# Simulation of canopy air temperature control in closed greenhouses modeled using coupled elliptic models

Rousseau Tawegoum

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Rousseau Tawegoum. Simulation of canopy air temperature control in closed greenhouses modeled using coupled elliptic models. CMSDA 2024 - 4th International Conference on Computational Modelling, Simulation and Data Analysis, Association for Computing Machinery New York NY United States, Dec 2024, Hangzhou (China), China. pp.809 - 813, 10.1145/3727993.3728126. hal-05245373

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# Simulation of canopy air temperature control in closed greenhouses modeled using coupled elliptic models

Rousseau Tawegoum

Institut Agro Rennes-Angers, center of Angers

Environmental Physics and Horticulture (EPHor) Research Unit

Angers, France

rousseau.tawegoum@agrocampus-ouest.fr

## Abstract

The microclimate that exists within the canopy and on the surface of plants can significantly impact both epidemiology of pathogens and the physiological functions of plants. Using the heat transfer theory and taking into account the canopy as a porous media, a coupled elliptic model in one-dimensional was carried out for a control objective to depict crop temperature and the air canopy. Existence and uniqueness of canopy air temperature optimal control were investigated. The finite difference scheme was used to derive the numerical model. Simulations were performed to assess the ability of the control to maintain canopy air temperature close to prescribed values with acceptable mean error for the apportionments of the two plants species in the considered greenhouse.

## CCS Concepts

* **Computing methodologies** → Modeling and simulation; Model development and analysis; Modeling methodologies; Computing methodologies; Modeling and simulation; Simulation evaluation; Mathematics of computing; Mathematical analysis; Mathematical optimization.

## Keywords

Coupled elliptic model, Temperature, Optimal control, Greenhouse

## ACM Reference Format:

Rousseau Tawegoum. 2024. Simulation of canopy air temperature control in closed greenhouses modeled using coupled elliptic models. In *2024 4th International Conference on Computational Modeling, Simulation and Data Analysis (CMSDA 2024), December 06-08, 2024, Hangzhou, China*. ACM, New York, NY, USA, 5 pages. https://doi.org/10.1145/3727993.3728126

## 1 Introduction

One of the main goals of greenhouses is to preserve environmental conditions near-optimal values for plants growth and development. For this purpose, one needs to control environmental variables such as humidity, air temperature, and the CO₂ concentration, jointly with soil-related factors and solar radiation. For the canopy part, these variables should be handled with the view to obtain a uniform distribution of temperature in order to have uniform growth and avert local condensation, holding the leaf temperature over the

dew point to avoid condensation on the plants, and minimizing the energy request as much as possible.

In a greenhouse, one can be observed different temperature. The climate outside the canopy is known as the ambient climate. The canopy climate is distinct to the temperature of the leaves. It deals with the climate conditions within the canopy and accounts for both the leaf and canopy air temperature. To measure the elements interacting in the greenhouse canopy system, [1] performed a path analysis. These authors observed that the canopy climate can be up to 5 °C colder than the surrounding air on days with high radiation, as opposed to 2°C cooler on cloudy days. The temperature of the leaves was either higher or lower than the temperature of the canopy air.

In conventional greenhouses, the open ventilation system leads to major heat losses, therefore a high demand of energy, particularly when the weather conditions are cool. The semi-closed or closed greenhouses approach seek to cover the cooling, heating, and dehumidification requirements of the greenhouse by utilizing reduced resource as possible [2]. The closed greenhouse is an original idea in sustainable energy management. In principle, it is carried out to maximize the use of solar energy through seasonal storage, lacking the need of ventilation windows. However, most the greenhouses climate control are based on finite dimensional framework. A short review can been found in [3]

Thanks to CFD techniques, greenhouse climate modeling has significantly expanded over the last 25 years, with an emphasis on comprehending the physical behavior of ventilation [4] and the influence of the canopy on the local climate. [5] developed a computational fluid dynamics model to forecast the distribution of CO₂, temperature and air vapor in a venlo-type semi-closed greenhouse with air conditioners using Fluent, a commercial software. With an air-leaf temperature gradient of roughly +1.5 °C in the upper layer of the canopy, +3.5 °C in the middle layer, and +2 °C in the lower layer, tomato crop leaf temperature were lower than air temperature. Navier-stokes k-epsilon Fluent software is usually employed for greenhouse design. It seems complex to be used for real-time control purposes owing to the high computation time it requires.

