Comparative analysis of recent metaheuristic algorithms for maximum power point tracking of solar photovoltaic systems under partial shading conditions

ABSTRACT

198 exception of conventional PSO techniques. This algorithm successfully handles changes that occur in irradiance conditions and is applicable in real-time. Other algorithms, such as adaptive radial movement optimization, grey wolf-assisted P&O, chaotic flower pollination, and hybrid PSO-P&O, are proposed for the PV systems to track the maximum power point at different operating conditions [25]- [28]. Another study presents a control scheme for grid-connected photovoltaic systems based on two different three-phase topologies of multilevel inverters. Based on the bifurcation method, the investigated topologies are divided into two basic groups: Topologies of DC-DC converter and cascade inverter grid-connected PV systems are designed to be sustainable by using particle swarm optimization, and a good solution is to use two multilevel converter topologies for maximizing PV generator energy yield and improving the system's ability to meet this same requirement for power generation delivered to the grid [29]. A comprehensive study was performed in this manuscript to analyze the obstacles in the 12 GMPPT methodologies for heuristic enhancement, notably manipulative and improvisational browsing capabilities. This paper discusses that variable vortex search (VVS)based GMPP tracking has more than 99% tracking accuracy in simulation and hardware tests. Furthermore, it diminishes computational complexity and is simple to implement. The most important existing heuristic optimization techniques are analyzed and discussed. The literary works processes and the new VVS-based GMPP strategy would then allow the researchers to pick the best strategy for their specific needs [30]- [33]. The researchers in the works [34], [35] allow us to understand the new algorithm used for partially shaded conditions based on humpback whale assisted and whale optimization for different partially shaded conditions.
A hybrid shuffled frog leaping algorithm with PSO, hybrid dragonfly and INC algorithm, hybrid P&O with ABC, and hybrid Jaya algorithm are proposed and are used to cover a large range in a shorter time period, resulting in reduced tracking time and power losses when compared to the conventional algorithms. The settling time is high, so the efficiency is low [36]- [39]. Hassan et al. [40] introduce novel models for predicting the power output of PV systems at various scales and in desert locations, incorporating a nonlinear autoregressive neural network to exogenous inputs with evolutionary algorithms for gradient-free training. Five models have been developed for each site, beginning with a free model and ending with the most expensive model. Hassan et al. [41] have proposed a new regression and ensemble-learning model for forecasting the performance of PV energy plants operating in desert regions, taking into account the innovative tools of the photovoltaic system and the character traits of the procedure settings and climatic conditions. Nwokolo et al. [42] developed and validated 294 physical models from six different PV power technologies using machine learning, Gumbel's probabilistic approach and hybridization of the two to aid in the possible determination of PV electric energy generation in the unique geographical and climatic environment of the experiment site.
The most valuable player (MVP) methodology [43], [44] is identified as a new algorithm and highly efficient in removing the drawbacks of traditional algorithms as well as other metaheuristic computational methods. There have been two of them working on components: the first phase is determining the best individual solution, and the next determines the most efficient approach among a group of the best possible remedies. As a result, the global peak is explored more quickly. In order to get this, a search space-limiting strategy is considered as a means of bringing down the interval of solutions. The MVP algorithm is also tested and validated using MPPT optimization problems [44]. The tracking speed, power fluctuations, and settling time are all minimized with this strategy. A new optimization technique known as artificial rabbit optimization is investigated in this work, in which the search space limitation technique from MPPT is combined with three strategies: detour foraging, random concealment, and resource dwindling [45]. The excursion foraging scheme explores all power peaks during partial shading conditions, the random hiding strategy determines the global peak, and the energy reduction framework improves its equilibrium among both the availability of the best possible duty ratio and, thus, tracking the maximum power. The following are the contributions of the paper: i) Development of MVP and artificial rabbit optimization (ARO) algorithms for MPPT applications; ii) Problem formulation to optimize the output power of the PV array; iii) Testing the algorithms using different case studies; and iv) Compare the performance of ARO and MVO with P&O and INC algorithms.
The paper is organized as follows. Section 2 discusses the mathematical modelling of the PV cell/module. In addition, the effects of the bypass diode on the PV array during partial shading conditions are also discussed. Section 3 deals with the operation of the traditional boost DC-DC converter. Section 4 discusses the formulation and mathematical modelling of MVP and ARO algorithms. Section 5 presents the simulation results under three different operating conditions. Section 6 concludes the paper.

