Experimental validation of a novel hybrid Equilibrium Slime Mould Optimization for solar photovoltaic system

Abstract

Maximizing Power Point Tracking (MPPT) is an essential technique in photovoltaic (PV) systems that guarantees the highest potential conversion of sunlight energy under any irradiance changes. Efficient and reliable MPPT technique is a challenge faced by researchers due to factors such as fluctuations in irradiance and the presence of partial shading. This paper introduced a novel hybrid Equilibrium Slime Mould Optimization (ESMO) MPPT-based algorithm combining the advantages of two recent algorithms, Slime Mould Optimization (SMO) and Equilibrium Optimizer (EO). The ESMO algorithm is compared with highly efficient MPPT-based techniques such as SMO, EO, Particle Swarm Optimization (PSO), Grey Wolf Optimization (GWO), and Whale Optimization Algorithm (WOA), both under a Simulink environment and a real-time experimental laboratory setup using a Dspace1104 controller and PV emulator. The comparison focuses on performance under several irradiance cases, including instant irradiance change, partial shading, complex partial shading, and dynamic partial shading. The key advantage of ESMO is the fact that it has a single tunable parameter, which makes implementation much easier and, at the same time, reduces the computational resources that are required by the control system. Extensive testing proves the superiority of ESMO over all other techniques, the average efficiency of which is 99.98% under all conditions. Additionally, ESMO provides fast average tracking times of 244 ms under simulation experiments and 200 ms for real-time experiments. These results show that ESMO can be very important for future implementation in large-scale solar PV systems.

Nomenclature

α

Ideality factor

${\mathit{\mu }}_{id}$

Inductor current ripple

${\mathit{\mu }}_{vd}$

Load voltage ripple

Cinput

Input capacitance

Coutput

Output capacitance

Dcycle

Duty cycle

fsw

Switching frequency

G

Irradiance

GSc

Standard test condition irradiance

Id

Diode current

$I_{mpp}$

Current at maximum power point

Iph

Photocurrent

Ipv

Photocurrent of a single cell

Isat

Dark saturation current

$I_{sc}$

Short-circuit current

K

Temperature coefficient of photocurrent

k

Boltzmann constant

L

Inductance

Np

Number of cells in parallel

Ns

Number of cells in series

Ncs

Number of cells in a module

$P_{max}$

Maximum power

q

Elementary charge

Rs

Series resistance

$R_{s,eq}$

Equivalent series resistance

$R_{sh,eq}$

Equivalent shunt resistance

Rsh

Shunt resistance

T

Temperature

TSc

Standard test condition temperature

Vinput

Input voltage

$V_{mpp}$

Voltage at maximum power point

$V_{oc}$

Open-circuit voltage

Voutput

Output voltage

VT

Thermal voltage

DCL

Duty Cycle

DCUT

Duty Cycle Update Timer

DPSC

Dynamic Partial Shading Condition

EO

Equilibrium Optimizer

ESMO

Equilibrium Slime Mould Optimization

GMPP

Global Maximum Power Point

I

Total current

P-V curve

Power-Voltage curve

PSC

Partial Shading Condition

PV

Photovoltaic

1. Introduction

Renewable energy sources, particularly solar energy (SE), are regarded as one of the foremost mechanisms humanity is looking to depend on during this advanced environmental pollution and climate change [1], [2]. It is generated by the sun through thermonuclear fusion reactions, and solar panels transform it into electricity for human usage. Photovoltaic (PV) systems play a crucial role in harnessing SE. Still, their efficiency can be impacted by diverse, dynamic environmental elements, such as fast irradiance changes (FIC) and partial shading conditions (PSC) [3], [4]. Such panels are created using advanced semiconductor technologies, which have become integral to electronic manufacturing in recent decades [5]. Solar panels exhibit a unique Power-Voltage curve that showcases the characteristics of the cell [6]. Despite the many benefits of SE, such as the fact that it is a clean, permanent renewable energy source with low maintenance costs, it still has drawbacks in power losses and manufacturing and installation costs [7]. This prompted researchers in renewable energies to dedicate considerable time and resources to enhance the solar cell's efficiency in manufacturing and energy production. One of these is the field of research into determining a cell's maximum possible power, mathematically represented by the maximum power point (MPP) on the PV curve.

Utilizing the power electronic converter, a pivotal component within the PV system, tracking and maintaining the reference voltage and current became necessary, guaranteeing the highest recorded power at the output [8]. A single peak is created in the PV curve in the case of uniform Irradiation. Still, the hotspots phenomenon is created by clouds, leaves, or any shaded object, resulting in current unmatching between modules. A bypass diode compensates for this problem, resulting in several peaks in the PV curve [9]. Researchers have been exploring advanced optimization techniques to overcome these challenges and improve the performance of PV systems. One encouraging approach involves the integration of modern Maximum Power Point Tracking (MPPT) techniques, EO, and Slime Mould Optimization (SMO) under the name Equilibrium Slime Mould Optimization (ESMO). The concept of ESMO aims to leverage the strengths of both EO and SMO to amplify the overall efficiency of PV systems in dynamic environments. By integrating the Equilibrium Pool tool of EO into one of the SMO equations, ESMO facilitates faster convergence to the optimum solution and helps overcome the issue of local optimum trapping [10].

