An Improved, Novel Musical Chairs Algorithm with Local Adaptive Exploration for MPPT of PV Systems

Abstract

Shadows falling on photovoltaic (PV) modules result in partial shading conditions (PSCs). These conditions affect the power generation of a PV system because of their varying nature. As a result of PSCs, multiple peaks are created; therefore, it is important to identify the global maximum power point (GMPP) for optimal output power. Several maximum power point tracking (MPPT) techniques have been proposed in the literature; however, they face challenges such as oscillation at steady state, long convergence time, high complexity, and low accuracy. In this study, an improved musical chairs algorithm with local adaptive exploration is proposed for MPPT of PV systems under partial shading conditions. The proposed method combines the population-based exploration capability of the musical chairs algorithm with a localized duty-cycle adjustment mechanism around the best operating point. Unlike an offline exhaustive scan, the proposed local exploration stage uses only a small set of neighboring duty-cycle candidates, making the method more suitable for online MPPT implementation. The results are analyzed using the MATLAB/Simulink tool for a 4 × 4 PV array under PSCs. The IMCA-LAE algorithm is compared against the perturb and observe (P&O) algorithm, the incremental conductance (INC) algorithm, the musical chairs algorithm (MCA), and the gray wolf and whale optimization algorithm (GWWA) to illustrate the effectiveness of the suggested hybrid MPPT approach. The efficacy is further examined regarding five performance criteria: generated output power, convergence time, mismatch power loss, efficiency, and fill factor. The proposed IMCA-LAE outperformed the other algorithms.

1. Introduction

Renewable energy systems, particularly PV arrays, have emerged as key technologies for sustainable energy production. Compared with non-renewable energy sources such as fossil fuels, PV energy is widely considered a viable alternative. It is environmentally friendly, installation costs are decreasing, it is inexpensive to maintain, and it is durable. Additionally, its source, the sun, is infinite. However, environmental factors such as irradiance and temperature have a substantial impact on PV system performance, reducing output power. PSCs are caused by shadows cast on the surface of the PV panel by obstructions such as trees, tall buildings, flying birds, and moving clouds. PSCs affect the output power generation of a PV array because irradiance and temperature vary across the shaded PV modules. The shadowed region of the PV module can generate no current at all, so the affected PV module does not significantly contribute to the total output power generated by the PV array. A PV array affected by PSCs can also lead to a hotspot problem, in which affected PV modules, rather than generating current, dissipate heat due to low solar irradiance incident on the surface of the PV module from the sun, which prevents current from being conducted. The hotspot problem can lead to physical damage to the solar panel. To ensure that the PV system operates at its peak power efficiently, it is crucial to extract the maximum power possible from the array under any given conditions. This task is achieved through a process known as MPPT. The PV module ideally allows the greatest power to be drawn at a single operating point at any given time. However, because there are multiple peaks under PSCs, it is crucial to monitor the global maximum power point (GMPP) rather than the local maximum power point (LMPP) to ensure that the PV module’s operating point is always at its highest. MPPT algorithms are control strategies designed to continually adjust the operating point of the PV system to track the GMPP. MPPT techniques can be characterized as conventional [1], intelligent, metaheuristic, and hybrid methods; these are depicted in

Table 1. Comparison of the categorized MPPT techniques.

MPPT Techniques	Advantages	Disadvantages
Conventional	Simple to implement, low cost, low computational requirements, and good accuracy under uniform conditions.	High oscillation around MPP, low accuracy under PSCs, high failure rate, high convergence speed, and high power loss.
Intelligent	Fast convergence time under PSCs, low oscillation at MPP, low failure rate, and high efficiency under PSCs.	High complexity, a large amount of data is required for training, and increased implementation cost.
Metaheuristic	High efficiency under PSCs, fast convergence speed, minimal oscillation at steady state, scalability for a large PV array, and low failure rate.	High complexity and increased implementation cost.
Hybrid	Low failure rate, very high efficiency under PSCs, convergence time might be prolonged, reduced oscillation at steady state, low computational burden, and good for large-scale PV arrays.	High complexity and high implementation cost.

alongside their advantages and disadvantages. The techniques vary in their approach, complexity, convergence speed, accuracy, and ability to handle dynamic situations. However, they are all driven by one goal: to track the GMPP for maximum output power generation.

Conventional MPPT methods such as perturb and observe (P&O) [2,3], incremental conductance (INC) [3,4,5], and hill climbing [1] have been suggested in the literature. Furthermore, constant voltage (CV) [6,7,8], open-circuit voltage (OCV) [9], short-circuit current (SCC) [10], adaptive reference voltage (ARV) [11,12], ripple correction control (RCC) [12], lookup table (LUT) [13,14,15], and 0.8 × $V_{oc}$ and skipping mechanisms [1] have been proposed. These techniques are all known for their simplicity of implementation and low cost. However, they are mostly faced with the challenge of a high failure rate and oscillation at steady state under PSCs, resulting in output power loss. Several improvements have been made in the literature to further enhance these conventional methods. An improved INC was proposed to enhance tracking efficiency under dynamic conditions [16]. Though it requires more computation in the controller, it has better tracking efficiency [17] under varying conditions and better convergence when compared with P&O [18]. The efficiency, response speed, and tracking efficiency of INC in the PV system [17] were improved. A modified P&O method (MP&O) was proposed in [19] by Ali et al., splitting the PV module into four operational regions based on the OCV estimate technique. To overcome setbacks encountered with conventional OCV, such as discontinuous power supply and low tracking efficiency [20], a fractional OCV approach (FOCV) was proposed by Jawad Ahmed to shorten the sampling interval and period of the PV array [21]. OCV with a switched semi-pilot cell to measure the OCV [22] was introduced by Baroi et al. In [23], an improved OCV method was proposed that directly measures $V_{mpp}$ without the need to measure $V_{oc}$ and constant k. A hybrid genetic algorithm (GA) and FOCV were proposed to further tackle these issues [20]. Asim et al. introduced an improved CV MPPT method to improve the efficiency of CV by using a pilot PV panel to measure $V_{oc}$ [24]. In [25], Lasheen et al. enhanced the performance of CV; proportional integral (PI) controller gains were actualized by the genetic algorithm (GA). An improved SCC was proposed in [26] based on a Takagi–Sugeno fuzzy model as an estimator of the PV array. A modified model adaptive reference voltage (MMARV) approach was proposed in [27]. Manna et al. [28] proposed a novel robust model reference adaptive (RMRA) MPPT controller [29] for PV systems. The conventional methods, despite having the advantages of simplicity, low cost, and being easy to implement [18,30] under uniform conditions [31], exhibit high oscillation around the MPP [13] and low tracking efficiency under rapidly changing environmental conditions [30], resulting in loss of power [32].