MODELLING OF PHOTOVOLTAIC CELL
The equivalent 5-parameter PV cell model circuit is shown in Figure 1. It is always necessary to extract the parameters of the PV cell for proper modelling of the PV systems [46]. The PV model has five parameters: Photocurrent (Iph), diode saturation current (Io), diode ideality factor (a), series ohmic resistance (Rs), and shunt ohmic resistance (Rp). The output current of the PV cell is presented in (1).
Where, V T denotes the thermal voltage due to temperature dependency, k denotes the Boltzmann constant, q denotes the electron charge, Tc denotes the cell temperature, V denotes the PV cell voltage, and I denotes the PV cell current.

Determination of
In standard testing conditions, the current output is as (3).
In (3) allows quantifying ℎ, which cannot be decided differently. The short-circuit current of the PV cell is calculated using (4).
However, this equation is only applicable in the ideal situation. As a result, equality is incorrect. Then, in (4) must be written as (5).
The photocurrent is impacted by both irradiation and temperature, as presented in (6).

Determination of I0
The shunt resistance is generally regarded as large, so the last term of the relationship should be ignored for the following approximation, By utilizing the three most notable points under typical test conditions: the open circuit voltage (I=0, = , ), the current during a short-circuit (V=0, = , ), as well as the voltage ( , ) and current ( , ) at MPP, the following relations are written [46].
The reverse saturation current is stated as (12).
Where, ɛG denotes material band-gap energy in eV, and D signifies the diode distribution coefficient.

Determination of
To make the suggested model more reasonable, are selected in such a way that the quantified maximum power is comparable to the experiment one , at STC. As a result, the following equation can be written [46].
The repetition process begins at = 0, which must rise to progress the MPP is getting closer to the experimental MPP. The corresponding is then measured. There is just a single pairing available ( ). To show that the proposed paradigm is more reasonable, are chosen in such a way that the computed maximal power is equivalent.

Short circuit current ( ) and open circuit voltage ( )
At normal sun irradiation levels, the short-circuit current is similar to the photocurrent ℎ , which is proportionate to the amount of solar energy G in 2 . The short-circuit current ( ) of the PV modules really aren't temperate-sensitive. It has the propensity to rise slightly when the module's temperature rises. This variance can be regarded as minor for modelling PV module performance. The short-circuit current can therefore be easily computed by employing the actual irritation condition.
Where, signifies PV module short-circuit current under the normal solar intensity. The open-circuit voltage at any particular time and place can be represented as (15).
Where, and 0 are the PV open-circuit voltages, denotes average sun illumination, 0 denotes the normal solar radiation, b denotes the PV module coefficient with no dimensions, c denotes the exponent considering all nonlinear effects of temperature, denotes the PV module temperature considering normal sun illumination, and 0 denotes actual temperature.

Impacts of bypass diodes in photovoltaic arrays
By forming a current channel around the defective cell, bypass diodes used in conjunction with either a single or a group of photovoltaic cells prevent current from passing from excellent, well-exposed to light, overheated PV cells and flaming out weak or partially shaded PV cells. Blocking diodes is not the same as power dissipation. Bypass diodes are often linked parallel to a PV cell or panels to shunt current around them while blocking diodes are typically connected series to the PV panels to keep current that passes from returning into existence. Blocking diodes differ from bypass diodes because, while the diode is physically the same in most circumstances, it is fitted separately and has a distinct purpose. Figure 2 shows the impact of the bypass diode on the I-V PV characteristics of two PV modules connected in series.

