Annual Operating Costs Minimization in Electrical Distribution Networks via the Optimal Selection and Location of Fixed-Step Capacitor Banks Using a Hybrid Mathematical Formulation

: The minimization of annual operating costs in radial distribution networks with the optimal selection and siting of ﬁxed-step capacitor banks is addressed in this research by means of a two-stage optimization approach. The ﬁrst stage proposes an approximated mixed-integer quadratic model to select the nodes where the capacitor banks must be installed. In the second stage, a recursive power ﬂow method is employed to make an exhaustive evaluation of the solution space. The main contribution of this research is the use of the expected load curve to estimate the equivalent annual grid operating costs. Numerical simulations in the IEEE 33- and IEEE 69-bus systems demonstrate the effectiveness of the proposed methodology in comparison with the solution of the exact optimization model in the General Algebraic Modeling System software. Reductions of 33.04% and 34.29% with respect to the benchmark case are obtained with the proposed two-stage approach, with minimum investments in capacitor banks. All numerical implementations are performed in the MATLAB software using the convex tool known as CVX and the Gurobi solver. The main advantage of the proposed hybrid optimization method lies in the possibility of dealing with radial and meshed distribution system topologies without any modiﬁcation on the MIQC model and the recursive power ﬂow approach.


Introduction
Electrical distribution networks are constantly growing due to the requirements of commercial, industrial, and residential users in urban and rural areas [1]. Most of the investments in electrical systems are condensed into medium-and low-voltage level applications to satisfy the growing electric demand [2]. For this reason, utilities are intended to improve the quality service in their grids in order to enable the interconnection of new users with minimum investment and operating costs [3]. One of the key aspects in the construction of electrical distribution grids is their topology. These networks are typically constructed with a radial configuration to minimize the investment in electrical infrastructure (conductors, isolators, protective devices, and so on) [4], but these investment reductions are counteracted by increments in the operating costs, since radial configurations have higher power losses in comparison with meshed grid configurations [5].
To deal with the costs of energy losses distribution networks, electric distribution companies typically employ shunt power compensation, i.e., with active and reactive formulation is that it ensures the optimal global optimal solution of the relaxed model. This implies that no statistical evaluations confirm the location of the capacitors, which is necessary for metaheuristic-based optimizers. Once the location of the fixed-step capacitor banks is defined by the MIQC model, a recursive power flow solution is implemented in order to evaluate all possible sizes (i.e., exhaustive evaluation of the solution space), which allows for determining the best possible sizes for these capacitors. It is worth mentioning that the exhaustive exploration of the solution space is possible since it has a few thousand options, which are easily assessed with any personal or desktop computer.
Note that the selection of the MIQC formulation to determine the set of nodes where the fixed-step capacitor banks will be located is motivated by the convexity of the solution space for each binary variable combination, which implies that, through the combination of a Branch and Bound method with the interior point approach, it is possible to ensure that the global optimum is found [28]. Even if some metaheuristic models perform efficiently in electrical engineering problems such as the reconfiguration of distribution grids including dispersed generation [29,30], more studies are required in this research field to make metaheuristics competitive against MIQC models.
The remainder of this article is organized as follows: Section 2 presents the proposed MIQC model that defines the optimal location of the fixed-step capacitor banks. This model corresponds to a quadratic objective function with linear integer constraints. Section 3 presents the recursive power flow solution method based on the successive approximation approach to define the optimal sizes of the fixed-step capacitor banks. Section 5 shows the main characteristics of the IEEE 33-and 69-bus systems. Section 6 presents the numerical results, their analysis, and discussions, as well as the comparison between the proposed two-stage solution methodology and the literature reports. Finally, Section 7 lists the main concluding remarks derived from this research, as well as some proposals for future work.