1.1. Literature review

According to the existing literature, MPPT is a crucial aspect of PV systems [11]. Researchers have invested considerable effort into studying and implementing various mathematical techniques to calculate and identify the MPP on the PV curve. Conventional algorithms like Perturb & Observe (P&O), Incremental Conductance (INC), and Hill Climbing (HC) have been proposed as techniques to track the MPP. However, these methods are limited to uniform irradiation conditions with a single peak on the PV curve, and they exhibit oscillation at the MPP [12], [13]. In scenarios involving PSC and the formation of multiple peaks, conventional techniques can become trapped at local points, leading to a substantial loss of system power [14], [15]. Researchers have explored solutions using soft computing and intelligent techniques to address this limitation. Methods such as Neural Networks MPPT algorithm [16] and Fuzzy Logic-based MPPT [17] Have been harnessed to determine the MPP. However, these approaches often require significant computational resources and may necessitate prior knowledge of the PV cell model [18]. Furthermore, a two-stage MPPT controller employing an ANFIS estimator and an MRAC has also been used in the literature for PV systems [19]. This approach effectively addresses dynamic environments and surpasses the conventional methods in convergence speed and tracking efficiency, but it is still not applicable under PSC. To overcome these challenges, metaheuristic algorithms have emerged, which do not rely on prior knowledge of the model and directly engage with the system. These algorithms leverage exploration and exploitation mechanisms to identify and approach the Global Maximum Power Point (GMPP) based on their mathematical behavior [20]. Metaheuristic algorithms can be categorized into evolutionary-based, physical-based, social-based, and swarm-based algorithms [21]. Some of the notable techniques used for MPPT on the PV curve include Short-Distance Running Algorithm (SDRA), Dragonfly Optimization (DFO), Social Ski Driver (SSD), Quantum Grover Search Algorithm, Salp Swarm Algorithm (SSA), Horse Herd Optimization Algorithm (HOA), Bat Algorithm (BA), Marine Predator Algorithm (MPA), Bald Eagle Search Optimization (BES), Harmony Search (HS) [22], SMO [23], and EO [24].

The presented overview in Table 1 summarizes recent techniques in the literature related to MPPT in PV systems. These metaheuristic algorithms hold promise for effectively tracking the MPP on the PV curve, offering potential solutions to the challenges posed by PSC scenarios.

Table 1.

Summarized overview of the recent technique.

Authors	Technique	Summary
Shiau et al. [25]	Fuzzy-Logic-Based	Standard MPPT technique based on fuzzy logic with a variety of inputs where the sum of the arctangent of conductance and the arctangent of INC with the lowest average reaching the time of 0.053 s.
Husain et al. [26]	Spider monkey	Spider Monkey Optimization technique, which uses the social structure principle of fission-fusion of spider monkeys, reduced steady-state, transient error, and settling time to 3 ms, compared to the conventional PSO and P&O methods.
Aljafari et al. [27]	Butterfly optimization	Hybrid Opposition-based reinforcement learning-butterfly optimization algorithm (OBR-BOA) record the highest average efficiencies with three different PV configurations under three different PSC scenarios with 99.8%,99.72%, and 98.50%, beaten the basic BOA, WOA, GWO, and PSO.
Ghazi et al. [28]	Circle Search Algorithm	A Circle Search Algorithm-Based Super Twisting Sliding Mode Control (CSA-STSMC) achieves an improvement between 0.1% and 0.5% of tracking the MPP under non-standard conditions, with the lowest ITAE of 0.89 of the regulation closed-loop compared to the other used methods CSA-PI, GWO-STSMC and InC-PI.
Xia et al. [29]	Ant-Fuzzy Optimization	Hardware dSPACE real-time test of the hybrid Ant colony and Fuzzy logic technique (AFO) reaches 98.7% accuracy with 0.9 s tracking time compared to basic ACO, FL and PSO under PSC.
Yuan et al. [30]	Hermite Interpolation Optimization	Hermite interpolation optimization (HPO) uses a fixed step on INC algorithm to improve tracking capability under PSC; it was able to achieve an average efficiency of 99.84% in 4 different scenarios against the conventional technique P&O and INC.
Qi et al. [31]	hybrid Cuckoo-Bee Colony	A hybrid of the well-known Cuckoo Search Algorithm and Artificial Bee Colony algorithm (CSA-ABC) reaches a surplus efficiency from 7-80%, outperforming the basic CSA, ABC, and PSO under different PSC cases.
Kumar et al. [32]	Adaptive Flower Pollination Algorithm	An improved version of the basic flower pollination algorithm (AFPA) produced energy up to 2.5% compared to FPA and reduced the tracking time by over 40% compared to PSO under uniform case, weak PSC, moderate PSC, and strong PSC
Yu et al. [33]	Quantum-behavior particle swarm optimization	Premature convergence issue of conventional QPSO was addressed by modifying the contraction–expansion coefficient. Improved QPSO reaches efficacies of 99.91% and 99.47% of single peak and multiple peaks, respectively, compared to QPSO, FA, and PSO under experimental results.
Ghazi et al. [34]	Dandelion optimizer	DO combined with Deep Reinforcement Learning (DRL) was used to regulate the voltage reference of MPPT. DO-DRL scores a high efficiency of 85.90% and 95% under summer day and winter day, respectively, compared to the other methods PSO-DRL, PSO-PI, DO-PI, and InC-PI.
Ha et al. [35]	Artificial Location Selection Optimization	The proposed ALSO reaches near perfect 100% reference MPP voltage tracking efficiency with simulation and over 95% with experimental setup assisted with Chroma 62100H PV emulator under five districts PSC curves.
Naser et al. [36]	Coot optimizer	Improved coot optimizer algorithm (ICOA) uses a new way to handle changes in power demand with only one tuning parameter, reaching an average efficiency of 99.94% and tracking time of 0.58 s under different PSCs with a SEPIC converter.
Chao et al. [37]	Intelligent bat algorithm	Improved Intelligent Bat Algorithm (IIBA) adjusts its search patterns based on the slope of the power curve, which allows it to outperform the CIBA, IFA, and MGWOA. IIBA scores by far the lowest average tracking time of 0.04 s compared to the other methods.
Chao et al. [38]	Cat swarm optimization	Modified cat swarm optimization (MCSO) adjusts its search speed based on two factors: the slope of the power output curve and an inertia weight. MCSO records the fastest average tracking time of 0.36 s and maximum oscillation amplitude of 9 W compared to the conventional CSO and the other 3 MCSO variants under 6 GMPP peak cases.

