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- Shahenda Sarhana
^{1,2,*}, Abdullah Shaheen^{3}, Ragab El-Sehiemy^{4}, and Mona Gafar^{5,6}

Human-centric Computing and Information Sciences volume **13**, Article number: 13 (2023)

Cite this article **1 Accesses**

https://doi.org/10.22967/HCIS.2023.13.013

Abstract

Optimal power flow (OPF) is a restricted optimization problem that requires optimizing control variable settings to reach the best collection of operating, technical, and secure restrictions for electric power systems. This paper proposes an enhanced artificial hummingbird optimizer (EAHO) with a linear control mechanism (LCM) and diverse territorial foraging strategies (TFSs) for handling the OPF problem. In the newly suggested solution, the LCM is integrated to enhance both global and local search capabilities. In addition to that, diverse TFSs are provided to support the exploration phase through different search directions. The ECHO incorporates the features of the AHO, LCM, and TFS in order to minimize the total cost of fuel (TCF), the entire transmission losses (ETLs), and the volume of environmental emissions (VEEs). The proposed method was scrutinized on IEEE, 30-, 57-, and 118-bus test grids. The results of the simulation indicate competition between the proposed schema and the state-of-the-art in terms of convergence rate and quality of the solution. The proposed schema achieves the minimum TCF $799.0878/hr, $41,678.25/hr, and $129,790.25/hr for the IEEE 30-, 57- and 118-bus grids, respectively, and the minimum ETLs 2.8571 and 9.872 and VEEs 0.204 and 1.0389 ton/hr for the IEEE 30- and 57-bus grids, respectively. Furthermore, a test was conducted to authenticate the statistical efficacy of the EAHO-inspired scheme. The EAHO presents a robust and straightforward solution for the OPF problem under diverse goal functions.

Keywords

Optimal Power Flow, Enhanced Artificial Hummingbird Optimizer, Linear Control Mechanism, Territorial Foraging Strategies, Total Cost of Fuel, Entire Transmission Losses, Environmental Emissions, Technical Economic Operation

Introduction

The optimal power flow (OPF) is a non-convex, non-differentiable, nonlinear, multi-modal and restricted optimization problem for which a series of operating, technical, and secure restrictions needs to be satisfied, and for which optimal control variable settings need to be selected, in order to achieve various optimization functions [1]. The goal functions include total cost of fuel (TCF), entire transmission losses (ETLs), volume of environmental emissions (VEEs), and better voltage improvement, while the control variables include generator voltage, generator power, reactive power of the compensators, and tap setting of the transformers [2]. To handle this challenging task, diverse conventional mathematical methodologies have been reported, such as a Newton-based approach [3], semi-definite programming [4], linear programming (LP) [5], fuzzy LP [6], nonlinear programming [7], and a sequential unconstrained minimization method [8]. However, to avoid becoming trapped in a local minimum, most traditional approaches necessitate an initial start close to the solutions. Furthermore, as the amount of control variables rises, the efficiency of the responses will be determined by the initial settings. Due to the obvious shortcomings of these methodologies and the rise of metaheuristic algorithms, power network issues have been resolved by them since they do not require derivative knowledge, can handle large non-linear problems, and are not prone to becoming stuck in a local minima. Furthermore, various metaheuristic optimizing methods have emerged as a result of continuous developments in computational intelligence. The majority of extant metaheuristic approaches are built on metaphors of diverse natural occurrences, animal habits, and musical instruments, such as the tree seed method [1], manta ray foraging optimization (MRFO) [9], the symbiotic organisms search optimizer [10], the multi-verse optimizer [11], the moth swarm optimizer [12], the electromagnetic field algorithm (EFA) [13], the moth-flame algorithm (MFA) [14], the coyote optimizer [15], the neural evolutionary algorithm [16], artificial neural synchronization using nature-inspired whale optimization [17], heuristic optimization of the multi-pulse rectifier [18] and the emergent MFA with water cycle [19]. The optimization applications are diverse, and researchers have tried to make the best use of optimization and machine learning algorithms. Among these applications are the multi-dimension clustering-based method [20], the evolutionary computation model [21], multi-timescale multi-dimension resource allocation [22], the predictive genetic algorithm (GA)-based model [23], the ensemble learning approach [24], the computational approach for chronic wound tissue [25], the generation and restoration of private face images [26], and the de-identification of facial images [27].

Related Work

This section includes a discussion of several enhancements that have been introduced in order to derive the best OPF solution, as in [28], a modified sine-cosine algorithm (MSCA) using the Levy flight strategy and changing agents number depending on the fitness value was provided only to minimize power losses without considering voltage levels. The authors of [29] presented a hybrid modified imperialist competitive algorithm and sequential quadratic programming (HMICA-SQP) and applied it to modified standard systems combining numerous solar and wind energy sources. In [30], a self-adaptive particle swarm optimization (PSO) was hybridized with DE in a fuzzy adaptable arrangement (SPSODE) in order to solve the OPF problem. In [31] and [32], an improved marine predators’ method was presented by integrating the possibilities of random situations occurrence with the aim of reducing fuel costs, power losses, and emission quantity, with the inclusion of the advanced technology of voltage source converters. Although this method considered multi-objectives, it was only tested on a small grid system, i.e., the IEEE 30. In [33], a modified social spider optimizer (MSSO) was used to solve the OPF problem by modifying the moving strategies of male and female spiders while maintaining an adequate female spider frequency. Although MSSO is considered a good solution for the OPF, its performance in minimizing TCF is questionable as one needs to wait for a long period of time to achieve a slightly significant reduction in fuel cost, while its performance in a large grid system such as IEEE 118 was insignificant.

In [34], a grey wolf algorithm (DGWA) was developed and used to solve the OPF problem by upgrading the location of the population in a spiral route around the global optimal and employing a randomized mutation operation to boost population variety. With the increasing number of iterations the exploration and exploitation of the DGWA became unstable and its search stability decreased. Furthermore, some studies have described a modified crow search optimizer (MCSO) worked by arbitrarily transitioning to local searching around the optimal crows’ location as a solution to the OPF problem. Although Shaheen et al. [35] considered AC power systems and the authors of [36] considered hybridized with multi-terminal DC systems, they did not consider constraints’ evaluations. Meanwhile, Meng et al. [37] implemented a crisscross search-based GWA (CS-GWA) to resolve the OPF by altering the hunting procedure in the GWA with a greedy process and including vertical and lateral crossover operations. In [38], an adaptable multiple teams perturbation-guiding Jaya (AMTPG-Jaya) optimizer was used to solve the OPF by investigating the search region with various populations, each led by a distinct exploratory track.

These techniques used evolutionary notions instead of deterministic models, did not use gradient computing, had a remarkable ability to avoid becoming caught in tricky local optima, and had the potential to tackle large-scale non-linear difficult optimization issues [39]. Irrespective of these advantages, several of these new procedures are gradually vanishing due to an absence of users, while others have attracted gained considerable interest. Additionally, such algorithms necessitate adequately adjusted algorithm-specific controlled elements, as the incorrect management of such components constrains the converging characteristic or results in a sub-optimal solution.

In this paper the authors propose the artificial hummingbird optimizer (AHO) [40] to solve the OPF problem. This entirely novel meta-heuristic technique simulates the amazing flying abilities and intelligent foraging methods of actual hummingbirds in the real world. Three different kinds of flying skills used in foraging methods are modelled, i.e., axial, diagonal, and omnidirectional movements. In particular, this study applies directed, territorial and migratory foraging tactics and constructs a visit table to mimic hummingbirds’ cognitive performance for sources of food. It is simple to use and contains a few basic parameters that may be changed. Every hummingbird in the AHO is assigned a specific source of food from which it can be nourished. A hummingbird can remember the place and rate at which nectar is replenished at this specific feeding source, and can also keep track of how long each source of food has gone without being examined. These one-of-a-kind skills equip the AHO with exceptional capability in the quest for the best options.

