An improved constant current step-based grey wolf optimization algorithm for photovoltaic systems

Dagal, Idriss; Akín, Burak; Dari, Yaya Dagal

doi:10.3233/JIFS-224535

An improved constant current step-based grey wolf optimization algorithm for photovoltaic systems

Article type: Research Article

Authors: Dagal, Idriss^{a; *} | Akín, Burak^{b; 1} | Dari, Yaya Dagal^{c; 2}

Affiliations: [a] Yildiz Technical University, Electrical Engineering, Istanbul Turkey | [b] Yildiz Technical University, Electrical Engineering, C-318 Davutpaşa Campus, Istanbul, Turkey | [c] Institut National Supérieurdes Sciences et Techniques d’Abéché (INSTA)

Correspondence: [*] Corresponding author. Idriss Dagal, Yildiz Technical University, Electrical Engineering, Istanbul Turkey. E-mail: [email protected]; https://orcid.org/0000-0002-2073-8956.

Note: [1] https://orcid.org/0000-0002-8647-1297.

Note: [2] https://orcid.org/0000-0003-0373-0394.

Keywords: Grey wolf optimization, metaheuristics, photovoltaics, maximum power point

DOI: 10.3233/JIFS-224535

Journal: Journal of Intelligent & Fuzzy Systems, vol. 46, no. 4, pp. 8441-8460, 2024

Published: 18 April 2024

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Abstract

In this paper, an improved constant current step based on the grey wolf optimization (CCS-GWO) algorithm for photovoltaic systems is investigated. The development of grey wolf optimization has been widely spread over photovoltaic applications. This method is one of the metaheuristic swarm optimization algorithm groups inspired by an optimum means of chasing prey by grey wolves. The proposed technique applies constant current steps to the pack of wolves (alpha, beta, and omega) by monitoring the average of the internal current step and external current step in order to target the leader alpha wolf position. Moreover, the proposed technique solves the convergence process issues, low convergence speed, and premature local optima problems of the traditional GWO algorithm. This CCS-GWO algorithm accurately tracks the maximum power point from the photovoltaic systems for load charging in different partial shading conditions (PSCs). A number of standard benchmark functions are presented with low average cost functions and their corresponding standard deviation values. The simulation results revealed that the proposed CCS-GWO approach outperforms the existing GWO and GA algorithms in terms of efficiency (98.55%) and tracking time (0.3 s).

1Introduction

1.1Background

The Grey wolf, salp swarm, artificial bees’ colony, ant colony, and particle swarm are metaheuristic swarm optimization types. They are developed based on the behavior of organic colonies such as birds, fishes, bacteria, locusts, insects, and animals. In order to find an optimal solution for the PV system [1] has developed a multi-grey wolf optimizer algorithm [2] has proposed a novel naturally inspired salp swarm-based grey wolf optimization maximum power point tracking (MPPT) to track rapidly and correctly the global maximum power point (GMPP) under stationary and varying partial shading conditions [3] has established a hybridization of grey wolf optimization-based fuzzy logic controller to catch and solve the GMPP and solve the oscillation problems around GMPP by tuning the generated output power at the GMPP [4] has exploited a new MPPT design utilizing the grey wolf optimization method for PV systems under partial shading conditions [5] proposed an improved grey wolf optimization algorithm with higher accuracy and faster convergence. For [6] in order to provide a better balance between exploration and exploitation has created a modified grey wolf algorithm. The literature [7] proposed a novel grey wolf optimization (NGOW) algorithm to solve non-constrained problems of the optimization. In [8] a fuzzy logic variant of the hierarchical operator is used to improve the performance of the hierarchical GWO algorithm [9] used evolutionary population dynamics (EPD) in the grey wolf optimizer to considerably enhance the performance of the GWO algorithm in terms of local optima evasion, local search, convergence rate, and exploitation. In [10]. The literature review results on GWO revealed that the opportunities for creating more robust and stable variants of GWO still exist and will prevail over the weakness of current variants [11]. However, the traditional GWO method is susceptible to falling into local optimum, impacting the performance of the algorithm, thus an equalized grey wolf optimizer is developed with opposite learning (REGWO) to solve this issue. In [12], a grey wolf optimizer based on the levy flight algorithm and mutation mechanism has been achieved and the results illustrated better effectiveness and accurateness of the calibration of the proposed algorithm compared to other existing techniques [13], an improved hybrid grey wolf optimizer sine cosine algorithm (IHGWOSCA) is proposed with both training and testing highest accuracies of 93.33% and the lowest accuracies of 76.75% and 81.52% respectively.