Intelligence methods include the Fibonacci series (FSs) [13], the Gauss–Newton (GN) algorithm, fuzzy logic controllers (FLCs) [4,6,33], artificial neural networks (ANNs) [34,35], and sliding mode controllers (SMCs). Ramaprabha et al. offered a better FS by integrating power ripples and a large search range in [36]. Pati and Sahoo in [37] introduced a modified FS to reduce the search area in each iteration. Mei Qin Yu in [38] introduced a modified GN that reduces the chance that the target value will increase with each iteration. The FLC, comprising fuzzification, inference, and defuzzification, was proposed in [20] to enhance convergence time, eliminate power loss, and prevent oscillation at the operating point. An improved ANN was designed employing real operating conditions in [39]. According to [35], ANN had a fast and precise response under varying solar irradiance. Intelligent strategies were developed to address the inadequacies of conventional methodologies to run the PV system at MPP, boost efficiency, and function effectively in rapidly changing environmental conditions. Nevertheless, these approaches have drawbacks, including high complexity and the need to analyze huge amounts of data for system training beforehand [13].

Metaheuristic approaches are optimization algorithm–based techniques, and variations including PSO [40], ant colony optimization (ACO) [41], GWO [42], cuckoo optimization (CO) [43], fish swarm optimization (FSO) [44], firefly optimization (FFO) [45], artificial bee colony optimization (ABCO) [46], grasshopper optimization (GHO) [47], Harris hawk optimization (HHO) [48], the musical chairs algorithm (MCA) [31], and more have all been proposed in the literature. Since the increment utilized for optimization is dependent on the location of search agents that are extremely close to one another, most optimization techniques are unaffected by high oscillation issues under steady state conditions [31]. These tactics are intended to increase the effectiveness of MPPT in both standard and demanding circumstances.

Hybrid techniques combine two or more techniques to achieve a more robust MPPT under PSCs. There are basic criteria to guide one’s decision in choosing the hybrid MPPT technique, such as analyzing PV system conditions (PSC, dynamic irradiance, and temperature), GMPP emergence, fast convergence time, computational resources (MCU/DSP/FPGA), oscillation under steady state constraints, robustness under dynamic changes, scalability for large PV arrays, parameter sensitivity, and tuning effort. Some examples include P&O-FSCC [49], FLC-IC [50], GA-P&O [51], GA-ANN [52], and GA-FLC [53]. The hybrid methods, though very effective, with a low failure rate, are often complex, and might take a longer time to converge; in addition, their hardware implementation cost is high.

Related Work

Several MPPT approaches have been proposed in the literature and can be grouped into conventional, intelligent, meta-heuristic, and hybrid categories [13,54,55]; this study focuses on a hybrid MPPT method. Conventional methods are highly affected by PSCs [55]; they often suffer from a high failure rate, getting trapped at LMPP rather than GMPP due to multiple peaks created in such conditions [56]. High convergence time is also experienced, with oscillation at the steady state as a result of fixed step sizes. Intelligent methods, although more effective and more accurate under PSCs than conventional methods, face challenges related to complexity and large-scale data processing [55]. In meta-heuristic techniques, the increment used for optimization depends on the position of the seeking agents, and all current meta-heuristic optimization algorithms operate relatively close to one another at steady state; as a result, most of them do not experience high oscillations at steady-state settings. In the hybrid approach, a combination of more than one technique is employed to enhance accuracy and eliminate oscillation at steady state. However, its biggest drawbacks are mainly high convergence time and complexity [54]. This study proposes IMCA-LAE as a hybrid MPPT technique with low convergence time and low computational burden.

The IMCA-LAE is compared with various other strategies to illustrate its usefulness in maximizing energy harvesting from PV systems while resolving significant problems such as partial shadowing, rapid environmental fluctuations, convergence time, oscillation at steady state, and failure rate. The gaps left by other MPPT techniques, as highlighted in

Table 2. A comparative analysis of some MPPT techniques.

MPPT Techniques	Optimization Tool	Convergence Speed	Tracking Efficiency	Oscillation at Steady State	Computation Complexity	Step Size	No. of Control Parameters
IMCA-LAE	MATLAB R2021a	very high	very high	no	low	dynamic	1
MCA [31]	MATLAB R2021a	very high	very high	very low	medium	dynamic	1
GWWA [57]	MATLAB R2021a	high	high	low	high	adjustable	2
Adaptive P&O-FLC [58]	MATLAB R2021a	high	high	no	high	adaptive	2
AFO [59]	MATLAB R2021a	medium	high	low	medium	adjustable	2
SSA-HC [60]	MATLAB R2021a	high	high	low	medium	dynamic	1
GW-EO [61]	MATLAB R2021a	high	high	low	medium	adjustable	2

, are addressed accordingly in this paper.

The meta-heuristic MPPT approaches, such as PSO, GWO, ACO, ABC, etc., are powerful and have gained a significant audience, demonstrating a unique ability to track GMPP under PSCs. However, they face challenges such as computational and implementation complexity, which increase processing time and complicate real-time implementation. Secondly, due to the large number of iterations required, the convergence time is slowed. Thirdly, oscillation may occur even after tracking the GMPP, resulting in a reduction in efficiency and system stability.

The proposed hybrid IMCA-LAE algorithm is designed to tackle these issues by minimizing the computational complexity, lowering the number of iterations for faster convergence, and mitigating oscillation at steady state. Partial shading effects, particularly moving clouds, remains a crucial challenge despite significant advancements in MPPT techniques.