**Research Novelty**
In this study, an EAHO was developed to enhance the performance of the AHO. First, the directed forage technique was improved by guiding the search paths, at each time, towards several directions around other hummingbirds besides the best solution, rather than using only the targeted location of the best hummingbird. Second, the territorial foraging technique was modified by incorporating diverse forms of territorial foraging by sharing different multi-information obtained from other hummingbirds, instead of depending on the individual experience of each bird. Additionally, a linear control mechanism was introduced to control the hummingbirds’ exploration and exploitation activities. This paper’s primary contributions are as follows:

The linear control mechanism enhances the AHO exploration and exploitation activities in order to solve the OPF problem.

Applications are examined on standard IEEE 30-bus, 57-bus and large-scale 118-bus systems.

Significant improvements are achieved from the technical and economic viewpoints.

Convergence and robustness demonstrate the superior capability of the EAHO in handling the OPF problem compared to the AHO.

EAHO shows more sensitive performance than the AHO due to the different variations of the controlling parameters.

The simulation results disclose the dominance of the suggested EAHO over the others in the literature.

The remainder of this work is divided into five sections: Section 2 introduces the related work; Section 3 formulates the OPF problem; Section 4 denotes the suggested EAHO configuration and OPF solution techniques; Section 5 clarifies and discusses the findings obtained by the suggested EAHO in comparison to the basic AHO and the newly developed approaches; and Section 6 presents a full conclusion to this study.

Problem Formulation

The fundamental goal of the OPF problem is to find the optimal control variables in optimizing a variety of specified objectives in electrical networks that are subject to multiple equality and inequality limitations.

**Problem Objectives**

Minimization of the TCF

In electrical network, total fuel cost refers to the accumulated costs of all committed generators. As a consequence, minimization of the TCF reflects the first target function ($F_1$) in $/hr and may be expressed as shown in (1) [41, 42]:

*(1)*

where $P_{Gi}$ represents the MW power output of every generator i, while the cost coefficients are symbolized by $a_i$, $b_i$, and $c_i$ for generator i.

Minimization of the ETLs

The second target function ($F_2$) is the ETLs through the transmission lines, which can be expressed as (2) [12, 42]:

*(2)*

where $G_{ij}$ refers to the line conductance linking buses i and j, Nb is the number of buses, V indicates the voltage, and θ the phase angle.

Minimization of the VEEs

The third objective function is the VEEs ($F_3$) in ton/hr, which may be formally expressed as:

*(3)*

where the emission coefficients for each generator i are indicated by $ζ_i$, $λ_i$, $α_i$, $β_i$, and $γ_i$.

The optimization model has equality constraints and inequality constraints, which are detailed below.

Equality constraints

The balancing power flows are usually defined as the equality constraints of both the active and reactive powers to fulfill the load demand and the power transmission losses, respectively [12, 42].

Inequality constraints

The generation constraints are expressed as follows:

*(4)*

*(5)*

*(6)*

*(7)*

*(8)*

The transformer constraints are expressed as follows:

*(9)*

The shunt VAR compensator constraints are expressed as follows:

*(10)*

Proposed EAHO for Economic Environmental OPF

**AHO Algorithm**

Each hummingbird in the AHO was assigned a specific source of food with which it could be nourished. A hummingbird could remember the place and rate at which nectar was replenished at this feeding site, and could also keep track of how long every source of food went without being revisited. These one-of-a-kind skills provide the AHO with exceptional capability in the quest for best choices. To begin, a hummingbird swarm with h_n numbers was randomized to h_n food sources, as follows:

*(11)*

To mimic the hummingbirds’ capacity to recall how long every source of food went without being revisited, a visiting table (VT) of sources of food was established as follows:

*(12)*

*(13)*

*(14)*

*(15)*

The AHO simulates the distinct foraging techniques of hummingbirds, i.e., directed, territorial, and migratory forms. The first two techniques are based on a random selection of one of the flight skills stated in Equations (13)–(15). Hummingbirds employ the directed strategy to investigate a specific source of food, culminating in the acquisition of a potential food supply that may be defined as follows:

*(16)*

*(17)*

The update process of the position of every source of food j is expressed in the following form for both strategies:

*(18)*

When there is food scarcity in a hummingbird's territory, it will typically migrate to a more distant food source on which to feed. To carry out this technique, the hummingbird moves to a separate source of food picked randomly from the whole seeking space if it is located at the food source with the lowest nectar-refill rate, as mathematically presented in Equation (11),

*(19)*

*(20)*

The visit table is a critical component of the AHO for storing information about food source visits. A hummingbird may discover its favorite source of food by inspecting it at each time. Equation (12) can be adjusted for every bird, as follows:

*(21)*

*(22)*

*(23)*

For the purposes of this study, an EAHO was developed with many adjustments with the aim of increasing the performance of the AHO.

First, the directed foraging technique was modified as follows:

*(24)*

*(25)*

Based on this modification, the directed foraging technique was improved by guiding the search paths, at each time, towards several directions around other hummingbirds besides the best solution, rather than by using only the targeted location of the best hummingbird. Thus, the exploitation behavior could be supported.

Second, the territorial foraging technique was modified by incorporating diverse forms of territorial foraging, as follows:

*(26)*

Additionally, a linear control mechanism was introduced using an adaptive parameter (ψ) as follows:

*(27)*

This parameter (ψ) provides a value that increases linearly with the increase in the iteration time, thus providing a method of controlling the hummingbirds’ exploration and exploitation activities. In the beginning, all the hummingbirds used the territorial foraging technique represented by Equation (26), i.e., 100% exploration behavior. Continuing to follow that, the exploration behavior via the territorial foraging strategy of Equation (26) decreases as the value of the parameter (ψ) grows, whereas the exploitation behavior via the directed foraging technique of Equation (24) increases.

Thus, the proposed EAHO was employed to address the OPF problem. Each hummingbird travels constantly within the specified search area, as shown by the method of verification provided with Equation (20). Consequently, the operational limitations were guaranteed for all control variables of the active power outputs in Equation (6), the voltages of the generation units in Equation (4), the transformer tap settings in Equation (9) and the shunt VAR compensation in Equation (10). Furthermore, the fitness function was modified with penalty terms to penalize violations of any restriction of the dependent variables, which represent the capability of the generators’ reactive power (Equation 5), the security constraints for the power flows in the transmission lines (Equation 7), and the voltages of a load buses (Equation 8). As a result, the fitness function was updated as follows:

*(28)*

*(29)*

*(30)*

*(31)*

In addition, Appendix (Table A1) contains a pseudo code which explains the code stages in detail. Fig. 1(a) shows the problem design model in which the control parameters are taken from the hummingbirds, and then the voltage settings of the generators, the output power settings of the generators, and the tap settings and reactive power settings of the reactive sources are defined. Then, the MATPOWER load flow is activated to run the load flow. It is implemented via the MATLAB interface, where the OPF is a real-time operational problem in power systems, to control the technological equipment of the generators, transformers and reactive power sources in order to support the decision maker, i.e., the operator of the power system, to find the optimal settings of the mentioned equipment for achieving economical, technical and environmental objectives. Therefore, the potential limitations of the system are determined based on the permissible range of the equipment. For the purpose of evaluation, the time complexity of the proposed EAHO was calculated as O(N2) based on the number of hummingbirds and the number of iterations, which was satisfied as all the state of art in the results section have the same complexity, so this study depended only on TCF, ELTs and VEEs when evaluating the proposed schema against the state-of-the-art.

Results of the Simulation

This section discusses the application of the AHO and proposed EAHO to the standard IEEE 30-bus, IEEE 57-bus, and IEEE 118-bus systems in order to solve the OPF issue. The three power grids are real-world case models, with the IEEE 30-bus network case study being a basic approximation of the American Electric Power network, and the IEEE 57-bus and IEEE 118-bus grids representing simple approximations of the American Electric Power system and Electric Grid Test Cases (https://electricgrids.engr.tamu.edu/electric-grid-test-cases/), respectively (in the Midwest of the United States). To establish the effectiveness and superior performance of the proposed EAHO, the three cases were studied as follows: Case 1 (minimizing the TCF), Case 2 (minimizing the ETLs), and Case 3 (minimizing the VEEs).