1.2Related work

The strong exploration technique of the grey wolf optimization (GWO) is used to update the followers’ position to enhance the variance of the population [14, 15] proposed a grey wolf optimizer based on a dimensional learning strategy that experimental results reveal that the DLGWO has a good performance in solving the problem of global optimization. Moreover [16] used a cross-mutation grey wolf algorithm mostly to reduce the complexity of solving the spacecraft attitude maneuver problem and lessen the convergence time [17] introduced an extended adaptive grey wolf optimization (AGWO) algorithm in order to adjust the convergence of the three-point fitness convergence parameters and demonstrated the better performance of AGWO compared to the conventional GWO by decreasing the required number of iterations, therefore, it outperforms the existing GWO in terms of variants. Similarly, [18] developed an improved quasi-opposition learning and dynamic search based on the grey wolf optimization technique in order to solve the unbalance exploration and exploitation problem of the traditional GWO. Isaac et al. [19] designed a GWO algorithm that yields the same specific result achieved by the iterative-based sizing technique, the proposed method is faster in convergence as compared to the iterative approach. On the other hand, [20] discussed a modified bio-inspired approach with GWO that improves the effectiveness of the instruction detection system (IDS) in sensing both normal and abnormal traffic in the network. In the work [21], a grey wolf optimizer with bubble-net predation (GWO-BP) is used to solve the dynamic characteristics problem of the ordinary modeling technique, additionally, the GWO-BP effectively balances the sensing and exploitation capabilities to quickly meet the optimum value and enhances the accuracy. Likely [22], combined GWO and an improved Gaussian diffusion method to enhance the accuracy of the source term estimation (STE) [22]. The feature selection for classification issues based on the wrapper approach was solved by a new grey wolf optimizer algorithm incorporated with a two-phase mutation [23, 24]. This research proposes a renowned nature-inspired flower pollination algorithm (FPA) which is intensely reviewed, modified, and integrated with the arbitrary walk filter to enhance its performance in terms of tracking speed, and efficiency [25]. The efficiency, accuracy and tracking speed of FPA algorithm is optimized. Evaluation of the proposed OFPA (Optimized Flower Pollination Algorithm) and the old FPAs technique is achieved for null shading condition, poor PSC, powerful PSC, and varying weather conditions [26] subsection adaptive hill climbing method (SSAHC), for getting the maximum power point (MPP) of a photovoltaic (PV) solar panel for any temperature and solar radiation level is proposed [27]. Reduce and Fix method is positively presented as enhancement in PAO algorithm to mitigate these issues of tracking speed and oscillations [28]. The proposed optimized hill climbing (OHC) algorithm achieves null steady-state oscillations without cooperating with the strength of the conventional hill climbing algorithm. They applied both algorithms to an off-grid PV system under constant and varying weather conditions, and the results show the superiority of the proposed OHC algorithm over the traditional HC algorithm. Numerous maximum power point tracking methods often stand out and set the partial shading conditions. However, depending merely on the algorithm reveals several advantages and shortcomings with problems like non-speedy tracking to global maximum power (GMP), failure of tracking path, extensive oscillations around GMP.

This paper is outlined as follows: Section 1 presents the modeling and description of the proposed CCS-GWO system; Section 2 discusses mathematical modeling and implementation of the proposed algorithm, while Section 3 includes the discussion and results of the simulation. Finally, Section 4 covers the conclusion including the findings of the study.

1.2.1Research gap and contribution

An improved GWO [52] is proposed to enhance wolf agents’ global and local research ability by using the Gaussian distribution function and nonlinear convergence technique [53] proposes an evolution and elimination method based on GWO in order to perform a good matching between the exploitation and exploration, further hasten the convergence and boost the conventional GWO accuracy. GWO has great potential for solving optimization issues in various disciplines of study [54]. However, It presents some shortcomings such as low convergence speed and local optima stagnation. Therefore, [55] suggests a new GWO with variable weights (VW-GWO) that reduces the possibility of being stuck in local optima, and [56] introduces a new 2DP colonies model embedded with a blackboard in order to empower the simulation of the grey wolf algorithm. In the same perspective, [57] works on the multi-objective grey wolf optimization by using several UCI datasets for attribute reduction. In [58], a fitness based on GWO is used to overcome the manual classification of spam reviews and obtain the optimal cluster heads [59] investigates an inspired GWO approach that expands the traditional GWO by means of a nonlinear adjustment method and a new position-updating equation. In order to improve the global research and local research ability of the existing GWO, [60] implemented a new algorithm with a convergence factor based on S-function change. Multi-objective optimization is used to increase the scheduling performance as compared to the single objective function [61].