The main contributions of this study are as follows:

• The implementation of a novel hybrid IMCA-LAE algorithm for MPPT. Examination of the concept of exploration and exploitation at the start and end of the optimization process, as well as the dynamic adaptive adjustable nature of LAE.
• Minimizing the complexity, convergence time, and oscillation at steady state and maximizing the accuracy of the proposed IMCA-LAE hybrid MPPT technique.
• Comparative analysis of INC, P&O, MCA, and GWWA with the proposed IMCA-LAE technique regarding five testing parameters: global maximum power point, convergence time, mismatch power loss, efficiency, and fill factor.

The remainder of this study is organized as follows. Section 2 presents the mathematical modeling. Section 3 covers the proposed IMCA-LAE technique. Section 4 captures the results and discussion. Lastly, a conclusion is made in Section 5.

2. PV System Architecture

Mathematical Modeling of PV Cells

Modeling a PV system is typically considered a method for estimating output power and analyzing its performance under different environmental circumstances. The PV cell, as the smallest unit in a PV system, converts energy from sunlight into electricity. Mathematical modeling of PV cells is significant because it enables the analysis of various PV cell characteristics under different environmental conditions. Accurately modeling a PV system with appropriate parameters is essential; failure to do so will result in unpredictable I-V and P-V characteristic curves.

In literature, one-diode, two-diode, and three-diode model have been used. This paper employed a one-diode model, which is simple and dependable, consisting of a single diode placed in parallel with the linear independent current source $I_{PV}$ which is obtained using Equation (1), a resistor in series $R_s$, and another in parallel ${ R}_P$, as illustrated in Figure 1. Table 1 describes the PV panel used in this study and its specifications. There are fewer parameters in the single diode model.

$I_{PV}$ is obtained using Equation (1), where $I_D$ is the diode saturation current (A), $I_P$ is the current (A) entering the parallel resistor ${ R}_P$, and I is the output current (A).

$$I_{PV}=I_D+I_p+I$$

(1)

$I_{PV}$ can also be obtained using Equation (2) as a function of incident solar irradiance and the cell’s temperature value, where ${\mathrm{I}}_{\mathrm{S}\mathrm{C}}$ is the short circuit current (A), ${\mathrm{K}}_{\mathrm{i}}$ is the short circuit coefficient, G is the incident solar irradiance ($\mathrm{W}/{\mathrm{m}}^2$), $G_{STC}$ is solar irradiance under standard test conditions (STCs) ($\mathrm{W}/{\mathrm{m}}^2$), T is the temperature, and $T_{STC}$ is the temperature under STCs.

$$I_{PV}=[I_{sc}+K_i(T-T_{STC}\left)\right]{\displaystyle \frac{G}{G_{STC}}}$$

(2)

The PV cell’s maximum output current (I) is obtained using Equation (3), where $I_{PV}$ is the photo-produced current (A), $I_0$ represents the diode saturation current (A), $q$ stands for electron charge $1.60217646\times {10}^{-19}$ C, V symbolizes the voltage across the cell (V), n is the diode ideality factor, k is the Boltzmann constant $1.389\times {10}^{-23} \mathrm{J}/\mathrm{K}$, T serves as the cell’s working temperature (K), and $R_s$ is the series resistance (Ω).

$$I=I_{PV}-I_0\left(exp{\displaystyle \frac{q\left(V+R_{S}I\right)}{nkT}}-1\right)-{\displaystyle \frac{\left(V+R_{s}I\right)}{R_S}}$$

(3)

Equation (4) is used to obtain the diode current $I_D$, where $I_{0,STC}$ signifies the saturation current under STC, $T_{STC}$ is the temperature under STC (K), T is the temperature (K), and $E_g$ is the band-gap energy of the semiconductor material, which is obtained using Equation (5).

$$I_D=I_{0,STC}{\left({\displaystyle \frac{T_{STC}}{T}}\right)}^{3}exp\left[{\displaystyle \frac{qEg}{nk}}\left({\displaystyle \frac{1}{T_{STC}}}-{\displaystyle \frac{1}{T}}\right)\right]$$

(4)

$$E_g=1.16-7.02\times {10}^{-4}\left({\displaystyle \frac{T^2}{T-1108}}\right)$$

(5)

$I_{0,STC}$ is obtained by using Equation (6), where $I_{SC}$ is the short circuit current, $V_{OC}$ is the open circuit voltage, and $V_t$ is the cell’s thermal voltage (V).

$$I_{0,STC}={\displaystyle \frac{I_{SC}}{\exp \left(\raisebox{1ex}{$V_{OC}$}\!\left/ \!\raisebox{-1ex}{$n{V}_t$}\right.\right)-1}}$$

(6)

The cell’s thermal voltage $V_t$ is actualized by employing Equation (7), where q represents the electron charge $1.60217646\times {10}^{-19}$ C, T is the working temperature of the cell (K), and k is the Boltzmann constant $1.389\times {10}^{-23} \mathrm{J}/\mathrm{K}$.

$$V_t={\displaystyle \raisebox{1ex}{$kT$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}$$

(7)

3. Methodology

3.1. Design Parameters

The PV system used in this study is off-grid, meaning it is not connected to any electric grid. The system comprises a PV array, each panel of which has the specifications shown in

Table 3. Details of the Aavid Solar ASMS-225M PV module.

Parameter	Value
Description	225 W monocrystalline module
Maximum power	225 W
Open circuit voltage $V_{oc}$	37 V
Short circuit current $I_{sc}$	8.17 A
Voltage at maximum power point $V_{mp}$	30 V
Current at maximum power point $I_{mp}$	7.5 A
Diode saturation current $I_D$	1.1344 × 10−10 A
Diode ideality factor	0.96161
Shunt resistance $R_p$	106.8072 $\Omega$
Series resistance $R_s$	0.34059 $\Omega$

, a DC-DC boost converter, and a load. The total-cross-tied (TCT) PV array connection method is used, which consists of a combination of 4 × 4 PV modules connected both in series and in parallel to extract the highest possible voltage and current under the existing PSCs, as shown in Figure 2. The PV array topology is the interconnection of PV modules for maximum power generation. The major topologies are series (S), parallel (P), series-parallel (SP), bridge-linked (BL), honeycomb (HC), and total-cross-tied (TCT). The S configuration is simple to implement, with no ties between the modules, resulting in minimal wiring and a higher open-circuit voltage output, but with a lower short-circuit current that remains the same across the entire module. It is more effective in small-scale PV arrays. The P configuration is also simple to implement, with strings of PV modules connected in parallel. It is slightly affected by PSCs compared to the S configuration, achieving the highest output current while its voltage is the lowest. However, this is not visible in practice. The SP configuration is commonly used, with strings of PV modules connected in series and linked in parallel. It is affected by distributed PSCs, is prone to power mismatch, and minimizes the overall performance of the PV system. The TCT uses the SP structure but with interconnection across all the nodes. Cross ties across the nodes provide an alternative route for current to flow under PSCs. It is considered the best topology under varying irradiance. The BL topology is similar to the TCT but with fewer interconnections between the nodes, and fewer wires are used. It is less effective under PSCs, resulting in low output power generation. The HC interconnection is similar to a hexagon in structure, with fewer wires connected across the nodes compared to TCT. However, its output power generation under PSCs is affected. Among the mentioned topologies, TCT is the most effective in terms of output power generation.