1.2. Motivation behind the work

In recent years, utilizing metaheuristic algorithms has experienced notable expansion, addressing diverse system problems and receiving positive feedback from researchers [39]. These techniques have been particularly leveraged for identifying the MPP on the PV curve due to their simplicity of implementation and low computational resource requirements [40], [41]. Efficiency, reaching time, convergence time, and the number of iterations are key factors used to compare the performance of different MPP tracking techniques. CPS scenarios, characterized by the formation of multiple nearby clusters of peaks, pose challenges for some algorithms in identifying the MPP, searching for the optimal solution more intricate [42], [43]. However, the Grey Wolf Optimization (GWO) algorithm has demonstrated superiority over other techniques in terms of efficiency, requiring a smaller number of iterations and faster convergence to the MPP [44]. Meanwhile, when navigating the search space, these algorithms face the issue of the optimal solution changing over time. In response, researchers have undertaken the re-implementation of arithmetic operations and the re-initialization of solutions within the algorithms, aimed at inducing deviation from the preceding optimal solution [45]. The behavior of the algorithms is influenced by intelligent random solution selection and updating based on their mathematical principles. However, randomness can cause large fluctuations after obtaining the optimal solution, particularly noticeable in the Particle Swarm Optimization (PSO) algorithm [46]. The study [47] compares the proposed improved MPPT technique combining PSO with P&O and a predefined search space optimized through Fuzzy Fokker Planck solutions with conventional PI, fuzzy, and PSO controllers, demonstrating superior algorithm convergence, reduced oscillation around the MPP, and faster response times. The hardware and simulation results show the method's robustness in handling conditions such as partial shading, dust, and GHG concentration, achieving a maximum efficiency of 99.23%. However, the reliance on fuzzy logic tuning adds complexity to the algorithm, which might increase computational overhead in real-time applications. Researchers have introduced expressions related to the iteration number to mitigate these issues and organize the search phase into exploration and exploitation stages, thereby balancing both phases. For instance, the SMO algorithm employs three sub-equations for process control during the exploration and exploitation phases. However, formulating these sub-equations can slow down the search process [48], warranting further enhancements and streamlining for optimal solution convergence. One of the attempts is in Xiao et al. paper [49] with a hybrid slime mold golden sine algorithm (SMGSA) to help enhance the performance of the SMA by modifying the third equation with the golden sine concept. However, the algorithm still needs more than one parameter for tuning. In this context, this paper puts forth an innovative approach, a hybrid algorithm with high efficiency and rapid convergence to the optimal global solution with just one tunable parameter, addressing the mentioned issues and advancing the field of MPP tracking in PV systems. The proposed approach offers promising potential for efficient and effective MPP tracking, particularly in challenging scenarios like CPSC.

To address the limitations of previous MPPT techniques concerning efficiency, tracking speed, convergence to the MPP, and the number of iterations required in various PV system scenarios, a new approach is proposed based on the hybridization of two modern algorithms, SMO and EO. The primary contributions of this research are as follows:

• •Study of the PV system using the novel hybrid ESMO approach, applied to three cases: FIC, partial shading, and complex CPSC.
• •Introduction of the equilibrium pool concept to address the slow convergence issue in the SMO algorithm, resulting in a hybrid technique that efficiently and rapidly identifies the MPP with just one tunable parameter.
• •Enhance efficiency and reliability through MPP identification in the PV as mentioned earlier system cases using the ESMO algorithm and comparison with various other techniques, including SMO, EO, GWO, WOA, and PSO.
• •Implementation of laboratory experiments to verify the algorithm's strength under two conditions: Partial shading and Dynamic partial shading conditions.

Subsequent sections of the study are structured as follows: Section 2 delves into PV system modeling under uniform and PSC. In Section 3, the Hybrid proposed MPPT technique is introduced. Section 4 showcases and examines the research findings. Finally, Section 5 summarizes the research conclusions and perspectives.

2. PV system modeling under uniform and PSC

A PV cell is a direct power source, converting sunlight into electrical power [50]. Several mathematical models can represent it, including the single diode model depicted as the blue circle in Fig. 1. This model is regarded as a perfect portrayal of the electrical performance of the PV cell. The output current is determined using Equation (1) and Equation (2), which express it as the sum of the diode current. However, the single diode model exhibits certain weaknesses when environmental conditions change, particularly in low-voltage scenarios [51]. To address this, the practical model incorporates resistors $R_s$ and $R_{sh}$ into the circuit [52]. It is represented by the red dashed circle in Fig. 1, and it is mathematically expressed by Equation (3).

$$I_{pv}=I_{ph}-I_d$$ (1)

$$I_{pv}=I_{ph}-I_{sat}[e^{\frac{V}{\alpha \cdot V_T}}-1]$$ (2)

$$I_{pv}=I_{ph}-I_{sat}[e^{\frac{V+I{R}_s}{\alpha \cdot V_T}}-1]-\frac{V+I{R}_s}{R_{sh}}$$ (3)

$$V_T=\frac{N_{s}kT}{q}$$ (4)

Figure 1.

Practical 1M5P PV model.

Several PV cells are connected in both parallel and series configurations to form a PV panel, facilitating increased power production at the output [53]. The mathematical relationship expressing the arrangement of Ns cells in series and Np cells in parallel is presented in Equation (5).

$$I=I_{ph}{N}_p-N_p{I}_{sat}[e^{\frac{V+I{R}_{s.eq}}{N_{cs}\alpha \cdot V_T}}-1]-\frac{V+I{R}_{s.eq}}{R_{sh.eq}}$$ (5)

The photoelectric current varies directly with the irradiance and exhibits an inverse relationship with the surrounding temperature. This relationship is mathematically described in Equation (6).

$$I_{pv}=(I_{pvSc}+K(T-T_{Sc}))\times \frac{G}{G_{Sc}}$$ (6)

Table 2 contains the electrical characteristics of the Sunpreme SNPM-GX-315 PV array used in this study. Fig. 2 shows the decrease in irradiance on the array leads to the formation of a uniform PV curve with a single power peak.