The following are the proposed CCS-GWO algorithm contributions:

- The proposed technique can be easily applied in PV solar applications where GMPP is required.
- In order to hunt the prey, the wolf’s mutation (movement steps) is improved, that is to say the exploration of the GWO becomes accurately effective.
- The proposed CCS-GWO method successfully tracks the required power with minor tracking time by using the adjacent or sequencing way of combined internal and external steps.
- The conventional GWO fails to yield during the converging process, and besides its regular problems such as convergence speed and local optimum stagnation, therefore, the proposed CCS-GWO series or adjacent combination of both internal and external step techniques mitigates the current problems mentioned.

1.3System modeling

1.3.1System description

The proposed CCS-GWO technique combines the PV-shaded arrays interfaced with the load through a boost converter. The boost converter is used to examine the PV array’s voltage. The CCS-GWO algorithm is used to check on the global power by controlling the boost converter’s duty cycle by means of the driver circuit Fig. 1.

Fig. 1

Circuit block diagram.

1.3.2System components modeling

1.3.2.1Modeling of photovoltaic solar panel

The equivalent circuit of a PV solar module consists of one of several diodes, parallel and series resistors [37]. The leakage current flow is minimized by the parallel resistor. The series resistor plays the role of measuring losses. PV solar cell presents two models: the single-diode model (Fig. 2) and the double-diode model. The single diode model has five parameters in Equation (1). The double diode model is more accurate as compared to the single diode model; however, it requires additional parameters in order to successfully model the PV solar cell. Due to its simplicity, the single diode model (Fig. 2) is used and presented some parameters such as photocurrent (I_L), output voltage (V), dark saturation current (I₀), electron charge (q), Boltzmann constant (K), ideality factor (A), absolute temperature (T), the series resistor (R_S) and the parallel resistor (R_SH). Equations (1), (2), (3), and (4) are approximative open circuit voltage, short circuit current, and open voltage for the PV solar cell I-V characteristics respectively [62-64].

Fig. 2

Circuit of Solar panel and its equivalent panels series.

(1)

I=IL-IO{exp[q(V+IRS)nkT]-1}-V+IRSRSH

Through, the semiconductor material and solar panel, the resistance R_S are offered. Complexity exists in describing the shunt resistance R_SH. For the p-n junction of non-ideal nature, a short circuit path is given around the junction due to the presence of the impurities located near the cell edges. In an ultimate case, R_S is 0, whereas R_SH is infinite. To improve the products, the manufacturers tried to minimize the resistance effect. Similarly, the ultimate setup is not possible.

The R_SH outcome is not considered. R_SH is infinite for simplification, thus Equation (1) the last term is abandoned.

1.3.2.2Open circuit voltage and short circuit current of PV solar panel

Dual significant current-voltage points are the short-circuit current I_SC, and the open-circuit voltage V_OC.

The power generation is 0 at both points. From (1) V_OC can be approximated when I = 0, the R_SH is discarded shown in Equation (2) the I_SC is at V = 0 and nearly the same as the light generated current I_L shown in Equation (3).

(2)

VOC≈nkTqln(ILIO+1)

(3)

ISC≈IL

By means of the solar circuit, the maximum power is generated at MPP and the current-voltage characteristics which are distinctive at various temperatures are shown in following Fig. 3.

1.3.2.3. Impact of irradiance and temperature

Fig. 3

Solar panel characteristics I-V (a) and P-V (b) curve at constant irradiation and varying temperatures.

The crucial environmental factors such as temperature and irradiance require paramount consideration. PV solar panel P-V and I-V characteristics subjected to these factors present a significant variation in terms of values. In daily climate change, the maximum power point changes, and due to that, some means of MPP tracking are continually obliged to guarantee the maximum requested power drawn from the PV solar panel. Figure 4 shows the PV panel’s V-P and V-I characteristics curves which are exposed to the effects of the irradiance.

Fig. 4

V-I (a) and P-V (b) curve at four irradiance variations and constant temperature.

Figure 4 depicted that by comparing with voltage, the change in current is greater. The dependency of voltage on irradiation is frequently abandoned. When the irradiation rises from the positive value of the current and the voltage, the power is also positive. Thus, more power is generated when more irradiation is available.

On the contrary, the voltage is affected mostly by temperature. The open-circuit voltage is linearly dependent on the temperature, as shown in Equation (4).

(4)

VOC=nkTqln(ISCIO+1)

Based on Equation (4) the temperature effects on V_OC show negative value since the voltage decreases when the temperature increases. The little increase in current doesn’t compensate for the voltage decrease due to the rise in temperature and thus a reduction in power is seen. When the temperature changes, the higher power point, short circuit current, and open-circuit voltage vary.

1.3.2.3. Boost converter

The conventional boost converter circuit diagram is shown in Fig. 5. The role of the boost converter is to step up the input voltage. It is comprised of the main power switch, a semiconductor diode switch, an inductor, and an output capacitor. The boost converter has two modes of operation.

