Machine learning-based thermo-electrical performance improvement of nanofluid-cooled photovoltaic

Abstract

Hybrid photovoltaic–thermal (hPVT) collectors are devices that allow the conversion of sun energy into useful thermal and electrical energy simultaneously. The power obtained from the photovoltaic (PV) module introduces random fluctuations into the system. While obtaining the data for PV power output in advance and for reducing the impact of random fluctuations, exact day-ahead PV power prediction is crucial. Machine learning algorithms have been proven an effective tool in PV technology for day-ahead prediction of PV-power output. This research employs the Gaussian process regression method using the Matlab environment for forecasting the hPVT collector's performance operating with pure water and Fe/water nanofluid. A one-year historical data pertaining to solar irradiance as well as ambient temperature for Roorkee (29.8543 °N, 77.8880 °E), India location has been used to validate the proposed model. This data is utilized for day-ahead forecasting of solar irradiance and ambient temperature. The outcome elucidates that as the mass-flow rate increases, the thermo-electric performance of the hPVT collector enhances. Raising the mass-flow rate of Fe/water nanofluid from 0.01 to 0.1 kg/s, the cell temperature decreases by 9.35 °C and 9.47 °C, respectively, for the actual and predicted data. The thermal, electrical, as well as overall efficiency of the hPVT collector, improves by 2.73%, 7.11%, and 9.84%, respectively, using Fe/water nanofluid (ϕ = 2%) in contrast to the water-cooled PVT system. Finally, results demonstrate that the outcomes obtained using the forecasted data closely follow the results obtained using the actual data. In conclusion, this analysis provides a comprehensive solution for utilizing nanofluids as a coolant in the most cost-effective ways.

Keywords

Hybrid photovoltaic–thermal nanofluids machine learning performance forecasting

Introduction

Nowadays, renewable energy technologies, particularly solar energy, have been proven a promising alternative to conventional energy sources (coal, natural gas, oil, nuclear), as it provides clean and low-carbon electricity. The electricity generation using solar energy is inexhaustible in nature, reliable, and does not produce any hazardous waste or greenhouse gases, as reported by Ağbulut et al.¹ Therefore, its utilization has been extensively increased globally. Solar energy can be extracted in the form of electricity and heat by the photovoltaic (PV) panel and solar thermal (ST) panel, respectively. The PV cell's conversion efficiency is usually very low (14–19%), and the remaining sun's energy is wasted as heat. Diwania et al.² mentioned that with the rising cell temperature, the PV electrical efficiency (η_EL) and cell lifetime reduces.

The PV and ST technologies can be integrated comprehensively using a hybrid PV–thermal (hPVT) approach in a single piece of equipment and provides electrical and thermal energy simultaneously. In the hPVT system, the PV panel is cooled by connecting a fluid stream (air, liquid) that absorbs the majority of the heat and cools down the PV panel, resulting in better η_EL. By using a fluid stream below the PV cell, the PV cell efficiency improves by 4–14%, as reported by Ramos et al.³ This permits for better overall efficiency (η_T) of hPVT collector as compared to PV and ST modules individually. Barbu et al.⁴ mentioned in their study that the PVT collector technology is a kind of micro-cogeneration technology capable of producing decentralized clean heat and energy for the end consumers and can be integrated into domestic homes in an efficient manner. Generally, water is proven to be a useful heat transfer fluid (HTF) due to its low cost, high thermal capacity, and non-toxic nature. However, it offers several shortcomings, such as freezing, rusting, and erosion, as reported by Hooshmandzade et al.⁵ Due to these shortcomings, the focus shifted toward the usage of nanofluids (NFs) as a coolant in hPVT systems. NFs are the mixture of nanoparticles (metal, metal-oxide) of nano-scale size with base fluid (water, ethylene glycol, oil). Figure 1 shows the schematic representation of a typical hPVT system.

Figure 1.

Schematic diagram of hybrid photovoltaic–thermal (hPVT) system with nanofluid.

Abbas et al.⁶ mentioned that nanoparticles could enhance the thermal conductivity, heat transfer capacity, and thermos-physical properties of conventional fluids such as water and oil. In recent times, NFs or nano-coolants are considered for efficiency improvement of hPVT systems and replace the conventional heat transfer fluids (glycol, ethylene, water, etc.). Nanoparticles can prove to be a better solution for removing rust and corrosion from the tube caused due to water cooling. The thermodynamic properties of NF are enhanced due to Brownian motion, transient effect, and electrical force between nanoparticles.

The thermo-electrical performance of various hPVT systems under the influence of various kinds of NFs has been investigated experimentally and numerically.⁷ Using nano-ferrofluids coolant, Ghadiri et al.⁸ reported an improvement of 79% in η_T as opposed to a conventional PVT system. Firoozzadeh et al.⁹ investigated the impact of NFs prepared by mixing carbon black nanoparticles into water. The reported output power for the proposed system was 7% and 5.4% higher than the water-cooled system and conventional PV system. Maadi et al.¹⁰ investigate the impact of different metallic and metalloid water-based NFs on the thermal and electrical performances of hPVT systems. The experimental and 2D numerical study shows that the ZnO–water NF helps in maintaining the optimal thermal exergy efficiency while SiO₂–water NF had the lowest thermal exergy efficiency. Hissouf et al.¹¹ performed a numerical investigation on hPVT systems using Cu and alumina nanoparticles in different proportions. The results obtained using numerical simulation were compared with experimental data. The hPVT system with Cu–water NF shows better performance. Using (2 vol.%) of Cu–water NF, the optimum η_EL and η _Thermal were 1.9% and 4.1%. Hader and Al-Kouz¹² performed a numerical study showing the impact of particle volume fraction (V_f) of Al₂O₃ nanoparticles on the thermal characteristics of hPVT systems. In another study, Rajaee et al.⁷ worked on an hPVT system integrated with thermoelectric (TE) and phase change material (PCM) to improve the overall yield. The outcome of the investigation reveals that the PVT system with NF alone shows better performance as compared to other configurations.

