Improving Module Temperature Prediction Models for Floating Photovoltaic Systems: Analytical Insights from Operational Data

Abstract

Floating photovoltaic (FPV) systems are gaining popularity as a valuable means of harnessing solar energy on unused water surfaces. However, a significant gap persists in our comprehension of their thermal dynamics and the purported cooling benefits they provide. The lack of comprehensive monitoring data across different climatic regions and topographies aggravates this uncertainty. This paper reviews the applicability of established module temperature prediction models, originally developed for land-based PV systems, to FPVs. It then details the refinement of these models using FPV-specific data and their subsequent validation through large-scale, ongoing FPV projects. The result is a significant improvement in the accuracy of temperature predictions, as evidenced by the reduced Mean Absolute Error (MAE) and improved R-squared ($R^2$) after parameter optimisation. This reduction means that the tailored models better reflect the distinct environmental influences and cooling processes characteristic of FPV systems. The results not only confirm the success of the proposed method in refining the accuracy of current models, but also indicate significant post-tuning changes in the parameters representing wind and convective effects. These adjustments highlight the increased responsiveness of FPVs to convective actions, especially when compared to ground-based systems, possibly due to the evaporative cooling effect of bodies of water. Through this research, we address a critical gap in our understanding of heat transfer in FPV systems and aim to enrich the knowledge surrounding the acknowledged cooling effect of FPVs.

1. Introduction

The escalating global demand for energy is driven by population growth, urbanisation, and economic development. This is expected to continue, with a projected annual increase of 1.3% by 2040 if no policy changes are made [1]. However, this increase in energy consumption is accompanied by a rise in greenhouse gas (GHG) emissions, which are a major contributor to climate change. Currently, energy-related emissions account for about 73% of global GHG emissions [2]. The Intergovernmental Panel on Climate Change (IPCC) emphasises that limiting global warming to 1.5 °C above pre-industrial levels is critical to avoid severe climate change impacts [3]. Recognising and addressing the current global energy challenge is essential to pave the way for a cleaner and more secure future [4]. Achieving this vital objective depends on the transition to cleaner energy sources, including renewables, and on improving energy efficiency.

In this context, renewable energy has emerged as a critical component in decoupling energy use from greenhouse gas emissions [1]. The renewable energy sector is expanding rapidly due to cost reductions and technological advances, with solar photovoltaics (PV) expected to dominate electricity generation by 2050. An estimated 20-fold increase in global PV capacity is projected [5]. However, large-scale PV installations can require significant land conversion, which can lead to conflicts with country-specific regulations, such as the Open Space Ordinance (FFÖ-VO), and may further limit land availability [6]. This is particularly challenging in regions where food production is prioritised, and competition for land is intensifying [7]. To address these issues, innovative technologies such as Integrated Photovoltaics (IPV)—which include applications like agri-photovoltaics (APV), building-integrated photovoltaics (BIPV), and floating photovoltaics (FPV)—have emerged. IPV aims to integrate PV systems into various environments in innovative ways to mitigate land use conflicts [6].

FPV installations have gained popularity as a solution that utilises unused water surfaces to accommodate solar panels, potentially adding significant capacity [8]. In addition, decreases in the efficiency of photovoltaic energy conversion in the temperature range, especially with extreme limits [9], and the fact that water-cooling the modules is a possibility, make it a very attractive potential advantage. On the other hand, there are other concerns with FPV installation, such as the mechanical stress on PV systems due to movement on water waves, which can affect their lifetime [10].

Despite the global expansion of FPV, a critical knowledge gap persists regarding the performance and reliability of these systems, with limited monitoring data available from different climates and technologies. For instance, studies such as [11] have compared air-cooled FPV modules with conventional land-based PV modules, revealing a performance advantage of 1.6–2.0% for FPV systems. Research by [12] investigated the effectiveness of water cooling versus air cooling in a pilot FPV system. Additionally, ref. [13] conducted a comprehensive field analysis on nine FPV systems, highlighting different thermal behaviours based on their design structures. The study by [14] classified FPV systems into categories such as ‘free-standing’, ‘small footprint’, ‘large footprint’, and ’insulated’, each with distinct characteristics. Furthermore, studies by [15,16] have emphasised the impact of wind cooling on the thermal management of FPV systems. The study by [17] confirmed the strong dependency of air-cooled FPV cell temperatures on wind speed, underscoring the significance of wind-induced cooling. Dörenkämper [18] compared modelled and measured operating temperatures of floating PV modules, highlighting discrepancies between models such as PVsyst and Sandia. They found that these models tend to overestimate module temperatures with increasing wind speed and proposed methods for determining the single heat loss coefficient tailored to specific sites. Ayyad and Golroodbari [19] enhanced the modelling of water cooling in FPV systems by defining the convective heat transfer coefficient as a linear function of wind speed. Another study by Micheli et al. [20] focused on predicting PV temperature in FPV conditions, accounting for the effect of humidity and cooling due to seawater flow and evaporation on modules. Their results showed an estimated increase in PV efficiency compared to inland conditions, with the expectation of further efficiency improvement under higher wind speeds typical of the sea environment. In a recent design study by Wang et al. [21], computational fluid dynamics (CFD) simulations were conducted to optimize the design of FPV systems. The study identified parameters such as temperature difference between wind and water, module height, and tilt angle as crucial factors influencing FPV performance.