Fig. 5

Boost converter.

In the on-state, the main power switch is closed or switched on, the diode switch is reverse-biased, and the induction is charged from the power supplied by the capacitor. At that state, no current flows to the load, and only the capacitor feeds the load.

In the off state, the main power switch is opened or switched off, the diode switch is forward-biased, and the capacitor gets fully charged by the power stored in the inductor via the diode switch [65, 66].

The input voltage and output voltage relationship

(5)

V0=11-DVin with D thedutycycle

The peak-to-peak inductor ripple current is deduced as follows:

(6)

ΔiL=D(1-D)V0fpL

The input current and output current relationship.

(7)

Iin=11-DI0

The peak-to-peak output ripple voltage is expressed as follows.

(8)

ΔV0=1fpCDI0

2.2.1Grey Wolf Optimizer (GWO)

Grey wolf optimization is predatory behavior of the population-based metaheuristic type algorithm. The GWO comprises four populations working on an orderly leadership hierarchy. In the pack, they collaborate together for three major steps such as hunting, encircling, searching, and attacking the prey. GWO population has four levels α, β, δ, and ω ranking first, second, third, and fourth respectively. In Fig. 6, the grey wolves are located as α, β, δ, and ω from the top layer to the bottom layer respectively in their hierarchical activity manner. The α grey wolf is the utmost chief of the pack, this grey wolf gives the most important decisions such as hunting, lunchtime, bedtime, rest time, wake-up time, and so on. The β grey wolf represents the deputy of the α grey wolf and acts as the adjunct of the alpha grey wolf and after the death of the α grey wolf, the β grey wolf can ensure the same responsibilities done by the alpha wolf. In the third hierarchy, the δ grey wolf is ordered to follow the instructions of the alpha and beta grey wolves. The δ grey wolf leads the omega grey wolf. The ω grey wolf is the last ordered grey wolf, which obeys hierarchically to all the grey wolves. GWO system position updating is depicted in Figs. 6 and 7 respectively.

Fig. 6

Hierarchy of wolves with dominance decreases from the top to down (α, β, δ, and ω).

Fig. 7

Position updating in grey wolf optimization (GWO) system [14].

2.2.2History of grey wolves live

2.2.2.1. Inspiration

They live in packs of four to nine groups and that can vary based on their ancestors. Wolves have body language communication like bees. Barking and howling are used for warning long-distance communication respectively. A wolf used to challenge its travel companion by growling or laying its ears back on its head. In the group, there are male and female hierarchies. The Grey wolves are of four different types namely alpha, beta, delta, and omega. The most dominant over the entire pack is the male alpha grey wolf type. The alpha male and male-only has babies that breed. At the age of three, the young wolf becomes a teenager and joins the pack or leaves for hundreds of miles to find its own land. At birth, their pups can’t see or hear and approximately weigh one pound. They work on a strict socially dominant hierarchy.

- The alpha is mainly chief for making decisions like hunting, sleeping, time to search for place and so for. The (alpha) wolves are not absolutely the strongest members of the pack, but they are the best in terms of controlling the pack.
- The beta wolves are second in the hierarchy of grey wolves; they are subordinate or deputy wolves after alpha wolves and help alpha wolves in decision-making or other events. Beta wolves can replace or lead if one of the alpha wolves passes away, becomes sick, is seriously wounded, or is too old. The respect is tensed up to alpha wolves, but they (beta wolves) give authority to other lower-level wolves as well. They are considered as counselors to the alpha wolves and punishment givers to the others.
- Omega are the lowest ranking wolves, act as scapegoats, they are the weakest wolves, and in a hierarchal manner, they eat after all the wolves that live in the pack, they are sometimes babysitters in the pack.
- The delta wolves are called subordinate because they must submit to alpha and beta wolves but, on other hand, they dominate omega wolves. The grey wolf and the prey are shown in Fig. 8 below.

Fig. 8

Physical images of the grey wolf and its prey.

2.2.2.2. Mathematical modeling and the proposed algorithm In this section, the mathematical modeling of the grey wolves’ hunting strategy, social behavior, chasing, encircling, tracking, and attacking prey are investigated in order to improve the implementation of the grey wolf optimizations.

2.2.2.3. Social hierarchy

In the mathematical modeling of the social behavior of wolves when applying GWO, we considered alpha as the first best solution. Beta and delta are the second and third-best solutions respectively. The omega wolf is the rest and the last candidate to be presented. The hunting event of the GWO is conducted by α, β, and δ whereas, the ω wolves follow these three wolves [67].

2.2.2.4. Encircling or blockading prey

The grey wolves encircle or blockade prey during the hunt. The following Equations are proposed for the mathematical modeling [68-70].

