Mod Tanh‐Activated Physical Neural Network MPPT Control Algorithm for Varying Irradiance Conditions

Abstract

ABSTRACT

The increasing adoption of solar photovoltaic systems necessitates efficient maximum power point tracking (MPPT) algorithms to ensure optimal performance. This study proposes a Mod tanh‐activated physical neural network (MAPNN)‐based MPPT control algorithm, which addresses inefficiencies in existing models caused by spectral mismatch and improper converter control. The proposed method incorporates beta‐distributed point estimation technique for mismatch factor correction and a Buck‐Boost converter with a feedback control using the Chinese Remainder Theorem – Puzzle Optimization Algorithm‐tuned PID controller. Simulations demonstrate an efficiency improvement of 98.42%, with a 4.54 dB reduction in total harmonic distortion and faster convergence compared to traditional methods such as ANN and LSTM. This system significantly enhances MPPT performance under dynamic irradiance conditions.

Introduction

Due to industrialization and construction, energy consumption has grown exponentially and largely. For larger energy amounts, high and continuous demand is created by this, and oils, natural gas, and coal are utilized for this purpose [1]. As per the estimation, soon these conventional sources will be almost exhausted, and it will be more and more difficult to guarantee enough supplies of these materials. In addition, pollution is caused by the exhaustion of gases from utilizing these fuels and leads to air pollution, ozone layer holes, global warming, and acid rain [2]. Thus, to overcome these issues, eco-friendly substitutes are needed. Thus, solar energy is preferred as one of the renewable energy sources, which can be harnessed through a PV system [3].

A combination of one or more solar panels (SPN) along with an inverter is termed PV. Mechanical and electrical hardware that is utilized for converting solar energy to electrical energy is also included by it [4]. Movements in the internal electric field of the cell causing electricity flow are made by electric charges, which are formed by the solar energy acquired from PV panels. When PV functions at its maximum power, named MPP, this PV panel remains nonlinear and has a specific optimal operating point [5]. The energy yielded from this PV panel is highly dependent on irradiance, load, and temperature. Over diverse times, these parameters generated an impact on MPP and assist in keeping the power maximal all the time [6]. Analysis of the PV system's energy conversion devices is required for maximizing solar energy capture. The converter is one of such main components. Strict requirements, namely, low output voltage ripple and high efficiency are placed at the converter design phase for ensuring maximum energy capture.

In a variety of locations, PV panels are operated and degradation over time is caused by these outdoor exposures of PV panels [7]. The PV panels' operation relies on weather conditions like temperature and sunshine; for understanding the PV panels' behavior, these phenomena's study is necessary. It is required to adapt the load with the PV generator by inserting a boost converter controlled by MPPT for extracting the maximum available PV power [8]. Diverse algorithms-centric MPPT types, namely, fuzzy logic control (FLC), particle swarm optimization (PSO), Perturb & Observe (P&O), and artificial neural network (ANN) have been utilized in the literature; however, these approaches suffer from total harmonic distortion (THD) [9–11]. Thus, a new approach has been proposed to overcome weather-negative impact and THD.

Machine-learning algorithms are increasingly applied to enhance MPPT algorithms, offering predictive capabilities and real-time adaptability. Yılmaz and Çorapsız proposed an ANN-based MPPT algorithm, showing its effectiveness in improving tracking accuracy and efficiency over traditional methods such as incremental conductance (INC) and P&O under various irradiance conditions. Their results underscore the potential of ANN models in overcoming the limitations of conventional approaches [12].

So far, several techniques have been proposed for MPPT control in changing weather conditions although they go through certain limitations that can be enlisted as: the tracking direction in a dynamic environment is lost, clouds influenced the PV panels' efficiency and the daily efficiency gets degraded; thus, spectral mismatch, temperature, and angle of-incidence effects are caused; and the slow transient response led to inaccurate operation because of the converter's improper control during the fast-changing atmospheric conditions, which is considered as a major drawback. A new technique has been proposed for overcoming these challenges and contributing to the succeeding purposes. The remainder of the article is organized as follows: the related works and their limits are examined in Section 2, the proposed control system is described in Section 3, the investigational results are propounded in Section 4, and lastly, the article is wrapped up in Section 5.

Literature Review

Numerous studies have explored methods to improve maximum power point tracking (MPPT) in photovoltaic systems, focusing on reducing power losses and enhancing efficiency. Sarang et al. compared traditional and AI-based MPPT techniques, demonstrating higher accuracy and faster response times with AI methods like PSO, Fuzzy Logic, and ANN, which achieved accuracies of over 98.8% [13]. Similarly, Bano et al. highlighted multi-level inverters combined with advanced control strategies, such as the APOD and POD, to mitigate THD, showcasing their potential in integrating PV systems with minimal losses [14].

Raj and Praveen [15] propounded a highly effective DC–DC Boost converter destined for utility-level PV models. Among other factors that minimized the oscillations close to MPPT utilizing the Advanced Perturb & Observation (APO) MPPT algorithm, it considered variations from shadows, ambient temperature, and inconsistencies of sunlight availability. An elevated efficiency of 3.21% is acquired by it; yet it could not determine while the algorithm detects the maximal power point.

Abouadane et al. [16] established the MPPT tracking approach, which utilized three consecutive measurements. The power difference between each of the two consecutive samples was analogized; moreover, it observed the voltage variation between the last two successive samples. These comparisons' outcomes exhibited that the approach implements the appropriate action like either decreasing or increasing the voltage. It acquired 98.6% efficiency. However, the quick variations were unable to be handled in dynamic conditions. The use of optimization algorithms in power electronics and renewable energy systems has gained significant attention. Ali et al. proposed hybrid evolutionary algorithms for solving multi-objective security-constrained unit commitment problems, integrating cost optimization, energy loss minimization, and voltage deviation [17]. The findings underline the relevance of hybrid evolutionary methods in addressing complex, nonlinear optimization challenges.