Nodal Selection Strategy
To select the nodes where the fixed-step capacitor banks will be installed, a mixedinteger quadratic convex (MIQC) formulation is proposed by using the equivalent linear power flow formulation for electrical distribution networks obtained by simplifying the MINLP model for distribution system reconfiguration reported in [31]. The simplifications made to the MINLP model in order to obtain an MIQC are the following: i.
All the voltage magnitudes are assumed to be known, i.e., these can be assigned as plane voltages (1∠0 • )or set as the power flow solution without capacitor banks (i.e., the benchmark case). ii. The magnitude of the currents through the distribution lines is mainly governed by the active and reactive power consumption, which implies that the effect of the second Kirchhoff law at each line is negligible in comparison with the first Kirchhoff law at each node.
By considering the aforementioned assumptions on the exact MINLP model proposed by [31], the MIQC model to define the set of nodes where the fixed-step capacitor banks will be installed is obtained (see Figure 1). This model is defined in Equations (1)-(5):  Obj. fun.: Subj. to.: Note that all the mathematical symbols, parameters, and variables are contained in the nomenclature list presented at the end of this document.
Note that the proposed MIQC formulation defined in Equations (1)-(5) has the following interpretation: Equation (1) defines the approximated annual grid operative costs of the network, which correspond to the sum of the expected costs concerning the energy losses and the installation of the fixed-step capacitor banks. Equation (2) defines the active power balance equilibrium at node j for each period of time h, which is linear when the voltages V r j,h and V i j,h are assumed to be known. Equation (3) defines the reactive power equilibrium at each node for each period of time. Inequality constraint (4) defines the possibility of installing as much as one fixed-step capacitor bank type at node j. Finally, inequality constraint (5) limits the maximum number of fixed-step capacitor banks that can be installed in the distribution grid.
In order to characterize the optimization model defined in Equations (1)-(5), the classification and type of variables are presented in Table 1, including the number and type of constraints. It is important to mention that, in this classification, n is the number of nodes, h represents the periods of time, c corresponds to the number of capacitor available, and l is the number of lines. To refine the solution provided by the MIQC model regarding the sizes of the fixedstep capacitor banks, the next section presents the methodology for finding the fixed-step capacitor banks, which is based on the recursive power flow solution for each possible fixed-step capacitor size combination.
It is important to mention that the dimension of the solution space defined by the binary variables in the MIQC model in Equations (1)-(5) takes a combination form, where it depends on the number of capacitors available and the number of nodes in the distribution network. Note that, for the IEEE 33-and 69-bus grids, these dimensions are 4960 and 50,116 when three capacitors are considered for installation [32].

Assigning the Optimal Sizes
Once the MIQC model presented in Section 2 has been solved, the values of the variables x jc are known. However, in the second stage of the proposed optimization methodology, the optimal sizes of the fixed-step capacitor banks are refined by fixing these capacitors in the j nodes.
Given that the optimization problem can take a maximum of N cap ava to be installed in the distribution network (one fixed-step capacitor bank per node), in this stage, an integer combination is proposed to represent each fixed-step capacitor size. Note that, if the small fixed-step capacitor bank is assigned to node i with 1 and the largest fixed-step capacitor bank for the k node is set as c, then the proposed codification for the selected nodes has the following structure: where x sol represents the vector associated with the selected nodes j, k, and m, along with their corresponding sizes (possible solution). It is important to mention that, for each solution in Equation (6), the total costs of the fixed-step capacitors are easily determined with the second component of the objective function (1).