(9)

D→=|C→.X→P(t)-X(t)→|

(10)

X→(t+1)=X→P(t)-A→.D→

Where t indicates the current position, X→P is the position of the prey, and X→ presents the position vector of a grey wolf. A→ and C→ are coefficient vectors calculated as follows [69, 71].

(11)

A→=2a.→r1→-a→

(12)

C→=2.r2→

Where components a→ are linearly decreased from 2 to 0 over the course of iterations and r1→ , r2→ are random vectors in [0,1].

(13)

a→=2-2(itmaxiterations)

2.2.2.5. Hunting

Grey wolves have a high ability to identify the prey’s position and encircle them in their location. The alpha grey is the guidance herd to lead others in their hunting activities. However, beta and delta wolves may also participate in hunting too. In the conceptual space, we do not have any remote idea about the location of the optimum prey. While mathematically mimicking the hunting behavior of the grey wolves, alpha, beta, and delta are deemed to have excellent knowledge about the probable location of the prey. Therefore, the first three best solutions have been found to date so far and kept. The rest of the search candidates, including omega wolves, are compelled to alter their positions corresponding to the position of the best search agent. These equations are proposed in order to achieve the hunting process [67, 72, 73].

(14)

D→α=|C1→.X→α-X→|,Dβ→=|C→2.X→β-X→|,D→δ=|C→3.X→δ-X→|

(15)

X→1=X→α-A1→.(Dα→),X→2=X→β-A2→.(Dβ→),X→3=X→δ-A3→.(Dδ→)

(16)

X→(t+1)=X→1+X→2+X→33

2.2.2.6. Attacking prey (exploitation)

The grey wolves start their mission of attacking when the prey stops moving. In order to mathematically achieve this step, we decrease the value a→ that makes the range of A→ also decreases Equation (11). Due to the range of A→ [- a→ , a→ ]. Therefore, the value of a→ is decreased from 2 to 0 over the iteration chain, and that makes the range of |A→|<1 , and forces the wolves to attack the prey.

2.2.2.7. Searching for prey (exploration)

The searching process of the wolves mostly depends on the movements or the position of alpha, beta, and delta wolves to look for the prey in a diverging manner and, they attack the prey in a converging way. In the divergence case, the mathematical modeling is done by using random values of |A→|>1 or |A→|<-1 to force the search agent to move away (diverge) from the prey. The exploration situation is applied in this case and allows the grey wolf optimization (GWO) algorithm to word widely explore the prey. C is another prominent factor in exploration. It is very helpful especially during the final iterations in case the local minimum stagnation problem occurs. C→ with values range in [0,2] can be considered as an obstruction effect to approach prey in the natural world. This obstacle or obstruction in nature generally happens during the hunting period of wolves and that precludes them from swiftly and accurately approaching their prey.

2.2.2.8. Proposed CCS-GWO Algorithm and its pseudo coding

The proposed CCS-GWO algorithm works based on the search agents’ or candidates’ movement or position steps. As far as the alpha grey wolf is the leading and decision maker herd, the step mutations are applied to alpha, beta, and delta successively in that order.

The descriptive explanation of the proposed CCS-GWO algorithm based on internal and external steps techniques is as follows.

The steps are divided into two steps:

- First step is called the internal step or lower step, and the second step is denoted by the external step or upper step.
- For |A→|<1 , where the wolves are converging toward the prey, the internal step is appropriately applied and the wolves achieve their mission whereas, the external step fails from being applied in this case.
- For |A→|>1 , where the wolves diverge away from the prey, the internal step fails from being applied and the wolves did not achieve their mission, whereas the external step is successfully applied, and the wolves reach their prey.
- Both |A→|<1 , and |A→|>1 cases: This case is called the hesitating or doubting step.

During the doubting step, when both are applied simultaneously in adjacent and parallel manners, the wolves fail to reach their target prey and the results are depicted in Fig. 17 (a) and 17 (b) respectively.

When both steps are applied intermittently in an adjacent manner, the wolves successfully reach their target prey, and the results are presented in Fig. 14 (a) and 14 (b).

If the intermittent adjacent manner is applied and the internal step is dominant, therefore, the result is shown in Fig. 15 (a).

If the intermittent adjacent manner is applied and the external step is dominant, therefore, the result is shown in Fig. 13 (a) and 13 (b).

The simultaneous parallel manner with the internal step dominant is illustrated in Fig. 16 (b).

The Parallel manner cannot be applied for both intermittent steps, only the adjacent manner is applicable in this case. Figure 9 gives an illustrative view of the hierarchical strategies of the proposed technique.

Fig. 9

Pictorial view of grey wolves’ hierarchy strategies.

The modified mathematical expressions of the new updated hunting process are expressed as follows.

