2.1.2. FPV Panel Data Collection

In a previous study, it was found that polyethylene (PE) foam was the most cost-effective way to add buoyancy to flexible solar modules [54]. This study uses this after-market conversion method to convert SunPower SPR-E-Flex PVs into FPVs [70]. The density of the green polyethylene 1.2 lb <sup>1</sup> 2 " (12.7 mm) was used to determine the area of foam needed to make the panel rise by approximately 10 mm above the water's surface [71] using the calculations detailed in [54]. The foam was cut into about 50 mm by 240 mm sections that were placed evenly on the backside of module. The sections were then adhered using 3 M 5200 fast-set waterproof adhesive. Each foam piece had a line of adhesive caulked onto its perimeter and through the center. Then, the foam piece was pressed on the surface of the panel to adhere it, see Figure 1. The FPV with PV control was deployed in Chassell Bay, MI during the summer of 2020 to determine operational temperature and performance. This resulted in the FPV floating directly above the water surface, but still enabling wave action to clear the modules as shown in Figure 2.

**Figure 1.** Cut away view showing adhesive underneath foam attached to c-Si-based flexible photovoltaic (PV) module: (**a**) top view and (**b**) orthogonal view.

The NanoDAQ monitoring board used in [54] was used in this study to measure module power and temperature of both the control (flat land-based mounted dry PV set at zero degree tilt angle) and wet FPV (floating on lake surface). The thermistors used for measuring temperature were held in place on the panels using 3M VHB tape. The air and water temperature were also measured with thermistors. The SunPower panels came with MC4 connectors installed on 12 AWG (2 mm2) wires. MC4 connectors were added to the 14 AWG (1.6 mm2) wires coming from the NanoDAQ, including the load wires. An additional hole was made in the NanoDAQ waterproof case and sealed using 3M 5200 to use the battery's USB port to power it. An AC load with a timer was used to drain the battery during mid-day to ensure there was a load to produce the power measurement. The schematic of the wiring diagram for the experimental set up is shown in Figure 3.

**Figure 2.** Closeup of floating photovoltaic/floatovoltaic (FPV) corner after deployment, showing water coverage from a modest wave (top left).

**Figure 3.** Wiring diagram for NanoDAQ monitoring board.

The FPV utilized mooring similar to that used in [54] except for the inclusion of a buoy. The wet FPV was moored by using an anchor and a towing ring on land. A rope was looped through the grommets in the solar PV and overhand loop knots were tied to secure the FPV in place. Energy generation of dry PV and wet FPV, temperature of air, water, PV, and FPV were recorded in 15 min increments.

### *2.2. Water Evaporation Modeling*

The Penman–Monteith model used in this study is a datum intensive water evaporation model because it requires the measurement of several weather data. Some of the data can be calculated, but the accuracy of the model is increased if they are measured. The Penman–Monteith model was originally designed to calculate the evapotranspiration losses from leaves' and canopies' surfaces [57]. However, the method has been adapted in several studies to estimate the evaporation of surface water [72–75]. One important thing to note regarding the use of the Penman–Monteith evapotranspiration model for lake evaporation is the use of water temperature instead of air temperature in some of the parameters' calculations: the outgoing longwave radiation, the partial vapor pressure at the water surface and slope of the temperature saturation water vapor curve. The original Penman–Monteith model estimates the evapotranspiration of crops; therefore, the model only uses the air temperature in its implementation. The use of water temperature instead of air temperature has been validated in several lake evaporation studies [72,74,75].

The Penman–Monteith [57] equation adapted to open water surfaces is [74,75]:

$$E\_L = \frac{1}{\lambda} \times \frac{\left(\Delta \times (R\_N - H\_S) + 86400 \times \rho\_a \times \mathbb{C}p\_a \times \frac{(P\_w - P\_d)}{r\_d}\right)}{\Delta + \gamma} \tag{1mm\text{-day}^{-1}} \tag{1}$$