DC-DC POWER CONVERTER
The boost converter is employed in order to enhance the direct current voltage in compliance with the duty ratio derived based on the output voltage circumstance. The structure of the boost converter includes an inductor, switch, diode and capacitor. The outcome enhanced voltage surpasses the source voltage. The boost converter circuit is provided in Figure 3. The operational modes of the boost converter are as follows:

Mode 1 (MOSFET ON)
The equivalent functional circuit for mode 1 is provided in Figure 4. In this mode, the switching pulse is provided as high for the switch, and the inductor starts getting charged during this period. During Mode 1, the capacitor keeps the output voltage at the desired level. The inductor voltage and output boosted voltage are given as:

Mode 2 (MOSFET OFF)
The identical operational circuit for Mode 2 is provided in Figure 5. In this mode, the switching pulse is provided as low for the switch, and the inductor starts getting discharged during this period. The inductor voltage and output boosted voltage are given as (18). The boost converter design equations are given as follows. The pulse width (D) of the boost converter is provided in (19).
The inductance is designed according to in (20).
The ripple current allowed across the inductor is provided in (21).
The output side capacitance value is calculated using in (22).
The ripple voltage allowed across the output side capacitance is provided in (23).

MAXIMUM POWER POINT TRACKING ALGORITHMS
In this study, 4 different algorithms are discussed, which include two classical algorithms, such as P&O and INC and two modern optimization algorithms, such as MVP and ARO.

Most valuable player algorithm
MVP algorithm is a type of metaheuristic algorithm which explores and exploits the solution, similar to other search-based stochastic algorithms [40]. The performance of the players (solutions) depends on individual skills, which resemble the dimensions of the optimization problem. The competition among players in a single team leads to the franchise player of a team, and competition among teams leads to the most valuable player (optimal solution). The individual player with various skill sets is provided in (24).
Where z is the number of the dimensions and Skll 1 , Skll 2 , Skll 3 … Skll are the skills of the th player. The th team, which consists of several players, is mentioned in (25).
Where, is the number of players for team , and the th team is provided in (26) by combining (24) and (25).
The squad size is based on the number of athletes and is chosen randomly so that the team size might be uneven. The team is formed as provided in the following steps: the teams are classified into mT 1 and mT 2 teams. The mP 1 (27) players are placed in mT 1 (29) and mP 2 (28) players are placed in mT 2 (30).
Where, is the number of participants, and is the total number of teams within a given tournament. The round-off function makes it an integer to the minimum value of the following integers. The tournament is started after the teams are formed.

Individual competition
The single-team players compete among themselves to select a franchise player. The player's skill or duty ratio is updated in (31).
Where, ℎ is the most valuable player on th team, MVP is by far the most valuable player in the entire competition, is the skill of a specific player among team participants, and rand will be termed as a random variable.

Team competition
Here, various teams compete in the tournament, and at any given contest, one team competes with the other; at the end, any team wins, and the fitness values are updated after the match. The normalized fitness equation for a particular team is provided in (32).
The following equation provides the possibility of an outcome for any match.
Prob{Team j Beat Team k } = 1 − (fit N (Team j )) p (fit N (Team j )) p + (fit N (Team j ))p If {Team j Team k } is higher than that of {Team k Team j } it means Team j wins. A reality factor is added to the equation mentioned above as, in real-time, the results vary even at the last moment; hence, a random variable is combined with the probability equation. If the random number is higher than 0.5, wins if not the wins and the players' skills in the competing teams are updated by (34) and (35) based on win or lose. Otherwise: after that, the greediness function is applied, where the fitness values of players before and after the tournament starts are compared, and higher values are adapted. If two players possess the same order of skills, then one player is replaced. The worst players are replaced with the best players' best solutions, and the

