# Machine Learning Performance Prediction of a Solar Photovoltaic-Thermoelectric System with Various Crystalline Silicon Cell Types

Authors: Hisham Alghamdi; Chika Maduabuchi; Abdullah Albaker; Abdulaziz Almalaq; Turki Alsuwian; Ibrahim Alatawi
Journal: International Journal of Energy Research
 (2023)

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## Abstract

Since the solar photovoltaic-thermoelectric (PV-TE) is an upcoming technology, the current literature on PV-TEs have failed to thoroughly investigate the effects of different solar cell types on the PV-TE’s performance. Such parametric study becomes necessary since the properties of the solar cell, characterized by the cell temperature coefficient and reference efficiency, determine the overall performance of the PV-TE. Further, the design information obtained from numerical solvers is minimal due to the high time and computational energy required to optimize the photovoltaic-thermoelectric (PV-TE) performance. This problem severely affects the ease with which useful design information can be drawn from current methods to design high-performance PVs. Additionally, to reduce the complexity of the existing numerical model, the previous models have introduced some principal assumptions that have significantly affected the accuracy of the numerical results. Finally, the numerical model has neglected several real-life operating conditions since they increase the model complexity. For the first time, deep neural networks are proposed to predict the photovoltaic-thermoelectric performance designed with 3 different crystalline solar cells as a perfect replacement for the inefficient numerical methods used to analyze the hybrid system. After that, the data generated by the numerical solver is fed to an optimum 3-layer deep neural network with 20 neurons per layer to forecast the hybrid system’s performance efficiently. The training of the neural network is governed by Bayesian regularization and the performance of the model is evaluated using the mean squared error, coefficient of determination, and training time. Results show that the deep neural network accurately learned the results that took the conventional numerical solver 1,600 mins to generate in just 12 min and 27 s, making the proposed network 128.51 times faster than the traditional numerical method. Furthermore, the accuracy of the network is demonstrated by the low mean squared error of3.34×10−8 and high training and testing regression of 100% and 99.98%, respectively.


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## Body

## 1. Introduction

### 1.1. Background

The need to transition from fossil fuel to clean energy-based systems has increased attention due to the adverse environmental impacts of toxic gas emissions in the atmosphere from various nonrenewable energy systems [1, 2]. Solar photovoltaics remains a viable option for renewable energy power generation, and their massive installation in different regions of the world is at an all-time high [3, 4]. More specifically, the crystalline silicon solar cell has dominated the solar photovoltaic market due to its readily available raw materials and easy production [5, 6]. However, the performance of the crystalline silicon solar cell degrades when exposed to high-temperature applications, which are available in concentrating solar systems [7, 8]. The photovoltaic-thermoelectric system was developed to harness the waste heat that heats the solar cell converting it to electrical power via thermoelectric semiconductors [9, 10]. The thermoelectric generator integrated into the solar photovoltaic system can utilize the solar spectrum components underutilized by the photovoltaic, converting it to valuable electrical energy, which adds up to the performance of the hybrid system [11, 12].


### 1.2. Photovoltaic-Thermoelectric Optimization

The development of photovoltaic-thermoelectric technology was necessitated by the need to reduce the overheating of solar panels when operated under concentrated solar irradiance. Two ways a thermoelectric generator can be attached to a photovoltaic module are by direct coupling [13] or spectral splitting [14]. In the direct coupling method, the thermoelectric module is attached to the backplate of the photovoltaic module using a thermal conductive paste. In this case, the excess heat accumulated by the solar panel is conducted to the hot junction of the thermoelectric module by conductive heat transfer. Meanwhile, the cold junction of the thermoelectric module is cooled using an appropriate cooling media like a fan or water jet. The temperature gradient between the thermoelectric generator necessitates a power generation which contributes to the increased power generated by the photovoltaic module [15–17]. The power and efficiency of the hybrid system becomes the sum of the powers and efficiencies of the individual photovoltaic and thermoelectric module. The case of the spectral splitting uses a beam splitter to separate the incoming solar irradiance into the ultraviolet and infrared components which are channeled to the photovoltaic and thermoelectric systems, respectively.Although the feasibility of the photovoltaic-thermoelectric system has been demonstrated using various theoretical and experimental methods [18, 19], there is still room for further enhancements in system efficiency [20–22]. The system’s performance can be further increased by optimizing the geometry of the thermoelectric generator. Lamba and Kaushik [23] developed a theoretical model using an iterative numerical approach in MATLAB software to assess the performance of a concentrated photovoltaic-thermoelectric system from energy and exergy perspectives. The optimized parameters included the fill factor and geometry parameters of the thermoelectric semiconductors. It was concluded that for a concentrated solar irradiance of 5.5 kW/m2, the optimum height for the thermoelectric semiconductors was 5 mm, generating a hybrid system power of 608.9 mW, which was 14% higher than that of the stand-alone photovoltaic system. Li et al. [24] investigated the best geometry configuration for maximum photovoltaic-thermoelectric performance. The study was conducted using a 3-dimensional finite element method developed in COMSOL Multiphysics, and the effects of different geometry configurations on the hybrid system performance were determined. The authors concluded that the optimum thermoelectric geometry configuration in a hybrid photovoltaic-thermoelectric system depended on the cell type and that the optimum geometry for a stand-alone thermoelectric is different from that for a photovoltaic-thermoelectric. Shittu et al. [25] utilized a 3-dimensional numerical method enacted in COMSOL Multiphysics software to investigate the optimum geometry needed to maximize the performance of photovoltaic-thermoelectric systems operating under transient weather conditions in the climate of the United Kingdom for a 24 h duration. It was concluded that the adverse impacts of fluctuating weather conditions on the power output of the hybrid system were reduced by 48.29% due to the optimization of the thermoelectric geometry. Ge et al. [26] used a one-dimensional mathematical model to optimize the structural parameter, the thermoelectric semiconductor area to length ratio, of thermoelectric generators directly coupled to photovoltaic cells for waste heat recovery and improvement of efficiency. An optimal thermoelectric semiconductor area-length ratio of 0.36 mm-1 obtained a maximum of 41.73% efficiency under simulated design conditions. Li et al. [27] provided a parametric geometry optimization of thermoelectric semiconductors in a photovoltaic-thermoelectric system using a finite element model carried out in COMSOL Multiphysics to uncover the mismatch between the optimal thermoelectric footprint in a stand-alone thermoelectric generator and a hybrid photovoltaic-thermoelectric. It was reported that the maximum overall performance was obtained when the area of the p- and n-type semiconductors was the same.Further efforts were made to optimize and assess the overall performance of solar photovoltaic-thermoelectric. Alnajideen and Min [28] carried out a practical demonstration of the higher performance of a spectrum splitting solar photovoltaic-thermoelectric system compared to a stand-alone solar photovoltaic module. The experimental results obtained showed that the stand-alone photovoltaic module efficiency was 15.9% while that of the hybrid system was 16.9%, indicating a 6.3% due to thermoelectric waste heat scavenging. Mahmoudinezhad et al., [29] experimentally assessed the performances of a stand-alone solar photovoltaic module, thermoelectric generator module, and a hybrid photovoltaic-thermoelectric system. The performance of the stand-alone photovoltaic and thermoelectric generator modules was compared to that of the beam splitting solar photovoltaic-thermoelectric system. It was concluded that utilizing the beam splitter diminished the power generation of the thermoelectric module while increasing the power generation of the hybrid solar system. Bamroongkhan et al., [30] experimentally demonstrated the feasibility of a solar parabolic dish concentrating hybrid photovoltaic-thermoelectric system to produce thermal and electrical energy. It was found that at ambient operating conditions of 37.8°C and 874 W/m2, the proposed system could produce hot water at a capacity of 80 L per day at a heat transfer rate of 535.5 W/day. Gupta et al., [31] conducted the performance analysis of a solar photovoltaic-thermoelectric integrated with a serpentine water collector for removing the waste heat from the back surface of the hybrid system. The performance of the proposed system was compared with that of a conventional photovoltaic-thermoelectric system using a different heat sink and the system’s electrical efficiency was improved by 5.72%. Yin and Li [32] compared the optimization parameters of the stand-alone photovoltaic module and the hybrid photovoltaic-thermoelectric. A multiobjective optimization algorithm was employed in optimizing the hybrid system’s performance and it was found that the thermoelectric figure of merit is only minimized by the solar concentration ratio under certain critical operating conditions. Khan et al., [18] experimentally assessed the performance of a hybrid photovoltaic-thermoelectric system cited in Lahore, Pakistan. Ten bismuth-telluride-based thermoelectric generators were lapped directly unto the rear contact of a 10 W crystalline Silicon-based module to harness the waste heat from the cell and convert it to electricity. The results showed that utilizing the thermoelectric modules decreased the photovoltaic cell temperature by 5.5%, indicating a reduction from 55°C to 52°C. Furthermore, the temperature reduction and additional power/energy generated by the thermoelectric module improved the stand-alone panel’s power output by 19%, signifying a rise from 8.78 W to 10.84 W. Finally, when the thermoelectric generators were added, the efficiency of the solar panel was increased from 11.6% to 14%, representing a 17% overall boost in the panel’s conversion efficiency.


