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This paper addresses the problem regarding the optimal placement and sizing of distribution static synchronous compensators (D-STATCOMs) in electrical distribution networks via a stochastic mixed-integer convex (SMIC) model in the complex domain. The proposed model employs a convexification technique based on the relaxation of hyperbolic constraints, transforming the nonlinear mixed-integer programming model into a convex one. The stochastic nature of renewable energy and demand is taken into account in multiple scenarios with three different levels of generation and demand. The proposed SMIC model adds the power transfer losses of the D-STATOMs in order to size them adequately. Two objectives are contemplated in the model with the aim of minimizing the annual installation and operating costs, which makes it multi-objective. Three simulation cases demonstrate the effectiveness of the stochastic convex model compared to three solvers in the General Algebraic Modeling System. The results show that the proposed model achieves a global optimum, reducing the annual operating costs by 29.25, 60.89, and 52.54% for the modified IEEE 33-, 69-, and 85-bus test systems, respectively.

Photovoltaic (PV) systems are a clean energy source that allows for power generation integration into electrical networks without destructive environmental effects. PV systems are usually integrated into electrical networks only to provide active power during the day, without taking full advantage of power electronics devices, which can compensate for the reactive power at any moment during their operation. These systems can also generate dynamic reactive power by means of voltage source converters, which are called PV-STATCOM devices. This paper presents a convex formulation for the optimal integration (placement and sizing) of PV-STATCOM devices in electrical distribution systems. The proposed model considers reducing the costs of the annual energy losses and installing PV-STATCOM devices. A convex formulation was obtained to transform the hyperbolic relation between the products of the voltage into a second-order constraint via relaxation. Two simulation cases in the two IEEE test systems (33- and 69-node) with radial and meshed topologies were implemented to demonstrate the effectiveness of the proposed mixed-integer convex model. The results show that PV-STATCOM devices reduce the annual cost of energy losses of electrical networks in a more significant proportion than PV systems alone.

This research addresses the efficient integration and sizing of flexible alternating current transmission systems (FACTS) in electrical distribution networks via a convex optimization approach. The exact mixed-integer nonlinear programming (MINLP) model associated with FACTS siting and sizing aims for the minimization of the expected annual operating costs of the network (i.e., energy losses and FACTS purchasing costs). The constraints of this problem include power equilibrium equalities, voltage regulation bounds, and device capacities, among others. Due to the power equilibrium constraints per node and period, the MINLP model is a non-convex optimization problem. To transform the exact MINLP model into a mixed-integer convex one, the approximation of the product between two variables in the complex domain is relaxed through its hyperbolic equivalent, which generates a set of convex cones. The main advantage of the proposed mixed-integer convex model is that it ensures the global optimum of the problem, even when considering objective multiplexes. Numerical simulations in the IEEE 33-, 69-, and 85-bus grids demonstrate the effectiveness and robustness of FACTS integration via the proposed convex approach in comparison with the exact solution of the MINLP model in the GAMS software as well as with combinatorial optimization algorithms (i.e., the black widow optimizer and the vortex search algorithm). All simulations were carried out in MATLAB with Yalmip optimization and the Gurobi and Mosek solvers. The simulation results show that, for a fixed operation of the FACTS devices (i.e., a VAR compensator) during the day, the annual operating costs are reduced by 12.63%, 13.97%, and 26.53% for the IEEE 33-, 69-, and 85-bus test systems, respectively, while for the operation variable, the reductions are by 14.24%, 15.79%, and 30.31%, respectively.

The optimal integration of photovoltaic (PV) systems and distribution static synchronous compensators (D-STATCOMs) in electrical distribution networks is important to reduce their operating costs, improve their voltage profiles, and enhance their power quality. To this effect, this paper proposes a mixed-integer convex (MI-Convex) optimization model for the optimal siting and sizing of PV systems and D-STATCOMs, with the aim of minimizing investment and operating costs in electrical distribution networks. The proposed model transforms the traditional mixed-integer nonlinear programming (MINLP) formulation into a convex model through second-order conic relaxation of the nodal voltage product. This model ensures global optimality and computational efficiency, which is not achieved using traditional heuristic-based approaches. The proposed model is validated on IEEE 33- and 69-bus test systems, showing a significant reduction in operating costs in both feeders compared to traditional heuristic-based approaches such as the vortex search algorithm (VSA), the sine-cosine algorithm (SCA), and the sech-tanh optimization algorithm (STOA). According to the results, the MI-convex model achieves cost savings of up to 38.95% in both grids, outperforming the VSA, SCA, and STOA.

The problem concerning the optimal placement and sizing of renewable energy resources and battery energy storage systems in electrical DC distribution networks is addressed in this research by proposing a new mathematical formulation. The exact mixed-integer nonlinear programming (MINLP) model is transformed into a mixed-integer convex model using McCormick envelopes regarding the product between two positive variables. Convex theory allows ensuring that the global optimum is found due to the linear equivalent structure of the solution space and the quadratic structure of the objective function when all the binary variables are defined. Numerical results in the 21-bus system demonstrate the effectiveness and robustness of the proposed solution methodology when compared to the solution reached by solving the exact MINLP model. Numerical results showed that the simultaneous allocation of batteries and renewable energy resources allows for the best improvements in the daily operating costs, i.e., about 53.29% with respect to the benchmark case of the 21-bus grid, followed by the scenario where the renewable energy resources are reallocated while considering a fixed location for the batteries, with an improvement of 43.33%. In addition, the main result is that the difference between the exact modeling and the proposed formulation regarding the final objective function was less than 3.90% for all the simulation cases, which demonstrated the effectiveness of the proposed approach for operating distributed energy resources in monopolar DC networks.

