Development of mathematical models for predicting vapour pressure deficit inside a greenhouse from internal and external climate

Vapour pressure deficit (VPD) inside protective structures under cropped conditions significantly affects the plant growth and productivity through its direct relationship with crop transpiration or irrigation management. Thus, monitoring VPD inside greenhouse during crop growth period becomes essential to limit it to a desired range. The present study was undertaken to develop mathematical models for predicting SVP, AVP and VPD inside a greenhouse independently using internal and external climatic parameters as inputs. The root mean square error (RMSE) was obtained in the range of 0.03-0.10 kPa and 0.27-1.03 kPa respectively for the models developed from internal and external climatic parameters as model inputs. The average model efficiency (neff) was computed to be 98.7 per cent, 92.2 per cent and 100.0 per cent respectively for SVP, AVP and VPD when predictions were made using internal climate as input. Similarly, for the models developed from external climate as model input, neff was worked out to be 96.7, 86.1 and 93.0 per cent for SVP, AVP and VPD respectively. The developed models presented a high degree of precision in predicting SVP, AVP and VPD with both internal and external climatic conditions as model inputs inside a naturally ventilated greenhouse under cucumber crop in soilless media.

Vapour pressure deficit (VPD) and leaf area index (LAI) are two key parameters which affect the greenhouse crop transpiration (Singh et al., 2017a) thereby the crop water requirement and productivity. Annu Priya et al. (2014) reported that increase in vapour pressure deficit resulted in increase in ET o during different seasons at Varanasi. The VPD between greenhouse air and crop affects the transpiration and accordingly the absolute air humidity.
VPD under cucumber crop in soilless culture is linearly related to transpiration even for higher values (>3.0 kPa) (Singh et al., 2017b). VPD negatively affects the mean fruit weight of cucumber with an increase in VPD under high relative humidity. According to Singh et al (2017b), VPD should lie in the range of 0.53-1.10 kPa for best possible growth and development of cucumber plant. Only a trivial effort has been made to model the vapour pressure (or VPD) inside a greenhouse under cucumber crop in soilless media. The present study was thus undertaken to develop mathematical models for predicting SVP, AVP and VPD inside a naturally ventilated greenhouse when cucumber crop was cultivated in soilless media separately from internal and external climate.

Description of study site
Experimental trials were carried out inside a 28.0 m×20.0 m (floor area 560.0 m 2 ) double-span naturally ventilated greenhouse oriented in North-South direction at Punjab Agricultural University, Ludhiana (latitude: 30° 56Ń , longitude: 75° 52´ E and altitude: 247.0 m above mean sea level). The entire surface area of the greenhouse floor was covered with a mat for avoidance of weed emergence. Cucumbers were cultivated in coco-peat growing media inside the greenhouse for two growing seasons

Measurement of dynamic parameters
The temperature and humidity sensors were installed (inside and outside the greenhouse) at an average height of 1.84 m in the plant community. The diurnal climatic data (temperature and humidity) was logged in the data logger (Delta-T Devices, UK) installed inside the greenhouse separately from inside and outside the greenhouse at an hourly interval on daily basis. Using the recorded data, the VPD and SVP inside the greenhouse were computed using methods reported in Arellano et al. (2006) and Sengar and Kothari (2008) respectively.

Saturation vapour pressure (SVP)
The vapour pressure indicates the evaporation rate of a liquid at a given temperature. In a closed system at a particular temperature, vapour pressure (or equilibrium vapour pressure) is the pressure exerted by a vapour in ther modynamic equilibrium with its solid or liquid phase. SVP is dependent on temperature and material under consideration. The observed data was analyzed critically through curve fitting and multiple regression analysis to form mathematical relationship between SVP and temperature. Mathematically, the developed model is given as. (1) The slope of SVP versus temperature curve is given by following equation.
(2) Where,  is the angle made by the tangent to the SVP versus temperature curve with horizontal surface.

Actual vapour pressure (AVP)
The AVP is dependent on relative humidity and temperature. (3)

Vapour pressure deficit (VPD)
The VPD was obtained by difference between SVP and AVP (4) Where, T apc is air temperature in plant community (°C), E apc is relative humidity in plant community (%), A (0.7392) and B (0.058) are empirical coefficients.

