## 2 Control Strategies Description
### Three-Phase Damping Control Strategy: Negative-Sequence Component Mitigation  
Usually the three-phase DERs are connected via a three-phase, three-wire connection and inject only positive-sequence currents. Therefore, by using such a system, only the negative-sequence component can be impacted. In addition, EN50160 [30] recommends only limits to the negative voltage unbalance factor, which is the ratio of the negative- to positive-sequence voltage components and no recommendation to the zero-sequence components are given whatsoever. In order to make the three-phase damping control strategy compliant to act resistively only to the negative-sequence voltage component, a modification in Equation (10) can be made so that only the negative-sequence component is impacted by the three-phase damping control strategy:  
\[\begin{bmatrix}\dot{i}_{a}\\ \dot{i}_{b}\\ \dot{i}_{c}\end{bmatrix}=T^{-1}\begin{bmatrix}0&0&0\\ 0&g_{1}&0\\ 0&0&g_{2}\end{bmatrix}T\begin{bmatrix}\underline{v}_{a}\\ \underline{v}_{b}\\ \underline{v}_{c}\end{bmatrix} \tag{15}\]  
By following the same procedure as the positive-sequence control strategy, the currents injected by the damping control strategy for negative-sequence component mitigation are obtained as follows.  
\[\dot{i}_{a}=\frac{1}{3}\bigg{\{}g_{1}\left[|\underline{v}_{a}|e^{\dot{i}\theta _{a}}+|\underline{v}_{b}|e^{\dot{i}(\theta_{b}+\frac{2\pi}{3})}+|\underline{v }_{c}|e^{\dot{i}(\theta_{c}-\frac{2\pi}{3})}\right]+g_{d2}\left[|\underline{v }_{a}|e^{\dot{i}\theta_{a}}+|\underline{v}_{b}|e^{\dot{i}(\theta_{b}-\frac{2 \pi}{3})}+|\underline{v}_{c}|e^{\dot{i}(\theta_{c}+\frac{2\pi}{3})}\right] \bigg{\}}\]  
\[\dot{i}_{b}=\frac{1}{3}\bigg{\{}g_{1}\left[|\underline{v}_{b}|e^{\dot{i}\theta _{b}}+|\underline{v}_{a}|e^{\dot{i}(\theta_{a}-\frac{2\pi}{3})}+|\underline{v }_{c}|e^{\dot{i}(\theta_{c}+\frac{2\pi}{3})}\right]+g_{d2}\left[|\underline{v }_{b}|e^{\dot{i}\theta_{b}}+|\underline{v}_{a}|e^{\dot{i}(\theta_{c}+\frac{2 \pi}{3})}+|\underline{v}_{c}|e^{\dot{i}(\theta_{c}-\frac{2\pi}{3})}\right] \bigg{\}} \tag{16}\]  
\[\dot{i}_{c}=\frac{1}{3}\bigg{\{}g_{1}\left[|\underline{v}_{c}|e^{\dot{i} \theta_{c}}+|\underline{v}_{a}|e^{\dot{i}(\theta_{c}+\frac{2\pi}{3})}+| \underline{v}_{b}|e^{\dot{i}(\theta_{b}-\frac{2\pi}{3})}\right]+g_{d2}\left[| \underline{v}_{c}|e^{\dot{i}\theta_{c}}+|\underline{v}_{a}|e^{\dot{i}( \theta_{c}-\frac{2\pi}{3})}+|\underline{v}_{b}|e^{\dot{i}(\theta_{c}+\frac{2 \pi}{3})}\right]\bigg{\}}\]The power balance of this adapted control strategy differs from the expression given in Equation (12). The fundamental input conductance, which includes the compensation only of the negative-sequence component is written as,  
\[g_{1}=\frac{3\ p_{\text{dc}}^{\bullet}}{\sum|\underline{v}_{x}|^{2 }+2\ \sum_{x\
eq y}|\underline{v}_{x}||\underline{v}_{y}|\cos(\theta_{\text{x}}- \theta_{y}-\frac{2\pi}{3})}\\ +g\underline{a}\frac{\sum|\underline{v}_{x}|^{2}+\sum_{x\
eq y}| \underline{v}_{x}||\underline{v}_{y}|\cos(\theta_{\text{x}}-\theta_{y}+\frac{ 2\pi}{3})}{\sum|\underline{v}_{x}|^{2}+2\ \sum_{x\
eq y}|\underline{v}_{x}|| \underline{v}_{y}|\cos(\theta_{\text{x}}-\theta_{y}-\frac{2\pi}{3})} \tag{17}\]  
Although Equation (17) differs from Equation (12), the same interpretation of Equation (12) can be applied for this power balance expression. The damping conductance \(g_{d2}\) in p.u. value can be calculated by using Equation (14).