Abstract
At present, Renewable Energy Sources (RES) utilization keeps on increasing because of their merits are more availability in the atmosphere, easy energy harvesting, less maintenance expenses, plus more reliability. Here, the solar power generation systems are utilized for supplying the energy to the local consumers. The accurate, and efficient solar power supply to the customers is a very important factor to meet the peak load demand. The accurate power generation of the sunlight system completely depends on its accurate parameters extraction. In this work, a Modified Raobased Dichotomy Technique (MRAODT) is introduced to identify the actual parameters of the different PV cells which are PWP 201 polycrystalline, plus RTC France. The proposed MRAODT method is compared with the other existing algorithms which are the teaching and learning algorithm, African vultures, plus tuna intelligence algorithm. Finally, from the simulation results, the MRAODT gives superior performance when associated with the other controllers in terms of parameters extraction time, accuracy in the PV cells parameters identification, plus convergence time of the algorithm.
Introduction
From the literature review, the usage of nonrenewable energy sources is decreasing drastically because their drawbacks are more atmospheric pollution, extreme greenhouse gas emissions, more catchment area needed for developing the power plants, plus the input fuel transportation is also very high^{1}. So, the researchers working on the development of renewable energy systems. The features of renewable power networks are low levels of atmospheric pollution, easy installation, fewer human sources required for maintenance, plus high robustness. Also, renewable energy is available in the environment very excessively. So, the input fuel supply cost is zero for RESs. The major renewable power systems are geothermal, wind, hydropower, tidal, plus sunlight networks. The hydropower stations work based on the water head. High water head gives more kinetic energy thereby extracting the hydro power is also more. Sometimes, the storage of water is absent then the water head level is reduced^{2}. As a result, the power utilization from the hydro system is limited.
So, the wind power stations are applied to the smart grid systems to enhance the functioning efficiency of the overall distribution power system. Here, the wind plants collect the wind kinetic energy by using the wind turbines. Wind flow over the blades generates the lift to rotate the wind blades^{3}. The wind blades are directly integrated with the rotor shaft for functioning the electric generator. The generator transfers the wind blades’ rotational energy into the useful power supply. The features of wind systems are less carbon dioxide emission, freely available on the earth, easy operation, very simple equipment, plus less effect on atmospheric conditions^{4}. However, this wind system should be located near remote locations because of its high noise creation. Also, this energy source is unpredictable plus highly dangerous to wildlife. So, the geothermal power supply strategy is used in automotive systems for continuous power supply to the battery charging application^{5}. In this geothermal network, the water content is transferred into the steam. The evaporated steam is directly sent to the steam turbine chamber to run the electrical generator. The merits of geothermal networks are very silent, always available, and have less impact on human beings, plus good energy density. However, it is less stable on the earth, plus more maintenance costs^{6}.
The fuel stackbased electrical energygenerated systems are utilized in the article^{7} for stabilizing the voltage of the automotive systems. From the literature study, the fuel stacks are suitable for stationary power generation, and transportation applications. The merits of fuel stack systems are more energy efficient, have zero emissions, are highly robust, more scalable, plus low operating costs^{8}. However, this system’s implementation cost is higher. So, the solar systems are integrated with the already existing power supply networks to meet the peak load consumer demand. A solar system is the most powerful and useful energy source for rural area people. The working structure of the sunlight system is provided in Fig. 1. Here, there are various categories of sunlight systems have existed in the literature which are thin film, siliconbased monocrystalline, copper indium gallium selenide, Cadmium telluride, plus Polycrystalline solar panels^{9}. The solar networks are developed by utilizing the 1diode, 2diodes, plus 3diodesbased solar cells. Here, the major issue of the sunlight system is nonlinear performance. As a result, the extraction of voltage from the sunlight system is quite a difficult task. In addition, the accurate parameter identification of solar cells is difficult^{10}. So, there are different categories of optimization methodologies are exist in the literature to obtain the accurate parameters of the solar cell thereby enhancing the efficiency of the overall system.
In Ref.^{11}, the authors explained that the total number of parameters required for the implementation of one diode solar cell is five which are named solar photocurrent (I_{ph}), ideality factor of the concern diode (ʎ), the saturation current of the circuit (I_{0}), shunt resistive element (R_{sh}), plus series placed resistive element (R_{s}). Here, the sunlight system performance and identification of suitable solar cell parameters are obtained by using its nonlinear V–I and P–V curves. The suitable parameters of the solar cell on the V–I curve are determined by applying various natural inspirationbased optimization algorithms^{12}. The classification of swarm intelligenceassociated algorithms and their application on solar parameters extraction is illustrated in Fig. 2. From Fig. 2, most of the research scholars worked out on the one diodebased sunlight system for enhancing its functioning efficiency by evaluating the accurate short circuit current, and series resistance of the circuit. The features of the 1diode circuit model are easy implementation, few factors required for the development, less mathematical computation is needed, plus the low cost of installation^{13}.
From the literature review, the solar cell parameters are determined by selecting the numerical, soft computing, and metaheuristic algorithms. Among all of the parameter extraction methodologies, metaheuristic techniques are the predominant methods when associated with the other methods because their merits are more accurate in the parameter selection of the solar cells, easy way to enhance the efficiency of the sunlight power system, plus more reliability. In Ref.^{14}, the authors estimated the 2diode model PV circuit variables which are named reverse saturation current of the diode one (I_{01}), saturation current of the diode (I_{02}), ideality factors of the diode’s variables (ʎ_{1} and ʎ_{2}), shunt resistive element (R_{sh}), photocurrent (I_{ph}), plus series placed resistive element (R_{s}).
