1. Introduction
An important part of efficient, affordable, and reliable power system operation, or in other words, optimal operation, is the optimal reactive power (Volt-VAR) control problem, which includes generator voltage, shunt compensator power, and the tap positions of tap changers [
1]. In a competitive and deregulated environment, the optimal Volt-VAR control issue is an important and efficient tool in electrical energy transmission networks. The main objective of solving the Volt-VAR control issue is to reduce network losses and, as a result, reduce the final cost of energy transmission in energy systems while satisfying a set of operational and physical constraints imposed by network and equipment limitations. The basic goal is to minimize essential key functions, such as the summation of bus voltage deviations and active power losses while also addressing several practical limitations [
2]. Because generator voltage is intrinsically continuous but shunt reactive power compensators and tap changer ratios are discrete variables, the optimal VAR control problem is viewed as a complex multimodal nonlinear optimization issue including discrete variables. This problem is also a multimodal, high-dimensional, and complex sophisticated problem with some nonlinear objective functions with multiple local minima and several nonlinear and discontinuous constraints [
1,
2,
3,
4].
Numerous methods, ranging from standard mathematical techniques to those related to artificial intelligence, have been proposed over the past years for the application of optimal VAR control problems. The introduction of the harmony search algorithm (HSA) is an example of development in this area [
5]. Additionally, Roy demonstrated the increased capability of the biogeography-based optimization (BBO) method for solving multi-constrained problems [
6]. In addition, differential evolution (DE) with dynamic multi-group self-adaptive operators (DMSDE) [
7], particle swarm optimization (PSO) based on multi-agent systems (MAPSO) [
8], fuzzy adaptive PSO (FAPSO) [
9], DE [
10], and comprehensive learning PSO (CLPSO) [
11] are other works in this area. Other approaches such as the seeker optimization algorithm (SOA) and a distributed Q-learning method were also discussed in [
12,
13], while in [
14] a stochastic problem was solved using quasi-oppositional teaching–learning-based optimization (TLBO) [
15], named QOTLBO.
To solve diversified integer nonlinear issues such as minimizing the
L index and power losses at the same time, chaotic enhanced PSO-based techniques such as MOCIPSO and MOIPSO were presented in [
16]. In Reference [
17], different ways of tackling the reactive power planning (RPP) problem were thoroughly explored. Ref. [
18] examined various significant practical limitations of the optimal power flow (OPF) issue, highlighting three in particular: the valve-point effect, the multi-fuel option, and, most critically, the forbidden operating zone. A soft computing technique based on differential evolution application of a new voltage stability index (NVSI) was developed in [
19] to detect weak buses in the RPP problem. To increase voltage stability, improve the voltage profile, and reduce network losses, Reference [
20] used chemical reaction optimization (CRO) to allocate a static synchronous compensator (STATCOM). The Gaussian bare-bones TLBO (GBTLBO) algorithm was presented in [
21] to address ORPD. Using a nature-inspired design termed the water cycle algorithm (WCA), the ORPD problem was resolved in [
22]. In [
23], to optimize the solution to the ORPD problem, moth-flame optimization (MFO) was successfully applied; this optimizer was inspired by moths’ night-time navigation method, in which they employ visible light sources for guidance. In [
24], to accomplish various goals of ORPD, an improved social spider optimization (ISSO) algorithm was recommended. Semidefinite programming (SDP) has recently received a lot of attention in the power system research community. Under certain technical constraints, a new SDP design was exploited in [
25] to propose a unique equivalent convex optimization formulation for the ORPD problem. Moreover, in [
26], the application of a well-known technique, i.e., grey wolf optimizer (GWO), was deployed to address the ORPD issue.
In addition to the approaches listed above, many other optimizers have been utilized to tackle optimal Volt-VAR control via various systems with single and multiple objectives. These methods include tight conic relaxation [
27], pseudo-gradient search based on PSO (PSO-IPG) [
28], hybrid DE and PSO [
29], a hybrid imperialist competitive algorithm (ICA) and PSO (HPSO-ICA) [
30], artificial bee colony (ABC) with chaotic (CABC) and DE (CABC-DE) [
31], a developed gravitational search algorithm (GSA) with conditional selection strategies (CSS) (IGSA-CSS) [
32], ant colony optimization (ACO) [
33], a modified stochastic fractal search (MSFS) [
34], improved ant lion optimization (IALO) [
35], the whale optimization algorithm (WOA) [
36], adaptive chaotic symbiotic organisms search (A-CSOS) [
37], a hybrid GSA and PSO (HPSO-GSA) [
38], the Gaussian bare-bones water cycle algorithm (NGBWCA) [
39], fractional-order Darwinian PSO (FO-DPSO) [
40], the exchange market algorithm (EMA) [
41], the differential search algorithm (DSA) [
42], ant lion optimization (ALO) [
43] and, new colliding bodies optimization (ICBO) [
44], JA (JAYA algorithm) [
45], the two-archive multi-objective GWO (MOGWA) [
46], ABC with firefly (ABC-FF) [
47], the crow search algorithm (CSA) [
48], a new version of PSO [
49], SARGA [
50], and a new version of DE [
51].