where *EL* (mm/day) is the daily evaporation rate and *Cpa* (kJ/kg/ ◦C) and ρ*<sup>a</sup>* (kg/m3) are, respectively, the heat capacity, and the density of air. The other parameters in the Penman–Monteith equation need to be calculated and depend on several weather data. The weather data needed to calculate these parameters are comprised of: the daily maximum (*Ta,max*) (◦C) and daily minimum (*Ta,min*) (◦C) air temperature; the daily maximum (*Tw,max*) (◦C), daily minimum (*Tw,min*) (◦C), and daily mean water temperature (*Tw*) (◦C); the daily maximum (*Rhmax*) (%), and daily minimum relative humidity (*Rhmin*) (%); the daily mean dew temperature (*Td*) (◦C), the daily mean atmospheric pressure (*P*) (kPa); the daily mean wind speed (*ws*) (m/s) at a height of 2 m above the water surface; and the daily incoming solar radiation (*RS*) (MJ/m2/day). The other parameters that are needed to calculate the components in the evaporation model of Penman–Monteith include: the altitude of the lake's location (*h*) (m); the surface area (*A*) (m2), and the effective depth (*dw*) (m) of the lake reservoir; and the latitude of the location of the water surface (φ) (rad).

When all the listed parameters are available, the computation of the lake water evaporation using the Penman–Monteith model starts with the calculation of the mean saturation vapor pressure (*Pw*) (kPa), and the actual vapor pressure of the air (*Pa*) (kPa) [58,67,73]:

$$P\_w = \frac{1}{2} \times 0.6108 \times \left( \exp\left(\frac{17.27 \times T\_{w,\text{max}}}{T\_{w,\text{max}} + 237.3}\right) + \exp\left(\frac{17.27 \times T\_{w,\text{min}}}{T\_{w,\text{min}} + 237.3}\right)\right) \tag{2}$$

$$P\_d = \frac{1}{2} \times 0.6108 \quad \times \begin{pmatrix} \frac{Rh\_{\text{min}}}{100} \times \exp\left(\frac{17.27 \times T\_{w,\text{max}}}{T\_{w,\text{max}} + 237.3}\right) + \frac{Rh\_{\text{max}}}{100} \\ \times \exp\left(\frac{17.27 \times T\_{w,\text{min}}}{T\_{w,\text{min}} + 237.3}\right) \end{pmatrix} \quad \text{(kPa)}\tag{3}$$

After the calculation of the two vapor pressures, the slope of the saturation vapor pressure curve (Δ) (kPa/ ◦C) is calculated [58,67,73]:

$$
\Delta = \frac{4096 \times P\_w}{\left(T\_w + 237.3\right)^2} \quad \left(\text{kPa} \cdot ^\circ \text{C}^{-1}\right) \tag{4}
$$

Then, the latent heat of vaporization (λ) (MJ/kg), which depends on the water temperature, is calculated [58,73]:

$$
\lambda = 2.501 - 0.002361 \times T\_w \quad \left(\text{kPa} \, ^\circ \text{C}^{-1}\right) \tag{5}
$$

From the latent heat of vaporization, the psychrometric constant (γ) (kPa/ ◦C) can be deduced [58,67],

$$\gamma = \frac{\mathbb{C}p\_a \times P}{\mathbb{R}\_{MW} \times \lambda} \quad \left(\text{kPa} \, ^\circ \text{C}^{-1}\right) \tag{6}$$

In Equation (6), *RMW* = 0.622 and is equal to the molecular weight of water vapor over the molecular weight of dry air.

After that, the wind function *fw* (MJ/m2/kPa/day) is needed to estimate the aerodynamic resistance of the water surface. The formula used to calculate the wind function is proposed by McJannet et al. [76]. The formula was found to work well with the Penman–Monteith evaporation model. The wind function calculation by McJannet's formula depends on the wind speed as well as on the surface area of the lake.

$$f\_w = (2.36 + 1.67 \times w\_s) \times A^{-0.05} \quad \text{(M)} \cdot \text{m}^{-2} \cdot \text{kPa}^{-1} \cdot \text{day}^{-1} \tag{7}$$

Once the wind function is known, a combination of the Penman–Monteith model equations presented in the works of Zotarelli et al. and Finch et al. gives the value of the aerodynamic resistance (s/m) [67,75]:

$$r\_a = \frac{\rho\_a \times \text{Cp}\_a \times 86400}{1000 \times \text{y} \times f\_{\text{w}}} \qquad \left(\text{s} \cdot \text{m}^{-1}\right) \tag{8}$$

The two remaining terms are the net solar radiation (*RN*) (MJ/m2/day) and the change in water heat storage flux (*HS*) (MJ/m2/day). The net solar radiation's calculation depends on the net longwave radiation (*RNL*) (MJ/m2/day) and the net shortwave radiation (*RNS*) (MJ/m2/day) [58,67,73].

