## 2 Modeling and Solution Strategies
### Multi-Objective Optimization Model
Daily generation curves for each primary energy source (solar, hydro, and wind) and daily load curves for the three consumer classes (residential, commercial, and industrial) assumed in this work for power flow simulations are presented in Figure 1. These curves are based on Brazilian DG plants and load characteristics and can be changed according to local characteristics, although generation curves were based on the capacity factors. Figure 1: Generation and load curves. Equality constraints (g) are referred to the power balance in system nodes, as shown in [27; 29; 31], and defined in this work as \(PSM_{\text{equations}}(x,p)\). Inequality constraints (\(h\)) usually represent inherent distribution systems characteristics or regulatory restrictions, such as voltage level constraints, cable capacity, and limits on long-time and short-time voltage variations. Due to the large number of constraints that can be modeled, three operational constraints were chosen: overvoltage, overcurrent, and penetration index. Overvoltage and overcurrent constraints are considered in the proposed model as they represent network limitations, which are generally set by local regulations. Concerning these constraints, the maximum voltage level (\(V_{\text{max}}\)) allowed is 1.05 p.u. and overcurrent is identified using the network section ampacity (\(A_{k-m}\)). Another constraint considered in the model is related to the fast voltage variation due to renewable DG intermittence, such that \(\Delta V_{k}=V_{k}^{\text{with }DG}-V_{k}^{\text{without DG}}\) is calculated every hourly power flow simulation and then a \(\Delta V_{\text{max}}\) equal to 0.05 p.u. is defined to verify the compliance with referred constraint. The penetration index (PI) represents a measure of impact (load relief) of DG plants on electrical systems. Various works found in the technical literature try to find or simply suggest a DG penetration value able to provide benefits to DisCo and DG owners without compromising the network operation. This index is calculated as shown in Equation (5) and a maximum penetration index is defined in each case study in this work. \[PI=\left(\frac{\sum\limits_{n=1}^{N}NDG_{n}\cdot P_{n}^{DG}}{\max\limits_{p=1,2,\ldots,N_{T}}\left(\sum\
olimits_{i\in\Omega_{B}}P_{i}^{p}\right)}\right) \tag{5}\]  
Therefore, the complete mathematical model is presented in Equation (6). \[\text{Minimize }\left\{ \begin{array}{c}f_{1}=C_{instal}=\sum\limits_{n=1}^{N} CO_{instal}^{n}\cdot NDG_{n}\\ f_{2}=LAE_{daily}=\sum\limits_{p=1}^{N_{T}}\sum\limits_{k-m}^{\Omega_{L}}r_{k- m}\cdot I_{k-m}^{2}T_{p}\\ f_{3}=EqCO_{2}E=\sum\limits_{p=1}^{N_{T}}\sum\limits_{n=1}^{N}cf_{n}.P_{n}^{ DG}.NDG_{n}.EqCO_{2}E_{n}T_{p}\\ \text{subject to }\left\{\begin{array}{c}PSM_{\text{equations}}(x,p)\\ PI\leq PI_{\text{max}}\\ k_{k-m}\leq A_{k-m}\\ V_{k}\leq V_{\text{max}}\\ \Delta V_{k}\leq\Delta V_{\text{max}}\end{array}\right\}\end{array} \right\} \tag{6}\]  
For each candidate solution, two types of penalization for constraint violations are adopted. When the PI is violated, the number of generation modules \(NGD_{n}\) of an \(n\)-th DG randomly chosen is decreased until the PI is attended. When any of the other constraints are violated, the objective function corresponding to the total installation cost is increased by an additional connection cost \(C_{con}\), aiming to simulate real situations where network reinforcements will be necessary. In this work, the connection cost is modeled as an exponential function multiplied by the installation cost, as given in Equation (7). \[C_{\text{con}}=\left\{ \begin{array}{c}C_{instal}\left[e^{\frac{\max(\sum \limits_{k-m}^{k}(t)-1)}{N_{B}}}-1\right]+\\ +C_{instal}\left[e^{\max(\frac{\max(\sum\limits_{k=1}^{N_{T}}V_{k}(t)-V_{\text{ max}})}{N_{B}}},\frac{\max(\sum\limits_{k=1}^{N_{T}}(t)-AV_{\text{max}})}{N_{B}})-1\right]\\ \forall V_{k}(t)>V_{\text{max}}\,;\;\text{where}\;k\in\Omega_{B}\\ \forall l_{k-m}(t)>A_{k-m}\,;\;\text{where}\;k-m\;\in\;\Omega_{L}\\ \forall t\in\Omega_{T}\end{array} \right. \tag{7}\]