Artificial rabbit optimization algorithm
Artificial rabbit optimization (ARO) is a unique biologically-inspired method that has been developed, and this algorithm is designed to give a solution for single-objective global optimization problems [45]. The artificial rabbit optimization algorithm is inspired by the rabbit's survivability tactics and is carefully 205 explored and quantitatively modelled. This algorithm is comprised of three searching modes first is detour foraging tactics, random hiding tactics, and energy shrink tactics come in the last. The detour foraging strategy gives credits to exploration; the random hiding strategy is devoted to utilization; and the power reduction method strengthens the balance between exploitation and exploration. ARO is used to extract maximum power from the photo voltaic module in typical circumstances, and various settings of intermittency provides the searching and hiding techniques of real rabbits and the energy shrink, which causes a transition between the techniques mentioned above.

Detour foraging
The rabbits each have their own grazing area, but it randomly consumes the grass in other areas and tends to perturb around the food source, which provides information about the position of other individual rabbits. This behavior is used in MPPT where the duty ratio denotes the rabbit, and it randomly checks for other duty ratios and hence updates the position of other local power peaks. As mentioned below, the mathematical model is proposed for the ARO algorithm's detour foraging step.
vi ⃗⃗⃗ (t+1)= (t)+ * ( ( ) − (t)) + (0.5 ⋅ (0.05 + 1 )) * 1 = * (37) Where, ( +1) is the position of th duty ratio at the time (t+1), ( ) is the position of th duty ratio at time , is the number panels, d signifies the dimensions, T denotes the highest iteration, ⌈⋅⌉ denotes the ceiling function which provides next possible integer, round denotes nearest possible integer, randperm allows for the random shuffling of numerals spanning l to d, 1 , 2 , and 3 are the random variables with the range of (0,1), L is the change in duty ratio or the gap to perform random, and 1 is normalized standard distribution. This guarantees the capability for a global range of searches of the ARO algorithm.

Random hide
The rabbit usually built various burrows for hiding purposes, and in this algorithm, for each iteration, a position of duty ratio is generated, and the current position is randomly chosen among the generated duty ratios. The th the tunnel of th rabbit gets delivered as: ⃗⃗⃗ , ( ) = ⃗⃗⃗⃗ ( ) + * * ⃗⃗⃗ ( ), = 1, … , 1, … , From (42), the d number of duty ratios is generated nearby to the original position of duty ratio. H is linearly decreased from 1 to / with several perturbations randomly for the total amount of repetitions in complete. The range of the generated duty ratios is reduced around the global peak of power as the iterations proceed gradually. The mathematical equations for the above-mentioned random hiding technique are provided as: ⃗⃗⃗ ( + 1) = ⃗⃗⃗ ( ) + * ( 4 * ⃗⃗⃗ , ( ) − ⃗⃗⃗ ( )), Where, ⃗⃗⃗ and provides the random locations of duty ratios and, 4 and 5 are two randomized factors with readings inside a certain limit of (0,1). As per in (46), the th search duty ratio updates its positioning about the randomly chosen place out of its d original position. The rabbit or duty ratio position is updated once detour hunting and haphazard concealing process are attained.

Energy shrink
In initial iterations, the ARO undergoes the detour foraging phase and searches for the local peaks, and in final iterations, random hiding is performed in which the current duty ratios shift around the global peak and update the positions. The transfer of phases is termed as energy shrinks as a source of energy to simulate the ARO switching process from the preliminary investigation to the defined stage. The power element is provided in (50).
Where, r is indeed the random variable whose range is within (0,1). All the update processes and computations are done until the end condition is satisfied and the best possible solution is retrieved. The pseudocode of ARO is presented in algorithm. The flowchart of ARO is shown in Figure 7.