Previous studies [18, 19] introduced a hybrid control model utilizing P&O and incremental conductance control by the graphical approach via meteorological data variation in the power supply context. For varying solar irradiation cases, it tracked MPPT utilizing active extraction of solar power in PV models. P&O exhibited superior performance in varying atmospheric conditions, yet the estimated energy output was low.

Siddique et al. [20] established a codesign approach that optimized design parameters for PV models. Because of the optimized duty cycle, it examined the boost converter's output voltage succeeding the reference voltage without any surge. A voltage source control (VSC) inverter was utilized to control the amplitude and stabilize the converted AC voltages' frequency. It acquired stable output power in the PV model. However, it rendered a few periodical cut-offs. Habib et al. applied the improved Whale Optimization Algorithm (IWOA) to automatic voltage regulators, emphasizing transient response and stability enhancement in power systems. The proposed approach demonstrated superior performance over traditional tuning methods, achieving faster settling times and reduced overshoots [21].

Ali et al. [22] presented a dynamic multi-objective optimization approach for grid-connected distributed resources (DR), incorporating Battery Energy Storage Systems (BESS) to enhance grid reliability and minimize voltage deviations under variable renewable energy conditions. The study highlights the synergistic benefits of combining distributed generation with advanced optimization algorithms.

Vaikundaselvan et al. [23] propounded P&O grounded MPPT algorithm for the solar system. The variation in maximum power led by the deviations in the weather conditions was tracked. It utilized a boost-type DC-DC converter as well as an MPPT control approach for activating the PV model. It doesn't correct atmospheric variations although it acquired 17.20% cell efficiency.

Hamouda et al. [24] established an adaptive neuro-fuzzy inference system (ANFIS)-centered MPPT controller joint with a PSO approach for tracking the PV generator's MPP. The MPP was recognized and tracked, and for an autonomous PV wire feeder unit, an accurate and fast response was provided. However, the predicting network resulted in convergence to a local minimum owing to the difficulty of training complex issues.

Ahmed et al. [25] explored a two-level inversion phase and filter-centered MPPT control approach. It can switch between the MPPT operation as well as the Constant Power Generation (CPG). To improve the PV model's transient behavior, the DC-link PI controller was replaced with an adaptive DC-link controller. A fast transient response was provided. However, in fast-changing atmospheric conditions, inaccurate operation was provided.

Previous studies [26, 27] introduced a multi-period approach for optimal Distributed Generation (DG) placement and network reconfiguration, achieving over 22% cost savings and 80% improvement in voltage deviation. The integration of DGs with grid optimization ensures efficient power distribution, especially in renewable-based systems.

Gonzalez-Castano et al. [28] propounded the Artificial Bee Colony (ABC) algorithm's use for the PV system's MPPT utilizing a DC–DC converter. Utilizing data values from the PV module, the ABC MPPT algorithm's procedure initiates; subsequently, it identified P–V characteristics and chose the optimal voltage. Next, the MPPT plan was applied and the voltage reference for the outer PI control loop was acquired, which in turn provided the current reference to the predictive digital current programmed control. Multi-peak I–V curves led by the bypass diode activation in partial shadowing operating conditions can't be dealt with.

Previous studies [29, 30] intended FLC for MPPT control, which aimed to control nonuniform and uniform changes in weather conditions. List of test cases, encompassing Partial Shading Condition (PSC), and load variation for catering as many of the realistic weather changes that could occur during the PV plants’ normal operation are extended. However, owing to lack in the conditions, it had weak reliability. Partial shading significantly impacts PV array performance by causing power losses and multiple peaks in the power–voltage (P–V) curve. To address this, Yousri et al. introduced the Argyle Puzzle reconfiguration method for PV arrays, which effectively disperses shading across sub-arrays. This novel approach consolidates power peaks into a single global maximum, enhancing the system's efficiency under PSCs [31].

Gundogdu et al. [32] established Adaptive Reference Voltage centered System Identification for a PV model under uniform irradiance. The acquired reference voltage has a polynomial structure. Assuming that the PV model was a nonlinear black-box type model, this polynomial system has been acquired. Subsequently, the PV model's mathematical system in polynomial structure providing the input–output relationship was acquired by utilizing these data with SI Toolbox. It remains stable for change in irradiance although it acquired better efficiency.

MPPT Control Approach in Solar PV for Varying Weather Conditions

Proposed Model

For domestic and industrial purposes, PV model usage has elevated dramatically; however, researchers have looked for ways for elevating their efficiency owing to their lower efficiency. In this study, a system for spectrum mismatch correction, improved converter control, and deep learning-centric MPPT control has been proposed for this purpose. In Figure 1, the proposed model's block diagram is displayed.

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Solar PV Module

PV cells convert light energy into electrical energy using semiconductor materials. When exposed to light, these materials absorb energy, exciting electrons and creating an electrical current. PV panels, typically mounted on roofs or the ground, provide electricity for various applications. Series and parallel resistances and a single-diode current source are comprised in the single-diode PV system. Generally, for preventing the backflow of current (unidirectional flow of current), the diodes are wielded in the PV cell; also, they are utilized as blocking devices. In addition, they are utilized as bypass devices for maintaining the whole solar power system's reliability in the event of a solar panel failure. Thus, the proposed technique uses the single-diode model, which is otherwise named the five-parameter model; also, its design comprises a parallel connection of the ideal diode and current source with bypassed shunt resistance. For enhancing the PV systems' performance, single-diode model solar cell parameters could be examined efficiently. Concerning the parameters, namely, minimum error I–V and P–V curves, the single-diode model has more benefits.

In Figure 2, the PV module's equivalent circuit is displayed.