Several other methods were adopted by researchers to improve the hPVT system's performance. Kolahan et al.¹³ performed a 3D numerical investigation on an hPVT system integrated with thermoelectric (PVT-TE) using Al₂O₃–water (0.2 wt.%) as NFs. The impact of various parameters such as mass flow rate (m_f), solar insolation, ambient temperature, and inlet temperature was investigated. The outcome shows that the integration of the thermoelectric unit with the hPVT system improves the η_T by 2.5–4%. Nazri et al.¹⁴ performed the experimental and 1D numerical study on the PVT-TE system and examined the impact of changing m_f on the exergetic efficiency of the system.

The inclusion of NFs as a coolant in hPVT systems offers several shortcomings, such as fluid instability due to agglomeration, corrosion, high system cost, and an increase in energy needed to force the nanofluid as a result of an increase in fluid thickness. Abbas et al.⁶ highlighted the parameters which affect the hPVT system's performance, such as the base fluid type, nanoparticles volume concentration in the base fluid, nanoparticles size, and the velocity of fluid flowing through the tube assembly.

As observed from the literature, the performance of hPVT system is better using NFs as a coolant in comparison to conventional HTFs. Recently, several researchers highlighted the capability of hybrid NFs in enhancing the overall performance of hPVT systems. Hybrid NFs outperform conventional NFs in heat transfer applications. Wole-Osho et al.¹⁵ experimentally investigated the impact of (Al₂O₃-ZnO) hybrid NFs on the overall performance of the hPVT system. The findings depict a 31% enhancement in overall η _Thermal in comparison to water cooling. Han et al.¹⁶ adopted merit function (MF) to examine the effectiveness of converting solar energy into thermo-electrical energy using Ag/COSO₄ NF. The results demonstrated that in comparison to the conventional fluid the MFR of Ag/COSO₄ NF achieved a 35.9% value. Further, the thermo-electrical performance of the hPVT system using Ag/COSO₄ NF is significantly better than the Ag/water NF. The impact of Ag–SiO₂ nanoparticles in base fluid on the performance of the hPVT system was investigated by Hjerrild et al.¹⁷ The result shows that using hybrid NFs (Ag–SiO₂), the η_EL of the hPVT system improves by 6.6% concerning the water cooling. The authors also investigate the V_f and MFR of fluid on the hPVT system performance. Okonkwo et al.¹⁸ performed a thermodynamic analysis investigation of the performance of the hPVT system using Al₂O₃/water and Al₂O₃ + Fe/water NFs. The impact of changing MFR, V_f of nanoparticles in water, and temperature were observed on the system's performance. According to the results, it has been concluded that hybrid NFs help in achieving better exergetic efficiency (η_Ex), while the η _Thermal is better in the case of mono-fluid.

Nevertheless, to say from the available literature that NFs impact the thermo-electrical performance of the hPVT system is still indecisive. There are some data that suggest the contrary in terms of electrical and thermal performance. Moreover, studies show that the basic assumptions do not apply to every NF, volume concentration, or inlet fluid velocity. Therefore, new studies are needed to probe the impact of variable parameters on the thermo-electrical performance of the hPVT system.

Predicting the performance of hPVT systems using artificial intelligence (AI) techniques has been a crucial option in recent decades to get rid of the lengthy mathematical modeling and its limitations. As a result, the benefits of lowering processing time and achieving high prediction accuracy are the motivation for using machine learning algorithms. Varol et al.¹⁹ forecast the thermal performance of PCM in solar collectors using ANN, ANFIS, and SVM methods. Ahmadi et al.²⁰ predict the thermo-electrical performance of the hPVT system using two models based on ANN and least square support vector machine (LV-SVM) methods. The system performs better using LV-SVM methods. Cao et al.²¹ employed a machine learning algorithm to simulate the η_EL of the hPVT system cooled by NFs. Six intelligent models, such as LS-SVR, ANFIS, and four ANN types were considered for the study. The optimum η_EL of the hPVT system is 27% using the ANFIS model.

Most of the studies used for forecasting the hPVT system performance using AI techniques use fewer samples in the training and validation process due to the lack of experimental data. In this study, the Gaussian process regression (GPR) method is used for day-ahead forecasting to investigate the performance of the hPVT collector utilizing pure water and Fe/water NF as coolant. A one-year historical data pertaining to solar irradiance as well as ambient temperature for Roorkee (29.8543 °N, 77.8880 °E), India location has been used to validate the proposed model. The methodology of the proposed work is shown in Figure 2. This overcomes the limitation of the studies mentioned in the literature, which use a limited number of samples to train the model.

Figure 2.

Proposed methodology.