Table 2.

PV array parameters.

Description	Values
Pmax	316 W
Vmpp	39.5 W
Impp	8 A
Voc	51.5 V
Isc	8.5 A
KV Temp-coefficient of Voc (%/deg.C)	-0.37
KI Temp-coefficient of Isc (%/deg.C)	0.04
α	1.1612
Ncs (Number of cells in a module)	72

Figure 2.

The uniformed PV curve during the uniform case.

As shown in Fig. 3, when clouds or tree leaves cast shade on a portion of the PV array, the current in the shaded cells decreases, while the current in the uncovered cells remains unchanged. This phenomenon triggers the “hot spot” effect, leading to overheating in the array [54]. To address this issue, crosswire-connected bypass diodes are utilized to dissipate the self-heating caused by mismatched currents [55], [56]. Consequently, multiple peaks appear along the PV curve, as illustrated in Fig. 4, showing a non-uniform PV curve with multiple local maximum power peaks (LMPP) and one global maximum power peak. Multiple peaks on the PV curve pose a challenge for most conventional algorithms when seeking the MPP, causing them to become trapped at the LMPP and hindering their ability to reach the GMPP [57], [58]. In such cases, using metaheuristic algorithms becomes necessary to navigate the search space, represented by the PV curve, and obtain multiple values that align with the peak achievable power level.

Figure 3.

PV array configurations with bypass diode during UC and PSC.

Figure 4.

The non-uniformed PV curve during the partial shading condition.

The DC-DC voltage boost converter regulates the PV voltage following the reference voltage corresponding to the MPP [59]. This control is achieved by manipulating the Duty Cycle (DCL), which spans from 0 to 1. Equation (7) details the formulas governing the boost converter.

$$(\begin{array}{c}\hfill V_{output}=\frac{V_{input}}{1-D_{cycle}}\hfill \\ \hfill D_{cycle}=\frac{t_{on}}{t_{sw}}\hfill \\ \hfill C_{input}=\frac{\mu i_d}{8\mu i_d{f}_{sw}}\hfill \\ \hfill \begin{array}{c}\hfill C_{output}=\frac{i_0{D}_{cycle}}{8\mu v_d{f}_{sw}}\hfill \\ \hfill L=\frac{V_{pv}{D}_{cycle}}{2\mu i_d{f}_{sw}}\hfill \end{array}\hfill \end{array}$$ (7)

$V_{\text{input}}$ and $V_{\text{output}}$ represent the input and output voltage related to the boost converter, respectively. In contrast, $C_{\text{input}}$ and $C_{\text{output}}$ denote the input and output capacitances, respectively. Additionally, L corresponds to the required inductance. The inductor ripples current and the load ripple voltage are denoted as $\mu I_d$ and $\mu v_d$, respectively, and the switching frequency is represented by $f_{\text{sw}}$.

3. The hybrid proposed MPPT

Metaheuristic algorithms provide a general search for the optimal solution within a given space, aligning with seeking the MPP on the PV curve. A new approach is proposed based on hybridizing two modern algorithms, SMO and EO. Under PSC, this hybrid algorithm effectively approaches the GMPP on the PV curve.

The two fundamental algorithms, SMA [23] and EO [24], are integrated to form the ESMO-based MPPT algorithm. The tracking mechanism is introduced as follows:

3.1. The Slime Mould Algorithm (SMA)

The slime mould “Physarum polycephalum” is an intelligent eukaryotic cell that utilizes a biological mechanism called bio-oscillation to guide cytoplasm diffusion through the veins formed by the group while searching for food. As the veins approach the food source, the cytoplasm flow speed increases, resulting in thickening of the veins. This mechanism is employed to ascertain the optimal path for reaching the food source, guided by the ambient environmental odor. This mechanism was mathematically modeled, formulating a novel concept centered on weight modulation for governing positive and negative influence within the slime mould propagation wave [23]. The flow chart of the SMA is presented in Fig. 5.

Figure 5.

The flowchart of SMA.

3.2. Equilibrium Optimizer (EO)

EO is an algorithm based on mass equilibrium models used in various engineering fields, including physical applications founded on the conservation of mass principle. This principle is formulated through the equation of mass change rate being equivalent to the inflow rate minus the outflow rate coupled with the generated mass [60]. In the EO algorithm, particles strive to reach equilibrium, representing the optimal solution attainable within the exploration space. The solution's position can be expressed as the concentration. The equilibrium concentration, also known as the equilibrium pool, comprises one of the best solutions achieved by combining five other candidate solutions [61]. The schematic diagram of the EO process is depicted in Fig. 6.

Figure 6.

The schematic diagram of the EO process.

3.3. The ESMO-based MPPT technique

The ESMO algorithm combines the two fundamental algorithms' best features, leveraging the EO's equilibrium pool feature to reduce stochasticity in the second equation. This integration enables the SMA to approach the best optimal solution more efficiently and rapidly [62]. The algorithmic outline of the ESMO approach is showcased in Algorithm 1.

Algorithm 1.

The pseudo-code of ESMO.

Where t denotes the ongoing iteration, and $MaxIter$ designates the maximum iteration. $U_b$ and $L_b$ signify the upper and lower confines of the exploration space. At iteration, k, $Y_A$, and $Y_B$ are designated as two random individuals drawn from the entire colony, and Z is equal to 0.03. $r_1$ and $r_2$ are random values between 0 and 1. $v_b$ is a uniform distribution within the range (-α, α), and $v_c$ is a uniform distribution within the range (-β, β), where α and β are equal to:

$$alpha=\arctan h(-(\frac{t}{MaxIter})+1)$$ (8)

and

$$beta=1-(\frac{t}{MaxIter})$$ (9)

The value of $p_i$ is presented in Equation (10).

$$p_i=\tanh |Z(i)-D{F}_{gbest}|$$ (10)

Where $Z(i)$ represents the fitness of $Y(i)$, and DF denotes the top fitness achieved throughout all iterations.