### 1.3. Machine Learning Algorithms for Photovoltaic-Thermoelectric Design

Despite the commendable laboratory results that have shown the higher performance of the proposed hybrid photovoltaic-thermoelectric system, the commercialization of this technology is far from being achieved because of the scanty design information present in the current body of literature. By design information, we mean optimum geometry dimensions and operating conditions of the thermoelectric semiconductors for maximum waste heat removal from the rear contact of the photovoltaic cell. A reasonable explanation for this information deficit is due to the inefficient finite element tools used to solve the coupled thermal-electrical equations for the hybrid photovoltaic-thermoelectric system. Due to the complexity of the 3-dimensional governing equations and the high-resolution meshing needed to obtain volume average solutions from over a million nodes, the speed of obtaining results is extremely low and the computational energy requirement is enormous. A solution to this challenge in thermoelectric generator design has been proposed using machine learning algorithms [33–35]. However, the information obtained from thermoelectric design cannot be used to design photovoltaic-thermoelectric since the coupling equations are different [36]. Up till date, no research attempt has been made to speed up the design process of hybrid photovoltaic-thermoelectric systems. Hence, the aim of this research work.


### 1.4. Scientific Novelty

The literature review has established the hybrid photovoltaic-thermoelectric system’s higher performance than the stand-alone photovoltaic module. However, the mass production of efficient solar photovoltaic-thermoelectric is limited by certain problems that have not been addressed. These problems are:(i)
The design process for the hybrid photovoltaic-thermoelectric is very rigorous and expensive. It is time-consuming to obtain optimization information from the simulation software to design high-performing devices. Furthermore, the complex system equations are mostly solved using Multiphysics solvers that require high and expensive computational memory. Since the solvers rely on the finite element method to solve the three-dimensional governing equations, the solid models are discretized into million mesh elements that take a long time to solve and provide design information. Up till date, no research attempt has been made to mitigate this stumbling block to the design of high-performing solar photovoltaic-thermoelectric devices. This work proposes to fill this knowledge gap by replacing the conventional numerical solvers with deep neural networks inspired by artificial intelligence. The proposed deep neural network is trained using finite element data and is deployed to optimize the hybrid device’s performance for operating conditions within or outside the finite element data. Hence, the deep neural network replaces the time-consuming and energy-intensive finite element method(ii)
Majority of the previous optimization studies on solar photovoltaic-thermoelectric were based on crystalline silicon solar cells. Nevertheless, the silicon solar cell type was neglected in the analysis by assuming a constant temperature coefficient and reference efficiency. Therefore, it can be inferred that the current optimization information governing the design of hybrid photovoltaic-thermoelectric cannot be generalized to all silicon solar cell types. However, photovoltaic modules are characterized by diverse cell types. Hence, it is important to factor the solar cell type when producing optimization information to produce hybrid photovoltaic-thermoelectric. This work seeks to fill this research gap by providing a comprehensive optimization study that factors the effect of three conventional solar cell types. The optimization study is also modified to simultaneously examine the impacts of diverse operating conditions on the solar cell type to provide a more realistic design guideline for the manufacture of high-performing hybrid systemsThe subsequent parts of this work are arranged in the following manner: Section2 explains the physical photovoltaic-thermoelectric model, while the equations used to describe the numerical model and its validation are provided in the Supplementary Material. Section 3 discusses the results of the numerically inspired optimization study and the subsequent coefficient of determination and mean squared error of the optimum regressive deep neural network model proposed to substitute the inefficient numerical process. Finally, the significant findings extracted from the results discussed in this work are highlighted in Section 4.


## 1.1. Background

The need to transition from fossil fuel to clean energy-based systems has increased attention due to the adverse environmental impacts of toxic gas emissions in the atmosphere from various nonrenewable energy systems [1, 2]. Solar photovoltaics remains a viable option for renewable energy power generation, and their massive installation in different regions of the world is at an all-time high [3, 4]. More specifically, the crystalline silicon solar cell has dominated the solar photovoltaic market due to its readily available raw materials and easy production [5, 6]. However, the performance of the crystalline silicon solar cell degrades when exposed to high-temperature applications, which are available in concentrating solar systems [7, 8]. The photovoltaic-thermoelectric system was developed to harness the waste heat that heats the solar cell converting it to electrical power via thermoelectric semiconductors [9, 10]. The thermoelectric generator integrated into the solar photovoltaic system can utilize the solar spectrum components underutilized by the photovoltaic, converting it to valuable electrical energy, which adds up to the performance of the hybrid system [11, 12].