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Output current formed under solar radiation and temperature is shown below, 1$I_{\mathrm{pho}}=I_{\mathrm{pv}}-I_{\mathrm{dio}}-I_r$ 2$I_{dio}=I_{\mathrm{revv}}\left(\text{exp}\left(\frac{e(V_{\mathrm{pv}}+R_{s\left(r\right)}{I}_{\mathrm{pho}})}{i_f{T}_{\mathrm{temp}}{K}_{\mathrm{Bz}}}\right)-1\right)$ 3$I_r=\left(\frac{V_{\mathrm{oc}}+I_{\mathrm{res}}{C}_{\mathrm{sc}}}{R_{\mathrm{shun}}}\right)$

Here, the current formed by the incident light is depicted as $I_{\mathrm{pho}}$, the PV current generated is signified as $I_{\mathrm{pv}}$, the diode current is depicted as $I_{\mathrm{dio}}$, the diode's reverse saturation current is given as $I_{\mathrm{revv}}$, shunt resistance is depicted as $R_{\mathrm{Shun}}$, the series resistance is indicated as $R_{s\left(r\right)}$, the diode ideality factor is depicted as $i_f$, the charge of an electron is depicted as $e$, the module's temperature is signified as $T_{\mathrm{temp}}$, the saturation current is indicated as $I_r$, and the current through the resistor is signified as $I_{\mathrm{res}}$. $V_{\mathrm{oc}}$is the open circuit voltage, and $C_{\mathrm{sc}}$ is the short circuit current. The Boltzmann constant is given as $K_{\mathrm{Bz}}=1.38\times 10^{-23}\mathrm{J}/\mathrm{kelvin}$. Boltzmann constant is a physical constant that relates the average light energy of solar with its temperature that helps to predict the likelihood of a system being in a particular state as a function of its energy. While the Maxwell–Boltzmann distributions give the probabilities of light energies of solar. These significantly help to express the thermal voltage in the semiconductor and produce current. From the generated current, the maximum output power was given as, 4$P_{\mathrm{out}}=V_{\mathrm{oc}}*C_{\mathrm{sc}}$

Solar PV module takes irradiance $\left(G_s\right)$ $(\mathrm{W}/{\mathrm{m}}^2)$ and temperature $\left(T_s\right)$ $(°\mathrm{C})$ as inputs. The PV module's characteristics curve was formed by varying $T_s$ and $G_s$. In the PV module, the P–V curve is one of the major characteristic curves, which exhibit a nonlinear relationship with temperature and irradiance. It demonstrates that the highest voltage occurs at the open-circuit condition and the current is zero and the short-circuit voltage is zero at the origin of the curve, however, the current is maximum. On this curve, some points have extreme power that is drawn from the PV array to the load. These points are called maximum power points (MPPs). These points’ locus is named the maximum power line. Thus, tracking this maximum power line is the extreme power point tracker's function, and the MPPT locus point was found from that. Now, spectral mismatch requires to be corrected over the MPPT locus point, which was performed by evaluating the error factor described in Section 3.1.1.

MMF Estimation

Irradiance values $\left(G_s\right)$ that were recorded periodically were utilized, and these values were taken precisely under varying outdoor conditions, such as sunny, rainy, windy, stormy, and cloudy. The orientation of the modules $\left({\theta }_{\mathrm{mod}}\left(\right)\right)$ was gauged and can be changed. Here, the MMF is evaluated based on the slight difference between the modules concerning their maximum output power. It is proved that the energy benefit could prevail over the mismatch losses in lower irradiance conditions. By estimating the MMF centered on BDPET, the MPP of the solar panels can be quickly tracked by the MPPT controller at the given time and convert the excess voltage into more current, boosting the charging efficiency and reducing the power losses. Therefore, the maximum power can be generated by MPPT. Average Photon Energy (APhE) was calculated using Equation (5). The APhE aims to the energetic distribution's characterization in an irradiance spectrum. It is attained by dividing the irradiance by the photon flux density. 5$\mathrm{APhE}=\left(\frac{{\int }_a^{b}F\left(\chi \right)d\chi }{{\int }_a^b\phi \left(\chi \right)d\chi }\right)$where minimum and maximum values of $\left(G_s\right)$ are depicted as $a\mathit{ }\mathrm{and}\mathit{ }b$, the wavelength is signified as $\chi$, the energy obtained is given as $F$, and photon flux density is indicated as $\phi$. Subsequent to this APhE calculation, PV modules'$I_{\mathrm{sc}}$ are analyzed and expressed as, 6$I_{\mathrm{sc}}=A_{\mathrm{pv}}\int F\left(\chi \right)R_{\mathrm{spec}}\left(\chi \right)d\chi$

