CEA

    controlled environment agriculture
GH

    greenhouse
OCP

    optimal control problem
PPFD

    photosynthetic photon flux density
OC

    optimal control

# Comparison of Dynamic Tomato Growth Models for Optimal Control in
Greenhouses

Michael Fink∗ Chair of Automatic Control Engineering
Technical University of Munich, Germany
<EMAIL_ADDRESS>Annalena Daniels∗ Chair of Automatic Control Engineering
Technical University of Munich, Germany
<EMAIL_ADDRESS>Cheng Qian  Chair of Automatic Control Engineering
Technical University of Munich, Germany
<EMAIL_ADDRESS>Víctor Martínez Velásquez  Chair of Automatic Control
Engineering
Technical University of Munich, Germany
<EMAIL_ADDRESS>Sahil Salotra  Chair of Automatic Control Engineering
Technical University of Munich, Germany
<EMAIL_ADDRESS>Dirk Wollherr  Chair of Automatic Control Engineering
Technical University of Munich, Germany
<EMAIL_ADDRESS>
###### Abstract

As global demand for efficiency in agriculture rises, there is a growing
interest in high-precision farming practices. Particularly greenhouses play a
critical role in ensuring a year-round supply of fresh produce. In order to
maximize efficiency and productivity while minimizing resource use,
mathematical techniques such as optimal control have been employed. However,
selecting appropriate models for optimal control requires domain expertise.
This study aims to compare three established tomato models for their
suitability in an optimal control framework. Results show that all three
models have similar yield predictions and accuracy, but only two models are
currently applicable for optimal control due to implementation limitations.
The two remaining models each have advantages in terms of economic yield and
computation times, but the differences in optimal control strategies suggest
that they require more accurate parameter identification and calibration
tailored to greenhouses.

###### Index Terms:

optimal control, greenhouse, tomato, vertical farm

**footnotetext: These authors contributed equally to this work.
© 2023 IEEE. This work was accepted for IEEE AGRETA2023. Personal use of this
material is permitted. Permission from IEEE must be obtained for all other
uses, in any current or future media, including reprinting/republishing this
material for advertising or promotional purposes, creating new collective
works, for resale or redistribution to servers or lists, or reuse of any
copyrighted component of this work in other works.

##  I Introduction

With a growing world population, the agricultural sector is challenged to
increase food production while minimizing the negative impact on the
environment and preserving biodiversity [1]. In addition, the availability of
agricultural land is decreasing due to factors including climate change and
geopolitical conflicts, highlighting the need for higher production density.
To achieve higher yields and quality while reducing cost and environmental
impact, there is a growing trend towards high-precision controlled environment
agriculture [2], of which greenhouses are a crucial element. GHs are partially
enclosed systems that regulate environmental variables like temperature,
humidity and CO2 concentration.

Despite their benefits for providing food all-year-round, GHs still face
challenges due to their high energy and cost requirements, making them less
profitable than arable farming during cropping seasons. Nonetheless, they
offer significant advantages for crops like tomatoes, which are difficult to
transport and store for extended periods. Maximizing the potential yield of
these crops while minimizing the economic and ecological costs can unlock the
full potential of GH tomato production.

The optimization of GH management for enhanced crop productivity and reduced
resource use and environmental impact requires the use of optimal control
(OC). OC is a mathematical technique that aims at finding the best control
strategy for a given system. It involves determining the optimal values of
control variables over a specified time horizon that maximize or minimize a
performance measure, subject to constraints. In the context of GH management,
OC can be used to optimize environmental factors such as temperature,
humidity, and lighting to enhance crop productivity while minimizing resource
use and environmental impact [3, 4, 5].