## 1.2. Photovoltaic-Thermoelectric Optimization

The development of photovoltaic-thermoelectric technology was necessitated by the need to reduce the overheating of solar panels when operated under concentrated solar irradiance. Two ways a thermoelectric generator can be attached to a photovoltaic module are by direct coupling [13] or spectral splitting [14]. In the direct coupling method, the thermoelectric module is attached to the backplate of the photovoltaic module using a thermal conductive paste. In this case, the excess heat accumulated by the solar panel is conducted to the hot junction of the thermoelectric module by conductive heat transfer. Meanwhile, the cold junction of the thermoelectric module is cooled using an appropriate cooling media like a fan or water jet. The temperature gradient between the thermoelectric generator necessitates a power generation which contributes to the increased power generated by the photovoltaic module [15–17]. The power and efficiency of the hybrid system becomes the sum of the powers and efficiencies of the individual photovoltaic and thermoelectric module. The case of the spectral splitting uses a beam splitter to separate the incoming solar irradiance into the ultraviolet and infrared components which are channeled to the photovoltaic and thermoelectric systems, respectively.Although the feasibility of the photovoltaic-thermoelectric system has been demonstrated using various theoretical and experimental methods [18, 19], there is still room for further enhancements in system efficiency [20–22]. The system’s performance can be further increased by optimizing the geometry of the thermoelectric generator. Lamba and Kaushik [23] developed a theoretical model using an iterative numerical approach in MATLAB software to assess the performance of a concentrated photovoltaic-thermoelectric system from energy and exergy perspectives. The optimized parameters included the fill factor and geometry parameters of the thermoelectric semiconductors. It was concluded that for a concentrated solar irradiance of 5.5 kW/m2, the optimum height for the thermoelectric semiconductors was 5 mm, generating a hybrid system power of 608.9 mW, which was 14% higher than that of the stand-alone photovoltaic system. Li et al. [24] investigated the best geometry configuration for maximum photovoltaic-thermoelectric performance. The study was conducted using a 3-dimensional finite element method developed in COMSOL Multiphysics, and the effects of different geometry configurations on the hybrid system performance were determined. The authors concluded that the optimum thermoelectric geometry configuration in a hybrid photovoltaic-thermoelectric system depended on the cell type and that the optimum geometry for a stand-alone thermoelectric is different from that for a photovoltaic-thermoelectric. Shittu et al. [25] utilized a 3-dimensional numerical method enacted in COMSOL Multiphysics software to investigate the optimum geometry needed to maximize the performance of photovoltaic-thermoelectric systems operating under transient weather conditions in the climate of the United Kingdom for a 24 h duration. It was concluded that the adverse impacts of fluctuating weather conditions on the power output of the hybrid system were reduced by 48.29% due to the optimization of the thermoelectric geometry. Ge et al. [26] used a one-dimensional mathematical model to optimize the structural parameter, the thermoelectric semiconductor area to length ratio, of thermoelectric generators directly coupled to photovoltaic cells for waste heat recovery and improvement of efficiency. An optimal thermoelectric semiconductor area-length ratio of 0.36 mm-1 obtained a maximum of 41.73% efficiency under simulated design conditions. Li et al. [27] provided a parametric geometry optimization of thermoelectric semiconductors in a photovoltaic-thermoelectric system using a finite element model carried out in COMSOL Multiphysics to uncover the mismatch between the optimal thermoelectric footprint in a stand-alone thermoelectric generator and a hybrid photovoltaic-thermoelectric. It was reported that the maximum overall performance was obtained when the area of the p- and n-type semiconductors was the same.Further efforts were made to optimize and assess the overall performance of solar photovoltaic-thermoelectric. Alnajideen and Min [28] carried out a practical demonstration of the higher performance of a spectrum splitting solar photovoltaic-thermoelectric system compared to a stand-alone solar photovoltaic module. The experimental results obtained showed that the stand-alone photovoltaic module efficiency was 15.9% while that of the hybrid system was 16.9%, indicating a 6.3% due to thermoelectric waste heat scavenging. Mahmoudinezhad et al., [29] experimentally assessed the performances of a stand-alone solar photovoltaic module, thermoelectric generator module, and a hybrid photovoltaic-thermoelectric system. The performance of the stand-alone photovoltaic and thermoelectric generator modules was compared to that of the beam splitting solar photovoltaic-thermoelectric system. It was concluded that utilizing the beam splitter diminished the power generation of the thermoelectric module while increasing the power generation of the hybrid solar system. Bamroongkhan et al., [30] experimentally demonstrated the feasibility of a solar parabolic dish concentrating hybrid photovoltaic-thermoelectric system to produce thermal and electrical energy. It was found that at ambient operating conditions of 37.8°C and 874 W/m2, the proposed system could produce hot water at a capacity of 80 L per day at a heat transfer rate of 535.5 W/day. Gupta et al., [31] conducted the performance analysis of a solar photovoltaic-thermoelectric integrated with a serpentine water collector for removing the waste heat from the back surface of the hybrid system. The performance of the proposed system was compared with that of a conventional photovoltaic-thermoelectric system using a different heat sink and the system’s electrical efficiency was improved by 5.72%. Yin and Li [32] compared the optimization parameters of the stand-alone photovoltaic module and the hybrid photovoltaic-thermoelectric. A multiobjective optimization algorithm was employed in optimizing the hybrid system’s performance and it was found that the thermoelectric figure of merit is only minimized by the solar concentration ratio under certain critical operating conditions. Khan et al., [18] experimentally assessed the performance of a hybrid photovoltaic-thermoelectric system cited in Lahore, Pakistan. Ten bismuth-telluride-based thermoelectric generators were lapped directly unto the rear contact of a 10 W crystalline Silicon-based module to harness the waste heat from the cell and convert it to electricity. The results showed that utilizing the thermoelectric modules decreased the photovoltaic cell temperature by 5.5%, indicating a reduction from 55°C to 52°C. Furthermore, the temperature reduction and additional power/energy generated by the thermoelectric module improved the stand-alone panel’s power output by 19%, signifying a rise from 8.78 W to 10.84 W. Finally, when the thermoelectric generators were added, the efficiency of the solar panel was increased from 11.6% to 14%, representing a 17% overall boost in the panel’s conversion efficiency.


## 1.3. Machine Learning Algorithms for Photovoltaic-Thermoelectric Design

Despite the commendable laboratory results that have shown the higher performance of the proposed hybrid photovoltaic-thermoelectric system, the commercialization of this technology is far from being achieved because of the scanty design information present in the current body of literature. By design information, we mean optimum geometry dimensions and operating conditions of the thermoelectric semiconductors for maximum waste heat removal from the rear contact of the photovoltaic cell. A reasonable explanation for this information deficit is due to the inefficient finite element tools used to solve the coupled thermal-electrical equations for the hybrid photovoltaic-thermoelectric system. Due to the complexity of the 3-dimensional governing equations and the high-resolution meshing needed to obtain volume average solutions from over a million nodes, the speed of obtaining results is extremely low and the computational energy requirement is enormous. A solution to this challenge in thermoelectric generator design has been proposed using machine learning algorithms [33–35]. However, the information obtained from thermoelectric design cannot be used to design photovoltaic-thermoelectric since the coupling equations are different [36]. Up till date, no research attempt has been made to speed up the design process of hybrid photovoltaic-thermoelectric systems. Hence, the aim of this research work.


## 1.4. Scientific Novelty

The literature review has established the hybrid photovoltaic-thermoelectric system’s higher performance than the stand-alone photovoltaic module. However, the mass production of efficient solar photovoltaic-thermoelectric is limited by certain problems that have not been addressed. These problems are:(i)
The design process for the hybrid photovoltaic-thermoelectric is very rigorous and expensive. It is time-consuming to obtain optimization information from the simulation software to design high-performing devices. Furthermore, the complex system equations are mostly solved using Multiphysics solvers that require high and expensive computational memory. Since the solvers rely on the finite element method to solve the three-dimensional governing equations, the solid models are discretized into million mesh elements that take a long time to solve and provide design information. Up till date, no research attempt has been made to mitigate this stumbling block to the design of high-performing solar photovoltaic-thermoelectric devices. This work proposes to fill this knowledge gap by replacing the conventional numerical solvers with deep neural networks inspired by artificial intelligence. The proposed deep neural network is trained using finite element data and is deployed to optimize the hybrid device’s performance for operating conditions within or outside the finite element data. Hence, the deep neural network replaces the time-consuming and energy-intensive finite element method(ii)
Majority of the previous optimization studies on solar photovoltaic-thermoelectric were based on crystalline silicon solar cells. Nevertheless, the silicon solar cell type was neglected in the analysis by assuming a constant temperature coefficient and reference efficiency. Therefore, it can be inferred that the current optimization information governing the design of hybrid photovoltaic-thermoelectric cannot be generalized to all silicon solar cell types. However, photovoltaic modules are characterized by diverse cell types. Hence, it is important to factor the solar cell type when producing optimization information to produce hybrid photovoltaic-thermoelectric. This work seeks to fill this research gap by providing a comprehensive optimization study that factors the effect of three conventional solar cell types. The optimization study is also modified to simultaneously examine the impacts of diverse operating conditions on the solar cell type to provide a more realistic design guideline for the manufacture of high-performing hybrid systemsThe subsequent parts of this work are arranged in the following manner: Section2 explains the physical photovoltaic-thermoelectric model, while the equations used to describe the numerical model and its validation are provided in the Supplementary Material. Section 3 discusses the results of the numerically inspired optimization study and the subsequent coefficient of determination and mean squared error of the optimum regressive deep neural network model proposed to substitute the inefficient numerical process. Finally, the significant findings extracted from the results discussed in this work are highlighted in Section 4.