Materials and methods

The study demonstrates the mathematical modeling of a nano-cooled PVT system and to implement the GPR method for day-ahead forecasting of irradiation intensity and ambient temperature.²² GPR has been extensively applied to predict the PV power output because it provides superior estimation accuracy as compared to parametric regression. Further, it is able to handle and process a wide variety of supervised data learning problems, even with a small set of available data.²³ A one-year historical data pertaining to solar irradiance and the ambient temperature has been considered for a day ahead forecasting using the GPR model, which is then utilized to calculate the η_EL and η _Thermal of hPVT collector using Fe/water NF and water as coolant. This analysis provides a comprehensive solution for utilizing the NFs in the most cost-effective ways. The performance of the hPVT system based upon forecasted data is compared with actual data so that the analysis can be done for improved performance as well as an economically viable system.

Heat balance

Radiation, convection, and conduction are the three forms of heat transfer processes through which the thermal energy circulates across the different hPVT layers. The heat transfer process that takes place in the layers of the hPVT module is represented in Figure 3. Radiation transmits heat without having physical contact. Any physical body generates radiation in the form of electromagnetic waves. These radiations can be absorbed, reflected, or transmitted when they hit a body. In the convection process, the energy transfers within a fluid from higher to lower temperature levels. Natural convection can happen as a result of differences in densities, while forced convection can be due to external physical movement used for circulation like a pump or the wind. Heat transfer through conduction takes place within a solid material or at the interface between two layers due to molecular collision. In PV modules, the conduction takes place due to thermal gradients in the direction from higher to lower temperatures.⁴

Figure 3.

Heat transfer mechanism in the layer of hybrid photovoltaic–thermal (hPVT).

The glass cover above the PV module receives direct and indirect solar radiations, of which a portion is reflected back, another portion is absorbed, and the rest of the radiations are transmitted to the solar cell. The radiation, conduction, and convection process happens across each layer of the hPVT system. The effects of heat loss from the edges are neglected.

A numerical model was developed in accordance with the heat transfer mechanism in the hPVT system. The following assumptions have been made to develop the mathematical model:

The impact of dust and partial shading has been neglected.

The elimination of fluctuations in the solar irradiance, ambient temperatures, and wind velocities facilitates the assumption of quasi-static for the smaller time periods (i.e. < 10 min).

Thermal resistance between the layers is considered negligible.

A constant thermal conductivity has been considered for each PVT component.

The system doesn't have any optical module surface losses. The coefficient of the transmittance–absorbance product (τ_α) is assumed to be unity.

The numerical model is developed using the assumptions stated above. The heat balance expression for each layer of the hPVT system is developed.

Glass layer (g)

When the solar irradiance falls on the glass cover, a certain amount of irradiance return back to the environment, and the remaining part gets absorbed by the module through heat transfer.²

I α_{g} = h_{g - a} (T_{g} - T_{a}) + U_{P V - g} (T_{g} - T_{P V}) = (3 v_{w} + 2.8) (T_{g} - T_{a}) + U_{P V - g} (T_{g} - T_{P V})

(1)

PV module (PV)

The PV module absorbed the solar radiation from the glass cover, as shown in the thermal resistance model in Figure 3. The major portion of energy received by the PV module is converted into electricity while the rest of the energy is absorbed by the absorber and tube assembly¹¹

I τ_{g} α_{P V} [1 - η_{E L}] = U_{P V - g} (T_{P V} - T_{g}) + U_{P V - A b} (T_{P V} - T_{A b}) + U_{P V - t} (T_{P V} - T_{t})

(2)

The thermal balance of the PV module consists of convective and radiative heat transfer coefficients to the PV layer and convective heat transfer to the absorber.

Thermal absorber (Ab)

As shown in Figure 3, the heat exchange takes place with the PV module, tube, and insulation layer in the thermal absorber.²⁴ The heat balance equation for the thermal absorber and tube assembly is shown in equations (3) and (4), respectively,

U_{P V - A b} (T_{P V} - T_{A b}) = U_{A b - t} (T_{A b} - T_{t}) + U_{A b - i n s} (T_{A b} - T_{i n s})

(3)

Heat transfer takes place in the tube with the absorber, working fluid, and layer of insulation.

U_{A b - t} (T_{A b} - T_{t}) + U_{P V - t} (T_{P V} - T_{t}) = U_{t - fluid} (T_{t} - \bar{T_{fluid}}) + U_{t - ins} (T_{t} - T_{ins})

(4)

Fluid (f)

The energy balance expression for fluid flowing in the tube takes into account the heat energy liberated by the absorber and heat collected by the fluid²⁴

U_{t - fluid} \cdot d \cdot A_{c} (T_{t} - \bar{T_{fluid}}) = m_{f} \cdot c_{p} \cdot d T_{fluid}

(5)

The NF temperature is computed as

T_{fluid - out} = T_{fluid} (L_{c}) = T_{fluid - in} A + T_{t} (1 - A)

(6)

The average temperature of NF is calculated as per equation (7). A and B are the factors that are used to simplify equations (6) and (7)

\bar{T_{fluid}} = \frac{1}{L_{c}} \int_{0}^{L_{c}} T_{fluid} (x) \cdot d x = T_{fluid - in} \cdot B + T_{t} \cdot (1 - B)

(7)

A = e^{- C L_{c}}

(8)

B = \frac{1 - A}{C \cdot L_{c}}

(9)

C = \frac{U_{t - f} \cdot W_{d}}{m_{f} \cdot c_{p}}

(10)

Layer of insulation (ins)

Heat exchange takes place between the thermal absorber, tubes, and atmosphere through the convective heat transfer¹¹

U_{A b - ins} (T_{A b} - T_{ins}) + H_{t - ins} (T_{t} - T_{ins}) = U_{ins - a} (T_{ins} - T_{a})

(11)

The heat transfer coefficients (HTCs) are the critical components in forming the thermal balance equations across the layers of the hPVT system as mentioned in the thermal circuit model. Therefore, the expression of HTCs present across each layer of the hPVT system is mentioned in Table 1.