(17)

WolfPosition(is,es)=(X1+X2+X33)is+(X1+X2+X33)es=(X1+X2+X33)(is+es)

(18)

X1=Xα-A1(Dαis+Dαes)

(19)

X2=Xβ-A1(Dβis+Dβes)

(20)

X3=Xδ-A1(Dδis+Dδes)

(21)

Dαi+Dαe=|C1.X1-WolfPosition(is,es)|

(22)

Dβis+Dβes=|C2.X2-WolfPosition(is,es)|

(23)

Dδis+Dδes=|C3.X3-WolfPosition(is,es)|

With is and es denoted by internal step and external step respectively.

The steps progress Fig. 10 of the proposed algorithm is described as follows.

Fig. 10

The proposed CCS-GWO algorithm MATLAB implementation.

Fig. 11

The Proposed CCS-GWO Algorithm and Pseudocode.

Step 1: Initialize the population positions of n grey wolves X (is, es)=1. . . . ., n.

Step 2: Initialize the parameters a, A, and C

Step 3: Compute the fitness of each grey wolf in internal and external step position.

Step 4: Assign the best grey wolves to Xalpha, Xbeta, and Xdelta respectively.

Step 5: Define t = 1

Step 6: While (t < iterations) for each grey wolf

Step 7: Update the position of the current grey wolf using Equation (17)

Step 8: End for

Step 9: Update a, A, and C

Step 10: While (1< |A|<1)

Step 11: If A < 1, the best agent converges toward the prey (internal step)

Step 12: If A > 1, the best agent diverges from the prey (external step)

Step 13: Check whether all the agents are evaluated.

Step 14: Verify whether the maximum iteration is reached.

Step 15: Update the first three grey wolves Xalpha, Xbeta, and Xdelta

Step 16: Apply internal step and external step mutation.

Step 17: Define i = i+1

Step 18: End while

Step 19: Return the best grey wolf Xalpha.

In order to understand some amalgams on the best fitness value of an algorithm, it is not obvious that they can be detected with a single objective function, this is the reason why we prefer to prove the accuracy and performance of the proposed CCS-GWO method by using different functions such as unimodal bench function, multimodal bench function, and fixed dimension multimodal bench function (Table 1). For each bench function, the number of search agents and the maximum number of iterations are 30 and 150 respectively. In Table 1, the proposed CCS-GWO algorithm presents the bench functions F5=(0.07265, 0.01993), F11=(0.03694, 0.01213), F19=(0.03653, 0.01331) as the average cost function and the corresponding standard deviation (std) results respectively. These bench functions permit to compute the exploitation capability of the proposed technique.

Table 1

Comparative results of GWO, hybridization and MPPT computing methods

MPPT	Tracking	Tracking	Oscillation	Tracking ability	Convergence	Implementation
Techniques	efficiency	Speed	around MPP	under PSCs	speed	Complexity
[29]	Medium	Fast	Small	High	Medium	Medium
[30]	Medium	Fast	Small	High	Medium	High
[31]	Medium	Fast	Small	High	Medium	High
[32]	Medium	Fast	High	Poor	High	Medium
[33]	High	Fast	Small	High	High	High
[34]	High	Fast	Small	High	Medium	High
[35]	Medium	Fast	Small	High	High	High
[36]	High	Fast	Small	High	Medium	High
[37]	High	Medium	Small	High	High	High
[38]	High	Fast	High	High	Very High	High
[39]	High	Fast	Small	High	High	Medium
[40]	High	Fast	Small	Poor	High	High
[41]	High	Fast	Small	Poor	High	Medium
[42]	Medium	Fast	Small	Poor	Medium	Medium
[43]	Medium	Fast	Small	High	High	Low
[44]	High	Fast	Small	High	High	Medium
[45]	High	Fast	Small	High	Medium	Medium
[46]	Medium	Fast	Small	High	Medium	High
[47]	High	Fast	Small	High	High	High
[48]	Medium	Fast	Small	High	High	Medium
[49]	Medium	Fast	Small	High	High	Medium
[50]	Medium	Fast	Small	High	High	Medium
[51]	High	Fast	Small	High	High	Medium

Table 2

Bench functions and parameters spaces of the proposed CCS-GWO Algorithm

Unimodal bench function
	Multimodal bench function	Fixed dimension multimodal bench function
f5=∑i=1n-1[100(xi+1-xi2)2+(xi-1)2]	f11=1400∑i=1nxi2-∏i=1ncosxii+1	f19=-∑i=14ciexp(-∑j=13aij(xj-pij)2)