Despite these valuable contributions, several studies have underscored a persistent gap, emphasising the necessity for further experimental and analytical research [22]. The objective of this study is to review the prevalent models in the literature and adapt them to FPV datasets from operational plants. This approach aims to fine-tune the model parameters so that they accurately reflect the operational conditions of FPV modules.

2. Methods

The study employs a data-driven methodology, as illustrated in Figure 1, beginning with an evaluation of the existing module temperature models that are commonly applied to ground-based PV systems. These models are subsequently tailored to FPV installations, with adjustments based on empirical data to better represent the distinct cooling properties of FPVs compared to land-based PV systems. The extensive datasets facilitate the fine-tuning of model parameters, ensuring they are calibrated to meet the specific conditions of FPV installations.

2.1. Overview of Module Temperature Models

The scientific literature presents a variety of models for estimating the temperatures of PV modules operating under real-world conditions. These models vary in complexity from simple equations with a few variables to sophisticated three-dimensional computational fluid dynamics (CFD) simulations that account for heat transfer within module layers and interaction with the environment. These advanced models consider both steady-state and dynamic energy balances affected by climatic variables such as irradiance and wind [23]. However, due to the difficulty in gathering comprehensive data and the computational intensity required, existing models typically rely on simplified assumptions. Comparative studies, which have primarily focused on ground-mounted systems, have scrutinised these models, underscoring their dependence on factors like in-plane irradiance, ambient temperature, and sometimes wind speed, to enhance precision [24,25]. This work aims to critically assess major module temperature prediction models, illuminating their efficacy and their reliance on various input parameters.

In the analysis of PV module temperature models, several common parameters are often considered. These parameters help in understanding and predicting the thermal behaviour of PV modules under various environmental conditions. Model-specific parameters are explained in detail in the next section alongside each model formulation.

The key parameters include the following:

• $T_{\mathrm{mod}}$ is the module temperature (°C);
• $T_{\mathrm{amb}}$ is the ambient temperature (°C);
• $G_T$ is the solar irradiance (W/m²);
• $WS$ is the wind speed (m/s).

• Faiman Model

The Faiman model, integrated into the IEC 61853 standard [26] and detailed in [27], simplifies the complex thermal dynamics of modules using coefficients $U_0$ and $U_1$. These coefficients represent the effects of irradiance and wind speed on temperature, respectively, and are crucial for accurately predicting module temperature. The model equation is given by

$$T_{\mathrm{mod}}=T_{\mathrm{amb}}+{\displaystyle \frac{G_T}{U_0+U_1·Ws}}$$

(1)

where

• $U_0$ combined heat loss coefficient. $(\mathrm{W}/{\mathrm{m}}^2\mathrm{K})$;
• $U_1$ combined heat loss factor influenced by wind. $(\mathrm{W}/{\mathrm{m}}^3·\mathrm{s}·\mathrm{K})$.

• PVsyst Model

The PVsyst model, based on the Faiman model, calculates the temperature of a photovoltaic cell using an empirical heat loss factor model. This model, commonly employed to estimate cell temperatures in photovoltaic systems, is implemented in the PVsyst software 7.4 [28]. The parameters $U_c$ and $U_v$ depend on the module construction and its mounting. For freestanding modules with rear surfaces exposed to open air, typical values are $U_c=29.0$ $(\mathrm{W}/{\mathrm{m}}^2\mathrm{K})$ and $U_v=0.0$ $(\mathrm{W}/{\mathrm{m}}^3·\mathrm{s}·\mathrm{K})$. The model equation is given by

$$T_{\mathrm{mod}}=T_{\mathrm{amb}}+{\displaystyle \frac{\alpha ·G_T·(1-{\eta }_m)}{U_c+U_v·WS}}$$

(2)

where

• $U_c$ is the constant heat transfer component $(\mathrm{W}/{\mathrm{m}}^2\mathrm{K})$;
• $U_v$ is the convective heat transfer component $(\mathrm{W}/{\mathrm{m}}^3·\mathrm{s}·\mathrm{K})$;
• $\alpha$ represents the solar irradiation absorption coefficient defined as $0.9$;
• ${\eta }_m$ denotes the PV efficiency, which, when possible, should be calculated according to the operating conditions of the module.

• Zenit Model

The Zenit model, developed by Fraunhofer ISE, is a Python 3 library for PV system simulation, as described in [29]. It includes yield assessment tools and is optimised for annual average simulations. While Zenit prioritizes fast simulations over precision, it offers a complete and highly adaptable simulation suite. The model equation is given by

$$T_{\mathrm{mod}}=T_{\mathrm{amb}}+T_s·{\displaystyle \frac{G_T}{1000\left({\mathrm{W}/\mathrm{m}}^2\right)}}$$

(3)

where

• $T_s$ (°C)—the temperature surplus includes the influence of wind and other environmental factors. It is derived from outdoor module temperature measurements analysed by Fraunhofer ISE at different sites with different wind conditions. It is independent of the PV module type. However, it depends on different PV configurations, such as ground-mounted, freestanding, and BIPV.