This paper addresses the problem concerning the efficient minimization of power losses in asymmetric distribution grids from the perspective of convex optimization. This research’s main objective is to propose an approximation optimization model to reduce the total power losses in a three-phase network using the concept of electrical momentum. To obtain a mixed-integer convex formulation, the voltage variables at each node are relaxed by assuming them to be equal to those at the substation bus. With this assumption, the power balance constraints are reduced to flow restrictions, allowing us to formulate a set of linear rules. The objective function is formulated as a strictly convex objective function by applying the concept of average electrical momentum, by representing the current flows in distribution lines as the active and reactive power variables. To solve the relaxed MIQC model, the GAMS software (Version 28.1.2) and its CPLEX, SBB, and XPRESS solvers are used. In order to validate the effectiveness of load redistribution in power loss minimization, the initial and final grid configurations are tested with the triangular-based power flow method for asymmetric distribution networks. Numerical results show that the proposed mixed-integer model allows for reductions of 24.34%, 18.64%, and 4.14% for the 8-, 15-, and 25-node test feeders, respectively, in comparison with the benchmark case. The sine–cosine algorithm and the black hole optimization method are also used for comparison, demonstrating the efficiency of the MIQC approach in minimizing the expected grid power losses for three-phase unbalanced networks.

With this study, we address the optimal phase balancing problem in three-phase networks with asymmetric loads in reference to a mixed-integer quadratic convex (MIQC) model. The objective function considers the minimization of the sum of the square currents through the distribution lines multiplied by the average resistance value of the line. As constraints are considered for the active and reactive power redistribution in all the nodes considering a 3×3 binary decision variable having six possible combinations, the branch and nodal current relations are related to an extended upper-triangular matrix. The solution offered by the proposed MIQC model is evaluated using the triangular-based three-phase power flow method in order to determine the final steady state of the network with respect to the number of power loss upon the application of the phase balancing approach. The numerical results in three radial test feeders composed of 8, 15, and 25 nodes demonstrated the effectiveness of the proposed MIQC model as compared to metaheuristic optimizers such as the genetic algorithm, black hole optimizer, sine–cosine algorithm, and vortex search algorithm. All simulations were carried out in MATLAB 2020a using the CVX tool and the Gurobi solver.

The rapid growth of the transmission networks has brought more uncertainties and new requirements in the transmission expansion planning (TEP) to the planners. The existing methods of solving TEP problem have a drawback since the DC load flow and the relaxed load flow models have been utilized to solve TEP problem. In this work, the TEP problem is solved based on mixed integer nonlinear non-convex programming model. A meta-heuristic algorithm by the means of differential evolution algorithm (DEA) is employed as an optimisation tool. An AC load flow model is used in solving the TEP problem, where accurate and realistic results can be obtained. Furthermore, the work considers the constraints checking and system violation such as real and power generation limits, possible number of lines added and bus voltage limits. The proposed technique is tested on Garver's 6 bus system and IEEE 24 bus system and has shown high capability in considering the active and reactive power in the same manner and solving the TEP problem. The method produced improved results for the test systems. In terms of minimising the cost and the solution quality, the proposed method obtained good and challenging results comparing to the previous works.

Mixed integer programming (MIP) is commonly used to model indicator constraints, i.e., constraints that either hold or are relaxed depending on the value of a binary variable. Unfortunately, those models tend to lead to weak continuous relaxations and turn out to be unsolvable in practice; this is what happens, for e.g., in the case of Classification problems with Ramp Loss functions that represent an important application in this context. In this paper we show the computational evidence that a relevant class of these Classification instances can be solved far more efficiently if a nonlinear, nonconvex reformulation of the indicator constraints is used instead of the linear one. Inspired by this empirical and surprising observation, we show that aggressive bound tightening is the crucial ingredient for solving this class of instances, and we devise a pair of computationally effective algorithmic approaches that exploit it within MIP. One of these methods is currently part of the arsenal of IBM-Cplex  since version 12.6.1. More generally, we argue that aggressive bound tightening is often overlooked in MIP, while it represents a significant building block for enhancing MIP technology when indicator constraints and disjunctive terms are present.

We consider nonlinear optimization problems that involve surrogate models represented by neural networks. We demonstrate first how to directly embed neural network evaluation into optimization models, highlight a difficulty with this approach that can prevent convergence, and then characterize stationarity of such models. We then present two alternative formulations of these problems in the specific case of feedforward neural networks with ReLU activation: as a mixed-integer optimization problem and as a mathematical program with complementarity constraints. For the latter formulation we prove that stationarity at a point for this problem corresponds to stationarity of the embedded formulation. Each of these formulations may be solved with state-of-the-art optimization methods, and we show how to obtain good initial feasible solutions for these methods. We compare our formulations on three practical applications arising in the design and control of combustion engines, in the generation of adversarial attacks on classifier networks, and in the determination of optimal flows in an oil well network.