A largest amount of genuine (3D) CFD simulations remain computationally too expensive to be performed in the real-time control context. A way to diminish this load is to construct approximation models that mirror the comportment of simulation models as closely as possible.

The control of incompressible N-S equations had been studied in the literature [6], [7], [8], [9], [10], [11]. However, the models used in these studies require to be refined in order to take plants and their

<figure><img src="image_2.png" /></figure>

This work is licensed under a Creative Commons Attribution International 4.0 License.
<figure><img src="image_3.png" /><figcaption>Figure 1: Overview of the system.</figcaption></figure>

environment into account with the objective to meet commercial software performance.

The objective of this paper is to investigate the optimal control law of canopy air temperature applicable for real-time control. The first section of the paper describes the system, the second deals with the analysis of the continuous optimal control problem. The third section provides simulation results.

## 2 System Description

An overview of the system, which includes a recirculating greenhouse and an air conditioning unit (ACU), is shown in figure 1. The fan controls the air inlet flow, and the ACU manages its temperature. The ACU [12] can provide the greenhouse with an air flow q at temperature $T_e$ using a passive mixing principle. The ACU recovers and treats the air that circulated through the greenhouse and that was removed from it at temperature $T_s$. Three different species of plants are arranged in the greenhouse based on their stage of growth, and they are illuminated by lamps at temperature $T_l$. The crops, lamps and canopy air all exchange sensible heat through the canopy which is assumed to be a heterogeneous medium.

Let $\Omega \subset \mathbb{R}$, and $\Omega = \Gamma_0 \cup \Gamma_1$. Considering that the flow is always laminar along the leaves and driven by convection forces, equations describing the system are given below :

$$ egin{cases} 
-\partial_x (K_c(x) \partial_x T_c) - \frac{\alpha_v(x)}{p} (T_c - T_a) = g(x) + \sigma \epsilon T_{cmoy} (T_1 - T_c) & \text{in } \Omega \\ 
-\partial_x (K_a(x) \partial_x T_a) - \rho v C_a(x) \partial_x T_a + \frac{\alpha_v(x)}{p} (T_c - T_a) = 0 & \text{in } \Omega \\ 
T_c(0) = T_e; & T_a(0) = T_e \\ 
K_c(L) \partial_x T_c = 0; & K_a(L) \partial_x T_a(L) = T_a - T_s 
\end{cases} 
\quad (1) $$

Where $T_c(x)$ crop temperature, $T_a(x)$ air temperature, $T_e(x)$ air input temperature, $T_l(x)$ light temperature, $T_{cmoy}$ mean leaf temperature, $\alpha_v(x)$ internal coefficient of heat exchange, $P(x)$ porosity of the canopy, $g(x)$ reaction term due to the crop metabolism,

$\epsilon$ emissivity of the leaf surface, $\sigma$ Stephan-Boltzmann constant, $K_a(x)$ air thermal conductivity, $K_c(x)$ crop thermal conductivity, $\rho$ air density, $v$ air speed, $C_a$ specific heat of the air under constant pressure.

$\alpha_v$ was defined as $\alpha_v(x) = \frac{2 \cdot \text{LAI} \cdot \rho \cdot C_s}{H \cdot r_a}$ and $r_a = 305(\frac{1}{V})^{0.5}$

Where H plant height, l the specific length of the plant, $r_a$ aerodynamic resistance, $C_s$ volumetric heat capacity and LAI Leaf Area Index.

## 3 Continuous Problem

### 3.1 Direct problem

The previous system can be rewritten in the more suitable following form:

$$ egin{cases} 
-\partial_x (K_c(x) \partial_x T_c) + aT_c + bT_a = g(x) + cT_1 & \text{in } \Omega \\ 
-\partial_x (K_a(x) \partial_x T_a) - d\partial_x T_a + bT_a - bT_c = 0 & \text{in } \Omega \\ 
T_c(0) = T_e; & T_a(0) = T_e \\ 
K_c(L) \partial_x T_c = 0; & K_a(L) \partial_x T_a(L) = T_a - T_s 
\end{cases} 
\quad (2) $$

where $a(x) = \frac{\alpha_v(x)}{p}$, $\sigma \epsilon T_{cmoy}$; $b(x) = \frac{\alpha_v(x)}{p}$; $c = \sigma \epsilon T_{cmoy}$ and $d(x) = \rho v C_a(x)$.