Table 1.

Heat transfer coefficients (HTCs).¹¹

Layer of hPVT system	Expression of heat transfer coefficient (HTCs)
PV–glass	$U_{P V - g} = \frac{1}{\frac{δ_{P V}}{λ_{P V}} + \frac{δ_{g}}{λ_{g}}}$ (12)
PV–thermal absorber	$U_{P V - A b} = \frac{λ_{a d}}{δ_{a d}} (1 - \frac{D_{o u t}}{W})$ (13)
PV−tube	$U_{P V - t} = \frac{δ_{P V}}{\frac{W^{2}}{8 λ_{P V}} + \frac{δ_{a d}}{λ_{a d}} \cdot \frac{δ_{P V} W}{D_{o u t}}}$ (14)
Thermal absorber−tube	$U_{A b - t} = \frac{8 λ_{A b}}{W - D_{o u t}} \cdot \frac{δ_{A b}}{W}$ (15)
Thermal absorber−insulation	$U_{A b - i n s} = \frac{1}{\frac{δ_{A b}}{λ_{A b}} + \frac{δ_{i n s}}{λ_{i n s}}}$ (16)
Tube−insulation	$U_{t - i n s} = \frac{2 λ_{i n s}}{δ_{i n s}} (\frac{Π}{2} + 1) \frac{D_{o u t}}{W}$ (17)
Tube−nanofluid	$U_{t - i n s} = \frac{1}{\frac{W}{h_{f} Π D_{i n}} + \frac{W}{λ_{a d}}}$ (18)
Insulation−ambient	$U_{i n s - a} = \frac{1}{\frac{δ_{i n s}}{2 λ_{i n s}} + \frac{1}{h_{g - a}}}$ (19)

PV: photovoltaic; hPVT: hybrid photovoltaic–thermal.

Thermophysical properties

In the current research work, the overall performance of the hPVT system is examined using iron (Fe) nanoparticles and water as basefluid. The properties of nanoparticles and basefluid at 298 K are illustrated in Table 2.

Table 2.

Thermo-physical properties at 298 K.

Thermo-physical property	Fe	Water
Density (kg/m³)	7800	997.1
Thermal conductivity (W/m K)	76.2	0.6130
Specific heat (J/kg K)	440	4179

After deciding the type of nanoparticles, the impact of change in volume concentration of nanoparticles (ϕ) and changing MFR have been examined over the performance of the hPVT system. While analyzing the heat transfer capacity of NF, the thermos-physical characteristics, such as density, thermal conductivity, and specific heat are the most important parameters to consider. Due to the large errors involved in measuring thermal conductivity, it is often considered to be the most difficult property to measure. Further, the thermos-physical properties considered for mathematical modeling of the hPVT system depend on the temperature. Table 3 presents the equations of thermos-physical parameters of NFs and base fluid that are utilized to solve thermal balance equations.

Table 3.

Thermo-physical properties of nanofluid and basefluid.¹¹

Thermo-physical property	Expression
Density	$ρ_{n f} = φ . ρ_{n p} + (1 - ϕ) ρ_{b f}$ (20) $ρ_{w a t e r} = - 0.03 X T^{2} + 1.505 X T + 816.781$ (21)
Heat capacity	$c_{p \cdot n f} = \frac{ϕ {(ρ \cdot c_{p})}_{n p} + (1 - ϕ) {(ρ \cdot c_{p})}_{b f}}{ρ_{n f}}$ (22) $c_{p \cdot n f} = ϕ c_{p \cdot n p} + (1 - ϕ) c_{p \cdot b f}$ (23) $c_{p, w a t e r} = - 0.0000463 X T^{3} + 0.0552 X T^{2} - 20.86 X T + 6719.637$ (24)
Viscosity	${\begin{matrix} υ_{n f} = (1 + 2.5 ϕ) υ_{b f}, ϕ < 0.05 \\ υ_{n f} = (1 + 2.5 ϕ + 6.5 ϕ^{2}) υ_{b f}, 0.05 < ϕ < 0.1 \end{matrix}}$ (25) $υ_{w a t e r} = 0.00002414 X 10^{\frac{247.8}{T - 140}}$ (26)
Thermal conductivity	$λ_{n f} = \frac{λ_{n p} + 2 λ_{b f} - 2 φ (λ_{b f} - λ_{n p})}{λ_{n p} + 2 λ_{b f} + φ (λ_{b f} - λ_{n p})} . λ_{b f} + \frac{ρ_{n p} φ c_{p b f}}{2} \sqrt{\frac{2 κ T_{n f}}{3 Π d_{n p} υ_{b f}}}$ (27)
Thermal conductivity	$λ_{w a t e r} = - 0.000007843 X T^{2} + 0.0062 X T - 0.54$ (28)

The utilization of NF enhances the thermo-electric performance of the PV/T hybrid system. In comparison with conventional cooling methods, NF might significantly heat transmission. NFs such as HTF can substantially contribute to enhancing the PV/T system performance by minimizing the PV module temperature and thus augmenting PV efficiency.