Table 3

Boost converter PV solar panel and algorithms parameters setting

Boost converter parameters	Selected PV solar panel parameters
Switching Frequency (Fsw = 50 KHz)	Pmax	Vmax	Voc	Imax	Isc
Inductance (L = 850μH)	60 W	17.1 V	21.1 V	3.5 A	3.8 A
Capacitance (C = 220 μF)
Duty cycle (Dmin = 0.1 &Dmax = 0.9)
Load (R = 10Ω)	Different MPPT Algorithms parameters
CCS-GWO	GWO	GA
Number of populations 200	Number of populations	Number of populations
Number of iterations 5	200	200
Convergence factor 2-0	Number of iterations 5	Number of iterations 5
Internal step [-1,1]	Convergence factor 2-0	Crossover probability 0.8
External step [-1,1]		Mutation probability 0.1

3System description under analysis

Different case studies have been conducted to assess the effects of uniform irradiance, fast varying irradiance, temperature, partial shading condition, load type and frequency. Besides, a comparative study of different algorithms was also presented.

3.1Results and analysis

In order to validate and prove the accuracy of the proposed study, we choose under partial conditions 4S2P for the MSX-60 W panel subjected to two patterns G1 = [1000, 1000, 1000, 1000] and G2 = [900, 850, 700, 300] at 25°C [74] Fig. 12 (a) and 12 (b) considered as the inputs of this study with global maximum power 345 W and current 7 A respectively.

Fig. 12

PV module P-V curve (a) and I-V curve (b) under partial shading conditions.

Based on the above Fig. 12 (b) input applied to the proposed CCS-GWO technique, we observe the followings results depicted Fig. 13 (a) and 13 (b). In Fig. 13 (a), the result shows that the CCS-GWO accurately tracked the current at a low tracking time of 0.5 s, moreover, the internal step and external step curves also follow-up with the CCS-GWO curve. In Fig. 12 (b), the result indicates that the genetic algorithm based constant current step (GA-CCS) tracks the current at a tracking time of 16,05 s and both internal and external steps fail to follow up with CCS-GA from 0 to 16,05 s.

Fig. 13

CCS-GWO, internal and external step curves (a) and CCS-GA, internal and external step curves (b) under partial shading conditions.

In this study, we presents two results:

For Fig. 15 (a), the Standard Test Conditions (1000 W/m², 25^∘C) is applied and the CCS-GWO, internal and external step power curves are shown. The adjacent internal and external steps method is used and CCS-GWO smoothly tracks the global maximum power of 345 W at a tracking time of 5 s.

For Fig. 15 (b), the CCS-GWO, internal and external step power curves are depicted and subjected to two patterns G1 = [1000, 1000, 1000, 1000] and G2 = [900, 850, 700, 300] at 25°C Fig. 12 (a) under partial conditions 4S2P. The result reveals that by using the adjacent internal and external steps method, the CCS-GWO curve starts from 01 s (due to step mutation) and tracks the global maximum power of 345 W at a tracking time of 5 s.

3.2Proposed CCS-GWO algorithm accuracy and efficiency

We analyze the efficiency and accuracy of CCS-GWO, GWO and GA under different partial shading conditions by altering the irradiance and temperature. The efficiency values as seen in Table 4 were calculated by means of Equation (24) which is defined as the ratio of the tracked output power to the existing maximum power of the system at the PV input. A comparation of the proposed algorithm with other existing techniques based on accuracy, efficiency, complexity, oscillation, and limitation is presented in Table 5.

Table 4

Efficient and powers of different algorithms under PSCs and STC

Solar irradiance [W/m²]	Temperature [°C]	Weather		Algorithms
		conditions		CCS-GWO		GWO		GA
		STC	PSC	P_GMPP	Eff	P_GMPP	Eff	P_GMPP	Eff
1000 900	25	✓		349	99.83	343	98.81	341	97.85
1000 850
1000 700			✓	340	99.77	336	98.74	326	97.73
1000 300
1000 900	45	✓		335	99.69	330	98.63	309	97.47
1000 850
1000 700			✓	325	99.51	321	98.44	302	97.33
1000 300

Table 5

Comparison with other algorithms

Type	CSS-GWO	GWO	GA
Tracking speed	Fast	Medium	Slow
Tracking accuracy	Accurate	Accurate	Low
Design complexity	Medium	Medium	Low
Steady state oscillation	Zero	Less	Less
Convergence/limitations	Global peak	Local peak	Local peak
Computational time/speed	1.94S	2.45S	2.89S

(24)

η=PouttrPinPV×100

With Pout_tr tracked output power and Pin_PV maximum available power to the PV input.

Figure 14 (a), (b) and (c) are histogram representation of worst, average and best cases of CCS-GWO, GWO and GA algorithm respectively.

Fig. 14

The CCS-GWO algorithm accuracy (a)GWO algorithm accuracy (b), GA algorithm accuracy (c).