The total voltage and current are received at the terminal of the PV array, and by applying Ohm’s law, the power is obtained. The current and voltage received at the PV array terminal are fed into the DC-DC boost converter to increase the DC output voltage, which is fed into the load. The MPPT is used to generate the duty cycle that controls the output voltage and, by extension, the output power.

As part of the design parameters, the solar panel type used and its descriptions are shown in Table 3. It is important to note these parameters before venturing into the design of the PV system.

It is important to determine the parameters before designing the DC-DC boost converter for effective performance and to avoid system failure. Continuous conduction mode (CCM) is utilized due to the close relationship between the parameters and the duty ratio. To obtain the DC ($V_{DC}$) output voltage, Equation (8) [31] is utilized.

$$V_{DC}={\displaystyle \frac{V_{PV}}{(1-d)}}$$

(8)

where ${\mathrm{V}}_{\mathrm{P}\mathrm{V}}$ is the PV array voltage, and d is the boost converter’s duty cycle, which shows the percentage of time the switch is turned on. The minimum inductance L required to operate the boost converter is shown in Equation (9) [31].

$$L\ge {\displaystyle \frac{V_{DC}}{2f_s.I_{PV}}} d(1-d)$$

(9)

where ${\mathrm{f}}_{\mathrm{s}}$ is the switching frequency, and ${\mathrm{I}}_{\mathrm{P}\mathrm{V}}$ is the terminal PV array output current.

To reduce fluctuation in the PV array’s output voltage, the input capacitance ${\mathrm{C}}_{\mathrm{i}\mathrm{n}}$ is utilized, as shown in Equation (10) [31].

$$C_{in}\ge {\displaystyle \frac{d_{max}}{8L.∆V_{mp}.f^2}}$$

(10)

where ${\mathrm{d}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ denotes the maximum duty ratio, and ${∆\mathrm{V}}_{\mathrm{m}\mathrm{p}}$ is the MPPT voltage ripple factor.

To guarantee that the DC link’s ripple voltage factor is as low as possible, the minimal capacitor is determined by Equation (11) [31].

$$C_{out}\ge {\displaystyle \frac{I_{PV}}{f_s{.V}_{DC}.∆V_{DC}}}d(1-d)$$

(11)

where ${\mathrm{I}}_{\mathrm{P}\mathrm{V}}$ is the PV array input current, and ${∆\mathrm{V}}_{\mathrm{D}\mathrm{C}}$ is the ripple voltage factor in the DC link.

The required parameters are first determined to enhance the effectiveness of the PV system. The PV panel parameters are shown in Table 3, and

Table 4. DC-DC boost converter design parameters.

Design Parameter	Value
PV array input voltage $V_{vp}$	120 V
Boost converter output voltage $V_{DC}$	350 V
Voltage $\Delta V_{DC}$ and $\Delta V_{mp}$	1%
Switching frequency	10 MHz
Inductor $L$	2.660 × 10−5 A
Input capacitor $C_{in}$	2.350 × 10−4 A
Output capacitor $C_{out}$	3.132 × 10−10 A
Simulation time $t_s$	0.1 s

contains the design parameters for the DC-DC boost converter. The ${\mathrm{V}}_{\mathrm{D}\mathrm{C}}$, L, ${\mathrm{C}}_{\mathrm{i}\mathrm{n}}$, and ${\mathrm{C}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ parameters in Table 3 were obtained by employing Equations (8)–(11).

3.2. Performance Parameter

All configurations’ performance is assessed through the calculation of criteria, including mismatch power loss ${∆\mathrm{P}}_{\mathrm{M}\mathrm{P}\mathrm{L}}$, fill factor (FF), and efficiency (η). The mismatch power loss ${∆\mathrm{P}}_{\mathrm{M}\mathrm{P}\mathrm{L}}$ is the ratio of the difference between the output power under no-shading conditions and the power under PSCs to the power under no-shading conditions, multiplied by 100 [62]. It is computed using Equation (12).

$$∆P_{MPL}={\displaystyle \frac{P_{UN}-P_{PSC}}{P_{UN}}}\times 100$$

(12)

where ${\mathrm{P}}_{\mathrm{U}\mathrm{N}}$ denotes the maximum output power under no-shading conditions, and ${\mathrm{P}}_{\mathrm{S}\mathrm{C}}$ is the maximum output power under PSC.

The fill factor (FF) is a crucial component in determining the PV array’s potential maximum power during PSCs [63]. A high FF value suggests that the PV array is effective. It simply calculates the ratio of global maximum power to the product of open circuit voltage and closed circuit current. To obtain the fill factor FF, Equation (13) is employed.

$$FF={\displaystyle \frac{P_{GMP}}{V_{OC}\times I_{SC}}}$$

(13)

where ${\mathrm{P}}_{\mathrm{M}\mathrm{P}\mathrm{L}}$ is the maximum power generated by the PV module or array, ${\mathrm{V}}_{\mathrm{O}\mathrm{C}}$ is the open circuit voltage, and ${\mathrm{I}}_{\mathrm{S}\mathrm{C}}$ is the short circuit current.

