## 5 Convex Reformulation
### Fixed Phase  
The PV injection maximization problem with fixed phase selection is formulated below. The objective function encapsulates the total sum of PV injection at all buses with PV, at any phase and any time, denoted in Equation (9).  
\[\max\sum_{i}\sum_{\phi}\sum_{t}I_{i,\phi,t} \tag{9}\]  
Subject to  
\[I_{i,t}^{R}=\sum_{\phi}\cos\theta_{\phi,t}I_{i,\phi,t} \tag{10}\]  
\[I_{i,t}^{l}=\sum_{\phi}\sin\theta_{\phi,t}I_{i,\phi,t}\]  
\[U_{j,t}^{R}=\sum_{i}\left[R_{ji}I_{i,t}^{R}-X_{ji}I_{i,t}^{I}\right]+U_{t}^{R, \text{bg}}\]  
\[U_{j,t}^{I}=\sum_{i}\left[R_{ji}I_{i,t}^{I}+X_{ji}I_{i,t}^{R}\right]+U_{t}^{I, \text{bg}} \tag{11}\]  
\[\sum_{\phi}I_{i,\phi,t}=\sum_{\phi}I_{k,\phi,t} \tag{12}\]  
\[U_{j,t}^{-}=(U_{j,t}^{R})^{2}+(U_{j,t}^{I})^{2} \tag{13}\]  
\[U_{j,t}^{-}\leq 2\%\ U_{t}^{+\text{bg}} \tag{14}\]  
\[I_{i,\phi,t}\leq I^{\max}D_{i,\phi,t}^{\text{fix}} \tag{15}\]  
\[j\in E_{\text{b}},\ (i,k)\in E_{\text{PV}},\ \phi\in\Phi,\ t\in T_{\text{w}} \tag{16}\]The PV-injected current is split into real and imaginary parts, as shown in (10). The real phase angle values \(\theta_{\phi,t}\) are given by measured phasor data and are fixed parameters during optimization routine. Thus, the cosine and sine functions are both constants, that are scaling the injected current value by the phase and angle of voltage phasors. In (11), the real and imaginary parts of the negative sequence voltage are calculated, reflecting the contribution of a single PV plant. The background-measured voltage component is added to the PV contribution. The condition of equal PV contribution is set in Equation (12). It is assumed that all PVs are in close vicinity and they are exposed to the same amount of irradiance. The real and imaginary parts of the negative sequence voltage are merged back in the quadratic equation (13). Convexity condition is granted by the linearity of voltage components. Real and imaginary parts of the negative sequence voltage are found as a linear function of the decision variable (PV-injected power). The sum of squared linear functions is convex, hence the problem is convex and the global solution can be found. The negative sequence voltage magnitude is limited to 2% of background positive sequence voltage in (14). To keep the convexity, measured data are utilized in the VU denominator. Such relaxation is fair, as the true positive sequence voltage tends to increase during PV network feed. The utilized values are smaller, thus staying on the safe side and underestimating available headroom for PV injection. The big-M concept is applied in (15) to limit injected current to zero in case of phases that have no PV plants. The \(I^{\max}\) is an arbitrary large positive value and is equal to 100 in current calculations. The sets are defined in (16) and are valid for all the equations presented above. All presented variables are non-negative.  
The VU minimization problem is based on the model presented above. All the constraints are the same, except for the objective function. As shown in Equation (17), the objective function is the sum of negative sequence voltages at every bus and all time instances.  
\[\min\sum_{j}\sum_{t}U_{j,t}^{-} \tag{17}\]  
Subject to (10)-(16)