## 2. Methodology

Figure1 shows the virtual models of the stand-alone and incorporated solar energy conversion systems considered during this research work. Figure 1(a) depicts the stand-alone solar photovoltaic system considered during this study. The solar cell is made from polycrystalline silicon, and its surface area is 40mm×40mm, according to the surface dimensions of the commercially available thermoelectric module surface dimensions [37, 38]. The module is exposed to concentrated solar irradiance from a compound parabolic concentrator with an optical concentrator of 10 [39, 40]. The global solar irradiance received by the optical concentrator is modeled using AM1.5 G conditions [41], which corresponds to 1 Sun. The back sheet of the photovoltaic module is cooled using a forced convective cooling mechanism via a water-cooling jet as specified in the virtual model using a convective heat transfer coefficient of 500 W/m2K [25]. The solar module is directly lapped onto an aluminum heat sink comprising 400 fins spread along the vertical and horizontal directions. The dimensions of the fins consisting of the heat sink are also depicted in Figure 1(a). The aluminum heat sink is attached to the photovoltaic module back sheet using a Dow Corning TC-5022 thermal conducting paste specified by a high thermal conductivity of 4 Wm-1 K-1 [42]. The incorporated system showing the solar photovoltaic module integrated with a thermoelectric module and the aluminum heat sink is shown in Figure 1(b). The thermoelectric module utilized in this work is a commercially available module produced by Melcor thermoelectric, HT 6-12-40, with geometry dimensions provided in our previous paper [43, 44]. The thermoelectric legs are bismuth-telluride semiconductors [45], which convert the thermal energy sapped from the solar photovoltaic module back sheet to electrical power. This study considers the temperature dependency of the thermoelectric material properties, measured from experiments [46], to ensure that the numerical results are accurate and verified. Polynomial expressions describing the effects of temperature on the semiconductor properties are available from reference [47]. Furthermore, the properties of other materials used to model the stand-alone and incorporated systems can be easily obtained from our previous work [48–54]. The photovoltaic-thermoelectric coupling equations, the numerical model verification, and the deep neural network equations are provided and well explained in the Supplementary Material accompanying this work.Figure 1
Representation of the PV-TE systems (a) stand-alone (b) hybrid.
(a)(b)


## 3. Results and Discussions

The deep neural network-inspired forecast integrated with the finite element method optimization technique has been implemented to predict the performance of the hybrid photovoltaic-thermoelectric system. The effects of various principal parameters on the performance parameters of the hybrid system were studied parametrically. These parameters were the concentrated solar irradiance, cold junction convective film coefficient, ambient temperature, wind speed, thermoelectric semiconductor height, andp-/n- semiconductor areas. The output parameters of the system were the photovoltaic cell temperature, thermoelectric generator temperature difference, thermoelectric power output, conversion efficiency, photovoltaic power and efficiency, and photovoltaic-thermoelectric power and efficiency. Three different photovoltaic cell types characterized by different temperature coefficients and reference efficiencies are included in the numerical model to determine their effects on the hybrid system’s performance. Finally, the performance, regression correlation, and computational time of the 20 deep neural networks with varying layers and hidden neurons are analyzed to select the best network configuration. The results of the numerical model validation, including mesh validation and comparison with experimental data, are provided in the Supplementary Information.


### 3.1. Numerical Optimization

#### 3.1.1. Photovoltaic-Thermoelectric with Cell 1

Figure2 presents the finite element optimization results of the hybrid system utilizing cell 1 as the effects of the principal operating parameters on the system performance are investigated. Overall, isolated variations of these parameters are observed to significantly affect the hybrid system performance, as reported in similar previous studies [31, 55]. Figures 2(a) and 2(b) depict the effect of concentrated solar irradiance on the system performance. It is noticed in Figure 2(a) that increasing the incident heat flux on the photovoltaic glass cover raises the temperature of the cell and the temperature difference between the thermoelectric generator’s junctions due to the increased thermal energy absorption by the materials that make up the system [56]. Consequently, the thermoelectric generator power and efficiency increased since a higher temperature difference benefit the electrical performance and conversion efficiency of the thermoelectric module following references [57, 58].Figure 2
Multiparametric optimization results of the hybrid photovoltaic-thermoelectric system made of cell 1 (a, b) concentrated solar irradiance (c, d) cold junction convective film coefficient (e, f) ambient temperature (g, h) wind flow speed (i, j) thermoelectric semiconductor height (k, l)n-thermoelectric semiconductor cross-sectional area (m, n) p-thermoelectric semiconductor cross-sectional area.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)On the other hand, Figure2(b) shows the reduction in the stand-alone photovoltaic and photovoltaic-thermoelectric power outputs and efficiencies due to an increase in concentrated solar irradiance. This reduction is because of the overheating of the solar panel due to the elevated cell temperature, which affects the material properties of the silicon semiconductor by increasing the energy of bound electrons, which implies the energy difference required to achieve excitation reduces. However, the power generation and efficiency of the photovoltaic module depend on the band gap between the excited and nonexcited states; hence, the stand-alone solar panel’s lower power output and efficiency at elevated temperatures [59]. However, the power and efficiency of the hybrid photovoltaic-thermoelectric system are slightly higher than that of the photovoltaic system. This disparity is because the power and efficiency obtained from the thermoelectric generator are not high enough to compensate for the rapidly falling stand-alone photovoltaic system performance. Also, only one thermoelectric module was used, and the concentrated solar irradiance was not very high (less than 20 Suns). Nevertheless, Khan et al. [18] reported that when ten thermoelectric modules were used under higher solar fluxes, the performance enhancement provided by the thermoelectric became more apparent. In all cases, the thermoelectric generator still contributed to the performance enhancement of the photovoltaic system.Figures2(c) and 2(d) depict the effect of convective film coefficients on the output performance parameters of the hybrid system. It is observed from Figure 2(c) that as the solar photovoltaic cell temperature drops, the other parameters increase with increasing cooling film coefficients, in agreement with previous results [25]. This increase is because better cooling film coefficients signify better heat removal capacity from the heat sink and higher system performance. Figure 2(d) also shows how the systems’ power and efficiency increase with better cooling coefficients, and the contribution of the power and efficiency to that of the hybrid system is made more evident due to the higher heat removal.Figures2(e) and 2(f) depict the effects of air temperature on the thermoelectric, photovoltaic, and photovoltaic-thermoelectric generating systems’ performances. It is observed in Figure 2(e) that, as the air temperature increases, the photovoltaic cell temperature increases rapidly while the thermoelectric temperature gradient rises slowly. The thermoelectric power and efficiency decrease because the system is operated in a hot environment [60]. The same trend is observed in solar photovoltaic and hybrid system performance, as portrayed in Figure 2(f), due to rising solar cell temperature [54].The effect of wind speed on the performance of the systems is graphically represented in Figures2(g) and 2(h). The plots show that the thermoelectric system performs worse in a windy environment, while the performance of photovoltaic and photovoltaic-thermoelectric systems thrives in a windy climate. This difference can be explained by considering that a windy environment contributes to the cooling of the photovoltaic system by reducing the heat retained at the top glass cover surface, consequently enhancing the system’s performance. However, for the thermoelectric generator system, the amount of heat that can be retained at the hot junction reduces, which causes the observed decrease in the thermoelectric temperature difference, and subsequent reduction in the system performance. The hybrid photovoltaic-thermoelectric system’s performance increases since it combines the larger photovoltaic and thermoelectric systems. Hence, this shows that the optimum operating conditions for the photovoltaic-thermoelectric and thermoelectric systems are different [33]; therefore, optimization studies like this are needed in designing high-performing hybrid systems.Figures2(i) and 2(j) illustrate the consequences of varying the thermoelectric semiconductor height on the system’s performance. Overall, it is observed that higher semiconductor heights are beneficial to the thermoelectric generator performance, while the photovoltaic-thermoelectric performs better with short semiconductors. The explanation for this observation is that longer semiconductors will develop a higher temperature difference due to the thermal spread between the heat retained at the device’s hot junction and the quantity of heat released at the cold junction. Nevertheless, in a short semiconductor, the thermal spread is greatly reduced because the thermal energy absorbed at the hot junction from the photovoltaic back sheet is immediately sent to the heat sink for heat dissipation and higher photovoltaic-thermoelectric performance. These findings are in perfect agreement with the work of the references [27, 61]. This result shows how the optimum operating conditions of the thermoelectric and photovoltaic-thermoelectric vary.The impacts of varying the cross-sectional areas on the performance are depicted in Figures2(k)–2(n). It was observed that changing either n- or p-semiconductors has the same effect on the system performance. This observation may be because of the bismuth-telluride semiconductors near similar electrical and thermal conductivities, as documented in reference [27]. Furthermore, it is noticed that the thermoelectric power generation and conversion efficiency are maximized for the prevalent operating conditions at optimum semiconductor cross-sectional areas of 0.7 mm2 and 0.56 mm2, respectively. This trend implies that the manufacturer’s specified area of 1.96 mm2 is 2.8 and 3.5 times higher than the area needed for maximum power and efficiency in a thermoelectric generator integrated into a photovoltaic-thermoelectric system, respectively. Hence, optimization studies like this need to facilitate the production of high-efficiency hybrid systems. Furthermore, it is observed in Figures 2(l) and 2(n) that, the performance of the photovoltaic and photovoltaic-thermoelectric systems increases as the semiconductor becomes thicker (increased area). This observation is because of the increased heat removal from the photovoltaic backplate due to the increased surface area contact with the thicker thermoelectric semiconductors. Therefore, the temperature of the solar cell drops with thicker semiconductors, as observed in Figures 2(k) and 2(m).