We assume that, $K_a$, $K_c$, a, b, c, d $\in L^\infty(\Omega)$; $g, T_1 \in L^2(\Omega)$; $q, T_e \in L^2(\Gamma_0 \in \partial\Omega)$; $T_s \in L^2(\Gamma_1 \in \partial\Omega)$; and suppose that $K_{c0}$, $K_{a0}$ exist such that

$$ K_c(x) \geq K_{c0} > 0; \quad K_a(x) \geq K_{a0} > 0 \quad (3) $$

$$ -\frac{1}{2} \partial_x d + b \geq 0 \quad (4) $$

We first recall the definition of space $H^2(\Omega)$ and Lax-Milgram Theorem.

$$ H^2(\Omega) = \{u \in L^2(\Omega), \partial_x, \partial_{xx} \in L^2(\Omega)\} $$

Theorem 1: Lax-Milgram [13]. "Let V be a real Hilbert space, and let $a_1(.,.)$ denotes a bilinear form. More over suppose that there exists positive constants $\alpha_0$ and $\beta_0$ such that the following conditions are satisfied for all $u, v \in V: |a_1(u, v)| \leq \alpha_0 ||u||_V ||v||_V;

$$ |a_1(u, u)| > \beta_0 ||u||_V^2 $$

Given a continuous linear form L on V, there exists a unique $u \in V$ such that $a_1(u, v) = L(v)$ for all $v \in V$.

Theorem 2: [3]. "If (3) and (4) hold, then (2) has a unique solution $(T_c, T_a) \in H^2(\Omega) \times H^2(\Omega)$.

### 3.2 Control problem

We first start by recalling the abstract setting of optimal control from the monographs [14] and [15]. These results will be applied later to our problem.

#### 3.2.1 Abstract setting. (a) Problem statement

Let V be a Hilbert space, Y a Banach space. J: V × H → R the objective function: convex and lower semicontinuous; A: V → Y a linear continuous operator. Consider the problem:

$$ (P) \quad \left\{ \begin{array}{ll} \min_{u \in V} & J(u, y) \\ & y = Au \\ u \in V_{ad}, y \in Y_{ad} \end{array} \right. \quad (5) $$
with $V_{ad} \subset V$: non-empty bounded set; $Y_{ad} \subset Y$: non-empty bounded set. Let $C = \{u \in U_{ad}, y \in Y_{ad}\}$, with $C \neq \emptyset$. One defines the reduced functional cost $F(u): V \to \mathbb{R}$. $F(u) = J(u, Au)$.

$$ (P) \Leftrightarrow \min_{u \in C} F(u) \qquad (6) $$

with the previous assumptions, C is convex, bounded and non-empty. F is convex, lower semicontinuous and proper.

Theorem 3: [14]. "If F is coercive on C, i.e. $F(v) \to +\infty$ when $v_V \to +\infty$ and $v \in C$, or if F is bounded, THEN (P) has at least one solution."

Theorem 4: [14]. If F is strictly convex then (P) has at most one solution.

### (b) Necessary optimality conditions

Let $V^*$ be the topological dual of V (i.e. the space of linear functional on V), $\langle \cdot, \cdot \rangle$, the bilinear canonical pairing over $V \times V^*$ and F.

#### Definition (subdifferentiability)

A function $F$ of $V \to \mathbb{R}$ is said to be subdifferentiable at the point $u \in V$ if it has a continuous affine minorant which is exact at $u$. The slope $u^* \in V^*$ of such a minorant is called a subgradient of $F$ at $u$, and the set of subgradients is called the subdifferential at $u$ and is denoted $\partial F(u)$.

One can obtain the characterization bellow:

$u^* \in \partial F(u)$ if and only if $F(u)$ is finite and $v-u, u^* \le F(v) \quad \forall v \in V$.

Remark: One can predict the role of subdifferentiation in optimization as a direct result of this definition:

$$ F(u) = \min_{v \in V} F(v) \text{ if and only if } 0 \in \partial F(u) $$

Proposition 1: [15]. "Let F be a convex function of $V \to \mathbb{R}$. If F is Frechet differentiable at $u \in V$, it is subdifferentiable at $u$ and $\partial F(u) = \{F'(u)\}$."