At which, the area of PV modules is depicted as $A_{\mathrm{pv}}$, and $R_{\mathrm{spec}}\left(\chi \right)$ is the spectral response. In addition, the mismatch factor (MMF) has been evaluated utilizing the equation, 7$\mathrm{MMF}=\left(\frac{\int E_{\mathrm{spr}}\left(\chi \right)R_{\mathrm{spec}}\left(\chi \right)d\chi \int E_{\mathrm{sii}}\left(\chi \right)R_{\mathrm{sample}}d\chi }{\int E_{\mathrm{sii}}\left(\chi \right)R_{\mathrm{spec}}\left(\chi \right)d\chi \int E_{\mathrm{spr}}\left(\chi \right)R_{\mathrm{sample}}d\chi }\right)$where spectral irradiance and spectral response are depicted as $E_{\mathrm{spr}}$ and $E_{\mathrm{sii}}$. Both the spectral response and spectral irradiance are functions of the photons' wavelength. Spectral response is one among the most significant parameters utilized for evaluating the quality of a detector to reflect its wavelength dependence. Spectral irradiance is the irradiance of a surface per unit frequency or else wavelength, relying on whether the spectrum is taken as a function of frequency or else of wavelength. At varying irradiance values, $R_{\mathrm{sample}}$ is gauged. The losses caused by slight differences in the PV modules' electrical characteristics, applied as a fixed percentage reduction of the system's DC power output are referred to as the Mismatch factor. For systems that have a wide error range on rated power, these losses are high. Therefore, estimating MMF is prominent in the proposed work. Then, utilizing the BDPET, these MMF values are estimated. The major characteristics of the proposed BDPET are Biasedness, consistency, sufficiency, mean square error, and relative efficiency, which help to improve the process of point estimation. The process of Beta-distributed point estimation involves using the value of a statistic that is attained from sample data for getting the best estimate of the MMF's corresponding unknown parameter. Point estimation is chosen since it is considered less biased and more consistent. Yet, a low index score was led by choosing random variables for covariates; thus, for following the beta distribution, it is modified. The beta distribution, which describes the possible outcomes of a random variable that ranges from 0 to 1, is a type of probability distribution. It has two parameters, namely, alpha and beta, that control its shape and location. These two parameters appear as random variable's exponents and manage the distribution's shape. Therefore, the possible random selection can be obtained very accurately, which helps to get a high index score. Therefore, the iterations of selecting MMF values are reduced with better outcomes. The steps are given below,

• ▪

  Select the parameter to be estimated and formulate the ideal point $\left(i_d\right)$

8$i_d={\mathrm{MMF}}^{\prime }\delta +{\tau }_i$

At which, the vector value of MMF is depicted as ${\mathrm{MMF}}^{\prime }$, $\delta$ is a parameter to be estimated, and the likelihood function is signified as ${\tau }_i$.

• ▪

  Calculating the expected value of log and posterior data values. Then, the likelihood of incoming values is voted on and expressed as,

9$E_{\mathrm{BDPET}}=\log (\phi -{\mathrm{MMF}}^{\prime }\delta )(1-\phi (C-{\mathrm{MMF}}^{\prime }\delta ))$

The cumulative density function for beta distribution is depicted as $\phi$, which is considered here to determine the likelihood that a random observation taken from the data values will be less than or equal to a particular value. The limits vary based on the region over which the distribution is defined. Hence, better probability data values can be obtained for selecting the random variables accurately. $C$ is the maximum log-likelihood function. The beta distribution is expressed as, 10$B\left(g,h\right)={\int }_0^{1}t\left(g-1\right)(1-t)(h-1)$where the distribution interval values are signified as $\mathit{gandh}$ and the time value is depicted as $t$.

• ▪

  Maximizing this function with respect to vote and effect parameters for estimation was performed utilizing standard maximation.

• ▪

  This step log calculation estimate was repeated till convergence.

Then, this estimation's output is given to the planar irradiance factor for considering $I_{\mathit{sc}}$ value that was given as $\left({\theta }_{\mathrm{mod}}\left(\right)\right)$ for PV array correction.

CRT-POA-PID-Based Buck-Boost Converter

For interfacing the PV panel to the load, a Buck-Boost converter is utilized as it maximizes the power transfer via impedance matching. This converter is a DC-to-DC converter. For increasing and decreasing the applied DC voltage's voltage level, it is utilized. Centered on the switching transistor's duty cycle, the output voltage is modifiable. This converter is otherwise named the step-up and step-down transformer, which came from the analogous step-up and step-down transformer. The input voltages are step-up/down to some level of more than or else lesser than the input voltage. The input power is equivalent to the output power by utilizing the lower conversion energy. A solid-state device, namely, metal-oxide-semiconductor field-effect transistor (MOSFET), inductor, diode (acts as the switch), capacitor, and DC voltage source are included in the basic Buck-Boost converter. Improvement of those inverters can be made through SiC-based gate drivers [33].

The CRT-POA-PID is utilized for feedback control purposes. In this, a conventional PID controller is chosen due to its low dynamics, and yet, it has a tuning issue. The Chinese Remainder Theorem- Puzzle Optimization Algorithm is utilized for PID tuning. This significantly reduces the issues by the optimized tuning and helps the equipment run longer and safer. Thus, by the optimal selection of the tuning factor, the output errors have been reduced abundantly. The CRT-POA-PID is utilized for feedback control purposes. In Figure 3, the Buck-Boost converter's schematic diagram is displayed. Vos is the output voltage. Vos shares the same role with V_o.

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The voltage relationship for the continuous conduction mode can be expressed as, 11$V_G=\frac{D}{1-D}$where the voltage gain and the duty ratio are depicted as $V_G$ and $D$, respectively. The output voltage $\left(V_o\right)$ can be expressed as, 12$V_0=\frac{V_{\mathit{in}}D}{1-D}-\frac{V_{\mathit{in}}(1-D)}{D}$where the input voltage is signified as $V_{\mathit{in}}$. Relationship of $V_0$ and $V_{\mathit{in}}$ is expressed as, 13$\frac{V_0}{V_{\mathit{in}}}=\frac{2D-1}{D\left(1-D\right)}$

By applying Kirchhoff's law, 14$V_a=\frac{V_{\mathit{in}}}{1-D}$ 15$V_b=-\frac{V_{\mathit{in}}}{D}$where the two voltage loops are indicated as $V_a$ and $V_b$. Here, the Kirchhoff law is used to conserve the electric charge. For any closed network, the voltage around a loop equals the sum of every voltage drop in the same loop and equals zero. The algebraic sum of every voltage in the loop must be equal to zero, and this property of Kirchhoff's law helps to conserve the electric energy. Then, $V_0$ is formulated as, 16$V_0=V_{\mathit{in}}\left(\frac{2D-1}{D\left(1-D\right)}\right)$ 17$I_a=\frac{\left(2D-1\right)V_{\mathit{in}}}{D{\left(1-D\right)}^2}$ 18$I_b=\frac{-\left(2D-1\right)V_{\mathit{in}}}{D^2\left(1-D\right)}$

