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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.

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

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].

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 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.

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 $γ_i$.

System Constraints
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)

The security constraints are expressed as follows:

(7)

(8)

where S_lj illustrates the power flows in the transmission line j, $V_{Lj}$ represents the voltage of a load bus j, and $N_{line}$ and NL indicate the number of system lines and load buses, respectively.
The transformer constraints are expressed as follows:

(9)

where $TP_i$ denotes the transformer tap settings and NT is their total number.
The shunt VAR compensator constraints are expressed as follows:

(10)

where $QC_i$ indicates the reactive output of the shunt compensators and NC is their total number.

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)

where, $Hb_j$ refers to the location of the j-th food source which denotes a solution vector containing the generators voltage, generators power, reactive power from the compensators, and tap setting of the transformers; Up and Lo are the bounds in their lower and upper values for the control variables of the OPF problem with dimension (dim); R is a randomized vector within the range [0, 1].
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)

where, $VT_{(j,k)}$ denotes the time intervals during which a hummingbird (j) did not visit the food source (k) where null denotes no value. Three flying skills were efficiently used and represented in the AHO during foraging: axial, diagonal, and omnidirectional movements [43]. They are expressed in the following Equations (13)–(15):

(13)

(14)

(15)

where, randperm and rand_i represent, respectively, the functions of integer permutations and randomized number creation for integers; $r_1$ is a random value following uniform distribution inside [0, 1]; rand is a random value which is arbitrary generated within the range [0,1] following the uniform distribution, and thus no fine tuning of this parameter was required.
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)

where $Hb_j$ (t) and $Hb_{(j,t arg⁡e t)}$ (t)) are the locations related to the existing and desired sources of food at time t; and N(0,1) indicates the Gaussian distribution function. The territorial foraging technique of the hummingbirds, in the second process, consists in seeking a newly developed supply of food inside the surrounding area within its neighborhood territory as follows:

(17)

where b indicates the territorial parameter.
The update process of the position of every source of food j is expressed in the following form for both strategies:

(18)

where TF() represents the objective value specified in Equation (1). According to this concept, if the nectar-refilling rate of the obtained source of food is greater than that of the existing one, the hummingbird determines to forsake the current one and feed at the resultant projected source of food.
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)

where $Hb_{worst}$ denotes the source with the weakest amount of nectar replenishment among the group. A method of verification must be created to ensure that each hummingbird is constantly traveling within the specified search area, in which any violating control variable would be returned to the search area border as:

(20)

Based on this equation, the basis for delimiting the range value is guaranteed when any of the control parameters fall outside their permissible range.
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)

Proposed EAHO for Solving the OPF Problem
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)

where, k is a random integer number and rand_i is a uniformly distributed pseudorandom integer function within the swarm size $h_n$.
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)

Based on this modification, the territorial foraging technique was improved by sharing the diverse information obtained from other hummingbirds instead of depending on the individual experience of each bird. Consequently, the hummingbirds’ skill in seeking out a potentially produced supply of food inside its neighborhood region could be improved. Thus, the exploration behavior could also be supported.
Additionally, a linear control mechanism was introduced using an adaptive parameter (ψ) as follows:

(27)

where, t is the current iteration time and MT is the maximum number of iterations.
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)

where, $ΔV_L$, $ΔQ_G$, and $ΔS_l$ are represented as:

(29)

(30)

(31)

whereas $TF_k$ indicates each objective function; and $ϕ_1$, $ϕ_2$, and $ϕ_3$ are the penalty coefficients for any violation in reactive power output from the generators, load voltage, and line flow. Also, when attempting to address the stated OPF issue, the equality and inequality requirements should be taken into account. In this study the Newton-Raphson method (NRM) was used to meet the equality criteria that define power flow balancing models. Consequently, the NRM was applied using MATPOWER, which constitutes a critical foundation for proving three-phase systems [44–47]. Fig. 1 denotes the key stages of the EAHO in solving the OPF problem.
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.

Fig. 1. Main steps of the proposed EAHO for solving the OPF problem: (a) problem design model and (b) main steps of the proposed EAHO.

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].

Fig. 2. IEEE 30 bus system.

Table 1. Cost coefficients for the IEEE 30-bus test system

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.

Table 2. Optimal results of the compared algorithms for Case 1

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.

Fig. 3. Convergence curves of the AHO and the proposed EAHO in Case 1.

Table 3. Comparison of minimization of the TCF in Case 1

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.

Fig. 4. Convergence curves of the AHO and the proposed EAHO in Case 2.

Table 4. Comparison of minimization of the ETLs in Case 2

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

Fig. 5. Convergence curves of the AHO and the proposed EAHO for Case 3.

Table 5. Comparison of minimization of the VEEs in Case 3

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.
Table 6. Statistical comparison of minimization of the TCF, ETLs, and VEEs for the AHO and the proposed EAHO (IEEE 30-bus system)

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.

Fig. 6. Analysis with parameter variation of the performance of the AHO and the proposed EAHO for (a) Case 1, (b) Case 2, and (c) Case 3.

Application to the IEEE 57-Bus System
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.

Fig. 7. Convergence curves of the AHO and the EAHO in Case 1.

Table 7. Comparison of minimization of the TCF in Case 1

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.

Fig. 8. Convergence curves of the AHO and the EAHO for Case 2.

Table 8. Comparison of minimization of the ETLs in Case 2

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

Fig. 9. Convergence curves of the AHO and the proposed EAHO in Case 3.

Table 9. Comparison of minimization of the VEEs in Case 3

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.

Fig. 10. Voltage magnitudes developed by the AHO and the proposed EAHO for the three cases of the IEEE 57-bus system.

Table 10. Statistical comparison of minimization of the TCF, ETLs, and VEEs for the AHO and the proposed EAHO (IEEE 57-bus system)

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.

Fig. 11. Analysis with parameter variations of the performance of the AHO and the proposed EAHO for (a) Case 1, (b) Case 2, and (c) Case 3.

Application to the Large-Scale IEEE 118-Bus System
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.

Fig. 12. Convergence curves of the AHO and the EAHO in minimizing the TCF for the IEEE 118-bus system.

Table 11. Comparison of minimization of the TCF for the IEEE 118-bus system

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.

Fig. 13. Voltage magnitudes developed by the AHO and the proposed EAHO in the 118-bus system.

Table 12. Comparison of the minimization of TCF and the monthly and annual savings of the studied systems

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

Fig. 14. Box plot of the AHO and the proposed EAHO in minimizing the TCF of the IEEE 118-bus system.

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.

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.

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.

The authors declare that they have no competing interests.

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

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