Proposition 2: [15]. If $F_1$ and $F_2 \in \Gamma(V)$, and if there is a point $\bar{u} \in \text{dom } F_1 \cap \text{dom } F_2$ where $F_1$ is continuous, one has: $\forall u \in F_1$, $\partial(F_1 + F_2)(u) = \partial F_1(u) + \partial F_2(u)$

Characterization of solutions:

Let $\tilde{F}(u) = F(u) + \chi_C(u)$ with $\chi_C$ the indicator function of C defined as

$$ \begin{cases} \chi_C(u) = 0 & \text{if } u \in C \\ \chi_C(u) = +\infty & \text{if } u \notin C \end{cases} \qquad (7) $$

$$ (P) \Leftrightarrow \min_{u \in V} \tilde{F}(u) \qquad (8) $$

Proposition 3: [15]. $\bar{u}$ is a solution of (P) iff $0 \in \partial \tilde{F}(\bar{u})$.

Theorem 5. $\bar{u}$ is a solution of (P) iff $\bar{u}$ solves the variational inequality or Euler inequality: $\langle \bar{u} - u, F'(u) \rangle_{V, V^*} \ge 0$.

Proof: The optimality condition $\Leftrightarrow 0 \in \partial(F + \chi_C)(\bar{u})$. Proposition 2 gives $0 \in \partial F(\bar{u}) + \partial \chi_C(\bar{u})$. By virtue of proposition 1 one has: $0 \in \partial F(\bar{u}) + \partial \chi_C(\bar{u}) \forall u \in V \Leftrightarrow -\tilde{F}'(\bar{u}) \in \partial \chi_C(\bar{u})$.

$$ \text{Either} \quad \begin{cases} \chi_C(u) < +\infty \\ \forall u \in V, \quad \chi_C(u) \ge \chi_C(\bar{u}) - (u - \bar{u}, F'(\bar{u}))_{V, V^*} \end{cases} \qquad (9) $$

$$ \Leftrightarrow \begin{cases} \bar{u} \in C \\ \forall u \in V, \quad \langle u - \bar{u}, -F'(\bar{u}) \rangle_{V, V^*} \le 0 \end{cases} \qquad (10) $$

$$ \Leftrightarrow \begin{cases} \bar{u} \in C \\ \forall u \in V, \quad \langle u - \bar{u}, F'(\bar{u}) \rangle_{V, V^*} \ge 0 \end{cases} \qquad (11) $$

For the Optimality system, by using the chain rule and the rules for total differential, one can determine the derivative of F' in (11), which gives:

$$ \begin{aligned} F'(u) h &= \partial_y J(A\bar{u}, \bar{u}) A^* h + \partial_u J(A\bar{u}, \bar{u}) \\ &= ((A^*)^{-1} \nabla_y J(\bar{y}, \bar{u}) + \nabla_u J(\bar{y}, \bar{u}), h) \end{aligned} \qquad (12) $$

The variational inequality is expressed as:

$$ \begin{cases} \bar{u} \in C \\ \forall u \in V, \quad \langle u - \bar{u}, (A^*)^{-1} \nabla_y J(\bar{y}, \bar{u}) + \nabla_u J(\bar{y}, \bar{u}) \rangle_{V, V^*} \ge 0 \end{cases} \qquad (13) $$

Defining a new variable $p = (A^*)^{-1} \nabla_y J(\bar{y}, \bar{u})$ called adjoint state, the necessary optimality condition reads:

$$ \begin{cases} y = Au \\ A^*p = \nabla_y J(y, u) \\ \langle p + \nabla_u J(y, u), u - \bar{u} \rangle_{V^*, V} \ge 0, \quad \forall u \in V, \end{cases} \qquad (14) $$

### 3.2.2 Canopy air temperature control.
As mentioned earlier, providing the appropriate microclimate for the crops' needs puts them in convenient growing conditions. Our control problem consists in maintaining the canopy temperature close to a desired profile $T_{aexp}$ by acting on the inlet air temperature $T_e$ and on the lighting temperature $T_l$.

Let define the following spaces: $V = L^2(\Gamma) \times L^2(\Omega)$; $Y = H^1(\Omega) \times H^1(\Omega)$; $V_{ad} = \{(T_e, T_l) \in V, T_e \ge 0, T_l \ge 0\}$; $Y_{ad} = Y$.