The results were compared to those obtained by the AHO. Furthermore, to validate the very efficient capability, different comparisons with numerous recently published findings based on recent algorithms were performed. The simulations were carried out with MATLAB 2017b. For the IEEE 30-bus and IEEE 57-bus systems, the number of hummingbirds in the swarm was set to 50 and 100, respectively, with a maximum number of 300 iterations. The swarm was set to 100 for the large-scale 118-bus system, with a maximum number of 600 iterations. To ensure a fair comparison, the AHO and the proposed EAHO completed thirty simulation runs [37].

**Application to the IEEE 30-Bus System**

IEEE 30-bus consists of 30 buses, six generators, 41 lines, nine reactive power sources, and four on-load tap changer transformers. The data for buses, the upper and lower requirements of reactive output generation, and the thermal limit of the system lines were collected from [48]. Fig. 2 displays the configuration of the IEEE 30-bus system. The limitations of the VAR injections were 5 MVA, while the highest and lowest voltages of the generator were 1.1 and 0.95 in per unit (p.u.), respectively. The cost and emission coefficients are tabulated in Table 1 [35].

Bus | a | b | c | γ | β | α | ξ | λ |

1 | 0 | 2 | 0.00375 | 4.091 | -5.554 | 6.49 | 0.0002 | 2.857 |

2 | 0 | 1.75 | 0.0175 | 2.543 | -6.047 | 5.638 | 0.0005 | 3.333 |

5 | 0 | 1 | 0.0625 | 4.258 | -5.094 | 4.586 | 0.000001 | 8 |

8 | 0 | 3.25 | 0.00834 | 5.326 | -3.55 | 3.38 | 0.002 | 2 |

11 | 0 | 3 | 0.025 | 4.258 | -5.094 | 4.586 | 0.000001 | 8 |

13 | 0 | 3 | 0.025 | 6.131 | -5.555 | 5.151 | 0.00001 | 6.667 |

Results for Case 1

In this case, the AHO and the proposed EAHO were applied for comparison with a number of recent optimization techniques such as the red fox optimizer (RFO) [49], the chimp optimization algorithm [50], golden eagle optimization (GEO) [51], and the dragonfly algorithm (DA) [52]. All these techniques were applied with the same number of 50 search agents and 300 iterations. Table 2 describes the optimal settings of the control parameters. As can be seen, there are 25 control parameters including the 6 voltage settings at the generators, 6 output power settings at the generators, and the 4 tap settings and 9 reactive power settings at the reactive sources. This table shows the higher ability of the proposed EAHO in minimizing the costs compared to the RFO, Chimp, GEO, and DA techniques with the least minimum, mean, maximum and standard deviations of 799.1306, 799.1803, 799.2566, and 0.0331, respectively. Also, Fig. 3 illustrates the convergence curves of the AHO, the proposed EAHO, RFO, Chimp, GEO, and DA for this case. As shown, the proposed EAHO demonstrates higher ability in minimizing the TCF compared to the others, and finally achieves a lower TCF objective. To validate the effectiveness of the proposed EAHO, it was compared with numerous recently published findings based on recent algorithms.

Variable | Variable | Interior point solver | AHO | Proposed EAHO | RFO | Chimp | GEO | DA |

Voltage setting of the generators (p.u) | ||||||||

Gen 1 | 1.1 | 1.0999 | 1.1 | 1.1 | 1.099243 | 1.098115 | 1.099854 | |

Gen 2 | 1.0893 | 1.0872 | 1.0876 | 1.087453 | 1.083498 | 1.087691 | 1.084286 | |

Gen 5 | 1.0647 | 1.0621 | 1.0609 | 1.060733 | 1.056954 | 1.040417 | 1.04816 | |

Gen 8 | 1.0738 | 1.0684 | 1.0685 | 1.068753 | 1.060704 | 1.046861 | 1.06958 | |

Gen 11 | 1.0998 | 1.0956 | 1.1 | 1.099077 | 1.074058 | 1.090666 | 1.047994 | |

Gen 13 | 1.0563 | 1.0994 | 1.1 | 1.088536 | 1.096935 | 1.09408 | 1.097682 | |

Output power of the generators (MW) | ||||||||

Gen 1 | 177.1444 | 177.1108 | 176.8249 | 177.6667 | 165.0934 | 803.627 | 177.5296 | |

Gen 2 | 48.7412 | 48.6063 | 48.5309 | 48.49249 | 55.97805 | 45.44967 | 48.00855 | |

Gen 5 | 21.3135 | 21.3632 | 21.3503 | 21.1841 | 20.44223 | 26.41867 | 21.84013 | |

Gen 8 | 21.2175 | 20.7633 | 21.3423 | 21.7659 | 24.24903 | 15.69192 | 21.06172 | |

Gen 11 | 11.943 | 12.1498 | 11.9545 | 11.0011 | 11.61763 | 10.89792 | 11.77368 | |

Gen 13 | 12.0006 | 12.0469 | 12.001 | 12 | 14.34276 | 16.95416 | 12.02488 | |

Tap setting of the transformers (p.u.) | ||||||||

Tr 6-9 | 1.078 | 1.0455 | 1.059 | 1.07898 | 1.038106 | 0.903541 | 0.952102 | |

Tr 6-10 | 1.069 | 0.9213 | 0.9004 | 0.913266 | 1.022167 | 1.1 | 1.002092 | |

Tr 4-12 | 1.032 | 1.0091 | 0.9937 | 0.993512 | 1.048784 | 1.021719 | 1.044547 | |

Tr 27-28 | 1.068 | 0.9707 | 0.9683 | 0.994722 | 1.014527 | 0.929453 | 1.00301 | |

Output reactive powers of the VAR sources buses (MVAR) | ||||||||

Bus 10 | 19 | 4.2321 | 4.8945 | 3.408829 | 4.198432 | 2.21381 | 1.991994 | |

Bus 12 | 0 | 4.0672 | 4.9912 | 3.932543 | 1.42175 | 0.65337 | 1.598165 | |

Bus 15 | 0 | 4.9936 | 4.9748 | 4.699829 | 1.268782 | 0.883118 | 4.247858 | |

Bus 17 | 0 | 4.7468 | 4.8217 | 3.443162 | 1.945214 | 1.474686 | 3.500174 | |

Bus 20 | 0 | 4.1779 | 4.2532 | 4.153516 | 3.095286 | 3.967784 | 3.023499 | |

Bus 21 | 0 | 4.993 | 4.9985 | 4.880616 | 2.520753 | 1.731144 | 2.661318 | |

Bus 23 | 0 | 2.9761 | 2.8133 | 3.401383 | 1.139229 | 3.946367 | 3.163069 | |

Bus 24 | 4.3 | 4.8674 | 4.9868 | 4.298713 | 0.647715 | 1.613684 | 1.491729 | |

Bus 29 | 0 | 2.786 | 2.5413 | 4.275768 | 3.979779 | 1.591782 | 2.344346 | |

Minimum cost_Pg ($/hr) | 800.25 | 799.1306 | 799.0878 | 799.2685 | 801.8666 | 803.627 | 799.8749 | |

Mean | - | 799.1803 | 799.1195 | 799.5011 | 803.8438 | 812.2798 | 801.6446 | |

Maximum | - | 799.2566 | 799.1825 | 800.0359 | 806.4305 | 819.0216 | 804.9345 | |

STd | - | 0.0331 | 0.026 | 0.176112 | 1.067645 | 3.763448 | 1.068357 |

From Table 2, the proposed EAHO sets the voltage at generators 1, 2, 5, 8, 11, and 13 as 1.1, 1.0876, 1.0609, 1.0685, 1.1, and 1.1 p.u., where their optimal output powers are 176.8249, 48.5309, 21.3503, 21.3423, 11.9545, and 12.001 MW, respectively. Also, it sets the tap settings at transformers 6–9, 6–10, 4–12, and 27–28 as 1.0590, 0.9004, 0.9937, and 0.9683 in p.u., respectively, and the reactive power sources at buses 10, 12, 15, 17, 20, 21, 23, 24, and 29 as 4.8945, 4.9912, 4.9748, 4.8217, 4.2532, 4.9985, 2.8133, 4.9868, and 2.5413 MVAR, respectively. On the other side, the interior point solver [44] in MATPOWER as an analytical technique achieved worse performance with costs of $800.25/hr compared to the AHO and the proposed EAHO. This is attributable to the lower capability of the analytical methods in achieving global optimal and avoiding local optimal.

