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This paper presents a comprehensive mathematical model to solve the restoration problem in balanced radial distribution systems. The restoration problem, originally modeled as mixed integer nonlinear programming, is transformed into a mixed integer second-order cone programming problem, which can be solved efficiently using several commercial solvers based on the efficient optimization technique family branch and bound. The proposed mathematical model considers several objectives in a single objective function, using parameters to preserve the hierarchy of the different objectives: 1) maximizing the satisfaction of the demand, 2) minimizing the number of switch operations, 3) prioritizing the automatic switch operation rather than a manual one, and 4) prioritizing especial loads. General and specialized tests were carried out on a 53-node test system, and the results were compared with other previously proposed algorithms. Results show that the mathematical model is robust, efficient, flexible, and presents excellent performance in finding optimal solutions.

In this paper, a two-stage procedure is proposed in order to solve the centralized self-healing scheme for electrical distribution systems. The considered self-healing actions are the reconfiguration of the distribution grid and, if needed, node and zone load-shedding. Thus, the proposed procedure determines the status of the switching devices in order to effectively isolate a faulty zone and minimize the number of de-energized nodes and zones, while ensuring that the operative and electrical constraints of the system are not violated. The proposed method is comprised of two stages. The first stage solves a mixed integer linear programming (MILP) problem in order to obtain the binary decision variables for the self-healing scheme (i.e., the switching device status and energized zones). In the second stage, a nonlinear programming (NLP) problem is solved in order to adjust the steady-state operating point of the topology found in the first stage (i.e., correction of the continuous electrical variables and load-shedding optimization). Commercial optimization solvers are used in the first stage to solve the MILP problem and in the second stage to solve the NLP problem. A 44-node test system and a real Brazilian distribution system with 964-nodes were used to test and verify the proposed methodology.

This paper presents a novel mixed-integer linear programming (MILP) model for the electric vehicle charging coordination (EVCC) problem in unbalanced electrical distribution systems (EDSs). Linearization techniques are applied over a mixed-integer nonlinear programming model to obtain the proposed MILP formulation based on current injections. The expressions used to represent the steady-state operation of the EDS take into account a three-phase representation of the circuits, as well as the imbalance of the loads, leading to a more realistic model. Additionally, the proposed formulation considers the presence of distributed generators and operational constraints such as voltage and current magnitude limits. The optimal solution for the mathematical model was found using commercial MILP solvers. The proposed formulation was tested in a distribution system used in the specialized literature. The results show the efficiency and the robustness of the methodology, and also demonstrate that the model can be used in the solution of the EVCC problem in EDSs. Index Terms-Electric vehicles charging coordination (EVCC), mixed integer linear programming (MILP), unbalanced electrical distribution systems (EDSs). NOMENCLATURE SetsF Sets of phases {A,B,C}. L Sets of circuits. N Sets of nodes. T Sets of time intervals. Constants α G n,tEnergy cost at node n in time period t. βElectric vehicle (EV) energy curtailment cost. t

This paper presents a new mixed-integer linear programming (MILP) model to solve the multistage long-term expansion planning problem of electrical distribution systems (EDSs) considering the following alternatives: increasing the capacity of existing substations, constructing new substations, allocating capacitor banks and/or voltage regulators, constructing and/or reinforcing circuits, and modifying, if necessary, the system's topology. The aim is to minimize the investment and operation costs of the EDS over an established planning horizon. The proposed model uses a linearization technique and an approximation for transforming the original problem into an MILP model. The MILP model guarantees convergence to optimality by using existing classical optimization tools. In order to verify the efficiency of the proposed methodology, a 24-node test system was employed.

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This paper presents a mixed-integer second-order cone programing (MISOCP) model to solve the optimal operation problem of radial distribution networks (DNs) with energy storage. The control variables are the active and reactive generated power of dispatchable distributed generators (DGs), the number of switchable capacitor bank units in operation, the tap position of the voltage regulators and on-load tap-changers, and the operation state of the energy storage devices. The objective is to minimize the total cost of energy purchased from the distribution substation and the dispatchable DGs. The steady-state operation of the DN is modeled using linear and second-order cone programing. The use of an MISOCP model guarantees convergence to optimality using existing optimization software. A mixed-integer linear programing (MILP) formulation for the original model is also presented in order to show the accuracy of the proposed MISOCP model. An 11-node test system and a 42-node real system were used to demonstrate the effectiveness of the proposed MISOCP and MILP models. Index Terms-Distributed generation, energy storage, mixed-integer linear programing (MILP), mixed-integer second-order cone programing (MISOCP), optimal operation of radial distribution networks, smart grid. ACRONYMS The abbreviations of common terms used in this paper are presented below. CB Fixed capacitor bank. DG Distributed generator. DN Distribution network. DSS Distribution substation. ESD Energy storage device. MILP Mixed-integer linear programing. MINLP Mixed-integer nonlinear programing. MISOCP Mixed-integer second-order cone programing. OLTC On-load tap-changer. OODN Optimal operation of DNs. OPF Optimal power flow. RS Renewable source.