of a Phase-Locked Loop algorithm (PLL).  
From the grid's current reference \(\mathbf{i}_{\mathbf{s}}^{*}\), it is possible to determine the converter space vector reference voltage \(\mathbf{v}_{\mathbf{r}}^{*}\) by means of a _minimum delay deadbeat control algorithm_. Denoting the sampling time interval with \(T_{s}\) and the generic sampling time instant with \(t_{m}\), the algorithm is based on the following equation:  
\[\mathbf{v}_{\mathbf{r}}^{*}\left(t_{m}\right)=\frac{L_{R}}{2T_{s}}\left[ \mathbf{i}_{\mathbf{s}}\left(t_{m}\right)-\mathbf{i}_{\mathbf{s}}^{*}\left(t_ {m}\right)\right]+\frac{\mathbf{v}_{\mathbf{s}}\left(t_{m-1}\right)+2\mathbf{v }_{\mathbf{s}}\left(t_{m}\right)+\mathbf{v}_{\mathbf{s}}\left(t_{m+1}\right)}{ 2}-\mathbf{v}_{\mathbf{r}}^{*}\left(t_{m-1}\right), \tag{28}\]As the grid voltages are repetitive, in a steady state condition the prediction of the voltage grid space vector \(\mathbf{v_{s}}\left(t_{m+1}\right)\) can be estimated by memorizing the \(\mathbf{v_{s}}\) values into a circular buffer; the minimum buffer size must be one period of \(\mathbf{v_{s}}\). Sudden perturbations of the grid voltage can be easily identified by comparing the sampled value with the expected one. In this case, the prediction can only be based on the most recent sample, i.e., \(\mathbf{v_{s}}\left(t_{m+1}\right)=\mathbf{v_{s}}\left(t_{m}\right)\).  
The total interleaved inverter space vector reference voltage value can be derived from Equation (28), which, according to Equation (25), is the semi-sum of the \(p\) and \(n\) converters' space vector reference voltages. In order to separately calculate the references for the upper and lower half bridges' converters, an optimum condition has to be added. In an ideal case, the common mode currents would be null in all the three phases:  
\[\left\{\begin{array}{l}\mathbf{i}_{s}^{\ast}\left(t_{m}\right)=0\\ \mathbf{i}_{s,0}^{\ast}\left(t_{m}\right)=0\end{array}\right., \tag{29}\]  
The differential voltage components can again be calculated by means of a deadbeat control:  
\[\left\{\begin{array}{l}\Delta\mathbf{v_{r}^{\ast}}\left(t_{m}\right)=\frac{ L_{p}}{T}\mathbf{i}_{s}^{\ast}\left(t_{m}\right)-\Delta\mathbf{v_{r}^{\ast}} \left(t_{m-1}\right)\\ \Delta\mathbf{v_{r,0}^{\ast}}\left(t_{m}\right)=\frac{L_{p}}{T_{s}}\mathbf{i}_ {s,0}^{c}\left(t_{m}\right)-\Delta\mathbf{v_{r,0}^{\ast}}\left(t_{m-1}\right) \end{array}\right., \tag{30}\]  
Based on the position in Equation (25), the reference space vector voltages for the two converters can be evaluated:  
\[\left\{\begin{array}{l}\mathbf{v_{r}^{\mathbf{p}^{\ast}}}\left(t_{m}\right)= \mathbf{v_{r}^{\ast}}\left(t_{m}\right)+\Delta\mathbf{v_{r}^{\ast}}\left(t_{m} \right)\\ \mathbf{v_{r}^{\mathbf{n}}}\left(t_{m}\right)=\mathbf{v_{r}^{\ast}}\left(t_{m} \right)-\Delta\mathbf{v_{r}^{\ast}}\left(t_{m}\right)\end{array}\right., \tag{31}\]