The capacity of a PV module to transform available solar radiation into electricity output is known as efficiency [64]. It can be obtained by taking the ratio of the global maximum power to the irradiance input power and the area of the PV module. Equation (14) is employed to calculate the efficiency.

$$\eta ={\displaystyle \frac{P_{GMP}/A_{PV}}{P_{IN}}}\times 100$$

(14)

where ${\mathrm{P}}_{\mathrm{G}\mathrm{M}\mathrm{P}}$ is the global maximum output power from the PV module or array, A_PV is the area of the PV module ($1.64 {\mathrm{m}}^2$) (which can also be derived by simply multiplying the length (1650 mm) and width (994 mm) of the PV module from the data sheet), and ${\mathrm{P}}_{\mathrm{I}\mathrm{N}}$ is the input power solar irradiance (1000$\mathrm{W}/{\mathrm{m}}^2$ at STC) received at the surface of the PV module.

3.3. The Improved Musical Chairs Algorithm (IMCA)

Eltamaly presented the musical chairs algorithm (MCA) as a novel MPPT algorithm in [31], and it was used in [65] to estimate PV cell parameters. The MCA mimics the popular musical chairs game, in which several players and chairs are arranged in a circle, with the number of players exceeding the number of chairs by one. The players go around sets of chairs at various intervals while music is played until the music stops, and each player tries to grab a chair to sit on. The player unable to secure a chair is considered the loser and eliminated from the game, with one chair removed for the next stage. The procedure continues until a winner is determined, who then sits in the last available chair. This concept is mathematically expressed in the MCA, where each participant signifies one solution, and the chair implies the most accurate solution next to each player. The principle facilitating the IMCA may be categorized into three sections: initialization of parameters, determination of the ideal solution, and evaluation of the region of interest [31]. In the proposed IMCA, numerous search agents are utilized in the initial exploration stage, and the number steadily decreases to enhance exploitation, eliminating inadequate results until an accurate and precise solution is obtained. This results in a fast convergence time, high accuracy, and the elimination of oscillations at steady state.

In the MCA, each player’s position is determined by their proximity to the nearest chair. The mathematical equation used is typically

$$P_i\leftarrow C_{nearest}$$

(15)

where $P_i$ is the new position of the $i^{th}$ player, and $C_{nearest}$ is the position of the nearest chair.

To calculate $C_{nearest}$, the equation finds the chair that minimizes the distance between the player’s position and each chair’s position:

$$C_{nearest}={argmin}_j|P_i-C_j|$$

(16)

where $C_j$ is the position of the $j^{th}$, and argmin finds the index j of the chair with the smallest absolute difference.

For each player, $P_i$, compute the distance to every chair $C_j$. The span that separates the $i^{th}$ player and $j^{th}$ chair $D_{ij}$ is given as

$$D_{ij}=|P_i-C_j|$$

(17)

To identify the chair with the smallest distance, known as the nearest chair index (NCI)

$$Nearest chair Index={argmin}_j{D}_{ij}$$

(18)

Fitness Evaluation: Power values at player positions are evaluated using the power obtained from the generated instantaneous PV array current and voltage.

Best Power Update: If the current iteration’s best player has higher power than the previous best, update the best power and position.

3.4. Local Adaptive Exploration (LAE)

The local adaptive exploration (LAE) stage is introduced to improve the online tracking capability of the proposed IMCA-based MPPT controller without requiring an exhaustive duty-cycle scan. In contrast to a classical brute-force search, which evaluates all possible solutions over a predefined search space [66,67], the proposed LAE stage evaluates only a small neighborhood around the current best duty cycle. This makes the method suitable for online implementation because the controller does not require prior offline knowledge of the complete PV power curve.

When the change in measured power remains below a predefined threshold for several consecutive iterations, the controller assumes that the search process has stagnated near the current operating region. A set of local candidate duty cycles is then generated around the current best duty cycle as

$$D_{LAE}\left(k\right)=clip \left(d_{ best}\right(k)+△d)$$

(19)

where:

$$△d\in \{-0.05,-0.02, 0, 0.02, 0.05\}$$

(20)

and the clipping function limits the duty cycle to the feasible range [0, 1]. One candidate is selected from this local neighborhood and applied to the PWM generator in the next sampling period. If the measured PV power improves, the best duty cycle and best power are updated. Therefore, the LAE stage provides local perturbation and diversity around the current best solution while avoiding the computational burden and real-time infeasibility.

3.5. The Computational Complexity and Feasibility of the IMCA-LAE

The complexity is analyzed with respect to time and space. The real-time feasibility of the IMCA-LAE is also considered, beginning with parameter initialization. The number of players can also be written as shown in Equation (21).

$$N_p=N_c+1$$

(21)

where $N_C$ is the maximum number of iterations, and $N_P$ is the number of chairs.

3.5.1. Time Complexity Analysis

The chair–player assignment and the local adaptive exploration stage determine the computational complexity of the proposed IMCA-LAE method. In each iteration, the IMCA assigns players to their nearest chairs, resulting in a computational cost, as shown in Equation (22).

$$O\left(N_p{N}_c\right)$$

(22)

where $N_P$ is the number of players and $N_C$ is the number of chairs. The fitness evaluation and best-solution update require

$$O\left(N_p\right)$$

(23)

When stagnation is detected, the LAE stage generates a fixed number of local duty-cycle candidates

$$K=5$$

(24)

Therefore, the LAE stage has constant complexity.

The fitness evaluation and best-solution update require a linear scan over the players, leading to a complexity as depicted in Equation (24), which is negligible compared to the chair assignment process because K is fixed and independent of the PV array size or duty-cycle resolution.

$$O\left(k\right)=O\left(1\right)$$

(25)

Over T iterations, if the LAE stage is triggered every B iterations, the total complexity becomes

$$O(T(N_p{N}_c+N_p)+{\displaystyle \frac{T}{B}}k)$$

(26)

Since K = 5, $N_p$ and $N_c$ are small in the implemented controller. Therefore, the proposed IMCA-LAE method has low computational cost and is more suitable for online MPPT implementation than a full brute-force duty-cycle scan.