Rao, Savsani, and Vakharia introduced the TLBO algorithm in 2011 [
15], which is based on teaching and learning operations. The optimal VAR control issue, on the other hand, includes the aforementioned features. As a result, there is a critical need for a sustainable global approach to power system optimization. The simulation results demonstrate that these improved
θ-self-adaptive teaching and learning (
θ-SATLBO) algorithms employing alternative distributions converge to more optimal solutions than previously published techniques.
In general, the author attempted to create better optimization algorithms for discovering better optimal solutions than earlier published methods. Almost all demonstrations are based on the quality of the effective solutions and the converging characteristics of the best run out of many runs. In the second section of this article, the standard formulation of the optimal VAR control issue is discussed, whereas in the third section, the arrangement of θ-SATLBO is explained. The next section summarizes the simulation results and compares and analyzes the methodologies utilized to address use cases of optimal VAR control problems. Finally, the concluding paragraph of this paper summarizes the implementation of the recommended algorithms.
The rest of this paper is organized as follows.
Section 2 presents the Volt-VAR control formulation for optimization.
Section 3 presents the new proposed algorithms for the optimal VAR control problem.
Section 4 shows the obtained optimal numerical results of the optimal VAR control problem. Finally,
Section 5 presents the conclusions.
4. Numerical Results of Optimal VAR Control Problem
The suggested procedures based on the optimal VAR control issue were evaluated on two standard power networks to verify their efficiency. TLBO optimizers were built in MATLAB 7.6 on a Pentium IV E5200 PC with 2 GB of RAM, and the simulation was performed. The chosen values of the final iterations () for two power systems, 30- and 57-buses of standard IEEE networks, were 100 and 150 with population sizes of 45 and 60, respectively.
There are discontinuous parameters with a step value of 0.01 p.u. for shunt compensators and transformer taps’ reactive powers, and penalty values in (16) are fixed at 500 [
12]. The following algorithm results represent the best possible solutions over 50 independent trails.
4.1. The First Test Network: IEEE 30-Bus Power Network (System 1)
In this part, simulation outcomes derived from the solution of the optimal VAR control issue using the provided techniques are discussed. The proposed new TLBO optimizers’ performance was evaluated using the IEEE 30-bus standard depicted in
Figure 2. Reference [
7] described the IEEE 30-bus network and its primary working limits and situations.
Table 1 shows the allowed ranges of decision variables. Six generators were situated on buses 1, 2, 5, 8, 11, and 13 in the IEEE 30-bus test system. Additionally, buses 3, 10, and 24 were designated as active compensatory shunt buses [
8].
The network loads were specified as:
Qload = 1.262 p.u., Pload = 2.834 p.u.
The entire primary units and network losses were defined as:
∑QG = 0.980199 p.u., ∑PG = 2.893857 p.u., Qlosss = −0.064327 p.u., Ploss =0.059879 p.u.
The proposed algorithms’ viability was evaluated using various goal functions on this test network, as explained below.
4.1.1. Minimization of Network Active Losses
The goal is to reduce total transmission losses to a minimum.
Table 2 summarizes 50 trials’ best optimal VAR control problem solutions for minimizing actual total transmission power losses using
θ-SAGTLBO. The results indicate that using
θ-SAGTLBO leads to active power losses of 0.0486217 p.u., which is smaller than the amount achieved using other methods. When evaluating convergence characteristics,
Figure 3 demonstrates that
θ-SATLBO optimizers achieve a better set of control parameters more quickly than other TLBO optimizers.
Table 3 compares the specifications of the ideal situations acquired by the suggested algorithm techniques after 50 runs to those obtained by the references. A summary of operation symbols, including the mean execution times, the best (
Best) and poorest (
Worst) real losses, the standard deviation (
Std.), the average real losses (
Mean), and loss saving percentage (
Psave) over 50 independent runs, are shown in the following table.
Table 3 shows that the
θ-SAGTLBO strategy reduces active power loss by 18.81%, the largest reduction in losses compared with other alternatives. According to the outcomes, the
θ-SAGTLBO algorithms outperform other algorithms in terms of resilience.