#### 3.1.2. Photovoltaic-Thermoelectric with Cells 2 and 3

The effects of the type of photovoltaic cell on the photovoltaic and photovoltaic-thermoelectric power and efficiency are shown in Figure3. This study is implemented by varying the previous input parameters while changing photovoltaic cells from types 2 and 3. In general, it is observed that the cell type plays a vital role in determining the performance of the hybrid system, which makes the system performance different from that of cell type 1. Figure 3(a) illustrates that as the concentrated solar irradiance is varied, the system efficiencies decrease linearly due to the temperature sensitivity of the silicon solar cell. Furthermore, it is observed that, for the same reasons, the system power output decreases parabolically as the solar irradiance increases. Beyond a critical concentrated solar irradiance of 14000 W/m2, the hybrid system’s power output gradually increases.Figure 3
Multi-parametric optimization results of multi-parametric optimization of hybrid photovoltaic-thermoelectric systems made of cells 2 and 3.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)On the other hand, Figure3(b) depicts that, for cell 3, the photovoltaic power and efficiency decrease with higher solar irradiances, and the hybrid system power and efficiency increase as the solar irradiance attains higher values. This disparity can be explained by considering that due to the inherent material properties of cell 3, the photovoltaic power output is relatively low compared to cells 1 and 2 and simply ranges from 2.5-2.95 W. However, the thermoelectric power, as plotted in Figure 2(a), is almost comparable to the photovoltaic power as it varies from 0-1.6 W. Therefore, the photovoltaic power of cell 3 does not exert a significant effect on the hybrid system’s capacity. This observation explains why the hybrid system’s power output still increases even though the photovoltaics power decreases.Figures3(c) and 3(d) present similar results as discussed in Figure 2(d) when the convective film coefficient was varied. This comparison implies that increasing the convective cooling film will always increase the system performance regardless of the type of solar cell. Nevertheless, the cell type determines the system’s performance since a visual inspection of Figures 2(d, 3(c), and 3(d) depict that cell 1 provides the highest efficiency and power while cell 3 provides minor performance. This observation is most likely due to the higher reference efficiency provided by cell 1 compared to other cell types, as found by reference [51]. Meanwhile, a similar trend is observed in Figures 2(f), 3(e), and 3(f), where the performance of all cell types considered in the analysis deteriorates with increasing air temperature. This deterioration is because of the same reasons explained in Section 3.1.1. However, cell 1 still provides the highest performance due to its relatively higher temperature coefficient and reference efficiency [62, 63].As the wind flow speed is increased in Figure3(g), it is observed that all system performance indices of cell 2 increase. However, Figure 3(h) depicts that increasing the flow speed in cell 3 decreases the photovoltaic system performance while increasing the photovoltaic-thermoelectric power and efficiency. This disparity is due to the higher contribution of the thermoelectric generator in cell 3. Again, it is observed that the trends in Figures 3(g) and 2(h) as the wind speed increases are similar because of the relatively higher temperature coefficients and reference efficiencies of cells 2 and 1, respectively, compared to cell 1.As the height of the thermoelectric semiconductor increases, the system power, and efficiencies of cells 1 and 2 depreciate, as depicted in Figures2(j) and 3(i). However, for cell 3 as illustrated in Figure 3(j), it is observed that the photovoltaic efficiency decreases as the semiconductor height increases. In contrast, the hybrid photovoltaic-thermoelectric power and efficiency appreciate with higher semiconductor heights. This abnormality is a significant result because, contrary to conventional knowledge, longer thermoelectric semiconductors degrade photovoltaic-thermoelectric performance [26, 27, 64]. However, cell 3 indicates that longer thermoelectric semiconductors are beneficial to the performance of the systems. Therefore, it cannot be generalized that shorter thermoelectric semiconductors are suitable for photovoltaic-thermoelectric performance. Instead, it is safe to say that it depends on the type of photovoltaic cell.The impacts of the thermoelectric semiconductor cross-sectional areas on the system’s performance for cells 2 and 3 are depicted in Figures3(k)–3(n). Overall, it is observed that the system’s parameters are very similar when either the n- or p-thermoelectric semiconductor cross-sectional areas are varied. Furthermore, the same trends were observed when cell 1 was used to design the system, and the cross-sectional areas were altered. This result demonstrates that thicker semiconductors will always perform better in a hybrid photovoltaic-thermoelectric system, irrespective of the solar cell type, than slimmer semiconductors.


### 3.2. The Optimal Deep Neural Network

After tuning the hyperparameters of the neural network in the Supplementary Information, it is concluded that the best network configuration to be used in learning the numerical-generated dataset is the 3-hidden layer neural network with 20 neurons per layer. This network feeds on the numerically generated dataset, and the network characteristics after training, testing, and deployment are provided in Figure4. The architecture of the optimal network is provided in Figure 4(a), and the network’s performance is depicted in Figure 4(b). From Figure 4(b) it is observed that the lowest mean squared error ever reported in this work (3.34×10−8) is obtained. This mean squared error is 99.67% more accurate than the default network (1 hidden layer with 10 neurons) as depicted in the Supplementary Information; hence, pointing out the necessity of the hyperparameter tuning carried out in this work. Furthermore, Figure 4(b) illustrates a perfect regression correlation between the optimum network forecasts and the numerically generated data set while recording 99.98% and approximately 100% regression correlations for the testing and all processes, respectively. Finally, Figure 4(d) points out that the entire numerical-generated dataset was perfectly learned in just 12 mins and 27 s. However, for the numerical solver, ANSYS software required an average computational time of 10 min to solve the complex system equations for just one step in one parameter. For 8 parameters and 20 substeps, this implies an average total computational time of 10min×8parameters×20substeps=1,600mins. Compared to the numerical process, it is observed that the optimum network is 128.51 times faster in accurately forecasting the photovoltaic-thermoelectric performance. This finding shows the necessity to rely on the deployed deep neural network regularized by the Bayesian algorithm to predict system performance since it is accurate, memory efficient, and faster than the computational time-consuming numerical method.Figure 4
Characteristics of the optimal deep neural network configuration (a) schematic network representation (b, c) network performance and regression (d) neural network training toolbox information.
(a)(b)(c)(d)


