A Mixed-Integer Conic Formulation for Optimal Placement and Dimensioning of DGs in DC Distribution Networks

Abstract

The problem of the optimal placement and dimensioning of constant power sources (i.e., distributed generators) in electrical direct current (DC) distribution networks has been addressed in this research from the point of view of convex optimization. The original mixed-integer nonlinear programming (MINLP) model has been transformed into a mixed-integer conic equivalent via second-order cone programming, which produces a MI-SOCP approximation. The main advantage of the proposed MI-SOCP model is the possibility of ensuring global optimum finding using a combination of the branch and bound method to address the integer part of the problem (i.e., the location of the power sources) and the interior-point method to solve the dimensioning problem. Numerical results in the 21- and 69-node test feeders demonstrated its efficiency and robustness compared to an exact MINLP method available in GAMS: in the case of the 69-node test feeders, the exact MINLP solvers are stuck in local optimal solutions, while the proposed MI-SOCP model enables the finding of the global optimal solution. Additional simulations with daily load curves and photovoltaic sources confirmed the effectiveness of the proposed MI-SOCP methodology in locating and sizing distributed generators in DC grids; it also had low processing times since the location of three photovoltaic sources only requires 233.16s, which is 3.7 times faster than the time required by the SOCP model in the absence of power sources.

Exact minimization of the energy losses and the CO2 emissions in isolated DC distribution networks using PV sources

“…The mixed-integer convex reformulation proposed in this research is a convex reformulation problem with integrality constraints on some variables [17]. Therefore, we take advantage of the fact that the mixed-integer convex reformulation proposed is a convex problem, which can be solved efficiently with some integer programming solvers such as the Branch & Bound (B&B) algorithm [34].…”

“…Equation ( 10) is a conic equality constraint that is still non-convex due to the presence of the equality symbol [17]. However, as described in [38], this symbol can be replaced by a low equal symbol, which allows transforming it into a convex constraint, as represented below.…”

“…The authors in [16] have presented optimization models to locate and size distributed generators in DC grids using mixed-integer nonlinear programming model, which is solved with optimization tools available in the general algebraic modeling system, i.e., GAMS, and heuristic optimization methods such as black hole optimizer, and population-based incremental learning. References [17] and [18] have proposed convex optimization models to place and size distributed generators in DC grids considering conic and semidefinite approximations considering the peak load conditions. For their part, authors in [12] and [19] have presented MINLP models to operate battery energy storage systems in DC grids, and their solutions are reached with heuristic algorithms and convex reformulations.…”

A Mixed-Integer Quadratic Formulation of the Phase-Balancing Problem in Residential Microgrids

“…This paper presents a mixed-integer quadratic formulation to this problem, which allows to find an optimal solution. In this way, by using the paradigm of disciplined convex programming, the non-linear integer stochastic problem is transformed into a tractable model [21,23].…”

“…Stochastic non-linear mixed-integer problems, such as (1) to (3) are highly complicated to solve, so heuristic algorithms are commonly used to solve them [16]. However, the proposed approximations transform the problem from a general non-linear mixed-integer problem to a convex-quadratic mixed-integer model, given by (16) to (23), since the objective function is quadratic and all the constraints are linear expressions with binary components; this type of problem may be solved in practice, using the Branch and Bound (B&B) algorithm [27]. The B&B algorithm departs from a relaxed problem; in this case, the convex quadratic programming problem QP.…”