• Sandia Model

The Sandia/King model, described in [30], estimates module temperature based on ambient temperature, wind speed, and solar irradiance. The Sandia/King model, established by King, also accounts for wind effects. Its parameters a and b are contingent upon the module’s materials, configuration, and mounting setup. a is a dimensionless parameter analogous to $U_0$ in the Faiman model, indicating irradiance impact, while b corresponds to $U_1$ in the Faiman model, representing wind influence.

The model equation is given by

$$T_{\mathrm{mod}}=T_{\mathrm{amb}}+G_T·e^{(a+b·WS)}$$

(4)

where

• a is a coefficient describing baseline heat loss from the module;
• b is a coefficient describing the effect of cooling by the wind.

• Kurtz Model

The Kurtz model, based on the previously mentioned Sandia model, provides different estimates for the coefficients a and b. This model does not distinguish between PV devices or technologies compared to the Sandia model. The model equation is given by

$$T_{\mathrm{mod}}=T_{\mathrm{amb}}+G_T·e^{(3.473+0.0594·WS})$$

(5)

• Risser and Fuentes

The Risser and Fuentes empirical model [31] employs a linear regression approach using three variables to predict the module’s back-surface temperature: ambient temperature ($T_{\mathrm{amb}}$), total irradiance ($G_T$), and wind speed ($WS$). This model was developed by fitting the model to a dataset of recorded module temperatures [31]. Like the model implemented in Zenit or Kurtz model, this one does not distinguish between different PV devices or technologies. Additionally, similar to the Kurtz model, it does not differentiate between different configurations such as freestanding or FPV. The model equation is given by

$$T_{\mathrm{mod}}=3.81+1.31·T_{\mathrm{amb}}+0.0282·G_T-1.65·WS$$

(6)

• Standard Model (Nominal Operating Cell Temperature (NOCT))

The Nominal Operating Cell Temperature (NOCT) is characterised by specific conditions: an irradiance of 800 W/m², an ambient temperature of 20 °C, and a wind speed of 1 m/s. The standard approach defines the operating temperature under NOCT conditions. This temperature, denoted as $T_{\mathrm{mod},\mathrm{NOCT}}$, typically falls within the range of 44 °C to 46 °C and is specific to the technology of the PV module.

The model equation under NOCT conditions can be expressed as follows:

$$T_{\mathrm{mod}}=T_{\mathrm{amb}}+{\displaystyle \frac{G_T}{G_{\mathrm{NOCT}}}}·(T_{\mathrm{NOCT}}-T_{\mathrm{amb},\mathrm{NOCT}})$$

(7)

where

• $G_{\mathrm{NOCT}}$ is the solar irradiance at NOCT (W/m²);
• $T_{\mathrm{NOCT}}$ is the nominal operating cell temperature (°C);
• $T_{\mathrm{amb},\mathrm{NOCT}}$ is the ambient temperature at NOCT (°C).

• Skoplaki Model

The Skoplaki model [32] calculates the module temperature using the ambient temperature, wind speed and solar irradiance. It uses a complex equation with various coefficients to model convective heat transfer and solar irradiance. Two different models developed by Skoplaki were considered. The first, referred to here as 1, is simpler and does not take into account specific parameters that depend on the PV device, so its temperature estimates are independent of technology and configuration as in Equation (8). The model equation is given by

$$T_{\mathrm{mod}}=T_{\mathrm{amb}}+{\displaystyle \frac{0.32}{8.91+2.0·WS}}·G_T$$

(8)

• Skoplaki 2 Model

The second model, labelled 2, is more complex and uses parameters specific to the device under test, such as the efficiency (${\mu }_{\mathrm{STC}}$) or the temperature coefficient of maximum power (${\beta }_{\mathrm{STC}}$), as shown in Equation (9). Another parameterization for this model is suggested in multiple research papers as in [33].

$$T_{\mathrm{mod}}=T_{\mathrm{amb}}+{\displaystyle \frac{G_T}{800}}·(T_{\mathrm{NOCT}}-20)·{\displaystyle \frac{h_{w,\mathrm{NOCT}}}{h_w}}·\left(1-{\displaystyle \frac{{\mu }_{\mathrm{STC}}}{0.9}}·\left(1-{\beta }_{\mathrm{STC}}·T_{\mathrm{STC}}\right)\right)$$

(9)

where

• $T_{\mathrm{NOCT}}$ is the nominal operating cell temperature (°C);
• $h_w$ is the wind heat transfer coefficient,

  $$h_w=5.7+2.8·WS;$$

  (10)
• $h_{w,\mathrm{NOCT}}$ is calculated using the wind speed at NOCT conditions (1 m/s);
• ${\mu }_{\mathrm{STC}}$ is the efficiency under standard test conditions (STC);
• ${\beta }_{\mathrm{STC}}$ is the temperature coefficient of maximal power under standard test conditions (STC);
• $T_{\mathrm{STC}}$ is the standard test condition temperature (°C).