Remark 2.
Due to the fact that the maximum size of the capacitor bank is assigned as c and the maximum number of devices installed is N cap ava , the size of the solution space for the optimal sizing problem of the fixed-step capacitor banks is c N cap ava . This number is 14 3 = 2774 for the studied set of alternatives regarding the fixed-step capacitor banks.
Considering a solution space with c = 14 and N cap ava = 3, the proposed methodology to determine the optimal sizes of these capacitor banks is exhaustive, which implies that the 2774 options are evaluated in a conventional power flow formula. The proposed power flow methodology is the successive approximation power flow method reported in [33].
The general recursive power flow formula for the successive approximation method is defined in Equation (7): where m is the iterative counter, V d is the vector that contains all the voltage variables in the complex domain for all the demand nodes in each period of time h, S cap,h is the complex vector that contains all the power outputs in the fixed-step capacitor banks in each period of time h (note that this vector is provided for each of the 2744 size combinations), S d,h is the complex demand vector with the active and reactive power consumption in the demand nodes for each period of time, V s,h is the complex voltage output at the substation bus, Y dd is a complex square matrix that contains all the admittances among the demand nodes, and Y ds is a rectangular complex matrix that contains the admittances between the demand and the substation buses. Note that diag(z) and z are a matrix with all the elements of the z at its diagonal and the conjugate operator of the complex vector z, respectively. It is important to emphasize that S cap,h is 0 for the set of nodes different from j, k, and m. Regarding the example codification presented in Equation (7), these nodes are listed below: The main characteristic of the recursive power flow Formula (7) is that its convergence to the power flow solution can be ensured by applying the Banach fixed-point theorem [34].
To determine if the power flow Formula (7) has converged, the difference between the voltage magnitudes between two consecutive iterations is used, i.e., where ε is the tolerance value, which is assigned as 1 × 10 −10 , as recommended in [33]. Note that, once the power flow problem is solved with Equation (7), the amount of power losses is calculated for each period of time, as presented in Equation (9): where V h is the vector that contains the substation and demand voltages, ordered as [V s,h V d,h ] ; and Y bus is the nodal admittance matrix of the distribution grid. With the power losses at each period of time, the exact operational costs of the distribution systems with fixed-step capacitor banks (i.e., min z costs ) can obtained as defined in Equation (10):

Summary of the Solution Methodology
The proposed solution methodology for locating and selecting fixed-step capacitor banks is summarized in Algorithm 1. Here, the problem of location is solved with a mixedinteger convex formulation, and the selection (i.e., sizing) is determined with the recursive evaluations of the power flow problem for each one of the N cap ava possible combinations.
Algorithm 1: Proposed two-stage solution methodology to select and locate fixed-step capacitor banks in distribution networks.
Data: Define the distribution network under study Obtain the per-unit equivalent of the distribution grid; Define the number of capacitor banks available for installation, i.e., N cap ava ; Solve the MIQC model (1)-(5); Select the positions where the capacitor banks will be located from the variable x jc ; for All the capacitor size combinations do Define the values for the vector S cap,h ; Solve the recursive power flow Formula (7); Calculate power losses at each period of time with Equation (9); Determine the objective function with Equation (10); end Order all the solutions in ascending form with the values of the objective function; Result: Report the optimal solution

Test Feeder Information
To validate the proposed two-stage optimization methodology, two classical test feeders known as the IEEE 33-and 69-bus systems are employed. The parametric information for these grids is presented below.

IEEE 33-Bus Grid
The IEEE 33-bus grid is an electrical network operated with 12.66 kV at the substation bus located at node 1. This system has a radial structure, i.e., 33 buses and 32 lines. The electrical configuration of this test feeder is presented in Figure 2a, and its electrical parameters are listed in Table 2.  4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

IEEE 69-Bus Grid
The IEEE 69-bus grid is an electrical network operated with 12.66 kV at the substation bus located at node 1. This system has a radial structure, i.e., 69 buses and 68 lines.
The electrical configuration of this test feeder is presented in Figure 2b, and its electrical parameters are listed in Table 3. Table 3. Parametric information for the IEEE 69-bus system.

Parameters for the Economic Assessment
To determine the annual grid operating costs in the network when the fixed-step capacitor banks are installed, the information regarding sizes and costs is presented in Table 4. Note that this information was adapted from [14].

Computational Implementation
The proposed two-stage optimization approach was implemented in the MATLAB software, version 2019b. The mixed-integer linear programming model was implemented using the CVX and the Gurobi solver. The recursive power flow solution was implemented with our own scripts, which used the successive approximation power flow formulation. All the simulations were run on a desktop computer with an Intel(R) Core(TM) i7-7700 2.8-GHz processor and 16.0 GB of RAM on a 64-bit version of Microsoft Windows 10 Home.