The weight factor in the current iteration is formulated by sorting the fitness values for a maximization problem. The weight $W_g$ is given by Equation (11):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [sortfit,sortIndex]=sort(fit)\hfill & \hfill fit=\{fi{t}_1,fi{t}_2,...,fi{t}_N\}\hfill \end{array}\hfill \\ \hfill Wg(sortIndex(j))=\left\{\begin{array}{cc}\hfill 1+r_3\cdot \log (\frac{f_{best}-sortfit(j)}{f_{best}-f_{worst}}+1)\hfill & \hfill 1\le j\le \frac{N}{2}\hfill \\ \hfill 1-r_3\cdot \log (\frac{f_{best}-sortfit(j)}{f_{best}-f_{worst}}+1)\hfill & \hfill \frac{N}{2}<j\le N\hfill \end{array}\right\}\hfill \end{array}$$ (11)

Where $r_3$ is a random number spans from 0 to 1, and the best and worst fitness values are denoted by $f_{best}$ and $f_{worst}$, which are equal to $sortfit(N)$ and $sortfit(1)$, respectively.

In the second equation of the Slime Mould Algorithm, the random terms $Y_A$ and $Y_B$ could be a factor in the slow and ineffective search process, leading to difficulties in recognizing the global solution in optimization problems. Although the paper [48] discussed positive results, there exists potential for additional enhancement in striving towards the optimal solution. Inspired by the EO, the equilibrium pool ($C_{eq}$) can be utilized to replace the random variable $Y_A$.

The slime mould individual of the equilibrium pool is presented as:

$$\begin{array}{c}\hfill Y_{eq(1)}=Y(sortIndex(N))\hfill \\ \hfill Y_{eq(2)}=Y(sortIndex(N-1))\hfill \\ \hfill Y_{eq(3)}=Y(sortIndex(N-2))\hfill \\ \hfill Y_{eq(4)}=Y(sortIndex(N-3))\hfill \\ \hfill Y_{avrg}=\frac{Y_{eq(1)}+Y_{eq(2)}+Y_{eq(3)}+Y_{eq(4)}}{4}\hfill \\ \hfill Y_{eq}=\{Y_{eq(1)},Y_{eq(2)},Y_{eq(3)},Y_{eq(4)},Y_{avrg}\}\hfill \end{array}$$ (12)

The new $i_{th}$ slime mould position is now expressed using the new Equation (13):

$$Y_i(t+1)=\left\{\begin{array}{c}\hfill r_1\cdot (Ub-Lb)+Lb,\hfill \\ \hfill Y_{Gbest}+v_b\cdot (W\cdot Y_{eq}-Y_B)\hfill \\ \hfill v_c\cdot Y_i(t)\hfill \end{array}\begin{array}{c}\hfill ,r_1<z\hfill \\ \hfill r_2<p_i(t),r_1\ge z\hfill \\ \hfill r_2\ge p_i(t),r_1\ge z\hfill \end{array}\right\}$$ (13)

Note that the new term helps strike a balance between the exploratory and exploitative phases, where the equilibrium pool array allows the process to gain great control during the search. The adjustment of the DCL is carried out following Equation (14).

$$D_i(t+1)=\left\{\begin{array}{c}\hfill r_1\cdot (Ub-Lb)+Lb,\hfill \\ \hfill D_{Gbest}+v_b\cdot (W\cdot D_{eq}-D_B)\hfill \\ \hfill v_c\cdot D_i(t)\hfill \end{array}\begin{array}{c}\hfill ,r_1<z\hfill \\ \hfill r_2<p_i(t),r_1\ge z\hfill \\ \hfill r_2\ge p_i(t),r_1\ge z\hfill \end{array}\right\}$$ (14)

Here, $D(t+1)$ denotes the revised DCL, $D_{Gbest}$ stands for the DCL of the global best, $D_B$ represents a randomly selected DCL from the colony $D_{eq}$ signifies a random DCL chosen from the Equilibrium pool consisting of the top five best DCLs achieved up to this point. The schematic depiction of the ESMO is displayed in Fig. 7.

Figure 7.

The schematic depiction of the ESMO.

Equation (15) represents the objective function employed by the proposed ESMO for converging toward the best DCL. This equation continuously compares the power generated by each individual ($P_i(t)$) with the previously identified global best power ($P_{Gbest}(t-1)$).

$$\begin{array}{c}\hfill if{P}_i(t)>P_{Gbest}(t-1)\hfill \\ \hfill P_{Gbest}(t)=P_i(t)\hfill \\ \hfill D_{Gbest}(t)=D_i(t)\hfill \end{array}$$ (15)

If the value of $P_i(t)$ surpasses the previously recorded $P_{Gbest}(t-1)$, the objective function also updates the current $D_i(t)$ as $D_{Gbest}(t)$ which guarantees continuously maximizing power generation.

Equation (16) guarantees that the search will be repeated if there is a change in the value of the irradiation or the shading condition (DPSC), which leads to a new GMPP value so that if this condition exceeds 3%, it will be triggered. In addition, this ensures that the search will not be repeated during the change in the load value.

$$\frac{|P_{Gbest}(t)-P_{Gbest}(t-1)|}{P_{Gbest}(t-1)}>0.03$$ (16)

3.4. Tracking mechanism

The ESMO algorithm derives the reference voltage aligned to the MPP from the PV curve, which is considered a non-linear search space. Within 40 milliseconds, four particles are randomly initialized in the exploration space, and the process continues as a value between 0 and 1 expressed in the DCL. The PV curve contains one GMPP with a value of 592.1 W and two LMPP with a value of LM1 =512 W and LM2 =501 W. The ESMO algorithm continues to search for the GMPP until it is identified. The ESMO algorithm moves on to the exploitation stage after identifying the GMPP, and the particles approach each other concerning the best particle until they reach the best voltage value corresponding to the highest power point. Fig. 8 illustrates the ESMO-based MPPT tracking mechanism.