Table 3 shows the comparisons for Case 1 with a symbiotic organism search (SOS) [10], a moth swarm algorithm (MSA) [12], MSCA [28], SPSODE [30], DGWA [34], MCSO [35], CS-GWA [37], AMTPG-Jaya [38], and JFS [53]. As can be seen, the proposed EAHO derives the minimum TCF compared to the others, indicating its superior performance not just against the AHO but also against the other current algorithms.

Method | TCF ($/hr) |

Proposed EAHO | 799.0878 |

AHO | 799.1306 |

SOS [10] | 801.5733 |

MSA [12] | 800.5099 |

MSCA [28] | 799.31 |

AMTPG-Jaya [38] | 800.1946 |

SPSODE [30] | 799.8066 |

DGWA [34] | 800.433 |

MCSO [35] | 799.3332 |

CS-GWA [37] | 799.9978 |

JFS [53] | 799.1065 |

Results for Case 2

The proposed EAHO demonstrates promising ability in minimizing the ETLs compared to the AHO, as illustrated in Fig. 4. Table 4 shows different comparisons with numerous recent algorithms such as the MSCA [28], SPSODE [30], MSSO [33], CS-GWA [37], and differential search algorithm (DSA) [39]. As shown, the proposed EAHO derives the minimum ETLs of 2.857 MW, while the AHO, MSCA [28], SPSODE [30], MSSO [33], CS-GWA [37], and DSA [39] obtain ETLs of 2.883, 2.93, 4.99, 2.8678, 3.086, and 3.0945 MW, respectively. Therefore, the proposed EAHO performs better than the other recent algorithms.

Results for Case 3

In this case, the proposed EAHO demonstrates promising ability in minimizing the VEEs compared to the AHO, as illustrated in Fig. 5. Table 5 shows comparisons with numerous recent algorithms such as the MRFO [2], MSSO [33], CSO [35]. MCSO [35], NBA [35], Krill herd algorithm (KHA) [54], Stud KHA [54], modified TLA [55] and adaptive real coded biogeography-based optimization (ARBO) [56]. As can be seen, the proposed EAHO derives the minimum VEEs of 0.204688 ton/hr, while the AHO, of MRFO [2], MSSO [33], CSO [35]. MCSO [35], NBA [35], KHA [54], Stud KHA [54], modified TLA [55], and ARBO [56] obtain VEEs of 0.204719, 0.204754, 0.20479, 0.2051355, 0.2048911, 0.2052063, 0.2048, 0.2049, 0.20493, and 0.2048 ton/hour, respectively. Therefore, the proposed EAHO performs better than the other recent algorithms.

Algorithm | ETLs (MW) |

Proposed EAHO | 2.857186 |

AHO | 2.883306 |

MSCA [28] | 2.93 |

SPSODE [30] | 4.9989 |

MSSO [33] | 2.8678 |

CS-GWA [37] | 3.0861 |

DSA [39] | 3.0945 |

Algorithm | VEEs (ton/hr) |

Proposed EAHO | 0.204688 |

AHO | 0.204719 |

MRFO [2] | 0.204754 |

MSSO [33] | 0.20479 |

CSO [35] | 0.2051355 |

MCSO [35] | 0.2048911 |

NBA [35] | 0.2052063 |

Stud KHA [54] | 0.2048 |

modified TLA [55] | 0.20493 |

ARBO [56] | 0.2048 |

Analysis with Statistical Results and Variation of Parameters

This section presents the statistical analysis of the three cases, which were studied by comparing the minimum, mean, maximum, standard deviation (STd) and standard error (STe) of the considered objective. For the AHO and the proposed EAHO, Table 6 tabulates the statistical indices for minimizing the TCF, ETLs and VEEs.

The EAHO attains the least minimum, average, and maximum objective values for the three cases studied.

It acquires the lowest STd values with 0.025953101, 0.025953101, and 3.17E-05 for Cases 1–3, respectively, while the AHO obtains counterparts of 0.033145185, 0.054081264, and 4.09E-05, respectively.

Additionally, it obtains the lowest STe values with 4.738E-03, 8.873E-03, and 5.787E-06 for Cases 1–3, respectively, while the AHO obtains counterparts of 6.0514E-03, 9.873E-03, and 7.467E-06, respectively.

Item | Case 1. TCF ($/hr) minimization | Case 2. ETLs (MW) minimization | Case 3. VEEs (ton/hr) minimization | |||

AHO | EAHO | AHO | EAHO | AHO | EAHO | |

Minimum | 799.1306403 | 799.0878159 | 2.883306319 | 2.857186261 | 0.204715 | 0.204688 |

Mean | 799.1802926 | 799.1194821 | 2.943384868 | 2.93440772 | 0.204777 | 0.204738 |

Maximum | 799.2566426 | 799.182546 | 3.064007198 | 3.046560285 | 0.204917 | 0.204806 |

STd | 0.033145185 | 0.025953101 | 0.054081264 | 0.048603203 | 4.09E-05 | 3.17E-05 |

STe | 6.05E-03 | 4.74E-03 | 9.87E-03 | 8.87E-03 | 7.47E-06 | 5.79E-06 |

Thus, the proposed EAHO shows greater robustness and efficacy than the basic AHO, and the proposed IHOA has stronger stability because it always achieves the lowest intended value. The AHO and the proposed EAHO are simple to use because they contain a few basic parameters that may be changed. Only the number of hummingbirds in the swarm ($h_n$) and the maximum number of iterations (MT) are their parameters. Thus, this study investigated the impacts of their variation on the performance of the AHO and the proposed EAHO by using swarms of 25 and 50 hummingbirds, respectively, with a maximum number of iterations of 200 and 300, respectively. For this purpose, the number of hummingbirds and the maximum number of iterations were set before the application of the AHO and EAHO. Then, they were applied, and the related objective outcomes were recorded and analyzed. Fig. 6 displays the results for the three cases studied in terms of the minimum, mean, maximum, and STd. The EAHO achieved the lowest minimum and mean objective values for the three cases studied with different parameter variations of the hummingbirds and maximum iterations. Thus, the EAHO was observed to perform better than the AHO.

It consists of 57 buses, 7 generators, 80 lines, 3 reactive power sources installed at buses 18, 25 and 53, and 17 on-load tap changer transformers. In this network, the system input are the data for buses, the upper and lower requirements of reactive output generation, and the thermal limit of the system lines were collected [55] as described in the flow chart in Fig. 1. Thus, there are 34 control parameters including the 7 voltage settings at the generators, the 7 output power settings at the generators, and the 17 tap settings and 3 reactive power settings at the reactive sources. The limitations of the VAR injections were 5 MVA, while the highest and lowest voltages of the generator were 1.1 and 0.95 p.u., respectively.

5.2.1 Results for Case 1

For the first case, the AHO and the proposed EAHO were applied to minimize the total costs, for which Fig. 7 shows the related convergence characteristics. The considered objective condition in this case is to minimize the fuel costs. Also, the optimal settings of both the AHO and the proposed EAHO minimize the costs to $41677.3/hr and $41665.74/hr, and there were no violations in the control and dependent variables of this system.

