1. Introduction
Electrical distribution networks represent the largest portion of power systems, which are entrusted with providing electrical energy from a transmission/sub-transmission substation to all end-users at medium- and low-voltage levels [
1,
2]. Owing to economic restrictions, investment distribution networks are typically constructed with radial configurations since this allows reducing inversion in the infrastructure, simplifying also the protective devices’ coordination [
3]. However, the radial structure of these systems implies that they experience high power and energy losses when compared with transmission systems, being that the energy losses in these grids are between 6% and 18% of the total power input at the substation bus, while in transmissions networks, these are between 1.5% and 2.0% [
4,
5].
Owing to the high energy losses in distribution networks, utilities implement different methodologies to reduce them as much as possible. Some recurrent strategies are the optimal location of fixed-step capacitor banks [
6,
7,
8]; optimal grid reconfiguration [
9,
10]; optimal siting and sizing of dispersed generators [
11,
12]; optimal siting and sizing of battery energy storage systems [
13,
14]; and reactive power compensation via distribution static compensators, i.e., STATCOMs [
15,
16,
17]. In the case of fixed-step capacitor banks, these are attractive technologies due to their low cost and high reliability; however, these devices inject reactive power in discrete steps, which implies that, due to the daily load variability, the total grid power losses are not completely minimized. The strategies that use the grid reconfiguration to minimize power losses are efficient regarding the percentage of energy losses reduction; however, these require important modifications of the grid topology, including new line construction and redesign of the protective devices schemes, which is more expensive when compared with shunt reactive power compensation. The optimal integration of dispersed generators based on renewables is undoubtedly the most efficient option to reduce the total energy losses of the network, especially when active and reactive power capabilities of these devices are used [
12]. However, from an economical point of view, the main application of the renewables is a reduction in greenhouse gas emissions and the reduction in the total energy purchasing costs in the substation buses. In the case of batteries, these devices help with the total energy losses reduction. However, their investment costs are bigger when compared with shunt reactive power compensation; the main application of the batteries is to extend the usability of the renewables in periods of time with high demand and low generation [
18]. Finally, STATCOMs are excellent devices to reduce the amount of grid energy losses costs, with the main advantage that these can inject variable reactive power as a function of the grid load behavior, even if these are more expensive that the capacitor banks; these allow important improvements in the total annual grid operation costs. Moreover, these devices present useful life times of about 25 years, with minimum maintenance costs, which make these appropriate devices for improving the grid performance regarding energy losses and voltage profiles [
19].
In the current literature regarding STATCOMs and their usage in distribution systems applications, some of the most relevant works are discussed below. Authors of [
16] presented the application of the discrete-continuous version of the vortex-search algorithm (DCVSA), where the discrete part is entrusted with determining the nodes where the STATCOMs must be located and the continuous part defines their optimal sizes. Numerical results demonstrated that the DCVSA approach reaches the best optimal solution in the IEEE 33- and 69-bus systems when compared with the GAMS optimization package. Authors of [
17] presented a complete revision of the strategies used to locate and size STATCOMs in distribution grids by making a classification in five categories, including analytical methods, artificial neural network-based approaches, metaheuristic methods, sensitivity approaches, and a combination of sensitivity approaches and metaheuristic methods. The main contribution of this research is the revision of the complete revision of the state of the art regarding distribution static compensators and their applications to improve the distribution system performance. In [
20] was proposed a hybrid optimization approach based on the classical Chu and Beasley genetic algorithm in the master optimization stage that determines the nodes where the STATCOMs must be installed, while the slave stage uses a second-order cone programming reformulation of the multiperiod optimal power flow problem to determine their optimal sizes. Numerical results demonstrate the efficiency of the proposed approach when compared with the GAMS solvers in the IEEE 33- and 69-bus systems. Castiblanco-Pŕez et al. in [
4] proposed the application of the discrete-continuous version of the Chu and Beasley genetic algorithm to locate and size STACOMs in distribution networks, with numerical results comparable with the DCVSA reported in [
16]. The main contribution of the authors was the usage of differentiated load zones classified in residential, commercial, and industrial users, including the possibility of having radial and mesh distribution system topologies. Computational validations were made in the IEEE 33-bus systems, and numerical results demonstrated the possibility of having high-quality solutions when compared with the MINLP solvers available in the GAMS optimization package.