## 3.1. Numerical Optimization

### 3.1.1. Photovoltaic-Thermoelectric with Cell 1

Figure2 presents the finite element optimization results of the hybrid system utilizing cell 1 as the effects of the principal operating parameters on the system performance are investigated. Overall, isolated variations of these parameters are observed to significantly affect the hybrid system performance, as reported in similar previous studies [31, 55]. Figures 2(a) and 2(b) depict the effect of concentrated solar irradiance on the system performance. It is noticed in Figure 2(a) that increasing the incident heat flux on the photovoltaic glass cover raises the temperature of the cell and the temperature difference between the thermoelectric generator’s junctions due to the increased thermal energy absorption by the materials that make up the system [56]. Consequently, the thermoelectric generator power and efficiency increased since a higher temperature difference benefit the electrical performance and conversion efficiency of the thermoelectric module following references [57, 58].Figure 2
Multiparametric optimization results of the hybrid photovoltaic-thermoelectric system made of cell 1 (a, b) concentrated solar irradiance (c, d) cold junction convective film coefficient (e, f) ambient temperature (g, h) wind flow speed (i, j) thermoelectric semiconductor height (k, l)n-thermoelectric semiconductor cross-sectional area (m, n) p-thermoelectric semiconductor cross-sectional area.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)On the other hand, Figure2(b) shows the reduction in the stand-alone photovoltaic and photovoltaic-thermoelectric power outputs and efficiencies due to an increase in concentrated solar irradiance. This reduction is because of the overheating of the solar panel due to the elevated cell temperature, which affects the material properties of the silicon semiconductor by increasing the energy of bound electrons, which implies the energy difference required to achieve excitation reduces. However, the power generation and efficiency of the photovoltaic module depend on the band gap between the excited and nonexcited states; hence, the stand-alone solar panel’s lower power output and efficiency at elevated temperatures [59]. However, the power and efficiency of the hybrid photovoltaic-thermoelectric system are slightly higher than that of the photovoltaic system. This disparity is because the power and efficiency obtained from the thermoelectric generator are not high enough to compensate for the rapidly falling stand-alone photovoltaic system performance. Also, only one thermoelectric module was used, and the concentrated solar irradiance was not very high (less than 20 Suns). Nevertheless, Khan et al. [18] reported that when ten thermoelectric modules were used under higher solar fluxes, the performance enhancement provided by the thermoelectric became more apparent. In all cases, the thermoelectric generator still contributed to the performance enhancement of the photovoltaic system.Figures2(c) and 2(d) depict the effect of convective film coefficients on the output performance parameters of the hybrid system. It is observed from Figure 2(c) that as the solar photovoltaic cell temperature drops, the other parameters increase with increasing cooling film coefficients, in agreement with previous results [25]. This increase is because better cooling film coefficients signify better heat removal capacity from the heat sink and higher system performance. Figure 2(d) also shows how the systems’ power and efficiency increase with better cooling coefficients, and the contribution of the power and efficiency to that of the hybrid system is made more evident due to the higher heat removal.Figures2(e) and 2(f) depict the effects of air temperature on the thermoelectric, photovoltaic, and photovoltaic-thermoelectric generating systems’ performances. It is observed in Figure 2(e) that, as the air temperature increases, the photovoltaic cell temperature increases rapidly while the thermoelectric temperature gradient rises slowly. The thermoelectric power and efficiency decrease because the system is operated in a hot environment [60]. The same trend is observed in solar photovoltaic and hybrid system performance, as portrayed in Figure 2(f), due to rising solar cell temperature [54].The effect of wind speed on the performance of the systems is graphically represented in Figures2(g) and 2(h). The plots show that the thermoelectric system performs worse in a windy environment, while the performance of photovoltaic and photovoltaic-thermoelectric systems thrives in a windy climate. This difference can be explained by considering that a windy environment contributes to the cooling of the photovoltaic system by reducing the heat retained at the top glass cover surface, consequently enhancing the system’s performance. However, for the thermoelectric generator system, the amount of heat that can be retained at the hot junction reduces, which causes the observed decrease in the thermoelectric temperature difference, and subsequent reduction in the system performance. The hybrid photovoltaic-thermoelectric system’s performance increases since it combines the larger photovoltaic and thermoelectric systems. Hence, this shows that the optimum operating conditions for the photovoltaic-thermoelectric and thermoelectric systems are different [33]; therefore, optimization studies like this are needed in designing high-performing hybrid systems.Figures2(i) and 2(j) illustrate the consequences of varying the thermoelectric semiconductor height on the system’s performance. Overall, it is observed that higher semiconductor heights are beneficial to the thermoelectric generator performance, while the photovoltaic-thermoelectric performs better with short semiconductors. The explanation for this observation is that longer semiconductors will develop a higher temperature difference due to the thermal spread between the heat retained at the device’s hot junction and the quantity of heat released at the cold junction. Nevertheless, in a short semiconductor, the thermal spread is greatly reduced because the thermal energy absorbed at the hot junction from the photovoltaic back sheet is immediately sent to the heat sink for heat dissipation and higher photovoltaic-thermoelectric performance. These findings are in perfect agreement with the work of the references [27, 61]. This result shows how the optimum operating conditions of the thermoelectric and photovoltaic-thermoelectric vary.The impacts of varying the cross-sectional areas on the performance are depicted in Figures2(k)–2(n). It was observed that changing either n- or p-semiconductors has the same effect on the system performance. This observation may be because of the bismuth-telluride semiconductors near similar electrical and thermal conductivities, as documented in reference [27]. Furthermore, it is noticed that the thermoelectric power generation and conversion efficiency are maximized for the prevalent operating conditions at optimum semiconductor cross-sectional areas of 0.7 mm2 and 0.56 mm2, respectively. This trend implies that the manufacturer’s specified area of 1.96 mm2 is 2.8 and 3.5 times higher than the area needed for maximum power and efficiency in a thermoelectric generator integrated into a photovoltaic-thermoelectric system, respectively. Hence, optimization studies like this need to facilitate the production of high-efficiency hybrid systems. Furthermore, it is observed in Figures 2(l) and 2(n) that, the performance of the photovoltaic and photovoltaic-thermoelectric systems increases as the semiconductor becomes thicker (increased area). This observation is because of the increased heat removal from the photovoltaic backplate due to the increased surface area contact with the thicker thermoelectric semiconductors. Therefore, the temperature of the solar cell drops with thicker semiconductors, as observed in Figures 2(k) and 2(m).


### 3.1.2. Photovoltaic-Thermoelectric with Cells 2 and 3

The effects of the type of photovoltaic cell on the photovoltaic and photovoltaic-thermoelectric power and efficiency are shown in Figure3. This study is implemented by varying the previous input parameters while changing photovoltaic cells from types 2 and 3. In general, it is observed that the cell type plays a vital role in determining the performance of the hybrid system, which makes the system performance different from that of cell type 1. Figure 3(a) illustrates that as the concentrated solar irradiance is varied, the system efficiencies decrease linearly due to the temperature sensitivity of the silicon solar cell. Furthermore, it is observed that, for the same reasons, the system power output decreases parabolically as the solar irradiance increases. Beyond a critical concentrated solar irradiance of 14000 W/m2, the hybrid system’s power output gradually increases.Figure 3
Multi-parametric optimization results of multi-parametric optimization of hybrid photovoltaic-thermoelectric systems made of cells 2 and 3.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)On the other hand, Figure3(b) depicts that, for cell 3, the photovoltaic power and efficiency decrease with higher solar irradiances, and the hybrid system power and efficiency increase as the solar irradiance attains higher values. This disparity can be explained by considering that due to the inherent material properties of cell 3, the photovoltaic power output is relatively low compared to cells 1 and 2 and simply ranges from 2.5-2.95 W. However, the thermoelectric power, as plotted in Figure 2(a), is almost comparable to the photovoltaic power as it varies from 0-1.6 W. Therefore, the photovoltaic power of cell 3 does not exert a significant effect on the hybrid system’s capacity. This observation explains why the hybrid system’s power output still increases even though the photovoltaics power decreases.Figures3(c) and 3(d) present similar results as discussed in Figure 2(d) when the convective film coefficient was varied. This comparison implies that increasing the convective cooling film will always increase the system performance regardless of the type of solar cell. Nevertheless, the cell type determines the system’s performance since a visual inspection of Figures 2(d, 3(c), and 3(d) depict that cell 1 provides the highest efficiency and power while cell 3 provides minor performance. This observation is most likely due to the higher reference efficiency provided by cell 1 compared to other cell types, as found by reference [51]. Meanwhile, a similar trend is observed in Figures 2(f), 3(e), and 3(f), where the performance of all cell types considered in the analysis deteriorates with increasing air temperature. This deterioration is because of the same reasons explained in Section 3.1.1. However, cell 1 still provides the highest performance due to its relatively higher temperature coefficient and reference efficiency [62, 63].As the wind flow speed is increased in Figure3(g), it is observed that all system performance indices of cell 2 increase. However, Figure 3(h) depicts that increasing the flow speed in cell 3 decreases the photovoltaic system performance while increasing the photovoltaic-thermoelectric power and efficiency. This disparity is due to the higher contribution of the thermoelectric generator in cell 3. Again, it is observed that the trends in Figures 3(g) and 2(h) as the wind speed increases are similar because of the relatively higher temperature coefficients and reference efficiencies of cells 2 and 1, respectively, compared to cell 1.As the height of the thermoelectric semiconductor increases, the system power, and efficiencies of cells 1 and 2 depreciate, as depicted in Figures2(j) and 3(i). However, for cell 3 as illustrated in Figure 3(j), it is observed that the photovoltaic efficiency decreases as the semiconductor height increases. In contrast, the hybrid photovoltaic-thermoelectric power and efficiency appreciate with higher semiconductor heights. This abnormality is a significant result because, contrary to conventional knowledge, longer thermoelectric semiconductors degrade photovoltaic-thermoelectric performance [26, 27, 64]. However, cell 3 indicates that longer thermoelectric semiconductors are beneficial to the performance of the systems. Therefore, it cannot be generalized that shorter thermoelectric semiconductors are suitable for photovoltaic-thermoelectric performance. Instead, it is safe to say that it depends on the type of photovoltaic cell.The impacts of the thermoelectric semiconductor cross-sectional areas on the system’s performance for cells 2 and 3 are depicted in Figures3(k)–3(n). Overall, it is observed that the system’s parameters are very similar when either the n- or p-thermoelectric semiconductor cross-sectional areas are varied. Furthermore, the same trends were observed when cell 1 was used to design the system, and the cross-sectional areas were altered. This result demonstrates that thicker semiconductors will always perform better in a hybrid photovoltaic-thermoelectric system, irrespective of the solar cell type, than slimmer semiconductors.