Additionally,
Figure 4,
Figure 5 and
Figure 6 illustrate the convergence graphs of control variable optimization generated by the
θ-SAGTLBO algorithm in terms of the number of generations required to achieve the best solution.
4.1.2. Improvement of the Voltage Profile
In this function, the goal function for the optimal VAR control issue is the minimization of voltage deviation (
SVD). The optimal control variable settings found using the various methods for case 2 are summarized in
Table 4. Each algorithm’s final solution and CPU time were monitored, and substantial statistical data are provided in
Table 5. As shown in
Table 4, the suggested
θ-SACTLBO and
θ-SAGTLBO algorithms produce an
SVD of 0.1233 p.u. In terms of the features of the solutions, the results clearly show that the presented SAGTLBO algorithms trump the other state-of-the-art methods. The convergence features of the voltage deviation minimization method using the TLBO algorithms are plotted in
Figure 7.
4.1.3. Improvement of the Network Voltage Profile with the Minimization of Active Losses
Instead of optimizing the
SVD and losses separately, the algorithms optimize both together.
Table 6 summarizes the optimal control variables,
SVD, and power losses associated with the methods. As can be seen from the data, the updated algorithms discovered the optimal tradeoff between active power losses and
SVD. The convergence rate of
SVD and loss minimization is presented in
Figure 8 for all TLBO optimizers. The active power losses in this scenario are greater than those in case 1 and less than those in case 2, although
SVD is superior to case 1 and inferior to case 2.
4.2. The Second Test Network: IEEE 57-Bus Power Network (System 2)
This system, as shown in
Figure 9, is presented as a large-scale network for the second step of the optimal VAR control issue to show the usefulness of the proposed algorithms in larger-scale systems. Eighty transmission lines with buses 18, 25, and 53, parallel reactive power generators, and seven generators on buses 1, 2, 3, 6, 8, 9, and 12, as well as fifteen load tap setting transformer branches, make up the test system being investigated. The bus statistics, the line data, and the allowed range of real power generation were obtained from Reference [
12], and the parameter limitations are shown in
Table 7.
The network loads are [
58]:
Qload = 3.364 p.u., Pload = 12.508 p.u.
The entire primary units and network losses obtained are [
58]:
∑QG = 3.4545 p.u., ∑PG = 12.7926 p.u., Qlosss = −1.2427 p.u., Ploss =0.28462 p.u.
4.2.1. Minimization of Network Active Losses
Table 8 presents the statistical information and CPU time of the ideal settings found using various methods. The
θ-SAGTLBO algorithm determined the optimum solution after 50 trial runs. The active power losses produced by the
θ-SAGTLBO algorithm are shown to be 0.2372619 p.u. In this table, we can see that the
θ-SAGTLBO method achieves a 16.64 percent reduction in power loss, which is greater than the other alternatives. The assessment of the resilience of the suggested simulation methodology is based on data from 50 separate runs with diverse initial populations. Obviously,
θ-SAGTLBO shows a more robust and effective performance than other methods. To ensure a close-optimal response in any randomized attempt, the
Std. index across several trials must also be extremely low.
Figure 10 depicts the convergence rates for network losses as a function of iteration number.
4.2.2. Improvement of the Voltage Profile
This experiment assessed the objective function of
SVD reduction for this network.
Table 9 illustrates the statistical information and CPU time for the various algorithms. The
SVD obtained by the
θ-SAGTLBO method is the best result for this case, as depicted in
Table 9. The algorithm convergence rate of voltage deviation minimization is illustrated in
Figure 11.
4.2.3. Improvement of the Network Voltage Profile with the Minimization of Active Losses
Rather than optimizing the
SVD and active power losses separately in this work, both objective functions are optimized simultaneously utilizing the updated methods for this popular standard network.
Table 10 summarizes the optimal control variables,
SVD, and network losses obtained with previous and TLBO optimizers. The presented optimizers identified the optimal tradeoff solutions for active power losses and
SVD. The optimal Volt-VAR control problem reveals that, in scenario 3 for this popular standard network, both
SVD and power losses cannot be further decreased without the other deteriorating. The convergence characteristics for network loss minimization and
SVD minimization are presented in
Figure 12 for all TLBO optimizers.
In short, θ-SATLBO algorithms, as novel efficient optimization algorithms, confirmed their superior efficiency and reliability in finding the optimal solutions to several optimal Volt-VAR control issues over other well-known search approaches. Therefore, we can conclude that θ-SATLBO algorithms are suitable and powerful optimizers for optimizing real-world contemporary issues. Hence, those interested in other fields can effectively use this method in their field of work.