IEEE 33-Bus Grid
For this test feeder, considering that, throughout the year, the system operates under peak load conditions, as analyzed by [14], the first stage of our proposed optimization method identifies nodes 13, 24, and 30 as the optimal location for the capacitor banks. Now, by fixing the sizes provided by the MIQC model and applying the refinement stage of our proposed optimization model, the optimal sizes for these locations are 450, 450, and 1050 kvar. Note that this solution has an expected annual operative cost of US$/year 23,747.317, with an investment of US$/year 467. 10. These values imply a reduction of 33.04% with respect to the annual cost of the benchmark case (i.e., US$/year 35,445.909 without capacitors). Table 5 presents a comparison with the solution of the exact MINLP model in the GAMS software and the best three solutions reported for our two-stage proposed optimization approach. The most important result in Table 5 is that, with the proposed two-stage optimization method, it is possible to generate a list with the alternative solutions. Note that, in this list, there are three solutions with better final objective function value than the solution obtained with the GAMS software.

IEEE 69-Bus Grid
For this test feeder, considering that, throughout the year, the system operates under peak load conditions, as analyzed by [14], the first stage of our proposed optimization method identifies nodes 11, 21, and 61 as the optimal location for the capacitor banks. In addition, by fixing the sizes provided by the MIQC model and applying the refinement stage of our proposed optimization model, the optimal sizes for these locations are 450, 150, and 1200 kvar. Note that this solution has an expected annual operative cost of US$/year 24,845.246, with an investment of US$/year 392.85. These values imply a reduction of 34.29% with respect to the annual cost of the benchmark case (i.e., US$/year 37,812.056 without capacitors). Table 5 presents a comparison with the solution of the exact MINLP model in the GAMS software and the best three solutions reported for our two-stage proposed optimization approach.
Note that the main result in Table 6 is that the proposed two-stage optimization method allows for identifying at least four solutions with better objective function values than the solution reached with the GAMS software.  Table 6 confirms that performing an exhaustive exploration once the nodes where the fixed-step capacitor banks will be installed have been identified enables the identification of additional solutions that cannot be found by means of commercial approaches. This situation is particularly important for utilities since additional investment alternatives can be identified prior to making the final decision regarding installation in the grids.

Numerical Results Considering Daily Load Variations
To verify the effectiveness and robustness of the proposed optimization approach, in this simulation scenario, the daily variations of the active and reactive power curve in 30-min periods were considered [35]. The active and reactive power variations are listed in Table 7. Considering the daily information in Table 7 for the active and reactive power demands, in the case of the IEEE 33-bus grid, the best solution provided by the proposed two-stage optimization method corresponds to locating the fixed-step capacitor banks in nodes 2, 7, and 30, with sizes of 150, 450, and 450 kvar, respectively. These capacitor banks have an investment cost of US$/year 302.70, with total annual operative costs of US$/year 12,763.112. This value corresponds to a reduction of 17.95% with respect to the benchmark case (i.e., US$/year 15,555.063 without installing fixed-step capacitor banks).
For the IEEE 69-bus system, the proposed methodology identifies the nodes 11, 24, and 61 and fixed-step capacitor banks with sizes of 150, 150, and 600 kvar, respectively. These capacitor banks have an investment cost of US$/year 282, with total annual operative costs of US$/year 13,141.378. This implies a reduction of 20.44% with respect to the benchmark case (i.e., US$/year 16,517.383 without installing fixed-step capacitor banks).

Applicability in Meshed Distribution Networks
To demonstrate the applicability of the proposed hybrid optimization model to deal with the problem regarding the optimal placement and sizing of fixed-step capacitor banks in electrical distribution grids with meshed configurations, in this simulation scenario, the meshed configuration of the IEEE 33-bus grid was considered with all the tie-lines in the reconfiguration problem studied in [36] permanently closed. For the sake of completeness, the lines added to the IEEE 33-bus system are listed in Table 8. For the IEEE 33-bus system, where the distribution lines in Table 8 are added to the distribution system topology presented in Figure 2a and the daily active and reactive power behavior in Table 7 is considered, the solution of the MIQC model (1)-(5) defines the location of the fixed-step capacitor banks at nodes 2, 8, and 30, respectively. In addition, by fixing these sizes in the recursive power flow solution methodology, the optimal sizes assigned for these nodes are 150, 300, and 600 kvar, respectively. When these fixed-step capacitor banks are installed, the annual grid operative costs take a value of US$/year 7927.316, i.e., a reduction of 14.88% with respect to the benchmark case.
Note that the most important result in the meshed scenario is as expected: the annual grid operating costs is low in comparison with the radial operative case, since the benchmark case in the radial scenario was US$/year 15, 555.063, while, in the full meshed operation scenario, this value decreased to US$/year 9313.495. This is due to the presence of meshes in the distribution network allowing for a better power flow distribution and improving voltage regulation in the network, which are directly related with the reductions in the total energy losses of the network compared to the radial topology.