System performance

The energy performance of the hPVT system can be calculated in terms of thermal, electrical, and overall efficiencies (η _Thermal , η_EL, and η_T). The η_EL depends upon the standard electrical efficiency (η_ST), temperature coefficient (β_o), and the cell temperature (T_PV)

η_{E L} = η_{S T} [1 - β_{o} (T_{P V} - T_{a i r})]

(29)

The typical thermal efficiency (η _Thermal ) equation is expressed as the ratio of the quantity of thermal energy generated by the system (Q_Thermal) to the incident solar radiations on the surface of a conductor

η_{T h e r m a l} = \frac{Q_{T h e r m a l}}{A I_{r r}} = \frac{m_{f} c_{p} (T_{n f . o} - T_{n f . i})}{A I_{r r}}

(30)

Bombarda et al.²⁵ suggested another expression for η _Thermal that separates both power and heat transfer rate and the electric power generated (P_EL) is subtracted from the solar radiation. This definition allows a fair and careful comparison of solar thermal panels and PVT

η_{T h e r m a l} = \frac{Q_{T h e r m a l}}{A I_{r r} - P_{E L}} = \frac{m_{f} c_{p} (T_{n f \cdot o} - T_{n f \cdot i})}{A I_{r r} - P_{E L}}

(31)

The efficiency of a system is determined by the energy input and the work performed by that energy input. PVT systems are hybrid systems. Thus, the overall efficiency (η_T) is the combination of electrical and thermal efficiencies in accordance with the first law of thermodynamics

η_{T} = η_{T h e r m a l} + η_{E L}

(32)

Exergy efficiency (ɳ_Ex) measures the effectiveness of the system in reversible conditions. As per the second law of thermodynamics, the ɳ_Ex is defined as the ratio of potentially useful output to the potentially useful input²

η_{E x} = \frac{\sum \dot{E} x_{o u t}}{\sum \dot{E} x_{i n}}

(33)

\sum \dot{E} x_{i n} = I \cdot A_{c} \cdot [1 - \frac{4}{3} \cdot (\frac{T_{a i r}}{T_{S}}) + \frac{1}{3} {(\frac{T_{a i r}}{T_{S}})}^{4}]

(34)

\sum \dot{E} x_{i n} - \sum \dot{E} x_{o u t} = \sum \dot{E} x_{d e s t}

(35)

\sum \dot{E} x_{i n} - \sum (\dot{E} x_{T h} + \dot{E} x_{e l}) = \sum \dot{E} x_{d e s t}

(36)

Machine learning-based prediction

Machine learning techniques have been widely employed in various domains linked with data-driven challenges. Machine learning approaches attempt to establish a meaningful input and output data relationship, using or without mathematical forms of issues. After the training dataset has well-trained the machine learning models, decision-makers may acquire pleasing forecasting output values by feeding the forecasted input data into the well-trained models. The pre-processing data approach is critical in machine learning and may significantly increase its performance. Machine learning has played a critical role in artificial intelligence fields such as lithology classification, signal processing, and medical picture analysis. Machine learning technology primarily employs three learning methods, as given in Figure 4.

Figure 4.

Classification of machine learning methods.

Supervised regression learning problem wherein training dataset comprised of input–output pairs is used to fit a function mapping a given set of input–output pairs from empirical data, and has become an essential component of machine learning and statistics, either as a sub-objective of a larger problem or for data analysis. Traditionally, parametric modeling has been employed for such cases, since it includes benefits in terms of interpretability. However, such parametric models when required to implement for complex data sets suffer from poor expressive power, and further, it is tedious or difficult to deal with their more complicated counterpart equivalents (such as feed-forward neural networks). With the emergence of kernel machines such as Gaussian processes, and support vector machines (SVMs), the prospect of flexible, workable models has emerged.

Gaussian process regression

GPR was performed by considering inference directly in function space. The GPR can be interpreted as a distribution over and inference occurring in the space of function from the function-space view. Consider a function to be an endlessly long vector, with each element representing an instance of the function at an input “x.” The Gaussian process may deduce the characteristics of the function from a finite number of points by treating the instances in the vector to be attributes of a stochastic process. To do this, the mean and covariance of a particular training set are computed as functions of a data point's position inside the data set. The covariance is represented using a suitable kernel that leverages hyperparameters to characterize the connection between data points in the collection. When interpolation and prediction are performed, the learned mean and covariance are utilized to provide a distribution of potential outputs for a certain input given the training set (x, y). Figure 5 shows an illustration of the GPR technique.

Figure 5.

Schematic illustration of Gaussian process regression (GPR).

In Figure 5, input 1 to input n and output 1 to output n constitutes the training set, while the test set is formed using X^∗ & Y^∗. The Gaussian distribution over functions is represented by the Gaussian field. Because of the Gaussian process's probabilistic structure, the training data set involves a measure of uncertainty that is clearly specified. Interpolation problems, on the other hand, might be due to modeling issues. A high condition number for the covariance matrix might result in a matrix computing inaccuracy.