• Mattei Model

The Mattei model [34] estimates module temperature using ambient temperature, wind speed, solar irradiance, and various parameters such as the nominal operating cell temperature (NOCT) and standard test conditions (STC). The model equation is given by

$$T_{\mathrm{mod}}={\displaystyle \frac{T_{\mathrm{amb}}·U_{PV}+G_T·\left(0.81-{\mu }_{\mathrm{STC}}·(1-{\beta }_{\mathrm{STC}}·T_{\mathrm{STC}})\right)}{U_{PV}+G_T·{\beta }_{\mathrm{STC}}·{\mu }_{\mathrm{STC}}}}$$

(11)

where

• $U_{PV}$ is the thermal losses coefficient from module to the surroundings $(\mathrm{W}/{\mathrm{m}}^2\mathrm{K})$

  $$U_{PV}=26.6+2.3·WS;$$

  (12)
• $T_{\mathrm{STC}}$ is the standard test condition temperature (°C);
• ${\mu }_{\mathrm{STC}}$ is the efficiency under standard test conditions (STC);
• ${\beta }_{\mathrm{STC}}$ is the temperature coefficient of maximal power under standard test conditions (STC).

2.2. Overview of Datasets

The datasets originate from two operational FPV systems. The first serves as the Training FPV dataset and is situated in a quarry pond, where it occupies less than 10% of the water surface. It comprises two arrays, each with a capacity of under 1000 kWp. This FPV system is engineered with high-efficiency mono-PERC solar modules arranged at a 12° south-facing angle. The modules are installed on a floating aluminium pontoon structure, buoyed by a network of hollow floats. On-site sensors monitor parameters like irradiance and module temperature, while satellite data provide ambient temperature and wind speed readings for system performance assessment. The dataset encompasses the summer months of 2022.

The second system, used as the Testing FPV dataset, is located on an artificial lake within an old quarry and covers over 30% of the lake’s surface. This larger power plant has a capacity exceeding 10 MW and uses monocrystalline PERC double glass modules. Data for this system cover the summer months of 2020. Figure 2 illustrates the design configurations of the investigated FPV systems, presenting both top and rear views of the setups analysed.

Data Analysis

When analysing the data, both sets were synchronised to 15 min intervals to match externally sourced meteorological data. The original wind speed measurements, taken at a height of 10 m, were adjusted to reflect conditions at 2 m, the assumed height of the PV modules. This adjustment was made using the wind profile power law, as represented by Equation (13).

$$v_2=v_1{\displaystyle \frac{ln\left(\frac{h_2}{z_0}\right)}{ln\left(\frac{h_1}{z_0}\right)}}$$

(13)

In flat terrain and with neutral atmospheric stratification, the logarithmic wind profile is a good approximation of the vertical wind speed [35,36]. As shown in Equation (13), the reference wind speed corresponds to $v_1$ measured at the reference height $h_1$, $v_2$ is the wind speed at the height $h_2$, and $z_0$ is the roughness length as given in

Table 1. Classification of terrain roughness and corresponding lengths ($z_0$), along with examples of typical terrain surfaces [36].

Roughness Class	Roughness Length ${\mathit{z}}_0$	Types of Terrain Surfaces
0	0.0002 m	Water surfaces: sea and lakes
0.5	0.0024 m	Open terrain with a smooth surface
1	0.03 m	Open agricultural land without fences and hedges
1.5–2.5	0.055–0.2 m	Agricultural land varies depending on the amount of houses, hedges, bushes, and plants.
3	0.4 m	Villages, small towns, agricultural terrain with many or high hedges, forests, and very rough and uneven terrain.
3.5	0.6 m	Larger cities with tall buildings
4	1.6 m	Big cities with tall buildings and skyscrapers

.

Following the above wind speed adjustment, the datasets were cleaned to remove missing data and outliers and then filtered to ensure stable operation, as shown in the following list:

• Low irradiance filter (<100 W/m²) to exclude extraneous low irradiance;
• Low wind speed filter (<1 m/s) helps to mitigate localised temperature variations caused by stagnant air conditions, allowing more accurate analysis of surface temperature dynamics;
• High wind speed filter (>8 m/s) to exclude irregular gusts from the dataset, ensuring a more stable representation of wind conditions.;
• Low module temperature filter (<1 °C) to mitigate snow/frost effects.

Figure 3 shows inverted pyramid plots representing the reduction in data points due to filtering. In the Training FPV dataset, the initial 12,745 data points are reduced to 5886 after the irradiance filter (46%), and further to 5055 (39.7%) after all other filters. In the Testing FPV dataset, the initial 6122 data points are reduced to 2079 after the irradiance filter (34%) and further to 1429 (23.3%) after all other filters.

