# Solar Photovoltaic Integration in Monopolar DC Networks via the GNDO Algorithm
## 2 Mathematical Formulation
\bigg{\}}, \tag{5}\]  
\[v_{j,h}^{2}=v_{j,h}^{2}-2R_{ij}p_{ij,h}+R_{ij}^{2}i_{ij,h}^{2},\{\forall ij\in \mathcal{L},\,\forall h\in\mathcal{H}\}, \tag{6}\]  
\[p_{ij,h}=v_{j,h}i_{ij,h},\ \{\forall ij\in\mathcal{L},\ \forall h\in\mathcal{H}\} \tag{7}\]  
\[x_{j}p_{pv}^{\min}\leq p_{j}^{pv}\leq x_{j}p_{pv}^{\max}\ \{\forall j\in \mathcal{N}\}, \tag{8}\]  
\[v_{j}^{\min}\leq v_{j,h}\leq v_{j}^{\max}\ \{\forall j\in\mathcal{N},\ \forall h\in \mathcal{H}\}, \tag{9}\]  
\[-i_{ij}^{\max}\leq i_{ij,h}\leq i_{ij}^{\max}\ \{\forall ij\in\mathcal{L},\ \forall h\in \mathcal{H}\}, \tag{10}\]  
\[\sum_{i\in\mathcal{N}}x_{j}\leq N_{pv}^{\max}, \tag{11}\]  
\[x_{j}\in\{0,1\},\ \{\forall i\in\mathcal{N}\}, \tag{12}\]  
where \(z_{\text{costs}}\) corresponds to the value of the objective function that defines the total annual grid operating costs; \(z_{1}\) is the component of the objective function associated with the total costs of energy purchasing in the slack source; \(z_{2}\) represents the value of the objective function related with the total annual investments in PV generation plants; \(z_{3}\) defines the variable costs of the PV generators, which is calculated as a function of the total expected energy generated during the planning period; \(C_{\text{kWh}}\) represents the expected energy costs of energy at the substation bus; \(T\) is the mean duration of an ordinary year; \(r_{e}\) and \(r_{a}\) are the interest rates regarding the possible increments of the energy service and the expected return of the utility company that will invest in PV solutions for its grids; \(p_{ij,h}^{sub}\) is the power output at the substation terminals for each period of time; \(\Delta h\) is a period of time defined as 1 h for daily operative scenarios; \(C_{pv}\) is a linear cost factor that defines the investment value of a kWp of PV generation installed in the distribution grid; \(y\) is the total number of years of the planning period; \(p_{j}^{pv}\) is a decision variable that determines the optimal size of the PV source connected at node \(i\); \(C_{O\&M}\) is a constant factor that defines the administration, operation, and maintenance costs of the PV generation units per unit of energy generated; \(G_{h}^{pp}\) corresponds to the per-unit expected generation curve that is used to implement a maximum power point tracking strategy in the PV inverters; \(p_{ij,h}\) (\(p_{jk,h}\)) represents the power sent from node \(i\) (\(j\)) to node \(j\) (\(k\)) in the period of time \(h\); \(R_{ij}\) means the resistance value of the branch in the route \(ij\); \(i_{ij,h}\) is the current flow in the branch that connects node \(i\) and node \(j\) in the period of time \(h\); \(p_{jk}^{d}\) represents the total constant power consumption at node \(j\) in the period of time \(h\); \(v_{i,h}\) (\(v_{j,h}\)) is the voltage value at node \(i\) (\(j\)) for the period of time \(h\); \(x_{j}\) is the decision variable that determines whether a PV source is installed at node \(j\) (\(x_{j}=1\)) or not (\(x_{j}=0\)); \(p_{pv}^{\min}\) and \(p_{pv}^{\min}\) correspond to the lower and upper limits regarding the admissible PV source sizes that can be installed in the distribution network; \(v_{j}^{\min}\) and \(v_{j}^{\max}\) are the minimum and maximum voltage regulation bounds admissible in the daily operation for each node \(j\) at any given time; \(i_{ij}^{\max}\) corresponds to the thermal bound associated with the conductor installed in route \(ij\); and \(N_{pv}^{\max}\) means the maximum number of PV sources that can be installed in the distribution grid. The interpretation of the mathematical model is as follows. Equation (1) corresponds to the objective function formulation comprising the operating cost of the network, which is associated with the acquisition of energy at the substation bus during the planning period, and the investment and operation costs of the PV sources (see Equations (2)-(4), respectively).