Even if the previous optimization methodologies are efficient to solve the problem addressed in this research, these have two main problems, which are (i) the usage of combinatorial methods to locate and size the STATCOMs since their random exploration and exploitative nature through the solution space makes it impossible to ensure the global optimum finding, and (ii) in the case of the exact MINLP approaches, the main complication is associated with the non-convexity of the solution space in their continuous component that highly increases the possibility of staying stuck in local optimal solutions. To deal with these problems, this research proposes a new methodology based on the hybridization of two convex-based models. The proposed methodology is completely described below.
Unlike aforementioned works regarding the optimal siting and dimensioning of STATCOMs in distribution networks, in this research, we propose a new optimization methodology based on mixed-integer convex optimization that decouples the location problem from the sizing problem. To solve the location problem, a mixed-integer quadratic formulation is employed considering that the voltage profiles are equal to
. This becomes the power balance equations in a set of linear equations that becomes the location problem into a mixed-integer approximation that ensures the optimality of the solution. The solution of the location problem allows fixing the values of the binary variables in the sizing problem. The assignation of the binary variables allow solving the sizing problem of the STATCOMs via the second-order cone programming reformulation proposed in [
20] with the main advantage that the global optimum finding is ensured via convex optimization. Numerical results in the IEEE 33- and 69-bus systems demonstrate the effectiveness and robustness of the proposed optimization approach when compared with the DCVSA and the MINLP solvers in GAMS.
The main contributions of this research are the following:
The presentation of a new optimization methodology to solve separately the problems of optimal location and sizing of STATCOMs in distribution networks to reduce the annual operative costs of the network, where the location problem is solved through a mixed-integer quadratic formulation, and the sizing stage is addressed with a second-order cone programming equivalent.
The validation of the proposed methodology in two classical test feeders composed of 33 and 69 nodes with better results than the best current approach reported in the current literature, i.e., the discrete-continuous version of the vortex search algorithm.
It is worth mentioning that in the scope of this research, distribution networks with meshed structures are not considered since the second-order cone programming reformulation of the optimal power flow problem proposed in [
21] only deals with purely-radial distribution grids; however, this can be taken as a opportunity of research in future works.
The proposed decoupled optimization approaches to solve the problem of the location and sizing of STATCOMs in distribution networks based on mixed-integer convex and conic programming was not previously proposed in the current literature. This was identified as a gap in the current literature that this research work is aiming to fulfill.
This document is structured as follows:
Section 2 presents the general optimization problem regarding the optimal placement and sizing of STATCOMs in distribution grids with a radial structure.
Section 3 presents the main aspects of the proposed solution methodology based on decoupling the location from the sizing problem via mixed-integer quadratic and second-order cone programming.
Section 4 shows the main characteristics of the IEEE 33- and 69-bus systems, including the parametrization of the costs function of the STATCOMs.
Section 5 presents the main numerical results of the proposed solution methodology and their comparison with recent literature reports.
Section 6 lists the main conclusions obtained from this research and defines some possible future works.
6. Conclusions and Future Works
The problem of the optimal siting and sizing STATCOMs in radial distribution grids to reduce the annual grid operative costs was addressed in this research from the point of view of mixed-integer convex optimization. The exact MINLP model was decoupled into two subproblems, named the location problem and the sizing problem. The former problem was formulated with a mixed-integer quadratic programming model that allows determining the optimal location of the STATCOMs. These locations are provided in the latter stage (sizing problem), where a second-order cone programming model was used to solve the multiperiod optimal power flow problem. To obtain these equivalent models, the component of the objective function regarding the annualized costs of the STATCOMs was linearized through Taylor’s series expansion. Numerical results demonstrated that the proposed decoupled approach improves the solutions reached by the DCVSA and the BONMIN and COUENNE solvers available in the GAMS optimization package for both test feeders under study.
In the case of the IEEE 33-bus system, the general improvement with respect to the base case was , which implied a reduction of about USD 15,973.21 per year of operation; for the IEEE 69-bus system, this reduction was , i.e., USD 18,908.72 per year of operation. These results improved the best numerical solution reported in the literature with the DCVSA of about USD 516.80 for the IEEE 33-bus system and USD 592.98 in the case of the IEEE 69-bus system, which confirmed the effectiveness and robustness of the proposed approach regarding literature reports.
As future works, it will be possible to develop the following research studies: (i) to extend the proposed decoupled optimization methodology to integrate renewable sources (i.e., solar photovoltaic or wind sources) to minimize the total grid generation costs; (ii) to extend the proposed formulation to distribution grids with meshed topologies; and (iii) to apply the optimization paradigm known as poly-optimization to solve problems regarding the optimal placement and sizing of shunt devices in distribution networks.