At which, the current loops are depicted as $I_a$ and $I_b$. The nonlinearity caused by the MOSFET switch has to be removed from these equations; so, the Laplace transform is applied to the acquired system model, from which a control transfer function was acquired. This makes it easier to analyze the acquired system model's stability, performance, and design controllers. Also, the acquired control transfer function describes the relationship between the system's input and output. This helps to understand the system's behaviors over time and also it responds to different inputs. 19$V_0\left(S\right)=\frac{D\left(1-D\right)C_0*\left(s+1\right)}{L{C}_0(1+{\alpha }_L)s^2+\frac{1}{1+{\alpha }_L}\left(\frac{1}{R_{\mathit{SE}}{C}_0}-\frac{R}{L}\right)*s+\frac{1}{L{C}_0{(1+{\alpha }_L)}^2}(\left(1-D^2\right))}$

Here, 20${\alpha }_L=\frac{R_L}{R_{\mathit{SE}}}$

At which, the inductor is depicted as $L$, resistors are signified as $R_L$and $R_{\mathit{SE}}$, the circuit's capacitor is depicted as $C_0$ and the voltage amplification in a steady-state is indicated as $s$. The basic control equation of CRT-POA-PID for the impact of the switching cycle is expressed as, 21$z\left(t\right)=Q_{p}e\left(t\right)+Q_i\int e\left(t\right)\mathit{dt}+Q_d\frac{\mathit{de}\left(t\right)}{\mathit{dt}}$where the drive coming from the Controller into the Process at a time$t$is depicted as$z\left(t\right)$, the proportional gain, integral gain, and derivative gain are signified as $Q_p,Q_i,\mathrm{and}\mathit{ }{Q}_d$, respectively. The difference between the set point and measured process variable at a time $t$ is depicted as $e\left(t\right)$. Utilizing an optimization approach, the approximate accuracy for undamped oscillations will be acquired by tuning the PID controller. Thus, the oscillatory period $T_{\mathrm{OP}}$ was gauged and given as a tuning parameter, 22$T_{\mathrm{OP}}=2\pi \sqrt{{\mathrm{LC}}_0}$

Fitness was set up utilizing the PID controller's overshoot $\left(O_s\right)$; so, $T_{\mathrm{OP}}$ value was acquired for low overshoot. 23$\mathrm{Fitness}\mathit{ }\left(F_g\right)={f\left(O_s\right)}_{\min }$

Utilizing the CRT-POA optimization algorithm, it was minimized. In this, the conventional puzzle optimization algorithm is chosen since it has no control parameters. Generally, the control parameter plays a significant role in determining the behavior of the population and regulating the population size in the optimization. However, the puzzle optimization approach, which is utilized in the proposed work, does not have any control parameter since it is a game-based approach. Thus, each puzzle is initialized with the same behavior within the infinite search space. Hence, the parameter setting is not needed, and yet, a very time-consuming process might result from choosing a random interval value at the updating stage. Thus, utilizing the CRT, it is chosen. This significantly eliminates the uncertainty caused by randomness and reduces the time complexity. Therefore, the number of iterations is reduced, and the selection time is increased. CRT-POA is initialized for the population as $T_{\mathrm{OP}}$ and chosen for minimum overshoot, and the process is mathematically given as, 24${\mathrm{WM}}_i=X_w,w\in \{1,2,3\text{…}.E\}$ 25$d_y=\left\{\begin{array}{l}\mathrm{\&}{\mathrm{WM}}_i-I\times y,F_g<F_i\\ \mathrm{\&}y-I\times {\mathrm{WM}}_i,\mathrm{else}\end{array}\right.$ 26$I=\mathrm{round}(1+\mathrm{round})$ 27$Y_i=y+r+\mathit{dy}$ 28$Y=\left\{\begin{array}{l}\mathrm{\&}{Y}_i,\mathrm{if}\mathit{ }{F}_i\\ \mathrm{\&}y,\mathrm{else}\end{array}\right.$where the guiding number of the $i\mathrm{th}$ puzzle is depicted as ${\mathrm{WM}}_i$, the changes of the $i\mathrm{th}$ puzzle is indicated as $y$, $r$is the random number in the interval $[0,1],$ the maximum number of fitness value is signified as $E$, a new status is given as $Y_i$, the obtained solution is depicted as $Y$, and the random number between 1 and 2 is indicated as $I$, which was chosen using CRT as given below, 29$I=\left(r\mathrm{mod}p\right)*\left(r\mathrm{mod}q\right)$

At which, the random prime number is depicted as $r$ and the minimum and maximum intervals of the distribution are signified as $p\mathit{ }\mathrm{and}\mathit{ }q$. Each member of the population starts to update the status utilizing puzzle pieces, and this was expressed as, 30$O_p=\mathrm{round}\left(0.5\times \left(1-\frac{h}{T}\right)\times O\right)$where the number of puzzle pieces is depicted as $O_p$, the iteration number counter is given as $h$, and the maximum number of iterations is signified as $T$. The update equation was expressed as, 31$Y=\left\{\begin{array}{l}\mathrm{\&}{Y}_i,\mathrm{if}\mathit{ }{F_i}^{*}<F_i\\ \mathrm{\&}{Y}_i,\mathrm{else}\end{array}\right.$

The iteration was done by updating the members of the population in the first and second stages, and the members' new status was determined. This solution was updated as $T_{\mathrm{OP}}$. The pulse width modulated (PWM) signal is formed by the controlled converter centered on the MPPT algorithm. Then, a gate signal is fed to the MOSFET switch in the converter. The Pseudo code for the CRT-POA algorithm is specified below,