Incremental conductance (INC) method
The INC algorithm detects the slope of the P-V curve, and the MPP is tracked by searching the peak of the P-V curve. This algorithm uses the instantaneous conductance and the incremental conductance

Perturb and observe algorithm (P&O)
The P&O algorithm detects and extracts the most power from the PV system. In this case, the photovoltaic voltage is only slightly perturbed, and the power P is evaluated. If P is positive, the PV voltage perturbation approaches towards the maximum power point. The outcome is that the perturbation is continued until P reaches the +ve zone. If P is negative, the PV voltage perturbation is pushed closer to MPP, and the Instability's orientation is inverted for MPP to be impacted by power.
Variables like dI and dV are +ve; hence, the increase in solar irradiation can also be recognized by an additional variable, dI. Hence, the duty ratio is modified in such a manner to minimize the operating voltage, in which dI and dV are +ve and avoid the control issue by modifying the switching indicated towards the direction of MPP.  Figure 10 shows the Simulink model of a PV system, where the extensively utilized DC-DC boost converter is developed and used for simulating and comparing all four algorithms for various testing conditions like standard testing, one-step iteration, and rapid testing. We have used PV arrays to change irradiance by shifting 4S configuration instances. The PWM generator generates a pulse and sends it to the boost converter. The proposed algorithms are introduced with the boost converter, and the result of the PV model is seen in scope.

SIMULATION RESULTS AND DISCUSSIONS
The MVP and ARO algorithms are tested on several PV configurations by contrasting the simulation data achieved by the classical algorithms, such as the P&O and INC algorithms. The suggested technique is built and evaluated using a typical boost converter under diverse insolation and partially shaded circumstances. The testing scenarios are shown in Table 1. Each shade pattern lasts 2 seconds, and many other factors determine the computation time. The algorithm parameters for all techniques are kept constant to evaluate the proposed methods' effectiveness.  Figure 10. Simulink model of PV system MATLAB/Simulink version 2018b is used to Simulate the specified photovoltaic system on a laptop with an Intel Core i5 AMD Ryzen 7 5800H with NVIDIA GeForce RTX Graphics 3.20 GHz and RAM of 16 giga bytes. In MATLAB, the solver is Dormand-Prince (ode45), and the scaling factor is configured to vary automatically. ARO's result is evaluated alongside other algorithms such as MVP, P&O, and INC. The number of repetitions and population size are critical metaheuristic algorithm constants. Therefore, these two variables were optimized based on the results of several trials and the information from the research.

Standard testing condition
The initial insolation is preserved at 1000 W/m 2 with a constant temperature of 25 ℃. The comparison is made between MVP, ARO algorithms and classical algorithms like P&O and INC for the standard testing condition results.  Figure 11 shows that the ARO algorithm can track better effectiveness in standard testing conditions than other algorithms. Figure 11 gives the standard testing conditions results of all four algorithms. Figure 11(a) shows the PV output power, Figure 11(b) shows the PV output voltage, Figure 11(c) shows the PV output current, and Figure 11(d) shows the comparison of the duty cycle of all four algorithms.

One-step irradiance condition
The initial solar insolation is preserved at 745 W/m 2 in the first instance and then increased to 850 W/m 2 in the second instance after 2 seconds. Figure 12 shows that the MVP algorithm can find the MPP at 60.93 W with 93.03% efficiency during the first two seconds of the first interval and at 41 Figure 12 gives the one-step irradiance condition results of all four algorithms. Figure 12(a) shows the PV output power, Figure 12