Figure 4 shows a 3D plot illustrating the relationship between module temperature, wind speed, and solar irradiance for Training FPV dataset. As can be clearly seen, the solar irradiance is plotted along a range from 0 to 1200 W/m², while the wind speed is plotted from 0 to 9 m/s. Different ambient temperatures, ranging from 0 to 40 °C, are represented by colour-coded dots. There are clear trends: higher solar radiation and higher ambient temperatures both increase module temperatures and have a dominating effect, while higher wind speed somewhat reduces module temperatures. The interaction of these factors is complex and has an important impact on module temperature, emphasising the importance of considering all of these variables in the assessment of FPV module temperature.

The box plots in Figure 5 illustrate the distributions of ambient temperature, irradiance, and wind speed for both FPV sites. These plots reveal that ambient temperatures follow similar patterns at both sites, with the Training FPV site generally experiencing slightly cooler conditions than the Testing FPV site. Although irradiance levels are comparable between the two sites, the Testing FPV site is subject to higher wind speeds, with peaks reaching up to 12 m/s. Given that higher wind speeds can enhance the cooling of PV modules and potentially improve their efficiency, this is a notable difference between the sites.

In essence, the climatic conditions at Training FPV and Testing FPV are sufficiently analogous to facilitate a meaningful comparison of the two systems’ performance. The systems also share other critical design features, such as the type of floats; narrow tilt angles; and the use of monocrystalline PV modules, which favour a controlled study approach. By tuning parameters with one dataset and validating with another, the study can leverage these similarities to enhance the reliability of its findings.

Figure 6 displays a heat map of the Pearson correlation coefficients (PCCs) that quantifies the relationships between data variables from the Training FPV system. Notably, module temperature exhibits a strong positive correlation with both irradiance and ambient temperature, as indicated by PCC values exceeding 0.7.

This indicates that increases in irradiance or ambient temperature typically result in higher module temperatures. These two variables also show a positive correlation with each other, reflecting their combined effect on module temperature. Consequently, irradiance and ambient temperature are the most frequently used variables for estimating module temperature under actual operating conditions. The heat map further reveals a negative correlation between module temperature and both wind speed and humidity, which serve as cooling factors. This cooling effect may be more pronounced in FPV installations due to their unique environment.

To capture the cooling influence of the FPV setting, some models incorporate additional variables such as wind speed, wind direction, humidity, and water temperature. Although the impact of wind speed and direction is generally considered less primary than irradiance and ambient temperature, with a PCC of approximately −0.24, their inclusion in models is important for quantifying cooling effects and optimising design parameters. Module temperature is also influenced by factors like the optical properties of the components, the thermal insulation on the backside, and the electrical efficiency of the cells.

2.3. Fitting the Model Parameters

To calibrate the coefficients of the analysed models, a parameter fitting approach is employed with the objective of minimising the discrepancy between the models’ predicted temperature values and the actual measured temperatures. An optimisation algorithm utilizes a minimisation function to determine the optimal values for the calibration coefficients. This function iteratively adjusts the coefficients and computes the ‘residual’ function, which is defined as the mean squared error (MSE) between the predicted and measured temperatures. The optimisation method updates the vertices of a simplex in the parameter space iteratively until convergence. For each model, depending on its equation and the parameters to be optimised, an appropriate minimisation function was used for tuning.

For the Faiman, Zenit, and PVSYST models, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) [37] method was used because it is effective with smooth differentiable functions to iteratively find the minimum. For the Kurtz and Sandia models, which are exponential functions, the Trust Region Reflective (TRF) [38] method was used. This is commonly used to solve nonlinear equation systems [39].

Looking at the other models, the RandF model requires four parameters to be fitted, Skoplaki 1 requires three parameters to be fitted, and the Skoplaki 2 and Mattei models also have complex equations requiring the flexible approach provided by the Nelder–Mead method [40]. The RandF, Skoplaki, Mattei, and Skoplaki 2 models use the Nelder–Mead method, which works well even with functions that are not smooth, proving that the method is well suited to the various optimisation challenges.

2.4. Evaluation Metrics for the Prediction Models

Several evaluation metrics are used to assess the performance and accuracy of the prediction models used in this study. These metrics provide quantitative measures of how well the models fit the data and allow comparisons between different models. The metrics used are as follows:

• Mean Squared Error (MSE): A commonly used measure of the accuracy of a prediction model. It is calculated as the average of the squared differences between the predicted values and the actual values in the dataset. The MSE is used in the fitting process to calibrate the coefficients of the prediction models.

  $$\mathrm{MSE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-\widehat{y_i})}^2$$

  (14)

  where $y_i$ is the actual value of the i-th data point and $\widehat{y_i}$ is the predicted value of the i-th data point.
• Normalised Root Mean Squared Error (NRMSE): A metric that represents the root mean squared error normalised by the range of the data. It provides a measure of the accuracy of the model relative to the range of the observed data. NRMSE is calculated as follows:

  $$\mathrm{NRMSE}={\displaystyle \frac{\sqrt{\mathrm{MSE}}}{\max \left(y\right)-\min \left(y\right)}}$$

