1. Introduction

2. Agrivoltaic Agrotunnel

3. Methodology

3.1. Thermal Model

The load calculations of the agrotunnel are estimated using the following thermal model developed. It is assumed that all the lights are on and planted crops are mature enough. Meanwhile, one person is doing some non-strenuous work such as planting, harvesting, watering, etc. in the building. The following sub-models are developed based on the load calculation principles [41].

3.1.1. External Loads

CLTD/CLF method, described in the 1997 AHSRAE Fundamentals [42], is used to calculate the transmission load of the agrotunnel, which is the most practical method for sizing the HVAC equipment while being simple.

Equation (1) is used for calculating the heat gains through the walls, the roof, and the floor. Cooling load temperature difference (CLTD) considers the collective impact of variations in indoor and outdoor temperatures, daily temperature fluctuations, solar radiation, and heat retention within the building structure. The values are determined using the tables provided in Chapter 28^th of ASHRAE Fundamentals Handbook [42].

$${\dot{Q}}_{cc}=\sum \frac{A_k}{R_{tot,k}}CLTD+\frac{A_{floor}}{R_{tot,floor}}(T_o-T_i),$$

(1)

where, ${\dot{Q}}_{cc}$ represents the heat losses due to the conduction and convection transmission through the envelope. $A_k$ is the surface area of each part of the envelope (m² or ft²) except for the floor and $A_{floor}$ stands for the surface area of the floor. $R_{tot,k}$ corresponds to the conduction and convection thermal resistance of each part of the envelope (m².K/W or hr. ft².F/Btu), which can be calculated by Equation (2) [43]:

$$R_{tot,k}=R_{conv,o}+R_{cond}+R_{conv,i},$$

(2)

$T_i$ and $T_o$ are the indoor design temperature and the outdoor temperature (°C or °F), respectively. As the ASHRAE tables offer hourly CLTD values based on a standard scenario, such as an outdoor maximum temperature of 95 °F, mean temperature of 85 °F, and a daily temperature range of 21 °F, Equation (1) is modified through Equations (3-5) to accommodate correction factors for situations deviating from this base case. CLTD values of the base case for the material used in the envelope are tabulated in Table 1.

$${\dot{Q}}_{cc}=\sum \frac{A_k}{R_{tot,k}}{CLTD}_{corrected}+\frac{A_{floor}}{R_{tot,floor}}(T_o-T_i),$$

(3)

$${CLTD}_{corrected}=CLTD+\left(78-T_i\right)+(T_{o,mean}-85),$$

(4)

$$T_{o,mean}=T_{o,max}-\frac{DRT}{2},$$

(5)

the daily range of temperature (DRT) indicates the difference between the maximum and the minimum temperature within 24 hours. For the floor transmission, ($T_o-T_i$) will be simply used.

The sensible and latent loads caused by infiltration can be calculated using Equations (6) & (7), respectively.

$${\dot{Q}}_{sens,inf}=n_{ACH}{V}_{AT}{\rho }_a{Cp}_a\left(T_o-T_i\right),$$

(6)

$${\dot{Q}}_{lat,inf}=n_{ACH}{V}_{AT}{\rho }_a\left({\omega }_o-{\omega }_i\right)h_{fg},$$

(7)

here, $V_{AT}$ is the total volume of the agrotunnel (m³ or ft³). ${\rho }_a$ and ${Cp}_a$ are the air density (kg/m³ or lb_m/ft³) and specific heat capacity (kJ/kg.K or Btu/lb_m.F), respectively. $n_{ACH}$ indicates the infiltration rate in ACH (hr^-1), which usually is 0.25 for a newly constructed and well-insulated building [44]; for this specific building without any windows and completely sealed walls, however, 0.1 will be a rational assumption. Also, ${\omega }_i$ and ${\omega }_o$ denote the indoor and outdoor humidity ratio in (kg_w/kg_da or lb_m,w/lb_m,da), respectively, and $h_{fg}$ corresponds to the enthalpy of vaporization of water in (kJ/kg or Btu/lb_m).

3.1.2. Lights

The waste heat dissipated by the LED lights mainly depends on their power and the cooling load factor ($F_{CLF,light}$), which can vary within a day based on the operating schedule of lights. $F_{CLF,light}$ would be 1 for the case lights are on for 24 hours. Otherwise, Table 2 will be used for one story building [42].