In Ref.^{15}, the authors studied all over three solar cell technologies in terms of parameter extraction, and operating efficiency. From the simulative study, the authors decided that the single plus double diode methodologies do not provide accurate results when associated with the triple diode cell technology. In this triplediode circuit, the leakage currents of the PV cell are also considered for the designing of the threediodebased sunlight system, and its related implementation data is collected from the datasheet given by the manufacturers at Standard Test Condition (STC). The major variables utilized to extract any solar cell parameters are the open circuit voltage point on the V–I curve, peak power, peak current available from the sunlight system, plus shortcircuited current point^{16}. The analytical methods give lowlevel accurate PV system parameters when associated with the metaheuristic, plus arithmetical methods. Suppose, any changes in the system conditions then the evaluation of the parameters may not be proper and the solar cells’ mathematical equations are developed by approximating the various sunlight conditions.
In Ref.^{17}, the authors utilized curve fitting, Particle Swarm Intelligence (PSI), Jaya, Flower Pollicization (FP), High Convergence PSO, and Simulated Annealing Algorithms for the identification of solar cell parameters with variable step sizes. However, these methods have the drawback of high complexity in development. So, the stochastic methodologies are involved with the numerical techniques for enhancing the functioning efficiency of the sunlight systems. Here, the hybrid methods evaluate the global solution instead of local optima, and it gives a wide range of solutions for the parameter extraction of the sunlight system. The NewtonRapson concept is integrated with the chaotic methods in Ref.^{18} to reduce the overall controller iteration value. As a result, the system works effectively with an optimal number of iterations.
In the article^{19}, the authors discussed the Levenberg, plus Wind Driven Optimization (WDO) methodologies for reducing the fluctuations in the system performance. The disadvantage of WDOdependent methodologies is slow convergence rate for identifying the global optima. So, the Elephant Herd Algorithm (EHA) is proposed in Ref.^{20} for verifying the parameters extraction of the sunlight system by using the Genetic Optimization Grey Wolf method. However, the limitations of the above methods are low accuracy levels, more fluctuations in the system behavior, plus less suitability for the three diodebased sunlight systems. Here, the modified Rao methodology is utilized for the suitable parameters.
Mathematical development of solar PV cells
From the solar cell manufacturers, the solar system is developed by placing the photocurrent in parallel with the diode. Here, the photocurrent is obtained by combining the two silicon materials which are Ptype, plus Ntype materials. Whenever the sunlight hits the power semiconductor materials then the freely running electrons in the materials observe the sunlight insolations for moving from one side layer of the PN diode to another layer^{21}. Here, the diode works to eliminate the reverse leakage currents of the sunlight network which is discussed in Fig. 2. From Fig. 3, the diode (D_{j}) improves the sunlight system efficiency by limiting the overcurrent flow in the network. The parameters I_{Dj}, I_{Ph}, I_{St}, I_{Ss}, plus I_{Lk} are the diode current, photocurrent, shunt current flow, series current flow, plus load current flow. The elements R_{St}, plus R_{Ss} are the shunt connected and series connected resistances.
The available photocurrent from the circuit is discussed in Eq. (1). Suppose, there are multiple cells considered in the sunlight system then Eq. (3) is applied to determine the sunlight current. Here, Eq. (8) is considered for evaluating the sunlight current by neglecting the shunt resistance of the PV circuit. Finally, the PV circuit involves all the resistive elements then the available PV module current is obtained by selecting Eq. (12), plus Eq. (13). Similar to the 1diode model, the 2diode model, plus 3diode model sunlight systems PV circuits are illustrated in Fig. 4a, plus b. From Fig. 4a, the extra diode is placed in parallel to the photocurrent for working the solar module with high accuracy at very low irradiation conditions. The addition of a diode with the existing 1diode circuit PV system creates complexity in the system. Also, its installation, and manufacturing costs are increased. So, there are various advanced PV technologies available in the present market for optimizing the cost by identifying accurate solar system parameters.
In the twodiode circuit, there are four junctions are existed to enhance the power supply rating of the sunlight system^{22}. Here, the band gap energy plays a major role in transferring the electrons from one direction to another direction. The required variables for the design of the 3diode circuit sunlight system are I_{Pg}, R_{St}, R_{Ss}, I_{Dj}, I_{Dk}, ʎ1, plus ʎ2 respectively. The major issue of one diode PV module is junction recombination losses which are limited by using the 2diode model sunlight system. In the 2diode PV circuit, there is a leakage in grain boundaries. So, one more diode is included in the twodiode circuit to form the triplediode sunlight system which is explained in Fig. 4a,b. The overall parameters evaluated in this system are nine which are named as I_{Pg}, R_{St}, I_{DL}, R_{Ss}, I_{Dj}, I_{Dk}, ʎ1, ʎ2, plus ʎ3.
Process involved in the solar PV cell parameters extraction
The process involved in the parameters evaluation of the sunlight system are collection of a set of experimental data by utilizing the P–V and I–V curves^{23}. Here, the fitness function is defined on the nonlinear characteristics of the sunlight system and it gives either the local maxima or the local minima. The main objective of this article is the identification of error constraints between the evaluated data, plus experimental setup data. The utilized fitness function of this proposed sunlight system is illustrated in Fig. 5. Based on Eq. (17), the RMSE value is determined with the help of actual experimental data and calculated data. The objective function of the proposed system is defined in Eq. (18) which is called a transcendental equation. From the literature study, many researchers represented the issue of the transcendental equation on solving the sunlight parameters extraction. Here, this issue is limited by using the dichotomy method.
Proposed technique for parameters extraction