In this study, a GPR model, a supervised learning approach, is used to train a model for each cluster. The GPR method can be conceptualized by learning mean and co-variance functions of realization of the Gaussian process (GP) at “ $a$ ” and is denoted by $f (a)$ .

f (a) \approx G P (m (a), k (a, a^{'}))

(37)

where f(a) is the function sampled from the GP, m(a) is the mean function, k(a, a^′) is a co-variance function which is a sub-class of kernel functions

m (a) = E [f (a)]

(38)

k (a, a^{'}) = E [(f (a - m (a)) f (a^{'} - m (a^{'}))]

(39)

Given a training data set (D_train) of M observations and the test data set (D_test) consisting of M^′ points

D_{t r a i n} = (A, B) = (A_{i}, B_{i})_{i = 1}^{M}, a_{i} \in R^{d}, b \in R

(40)

D_{t e s t} = A_{*} = (a_{*} i)_{i = 1}^{M^{'}}, a_{*}, i \in R^{d}

(41)

Accuracy assessment

In this investigation, four different metrics were discussed to evaluate the success of the prediction model. These matrices are root mean squared error (RMSE), coefficient of determination (R²), mean squared error (MSE), and mean absolute error (MAE). The description of these matrices is presented in Table 4. In the expressions of accuracy assessment parameters, x_i and y_i are the measured and predicted values, and n represents the number of observations.

Table 4.

Expression and description of performance assessment matrices.

Matrices	Expression	Description
RMSE	$\sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(y_{i} - x_{i})}^{2}}$	RMSE is the standard deviation of the predicted errors. It measures the overall accuracy of the predicted models. The value of RMSE is always positive and for the ideal case, it is nearly equals to 0.²⁶
R ²	$1 - \frac{\sum {(y_{i} - x_{i})}^{2}}{\sum {(x_{i} - \bar{x})}^{2}}$	The coefficient of determination (R²) is a statistical measurement used to assess how variations in one variable may be explained by changes in a second variable. Its value ranges between 0 and 1. Ideally, its value is nearly equals 1.²⁷
MSE	$\frac{1}{n} \sum_{i = 1}^{n} {(y_{i} - x_{i})}^{2}$	MSE measures the average of the square of errors. Its value is always positive.
MAE	$\frac{\sum_{i = 1}^{n} \| y_{i} - x_{i} \|}{n}$	MAE is a measure of errors among the paired observations. The smaller value of MAE indicates better performance of the model.²⁷

RMSE: root mean squared error; R²: coefficient of determination; MSE: mean squared error; MAE: mean absolute error.

Results and discussion

The present study aims to propose a reliable machine learning model for estimating the thermo-electrical performance of the hPVT system cooled utilizing NFs. In this section, the effect of mass-flow rate (MFR) of fluid and nanoparticle V_f was investigated on the thermal and electrical performances of the hPVT system. The hPVT system's performance obtained on the actual data for a particular day was compared with the predicted data using GPR. The yearly solar irradiance and ambient temperature data for the climatic conditions of Roorkee (29.8543 °N, 77.8880 °E), India,²⁸ have been used to demonstrate the efficacy of the suggested model. The yearly solar irradiance data is shown in Figure 6.

Figure 6.

Yearly solar irradiance data.²⁸

As observed in Figure 6, the solar irradiance level is higher from March to June as the summer season begins in the month of March. The rainy season begins in the month of July; hence, the solar irradiance level is low during the month of July and August. As reported in the literature, the performance of the hPVT system is considerably influenced by the temperature. Therefore, the surface temperature plays a vital role in the efficiency of the hPVT system. Figure 7 represents the annual surface temperature data. The surface temperature reaches its peak value in the month of May and June as the solar irradiance level is higher during this period. The surface temperature is at its lowest level in the winter season, usually from November to February.

Figure 7.

Yearly surface temperature data.²⁸

In this study, the thermo-electrical performance of the hPVT system is investigated using the actual solar irradiance and surface temperature data of a particular day and the forecasted data using the GPR technique, as shown in Figures 8 and 9, respectively. The η_EL, η _Thermal , and η_T were evaluated on the basis of actual and predicted data in order to observe the accuracy of forecasted data using machine learning.

Figure 8.

Actual and predicted solar irradiance data with time.

Figure 9.

Actual and predicted values of surface temperature with time.

In this work, a total of four samples for solar irradiance and the ambient temperature are taken in an hour which gives a total of 96 samples a day. For ease of implementation and to reduce the computational burden, the per-day samples are assumed to be the average sample for the complete month. It means each month has 96 data samples, which implies 1152 data samples in a year. The data samples are trained by four different ML methods such as GPR, SVM, fine tree, and linear regression. The considered forecasted model has the following profile for solar irradiance data: RMSE: 23.12, R² value: 0.96, MSE: 475.10, MAE: 11.6, and training time: 70.1 s, and is compared with other prediction models as shown in Table 5. It can be seen from the table that GPR has the minimum error profile as compared to other models such as SVM, fine tree, and linear regression.

Table 5.

Performance matrices of various ML algorithms for solar irradiance.

Prediction approach	RMSE	R²	MSE	MAE	Time taken (s)
GPR	23.12	0.96	475.10	11.6	70.1
SVM	35.21	0.87	550.14	17.2	72.1
Fine tree	45.21	0.83	573.12	21.1	80.2
Linear regression	46.81	0.78	580.54	26.4	81.5

ML: machine learning; GPR: Gaussian process regression; SVM: support vector machine; RMSE: root mean squared error; R²: coefficient of determination; MSE: mean squared error; MAE: mean absolute error.

Similar performance matrices were calculated for surface temperature data as mentioned in Table 6. The observations show that the considered prediction model (GPR) has the minimum error for ambient temperature data as compared to other machine learning algorithms.