Figure 8.

The tracking mechanism of the ESMO-based MPPT.

4. Results and discussion

The results were executed under the MATLAB/SIMULINK environment, with a simulation time step of 1e-5 s. The detailed implementation of the proposed ESMO-based MPPT technique is depicted in Fig. 9. The simulations were conducted using MATLAB R2016a version (The MathWorks, Inc., Natick, MA, USA).

Figure 9.

ESMO-based MPPT Simulink Block.

The efficacy of the new ESMO technique was tested in three different cases:

• •Case (1): FIC.
• •Case (2): PSC.
• •Case (3): CPSC.

The ESMO technique was compared to the two fundamental techniques, SMO and EO, and other well-known techniques like GWO, WOA, and the classical PSO.

4.1. Under FIC

The study was conducted under the scenario of fast changes in Irradiation. The modules received uniform values of Irradiation in successive time intervals. The results obtained were as follows: Fig. 10(a) shows the power response during the three fast Irradiation changes in the mentioned intervals. The ESMO technique achieved the lowest reaching time of 0.25 s and settling time of 0.4 s in the first interval, with a maximum power tracked at 157.8 W. In comparison, SMO, EO, GWO, WOA, and PSO achieved convergence times of 0.35 s, 0.4 s, 0.37 s, 0.3 s, and 0.4 s, settling times of 0.4 s, 0.5 s, 0.4 s, 0.32 s, and 0.6 s, and maximum power tracked of 157.75 W, 156 W, 157.35 W, 157.79 W, and 156.6 W, respectively. Fig. 10(b) illustrates the searching process of the DCL. In the second interval, the ESMO technique achieved a 0.1 s rapid convergence and stabilization duration of 0.12 s, with a maximum power tracked at 61.5 W. In contrast, SMO, EO, GWO, WOA, and PSO achieved convergence times of 0.22 s, 0.4 s, 0.2 s, 0.16 s, and 0.3 s, settling times of 0.25 s, 0.42 s, 0.21 s, 0.2 s, and 0.55 s, and maximum power tracked of 61.4 W, 60 W, 60.5 W, 61.48 W, and 60 W, respectively. The effectiveness of the ESMO technique was evident in these results. Similarly, in the third interval, the ESMO technique achieved the lowest reaching time of 0.205 s and settling time of 0.212 s, with a maximum power tracked at 110.06 W. In comparison, SMO, EO, GWO, WOA, and PSO achieved reaching times of 0.21 s, 0.213 s, 0.212 s, 0.206 s, and 0.21 s, settling times of 0.23 s, 0.22 s, 0.22 s, 0.209 s, and 0.215 s, and maximum power tracked of 109.7 W, 110 W, 110 W, 110 W, and 108.6 W, respectively. Fig. 10(c) and Fig. 10(d) illustrate the searching process for the optimal voltage and current, respectively.

Figure 10.

Case 1 FIC: (a) Power output comparison of the techniques; (b) DCLs comparison of the techniques; (c) PV voltage comparison of the techniques; (d) PV current comparison of the techniques.

4.2. Partial shading conditions (PSC)

PSC is studied using the 5SP1 configuration under the irradiance values shown in Table 3. The PV curve pattern obtained during PSC has a GMPP value of 321.8 W, shown in Fig. 11. The proposed ESMO technique successfully identified the GMPP with high efficiency, achieving the shortest adaptation period of 0.25 s, the shortest stabilization time of 0.28 s, and the highest power value estimated at 321.77 W with an impressive efficiency of 99.99% in only seven iterations. Fig. 12(a) presents the power response during the PSC, highlighting the superiority of ESMO over the other techniques, including SMO, EO, GWO, WOA, and PSO. These techniques obtained power values of 321.7 W, 321.7 W, 320 W, 321.5 W, and 318 W, with convergence times equal to 0.4 s, 0.41 s, 0.55 s, 0.45 s, and 1.1 s, and settling times equal to 0.45 s, 0.44 s, 0.62 s, 0.5 s, and 1.3 s, respectively, over several iterations ranging from 11 to 21. Fig. 12(b) represents the DCL of the techniques used in this work. The PV voltage and the PV current are illustrated in Fig. 12(c) and Fig. 12(d), respectively, showcasing the efficient tracking of the GMPP by the proposed ESMO technique in the presence of PSC.

Table 3.

Irradiance pattern for PSC case 2.

Case No.	Irradiance (W/m2)	GMPP(W)
Case 2 Partial Shading condition	PV1= 310 PV2= 190	321.8
	PV3= 510 PV4= 440
	PV5= 120

Figure 11.

PV curve pattern during PSC condition.

Figure 12.

Case 2 PSC: (a) Power output comparison of the techniques; (b) DCLs comparison of the techniques; (c) pv voltage comparison of the techniques; (d) pv current comparison of the techniques.

4.3. CPSC

In many PV systems, multiple modules are utilized to produce a higher power output. However, in scenarios where shading occurs, a complex PSC state is formed. This results in the PV curve having several peaks and clusters. Fig. 13 illustrates the PV curve during the CPSC. To simulate the CPSC, specific irradiance values are applied, as shown in Table 4. Additionally, Fig. 13 portrays the PV curve during CPS condition with cluster formation.

Figure 13.

PV curve during CPS condition with cluster formation.

Table 4.

Irradiance pattern for CPSC case 3.