Fig. 7 attests to the better convergence feature of the proposed EAHO, as it shows the proposed EAHO’s superior ability to minimize the TCF compared to the AHO. Moreover, Table 7 shows a comparison for handling this case between the proposed EAHO, AHO, MSA [12], MSCA [28], EFA [13], HMICA-SQP [29], NBA [42], ARBO [56], salp swarm algorithm (SSA) [57], and social network search (SNS) [58]. As can be seen, the proposed EAHO derives the minimum TCF of $41665.74/hr, while the AHO, MSA [12], MSCA [28], EFA [13], HMICA-SQP [29], NBA [42], ARBO [56], SSA [57], and SNS [58] obtain TCFs of $41677.3/hr, $41673.7231/hr, $41715.7101/hr, $41881.66/hr, $41686.82/hr, $41686/hr, $41672.3/hr, $41685.50/hr, and $41901.9977/hr, respectively. As shown, the proposed EAHO derives a lower TCF than all the others, indicating that it performs better than not only the AHO but also the other current algorithms.

Method | TCF ($/hr) |

Proposed EAHO | 41665.74 |

AHO | 41677.3 |

MSA [12] | 41673.7231 |

EFA [13] | 41715.7101 |

HMICA-SQP [29] | 41881.66 |

SSA [57] | 41672.3 |

SNS [58] | 41685.5 |

NBA [42] | 41686.82 |

ARBO [56] | 41686 |

Improved GA [59] | 41796.84 |

Results for Case 2

In this case, the EAHO demonstrated promising ability in minimizing the ETLs compared to the AHO, as illustrated in Fig. 8. For this case, Table 8 shows different comparisons with numerous algorithms such as the SPSODE [30], SSO [33], MSSO [33], CS-GWA [37], Stud KHA [54], SSA [57], SNS [58], GA [59], and improved GA [59]. As can be seen, the proposed EAHO derived the minimum ETLs of 9.713176 MW, while the AHO, SPSODE [30], SSO [33], MSSO [33], CS-GWA [37], Stud KHA [54], SSA [57], SNS [58], GA [59], and improved GA [59] obtained ETLs of 10.13547, 11.7328, 10.6144, 9.98433, 9.7809, 10.6877, 11.32, 10.1952, 11.814, and 10.516 MW, respectively. Therefore, the proposed EAHO performs better than the other recent algorithms.

Results for Case 3

The convergence curves of the AHO and the proposed EAHO are illustrated in Fig. 9. Table 9 shows different comparisons with numerous recent algorithms including the SSO [33], MSSO [33], TLA [59], GA [59], and Improved GA [59]. As shown, the proposed EAHO derived the minimum VEEs of 1.037015 ton/hr, while the AHO, SSO [33], MSSO [33], TLA [55], GA [59], and improved GA [59] obtained VEEs of 1.039293, 1.7024, 1.0393, 1.1724, 1.121, and 1.083 ton/hr, respectively. Therefore, the proposed EAHO performs better than the other recent algorithms in this case too.

Algorithm | ETLs (MW) |

Proposed EAHO | 9.713176 |

AHO | 10.13547 |

SPSODE [30] | 11.7328 |

SSO [33] | 10.6144 |

MSSO [33] | 9.98433 |

CS-GWA [37] | 9.7809 |

Stud KHA [54] | 10.6877 |

SSA [57] | 11.32 |

SNS [58] | 10.1952 |

GA [59] | 11.814 |

Improved GA [59] | 10.516 |

Algorithm | VEEs (ton/hr) |

Proposed EAHO | 1.037015 |

AHO | 1.039293 |

SSO [33] | 1.7024 |

MSSO [33] | 1.0393 |

TLA [55] | 1.1724 |

GA [59] | 1.121 |

Improved GA [59] | 1.083 |

Fig. 10 displays the voltage magnitudes developed by the AHO and the proposed EAHO for the three cases of the 57-bus system compared to the initial case. As shown in Fig. 11, both the AHO and the proposed EAHO keep all voltage magnitudes within acceptable bounds in all three cases of this system. The figure clearly shows that all voltage magnitudes are within normal bounds, with no violations, thus enhancing the powerful performance of the AHO and the proposed EAHO in dealing with one of the most important inequality constraints of the OPF problem.

Item | Case 1. TCF ($/hr) minimization | Case 2. ETLs (MW) minimization | Case 3. VEEs (ton/hr) minimization | |||

AHO | EAHO | AHO | EAHO | AHO | EAHO | |

Minimum | 41683.53115 | 41678.25368 | 10.17807051 | 9.872638166 | 1.042887468 | 1.038952812 |

Mean | 41718.38246 | 41709.28547 | 10.81206553 | 10.57936177 | 1.049798633 | 1.043468666 |

Maximum | 41756.96678 | 41744.92682 | 11.53936925 | 11.10101034 | 1.08249053 | 1.057330228 |

STd | 19.07785184 | 15.96427124 | 0.350907367 | 0.283139817 | 0.007097947 | 0.004299523 |

STe | 3.4829 | 2.9146 | 0.06406 | 0.05169 | 1.30E-03 | 7.85E-04 |

Analysis with Statistical Results and Variation of Parameters

For this system, Table 10 tabulates the statistical indices of the minimum, mean, maximum, STd and STe or minimizing the TCF, ETLs and VEEs using the AHO and the proposed EAHO. As shown, the EAHO obtained the lowest minimum, average and maximum objective values for the three cases studied. It also obtained the lowest STd values at 0.025953101, 0.025953101, and 3.17E-05 for Cases 1–3, respectively, while the AHO obtained counterparts of 0.033145185, 0.054081264, and 4.09E-05, respectively. Additionally, it acquired the lowest STe values with 4.738E-03, 8.873E-03, and 5.787E-06 for Cases 1–3, respectively, while the AHO obtained counterparts of 6.0514E-03, 9.873E-03, and 7.467E-06, respectively. Thus, the proposed EAHO has higher robustness and efficacy than the basic AHO, and the proposed IHOA has stronger stability because it always reaches the lowest intended value. On the other side, the impact of the variations in the parameters on the performance of the AHO and the proposed EAHO were investigated by using a swarm of 50 and 100 hummingbirds respectively, with a maximum number of iterations of 200 and 300, respectively. Fig. 11 displays the results for the three cases studied in terms of the minimum, mean, maximum and STd. As can be seen, the EAHO achieved the lowest minimum, mean, maximum and STd objective values for the three cases studied with different numbers of hummingbirds and maximum iterations. Thus, the EAHO was observed to perform better than the basic AHO even for different parameters variation.

This system comprises 118 buses, 9 on-load tap changing transformers, 186 lines, 14 capacitive sources, and 54 generators. To cope with the application of the AHO and EAHO, the position of each hummingbird took into account 130 control variables, which include the voltage of 54 generators, the real power output of 53 generators (with the exception of the slack unit at bus 69), the reactive output of 14 capacitor sources, and the tap setting of 9 transformers. Regarding the application of the AHO and the EAHO techniques to the IEEE 118-bus network, the results thus obtained for the related convergence curves are shown in Fig. 12. Table 11 shows a comparison with the techniques reported in the literature regarding the same case. It can be clearly seen that the TCF of $129716.5929/hr obtained by the proposed EAHO is lower than that obtained by the SSO [33], MSSO [33], CBO [41], MSA [60], and enhanced MSA [60], which obtained by and of $132080.4118/hr, $129879.4536/hr, $135297.22/hr, $135310.84/hr, and $135262.57/hr, respectively. Thus, the proposed EAHO yielded more economical results than the AHO and other previously described approaches. Otherwise, Fig. 13 displays the voltage magnitudes developed by the AHO and the proposed EAHO for the 118-bus system compared to the initial case. As can be seen, both the AHO and the proposed EAHO improved all the voltage magnitudes, keeping them within the acceptable bounds.