  (15)

  where MSE is the mean squared error and y is the actual values of the data.
• Mean Absolute Error (MAE): Another commonly used measure of the accuracy of a prediction model. It is calculated as the average of the absolute differences between the predicted values and the actual values in the dataset. The MAE is a measure of how well the model fits the data, with a lower MAE indicating a better fit.

  $$\mathrm{MAE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n|y_i-\widehat{y_i}|$$

  (16)
• The R-squared ($R^2$) value: Another commonly used measure to evaluate the performance of a prediction model. The $R^2$ value represents the proportion of variance in the dependent variable that is accounted for by variation in the input variable. It is a statistical measure that ranges from 0 to 1, with a higher value indicating a better fit of the model to the data.

  $$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{(y_i-\widehat{y_i})}^2}{{\sum }_{i=1}^n{(y_i-\overline{y})}^2}}$$

  (17)

The calibration process uses the measured module temperatures and other meteorological parameters from the Training FPV dataset, as described in Section 2.2. After calibration, the performance of the trained models is evaluated using the Testing FPV dataset to verify the accuracy of the refined module temperature predictions. This evaluation serves as an indicator of the models’ responsiveness to new data and provides a measure of their predictive accuracy.

3. Results

Before fitting the models, we first compared the predictions of the ten different models against the measured module temperature. The models evaluated were Faiman, PVsyst, Zenit, Sandia, Kurtz, Risser and Fuentes (R&F), Standard, Skoplaki, Skoplaki 2, and Mattei. Before parameter tuning, the performance of these models varied considerably, as shown in

Table 2. Comparison of model evaluation metrics before and after parameter tuning.

Model	Before Tuning			After Tuning
	NRMSE (%)	R2	MAE	NRMSE (%)	R2	MAE
Faiman	11.34	0.75	3.93	5.85	0.93	1.88
Zenit	15.19	0.56	5.41	8.78	0.85	2.97
Sandia	15.71	0.53	5.93	5.87	0.93	1.82
R&F	33.53	−1.16	12.71	6.32	0.92	2.13
Skoplaki	10.89	0.77	3.78	5.82	0.93	1.87
PVsyst	21.62	0.1	7.88	5.85	0.93	1.88
Mattei	9.01	0.84	3.12	5.87	0.93	1.88
Skoplaki 2	7.21	0.9	2.39	5.68	0.94	1.73
Standard	17.77	0.39	6.4	8.81	0.85	2.98
Kurtz	20.08	0.22	7.61	5.87	0.93	1.82

. In particular, the Skoplaki 2 model demonstrated superior prediction accuracy and model fit, with the lowest NRMSE of 7.21% and the highest $R2$ value. After Skoplaki 2, the Mattei model ranked second in performance, followed by Faiman and the Skoplaki 1. The Zenit, Sandia, and Standard models showed moderate performance. The least accurate models before tuning were the Kurtz, PVsyst, and R&F models.

However, all models consistently overestimated the measured module temperatures under different conditions. A plot was generated to visualise the discrepancy between the predicted module temperatures from the untuned models and the measured module temperatures shown in Figure 7. This plot clearly shows that all models consistently overpredicted the measured module temperatures. This discrepancy indicates that the existing models commonly used for land-based photovoltaic systems are not suitable for accurately predicting temperatures in FPV systems. The observed overestimation suggests the presence of a cooling effect unique to FPV systems that is not adequately accounted for in existing models. These results highlight the importance of developing and refining models specifically tailored to the unique characteristics of FPV systems.

The results of the model fitting process are first illustrated in Figure 8, which demonstrates the effectiveness of the parameter tuning procedure. Of significance is the consistent reduction in MAE and NRMSE after refinement, coupled with increased $R^2$ values, as summarised in Table 2, indicating improved accuracy in predicting operating module temperatures. This improvement highlights the power of the parameter tuning methodology employed. Additionally, the specific adjustments made to the parameters of each model are detailed in

Table 3. Summary of Model Equations with Default and Tuned Parameter Values.

