This paper presents a stochastic mixed-integer nonlinear programming (MINLP) model for the optimal operation of islanded microgrids in the presence of stochastic demands and renewable resources. In the proposed formulation, the microgrid is modeled as an unbalanced three-phase electrical distribution system comprising distributed generation (DG) units with droop control, battery systems (BSs) and wind turbines (WTs). The stochastic nature of the consumption and the renewable generation is considered through a scenario-based approach, which determines the optimal values of the decision variables that minimize the average operational cost of the microgrid. A set of efficient linearizations are used to transform the proposed MINLP model into an approximated convex model that can be solved via commercial solvers. In order to assess the effectiveness of the obtained solution, Monte Carlo simulations (MCS) are carried out. Results show that the proposed model considers the uncertainty while reducing the average operational costs and load curtailments, when compared with a deterministic model.@en

This paper proposes a two-stage stochastic market clearing (SMC) model based on a semidefinite programming (SDP) relaxation. The SMC model aims at determining the day-ahead schedule (DA) and the real-time (RT) balance settlement that minimize the total expected production cost. The network capacity constraints are considered in the proposed model through an AC power flow formulation, while the uncertainty in the renewable-based generation is taking into account using a set of stochastic scenarios. In order to solve the proposed non-linear programming model, a SDP relaxation is used. An illustrative example (3-bus test system) and the IEEE Reliability 24-bus test system are used to show the effectiveness and accuracy of the proposed model. Results shown that the proposed SDP relaxation introduce a negligible error, when compared with the solution after solving the original non-linear model.

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This paper presents a new mixed-integer nonlinear programming (MINLP) model for the optimal operation of unbalanced three-phase droop-based microgrids. The proposed MINLP model can be seen as an extension of an optimal power flow for microgrids operating in islanded mode, that aims to minimize the total amount of unsupplied demand and the total distributed generator (DG) generation cost. Since the slack bus concept is not longer valid, the proposed model considers the frequency and voltage magnitude reference as variables. In this case, DGs units operate with droop control to balance the system and provide a frequency and voltage magnitude reference. Additionally, a set of efficient linearizations are introduced in order to approximate the original MINLP problem into a mixed-integer linear programming (MILP) model that can be solved using commercial solvers. The proposed model has been tested in a 25-bus unbalanced three-phase microgrid and a large 124-node grid, considering different operational and time-coupling constraints for the DGs and the battery systems (BSs). Load curtailment and different modes of operation for the wind turbines have also been tested. Finally, an error assessment between the original MINLP and the approximated MILP model has been conducted.

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This paper presents a mixed integer non-linear programming model to optimize, in a centralized fashion, the operation of multiple buildings in a microgrid. The proposed model aims to minimize the total cost of the energy imported from the main grid at the interconnection point, managing the power demand and generation of buildings, while operational constraints of the electrical grid are guaranteed. This approach considers the management of heating, ventilation, and air conditioning units, lighting appliances, photovoltaic generation and energy storage system of each building. Comfortable indoor conditions for the occupants are kept by a set of mathematical constraints. Additionally, a strategy that simplifies the original model is presented, based on a set of linearization techniques and equivalent representations, obtained through a pre-processing stage executed in EnergyPlus software. This strategy allows approximating the proposed model into a mixed integer linear programming formulation that can be solved using commercial solvers. The proposed model was tested in a 13-bus microgrid for different deterministic cases of study with non-manageable loads and smart buildings. A large-size test case is also considered. Finally, a rolling horizon strategy is proposed with the aim of addressing the uncertainty of the data, as well as reducing the amount of forecasting data required.

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In this paper, a new and generalized model for the optimal operation of microgrids is presented. The proposed mathematical model considers both the grid-connected (GC) and islanded (IS) operational modes. First, a mixed integer non-linear programming (MINLP) formulation is introduced, modeling the microgrid as an unbalanced ac three-phase electrical distribution system, comprising distributed generator (DG) units, battery systems and wind turbines. In GC mode, the frequency and the voltage magnitude references are imposed by the main grid at the point of common couple, while in IS mode, it is assumed that the DG units operate with droop control. Additionally, a set of convexification procedures are introduced in order to approximate the original MINLP model into a new convex formulation that can be solved using commercial solvers. The proposed model has been tested in a 25-bus microgrid for different scenarios, including one where a degradation of the voltage magnitude reference is observed. Results show that the proposed model is able to properly define the operational mode of the microgrid, based on the technical constraints of the system.