Method | Cost ($/hr) |

Proposed EAHO | 129790.25 |

AHO | 129871.31 |

SSO [33] | 132080.4118 |

MSSO [33] | 129879.4536 |

CBO [41] | 135297.22 |

MSA [60] | 135310.84 |

Enhanced MSA [60] | 135262.57 |

SSO [33] | 132080.4118 |

MSSO [33] | 129879.4536 |

CBO [41] | 135297.22 |

To explain the improvement of the proposed EAHO against the standard AHO, a close accord was executed in order to minimize the TCF and improve its monthly and annual savings for the studied systems, as shown in Table 12 and Fig. 14. As can be seen, the EAHO achieved slight improvements of 0.0053%, 0.0114%, and 0.0623% over the AHO regarding the IEEE 30-, 57- and 118-bus systems, respectively. However, these small values of percentage, it shows great savings over the year, which are significantly increased with the bigger systems. For the IEEE 30-bus system, the annual savings are $369.792, while the savings reached $45603.216 per year for the IEEE 57-bus system. For the large-scale IEEE 118-bus system, the savings reached $700358.4 per year. These savings show the enhanced performance of the proposed EAHO.

Method | AHO | EAHO | %Improve | Savings ($) | |

Monthly | Annual | ||||

IEEE 30-bus system | 799.0878 | 799.1306 | 0.0053 | 30.816 | 369.792 |

IEEE 57-bus system | 41683.53115 | 41678.25368 | 0.0114 | 3800.26 | 45603.216 |

IEEE 118-bus system | 129871.31 | 129790.25 | 0.0623 | 58363.2 | 700358.4 |

Conclusion and Future Work

This paper attempts to address the OPF problem by proposing a linear control mechanism (LCM) and diverse territorial foraging strategies (TFSs). The LCM is integrated into the AHO to enhance the latter’s global and local search capabilities. It introduces diverse TFSs to support the exploration phase through different search directions, and implements a new application of the proposed EAHO to resolve the OPF problem. Three objectives’ functions are considered with the aim of achieving technical and economical operation of power systems. Specifically, the first two objectives are to minimize the total fuel cost and the entire real power losses, respectively, while the third objective is to preserve and maintain the voltage profile at accepted levels. Numerical simulations were carried out on three IEEE Standard test systems to represent the scalability of the EAHO. The salient features of the proposed method involve the following: the EAHO algorithm promises improved effectiveness with considerable savings throughout the year in terms of minimizing fuel expenditures, which are greatly enhanced with larger systems. The yearly savings for the IEEE 30-bus system are $369.792, whereas the savings for the IEEE 57-bus system are $45603.216 per year. The savings for the large-scale IEEE 118-bus system amount to $700358.4 per year. Furthermore, the greater capability of the proposed EAHO makes it a more economical solution compared to other recent, different methods of optimization. Based on the high level of accuracy of the proposed EAHO in the abovementioned experiments, the authors intend to continue with this study in the future by applying the proposed schema to solve different unconstrained and constrained engineering design problems. In addition to that, the proposed algorithm is recommended to test its adequacy and limitations in solving the OPF problem with a high penetration of renewable energy resources in power transmission and distribution grids. Finally, it could also be developed for AC-DC power networks with the integration of advanced voltage source converters.

Author’s Contributions

Conceptualization, AS, MGG. Investigation and methodology, SS, MGG, AS. Project administration, RAE. Resources, MGG. Supervision, AS. Writing of the review and editing, SS, MGG. Software, AS, MGG. Validation, SS, RAE. Funding acquisition, SS.

Funding

This research work was funded by Institutional Fund Projects under Grant No. IFPIP-598-612-1443. The authors gratefully acknowledge technical and financial support provided by Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.

Competing Interests

The authors declare that they have no competing interests.

Appendix

Table A shows the pseudo code of the proposed EAHO and its steps in detail.
**Table A1. **Pseudo-code of the proposed EAHO

Author Biography

**Name: ** Dr. Shahenda Sarhan

**Affiliation: ** Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, KSA
Faculty of Computers and Information Sciences, Mansoura University, Mansoura, Egypt

**Biography: ** SHAHENDA SARHAN received the B.Sc., M.Sc., and Ph.D. degrees in computer sciences from Mansoura University, Egypt in 2003, 2009 and 2012 respectively. Since 2022, she has been a Professor in computer science with the Faculty of Computers and Information, Mansoura University. She is currently an Associate Professor with the Faculty of Computing and Information Technology, King Abdulaziz University. Her research interests include artificial intelligence and computer networks

**Name: ** Dr. Abdullah Shaheen

**Affiliation: ** Department of Electrical Engineering, Faculty of Engineering, Suez University, Suez 43533, Egypt.

**Biography: ** Dr. Abdullah M. Shaheen was born in Tanta, Egypt, in 1985. He received the B.Sc. degree from Suez Canal University, Port-Said, Egypt, in 2007, and the M.Sc. and Ph.D. degrees from Menoufia University, Shebin El-Kom, Egypt, in 2012 and 2016, respectively. He is currently with the Department of Electrical Engineering, Faculty of Engineering, Suez University, El-Suweis, Egypt. His research interests include power system operation, control, and planning, the applications of optimization algorithms in electric power systems, renewable integration, and smart grids.

**Name: ** Prof. Ragab A. El-Sehiemy

**Affiliation: ** Electrical Engineering Department-Faculty of Engineering, Kafrelsheikh University, Egypt

**Biography: ** Ragab A. El-Sehiemy (SMIEEE) gained his B.Sc., M.Sc., and Ph.D. degrees in Electrical Power Systems from Menoufia University, Egypt, in 1996, 2005, and 2008, respectively. He has worked with the Arab Contractor Company as an electrical engineer for ten years. He joined the Faculty of Engineering at Kafrelsheikh University as an Assistant Researcher in 2009. Currently, he is a Full Professor of Electrical Power Systems at Kafrelshiekh University, Egypt. He was a recipient of the Prof. Mahmoud Khalifia Award in Power System Engineering from the Academy of Research and Technology in 2016 and the Computer and Information in Industry Award from the Academy of Research and Technology in 2021. In 2021, Prof. El-Sehiemy was among the world’s top 2% high citations. Prof. El-Sehiemy is also a member of the Electricity and Energy Research Committee at the Academy of Research and Technology, Egypt. Finally, he was selected as a member of the arbitration committees in the scientific committees for the promotion of professors and assistant professors. His research interests include power-systems operation, planning and control, smart grids, renewable energy, AI and its application to power systems.

**Name: ** Dr. Mona Gamal Gaffer

**Affiliation: ** Department of Computer Science, College of Science and Humanities in Al-Sulail, Prince Sattam bin Abdulaziz University, Kharj, Saudi Arabia.
Machine Learning and Information Retrieval Department, Artificial Intelligence, Kafrelsheikh University, Egypt.

**Biography: ** MONA G. GAFAR received the B.Sc., M.Sc., and Ph.D. degrees from the Faculty of Computers and Information, Mansoura University, in 2006, 2010, and 2014, respectively. She is currently an Assistant Professor with the Department of Computer Science, College of Science and Humanities in Al-Sulail, Prince Sattam bin Abdulaziz University, Saudi Arabia. In addition, she is an associate professor with the Department of Machine Learning and Information Retrieval Department, Faculty of Artificial Intelligence, Kafrelsheikh University, Egypt. Her research interests involve artificial intelligence, data mining, software engineering and applications of modern optimization techniques

References

[1]
A. A. El-Fergany and H. M. Hasanien, “Tree-seed algorithm for solving optimal power flow problem in large-scale power systems incorporating validations and comparisons,” Applied Soft Computing, vol. 64, pp. 307-316, 2018.

[2]
E. E. Elattar, A. M. Shaheen, A. M. Elsayed, and R. A. El-Sehiemy, “Optimal power flow with emerged technologies of voltage source converter stations in meshed power systems,” IEEE Access, vol. 8, pp. 166963-166979, 2020.

[3]
A. J. Santos and G. R. M. Da Costa, “Optimal-power-flow solution by Newton's method applied to an augmented Lagrangian function,” IEE Proceedings-Generation, Transmission and Distribution, vol. 142, no. 1, pp. 33-36, 1995.

