## 2 Control Strategies Description
### Positive-Sequence Control Strategy  
The control strategies for the majority of the commercially available three-phase, inverter-connected DERs is based on positive-sequence current injection. This is because of the fact that the most common three-phase angle detection techniques are based on phase locked loop (PLL), which use a synchronous reference frame [32]. In other words, the three-phase PLLs use a coordinate transformation from \(abc\) to \(dq\)[12] and, therefore, the output signals of these PLLs have phase angles corresponding to the positive-sequence component of the grid voltage. If it is assumed that only fundamental components are used, the analytical model of the positive-sequence control strategy can then be written as  
\[\underline{v}_{x}=|\underline{v}_{x}|\exp(j\theta_{x}) \tag{3}\]  
where \(\underline{v}_{x}\) is the rms value of the phase voltage, \(|\underline{v}_{x}|\) its magnitude, and \(\theta_{x}\) its respective phase angle. Then, the exchanged sequence currents \(\underline{i}_{0}\), \(\underline{i}_{1}\), and \(\underline{i}_{2}\) can be written as  
\[\begin{bmatrix}\underline{i}_{0}\\ \underline{i}_{1}\\ \underline{i}_{2}\end{bmatrix}=\begin{bmatrix}0&0&0\\ 0&g_{1}&0\\ 0&0&0\end{bmatrix}\begin{bmatrix}\underline{v}_{0}\\ \underline{v}_{1}\\ \underline{v}_{2}\end{bmatrix} \tag{4}\]  
where \(\underline{v}_{0}\), \(\underline{v}_{1}\) and \(\underline{v}_{2}\) are the zero-, positive-, and the negative-sequence voltage components. Parameter \(g_{1}\) is the fundamental input conductance, which is described later in this section. Equation (4) can be transformed into phase quantities by using  
\[\begin{bmatrix}\underline{i}_{a}\\ \underline{i}_{b}\\ \underline{i}_{c}\end{bmatrix}=T^{-1}\begin{bmatrix}0&0&0\\ 0&g_{1}&0\\ 0&0&0\end{bmatrix}T\begin{bmatrix}\underline{v}_{a}\\ \underline{v}_{b}\\ \underline{v}_{c}\end{bmatrix} \tag{5}\]  
where \(T\) is the transformation matrix from phase quantities \((a,b,c)\) to symmetrical components (_0,1,2_). Consequently, the following equations for the phase currents are obtained,  
\[\underline{i}_{a} =\frac{1}{3}\delta_{1}\{\underline{v}_{a}+a\underline{v}_{b}+a^{ 2}\underline{v}_{c}\}\] \[\underline{i}_{b} =\frac{1}{3}\delta_{1}\{a^{2}\underline{v}_{a}+\underline{v}_{b}+a \underline{v}_{c}\}\] \[\underline{i}_{c} =\frac{1}{3}\delta_{1}\{a\underline{v}_{a}+a^{2}\underline{v}_{b} +\underline{v}_{c}\} \tag{6}\]  
with \(a=\rho^{2\pi/3}\).  
\[\underline{i}_{a} =\frac{1}{3}\bigg{\{}g_{1}\left[|\underline{v}_{a}|e^{\theta_{a}} +|\underline{v}_{b}|e^{|(\theta_{b}+\frac{2\pi}{3})}+|\underline{v}_{c}|e^{|( \theta_{c}-\frac{2\pi}{3})}\right]\bigg{\}}\] \[\underline{i}_{b} =\frac{1}{3}\bigg{\{}g_{1}\left[|\underline{v}_{b}|e^{\theta_{b}} +|\underline{v}_{a}|e^{|(\theta_{a}-\frac{2\pi}{3})}+|\underline{v}_{c}|e^{| (\theta_{c}+\frac{2\pi}{3})}\right]\bigg{\}} \tag{7}\] \[\underline{i}_{c} =\frac{1}{3}\bigg{\{}g_{1}\left[|\underline{v}_{c}|e^{\theta_{c} }+|\underline{v}_{a}|e^{|(\theta_{a}+\frac{2\pi}{3})}+|\underline{v}_{b}|e^{| (\theta_{b}-\frac{2\pi}{3})}\right]\bigg{\}}\]  
where \(\underline{v}_{a}\), \(\underline{v}_{b}\), \(\underline{v}_{c}\), \(\theta_{a}\), \(\theta_{b}\), and \(\theta_{c}\) are the respective phase voltages and angles, while the fundamental input conductance \(g_{1}\) ensures the power balance between the ac and the dc side and it is calculated by using the following equation.  
\[g_{1}=\frac{3\ p_{\rm dc}^{\bullet}}{\sum|\underline{v}_{x}|^{2}+2\ \sum_{x\
eq y}| \underline{v}_{x}|\underline{v}_{y}|\cos(\theta_{\rm x}-\theta_{y}-\frac{2 \pi}{3})} \tag{8}\]Additional information about the fundamental conductance and its mathematical extraction can be found in [12; 26].