## 3.1.1. Photovoltaic-Thermoelectric with Cell 1

Figure2 presents the finite element optimization results of the hybrid system utilizing cell 1 as the effects of the principal operating parameters on the system performance are investigated. Overall, isolated variations of these parameters are observed to significantly affect the hybrid system performance, as reported in similar previous studies [31, 55]. Figures 2(a) and 2(b) depict the effect of concentrated solar irradiance on the system performance. It is noticed in Figure 2(a) that increasing the incident heat flux on the photovoltaic glass cover raises the temperature of the cell and the temperature difference between the thermoelectric generator’s junctions due to the increased thermal energy absorption by the materials that make up the system [56]. Consequently, the thermoelectric generator power and efficiency increased since a higher temperature difference benefit the electrical performance and conversion efficiency of the thermoelectric module following references [57, 58].Figure 2
Multiparametric optimization results of the hybrid photovoltaic-thermoelectric system made of cell 1 (a, b) concentrated solar irradiance (c, d) cold junction convective film coefficient (e, f) ambient temperature (g, h) wind flow speed (i, j) thermoelectric semiconductor height (k, l)n-thermoelectric semiconductor cross-sectional area (m, n) p-thermoelectric semiconductor cross-sectional area.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)On the other hand, Figure2(b) shows the reduction in the stand-alone photovoltaic and photovoltaic-thermoelectric power outputs and efficiencies due to an increase in concentrated solar irradiance. This reduction is because of the overheating of the solar panel due to the elevated cell temperature, which affects the material properties of the silicon semiconductor by increasing the energy of bound electrons, which implies the energy difference required to achieve excitation reduces. However, the power generation and efficiency of the photovoltaic module depend on the band gap between the excited and nonexcited states; hence, the stand-alone solar panel’s lower power output and efficiency at elevated temperatures [59]. However, the power and efficiency of the hybrid photovoltaic-thermoelectric system are slightly higher than that of the photovoltaic system. This disparity is because the power and efficiency obtained from the thermoelectric generator are not high enough to compensate for the rapidly falling stand-alone photovoltaic system performance. Also, only one thermoelectric module was used, and the concentrated solar irradiance was not very high (less than 20 Suns). Nevertheless, Khan et al. [18] reported that when ten thermoelectric modules were used under higher solar fluxes, the performance enhancement provided by the thermoelectric became more apparent. In all cases, the thermoelectric generator still contributed to the performance enhancement of the photovoltaic system.Figures2(c) and 2(d) depict the effect of convective film coefficients on the output performance parameters of the hybrid system. It is observed from Figure 2(c) that as the solar photovoltaic cell temperature drops, the other parameters increase with increasing cooling film coefficients, in agreement with previous results [25]. This increase is because better cooling film coefficients signify better heat removal capacity from the heat sink and higher system performance. Figure 2(d) also shows how the systems’ power and efficiency increase with better cooling coefficients, and the contribution of the power and efficiency to that of the hybrid system is made more evident due to the higher heat removal.Figures2(e) and 2(f) depict the effects of air temperature on the thermoelectric, photovoltaic, and photovoltaic-thermoelectric generating systems’ performances. It is observed in Figure 2(e) that, as the air temperature increases, the photovoltaic cell temperature increases rapidly while the thermoelectric temperature gradient rises slowly. The thermoelectric power and efficiency decrease because the system is operated in a hot environment [60]. The same trend is observed in solar photovoltaic and hybrid system performance, as portrayed in Figure 2(f), due to rising solar cell temperature [54].The effect of wind speed on the performance of the systems is graphically represented in Figures2(g) and 2(h). The plots show that the thermoelectric system performs worse in a windy environment, while the performance of photovoltaic and photovoltaic-thermoelectric systems thrives in a windy climate. This difference can be explained by considering that a windy environment contributes to the cooling of the photovoltaic system by reducing the heat retained at the top glass cover surface, consequently enhancing the system’s performance. However, for the thermoelectric generator system, the amount of heat that can be retained at the hot junction reduces, which causes the observed decrease in the thermoelectric temperature difference, and subsequent reduction in the system performance. The hybrid photovoltaic-thermoelectric system’s performance increases since it combines the larger photovoltaic and thermoelectric systems. Hence, this shows that the optimum operating conditions for the photovoltaic-thermoelectric and thermoelectric systems are different [33]; therefore, optimization studies like this are needed in designing high-performing hybrid systems.Figures2(i) and 2(j) illustrate the consequences of varying the thermoelectric semiconductor height on the system’s performance. Overall, it is observed that higher semiconductor heights are beneficial to the thermoelectric generator performance, while the photovoltaic-thermoelectric performs better with short semiconductors. The explanation for this observation is that longer semiconductors will develop a higher temperature difference due to the thermal spread between the heat retained at the device’s hot junction and the quantity of heat released at the cold junction. Nevertheless, in a short semiconductor, the thermal spread is greatly reduced because the thermal energy absorbed at the hot junction from the photovoltaic back sheet is immediately sent to the heat sink for heat dissipation and higher photovoltaic-thermoelectric performance. These findings are in perfect agreement with the work of the references [27, 61]. This result shows how the optimum operating conditions of the thermoelectric and photovoltaic-thermoelectric vary.The impacts of varying the cross-sectional areas on the performance are depicted in Figures2(k)–2(n). It was observed that changing either n- or p-semiconductors has the same effect on the system performance. This observation may be because of the bismuth-telluride semiconductors near similar electrical and thermal conductivities, as documented in reference [27]. Furthermore, it is noticed that the thermoelectric power generation and conversion efficiency are maximized for the prevalent operating conditions at optimum semiconductor cross-sectional areas of 0.7 mm2 and 0.56 mm2, respectively. This trend implies that the manufacturer’s specified area of 1.96 mm2 is 2.8 and 3.5 times higher than the area needed for maximum power and efficiency in a thermoelectric generator integrated into a photovoltaic-thermoelectric system, respectively. Hence, optimization studies like this need to facilitate the production of high-efficiency hybrid systems. Furthermore, it is observed in Figures 2(l) and 2(n) that, the performance of the photovoltaic and photovoltaic-thermoelectric systems increases as the semiconductor becomes thicker (increased area). This observation is because of the increased heat removal from the photovoltaic backplate due to the increased surface area contact with the thicker thermoelectric semiconductors. Therefore, the temperature of the solar cell drops with thicker semiconductors, as observed in Figures 2(k) and 2(m).