Equation (8) gives the waste heat dissipated by lights in (W or Btu/hr). $n_{lights}$, $P_{lights}$, $F_{lu}$, and $F_a$ are the number of lights used, the power of lights used in (W), the lights use factor, which can be assumed to be 1, and the ballast allowance factor, which can be assumed 1 for LED lights [45].

$${\dot{Q}}_{lights}=n_{lights}{P}_{lights}{F}_{lu}{F}_a{F}_{CLF,light},$$

(8)

3.1.3. People

Each person working in agrotunnel brings in both sensible and latent loads. Sensible and latent heat gained by a person who is doing sedentary (standing or walking slowly) or some light works are 92.4 W (315 Btu/hr) and 95.3 W (325 Btu/hr), respectively [42]. Total cooling load caused by the presence of the people in agrotunnel can be calculated using Equation (9):

$${\dot{Q}}_{people}={\dot{Q}}_{sens,people}+{\dot{Q}}_{lat,people}=n_{people}\times (315\times F_{CLF,people}+325\times F_{CLF,lat,people}),$$

(9)

where, $n_{people}$ represents the total number of people who are working in agrotunnel, and $F_{CLF,people}$ stands for the cooling load factor of people, based on the number of hours they work each day. In the basic scenario in the present study, it is assumed that one person is working 8 hours 7 days of week from 9 AM to 5 PM. Therefore, using Table 3, the $F_{CLF,people}$ will be extracted. $F_{CLF,lat,people}$ includes either 0 or 1 to consider the effect of the grower’s presence.

3.1.4. Evapotranspiration of Plants

There are many models of evapotranspiration [46,47], which incorporate the effect of various parameters, interpretation of which is beyond the scope of this study. In this case 8 walls each with 720 pots for a total of 5,760 all of which are being watered aeroponically at potentially different stages of growth and different species present an enormously complicated system. A more straightforward way to estimate the evapotranspiration load in a grow room involves calculating the net water consumption of plants. This entails measuring the volume of irrigation water introduced into the agrotunnel and subtracting the volume of water drained away [48]. The average amount of water consumed (${\dot{m}}_{water}$) can be estimated in (kg/s or lb_m/hr). Hence, the heat of vaporization will be calculated using Equation (10):

$${\dot{Q}}_{evap}={\dot{m}}_{water}{h}_{fg}$$

(10)

During this process, the moist of soil and plants’ leaves absorb heat from the environment, causing a temperature drop (sensible heating load), then evaporate. The energy absorbed is converted to latent heat of vapor in the air, which causes an increase in the humidity level of the microclimate (the latent cooling load). This transfer in energy leads to a constant enthalpy process, which is shown in Figure 6. Since the evapotranspiration of mature plants in a fully planted agrotunnel can be significant, it is assumed that the sensible heating load caused by the evapotranspiration of plants on the warmest day of the year will be met by the HP and the latent part is met by the dehumidifier. This is a rational assumption according to the experience of growers in similar facilities [49].

Total sensible and latent thermal loads of the agrotunnel will be calculated using Equations (11) & (12), respectively.

$${\dot{Q}}_{tot,sens}={\dot{Q}}_{cc}+{\dot{Q}}_{sens,inf}+{\dot{Q}}_{lights}+{\dot{Q}}_{sens,people}-{\dot{Q}}_{sens,evap}$$

(11)

$${\dot{Q}}_{tot,lat}={\dot{Q}}_{lat,inf}+{\dot{Q}}_{lat,people}+{\dot{Q}}_{lat,evap}$$

(12)

The following assumptions are taken into account to run the simulation:

• It is assumed that the growing room and the monitoring room are at the same temperature such that no heat transmission occurs between them.
• It is assumed that the air inside the agrotunnel is uniformly mixed, which is appropriate as the HP locations and clearance over the grow walls ensure paths of airflow.
• The design conditions of agrotunnel are the temperature of 22 °C, and the relative humidity (RH) of 68%.
• The thermal resistance of the still air ($R_{conv,i}$) on the vertical walls is 0.12 m².K/W [44].
• The thermal resistance of the still air ($R_{conv,i}$) on the ceiling is 0.11 m².K/W whenever the indoor temperature is higher than the outdoor temperature, and is 0.16 m².K/W when the indoor temperature is lower than the outdoor temperature [44].
• The thermal resistance of the still air ($R_{conv,i}$) on the floor is 0.16 m².K/W whenever the indoor temperature is higher than the outdoor temperature, and is 0.11 m².K/W when the indoor temperature is lower than the outdoor temperature [44].
• The thermal resistance of the outdoor moving air ($R_{conv,o}$) is 0.029 m².K/W [44].