Model development with external climate and time as inputs
Saturation vapour pressure (SVP) Where, T aos is outside air temperature, E apc is relative humidity (%) of outside air at time t, t (0,1, 2, 3…) is the time (hour). P, Q, R ,S and T are empirical coefficients such that P= 0.7392, Q = 0.06264, R = 0.0019, S = 0.8427 and T = ) 100 1 ( 0.00021. The above equations (equations 7, 8 and 9) can be used to predict SVP, AVP and VPD from external climate (temperature and humidity) and time as inputs.

Statistical analysis
The statistical parameters viz. standard deviation (), coefficient of determination (R 2 ), root mean square error (RMSE) and model efficiency (n eff ) were estimated to evaluate the performance of developed models (Table 1). The model efficiency was calculated using equation given by Nash and Sutcliffe (1970).

Variation of temperature in the plant community
Air temperature in plant community (T apc ) was certainly affected with height in plant community during both cropping seasons ( Fig. 1 and 2). During season 1 (Sept.'16 to Jan.'17), the average rise in air temperature was recorded in the range of 22.9-24.6 °C from 0.5 m to 1.8 m height. Similarly, during season 2 (Feb. to May'17), the average rise in air temperature was found in the range of 30.9-33.5 °C with height from 0.5 m to 1.8 m height. It is therefore important to train the plant to an optimum height in relation to the temperature variation in the vertical profile in greenhouses, especially during summer.

Model validation from internal climate
The models were validated through comparison between the actual and predicted data of SVP, AVP and VPD for selected period i.e. during 28 th December 2016 to 2 nd January 2017. The statistical parameters estimated for evaluation of model performance are presented in Table 1. The developed model of VPD was validated for a period of six days from December 28, 2016 to January 02, 2017 during season 1.
For the air temperature in plant community, the mean standard deviation for observed and predicted data was 4.4°C and 5.1°C respectively. The average root mean square error (RMSE), coefficient of determination (R 2 ) and model efficiency (n eff ) were computed to be 1.7 °C, 0.95 and 88.3 per cent respectively. However, for relative humidity of air within plant community, the mean standard deviations for Where,  obs = observed or computed standard deviation,  pre = predicted standard deviation, R 2 = coefficient of determination, RMSE = root mean square error and n eff = model efficiency. observed and predicted data were 20.7 per cent and 21.3 per cent respectively. RMSE, R 2 and n eff were obtained as 8.8, 0.95 and 81.7 per cent respectively.
During season 2, the developed model of VPD was also validated for a period of six days from March 28, 2017 to April 02, 2017. For VPD, the mean standard deviation for actual and predicted data was 0.86 and 0.88 kPa respectively. The average RMSE, R 2 and n eff were computed to be 0.02 kPa, 1.0 and 100.0 per cent respectively. For SVP, the mean standard deviation for actual and predicted data was 1.09 and 0.86 kPa respectively. The average RMSE, R 2 and n eff were obtained to be 0.07 kPa, 1.0 and 98.7 per cent respectively (Table 1).

Model validation from external climate
The developed models were validated for diurnal variation in SVP, AVP and VPD through a comparison between the actual and predicted data. Fig. 3 and 4 demonstrate the relationship between predicted and actual data VPD for the models developed to make predictions using internal and external climatic data as model inputs during season 1 and 2 respectively. For VPD, the mean standard deviation for observed and predicted data was computed as 2.16 and 2.41 kPa respectively. The average RMSE, R 2 and n eff were 0.56 kPa, 0.94 and 93.0 per cent respectively.
For SVP the mean standard deviations for observed and predicted data were 4.14 and 3.83 kPa respectively. The average RMSE, R 2 and n eff were computed to be 0.69 kPa, 0.99 and 96.7 per cent respectively. For AVP the mean standard deviations for observed and predicted data were 2.11 and 1.57 kPa respectively. The average RMSE, R 2 and n eff were obtained to be 0.67 kPa, 0.93 and 86.1% respectively.
The statistical analysis of simulation results of the model as shown in Table 1 indicated that the above models can be applied at any compartment of heated (naturally) or unheated greenhouses, may or may not provide with natural ventilation. However, variation may arise with respect to time and regional climatic conditions.

CONCLUSIONS
Having known the importance of VPD for plant growth and productivity under a protective structure, models were developed for predicting SVP, AVP and VPD independently with internal and external climatic parameters as model inputs. The statistical comparisons indicated that the developed models were sufficiently accurate to simulate the SVP, AVP and VPD. Thus, the developed models can be adopted to predict the SVP, AVP and VPD inside a greenhouse under cropped conditions independently with internal and external climatic conditions as model inputs.