3.5.2. Space Complexity Analysis

The memory requirement is limited to storing the player positions, the chair positions, the best duty cycle, the best measured power, and a few scalar control variables. Since no complete PV power array is stored, the space complexity is

$$O(N_p+N_c+K)$$

(27)

With fixed K = 5, this becomes

$$O(N_p+N_c)$$

(28)

3.5.3. Real-Time Implementation Feasibility

The proposed IMCA-LAE algorithm is computationally efficient and well-suited for real-time MPPT applications. The number of chairs ${\mathrm{N}}_{\mathrm{C}}$ is typically small (5–15), and the algorithm relies exclusively on simple arithmetic operations such as subtraction, comparison, and indexing. The brute-force assistance is invoked intermittently and does not impose a continuous computational burden. Moreover, IMCA-LAE does not require gradient computation, matrix operations, or complex probability distributions, which significantly reduces execution time and numerical instability. For typical MPPT configurations ($\mathrm{I}\le 100$, ${\mathrm{N}}_{\mathrm{C}}=10$, $\mathrm{B}=20$), the algorithm converges well within a single MPPT sampling period (≤0.1 s), as demonstrated in the simulation results. IMCA-LAE is therefore thought to be very appropriate for real-time implementation on inexpensive microcontrollers, DSPs, and FPGA-based MPPT systems. Nevertheless, no hardware experiment or hardware-in-the-loop (HIL) was carried out to support the assertion.

Unlike conventional metaheuristic MPPT techniques that rely solely on probabilistic convergence, the proposed IMCA-LAE combines fast local exploitation with deterministic brute-force verification. This hybrid strategy ensures GMPP tracking under simulated partial shading conditions while maintaining low computational complexity and negligible steady-state oscillations. As shown in

Table 5. Computational complexity comparison of MPPT algorithms.

MPPT Algorithm	Time Complexity	Space Complexity	GMPP Guarantee	Steady State Oscillation	Real-Time Suitability
P&O	$O\left(I\right)$	$O\left(1\right)$	No	High	High
INC	$O\left(I\right)$	$O\left(1\right)$	No	Medium	High
PSO	$O(I.N_p)$	$O\left(N_p\right)$	Probabilistic	Medium	Moderate
GWO	$O(I.N_p)$	$O\left(N_p\right)$	Probabilistic	Low	Moderate
WOA	$O(I.N_p)$	$O\left(N_p\right)$	Probabilistic	Low	Moderate
MCA	$\mathrm{O}(\mathrm{I}.{N_c}^2)$	$O\left(N_c\right)$	Partial	Low	High
IMCA-LAE (Proposed)	$\mathrm{O}(\mathrm{T}({\mathrm{N}}_{\mathrm{p}}{\mathrm{N}}_{\mathrm{c}}+{\mathrm{N}}_{\mathrm{p}})+{\displaystyle \frac{\mathrm{T}}{\mathrm{B}}}\mathrm{k})$	$\mathrm{O}({\mathrm{N}}_{\mathrm{p}}+{\mathrm{N}}_{\mathrm{c}}+\mathrm{K})$	Validated under simulated PSCs	Very low	High

, the proposed IMCA-LAE achieves guaranteed GMPP tracking with lower oscillation and competitive computational complexity compared to population-based meta-heuristic MPPT algorithms.

3.6. Algorithm and Flowchart

The proposed IMCA-LAE algorithm, as shown in

Table 6. The proposed IMCA-LAE implementation process.

Step	Algorithm Stage	Description/Operation
1	Start	Begin IMCA-LAE MPPT process
2	Input	Vpv (PV voltage), Ipv (PV current)
3	Parameter initialization	Set number of chairs Nc, players Np = Nc + 1, maximum iterations MaxIter, local adaptaive exploration interval LAE_int, Initialize players di ∈ [0, 1], Initialize chairs cj ∈ [0, 1], Set dbest = d1, Pbest = −∞, iter = 0, stagnationCount = 0, Pprev = 0
4	Main procedure	Measure Vpv and Ipv, Compute P = Vpv × Ipv, iter = iter + 1
5	IMCA assignment	For each player i:      - Find nearest chair j*      - di = cj*
6	Evaluation	dcurrent = dbest, Pcurrent = P If Pcurrent > Pbest:      - Update Pbest and dbest
7	Exploitation:	di = dbest + 0.02 × randn   Limit di ∈ [0, 1]
8	Stagnation check	If |Pcurrent − Pprev| < 1 × 10−4:      - stagnationCount++     Else:      - stagnationCount = 0 Pprev = Pcurrent
9	Local adaptive exploration (LAE)	If stagnationCount > 10:      - Generate candidates around dbest      - Limit candidates to [0, 1]      - Select one randomly      - Update dbest      - Reset stagnationCount
10	Exploration enhancement	Randomly reinitialize chairs
11	Output	d = dbest Return d

, begins by indicating the inputs in step 2. Subsequently, step 3 involves the initialization of the search agents, number of chairs and players, chair positions, best duty cycle, and observing variables during the first execution. In step 4, at each sampling time, the instantaneous voltage and current are read from the PV array to determine the power by simply multiplying current by voltage. In step 5, IMCA begins the player-to-chair assignment, with each player representing a candidate duty circle position. The instantaneous measured power is compared with the best previously recorded power, and the best duty cycle is updated whenever there is an improvement under evaluation in step 6. In step 7, a local exploitation step then produces new player placements around the current best duty cycle using a modest random perturbation. In step 8, stagnation is checked to determine if the change in power is extremely small. If the difference is less than the threshold, the controller assumes there is no significant change and concludes that the search is stagnant. The algorithm keeps an eye on the power change between iterations to prevent premature convergence. In step 9, an LAE-inspired adaptive exploration stage is triggered if the power variation stays below a predetermined threshold for multiple repetitions. During this stage, neighboring candidate duty cycles are created, and one is chosen at random. In step 10, the optimal duty cycle is then applied to the PWM controller for the subsequent sample period, and the chair positions are reinitialized to increase diversity.

The summarized proposed IMCA-LAE MPPT process can be seen in the flowchart shown in Figure 3.

4. Results and Discussion

The structure of the TCT-connected PV array, which is simulated in MATLAB/SIMULINK R2021a, is shown in Figure 4, with a total of 16 PV panels. Each panel comprises two input ports for both solar irradiance and temperature, respectively. Two spreadsheet blocks are utilized to extract the dynamic solar irradiance and temperature and transmit them to each panel. A bypass diode is attached to each panel to tackle issues of hotspots under PSCs; the diode provides a pathway for the current to bypass the affected PV panel to prevent the effect from affecting the overall current in series along that column. The total current and voltage are collected at the two terminals of the entire PV array; subsequently, the power is obtained by multiplying the current and voltage. The PV array current and voltage are fed into the algorithm via a MATLAB function block as inputs. The output from the algorithm is linked to the pulse width modulation (PWM) generator DC-DC to obtain the duty cycle, which is fed into the gate (g) port of the insulated gate bipolar transistor (IGBT)/diode. Through the collector (c), the IGBT is connected to the positive part of the PV system, while the emitter (e) is connected to ground. Figure 4 depicts the PV system model, DC-DC converter, and IMCA-LAE MPPT control.