Input: Oscillatory period $T_{\mathit{OP}}$

Output: Tuned $T_{\mathit{OP}}$

Begin

Initialize Population size, maximum iteration $\left(T\right)$

Set initial iteration $h=1$

While $\left(\mathit{h}\mathit{\le }\mathit{T}\right)$do

Evaluate $\mathit{Fitness}$

If $\left(\mathit{i}\mathit{!}\mathit{=}\mathit{O}\right)${

Update best member with ${f\left(O\right)}_{\min }$

Select $\mathit{W}{\mathit{M}}_{\mathit{i}}$

Estimate ${\mathit{d}}_{\mathit{y}}$ with $I,y$

Determine $I$ with CRT

Update ${\mathit{Y}}_{\mathit{i}}$

} Else if $\left(\mathit{i}\mathit{=}\mathit{=}\mathit{O}\right)${

Select as best solution

}

End If

$\mathit{h}\mathit{=}\mathit{h}\mathit{+}\mathit{1}$

Return ${\mathit{T}}_{\mathit{OP}}$

End while

End

MPPT Control

MPPT has a controller and concerning the variation of solar irradiance and temperature, a converter is utilized to operate PV at MPP. MPPT demands the DC–DC converter in between the PV array and load for keeping the PV array voltage or current at the MPP point for a specific condition of irradiation and temperature by controlling the converter's duty ratio, which is obtained using modified PNN. The proposed technique has the advantage that it does not require knowledge of the PV modules' operational characteristics encompassing the PV system, or else the PV array structure. Moreover, it could detect the MPP in significantly fewer search steps owing to its inherent learning ability. In this, the MAPNN technique receives temperature and irradiance as inputs in the form of voltage $\left(V_{\mathit{pv}}\right)$ and current $\left(i_{\mathit{pv}}\right)$, and subsequently, the optimal duty cycle at which the PV array has maximum power is offered. Hence, for controlling the duty cycle $\left(D\right)$ signal, MAPNN's output is used; thus, the converter's parameters are adjusted for transferring the maximum power from the PV source for charging the battery and converting it to the best voltage for getting the maximum current into the battery. Moreover, it could supply power to a DC load, which is connected directly to the battery.

Proposed Method for MPPT Control: Mod Tanh-Activated Physical Neural Network

MAPNN has been utilized for MPPT control purposes. The PNN is chosen for its best performance in analog computation forms. Also, it has a faster training process, and there is no local minima issue. Since the training set's size increases, it has guaranteed coverage to an optimal classifier. Their estimated predicted probabilities yielded by the Softmax layer's output utilizing Softmax as activation suffered due to inappropriate activation of neurons. It leads to more unstable and slower convergence compared to cross-entropy. It can't be utilized reliably as the true probabilities; thus, it is modified to the Mod tanh activation function. The Mod tanh activation function's output is zero centered; thus, mapping the output values as strongly negative, neutral, or else strongly positive can be done easily. This helps to determine whether a neuron should be activated. Hence, the complexity of the proposed model is reduced abundantly.

The strategy and factors of a Mod tanh activation function are to be bounded, parameterized, and, generally, nonlinear. Together with these, when utilized in a PNN, the activation function must increase monotonically; thus, the most often wielded functions are the nonsymmetric and antisymmetric sigmoid – and just for the output layer – a linear mapping. Also, the weights of every single neuron are updated via backpropagation, which helps to minimize the classification error of the network. Moreover, by the modified PNN, the accurate control parameter can be classified. For estimating the PV module's maximum voltage with the inputs, namely, temperature, and irradiance, MAPNN is utilized. In general, the maximum voltage is obtained by adjusting the duty cycle of the MPPT controller. Therefore, the optimal duty cycle range is obtained by the trained MAPNN model. During the training process, the neuron's weight value is updated based on the obtained duty cycle in the previous iteration. This process is continued till obtaining the appropriate duty cycle. By the trained model, the duty cycle of the MPPT controller is adjusted depending on the voltage and current.

A deep learning algorithm in which each layer is made up of nodes named neurons is termed PNN. It comprises node layers, namely, the output layer, hidden layer, and input layer. Every single neuron in the layer is linked to another, and all connections are associated with weight values. In Figure 4, the network architecture is displayed. The input layer accepts the data first and hands over the input to the subsequent layers for further processing. Subsequently, the weighted input data is gathered by the hidden layer and the output is delivered via the activation function's utility. The activation function is for introducing nonlinearity into the network. The hidden layer's output $h{d}_n$ is given below, 32$h{d}_n={\Psi }_{\mathit{af}}\left(\sum A_{\mathit{VI}}{\theta }_n+k_n\right)$

[IMAGE OMITTED. SEE PDF]

Here, the values of $V_{\mathit{PV}}$ and $i_{\mathit{pv}}$ consideration are depicted as $A_{\mathit{VI}}$, the Mod tanh activation function is declared as ${\psi }_{\mathit{af}}$, the network weights are signified as ${\theta }_n$, and the bias of the hidden neurons is indicated as $k_n$. Afterward, for the given inputs, the output layer produces the final outcome. The activation function is expressed as, 33${\psi }_{\mathit{af}}=‖\frac{\left(e^u-e^{-u}\right)}{e^u+e^{-u}}‖$where a mod function is depicted as $\Vert .\Vert$, the weight values of the hidden layer is signified as $u$. Then, the error function, which is the difference between the network outputs and desired outputs, is evaluated. It can be given as, 34$\mathrm{EF}=\frac{1}{M}{|f^d(C_{\max \left(I\right)},V_{\max \left(I\right)})-Z^d(h{d}_n,o{t}_n,m)|}^2$ 35${(h{d}_n,o{t}_n,m)}^{*}=\underset{(h{d}_n,o{t}_n,m)}{\text{arg}\min }\mathrm{EF}(h{d}_n,o{t}_n,m)$

