## 2 Modeling and Solution Strategies
### Multi-Objective Optimization Model  
The DG sizing and siting problem addressed in this work assumes that the locations of accessible primary energy sources and the maximum generation capacity available for DG projects are previously known. Each DG plant \(n\) for a total number of \(N\) plants can install \(NDG_{n}\) modules for a total number of \(NDG_{n}^{max}\) modules, each one with capacity \(P_{n}^{DG}\), such that the power output of each DG plant \(n\) is given by \(NDG_{n}.P_{n}^{DG}\). Therefore, decision variable vector \(x\) of the optimization problem is composed by network buses and number of modules of each DG plant \(n\), with total number \(2\cdot N\) (size and site) of variables. The vector of multi-objective functions (\(F\)) of the proposed modeling is formed by three objective functions (\(f_{1}\), \(f_{2}\), \(f_{3}\)). The first objective-function (\(f_{1}\)) represents an economical viewpoint and corresponds to minimize the total installation cost of DG plants (\(C_{instal}\)), obtained by Equation (2), where \(CO_{instal}^{n}\) is the investment cost of the \(n\)-th DG plant, \(NDG_{n}\) is the quantity of generation modules of the \(n\)-th DG plant to be installed and \(N\) is the total number of DG plants to be connected in the network. In this paper, installation cost data are mean overnight cost: 1497 USD/kWe for onshore wind, 1210 USD/kWe for photovoltaic plants, and 1492 USD/kWe for hydro [25]. \[C_{instal}=\sum_{n=1}^{N}CO_{instal}^{n}\cdot NDG_{n} \tag{2}\]The second objective function (\(f_{2}\)) corresponds to the minimization of daily active power losses (\(LAE_{daily}\) in MWh), which is one of the objectives most commonly considered in DG sizing and siting planning [26]. The required data to calculate branch currents are obtained by a backward/forward sweep load flow known by Power Summation Method (PSM) and the daily energy losses are calculated by Equation (3), where \(r_{k-m}\) and \(I_{k-m}\) are the resistance and current of the k-m branch, respectively; \(N_{T}\) is the number of time intervals considered in the daily analysis, \(\Omega_{L}\) is the set of distribution network branches, and \(T_{p}\) is the duration of the time interval. In this case, time intervals of one hour are considered, aiming at a better discretization of daily consumption, although other time intervals can be considered in the modeling without significant changes in results. \[f_{2}=LAE_{daily}=\sum_{p=1}^{N_{T}}\sum_{k-m}^{\Omega_{L}}r_{k-m}\cdot I_{k-m} ^{2}T_{p} \tag{3}\]  
The third objective function (\(f_{3}\)) reflects environmental concerns and corresponds to the minimization of daily amount of equivalent CO\({}_{2}\) emission \(EqCO_{2}E\), given by Equation (4). CO\({}_{2}\) emission factors \(EqCO_{2}EF\) are based on [27; 28; 29; 30]: 21 gCO\({}_{2}\)eq/kWh for onshore wind, 17 gCO\({}_{2}\)eq/kWh for photovoltaic and 105 gCO\({}_{2}\)eq/kWh for hydro, while adopted capacity factors \(cf\) are [26]: 0.18 for photovoltaic, 0.34 for onshore wind and 0.47 for hydro. In this case, \(cf_{n}\) is the capacity factor of the \(n\)-th DG plant and \(P_{n}^{DG}\) is the active power capacity for each module of the \(n\)-th DG plant. \[EqCO_{2}E=\sum_{p=1}^{N_{T}}\sum_{n=1}^{N}cf_{n}.P_{n}^{DG}.NDG_{n}.EqCO_{2}EF _{n}T_{p} \tag{4}\]  
It should be noted that a greater number of DG modules results in higher installation costs and CO\({}_{2}\) emissions; nonetheless, dispersed generation in distribution systems represents load reductions and, therefore, provides power loss reductions. Thus, minimization of installation costs and CO\({}_{2}\) emissions are conflicting with minimization of power losses.