Case No.	Irradiance (W/m2)	GMPP(W)
Case 3 Complex Partial Shading conditions	PV1= 300 PV2= 60	873.83
	PV3= 505 PV4= 345
	PV5= 905 PV6= 225
	PV7= 435 PV8= 315
	PV9= 660 PV10= 830

In this case, the PV curve displays two discernible groupings, denoted as Cluster 1 and Cluster 2. Cluster 1 includes LM4 = 704 W, LM5 = 773.3 W, and LM6 = 754.6 W, while Cluster 2 consists of LM7 = 805.2, LM8 = 873.83 W, and LM9 = 849 W. The cluster head maxima (CHM) represents the MPP within each cluster, with CHM1 = LM5 and CHM2 = LM8. GMPP corresponds to the overall peak on the PV curve, and its value is GMPP = LM8 = 873.83 W. Fig. 14(a) presents the power response comparison among the techniques. The proposed ESMO technique outperformed the others, achieving the shortest response time of 0.21 s and reaching the time of 0.22 s, and required the fewest iterations (6) to identify the GMPP. On the other hand, SMO, EO, GWO, WOA, and PSO obtained convergence times of 0.32 s, 0.37 s, 0.37 s, 0.45 s, and 0.8 s, respectively, with reaching times of 0.4 s, 0.38 s, 0.38 s, 0.55 s, and 0.85 s, along with a higher number of iterations (ranging from 7 to 18). Fig. 14(b) presents the DCL comparison among the techniques. The hybridization of the two basic techniques in the proposed ESMO played a critical role during the exploratory and exploitative phases, enabling it to identify the GMPP even under CPSC efficiently. The PV voltage and current for the fourth case are illustrated in Fig. 14(c) and Fig. 14(d), respectively. Table 5 presents a comprehensive overview of all the examined scenarios.

Figure 14.

Case 3 CPSC: (a) Power output comparison of the techniques; (b) DCLs comparison of the techniques; (c) pv voltage comparison of the techniques; (d) pv current comparison of the techniques.

Table 5.

In-depth comparison of all the studied cases.

Tech.	Case no.	Converge time (s)	Settling time (s)	GMPP (W)	Power tracked (W)	Iteration Number	Avg Efficiency (%)
ESMO	Case 1 Case 2 Case 3	0.25, 0.1, 0.205 0.25 0.21	0.4, 0.12, 0.212 0.28 0.22	157.8, 61.5, 110.06 321.8 873.83	157.8, 61.5, 110.06 321.77 873.5	10 7 6	99.99 99.99 99.96
SMO	Case 1 Case 2 Case 3	0.35, 0.22, 0.21 0.4 0.32	0.4, 0.25, 0.23 0.45 0.4	157.8, 61.5, 110.06 321.8 873.83	157.75, 61.4, 109.7 321.7 870.1	18 13 8	99.82 99.96 99.57
EO	Case 1 Case 2 Case 3	0.4, 0.4, 0.213 0.41 0.37	0.5, 0.42, 0.22 0.44 0.38	157.8, 61.5, 110.06 321.8 873.83	156, 60, 110 321.7 871.2	15 12 9	98.78 99.96 99.69
GWO	Case 1 Case 2 Case 3	0.37, 0.2, 0.212 0.55 0.37	0.4, 0.21, 0.22 0.62 0.38	157.8, 61.5, 110.06 321.8 873.83	157.35, 60.5, 110 320 869	18 15 7	99.34 99.44 99.44
WOA	Case 1 Case 2 Case 3	0.3, 0.16, 0.206 0.45 0.45	0.32, 0.2, 0.209 0.5 0.55	157.8, 61.5, 110.06 321.8 873.83	157.79, 61.48, 110 321.5 865	13 11 12	99.96 99.90 98.98
PSO	Case 1 Case 2 Case 3	0.4, 0.3, 0.21 1.1 0.8	0.6, 0.55, 0.215 1.3 0.85	157.8, 61.5, 110.06 321.8 873.83	156.6, 60, 108.6 318 858.28	20 21 18	98.48 98.81 98.22

4.4. Experimental validation

The proposed algorithm was thoroughly examined to validate its superiority. Initially, the ESMO algorithm's theoretical performance was assessed through MATLAB/Simulink simulations. Following this theoretical verification, the ESMO algorithm efficiency was experimentally evaluated within a laboratory setup. Fig. 15 shows the experimental setup of the PV system. The experimental system schema, including the MATLAB blocks and the controlling principal of the PV emulator, is presented in Fig. 16, and The electrical parameters are shown in Table 6.

Figure 15.

Image of the experimental setup (1—PV emulator, 2—dSPACE 1104, 3—DC-DC converter, 4—The Host computer, 5—resistor load, 6—oscilloscope, 7—current sensors, 8—voltage sensors).

Figure 16.

Experimental schema block for proposed ESMO-based MPPT.

Table 6.

The electrical parameters of the experimental setup.

Description	Values
Inductance (L)	30 mH
capacitance (C)	1100 uF
Load Resistance (RL)	30 Ω
Switching freq (f)	20 kHz
Duty Cycle Update Timer (DCUT)	40 Hz

To accurately emulate the output of the PV array under PSC and DPSC, a programmable power supply, specifically the GWINSTEK-APS1102A, was employed. This power supply facilitated precise simulation, mimicking the behavior of the PV array. It was crucial to experimentally verify the output of the programmable power supply to ensure its conformity with the PV array's output. Fig. 17 demonstrates the P-V/I-V curve pattern employed for this verification, and Fig. 18 shows the output of the power supply on specific points on the I-V curve. The accuracy of the power supply can be observed through its ability to accurately emulate the behavior of the PV array, specifically in replicating the non-linear characteristics of the cell's equation. This was achieved by measuring multiple random points along the I-V curve and confirming that the power supply's output aligns with the expected measurement.

Figure 17.

P-V and I-V curve of the verification.

Figure 18.

The output of the power supply on specific points on the I-V curve.

Furthermore, the PV array emulator was controlled by the DCL of the DC-DC boost converter, which was effectively connected to the dSPACE1104 R&D Controller Board. A sampling frequency of 20 kHz was adopted to maintain accurate control over the input. The DCL Update Timer (DCUT) was set to 40 Hz, equivalent to 25 ms. This particular time interval was determined to facilitate the transition of DCL values from one state to another within the control algorithm. This value was selected based on careful observation and analysis of the algorithm's control state as executed by the DSpace board. The objective behind this choice was to achieve a steady state condition that ensures the code execution does not overrun or result in divergent behavior. By carefully tuning the DCUT to 40 Hz, the system effectively maintains stability and enhances the overall performance of the control process.