Conclusions and Future Work
The problem regarding the optimal placement and sizing of fixed-step capacitor banks in electrical distribution networks with radial structure was addressed in this research via the application of a two-stage optimization methodology. The first stage of the proposed optimization approach dealt with selecting the nodes where the fixed-step capacitor banks would be installed through the implementation of an MIQC model. The second stage corresponded to an exhaustive assessment of the solution space by using the successive approximation power flow method for each possible size combination of the fixed-step capacitor banks on the nodes, which was provided by the MIQC model.
The numerical results in the IEEE 33-and 69-bus grids showed that the proposed two-stage optimization approach finds better objective function values than the GAMS software with its exact MINLP solvers. For the IEEE 33-bus grid, the difference between the proposed method and the GAMS software was US$/year 111.996. In addition, for this test feeder, the proposed MIQC model found two additional solutions with better objective function values when compared to the GAMS optimal solution. In the case of the IEEE 69-bus system, the proposed two-stage approach found an additional gain of US$/year 31.664. In addition, the proposed MIQC model found three alternative solutions with better objective function values in comparison with the GAMS optimal solution.
As for the reductions with respect to the benchmark case, in the scenario involving a year-long operation under peak load conditions, for the IEEE 33-bus, the annual expected improvement was about 33.04%, and, for the IEEE 69-bus system, it was 34.29%. However, when the daily active and reactive power consumption was considered, the expected improvement was about 17.95% for the IEEE 33-bus grid, and, for the IEEE 69-bus grid, it was about 20.44%. Note that these behaviors were expected in the context of this simulation, given that the daily behavior of the active and reactive power curves corresponded to a realistic operative scenario, whereas the peak load operation represented a theoretical operative scenario that only served to validate new solution methodologies with respect to literature reports.
The main limitation of the proposed optimization method is related to the possibility of evaluating the entirety of the solution space regarding the possible sizes of the fixed-step capacitor banks, since, for large test feeders where more than three capacitor banks are available, the proposed recursive solution methodology entails long processing times. Note that, for a system with an availability of 7 capacitor banks and 14 possible sizes, the dimension of the solution space is 105,413,504, i.e., more than 105 million possible solutions. Therefore, for large solution spaces, it is recommended to replace the recursive power flow evaluation method with a specialized metaheuristic optimization technique that can efficiently deal with discrete variables with reduced computational effort.
As future work, it will be possible to conduct the following studies: (i) to propose a mixed-integer conic model that allows for integrating fixed-step capacitor banks with the optimal grid reconfiguration of the network while considering daily active and reactive power variations; (ii) to extend the proposed MIQC model in order to locate and size renewable energy resources and batteries in distribution networks; and (iii) to consider more realistic models to represent capacitor banks, including energy losses and reactive power injection variability as a function of the voltage at the nodes where the banks are connected.   A jl Component of the node-to-branch incidence matrix that associates node j with line l. C Set that contains all fixed-step capacitor bank types available for installation in the distribution grid. H Set that contains all hours of the operation period (typically 24 h). L Set that contains all distribution lines of the network. N Set that contains all the nodes of the network.   x sol Solution vector that contains the nodes where the fixed-step capacitor banks will be located along with their possible sizes. x jc Binary variable associated with the installation (x jc = 1) or not (x jc = 0) of a fixed-step capacitor bank type c at node j. z approx Approximate objective function value associated with the expected annual grid operating costs (US$). z costs Expected annual operating costs of the network (US$).