*2.1. Second-Order Cone Programming (SOCP) Approximations*

The SOCP approximations aim to transform the nonlinear expressions, which are present in Equations (1) and (2) by the following substitutions:

$$L\_{ij,s} = \frac{p\_{ij,s}^2 + q\_{ij,s}^2}{v\_{i,s}^2}, \quad (i, \ j) \in \mathsf{W} \backslash \mathsf{W}^S$$

$$L\_{ij,s} = \frac{p\_{ji,s}^2 + q\_{ji,s}^2}{v\_{j,s}^2}, \quad (i, \ j) \in \mathsf{W} \backslash \mathsf{W}^S \tag{9}$$

$$u\_{j,s} = v\_{i,s'}^2 \quad i \in B$$

Variables defined within equation set (9) are replaced into equation sets (2) and (3):

$$\begin{aligned} \sum\_{\substack{i:(j)\in\mathcal{W}\end{subarray}} p\_{ij,s} &= \sum\_{\substack{(ij)\in\mathcal{W}, \ s\in\mathcal{S}}} r\_{ij} \cdot L\_{ij,s} + P^L\_{j,s} - P^{\text{gen}}\_{j,s} - P^{DG\_{\text{int}}}\_{j,s} + \sum\_{\substack{k:(jk)\in\mathcal{W}, \ s\in\mathcal{S}}} p\_{jk'} \quad j\in\mathcal{B}\backslash\mathcal{S}^F\\ \sum\_{\substack{i:(j)\in\mathcal{W}, \ s\in\mathcal{S}}} q\_{ij,s} &= \sum\_{\substack{(ij)\in\mathcal{W}, \ s\in\mathcal{S}}} x\_{ij} \cdot L\_{ij,s} + Q^L\_{j,s} - Q^{\text{gen}}\_{j,s} - Q^{DG\_{\text{int}}}\_{j,s} + \sum\_{\substack{k:(jk)\in\mathcal{W}, \ s\in\mathcal{S}}} q\_{jk'} \quad j\in\mathcal{B}\backslash\mathcal{B}^F \end{aligned} \tag{10}$$

$$\begin{aligned} \mathbf{u}\_{j,s} - \mathbf{u}\_{i,s} &\le r\_{ij} \left( p\_{ji,s} - p\_{ij,s} \right) + \mathbf{x}\_{ij} \left( q\_{ji,s} - q\_{ij,s} \right) + M \mathbf{1} - y\_{ij} \right), \ i, j \in \mathcal{W} \\\ \mathbf{u}\_{j,s} - \mathbf{u}\_{i,s} &\ge r\_{ij} \left( p\_{ji,s} - p\_{ij,s} \right) + \mathbf{x}\_{ij} \left( q\_{ji,s} - q\_{ij,s} \right) - M \mathbf{1} - y\_{ij} \right), \ i, j \in \mathcal{W} \end{aligned} \tag{11}$$

Furthermore, SOCP constraints are given with a set of formulation below:

$$\begin{aligned} \|L\_{ij,s} \ge \frac{p\_{ij,s}^2 + q\_{ij,s}^2}{v\_{i,s}^2 - u\_{i,s}} & \implies \left\| \begin{array}{l} 2p\_{ij,s} \\ 2q\_{ij,s} \\ L\_{ij,s} - u\_{i,s} \end{array} \right\|\_{2} \le L\_{ij,s} + u\_i = \succ \quad p\_{ij,s}^2 + q\_{ij,s}^2 \le L\_{ij,s} \cdot u\_{i,s} \\\ L\_{ij,s} \ge \frac{p\_{ji,s}^2 + q\_{ij,s}^2}{v\_{j,s}^2 - u\_{j,s}} & \implies \left\| \begin{array}{l} 2p\_{ji,s} \\ 2q\_{ji,s} \\ L\_{ij,s} - u\_{j,s} \end{array} \right\|\_{2} \le L\_{ij,s} + u\_{j,s} = \succ \quad p\_{ji,s}^2 + q\_{ji,s}^2 \le L\_{ij,s} \cdot u\_{j,s} \end{aligned} \tag{12}$$

Additional constraints are introduced to reflect restrictions related to power line capacity and bus voltage limits:

$$L\_{ij,s} \le y\_{ij} \cdot s\_{ij}^{\max}, \ i, j \in \mathcal{W} \tag{13}$$

$$\begin{aligned} u\_i &= \left(\upsilon\_i^{\text{set}}\right)^2, \ i \in B^F\\ \left(\upsilon\_i^{\text{MIN}}\right)^2 &\le u\_{i,s} \le \left(\upsilon\_i^{\text{MAX}}\right)^2, \ i \in B \backslash B^F \end{aligned} \tag{14}$$

where:


This paper did not consider voltage improvements as the primary objective function but includes the bus voltage constraints to ensure the required voltage profile and power quality (Equations (11) and (14)) under a different set of operating scenarios. The voltage constraints are modeled as hard constraints that maintain normal voltage conditions for all considered operating scenarios regarding the different network load and DG production levels. The proposed model does not consider power quality issues related to harmonics injected by DG units and power electronics at grid interfaces. This factor can also be a limiting factor especially in weak distribution networks in which THD can be significantly affected by DG connection. To tackle THD and limit the levels at each system bus according to power quality standards, the model should include harmonic power flow calculation, which would complicate the model and significantly prolong the computational time.