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In this paper, the economic impact of the active power droop gain in droop-based microgrids has been studied. To accomplish this, a theoretical analysis is presented first. This analysis is used to study a simple case in which the solution of the economic dispatch problem is compared with the dispatch obtained after applying the standard definition of the droop gains for the droop-controlled DGs, given by the Standard IEEE 1547.7. Then, the mathematical model for the optimal power flow of droop-based islanded microgrids is used to simulate and study a real unbalanced three-phase system. Results have shown that the standard definition guarantees a proper active power sharing among all the DG units, independent of the load level. However, as shown by the simulations, the standard definition does not necessarily correspond to the solution with minimum active power losses and can have a significant economic impact when compared with the optimal economic dispatch solution.

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An extended dynamic programming (EDP) approach is developed to optimize the ac steady-state operation of radial electrical distribution systems (EDS). Based on the optimality principle of the recursive Hamilton-Jacobi-Bellman equations, the proposed EDP approach determines the optimal operation of the EDS by setting the values of the controllable variables at each time period. A suitable definition for the stages of the problem makes it possible to represent the optimal ac power flow of radial EDS as a dynamic programming problem, wherein the 'curse of dimensionality' is a minor concern, since the number of state and control variables at each stage is low and the time complexity of the algorithm grows linearly with the number of nodes of the EDS. The proposed EDP is applied to solve the economic dispatch of the DG units installed in a radial EDS. The effectiveness and the scalability of the EDP approach is illustrated using real-scale systems and comparisons with commercial programming solvers. Finally, generalizations to consider other EDS operation problems are also discussed.

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In this chapter the most significant characteristics and functionalities of an energy management system (EMS) for microgrids are introduced. For this, the definitions of hierarchical control layers are considered. First, the main concepts and modules of the hierarchical control structure of a generalized EMS are presented. Then, energy management function is represented as an optimization problem, described as the simultaneous solution of both, a unit commitment problem and an economic load dispatch problem. An extension of the energy management problem is also formulated based on an optimal power flow. Second, the advantages and disadvantages of using either a centralized or a decentralized EMS approach are discussed. Finally, since the energy management problem is represented as an optimization problem, the most common methodologies and solution algorithms used in the specialized literature are discussed, including metaheuristics, mixedinteger linear approximations, and nonlinear approaches, as well as software tools for implementing models and simulations.

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This paper presents a mixed-integer linear programming (MILP) model for the optimal energy management of residential microgrids, modeled as unbalanced, three-phase, electrical distribution system (EDS). Initially, the problem is formulated as a mixed-integer nonlinear programming (MINLP) problem. Then, a set of linear approximations and equivalent mathematical representations are used to obtain a precise MILP model. The proposed formulation considers three-phase generation units (GU), single-phase photovoltaic (PV) resources, and single-phase energy storage systems (ESS), as well as load management. The aim of the proposed model is to minimize the final operational costs of the microgrid while considering operational constraints of the EDS and an unexpected outage of the main grid through a security-constrained set of equations. The optimal solution of the MILP model is found using commercial convex optimization solvers. The proposed model was tested in a residential, three-phase EDS. Results show that the proposed linearizations and approximations produce accurate solutions when compared with a nonlinear three-phase OPF formulation, with an error in the objective function near to 2% and a maximum error in the voltage near to 1%. Efficiency and flexibility of the proposed methodology are also discussed.@en

This paper presents a new mixed integer linear programing (MILP) model for the management of energy consumption and comfort in smart buildings. Initially, a detailed mixed integer non-linear programming (MINLP) model is formulated. The approach considers the management of heating, ventilation and air conditioning (HVAC) units, lighting appliances, photovoltaic generation (PV) and energy storage system (ESS). Then, a set of linear and equivalent representations are used to approximate the problem by an MILP model. The aims of the proposed model is to minimize the electricity bill by managing the loads, as well as the schedule of the ESS, meanwhile comfortable indoor conditions are ensured by a set of mathematical constraints. A commercial MILP solver was used to guarantee optimality. The strategy was tested in an university building with multiple zones. Comparisons between the proposed MILP model and simulations in EnergyPlus were used to validate the results.

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