Model	Equation	Parameters Before	Parameters After
Faiman	$T_{\mathrm{mod}}=T_{\mathrm{amb}}+{\displaystyle \frac{G_T}{U_0+U_1·Ws}}$	$U_0=25$$(\mathrm{W}/{\mathrm{m}}^2\mathrm{K})$, $U_1=6.84$$(\mathrm{W}/{\mathrm{m}}^3·\mathrm{s}·\mathrm{K})$	$U_0=37.61$$(\mathrm{W}/{\mathrm{m}}^2\mathrm{K})$, $U_1=8.32$$(\mathrm{W}/{\mathrm{m}}^3·\mathrm{s}·\mathrm{K})$
PVsyst	$T_{\mathrm{mod}}=T_{\mathrm{amb}}+{\displaystyle \frac{\alpha ·G_T·(1-{\eta }_m)}{U_c+U_v·WS}}$	$U_c=29.0$ $(\mathrm{W}/{\mathrm{m}}^2\mathrm{K})$, $U_v=0.0$ $(\mathrm{W}/{\mathrm{m}}^3·\mathrm{s}·\mathrm{K})$	$U_c=37.61$$(\mathrm{W}/{\mathrm{m}}^2\mathrm{K})$, $U_v=8.32$$(\mathrm{W}/{\mathrm{m}}^3·\mathrm{s}·\mathrm{K})$
Zenit	$T_{\mathrm{mod}}=T_{\mathrm{amb}}+T_s·{\displaystyle \frac{G_T}{1000}}$	$T_s=23$ °C	$T_s=17.6$ °C
Sandia	$T_{\mathrm{mod}}=T_{\mathrm{amb}}+G_T·e^{a+b·\mathrm{Ws}}$	$a=-3.56$, $b=-0.075$ (s/m)	$a=-3.71$, $b=-0.143$ (s/m)
Kurtz	$T_{\mathrm{mod}}=T_{\mathrm{amb}}+G_T·e^{a+b·\mathrm{Ws}}$	$a=-3.473$, $b=-0.0594$ (s/m)	$a=-3.71$, $b=-0.143$ (s/m)
R& F	$T_{\mathrm{mod}}=3.81+1.31·T_{\mathrm{amb}}+0.0282·G_T-1.65·WS$	$a=3.81$, $b1=1.31$, $b2=0.0282$, $b3=-1.65$	$a=4.22$, $b1=0.98$, $b2=0.02$, $b3=-0.95$
Standard	$T_{\mathrm{mod}}=T_{\mathrm{amb}}+{\displaystyle \frac{\left(factor\right)G_T}{G_{\mathrm{NOCT}}}}·(T_{\mathrm{NOCT}}-T_{\mathrm{amb},\mathrm{NOCT}})$	Factor 1	Factor: 0.71
Skoplaki	$T_{\mathrm{mod}}=T_{\mathrm{amb}}+{\displaystyle \frac{0.32}{8.9+2·WS}}·G_T$	$a=0.32$, $b1=8.9$, $b2=2$	$a=0.26$, $b1=9.82$, $b2=2.17$
Skoplaki 2	$T_{\mathrm{mod}}=T_{\mathrm{amb}}+{\displaystyle \frac{G_T}{800}}·(T_{\mathrm{NOCT}}-20)·{\displaystyle \frac{h_{w,\mathrm{NOCT}}}{h_w}}·(1-{\displaystyle \frac{{\mu }_{\mathrm{STC}}}{0.9}}·\left(1-{\beta }_{\mathrm{STC}}·T_{\mathrm{STC}}\right))$	$h_w=U_c+U_v·WS$ $U_c=5.7$, $U_v=2.8$	$U_c=6.35$, $U_v=2.52$
Mattei	$T_{\mathrm{mod}}={\displaystyle \frac{T_{\mathrm{amb}}·U_{PV}+G_T·\left(0.81-{\mu }_{\mathrm{STC}}·(1-{\beta }_{\mathrm{STC}}·T_{\mathrm{STC}})\right)}{U_{PV}+G_T·{\beta }_{\mathrm{STC}}·{\mu }_{\mathrm{STC}}}}$	$U_{pv}=U_c+U_v·WS$ $U_c=26.6$, $U_v=2.3$	$U_c=24.27$, $U_v=5.25$

.

In particular, the Skoplaki 2 model emerged as the best performer; closely followed by the Sandia and Kurtz; and then the Faiman, Pvsyst, and Mattei models. The Zenit and RandF models also showed significant improvements, although their performance lagged behind that of the previously mentioned models. The standard model showed the least improvement, despite incorporating adjustments for irradiance and attempts to refine its fit.

To give an example of the tuning process used for all ten models, we zoom in on one model, the Faiman model. Initially, we employed standard parameters with values of $U_0=25$ $(\mathrm{W}/{\mathrm{m}}^2\mathrm{K})$ and $U_1=6.84$ $(\mathrm{W}/{\mathrm{m}}^3·\mathrm{s}·\mathrm{K})$. However, after tuning, these parameters were refined to $U_0=37.61$ $(\mathrm{W}/{\mathrm{m}}^2\mathrm{K})$ and $U_1=8.32$ $(\mathrm{W}/{\mathrm{m}}^3·\mathrm{s}·\mathrm{K})$.

Figure 9 provides a visualisation of the performance of the Faiman model before and after tuning, spanning three days in July 2022. The average temperature differences for three days before tuning were as follows: Day 1—4.38 °C, Day 2—3.66 °C, and Day 3—5.13 °C. After tuning, the average differences were reduced to the following: Day 1—2.00 °C, Day 2—1.66 °C, and Day 3—2.34 °C. This example demonstrates the effectiveness of the parameter tuning approach across all models, resulting in improved accuracy in predicting operating module temperatures.

4. Discussion

In the scientific literature, the cooling effect observed in FPV systems is often attributed primarily to wind speed, which is an oversimplification. Convective cooling is influenced not only by wind speed but also by its moisture content. If wind speed were the only factor, then land-based systems experiencing wind speeds comparable to those over large bodies of water would exhibit similar temperature predictability to FPV systems, which is not the case. It is well documented that bodies of water are in a constant state of evaporation, contributing to higher humidity levels and a reduction in the temperature of the air above them [22,41]. This complex psychrometric interaction requires more targeted data analysis to improve the predictive capabilities of existing module temperature models or to develop new, more sophisticated models. Such advances are likely to follow from the findings presented in this research.