$$R\_N = R\_{NS} - R\_{NL} \quad \text{ (MJ} \cdot \text{m}^{-2} \cdot \text{day}^{-1}) \tag{9}$$

The net shortwave radiation is calculated using the albedo (*a*) and the measured incoming solar radiation (*RS*) (MJ/m2/day) [58,67,73–75]:

$$R\_{NS} = (1 - a) \times R\_S \qquad \left(\text{MJ} \cdot \text{m}^{-2} \cdot \text{day}^{-1}\right) \tag{10}$$

The net longwave radiation is calculated by taking the difference between the outgoing longwave radiation (*ROL*) (MJ/m2/day) and the incoming longwave radiation (*RIL*) (MJ/m2/day). The incoming longwave radiation is given by the Equation (11) [77,78]

$$R\_{\rm IL} = \sigma \left( \mathbf{C}\_f + \left( 1 - \mathbf{C}\_f \right) \left( 1 - \left( 0.261 \times \exp\left( -7.77 \times 10^{-4} T\_a^2 \right) \right) \right) \left( T\_d + 273.15 \right)^4 \qquad \left( \text{MJ} \cdot \text{m}^{-2} \cdot \text{day}^{-1} \right) \tag{11}$$

In Equation (11), σ [MJ/m2/T4/day] is the Stefan–Boltzmann's constant, *Ta* is the daily mean air temperature and *Cf* is the cloud coverage fraction that has been estimated as follows [79]:

$$\begin{array}{l} \mathcal{C}\_{f} = 1.1 - R\_{Ratio} \text{ ; } R\_{Ratio} \le 0.9\\ \mathcal{C}\_{f} = 2(1 - R\_{Ratio}) \text{ ; } R\_{Ratio} > 0.9 \end{array} \tag{12}$$

The parameter *RRatio* is the ratio of the incoming solar radiation (*RS*) to the clear sky radiation *RCS* (MJ/m2/day). The clear sky radiation is calculated using Equation (13) [75,78,79]:

$$R\_{\rm CS} = \left(0.75 + 2 \cdot 10^{-5} \times h\right) \times R\_{\rm EX} \qquad \left(\text{MJ} \cdot \text{m}^{-2} \cdot \text{day}^{-1}\right) \tag{13}$$

The extraterrestrial radiation *REX* (MJ/m2/day) depends on the latitude of the lake, the sunset hour angle, the solar declination angle, the solar constant, and the inverse relative distance from the sun to earth. This calculation is a well-known procedure that has been detailed in the guide for crop evapotranspiration calculations by the FAO [58]. The outgoing longwave radiation depends on the water surface temperature and is calculated as:

$$R\_{OL} = \varepsilon \times \sigma \times \left(T\_W + 273.15\right)^4 \qquad \left(\text{MJ} \cdot \text{m}^{-2} \cdot \text{day}^{-1}\right) \tag{14}$$

*Tw* ( ◦C) is the mean daily water temperature and ε is the emissivity of the water surface. The emissivity of water surface varies between 0.95 and 0.99 for water temperatures below 55 ◦C [80]. An average value of ε = 0.97 has been used in this study. The net longwave radiation is therefore:

$$R\_{\rm NL} = R\_{\rm IL} - R\_{\rm OL} \qquad \left(\text{MJ} \cdot \text{m}^{-2} \cdot \text{day}^{-1}\right) \tag{15}$$

The water heat storage flux (*HS*) (MJ/m2/day) expresses the change in the heat stored in the water from one day to another. The heat storage flux calculation methods used in two different studies by Abtew et al., and Finch et al. are suitable for shallow water bodies evaporation [73,75]. Since Lake Mead is a deep lake, the equilibrium temperature approach proposed by de Bruin has been used instead. In this approach, an equilibrium temperature is used to estimate a mean daily uniform temperature of the water body for each day [81]. The heat storage flux's formula using de Bruin's method is [78,81–83]:

$$\mathbf{M}\_S = \rho\_w \mathbf{C} p\_w d\_w \times \frac{\left(T\_{uw,j} - T\_{uw,j-1}\right)}{\Delta t} \qquad \qquad \left(\mathbf{M} \mathbf{j} \cdot \mathbf{m}^{-2} \cdot \mathbf{day}^{-1}\right) \tag{16}$$