[4]
X. Bai and H. Wei, “A semidefinite programming method with graph partitioning technique for optimal power flow problems,” International Journal of Electrical Power & Energy Systems, vol. 33, no. 7, pp. 1309-1314, 2011.

[5]
O. Crisan and M. A. Mohtadi, “Efficient identification of binding inequality constraints in the optimal power flow Newton approach,” IEE Proceedings C (Generation, Transmission and Distribution), vol. 139, no. 5, pp. 365-370, 1992.

[6]
R. El-Sehiemy, A. A. Abou El Ela, and A. Shaheen, “A multi-objective fuzzy-based procedure for reactive power-based preventive emergency strategy,” International Journal of Engineering Research in Africa, vol. 13, pp. 91-102, 2014.

[7]
H. W. Dommel and W. F. Tinney, “Optimal power flow solutions,” IEEE Transactions on Power Apparatus and Systems, vol. 87, no. 10, pp. 1866-1876, 1968.

[8]
M. Rahli and P. Pirotte, “Optimal load flow using sequential unconstrained minimization technique (SUMT) method under power transmission losses minimization,” Electric Power Systems Research, vol. 52, no. 1, pp. 61-64, 1999.

[9]
A. M. Shaheen, R. A. El‐Sehiemy, A. M. Elsayed, and E. E. Elattar, “Multi‐objective manta ray foraging algorithm for efficient operation of hybrid AC/DC power grids with emission minimization,” IET Generation, Transmission & Distribution, vol. 15, no. 8, pp. 1314-1336, 2021.

[10]
S. Duman, “Symbiotic organisms search algorithm for optimal power flow problem based on valve-point effect and prohibited zones,” Neural Computing and Applications, vol. 28, pp. 3571-3585, 2017.

[11]
A. M. Shaheen and R. A. El-Sehiemy, “Application of multi-verse optimizer for transmission network expansion planning in power systems,” in Proceedings of 2019 International Conference on Innovative Trends in Computer Engineering (ITCE), Aswan, Egypt, 2019, pp. 371-376.

[12]
A. A. A. Mohamed, Y. S. Mohamed, A. A. El-Gaafary, and A. M. Hemeida, “Optimal power flow using moth swarm algorithm,” Electric Power Systems Research, vol. 142, pp. 190-206, 2017.

[13]
H. Bouchekara, “Solution of the optimal power flow problem considering security constraints using an improved chaotic electromagnetic field optimization algorithm,” Neural Computing and Applications, vol. 32, no. 7, pp. 2683-2703, 2020.

[14]
A. Dabba, A. Tari, and S. Meftali, “Hybridization of Moth flame optimization algorithm and quantum computing for gene selection in microarray data,” Journal of Ambient Intelligence and Humanized Computing, vol. 12, no. 2, pp. 2731-2750, 2021.

[15]
A. A. A. El-Ela, R. A. El-Sehiemy, A. M. Shaheen, and A. R. Ellien, “Optimal allocation of distributed generation units correlated with fault current limiter sites in distribution systems,” IEEE Systems Journal, vol. 15, no. 2, pp. 2148-2155, 2021.

[16]
Y. Shao, J. C. W. Lin, G. Srivastava, D. Guo, H. Zhang, H. Yi, and A. Jolfaei, “Multi-objective neural evolutionary algorithm for combinatorial optimization problems,” IEEE Transactions on Neural Networks and Learning Systems, 2021.
https://doi.org/10.1109/TNNLS.2021.3105937

[17]
A. Sarkar, M. Z. Khan, M. M. Singh, A. Noorwali, C. Chakraborty, and S. K. Pani, “Artificial neural synchronization using nature inspired whale optimization,” IEEE Access, vol. 9, pp. 16435-16447, 2021.

[18]
M. Wozniak, A. Sikora, A. Zielonka, K. Kaur, M. S. Hossain, and M. Shorfuzzaman, “Heuristic optimization of multipulse rectifier for reduced energy consumption,” IEEE Transactions on Industrial Informatics, vol. 18, no. 8, pp. 5515-5526, 2022.

[19]
S. Khalilpourazari and S. Khalilpourazary, “An efficient hybrid algorithm based on water cycle and moth-flame optimization algorithms for solving numerical and constrained engineering optimization problems,” Soft Computing, vol. 23, pp. 1699-1722, 2019.

[20]
A. Liu, W. Miller, M. E. Cholette, G. Ledwich, G. Crompton, and Y. Li, “A multi-dimension clustering-based method for renewable energy investment planning,” Renewable Energy, vol. 172, pp. 651-666, 2021.

[21]
J. C. W. Lin, Y. Djenouri, G. Srivastava, and P. Fourier-Viger, “Efficient evolutionary computation model of closed high-utility itemset mining,” Applied Intelligence, vol. 52, no. 9, pp. 10604-10616, 2022.

[22]
H. Yu, Z. Zhou, Z. Jia, X. Zhao, L. Zhang, and X. Wang, “Multi-timescale multi-dimension resource allocation for NOMA-edge computing-based power IoT with massive connectivity,” IEEE Transactions on Green Communications and Networking, vol. 5, no. 3, pp. 1101-1113, 2021.

[23]
J. C. W. Lin, Y. Djenouri, G. Srivastava, U. Yun, and P. Fournier-Viger, “A predictive GA-based model for closed high-utility itemset mining,” Applied Soft Computing, vol. 108, article no. 107422, 2021. https://doi.org/10.1016/j.asoc.2021.107422

[24]
A. Abbasi, A. R. Javed, C. Chakraborty, J. Nebhen, W. Zehra, and Z. Jalil, “ElStream: an ensemble learning approach for concept drift detection in dynamic social big data stream learning,” IEEE Access, vol. 9, pp. 66408-66419, 2021.

[25]
C. Chakraborty, “Computational approach for chronic wound tissue characterization,” Informatics in Medicine Unlocked, vol. 17, article no. 100162, 2019.
https://doi.org/10.1016/j.imu.2019.100162

[26]
J. Yang, J. Liu, R. Han, and J. Wu, “Generating and restoring private face images for internet of vehicles based on semantic features and adversarial examples,” IEEE Transactions on Intelligent Transportation Systems, vol. 23, no. 9, pp. 16799-16809, 2022.

[27]
J. Yang, W. Zhang, J. Liu, J. Wu, and J. Yang, “Generating de-identification facial images based on the attention models and adversarial examples,” Alexandria Engineering Journal, vol. 61, no. 11, pp. 8417-8429, 2022.

[28]
A. F. Attia, R. A. El Sehiemy, and H. M. Hasanien, “Optimal power flow solution in power systems using a novel sine-cosine algorithm” International Journal of Electrical Power & Energy Systems, vol. 99, pp. 331-343, 2018.

[29]
J. B. Hmida, T. Chambers, and J. Lee, “Solving constrained optimal power flow with renewables using hybrid modified imperialist competitive algorithm and sequential quadratic programming,” Electric Power Systems Research, vol. 177, article no. 105989, 2019. https://doi.org/10.1016/j.epsr.2019.105989

[30]
E. Naderi, M. Pourakbari-Kasmaei, F. V. Cerna, and M. Lehtonen, “A novel hybrid self-adaptive heuristic algorithm to handle single-and multi-objective optimal power flow problems,” International Journal of Electrical Power & Energy Systems, vol. 125, article no. 106492, 2021.
https://doi.org/10.1016/j.ijepes.2020.106492

[31]
A. M. Elsayed, A. M. Shaheen, M. M. Alharthi, S. S. Ghoneim, and R. A. El-Sehiemy, “Adequate operation of hybrid AC/MT-HVDC power systems using an improved multi-objective marine predators optimizer,” IEEE Access, vol. 9, pp. 51065-51087, 2021.