At which, the loss function is depicted as $\mathrm{EF}$, the target output is signified as ${(C_{\max \left(I\right)},V_{\max \left(V\right)})}_{\mathit{tar}}$, ${(C_{\max \left(I\right)},V_{\max \left(V\right)})}_{\mathit{out}(h{d}_n,o{t}_n)}$ depicts the actual output with optimized variables $(h{d}_n,o{t}_n,m)$, which are the number of layers and neurons, and the weights and bias trained are signified as ${(h{d}_n,o{t}_n,m)}^{*}$. The training data and the neural network's structure field are depicted as $f^d$ and $Z^d$, respectively. The duty ratio $\left(D\right)$ is calculated, and MAPNN is trained for the different combinations of $\left(V_{\mathrm{pv}}\right)$ and Current $\left(i_{\mathrm{pv}}\right)$ value. The neural network's training means adjusting the layers' weights for getting the target values. Weights are adjusted for tracking the target values with minimum error throughout the training procedure. The proposed MAPNN's pseudo code is

Input: Voltage $\left(V_{\mathrm{pv}}\right)$ and Current $\left(i_{\mathrm{pv}}\right)$

Output: Duty cycle $D$

Begin

Initialize $A_{\mathit{VI}}$, $u$, $h{d}_n$, Iteration $\left(\mathit{iter}\right)$

Evaluate network weights

For $\mathit{iter}=1\mathrm{to}\mathit{ }N$

While $\mathit{iter}=1$, do

Inject input through the input layers

Estimate weight value for $\left(V_{\mathit{pv}}\right)$ and $\left(i_{\mathit{pv}}\right)$

Activate nonlinearity function//using Mod tanh activation function

Compute bias of the hidden neuron

Activate hidden layer $h{d}_n$

End while

End for

Summing all the layers

If $(\mathrm{EF}==D)${

Stop criteria

} else {

Set $\mathit{iter}=\mathit{iter}+1$

} End if

Return $D$

End

Result and Discussion

The proposed system's performance is investigated by analogizing its outcomes with other existing systems in this section. This model is implemented in the working platform of MATLAB. The performance and comparative analysis are performed to state the proposed study's efficiency.

Performance Analysis of PV Characteristics

Here, by varying the temperature level, such as $25^{\circ }\mathrm{C},50^{\circ }\mathrm{C}$, and $75^{\circ }\mathrm{C}$, the PV model's characteristics are examined.

Figure 5 showed that there is no change in the current when the temperature increases and the voltage alone decreases. The net module output power is decreased owing to a decrease in voltage. Hence, an inverse relationship is exhibited by the PV module with temperature. The PV system's characteristics have been clearly understood from this analysis and used for further analysis.

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Performance Analysis of BDPET

The proposed BDPET algorithm's performance is analogized with the existing approaches, namely, PET, Baysier, and the top-down technique centered on the convergence rate, error, and accuracy.

The proposed system's accuracy and error rate comparison with the traditional systems is represented in Table 1. The use of a single statistic (point computed from the data in a sample) as an estimate of a parameter (a number computed from all of the measurements in a population) is referred to as PET. The Bayesian technique allows the use of objective data in specifying a previous distribution. With the Bayesian technique, various individuals may specify different previous distributions. In the top-down approach, the numbers of distributed points are divided into smaller groups. From each group, the random point is estimated based on the high vector value. For making an efficient solar PV, accuracy is gauged by acquiring the optimum MMF value. The wrongly estimated MMF value is termed the error rate. The recommended system's better performance is displayed by a higher value of accuracy and a lower value of error for the proposed system. The proposed system's accuracy is 98.64%, which is enhanced by 3.64% and 4.88% than the existing PET and Baysier, respectively. Similarly, the proposed system's Error rate is 3.64% and 8.43% decreased than the existing PET and top-down approach, respectively. Hence, the proposed BDPET system with load has better and more efficient performance obviously.

Table 1 Performance analysis based on accuracy and error.

Techniques/Metrics	Accuracy (%)	Error (%)
Proposed BDPET	98.64	1.36
PET	95.00	5.00
Baysier	93.76	6.24
Top-down approach	90.21	9.79

The convergence rate analysis of the proposed BDPET and the existing algorithms is demonstrated in Figure 6. Centered on the system's MMF values concerning the number of irradiances, the convergence analysis is performed. In this, for the analysis, 1.84–1.96 irradiances were taken. Based on the maximum power generation, the MMF value is evaluated. Thus, for better performance, the MMF value should be higher. The statement shows that when analogized to the existing approaches, the MMF value achieved by the proposed system is higher for all iterations. Furthermore, when compared to all other approaches, the proposed BDPET has acquired faster convergence. Thus, as per the analysis, the enhanced design is better than the existing techniques.

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Performance Analysis of MAPNN

Here, centered on the output power and convergence rate, the proposed MAPNN's performance is examined and analogized with the existing long short-term memory (LSTM), ANN, and PNN. The PV model is simulated concerning the time responses for the changes in solar irradiance and temperature.

The proposed and existing technique's output power acquired for the constant solar irradiance and temperature is exhibited in Figure 7. LSTM is not considered in the evaluation. The proposed technique's better performance when analogized to the existing techniques is displayed by analyzing the graph. By reaching beyond 500 W in the graph, the proposed technique displayed its efficiency. Thus, the analysis concludes that the proposed system's performance is the higher efficient classifier for tracking the optimal power.