Table 6.

Performance matrices of various ML algorithms for ambient temperature.

Prediction approach	RMSE	R ²	MSE	MAE	Time taken (s)
GPR	6.86	0.87	51.2	5.3	48
SVM	7.81	0.81	61.5	10.52	51
Fine tree	10.21	0.785	70.2	15.14	61.2
Linear regression	17.81	0.69	80.5	26.41	80.5

Of the total considered samples, 80% are used for training purposes, and the rest for testing and validation purposes. Once the training is completed, the required next-day forecasted solar irradiance and the ambient temperature are obtained. The values of design parameters considered in the simulation are shown in Table 7.

Table 7.

Values of parameters used for simulation.

Parameter	Value
Absorptance	0.04
Thickness of glass cover	0.004 m
Transmittance	0.92
Thermal conductivity of glass layer	0.7 W/m⁻¹K⁻¹
PV module temperature coefficient	0.0045 K⁻¹
PV cell efficiency at STC	0.18
Absorptance factor of PV module	0.9
Thermal conductivity of PV module	100 W/m⁻¹K⁻¹
Cell temperature at STC	298 K
Absorber thickness	0.0005 m
Number of tubes	10
Distance between tubes	0.1 m
Tube length	2 m
Outer tube diameter	0.010 m
Inner tube diameter	0.006 m
Thickness of tube	0.00012 m
Thickness of insulation	0.05 m
Thermal conductivity of insulation layer	0.034 W/m⁻¹K⁻¹
Collector slope	30°
Packing factor	0.9

PV: photovoltaic; STC: standard test condition.

In order to acquire the day-ahead forecasted data, the obtained data of solar irradiance and ambient temperature are desampled as presented in Figures 8 and 9, accordingly.

As observed from Figure 8, the solar irradiance increases with time, reaches its maximum value, and then decreases. The surface temperature also increases with an increase in solar irradiance level. The actual and predicted value of surface temperature with time is depicted in Figure 9.

Figure 10 illustrates the impact of increasing the MFR on the PV module temperature. It is identified that with an increase in MFR the temperature of the PV module temperature reduces. Further, it is observed that for the NF-cooled system, the PV module temperature is lower as compared to the water-cooled system. Hence, Fe/water NF (ϕ = 2%) helps in reducing the cell temperature, thereby enhancing the effectiveness of the considered hPVT system.

Figure 10.

PV cell temperature for various MFR.

For Fe/water NF, the PV cell temperature reduces to 32.4 °C from 41.75 °C when an MFR of 0.1 kg/s in comparison to 0.01 kg/s was used. Similar findings were obtained for PV cell temperature based on predicted data obtained using GPR. For a water-cooled hPVT system, the PV cell temperature reduces from 39.5 °C to 31.4 °C when an MFR of 0.1 kg/s is used as compared to 0.01 kg/s. This reduction in PV cell temperature occurs because the Reynolds number increases with an increase in MFR, which eventually cause an increase in the heat transfer coefficient (HTC) of the tubes. Thus, within the range considered, higher flow rates result in more heat being removed from the cell.

Figure 11 represents the effect of MFR on the η_EL of the hPVT system for both nano-cooling and water-cooling cases. For the water-cooled hPVT system, the highest η_EL obtained were 14.23% and 14.34% at 0.1 kg/s MFR for actual data and predicted data, respectively. For Fe/water NF (ϕ = 2%), the highest η_EL obtained were 15.6% and 15.45% for predicted data and actual data, respectively. The improvement in η_EL is mainly due to a reduction in PV cell temperature caused by an increase in MFR.

Figure 11.

Effect of mass flow rate (MFR) on electrical efficiency.

Ambient temperature, MFR, inlet and outlet temperature of cooling tubes, solar irradiance, and thermo-physical properties are the factors that might impact the η_Thermal of the hPVT system. Figure 12 reports the variation in η_Thermal with MFR for both nano-cooling and water-cooling cases. An increase in MFR results in an increase in η_Thermal of the hPVT system. The increment is higher for the nano-cooled hPVT system in contrast to the water-cooled system. The highest η_Thermal obtained for Fe/water NF (ϕ = 2%) system was 40.12% and 39.86% for actual data and predicted data, respectively. For the water-cooled system, the highest numerical value of η_Thermal obtained was 38.75% and 37.43% for actual data and predicted data, respectively, at the MFR of 0.1 kg/s. Hence, the η_Thermal increases around 3.414% and 6.09% for actual and predicted data using Fe/water NF when compared with a water-cooled hPVT system. This concludes that the outcomes of the hPVT system obtained using predicted data using a machine learning algorithm closely follow the outcomes obtained using actual data. Further, it has been recorded that the η_Thermal increases slightly beyond 0.04 kg/s MFR. It shows that the rise in MFR beyond 0.04 kg/s has a very small influence on the η_Thermal of the hPVT system.

Figure 12.

Effect of mass flow rate (MFR) on thermal efficiency.

Figure 13 depicts the variation in η_T of the hPVT system with MFR for nano-cooling and water-cooling cases. The η_T of the hPVT system is the sum of η_Thermal and η_EL. Figure 13 clearly shows a good correspondence between the overall output of the hPVT system obtained using actual and predicted data. The highest η_T for the actual and predicted data using Fe/water NF (ϕ = 2%) was 55.23% and 54.87%, respectively. For the water-cooled hPVT system, the highest η_T for the actual and predicted data was 49.34% and 49.10%, respectively.