Rapid testing conditions
Solar irradiance changes quite drastically during overcast periods of the season, impacting the effectiveness of PV systems. A study presented characterized a rapid change in solar irradiance as also being considered to boost the effectiveness of the offered techniques. As a result, the recommended algorithms are evaluated under rapid increases in radiation exposure. The simulation lasted for 12 seconds with six intervals of 2 seconds each, with the sun's insolation changing dramatically every 2 seconds. Figure 13 gives the rapid testing condition results of all four algorithms. Figure 13(a) shows the PV output power, Figure 13 Figure 13 shows that the simulation began with 817 W/m 2 during 0-0.2 seconds in the first interval, then changed to 742 W/m 2 during 0.2-0.4 seconds in the second interval, 900 W/m 2 during 0.4-0.6 seconds in the third interval, 520 W/m 2 during 0.6-0.8 seconds for the fourth interval, 930 W/m 2 during 0.8-1 second for the fifth interval, and 1000 W/m 2 during the last instance. The simulation results shown in Figure 13 shows that the MVP method functions very well during an extreme shift in insolation. It is stated that the MVP algorithm can track the MPP at 41.8 W with 56% efficiency from 0 to 2 seconds in the first interval, 60.9 W with 93.64% efficiency in the second interval from 2 to 4 seconds, and 37.6 W with 70.90% efficiency in the third interval from 4 to 6 seconds. 41.8 W with 56% efficiency from 6 to 8 seconds in the fourth interval; 60.9W with 93.64% efficiency in the fifth interval from 8 to 10 seconds; and 37.6 W with 70.90% efficiency in the last interval from 10 to 12 seconds.
It has also been observed that classic P&O and INC algorithms fail to monitor the MPP effectively continuously, and the algorithms slip into the zone under rapid changes in irradiation. According to the results of numerous simulations, variable step size in conventional MPPT techniques and the ARO algorithm cannot increase tracking speed, accuracy, or efficiency during an extreme shift in insolation as much as the MVP algorithm tracks.  Table 2 gives the detailed performance of all four algorithms when algorithms are proposed under standard testing conditions. As tabulated, the MVP algorithm locates MPP at 86.25 W with 98.74% efficiency, whereas the ARO algorithm locates MPP at 86.83 W with an efficiency of 99.37% and classical algorithms such as P&O and INC tracks MPP at 69.02 W and 78.6 W respectively with an efficiency of 79.01% and 89.98%, hence considering these performance results of all four algorithms, we suggest that the ARO algorithm performs better in standard testing conditions. Figure 14 shows a pictorial representation of the comparison.  Table 3 gives the detailed performance of all four algorithms when proposed under one-step irradiance testing conditions. As evaluated in Table 3 38.312 W in the first instance and 50.1 W in the second interval, with an efficiency of 58.58% and 67.47%. Hence, considering the performance results of all four algorithms, we suggest that the MVP algorithm performs better in one-step iteration testing conditions. Figure 15 shows a pictorial representation of the comparison (Case 2).   Table 4 gives the detailed performance of all four algorithms when algorithms are proposed under rapid iteration testing conditions. As tabulated in Table 4, the MVP algorithm locates MPP at an average of 78.23% efficiency under rapid irradiation change. At the same time, the ARO algorithm locates MPP at an average of 69.55% efficiency, and classical algorithms such as P&O and INC track MPP at an average of 43.87% and 47.69% efficiency, respectively. From Table 4, it is evident that the ARO algorithm and classical algorithms fail to track better efficiency, whereas the MVP algorithm is slightly better in terms of efficiency under severe shading conditions. Figure 16 shows a pictorial representation of the comparison (Case 3).  Figure 15. Pictorial representation performance metrics for Case 2 Figure 15. Pictorial representation performance metrics for Case 3

CONCLUSION
This study uses an extensive comparative analysis by considering recently reported metaheuristic algorithms, such as MVP algorithm and ARO for MPPT of PV systems. PV system with boost converter was subjected to recently proposed algorithms like MVP algorithm, and ARO algorithm is compared with classical algorithms like INC and P&O based techniques under various testing circumstances, such as standard testing method, one-step iteration testing method and rapid testing method. The system's efficiency with the four algorithms is compared and analyzed along with tracking speed and efficiency under all irradiance circumstances. The simulated results show that the ARO algorithm, which locates MPP at 86.8 W with 99.86% efficiency, is better among all the other algorithms under no shading conditions. The MVP algorithm tracks slightly better results in slightly shaded and rapidly shaded conditions in terms of efficiency. So, this study suggests that the ARO algorithm for standard testing conditions and based on the results obtained from Case 2 and Case 3, the MVP algorithm is a better option for change in operating conditions. Finally, this study concludes that the MVP algorithm is better in all aspects and can be an alternative tool for MPPT applications.