3.1.5. Maximum Heating Load Model

In order to size HP, the maximum cooling load obtained by CLTD method will be compared to the maximum heating load under the worst-case scenario. The worst-case scenario for heating load estimations is defined for an evacuated agrotunnel, where conduction and convection heat transfer along with the infiltration losses are the only factors included. Total sensible heat loss from the agrotunnel is calculated by Equation (13) in (W or Btu/hr) [44]:

$${\dot{Q}}_{sens,heat}={\dot{Q}}_{cc,heat}+{\dot{Q}}_{sens,inf,heat}=\sum \frac{A_k}{R_{tot,k}}\left(T_i-T_o\right)+n_{ACH}{V}_{AT}{\rho }_a{Cp}_a\left(T_i-T_o\right),$$

(13)

Total latent heat loss through infiltration is calculated using Equation (7). Accordingly, the maximum total heating load will be calculated by Equation (14):

$${\dot{Q}}_{tot,heat}={\dot{Q}}_{cc,heat}+{\dot{Q}}_{sens,inf,heat}+{\dot{Q}}_{lat,inf,heat}$$

(14)

3.1.6. Heat Pump Model

The black box heat pump model is created by utilizing the performance datasheet of GSZ16 Goodman Air-Air Split heat pump (16 SEER and 9 HSPF) for both heating and cooling functions [50]. A supervised regression approach is implemented in Python to derive mathematical functions for the COP (Coefficient of Performance) of heat pumps within a nominal capacity range of 18,000-60,000 BTU/hr. Drawing from Li et al.'s correlations for load demand and COP of an air-conditioning heat pump [51], it is anticipated that the COP regressions would follow a second-degree polynomial pattern based on indoor and ambient temperatures.

3.1.6.1. Heating Mode

The correlation of COP is dependent on the ambient temperature (T_o) in (°C) as Equation (15):

$${COP}_{heating}=b_0+b_1{T}_o+b_2{T}_o^2,$$

(15)

the coefficients $b_0-b_2$ are extracted using the polynomial curve fitting technique and are shown for all rated capacities of GSZ16 Goodman Heat Pumps in Table 4 for heating mode. The fitting curve of heating COP for 18,000 BTU/hr heat pump is shown in Figure 7a.

3.1.6.2. Cooling Mode

The manufacturer's datasheet for cooling mode were created considering fluctuations in 3 variables (ambient temperature, indoor dry bulb temperature, and indoor wet bulb temperature) [50]. Hence, the regression is carried out in two steps: first, fitting a 2D curve as a function of indoor dry bulb and wet bulb temperatures, second, fitting a 3D surface as a function of first formulation and the outdoor temperature. Figure 6b depicts the fitting surface for the COP of an 18,000 Btu/hr HP in cooling mode. The COP for the cooling mode is determined using Equation (16).

$${COP}_{cooling}=b_o+b_1{T}_i+b_2{T}_{iwb}+b_3{T}_o+b_4{T}_i{T}_{iwb}+b_5{T}_i{T}_o+b_6{T}_{iwb}{T}_o+b_7{T}_i^2+b_8{T}_{iwb}^2+b_9{T}_o^2+b_{10}{T}_i^2{T}_o+b_{11}{T}_{iwb}^2{T}_o+b_{12}{T_i{T}_{iwb}T}_o$$

(16)

Table 5. Coefficients of polynomial correlations of COP in cooling mode.

Table 5. Coefficients of polynomial correlations of COP in cooling mode.