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The comparison between the proposed and conventional systems centered on THD, Mean Power Tracked, Power at Global maximum, and Tracking factor is illustrated in Table 2. The measure of the voltage or current waveform deviation from the ideal sinusoidal shape is termed THD. 4.54 dB is the proposed system's THD, which is highly enhanced by 3.24 dB than the existing PNN, 5.55 dB than ANN, and 8.46 dB than LSTM. According to the output power level, mean power tracking adjusts the power's supplied voltage. The maximum power extraction from PV modules is termed Global MPP. The proposed system is examined and analogized with the existing systems centered on these metrics. When compared to the existing systems, the proposed system exhibits higher improvement. Thus, the improved design is better than the existing approaches as per the analysis.

Table 2 Performance analysis of proposed and existing models.

Techniques/metrics	Total harmonic distortion (THD) (dB)	Mean power tracked (W)	Power at global maximum (W)	Tracking factor (%)
MAPNN	4.54	45	65.1	98.42
PNN	7.78	40	61.01	95.29
ANN	10.09	39	56	92.44
LSTM	13	35	53	90.31

The convergence of the proposed MAPNN and existing systems is displayed in Figure 8. Centered on attaining the proposed system's low error rate, the fitness value is evaluated. Thus, the graph is developed centered on the percentage of the above three conditions’ efficiency. From the graph analysis, the proposed MAPNN's efficiency for 10–20 iterations, 80–90 iterations, and 100 iterations are 2.3%, 0.8%, and 0.6%, which are lower than the existing approaches. The analysis shows that the proposed system is superior to the existing techniques.

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Performance Analysis of CRT-POA-PID

The proposed CRT-POA-PID algorithm's performance is analogized with the existing techniques, such as PID, proportional integral (PI), and programmable logic controller (PLC) centered on the Settling time, Overshoot, and Output voltage (V_out).

The comparison between the proposed and the traditional model based on settling time, overshoot, and V_out is illustrated in Table 3. The time needed for a response for becoming steady in the proposed controller is named settling time, while an output exceeding its final, steady-state value is termed overshoot. As per this state, the settling time and overshoot for the proposed system are evaluated and analogized with the existing system. The proposed system's settling time is 2.3 ms, which is enhanced by 2.7 ms than the existing PID, 4.61 ms than PI, and 6.7 ms than PLC. The proposed system's overshoot is highly enhanced by 11% than the existing PID. Similarly, the proposed system's Vout achieved 12 v. Hence, the proposed system's superiority is clearly shown by the analysis.

Table 3 Performance analysis of proposed and existing models.

Techniques/metrics	Settling time (MS)	Overshoot (%)	Vout (V)
Proposed CRT-POA-PID	2.3	120	12
PID	5	109	10
PI	6.91	100	7
PLC	9	92	5

The performance of the proposed and existing systems, namely, voltage mode control (VMC), FLC, and PID is shown in Figure 9. The controller voltage has been raised when the time increases. In this, voltage and time are linearly proportional. The proposed system's efficient performance is exhibited by reaching the proposed system's higher voltage level. In this, the proposed system maintains its constant voltage level beyond the voltage of 10 V. This graphical analysis concludes that the proposed system outperforms the prevailing techniques. Partial shading scenarios in photovoltaic (PV) systems occur when solar panels experience uneven irradiance due to obstructions such as trees, buildings, or clouds. This results in multiple power peaks and reduced system efficiency, as some panels operate sub-optimally. Traditional MPPT algorithms, like P&O, often fail to identify the global maximum power point under such conditions, becoming trapped in local peaks.

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Advanced methods like the dynamic super-twisting sliding mode controller (STSMC) integrated with Gaussian Process Regression (GPR) have been shown to effectively mitigate these challenges. By predicting optimal voltage references and minimizing oscillations, these approaches achieve faster convergence and higher efficiency under partial shading conditions compared to classical or metaheuristic algorithms such as Cuckoo Search (CSA) [11].

CRT-POA-PID tracking of one of the local peaks for each of the irradiance patterns described is displayed in Figure 10. However, under the irradiance pattern in Case 2, a considerable difference in power tracked is observed. Thus, in Figure 10, Case 3 is not considered. By tracking the second local (Case 3) at 12 V, CRT-POA-PID exhibited a small efficiency.

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Comparative Analysis of the Proposed and Previous Approaches

In this, the proposed PV technique's performance is analogized with existing hybrid techniques built by [25, 28, 29, 32, 34–37] centered on their high-power efficiency. The proposed technique's performance measure centered on its efficiency is shown in Figure 10. When analogized to the hybrid system presented by the mentioned researchers, the proposed model exhibited greater efficiency. The proposed system has been examined under several conditions and it displayed superiority in all conditions. The existing systems built by Ahmed et al., which is Finite Set Model Predictive Control (FS-MPC) [25], and Gonzalez-Castano et al., which is Digital Signal Controller (DSC) [28]. However, in maximum power tracking, the proposed CRT-POA-PID displayed greater performance; this is due to the CRT-POA-PID approach usage for optimal step selection in MPPT. Hence, the proposed technique's performance is superior to the existing approaches (Figure 11).

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Conclusion

For enhancing the PV system's efficiency, the MPPT controller is necessary. The CRT-POA-PID controller is deployed in the PV model for elevating power efficiency in the proposed work. Improving efficiency by utilizing a properly modified converter is this study's major goal. A Buck-Boost converter, Feedback control, MPPT control, a plane of array irradiance, and MMF estimation are comprised in the proposed model. For evaluating the proposed model's performance under several conditions, the performance analysis is done. The output power acquired by the proposed PV system in several time responses is greater than the PV system. The power supply has been maximized to a greater extent by the proposed CRT-POA-PID controller. Hence, the proposed PV system's superior performance is verified by the experimental simulations. Efficiency improvement is focused mainly on not gaining margin. Thus, the system can be built to elevate the Solar PV panels' gain margin in the future.

Acknowledgments

Open access publishing facilitated by RMIT University, as part of the Wiley – RMIT University agreement via the Council of Australian University Librarians.

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