Figure 13.

Effect of mass flow rate (MFR) on overall efficiency.

The V_f of nanoparticles (ϕ) in the basefluid is also a major factor that impacts the overall performance of the hPVT system. As mentioned in the literature, the viscosity and thermal conductivity of the fluid increase with an increase in V_f of nanoparticles, but at the same time, it leads to lower thermal conductivity. The random flow of nanoparticles is the main reason behind the enhanced thermal conductivity at a higher concentration of nanoparticles in the basefluid. Figure 14 clearly shows that the PV cell temperature reduces with an increase in V_f of nanoparticles in the basefluid. The enhanced thermal conductivity increases the HTCs, which helps in lowering the PV temperature.

Figure 14.

Influence of nanoparticle V_f on T_PV .

The cell temperature reduces from 31.12 °C to 28.34 °C and 30.76 °C to 28.21 °C for the actual and predicted data when the V_f of nanoparticles increases from 0% to 2%.

Figure 15 represents that the η_EL increases from 8.12% to 15.23% and 8.04% to 15.10% for the predicted data and actual data, respectively when the V_f of nanoparticles increases from 0% to 2%. The decrement in PV cell temperature leads to enhance the η_EL. The impact of nanoparticles V_f on the η_Thermal is depicted in Figure 16. The η_Thermal increases with an increase in particle V_f of nanoparticles. However, the impact of V_f is much more significant on the η_EL.

Figure 15.

Influence of nanoparticle V_f on η_EL.

Figure 16.

Influence of nanoparticle V_f on η_Thermal .

Figure 17 clearly shows that η_T of the hPVT system increases from 47.16% to 57% and 44.95% to 54.8% for the actual and predicted solar irradiance and surface temperature data, respectively, when the V_f of Fe nanoparticles in basefluid enhances from 0% to 2%.

Figure 17.

Influence of nanoparticle V_f on η_T.

Because of high specific heat and good thermal conductivity, the heat transfer rate increases as the V_f of nanoparticles increases. This leads to more heat absorption from the panel. Apart from lowering the solar cell temperature, absorbing a considerable quantity of PV panel thermal energy boosts its η_EL. At the same time, the absorption of more heat causes a rise in the working fluid outlet temperature. Hence, the inlet and outlet temperature difference increases. As a result, the η_Thermal improves. These factors help in enhancing the overall efficiency of the hPVT system.²⁹

Conclusion

The study analyzed the thermo-electrical performance of the hPVT system using the Fe/water NF system and water-cooled system based on actual input data (solar irradiance and surface temperature) and a day ahead forecasted data obtained using the GPR technique. The performance is comprehensively explored under the changing MFR and V_f of nanoparticles in the basefluid. The following conclusions were obtained:

The thermal, electrical, and overall efficiency results showed that their highest value occurred when Fe/water NF was used as a coolant, and the lowest level was obtained for the water-cooling case.

Compared to the water-cooling case, the η_EL, η_Thermal, and η_T of the nano-cooled system (ϕ = 2%) increased by 9.627% and 7.184%, 3.414% and 6.09%, 10.66% and 10.515% for the actual data and forecasted data, respectively, at the MFR of 0.1 kg/s.

The V_f of nanoparticles is a critical factor that impacts the overall η_T of the hPVT system. The η_T increased by 9.84% and 9.85% for actual and forecasted data, respectively, when the V_f of Fe nanoparticles in the basefluid increased from 0% to 2%.

The outcomes of the hPVT system obtained using day-ahead forecasted data closely follow the outcomes obtained using actual data. From this perspective, we can easily predict the day-ahead performance of the hPVT system.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Rajeev Kumar

Sourav Diwania is currently working as an Assistant Professor in the Department of Electrical and Electronics Engineering, KIET Group of Institutions Ghaziabad. His area of interest is Hybrid-photovoltaic Thermal systems.

Maneesh Kumar is currently working as a project associate in the Department of Hydro and Renewable Energy in Indian Institute of Technology Roorkee India. His area of interest is microgrid and grid connected power systems.

Rajeev Kumar is currently working as an Assistant professor in the Department of Electrical and Electronics Engineering KIET Group of Institutions Ghaziabad. His area of interest is Power system Dynamics and control with Soft computing techniques.

Arun Kumar is currently working as an Assistant professor in the Department of Electrical and Electronics Engineering KIET Group of institutions Ghaziabad. His area of interest is Artificial Intelligence and machine learning.

Varun gupta is currently working as an Associate Professor in the Department of Electrical and Electronics Engineering KIET Group of Institutions Ghaziabad. His main area of interest is Biomedical Engineering and Instrumentation.

Pavan Khetrapal is currently working as an Associate Professor and Head of the Electrical and Electronics Engineering Department in SRM University Delhi NCR Campus Modinagar India. His main area of interest includes Performance Benchmarking of Electricity utilities.

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Machine learning-based thermo-electrical performance improvement of nanofluid-cooled photovoltaic–thermal system

Abstract

Keywords

Introduction

Materials and methods

Heat balance

Glass layer (g)

PV module (PV)

Thermal absorber (Ab)

Fluid (f)

Layer of insulation (ins)

Thermophysical properties

System performance

Machine learning-based prediction

Gaussian process regression

Accuracy assessment

Results and discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Appendix

References