		Rated Capacity (Btu/hr)
		18,000	24,000	30,000	36,000	42,000	48,000	60000
Coefficients of COP	b0	10.45648	9.14469	10.80332	10.66776	11.1678	10.7799	10.48214
	b1	-0.11064	0.012049	-0.13687	-0.12536	-0.14811	-0.12931	-0.12989
	b2	-0.21069	-0.21788	-0.18697	-0.19849	-0.208	-0.1978	-0.18888
	b3	-0.17693	-0.15915	-0.18717	-0.18142	-0.19111	-0.18515	-0.18156
	b4	-0.00038	-5E-05	-0.00024	-0.00012	8.46E-05	-0.00013	-0.00014
	b5	0.001363	-0.00014	0.001632	0.001463	0.001723	0.001535	0.001532
	b6	0.002596	0.002523	0.002229	0.002317	0.002419	0.002348	0.002227
	b7	0.002739	-0.0001	0.003153	0.002864	0.003197	0.002939	0.00295
	b8	0.008582	0.008251	0.007745	0.007893	0.00789	0.00791	0.007562
	b9	0.000623	0.000609	0.000722	0.000671	0.000727	0.000694	0.000706
	b10	-3.4E-05	1.16E-06	-3.8E-05	-3.3E-05	-3.7E-05	-3.5E-05	-3.5E-05
	b11	-0.00011	-9.6E-05	-9.2E-05	-9.2E-05	-9.2E-05	-9.4E-05	-8.9E-05
	b12	7.73E-06	9.2E-07	4.63E-06	2.34E-06	-1.6E-06	2.54E-06	2.61E-06
	R2	99.96	99.98	99.97	99.98	99.98	99.98	99.98

When the maximum heating and cooling load of the agrotunnel on the coldest/warmest day of the year is calculated, the size of HP can be determined. After selecting the HP, its COP correlation is used to calculate the power consumption in kW.

3.3. Solar PV Sizing

PV system sizing was carried out by determining the 1-kW PV potential using the open-source System Advisor Model (SAM) [55] for each of the six locations: Albany, Edmonton, Houston, London, Montreal and San Francisco. Using the total annual electrical energy required for agrotunnel operation (E_{Annual load} in (kWh)) in each location and the annual energy potential of 1-kW PV system determined from SAM (E_{Annual load, 1kW} in (kWh)), the total PV system size (PV_Sys.Capacity) required to meet the annual electrical load is ascertained. The formula for the determining the PV size in (kW) is given by Equation (17):

$${PV}_{Sys.Capacity}=\frac{E_{AnnualLoad}}{E_{AnnualLoad,1kW}}$$

(17)

The PV system capacity determined through the estimate calculation is then used as input in SAM to run a detailed simulation at the appropriate scale. The simulations are run for each location and double checked if they meet the annual electrical load of the agrotunnel. Additionally, the land area required for PV systems, considering their modules’ transparency levels ranging from 0% (conventional silicon modules) to 69% (semi-transparent modules) will be reported. Table 7 details the assumptions used to perform the final modelling in SAM. A novel fixed-tilt racking design, adaptable into a variable-tilt racking system [15], is considered for use in this research. Soiling losses, DC power losses and AC power losses are kept as default.

Self-consumption (SC) and self-sufficiency (SS) are important metrics for assessing the energy performance of distributed energy systems at the local level [56]. SC measures the proportion of locally produced energy that is consumed locally, calculated as the energy generated by the PV system (E_PV) minus the energy sent to the grid (E_togrid), divided by the total energy generated by the PV system (E_PV) as seen in Equation 18. SS, on the other hand, indicates the extent to which local energy production meets local energy demand, calculated as the energy generated by the PV system minus the energy sent to the grid, divided by the local energy consumption (E_load) (Equation 19). Both metrics exclude nighttime hours when the PV system is not operational, and all energy is sourced from the grid, thus SC and SS are set to zero during these times. These metrics can be formulated through Equations (18) & (19) [57,58].

$$SC=\frac{E_{PV}-E_{togrid}}{E_{PV}}\times 100$$

(18)

$$SS=\frac{E_{PV}-E_{togrid}}{E_{Load}}\times 100$$

(19)

Table 7. Input parameter for SAM for PV system sizing for the agrotunnel.

Table 7. Input parameter for SAM for PV system sizing for the agrotunnel.

Parameters	London	Albany	Edmonton	Houston	Montreal	San Francisco
PV module	Heliene 144HC-460 Bifacial [59]
Module type	Mono Crystalline Silicon - Bifacial
Tilt angle	34o [60]	34o [61]	53o [62]	27o [63]	30o [64]	32o [65]
Azimuth	180o
Soiling losses	5%
DC Power losses	4.44%
AC Power losses	1%

3. Results

4. Discussion

5. Conclusions

Institutional Review Board Statement Not applicable.

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