## 2 Control Strategies Description
### Three-Phase Damping Control Strategy  
As mentioned earlier, voltage unbalance is the second power quality problem in LV grids, which appears in areas with high penetration of DERs. In order to address this issue, the three-phase damping control strategy can be used. This control strategy is examined in [26; 33] and is able to mitigate the voltage unbalance at the point of common coupling (PCC). The reaction of the damping control strategy is to behave resistively towards the zero- and negative-sequence voltage components [12; 26; 34]. Therefore, the exchanged symmetrical component currents can be expressed as follows,  
\[ \begin{bmatrix}\dot{i}_{0}\\ \dot{i}_{1}\\ \dot{i}_{2}\end{bmatrix} = \begin{bmatrix}g_{d}&0&0\\ 0&g_{1}&0\\ 0&0&g_{d}\end{bmatrix}\begin{bmatrix}\underline{v}_{0}\\ \underline{v}_{1}\\ \underline{v}_{2}\end{bmatrix}  \tag{9}\]  
where \(g_{d}\) is the fundamental damping conductance of the inverter and it sets the resistive behavior towards both the zero- and negative-sequence components. By further developing Equation (9) to phase quantities, the following expression is obtained. \[ \begin{bmatrix}\dot{i}_{a}\\ \dot{i}_{b}\\ \dot{i}_{c}\end{bmatrix} =T^{-1} \begin{bmatrix}g_{d}&0&0\\ 0&g_{1}&0\\ 0&0&g_{d}\end{bmatrix} T \begin{bmatrix}\underline{v}_{a}\\ \underline{v}_{b}\\ \underline{v}_{c}\end{bmatrix}  \tag{10}\]  
By following the same procedure as the positive-sequence control strategy, the currents injected by the damping control strategy can be described mathematically by the following set of equations. \[\dot{i}_{a}=\frac{1}{3}\bigg{\{}g_{1}\left[|\underline{v}_{a}|e^{\dot{i}_{a}}+ |\underline{v}_{b}|e^{\dot{i}(\theta_{b}+\frac{2\pi}{3})}+|\underline{v}_{c}| e^{\dot{i}(\theta_{b}-\frac{2\pi}{3})}\right]+g_{d}\left[2|\underline{v}_{a}|e^{ \dot{i}_{a}}-|\underline{v}_{b}|e^{\dot{i}(\theta_{b}+\frac{2\pi}{3})}-| \underline{v}_{c}|e^{\dot{i}(\theta_{b}-\frac{2\pi}{3})}\right]\bigg{\}}\]  
\[\dot{i}_{b}=\frac{1}{3}\bigg{\{}g_{1}\left[|\underline{v}_{b}|e^{\dot{i}_{b}}+ |\underline{v}_{a}|e^{\dot{i}(\theta_{b}-\frac{2\pi}{3})}+|\underline{v}_{c}| e^{\dot{i}(\theta_{c}+\frac{2\pi}{3})}\right]+g_{d}\left[2|\underline{v}_{b}|e^{ \dot{i}_{b}}-|\underline{v}_{a}|e^{\dot{i}(\theta_{b}-\frac{2\pi}{3})}-| \underline{v}_{c}|e^{\dot{i}(\theta_{c}+\frac{2\pi}{3})}\right]\bigg{\}} \tag{11}\]  
\[\dot{i}_{c}=\frac{1}{3}\bigg{\{}g_{1}\left[|\underline{v}_{c}|e^{\dot{i}_{c}} +|\underline{v}_{a}|e^{\dot{i}(\theta_{c}+\frac{2\pi}{3})}+|\underline{v}_{b} |e^{\dot{i}(\theta_{b}-\frac{2\pi}{3})}\right]+g_{d}\left[2|\underline{v}_{c}| e^{\dot{i}_{c}}-|\underline{v}_{a}|e^{\dot{i}(\theta_{c}+\frac{2\pi}{3})}-| \underline{v}_{b}|e^{\dot{i}(\theta_{b}-\frac{2\pi}{3})}\right]\bigg{\}}\]  
When the three-phase damping control strategy is used in injection mode, parameters \(g_{1}\) and \(g_{d}\) have opposite signs. The positive-sequence power injection is determined by the terms related to \(g_{1}\). The fundamental input conductance is calculated by a dc-bus controller to maintain the power balance between the ac and dc sides. The zero- and negative-current injection is set by the terms related to \(g_{d}\). Additional information regarding the analytical expressions of the three-phase damping control strategy can be found in [26]. For the practical implementation of the control strategy, a dc-bus voltage controller is utilized to maintain the power balance between the dc and ac side. The output signal generated by this controller is the fundamental conductance \(g_{1}\) of the VSI [35]. The analytical representation of this parameter can be expressed in the following manner.