In the context of comparable simulation experiments, the proposed algorithm ESMO was empirically validated, demonstrating its superiority in two distinct P-V curve scenarios of the PV system: PSC and DPSC. These two curves, as depicted in Fig. 19, were employed, featuring two different global values of maximum power, namely 269 W, and 236 W, along with two corresponding maximum voltage values of 60.4 V and 60.8 V, and two maximum current values of 4.45 A and 3.85 A, respectively. In the first case, all algorithms successfully identified the global maximum value. However, the proposed algorithm ESMO exhibited a distinct advantage over others in terms of reaching speed and reaching time, as depicted by the red line in Fig. 20. Notably, ESMO achieved the fastest reaching time, recording an estimated time of 200 ms. In comparison, other algorithms, SMO, EO, GWO, WOA, and PSO, demonstrated longer reaching times, registering values of 350 ms, 425 ms, 575 ms, 350 ms, and 775 ms, respectively. The results highlight ESMO's superior performance, confirming its effectiveness in swiftly converging toward the GMPP.

Figure 19.

The two P-V curve pattern of case 1 and case 2.

Figure 20.

Case 1 PSC: The experimental results of the MPPT for (a) ESMO, (b) SMO, (c) EO, (d) GWO, (e) WOA, and (f) PSO.

In the second case, involving DPSC, the algorithms' capacity to re-identify the new maximum value estimated at 236 W was effectively evaluated following a modification in the external perimeter of the PV system. The experimental results of the second case are shown in Fig. 21. In this context, the proposed algorithm demonstrated the swiftest response time, estimated at 200 ms, showcasing its efficient and rapid adaptability. Conversely, other algorithms, namely SMO, EO, GWO, WOA, and PSO, exhibited longer response times, recording values of 325 ms, 500 ms, 450 ms, 400 ms, and 950 ms, respectively. The results unequivocally validate the proposed algorithm's superior performance in swiftly adjusting to the dynamic changes in the system, ensuring quicker convergence towards the new maximum value of 236 W under these varying conditions. Table 7 presents a comparison of experimental results.

Figure 21.

Case 2 DPSC: The experimental results of the MPPT for (a) ESMO, (b) SMO, (c) EO, (d) GWO, (e) WOA, and (f) PSO.

Table 7.

Comparison of experimental results.

Tech.		ESMO	SMO	EO	GWO	WOA	PSO
Response time (ms)	PSC DPC	200 200	350 325	425 500	575 450	350 400	775 950

5. Conclusions and perspectives

The research paper introduces a novel Equilibrium ESMO technique for tracking the MPP in solar systems. ESMO combines two new techniques and models them as mathematical equations to serve as an MPPT-based control approach. The proposed technique is extensively tested in three distinct scenarios of PV systems, including FIC, PSC, and CPSC. Comparisons are made between ESMO and other MPPT techniques, namely Slime Mould Optimization (SMO), EO, Grey Wolf Optimization (GWO), Whale Optimization Algorithm (WOA), and PSO. The ESMO technique exhibits increased output power efficiency and reduced power losses in all cases. Simulation using MATLAB/SIMULINK and a comparative statistical study underscore the efficacy and robustness of ESMO, with an impressive average efficiency of up to 99.98 percent in all scenarios, including CPSC, where some techniques struggle with local maxima trapping. Moreover, the proposed algorithm's strength is further verified through experimental testing in two PV system cases, PSC and Dynamic Partial shading conditions (DPSC). ESMO recorded the fastest response times, identifying the two GMPPs within 200 ms. These experimental results affirm ESMO's effectiveness in efficiently handling varying conditions of PV systems. It is noteworthy that ESMO effectively addresses the slow convergence observed in the SMO algorithm by incorporating the equilibrium pool feature of EO. This unique combination enhances the rapid adaptability and performance of ESMO under challenging conditions. Despite the proposed ESMO algorithm's success in obtaining high-efficiency rates for tracking GMPP during the majority of irradiation scenarios, applying the proposed algorithm under large-scale scenarios requires more studies and considerations regarding data storage, computational resources, and implementation complexity. Since huge irradiation and temperature data will certainly be captured, searching for solutions to store and process this data is essential. The computational requirements of the algorithm could potentially limit its ability to identify the GMPP accurately under rapidly changing irradiation patterns. Future research will investigate optimization strategies to reduce the computational complexity of the ESMO algorithm while maintaining its effectiveness in PSC. Additionally, exploring hardware acceleration techniques could be beneficial for real-world deployments, especially in large-scale PV systems encountering PSC.

CRediT authorship contribution statement

Djallal Eddine Zabia: Writing – original draft, Software, Formal analysis, Data curation, Conceptualization. Hamza Afghoul: Writing – original draft, Software, Methodology, Formal analysis, Data curation. Okba Kraa: Writing – original draft, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Yassine Himeur: Writing – review & editing, Visualization, Supervision, Resources, Project administration, Funding acquisition. Haitham S. Ramadan: Writing – review & editing, Visualization, Supervision, Project administration, Formal analysis. Istemihan Genc: Writing – review & editing, Visualization, Supervision, Project administration, Investigation. Abdoulkader I. Idriss: Writing – review & editing, Supervision, Project administration, Investigation. Sami Miniaoui: Writing – review & editing, Visualization, Supervision, Investigation, Funding acquisition, Formal analysis. Shadi Atalla: Writing – review & editing, Visualization, Validation, Supervision, Resources, Project administration. Wathiq Mansoor: Writing – review & editing, Visualization, Supervision, Resources, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability statement

The data that support the findings of this study are available upon reasonable request.