*Energies* **2020**, *13*, 6285

The constants' values ρ*<sup>w</sup>* (kg/m3), *Cpw* (MJ/kg/ ◦C), *dw* (m) are, respectively, the density of water, the heat capacity of water, and the depth of the lake. *Tuw,j* and *Tuw*,*j*−<sup>1</sup> are, respectively, the mean uniform water temperature for day (*j*), and day (*j* − 1). Δ*t* is the time step for the temperature estimation. The mean uniform water temperature (*Tuw,j*) depends on the equilibrium temperature (*Te*) (◦C) and the time constant (τ) (day):

$$T\_{\rm unv,j} = T\_\varepsilon + \left(T\_{\rm unv,j-1} - T\_\varepsilon\right) \times \exp\left(\frac{-1}{\tau}\right) \qquad (^\circ \text{C}) \tag{17}$$

$$T\_{\rm \varepsilon} = T\_{\rm wb} + \frac{R\_{\rm N,wb}}{4 \times \sigma \times \left(T\_{\rm wb} + 273.15\right)^3 + f\_{\rm w} \times \left(\Lambda\_{\rm wb} + \gamma\right)}\tag{18}$$

$$\tau = \frac{\rho\_w \times \text{Cp}\_w \times d\_w}{4 \times \sigma \times \left(T\_{wb} + 273.15\right)^3 + f\_w \times \left(\Delta\_{wb} + \gamma\right)} \qquad \text{(day)}\tag{19}$$

*RN,wb* (MJ/m2/day), *Twb* ( ◦C), and Δ*wb* (kPa/K) are, respectively, the net radiation at the wet-bulb temperature, the wet-bulb temperature, and the slope of the saturation vapor pressure curve at the wet-bulb temperature. The wet-bulb temperature (*Twb*) is calculated using the following equation [78,83]:

$$T\_{\rm ub} = \frac{0.00066 \times 100 \times T\_a + \frac{\left(4098 \times P\_b \times T\_d\right)}{\left(T\_d + 237.3\right)^2}}{0.00066 \times 100 + \frac{\left(4098 \times P\_b \times T\_d\right)}{\left(T\_d + 237.3\right)^2}} \qquad \text{( $^\circ$ C)}\tag{20}$$

The saturation vapor pressure curve at the wet-bulb temperature Δ*wb* (kPa/K) is calculated by:

$$
\Delta\_{wb} = \frac{4096 \times 0.6108 \times \exp\left(\frac{17.27 \times T\_{wb}}{T\_{wb} + 237.3}\right)}{\left(T\_{wb} + 237.3\right)^2} \qquad \left(\text{kPa} \cdot \text{K}^{-1}\right) \tag{21}
$$

The net radiation (*RN,wb*) at the wet-bulb temperature is:

$$R\_{N,wb} = (1 - a) \times R\_S + \left(R\_{IL} - R\_{OL,wb}\right) \qquad \left(\text{MJ} \cdot \text{m}^{-2} \cdot \text{day}^{-1}\right) \tag{22}$$

In Equation (22), *ROL,wb* (MJ/m2/day) is the outgoing longwave radiation at the wet-bulb temperature and is calculated by:

$$R\_{\rm OL,ub} = \mathbb{C}\_f \times \sigma \times \left( \left( T\_4 + 273.15 \right)^4 + 4 \times \left( T\_4 + 273.15 \right)^3 \times \left( T\_{ub} - T\_4 \right) \right) \quad \text{(MJ:m}^{-2}\text{-day}^{-1}\text{)}\tag{23}$$

After the calculation of all parameters, the lake evaporation's value (*EL*) can be calculated using Equation (1).

#### *2.3. Energy Production Modeling*

The power output of a PV module (*Pout*) (W) is calculated by applying different losses to the incoming solar irradiance and is given by:

$$P\_{\rm out} = I\_{\rm S} \times Ap \times \eta p \qquad \text{ (W)}\tag{24}$$

where *IS* (W/m2) is the incoming solar irradiance, *AP* (m2) is the effective area of the solar panel, and η*<sup>P</sup>* (%) is the efficiency of the PV system. In this study, the efficiency of the system includes the electrical efficiency of the module, which is dependent on the operating temperature, the shading losses, the soiling and hotspot losses, and the mismatch losses. Additionally, the solar irradiation component used is the global horizontal irradiation because the inclination of the panels is 0◦. The power output is calculated hourly and summed up to determine the energy production of the system over a year.