[32]
M. Al Harthi, S. Ghoneim, A. Elsayed, R. El-Sehiemy, A. Shaheen, and A. Ginidi, “A multi-objective marine predator optimizer for optimal techno-economic operation of AC/DC grids,” Studies in Informatics and Control, vol. 30, no. 2, pp. 89-99, 2021.

[33]
T. T. Nguyen, “A high performance social spider optimization algorithm for optimal power flow solution with single objective optimization,” Energy, vol. 171, pp. 218-240, 2019.

[34]
M. Abdo, S. Kamel, M. Ebeed, J. Yu, and F. Jurado, “Solving non-smooth optimal power flow problems using a developed grey wolf optimizer,” Energies, vol. 11, no. 7, article no. 1692, 2018. https://doi.org/10.3390/en11071692

[35]
A. M. Shaheen, R. A. El-Sehiemy, E. E. Elattar, and A. S. Abd-Elrazek, “A modified crow search optimizer for solving non-linear OPF problem with emissions,” IEEE Access, vol. 9, pp. 43107-43120, 2021.

[36]
A. M. Shaheen, A. M. Elsayed, and R. A. El-Sehiemy, “Optimal economic-environmental operation for AC-MTDC grids by improved crow search algorithm,” IEEE Systems Journal, vol. 16, no. 1, pp. 1270-1277, 2022.

[37]
A. Meng, C. Zeng, P. Wang, D. Chen, T. Zhou, X. Zheng, and H. Yin, “A high-performance crisscross search based grey wolf optimizer for solving optimal power flow problem,” Energy, vol. 225, article no. 120211, 2021.
https://doi.org/10.1016/j.energy.2021.120211

[38]
W. Warid, “Optimal power flow using the AMTPG-Jaya algorithm,” Applied Soft Computing, vol. 91, article no. 106252, 2020.
https://doi.org/10.1016/j.asoc.2020.106252

[39]
K. Abaci and V. Yamacli, “Differential search algorithm for solving multi-objective optimal power flow problem,” International Journal of Electrical Power & Energy Systems, vol. 79, pp. 1-10, 2016.

[40]
W. Zhao, L. Wang, and S. Mirjalili, “Artificial hummingbird algorithm: a new bio-inspired optimizer with its engineering applications,” Computer Methods in Applied Mechanics and Engineering, vol. 388, article no. 114194, 2022.
https://doi.org/10.1016/j.cma.2021.114194

[41]
H. R. Bouchekara, A. E. Chaib, M. A. Abido, and R. A. El-Sehiemy, “Optimal power flow using an improved colliding bodies optimization algorithm,” Applied Soft Computing, vol. 42, pp. 119-131, 2016.

[42]
S. Gupta, N. Kumar, and L. Srivastava, “Bat search algorithm for solving multi-objective optimal power flow problem,” in Applications of Computing, Automation and Wireless Systems in Electrical Engineering. Singapore: Springer, 2019, pp. 347-362.

[43]
M. A. Hamida, R. A. El-Sehiemy, A. R. Ginidi, E. Elattar, and A. M. Shaheen, “Parameter identification and state of charge estimation of Li-Ion batteries used in electric vehicles using artificial hummingbird optimizer,” Journal of Energy Storage, vol. 51, article no. 104535, 2022.

[44]
MATPOWER Software [Online]. Available: https://matpower.org.

[45]
A. M. Shaheen, E. E. Elattar, R. A. El-Sehiemy, and A. M. Elsayed, “An improved sunflower optimization algorithm-based Monte Carlo simulation for efficiency improvement of radial distribution systems considering wind power uncertainty,” IEEE Access, vol. 9, pp. 2332-2344, 2021.

[46]
A. Shaheen, A. Elsayed, A. Ginidi, R. El-Sehiemy, and E. Elattar, “Reconfiguration of electrical distribution network-based DG and capacitors allocations using artificial ecosystem optimizer: practical case study,” Alexandria Engineering Journal, vol. 61, no. 8, pp. 6105-6118, 2022.

[47]
A. M. Shaheen, A. M. Elsayed, R. A. El-Sehiemy, S. Kamel, and S. S. Ghoneim, “A modified marine predators optimization algorithm for simultaneous network reconfiguration and distributed generator allocation in distribution systems under different loading conditions,” Engineering Optimization, vol. 54, no. 4, pp. 687-708, 2022.

[48]
Y. Liu, D. Gong, J. Sun, and Y. Jin, “A many-objective evolutionary algorithm using a one-by-one selection strategy,” IEEE Transactions on Cybernetics, vol. 47, no. 9, pp. 2689-2702, 2017.

[49]
D. Polap and M. Wozniak, “Red fox optimization algorithm,” Expert Systems with Applications, vol. 166, article no. 114107, 2021.
https://doi.org/10.1016/j.eswa.2020.114107

[50]
M. Khishe and M. R. Mosavi, “Chimp optimization algorithm,” Expert Systems with Applications, vol. 149, article no. 113338, 2020.
https://doi.org/10.1016/j.eswa.2020.113338

[51]
A. Mohammadi-Balani, M. D. Nayeri, A. Azar, and M. Taghizadeh-Yazdi, “Golden eagle optimizer: a nature-inspired metaheuristic algorithm,” Computers & Industrial Engineering, vol. 152, article no. 107050, 2021.
https://doi.org/10.1016/j.cie.2020.107050

[52]
S. Mirjalili, “Dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems,” Neural Computing and Applications, vol. 27, pp. 1053-1073, 2016.

[53]
A. M. Shaheen, R. A. El-Sehiemy, M. M. Alharthi, S. S. Ghoneim, and A. R. Ginidi, “Multi-objective jellyfish search optimizer for efficient power system operation based on multi-dimensional OPF framework,” Energy, vol. 237, article no. 121478, 2021.
https://doi.org/10.1016/j.energy.2021.121478

[54]
H. Pulluri, R. Naresh, and V. Sharma, “A solution network based on stud krill herd algorithm for optimal power flow problems,” Soft Computing, vol. 22, pp. 159-176, 2018.

[55]
A. Shabanpour-Haghighi, A. R. Seifi, and T. Niknam, “A modified teaching–learning based optimization for multi-objective optimal power flow problem,” Energy Conversion and Management, vol. 77, pp. 597-607, 2014.

[56]
A. R. Kumar and L. Premalatha, “Optimal power flow for a deregulated power system using adaptive real coded biogeography-based optimization,” International Journal of Electrical Power & Energy Systems, vol. 73, pp. 393-399, 2015.

[57]
A. A. El-Fergany and H. M. Hasanien, “Salp swarm optimizer to solve optimal power flow comprising voltage stability analysis,” Neural Computing and Applications, vol. 32, pp. 5267-5283, 2020.

[58]
R. El-Sehiemy, A. Elsayed, A. Shaheen, E. Elattar, and A. Ginidi, “Scheduling of generation stations, OLTC substation transformers and VAR sources for sustainable power system operation using SNS optimizer,” Sustainability, vol. 13, no. 21, article no. 11947, 2021.
https://doi.org/10.3390/su132111947

[59]
J. Zhang, S. Wang, Q. Tang, Y. Zhou, and T. Zeng, “An improved NSGA-III integrating adaptive elimination strategy to solution of many-objective optimal power flow problems,” Energy, vol. 172, pp. 945-957, 2019.

[60]
B. Bentouati, A. Khelifi, A. M. Shaheen, and R. A. El-Sehiemy, “An enhanced moth-swarm algorithm for efficient energy management based multi dimensions OPF problem,” Journal of Ambient Intelligence and Humanized Computing, vol. 12, pp. 9499-9519, 2021.

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Shahenda Sarhana^{1,2,*}, Abdullah Shaheen^{3}, Ragab El-Sehiemy^{4}, and Mona Gafar^{5,6}, Optimal Multi-dimension Operation in Power Systems by an Improved Artificial Hummingbird Optimizer, Article number: 13:13 (2023) Cite this article 1 Accesses

**Received**8 March 2022**Accepted**22 April 2022**Published**30 March 2023

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