In the dichotomy approach, the number of iterations required for the identification of suitable sunlight parameters is much less, and it needed less convergence time for developing the solar cells. Here, the intermediate theorem is applied to the solar system by selecting a function f(n). The function is working under continuous conditions in the time duration [u, v] then the variables f(u), plus f(v) have opposite signs. As a result, the parameter “w” exists in the middle of the parameters U, plus V. With the help of the above information, the sunlight parameters are obtained by utilizing the metaheuristic technique. In this metaheuristic method, at starting, the unknown parameters are determined and which are applied to the fitness function to check the output of the algorithm. In the second state, the adaptive modified Rao algorithm is selected for extracting the sunlight system values. In the third state, the dichotomy concept is included with the required objective function. In the fourth state, the allPV cell circuits are used to identify their parameters with unique changes in the proposed algorithm to achieve the optimal RMSE value. The proposed modified Rao algorithm output parameters are compared with other algorithms at benchmark conditions.
The functioning flow of the modified Rao algorithm is illustrated in Fig. 5. From Fig. 5, it is identified that the Rao concept doesn’t involve any data sets for extracting the sunlight system parameters, and it works in a straightforward situation without any uncertainties. In the first iteration of the Rao algorithm, the global and worst solutions are evaluated by applying Eq. (19). In Ref.^{24}, the authors used the selfmodified Rao concept which works quite similarly to the Rao. In this modified Rao method, all the populations are split into various groups to obtain the different solutions of the PV parameters. Here, all the particles search the entire utilized region to get the best optimal solution. The presently available solutions are crossverified with the already available solutions to improve the accuracy of the sunlight system.
Ethical approval
This paper does not contain any studies with human participants or animals performed by any of the authors.
Discussion of results
1st case study: 2diode PV circuit of RTC—France
In this case, the 2diode model, and 3diode model sunlight systems are selected for extracting the efficiency of the modified Rao method. Here, a 58 mm diameter solar system is selected which is made up of silicon material for commercial applications, and it is an RTC France cell. The overall experimental investigation has been done at 35 °C and 1000 W/m^{2} irradiation value. The RTC France solar cell data sheet is given in Table 1. The available parameters of the sunlight systems by applying the different algorithms are mentioned in Table 2, plus Table 4. The obtained RMSE value for the seven parametersbased 2diode circuit sunlight system is 7.33167 × 10^{–4}. Also, the ten parameters were determined for the sunlight system with an accuracy of 7.33167 × 10^{–4}. The determined RMSE values for the solar cells are given in Table 4. The proposed method evaluated theoretical and experimental RTCFrance solar PV parameters are indicated in Fig. 6, plus Fig. 7. From Fig. 6, and Fig. 7, it is identified that the RMSE value is much less for both the theoretical and experimental RTCFrance Solar Cells.
The total number of iteration counts required for the evaluation of the parameter RTC France sunlight system cell by using MRAODT is less, and its convergence time is also low when associated with the other natureinspired metaheuristics. Finally, the RMSE value of the modified Raobased DT controller is low and it takes 100 iterations to achieve the highly accurate solar system parameters. The required iterations, plus the population value summary of the proposed method along with the slider mode optimization are illustrated in Tables 2 and 3. Table 3, it evaluated that the population and iteration values of the optimization algorithm increased then the RMSE value was reduced. Finally, the convergence time of the optimization method is increased then the error value is reduced.
2nd case study: 3diode and 1diode photo cell watt PWP207
Here, the Photo 3diode, plus 1diode Cells are utilized for commercial application and it is indicated as Watt PWP207. The threediode sunlight system’s objective function is to determine the seven parameters and five variables for a 1diode sunlight system. The utilized cells for both the systems are 36 which are working at 45 °C temperature and 1000 W/m^{2} irradiance. The evaluated proposed method RMSE is equated with the already existing optimization algorithms as shown in Tables 4 and 5.
3rd case study: experimental analysis of 1diode, and 3diode sunlight systems
The proposed method is investigated by selecting the largescale system. The evaluated experimental waveforms for the 1Diode system are explained in Fig. 8. Here, “3” solar strings are utilized, and those strings are consisted of “6” modules. The model GLM303 monocrystalline PV module along with “36” works as a local consumer application. The model PROVA1011 is used for investigating the nonlinear curves of the sunlight system at multiple sunlight temperatures and irradiation conditions. The data sets provide the electrical characteristics of sunlight systems. In this case, the proposed technique is analysed for a 1Diode cell for estimating “5” parameters of the sunlight system as given in Table 6. The nonlinear curves of the LM100 model PV array are illustrated in Fig. 9. From Table 6, and Fig. 10, the proposed modified Rao method predicts the sunlight parameters accurately under various environmental conditions. Finally, the 3diode sunlight system experimental parameters are given in Table 7.
Conclusion
The solar power supply system efficiency is majorly dependent on its accurate PV module design, and the nonlinear characteristics of the sunlight system. Here, the modified Rao with Dichotomy technique is applied to the 1diode, 2diode, plus 3diode solar cell systems to identify their suitable parameters to supply the peak power to the consumers. The total utilized PV modules in this work are eighteen, and the RMSE value is determined for FranceRTC, Watt PWP207. From the above simulation and experimental investigation, the proposed MRAODT algorithm extracts the sunlight system parameters with more accuracy under multiple sun temperatures, plus irradiation conditions. The modified Rao algorithm reaches the convergence speed at 489 iterations. The proposed MRAODT method advantages are fast convergence speed, needed lowlevel iterations for identifying the suitable PV cell variables, more suitable quick changes of sunlight conditions, plus easy adaptability.
Data availability
The data used to support the findings of this study are included in the article.
Abbreviations
 MRAODT:

Modified Rao dichotomy technique
 RES:

Renewable energy source
 RMSE:

Root mean square error
 MPSO:

Modified particle swarm optimization
 HCSPSO:

High convergence speed particle swarm optimization
 AMPSO:

Adaptive modified particle swarm optimization
 CSABT:

Cuckoo search algorithmbased biogeography technique
 EJA:

Enhanced Jaya algorithm
 ASOA:

Adaptive sequential optimization algorithm
 MGBA:

Modified gradientbased algorithm
 TLBOT:

Teaching learning based optimization algorithm
 ACSA:

Adaptive cuckoo search algorithm
 MOWOA:

Modified oppositionbased whale optimization algorithm
 AMFA:

Adaptive moth flame algorithm
 WDA with PSA:

Wind driven algorithm with particle swarm algorithm
 HCPSA:

High convergence particle swarm algorithm
 VAPPSO:

Variable accelerating parameters of particle swarm optimization
 ASDT:

Adaptive successive discretization technique
 MHNMT:

Modified hybrid NelderMead technique
 ABCTLA:

Artificial bee colony teaching learning algorithm
 WOBDEA:

Whale optimizationbased differential evolutionary algorithm
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Kumar, S.S., Balakrishna, K. A novel optimal identification of various solar PV cell parameters by using MRDT controller.
Sci Rep 14, 10467 (2024). https://doi.org/10.1038/s4159802461359x

Received: 07 April 2024

Accepted: 06 May 2024

Published: 07 May 2024

DOI: https://doi.org/10.1038/s4159802461359x
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