Abstract
This paper deals with the problem of the optimal selection of capacitor banks in electrical AC distribution systems for minimizing the costs of energy losses during a year of operation through a discrete version of the vortex search algorithm (DVSA). This algorithm works with a hypersphere with a variable radius defined by an exponential function where a Gaussian distribution is used to generate a set of candidate solutions uniformly distributed around the center of this hypersphere. This center corresponds to the best solution obtained at the iteration t, which is initialized at the center of the solution space at the iterative search beginning. The main advantage of combining the exponential function with the Gaussian distribution is the correct balance between the exploration and exploitation of the solution space, which allows reaching the global optimal solution of the optimization problem with a low standard deviation, i.e., guaranteeing repeatability at each simulation. Two classical distribution networks composed of 33 and 69 nodes were used to validate the proposed DVSA algorithm. They demonstrated that the DVSA improves numerical reports found in specialized literature regarding the optimal selection and location of fixed-step capacitor banks with a low computational burden. All the simulations were carried out in MATLAB software.
Highlights
Electrical distribution networks are responsible for transferring energy from the transmission system to industrial, commercial, and domestic users [1]
The problem of the optimal location and selection of fixed-step capacitor banks in AC distribution networks can be represented with an mixed-integer nonlinear programming (MINLP) model, where (i) the integer characteristic is defined by the possibility of locating capacitor bank k into an arbitrary node i, which is defined with the integer variable xi,k ; (ii) the mixed nature of the model is giving by the presence of continuous variables such as power injection in slack nodes, and voltage magnitudes vi and angle θi in all the nodes; (iii) the nonlinear structure is defined by the presence of a trigonometric function in the power balance equations as well as by the products between voltage magnitudes in different nodes
X Note that the proposed discrete version of the vortex search algorithm (DVSA) for optimal location and selection of fixed-step capacitor banks in distribution networks are extended to multiple periods of analysis by adding the sub-index t in all the voltages, angles, and power in the mathematical model (1)–(6) (see optimization model (A1)–(A6) reported in Appendix B) since the optimal location and size of the capacitors is uncoupled in time
Summary
Electrical distribution networks are responsible for transferring energy from the transmission system to industrial, commercial, and domestic users [1]. After a careful review of the state of the art, we do not find evidence regarding a discrete version of the VSA applied to mixed-integer nonlinear optimization problems This is a gap that this research tries to fill in the electrical engineering area by applying this variant of the VSA applied to the issue of the optimal location and selection of fixed-step capacitor banks in distribution networks. X The use of a discrete codification implements integer numbers as decision variables that simplify the dimension of the classical binary vectors used in the literature to represent this optimization problem This codification compacts in a unique stage, the location and sizing problems of capacitor banks, which substantially reduces the processing times.
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