The analysis presented in this study uses a purely data-driven approach and clearly demonstrates the underperformance of conventional models when applied to FPV systems. Key environmental variables that have a significant impact on performance include solar irradiance, wind speed, ambient temperature, and humidity. The research shows that solar irradiance and ambient temperature are the primary determinants of module temperature, with a strong positive correlation. Conversely, humidity and wind speed are negatively correlated with module temperature due to their role in facilitating convective cooling.

The evaluation of standard models, typically used for predicting module temperature in terrestrial photovoltaic PV systems, was extended to FPV counterparts. The models’ predictions were compared with actual module temperature measurements from an FPV system. The findings consistently revealed an overestimation of module temperature by all models, lending strong credence to the hypothesis that the proximity to water exerts a cooling effect on FPV modules, no matter the claimed driving mechanism.

To improve the accuracy of the module temperature prediction, the model parameters were fine-tuned using a designated Training FPV dataset and subsequently verified using a Testing FPV dataset. The climatic conditions at both the Training FPV and Testing FPV sites are comparable, among other critical similarities, making them an appropriate comparative framework for assessing the performance of the two FPV systems. After calibration, there was a noticeable improvement in the performance of the models, as evidenced by a reduction in MAE and NRMSE, indicating a more accurate prediction of module temperatures.

The best-performing model was Skoplaki 2, known for its complexity and incorporation of device-specific parameters such as efficiency (${\mu }_{\mathrm{STC}}$) and temperature coefficient of maximum power (${\beta }_{\mathrm{STC}}$). By accounting for wind effects through the wind heat transfer coefficient ($h_w$), Skoplaki 2 showed superior performance with an MAE of approximately 1.6 °C and an $R^2$ value of 0.94. After Skoplaki 2, the Faiman and PVsyst models showed significant improvements after tuning. It is worth noting that the Faiman, PVsyst, and Skoplaki 1 models essentially represent the same underlying model structure, differing primarily in their initial parameter values. Similarly, the Sandia and Kurtz models share an identical equation, differing mainly in their parameterization. This underlines the interconnectedness and similarities between these models (Faiman, PVsyst, Skoplaki, Sandia, and Kurtz), despite their apparent differences in nomenclature and parameter values [42], thus explaining their similar performance after calibration together.

Other models such as R&F and Zenit also showed significant improvement after tuning, although their performance was comparatively less robust. Zenit’s performance was hampered by the fact that it did not consider the effect of wind separately. The R&F model, being an empirical model based on linear regression analysis, may have been susceptible to overfitting with certain training datasets. Furthermore, the standard model, despite attempts to improve its performance by adding a parameter factor to irradiance and refining this parameter, still exhibited poor performance post-tuning. This highlights the limitations of simplistic model structures in capturing the complexities of FPV system behaviour.

Interestingly, post-tuning, certain models, such as the Faiman, PVsyst, and Sandia models, showed a pronounced increase in the lumped convection coefficient or parameter. This underscores the significance of the synergistic cooling effects due to evaporative cooling from the water body and the convective forces generated by the wind as pivotal in the observed cooling phenomena in FPV systems.

However, it is important to acknowledge that while parameter tuning significantly improved model performance, this method may not completely eliminate the challenges associated with model calibration. As observed in previous studies, applying coefficients obtained from one dataset to another site may introduce biases in the estimated module temperature [43]. The dependence of model coefficients on dataset characteristics and time period further highlights the need for ongoing refinement and validation of model parameters.

5. Conclusions and Outlook

This research highlights the need for tailored modelling approaches to accurately predict module temperatures in FPV systems. It has been shown that traditional models designed for land-based PV systems often fail to capture the complex thermal dynamics inherent in FPV installations, resulting in overestimated module temperatures. However, by refining the model parameters using FPV-specific datasets, we have demonstrated significant improvements in prediction accuracy. The observed reduction in MAE after parameter tuning highlights the improved ability of the refined models to better reflect the cooling effect present in FPV systems. In particular, the significant increase in the coefficients associated with wind effects in models such as Faiman and Sandia underscores the critical importance of accounting for both evaporative cooling from water bodies and convective cooling from wind to ensure accurate module temperature predictions. There remains an urgent need for research efforts to focus on deriving insights that can simplify module temperature models, which play an integral role in the assessment of FPV solar power plant yields. Our study argues for continued research to improve modelling methods and integrate more diverse data. Long-term experimental studies in different geographical locations are essential. These studies should be complemented by advanced models that can accurately predict evaporation rates and track the microclimate both above and below the panels. Such a comprehensive analysis is essential to better understand and optimise the thermal dynamics of FPV systems over longer periods of time. In addition, adjustments to account for the unique geometries of FPV systems and localised wind speed variations could provide further valuable insights.