Considering the cost functional

$$ J_1(T_e, T_l) = \frac{1}{2} \int_{\Omega} (T_a - T_{aexp})^2 dx + \frac{\lambda_1}{2} \int_{\Omega} T_l^2 dx + \frac{\lambda_2}{2} T_e^2 \Big|_{\Gamma_0} \quad \lambda_i > 0 \qquad (15) $$

The boundary control and the distributed control problem can be expressed as:

$$ \min_{(T_e, T_l) \in V_{ad}} J_1(T_e, T_l) = \frac{1}{2} \int_{\Omega} (T_a - T_{aexp})^2 dx + \frac{\lambda_1}{2} \int_{\Omega} T_l^2 dx + \frac{\lambda_2}{2} T_e^2 \Big|_{\Gamma_0} \qquad (16) $$

Proposition 4. (16) has a unique solution $(T_e, T_l)$.

Proof: $J_1$ is strikely convex. By virtue of theorems 2, 3 and 4, existence and uniqueness are deduced.

The optimality system needs to define the adjoint system which can be obtained more easily from Lagrange technic. Considering the adjoint states $(w, z)$ the following adjoint states equations are derived from the stationarity of the Lagrangian with respect to the states.

$$ \begin{cases} K_c \partial_{xx} w - aw - bz = 0 & \text{in } \Omega \\ K_a \partial_{xx} z + d \partial_x z - bw - bz = T_a - T_{aexp} & \text{in } \Omega \\ w(0) = 0; \quad z(0) = 0 & \\ K_c(L) \partial_x w(L) = 0; \quad K_a \partial_x z(L) + dz(L) = 0 & \end{cases} \qquad (17) $$

The gradient of the cost functional with respect to the controls can be evaluated once the adjoint problem is solved. The equivalence between the constrained minimization of $J_1(T_e, T_l)$ and the stationarity of the Lagrangian gives:

$$ \partial_{Tl} J_1 = \lambda_2 T_e \Big|_{\Gamma_0} \qquad (18) $$

The control computation algorithm uses the conjugate gradient method, in which the gradient of the cost function is calculated thanks to the solution of the adjoint problem.
<figure><img src="image_4.png" /><figcaption>Figure 2: simulation results, control problem. canopy temperature target (solid line), canopy temperature measured (dotted line).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3: simulation results, perturbed control problem. canopy temperature target (solid line), canopy temperature measured (dotted line).</figcaption></figure>

# 4 Simulation Results

For the purpose of a numerical solution of (1) and the adjoint problem (17) we used the finite difference framework with the conductivities upwinding decentred and the forward difference quotients to approximate the derivatives. Matlab software was used to carry out simulations in order to assess the control's capacity to maintain canopy air temperature closed to the desired profile. We used some of the impatiens x novae-guinea crop data from the literature, including the porosity, specific length, and aerodynamic resistance. However, the thermal conductivity that was taken into account was comparable to that of plants that were much denser or drier. A greenhouse containing two plant species was considered and simulations corresponding to nocturnal and diurnal behaviour were carried out.

Simulation results of the direct problem can be found in [3].

Figure 2 shows the simulation results of canopy air temperature $T_a$ control in the greenhouse canopy. $T_{aexp}$ is the synthesized setpoint profile. In Figure 2a, starting from $T_e = 31$ °C and constant distribution of $T_l$, the controls make $T_a$ to converge to a profile similar to that of $T_{aexp}$, with an average error of less than 0.6°C. In Figure 2b, the initial value of $T_e = 11$ °C. The profile of $T_a$ has the same shape as that of the setpoint, with an average error of less than 0.3 °C.

In order to test the robustness of the control, we considered a perturbed set point profile by adding a standard normal distribution

noise of standard deviation 0.25 °C. Figure 3a shows the profile obtained with $T_e = 31$ °C and constant distribution of $T_l$, and figure 3b the profile obtained with a value $T_e = 11$ °C. As before, the canopy air temperature converges in the same zone as that the initial profile. As the temperature profiles obtained are smooth, the controls appear to be insensitive to variations due to noise in the setpoint profile. In both cases, the performance obtained is slightly downgraded than that of smooth setpoint profile of figure 2.

# 5 Conclusion

This work presents the use of optimal control to control air temperature in the canopy of a greenhouse. Using subdifferential notions, we have verified the existence and uniqueness conditions of boundary control and distributed control. The conjugate gradient method, where the gradient is deduced from the adjoint state, was used. For the cases treated, the solution lies in the same zone as that of the initial conditions, with improved accuracy when starting from a lower profile. Simulation for non-smooth setpoint profile gives results slightly less accurate than that of smooth setpoint profile. Further work will focus on the unsteady case.

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