1.1. Background
Optimization tools are used in various studies to make the best decisions possible in swarm-based problems [
1,
2,
3]. Researchers have been considering more details of the methods and datasets in order to obtain more reliable and fast solutions for their optimization problems. The number of calculations and computational time increase with the increasing scale of the network and the number and type of objectives. In some cases, simulations can take several days or even weeks to complete the process. The deployment of parallel computing to handle a large number of computations is the primary focus of this study. Moreover, we aim to analyze the relationship between the accuracy and the speed of the performed parallel computation since parallel computing may converge to different values than serial computing. In this regard, the results for both cases are compared, and the accuracy of parallel computing is evaluated comparatively.
DGs have been penetrating into the distribution network for energy loss minimization [
4,
5,
6], voltage profile improvement [
7,
8,
9], peak clipping, cost minimization [
10], and similar purposes through constrained optimization formulations. Therefore, the optimal DG siting and sizing problem involves solving algebraic or differential equations concerning the consolidation of high- and low-energy consumption points in the power grid. The number of calculations is also related to consumer behavior and the number of alternative locations [
11]. Most of the past studies have concentrated on the formulation of DG siting and sizing problems, while decreasing the computational time was relatively discarded. However, the computational time required for reliable solutions for the increasing number of inputs and objective functions is getting more important day by day, especially for operational problems involving real-time solutions.
This study is therefore devoted to decreasing the computational time of optimal DG planning problems. In this regard, we investigated the most efficient parallel computing scheme that reduces the computational time without scarifying the accuracy of the serial configurations. The related formulation is tested in known power distribution test systems.
1.2. Literature Review
Many formulations have been proposed for the siting and sizing of DGs, some focusing on the single-objective problem, others on multi-objectives aiming to reduce active losses [
12], improve the voltage profiles [
13], improve the reliability indices of the distribution network (DN) [
14], minimize the emission rate [
15], provide better load distribution by peak clipping [
16], and reduce the operation cost of micro-grids without bringing discomfort to the users [
17]. Among several alternatives, photovoltaics (PVs) and wind turbines (WTs) are the two most-common DG units that can be placed close to the load centers. In [
11], a nonlinear single- or multi-objective cost function was minimized in order to achieve optimal DG allocation and sizing, which was developed in accordance with the utilities’ interests and concerns. Then, optimization equations were solved using either analytical methods or heuristic approaches, depending on the problem types and desired computational speed. Due to the application ease of their derivative-free solution procedures for the formulations comprising the use of both the integer and the real control variables, heuristic approaches are often preferred.
The optimum sizing and siting of a single DG to minimize the daily energy losses and voltage violations in a DN was achieved by employing the meta-heuristic cuckoo search algorithm [
18]. Then, the resulting daily percentage energy losses were decreased from 3.46% to 0.92% and 3.75% to 1.01% without any voltage magnitude violations for the summer and winter seasons, respectively. Sultana et al. used the grey wolf optimizer (GWO) for multi-DG allocation and sizing in a distribution system to minimize the reactive power losses and improve the voltage profiles [
19]. Another study used an analytical method for expansion planning of the Addis North distribution network, considering the integration of the optimal sizes of distributed generations for the projected demand growths [
20]. The active and reactive power losses were minimized by 21.285% and 19.633%, respectively, and the voltage profiles were improved by 8.78%. A multi-objective ant lion optimization (ALO) algorithm was modified to determine the near-optimal numbers, sizes, locations, and types of the DG units for the objectives of voltage profile improvement and installation cost minimization [
21]. The resulting optimal values were then tested with the extreme monthly distributions to identify the impacts of load and generation volatilities on the voltage profiles. The enhanced multi-objective harmony search optimization algorithm reduced the power losses and improved the voltage profiles [
22]. Another paper used the same heuristic algorithm to determine the optimal state of switching devices (open or closed) in a given distribution network, aiming to minimize active power losses [
23].
The ALO algorithm and its multi-objective variation (MOALO) are becoming more well-known in recent times. It was used to address smart grid issues such as voltage profile improvement, cost optimization of distributed generation, and power loss reduction in [
24,
25]. In [
26], the authors found optimal distributed energy resource (DER) size configurations that aim to minimize DER losses. The MOALO algorithm was used in this work to determine the Pareto-front-optimal solutions to minimize installation costs and optimize voltage profiles.
There are several parallel computing applications in the power systems area. In [
27], the authors solved the optimal switching problem by parallel processing of mixed-integer linear programming. The authors of [
28] solved the reactive power optimization problem by using an adaptive differential evolution method. Several heuristic algorithms were utilized for parallel computing in solving bidding problems in local energy markets in [
29]. A parallel particle swarm optimization was used in [
30] to maximize the profits of the industrial customers that provide operational services to the power grid. The economic dispatch problem was solved using a parallel bat algorithm in [
31]. An optimal distributed reactive power supply was formulated and solved by using a parallel harmony search algorithm in [
32].
In most of the studies related to parallel computing, the main aim was to improve communication and cooperation among processors to increase computational speed and efficiency [
33,
34,
35,
36,
37,
38]. In [
33], the authors tried to explain how to achieve multi-processor operations with excellent efficiency by making better use of multiple processors. Three techniques, namely the parallel island model, parallel evaluation of a population, and parallel evaluation of a single solution were used. This paper will utilize the second method, which is a variation of the master/slave method.
Speedup and efficiency are the most popular parameters used to evaluate the computational speed in parallel processing. The speedup of an n-processor computation is the ratio of the time required to solve a problem with a single processor to the time required to solve the problem with n processors. It is obvious that an ideal system with n-processors has a speedup equal to n. However, this is not the case in practice since each processor spends some time on communication and cannot use 100% of its time for computation. Therefore, efficiency is used to measure the percentage of time that a processor uses for the computation task. In other words, the speedup compares how many times faster a problem can be solved as a function of the number of parallel units, and efficiency measures how much of that benefit is obtained per contributing processor unit [
39].
1.3. Contribution
With the advancement of technology and increase in the size of the network, it is necessary to address more complex issues in order to make the best decisions among several alternatives. This issue motivated us to gather a great deal of information in the relevant field, which increases the number of calculations required to arrive at the best decision. However, one of the most-significant considerations for researchers is the calculation time. The associated literature for such research reveals few investigations into time consumption, despite the fact that it must be included. In this regard, the computational time is handled as an additional objective in this paper, along with the other network objectives.
The following summarizes the major contributions of the study:
A parallel computation approach based on the master/slave method is applied to the optimal allocation and sizing of DG units in a DN, considering minimizing energy losses and DG costs.
The impacts of the number of parallel processors on the optimal control parameters, objective functions, and dependability of the method are determined.
The range of the optimal number of parallel processors providing better speedup and efficiency is determined.
Optimum solutions for the different number of processors are discussed with respect to three multi-objective optimization performance criteria.
The structure of the paper is as follows. The proposed problem’s formulation is covered in
Section 2. The implementation of the optimization algorithm and its paralleling solution are discussed in
Section 3. The test systems are detailed in
Section 4, and the results are analyzed in
Section 5. Conclusions are summarized in
Section 6.