## 3.1.2. Photovoltaic-Thermoelectric with Cells 2 and 3

The effects of the type of photovoltaic cell on the photovoltaic and photovoltaic-thermoelectric power and efficiency are shown in Figure3. This study is implemented by varying the previous input parameters while changing photovoltaic cells from types 2 and 3. In general, it is observed that the cell type plays a vital role in determining the performance of the hybrid system, which makes the system performance different from that of cell type 1. Figure 3(a) illustrates that as the concentrated solar irradiance is varied, the system efficiencies decrease linearly due to the temperature sensitivity of the silicon solar cell. Furthermore, it is observed that, for the same reasons, the system power output decreases parabolically as the solar irradiance increases. Beyond a critical concentrated solar irradiance of 14000 W/m2, the hybrid system’s power output gradually increases.Figure 3
Multi-parametric optimization results of multi-parametric optimization of hybrid photovoltaic-thermoelectric systems made of cells 2 and 3.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)On the other hand, Figure3(b) depicts that, for cell 3, the photovoltaic power and efficiency decrease with higher solar irradiances, and the hybrid system power and efficiency increase as the solar irradiance attains higher values. This disparity can be explained by considering that due to the inherent material properties of cell 3, the photovoltaic power output is relatively low compared to cells 1 and 2 and simply ranges from 2.5-2.95 W. However, the thermoelectric power, as plotted in Figure 2(a), is almost comparable to the photovoltaic power as it varies from 0-1.6 W. Therefore, the photovoltaic power of cell 3 does not exert a significant effect on the hybrid system’s capacity. This observation explains why the hybrid system’s power output still increases even though the photovoltaics power decreases.Figures3(c) and 3(d) present similar results as discussed in Figure 2(d) when the convective film coefficient was varied. This comparison implies that increasing the convective cooling film will always increase the system performance regardless of the type of solar cell. Nevertheless, the cell type determines the system’s performance since a visual inspection of Figures 2(d, 3(c), and 3(d) depict that cell 1 provides the highest efficiency and power while cell 3 provides minor performance. This observation is most likely due to the higher reference efficiency provided by cell 1 compared to other cell types, as found by reference [51]. Meanwhile, a similar trend is observed in Figures 2(f), 3(e), and 3(f), where the performance of all cell types considered in the analysis deteriorates with increasing air temperature. This deterioration is because of the same reasons explained in Section 3.1.1. However, cell 1 still provides the highest performance due to its relatively higher temperature coefficient and reference efficiency [62, 63].As the wind flow speed is increased in Figure3(g), it is observed that all system performance indices of cell 2 increase. However, Figure 3(h) depicts that increasing the flow speed in cell 3 decreases the photovoltaic system performance while increasing the photovoltaic-thermoelectric power and efficiency. This disparity is due to the higher contribution of the thermoelectric generator in cell 3. Again, it is observed that the trends in Figures 3(g) and 2(h) as the wind speed increases are similar because of the relatively higher temperature coefficients and reference efficiencies of cells 2 and 1, respectively, compared to cell 1.As the height of the thermoelectric semiconductor increases, the system power, and efficiencies of cells 1 and 2 depreciate, as depicted in Figures2(j) and 3(i). However, for cell 3 as illustrated in Figure 3(j), it is observed that the photovoltaic efficiency decreases as the semiconductor height increases. In contrast, the hybrid photovoltaic-thermoelectric power and efficiency appreciate with higher semiconductor heights. This abnormality is a significant result because, contrary to conventional knowledge, longer thermoelectric semiconductors degrade photovoltaic-thermoelectric performance [26, 27, 64]. However, cell 3 indicates that longer thermoelectric semiconductors are beneficial to the performance of the systems. Therefore, it cannot be generalized that shorter thermoelectric semiconductors are suitable for photovoltaic-thermoelectric performance. Instead, it is safe to say that it depends on the type of photovoltaic cell.The impacts of the thermoelectric semiconductor cross-sectional areas on the system’s performance for cells 2 and 3 are depicted in Figures3(k)–3(n). Overall, it is observed that the system’s parameters are very similar when either the n- or p-thermoelectric semiconductor cross-sectional areas are varied. Furthermore, the same trends were observed when cell 1 was used to design the system, and the cross-sectional areas were altered. This result demonstrates that thicker semiconductors will always perform better in a hybrid photovoltaic-thermoelectric system, irrespective of the solar cell type, than slimmer semiconductors.


## 3.2. The Optimal Deep Neural Network

After tuning the hyperparameters of the neural network in the Supplementary Information, it is concluded that the best network configuration to be used in learning the numerical-generated dataset is the 3-hidden layer neural network with 20 neurons per layer. This network feeds on the numerically generated dataset, and the network characteristics after training, testing, and deployment are provided in Figure4. The architecture of the optimal network is provided in Figure 4(a), and the network’s performance is depicted in Figure 4(b). From Figure 4(b) it is observed that the lowest mean squared error ever reported in this work (3.34×10−8) is obtained. This mean squared error is 99.67% more accurate than the default network (1 hidden layer with 10 neurons) as depicted in the Supplementary Information; hence, pointing out the necessity of the hyperparameter tuning carried out in this work. Furthermore, Figure 4(b) illustrates a perfect regression correlation between the optimum network forecasts and the numerically generated data set while recording 99.98% and approximately 100% regression correlations for the testing and all processes, respectively. Finally, Figure 4(d) points out that the entire numerical-generated dataset was perfectly learned in just 12 mins and 27 s. However, for the numerical solver, ANSYS software required an average computational time of 10 min to solve the complex system equations for just one step in one parameter. For 8 parameters and 20 substeps, this implies an average total computational time of 10min×8parameters×20substeps=1,600mins. Compared to the numerical process, it is observed that the optimum network is 128.51 times faster in accurately forecasting the photovoltaic-thermoelectric performance. This finding shows the necessity to rely on the deployed deep neural network regularized by the Bayesian algorithm to predict system performance since it is accurate, memory efficient, and faster than the computational time-consuming numerical method.Figure 4
Characteristics of the optimal deep neural network configuration (a) schematic network representation (b, c) network performance and regression (d) neural network training toolbox information.
(a)(b)(c)(d)


## 4. Conclusions

This study is aimed at providing a more efficient means of predicting the performance of a solar photovoltaic thermal power plant by replacing the current numerical methods with deep neural networks. The motivation for this replacement was that the deep neural network would predict the hybrid system’s performance much faster and more efficiently than the current numerical method. This work was the first to propose data-driven models to predict photovoltaic-thermoelectric performance. The data used to train the deep neural network was generated using the ANSYS numerical solver while considering several principal operating conditions that have been neglected in previous studies, making the numerical results very accurate. The data generated by the numerical solver were fed to a hyperparameter-tuned 3-layer deep neural network with 20 neurons per layer, and the system’s performance was forecasted with very high accuracies. In the end, the optimum parameters of the solar photovoltaic-thermoelectrics and the computational times of the numerical method and the deep neural network are compared. Based on the results obtained, the following conclusions are made:(i)
The effects of the different operating parameters on the system’s performance significantly differ when the impact of the solar cell type is considered in the analysis. The optimum operating conditions and geometry parameters for maximum performance using the three different types of solar cells were obtained(a)
For instance, although shorter thermoelectric semiconductors perform better in conventional cells 1 and 2, longer thermoelectric semiconductors perform better in cell 3(b)
Furthermore, although the concentrated solar irradiance diminished the hybrid system’s performance when cells 1 and 2 were used, it was observed that the combined system’s performance was better when higher concentrated solar irradiances were absorbed by cell 3(c)
Finally, while higher wind flow speeds were beneficial to the combined system’s performance when cells 1 and 2 were used, it was observed that higher wind speeds reduced the hybrid system power and efficiency when solar cell 3 was adopted(ii)
Regarding the optimum deep neural network obtained after hyperparameter tuning, the following conclusions are made:(a)
The optimum deep neural network model used to learn the numerically generated data improved the default network’s accuracy by 99.67%.(b)
Furthermore, the optimized deep neural network obtained the lowest mean squared error of3.34×10−8 reported in this work(c)
In addition, a high regression correlation of 100% was obtained between the numerical data set and the neural network’s prediction during the training process(d)
Finally, the optimized neural network learned the entire numerical dataset that took 1600 mins to generate in just 12 mins and 27 s, making it 128.51 times faster than the numerical method in forecasting the photovoltaic-thermoelectric performance(iii)
For a future study, the prototype of the photovoltaic-thermoelectric systems proposed in this work will be experimentally investigated to determine their performance. Additionally, the deep neural network model will be sold to solar cell manufacturing companies to speed up the design process of high-performing customized hybrid systems



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*Source: 1990593-2023-02-10.xml*