Under uniform conditions, Figure 5a shows the relationship between the current and voltage with maximum values of 30 A and 119.924 V, respectively. Figure 5b demonstrates the power-to-voltage relationship via the characteristic curve, with the maximum power at 3600 W. Figure 5c shows the convergence time of 0.0824 s at which the maximum power is actualized under uniform irradiance of $1000 \mathrm{W}/{\mathrm{m}}^2$, having just one peak, with the temperature at a standard test condition (STC) of 25 °C.

Figure 5d shows the DC input power and the maximum voltage generated by the PV array at 120 V. The effect of the DC boost converter is shown in Figure 5e, where the voltage increases from 120 V to 350 V, indicating that the duty ratio $d$ is approximately 65%.

Due to PSCs, the solar irradiance is not uniform throughout the PV array. To demonstrate the effect of PSCs, mainly due to passing clouds, a dynamic PV temperature and solar irradiance are used over a time frame, as shown in

Table 7. Dynamic temperature under PSCs.

Time	T1	T2	T3	T4	T5	T6	T7	T8	T9	T10	T11	T12	T13	T14	T15	T16
0	25	50	−2	22	35	11	25	41	−14	25	25	3	25	36	25	0
0.02	30	0	66	14	25	25	−5	12	16	6	25	45	9	43	4	−12
0.05	−10	20	9	1	−7	22	47	25	8	34	25	−3	34	28	33	3
0.1	2	25	15	40	18	9	13	32	26	11	25	22	30	30	16	7

. The column with headings T1 to T16 depicts the dynamic temperature under PSCs for each of the 16 PV arrays (4 × 4) over various time intervals, as shown in Table 7.

Table 8. Dynamic irradiance under PSCs.

Time	G1	G2	G3	G4	G5	G6	G7	G8	G9	G10	G11	G12	G13	G14	G15	G16
0	220	140	140	450	990	200	340	456	100	400	450	340	120	950	200	1000
0.02	100	940	450	560	321	340	220	400	200	300	200	320	420	230	300	100
0.05	250	650	350	910	245	110	120	890	1000	560	100	450	900	450	600	300
0.1	1000	800	600	340	100	300	500	900	450	200	400	130	100	600	100	400

depicts the dynamic solar irradiance under PSCs, where G1 to G16 indicate the individual solar irradiance as input to all the 16 PV arrays. The dynamic temperature and solar irradiance mimic a real-time scenario, although the data are self-generated.

Table 9. The GMPP in relation to the time of convergence.

MPPT Algorithm	Power (W)	Voltage(V)	Time (s)
IMCA-LAE	1504.77	122.178	0.08246
MCA	1464.26	122.781	0.08295
GWWO	1426.22	124.39	0.08402
INC	1395.43	125.284	0.08466
P&O	1375.43	125.953	0.08512

shows the output current, voltage, and power under dynamic solar irradiance and temperature conditions. Figure 6a,b show the effect of varying temperature and solar irradiance on the PV characteristic curve. Several contours can be seen representing local minimum points due to varying solar irradiance, as depicted in Figure 6a,b. In Figure 7a,b, the effect of temperature is observed, with high temperature causing the P-V and P-T curves to shift towards the left, lowering both the voltage and MPP, while low temperature causes the P-V and P-T curves to shift towards the right, raising both the operating voltage and the achievable peak power. In both cases, the position of the GMPP is affected. This is consistent with the thermal behavior of solar modules. Furthermore, aside from varying temperature and solar irradiance, aging due to material degradation, corrosion, or prolonged exposure to environmental stress can alter the electrical characteristics of PV modules over time, reducing output power.

The results in Figure 6a,b show the various MPPT techniques employed, the maximum output power generated, and the corresponding convergence time. Several local peaks emerge as a result of the partial shade effect and changing environmental parameters like temperature and irradiance. There is only one peak visible under uniform or no-shading conditions, as shown in Figure 5b,c. PSCs, on the other hand, produce several local maximum peaks with a single global maximum power point (GMPP) due to varying temperature and solar irradiance, as depicted in Figure 6a,b and Figure 7a,b. Therefore, the suggested IMCA-LAE is successfully implemented for precise GMPPT.

The proposed IMCA-LAE technique dominated in terms of both MPP and convergence speed, having the highest MPP and lowest convergence speed (of 0.08246 s) compared to other methods. Both figures have shown that some of the MPPT methods exceeded others in terms of MPP; however, their convergence time was lower. There is a sort of trade-off between MPP and convergence time in some cases.

The solar irradiance and temperature are considered two main factors affecting the output power of a PV system; they are also the two inputs to the solar panel. Here, the focus is on the effect of temperature.

Table 10. Temperature at STC (25 °C).

Time	T1	T2	T3	T4	T5	T6	T7	T8	T9	T10	T11	T12	T13	T14	T15	T16
0	25	25	25	25	25	25	25	25	25	25	25	25	25	25	25	25
0.02	25	25	25	25	25	25	25	25	25	25	25	25	25	25	25	25
0.05	25	25	25	25	25	25	25	25	25	25	25	25	25	25	25	25
0.1	25	25	25	25	25	25	25	25	25	25	25	25	25	25	25	25

shows the temperature at 25 °C STC for all 16 panels.

Table 11. High temperature above (25 °C).

Time	T1	T2	T3	T4	T5	T6	T7	T8	T9	T10	T11	T12	T13	T14	T15	T16
0	50	45	36	66	37	30	28	34	45	88	36	100	55	66	56	45
0.02	26	46	46	32	28	42	46	27	29	50	42	27	36	39	34	39
0.05	32	48	56	41	51	26	51	34	35	34	67	32	31	40	56	55
0.1	27	39	44	36	39	51	32	41	60	61	90	44	54	52	43	49

presents varying temperatures above the 25 °C STC.