Fig. 15

The CCS-GWO, internal and external step curves (a) under STC and the CCS-GWO, internal and external step curves (b) under partial shading conditions.

The mean accuracy of GA algorithm is 72%, the GWO has the mean accuracy of 78.7% whereas the proposed CCS-GWO outperforms the two other algorithms with a mean accuracy of 93%.

In this part, both Fig. 16 (a) and 16 (b) are simulated under complex partial shading conditions (CPSCs) and the results have proved that the proposed CCS-GWO algorithm demonstrates higher performance and good accuracy as compared to the conventional GWO with tracking times of 15 s and 17 s respectively.

Fig. 16

The CCS-GWO and GWO power curves (a) and the CCS-GWO and GWO current curves (b) under complex partial shading conditions (CPSCs).

Here, Fig. 17 (a) and 17 (b) show GWO same curves in different aspects:

Fig. 17

The CCS-GWO and internal, external step curves (a) at STC and the CCS-GWO, GWO and GWO-PSCs curves (b) under partial shading conditions.

Figure 17 (a) shows that the CCS-GWO is in dominance mode because by applying the adjacent internal and external step method, from the starting until the end of the path, the CCS-GWO curve is above that of internal-external step curves. In contrast, the non-dominance mode has been found in Fig. 15 (a) where the CCS-GWO curve is under that of internal-external steps one.

Figure 17 (b) presents the GWO curves under normal, partial shading and internal-external steps conditions. The CCS-GWO demonstrated a good tracking time as compared to the normal and partial shading conditions.

This part presents comparative curves of GA, GWO and CCS-GWO Fig. 18 (a) and CCS-GWO, internal step and external step curves Fig. 18 (b).

Fig. 18

GA, GWO and CCS-GWO curves (a) and CCS-GWO, internal step and external step curves (b).

Figure 17 (b) shows that the external step is dominant (adjacent mutation step mode) as compared to the internal step because the CCS-GWO curve is mostly close to the external step curve.

Figure 19 shows that both adjacent mutation step mode curve (a) and parallel mutation step mode (b) are dominant because internal-external step curves are above the CCS-GWO.

Fig. 19

Adjacent mutation step mode curve (a) and parallel mutation step mode (b) under partial shading conditions.

Fig. 20

GA, GWO and CCS-GWO powers curves at STC.

Nomenclature

Abbreviations

GWO Gmacr;rey wolf optimization

CCS greaterthan Constant current step

STE greaterthan Source term estimation

PSCs greaterthan Partial shading conditions

MPPT greaterthan Maximum power point tracking

GMPP greaterthan Global maximum power point

NGWO greaterthan Novel grey wolf optimization

AGWO Adaptive grey wolf optimization

R_s, R_sh Solar cell series and shunt resistance

DLGWO Dimensional learning grey wolf optimization

Is, es. Internal step and external step

q The charge of the electron (C)

T Absolute temperature (°K)

k Boltzmann constant (J/K)

fp Switching frequency

L Inductance

C Capacitance

4Conclusion

This paper suggested an improved constant current step based on the grey wolf optimization (GWO) algorithm abbreviated as CCS-GWO to enhance the inadequate exploration, exploitation, and premature optimum convergence drawbacks of the conventional GWO. A new algorithm is proposed in order to facilitate the mutation sequency and increased hunting performance of the best agents (wolves). The process is done in two steps where the first step (internal step) allows the wolves to converge toward the prey whereas the second step (external step) precludes the wolves from converging toward the prey. By combining these two steps simultaneously or /and in sequencing ways, therefore; convergence speed and efficiency are improved. By applying various climate changes inputs to the proposed CCS-GWO methods, we observed that the new technique outperforms the traditional GWO in terms of mutation accuracy, convergence and exploration (hunting) and performs a good efficiency of 98.55% as compared to 97.10%, and 96.85% for GWO and GA respectively. Besides, the results of the proposed CCS-GWO method for the benchmark functions F5, F11 and F19 presents low standard deviation values of 0.01993, 0.01213, and 0.01331 respectively. Finally, we summarize that the proposed CCS-GWO approach works accurately and tracks the internal steps during the converging process, external steps during the diverging process, combined adjacent internal and external steps, and combined parallel internal and external steps in general cases.

Declarations

Ethical approved

Not applicable for this manuscript.

Competing or conflict of interests

The authors declared no potential conflicts of interest with regards to research, authorship, and/or publication of the article.

Authors’ contributions

Idriss Dagal: Design; writing original draft; writing review & editing; funding. Burak Akín: Resources & supervision & Editing. Yaya Dagal Dari: Resources & supervision & Editing.

Data availability and materials

All data generated or analysed during this study are included in this published article.

Acknowledgment

The authors declared that this work does not receive any funding.

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