Table 12. Temperature below (25 °C).

Time	T1	T2	T3	T4	T5	T6	T7	T8	T9	T10	T11	T12	T13	T14	T15	T16
0	15	10	11	3	2	4	22	17	18	15	12	14	11	10	21	5
0.02	10	10	10	10	10	10	10	10	10	10	10	10	10	10	10	10
0.05	5	7	23	0	15	20	7	19	11	8	0	6	21	17	11	23
0.1	6	18	3	7	9	19	12	21	14	3	13	20	19	15	9	1

indicates temperatures below the 25 °C STC. The impact of these temperature variations on output power, voltage, and convergence time is shown in

Table 13. The effect of temperature of P-V and P-T characteristic curves.

IMCA-LAE	Power (W)	Voltage (V)	Time (s)
STC temperature	1495.42	121.612	0.08218
High temperature	1353.58	113.398	0.07666
Low temperature	1514.42	122.381	0.08273

. Higher temperatures reduce the maximum output power and voltage, shifting the P-V and P-T characteristic curves backward. Conversely, lower temperatures below the 25 °C STC shift the maximum output power and voltage forward along the curves. Temperature variations can affect the current or voltage in a solar cell, leading to a drop in the net output power. As shown in Figure 7a,b, temperature variations can continuously shift the position of the MPP. Furthermore, aging due to material degradation, corrosion, or prolonged exposure to environmental stress can alter the electrical characteristics of PV modules over time, reducing output power.

As initially stated, the performance of the PV system is examined regarding five performance criteria, namely maximum output power, convergence time, mismatch power loss, efficiency, and fill factor, as shown in

Table 14. Performance criteria.

MPPT Algorithm	Power (W)	Time (s)	Mismatch Power Loss (%)	Efficiency (%)	Fill Factor (%)
IMCA-LAE	1504.77	0.08246	58.200	5.735	41.800
MCA	1464.26	0.08295	59.326	5.580	40.674
GWWA	1426.22	0.08402	60.383	5.435	39.617
INC	1395.43	0.08466	61.124	5.318	38.762
P&O	1375.43	0.08512	61.794	5.242	38.206

. The maximum output power (W) and convergence time (s) are obtained directly from the simulation, while the mismatch power loss, fill factor, and efficiency are obtained using Equations (12)–(14), respectively. The five performance criteria are further illustrated in Figure 8a–e. Table 14 column 2 and Figure 8a highlight the maximum generated power under PSCs for the different MPPT techniques, with the proposed IMCA-LAE technique having the highest maximum power, followed by MCA, with P&O having the least power. As shown in Figure 8b and Table 14 column 3, the IMCA-LAE technique had the fastest convergence time at 0.08246 s, followed by the MCA at 0.08295 s, and the slowest time was recorded for P&O at 0.08512 s. The mismatch power loss demonstrates the deviation of the power under PSCs from the power under standard uniform irradiance. As recorded in Table 14, column 4, and also in Figure 8c, IMCA-LAE has the lowest power loss at 58.2%, followed by MCA at 59.326%, with the highest mismatch at 61.794% for P&O. Efficiency is a metric that quantifies how well a photovoltaic system can produce the most power under PSCs. As shown in Table 14, column 5 and Figure 8d, IMCA-LAE has the highest efficiency at 5.735%, followed by MCA at 5.580%; P&O recorded the lowest efficiency at 5.242%. In an ideal case under uniform conditions, the FF is approximately 74.4%; however, with the effect of PSCs, the FF drops. Table 14, column 6 and Figure 8e show the FF for different MPPT algorithms under PSCs. The higher the FF, the better and more efficient the system will be.

Limitations

The proposed IMCA-LAE MPPT technique exhibited significant improvement in terms of convergence speed, reduced oscillation, GMPP tacking, low mismatch power, high efficiency, and fill factor. However, there are still some limitations to this research, as it focused only on MATLAB/Simulink simulation results without experimental or HIL validation, and also the simulation time was set to 0.1 s. Additionally, real-time implementation may be constrained by hardware limitations.

To further examine the dynamic behavior of the proposed controller, the duty ratio generated by the IMCA-LAE algorithm was recorded during the sharp irradiance transitions listed in Table 8. As shown in Figure 9, the duty ratio changes immediately after each irradiance transition, indicating that the MPPT controller actively adjusts the operating point of the boost converter in response to the changing PV characteristics. The corresponding PV power and converter output voltage responses show that the converter follows the duty-cycle command without instability or sustained oscillation. After each transition, the duty ratio settles around a new operating value, allowing the PV array to operate close to the tracked maximum power point. This confirms that the proposed controller is not only able to identify a high-power operating point, but also to dynamically regulate the converter under rapidly varying partial shading conditions.

5. Conclusions

The reduction in output power generation resulting from PSCs has necessitated exploring alternative methods for tracking the maximum power point in a PV system. Various MPPT methods have been proposed in the literature and fall into four categories: conventional, intelligent, metaheuristic, and hybrid. All the proposed methods are focused on MPPT; however, they have drawbacks such as steady-state oscillations, slow convergence, high computational cost, numerous control parameters, high complexity, and high implementation cost. It is important to deploy a method that is able to track the GMPP within the shortest time, with low or no oscillation at steady state, fewer control parameters, and less complexity. The IMCA utilizes the concept of the MCG, where certain number of players go around a set of chairs; at intervals, a player is eliminated, and a chair is removed, until a winner emerges. The LAE dynamically adjusts the perturbation size, improving the duty cycle around the best solution around the optimal operating point. This idea is applied in MPPT to increase the PV system’s efficacy. The proposed IMCA-LAE technique is benchmarked with four other algorithms: the MCA, GWWA, INC, and P&O techniques. From the results obtained, the proposed IMCA-LAE technique attained the highest output power, fastest tracking time, highest efficiency, and highest fill factor, as well as the lowest mismatch power loss and oscillation at steady state. The superiority of the proposed IMCA-LAE technique over alternative approaches demonstrated the technique’s effectiveness in PV system MPPT.

Future Work

The suggested hybrid IMCA-LAE technique showed its efficiency in MPPT, fast convergence time, and reduced complexity. However, there is still room for enhancement, including real-time data, experimental testing, or HIL validation, as well as comparison with other PV technology types.