OC requires accurate models of the GH building environment and the crop
growth. Crop models can offer insights into how various factors influence crop
growth and yield. Existing OC approaches [6, 7] use models customized for
specific crops and problems, which necessitates extensive experimentation for
each adaptation and limits their general applicability. In contrast, numerous
models have been developed for the general description of plants in
agriculture and biology. Among them are several tomato models developed in the
last 30 years [8, 9, 3, 10, 11, 12, 13], indicating the continuous interest of
research and industry in more accurate models for various applications.
However, control engineering requires a mathematically analytical description
of models and most of these models were not designed for that use-case and are
not given as a set of equations but as a defined combination of look-up
tables. Hence, they are often implemented as a pure simulation black box model
with no interfaces to the inside of the models, and the models themselves
frequently include discontinuities, and only guarantee good performance within
unknown boundaries. As a result, selecting a model that is accurate,
computationally efficient, and appropriate for the use in OC remains a
challenge.

Previous studies have used different approaches to select or synthesize crop
models for control engineering, including a model comparison [14] of the
TomSim [9] and reduced TOMGRO model [15] and combining components from
existing models [16, 17, 18]. While previous studies have used a common model
structure to combine model components into new models [18], we do not use this
approach since the individual components lack modularity and explainability.
Instead we compare existing models regarding their performance and
applicability in OC. To the best of our knowledge, there is a lack of
comparative studies on tomato crop models that include performance metrics
regarding OC results, computational time, and applicability.

In this paper, we compare three established crop models. The SIMPLE crop model
by Zhao et al. [19] is a simple but generic crop model that can be used for
various crops. The reduced TOMGRO model by Jones et al. [15] is designed for
tomatoes only, and agricultural scientists and farmers commonly use the tomato
model of the DSSAT toolbox [20]. The comparison evaluated the model’s
performance when applied to OC approaches in a GH environment. We addressed
the remaining challenges that may impact the implementation of these models in
real-world agricultural settings.

Our key contributions can be summarized as follows:

  * •

Comparison of the accuracy of SIMPLE, reduced TOMGRO, and DSSAT, using a GH
data set from the Autonomous GH Challenge in the Netherlands [21, 22].

  * •

Comparison of the structure and applicability of these models in OC and re-
formulation of two models as state-space models to be used in control theory.
Combination of the models with a GH environment model.

  * •

Incorporation of the GH-crop-model into an OC approach, and analysis of the
results for their performance and meaning with recommendations for future
users.

The paper is organized as follows. Sec. II provides an overview of the tomato
models and the GH environment. In Sec. III, we introduce the OC approach. Sec.
IV discusses the validation of the models and presents the comparison results.
The ensuing Sec. V delves into the implications of our findings and explores
the limitations of this study and future research directions. Finally, in Sec.
VI, we conclude the paper.

##  II Models for the Growth of Tomatoes

Before a comparison can be conducted, an introduction to the three crop models
and the GH model is given.

###  II-A SIMPLE Model

The parameters of the SIMPLE model [19] have been calibrated carefully using a
large arable farming experimental data set for various crops. The model
represents up to 14 crops, including tomatoes, and 22 cultivars with the
modification of just 13 crop parameters, of which four are cultivar-specific
and nine are species-specific. Although the model takes into account various
factors like plant phenology, the impact of photosynthesis on growth, the
influence of CO2 concentration, drought stress, and radiation interception, it
has some limitations, including the exclusion of vernalization effects and the
lack of nutrient dynamics. The state space-model for day $i$ can be described
as

${\boldsymbol{x}}_{\text{s},i+1}={\boldsymbol{f}}_{\text{s}}({\boldsymbol{x}}_{\text{s},i},{\boldsymbol{u}}_{\text{s},i})$
(1)

with the state of the SIMPLE model

${\boldsymbol{x}}_{\text{s},i}=\begin{bmatrix}m_{B,i}&\tau_{i}&I_{50B,i}\end{bmatrix}^{\mathsf{T}}$
(2)

where $m_{B,i}$ is the tomato biomass in
$\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}$, $\tau_{i}$ is the cumulative
temperature in $\frac{\mathrm{\SIUnitSymbolCelsius}\text{\,}\mathrm{d}}{}$
(temperature integrated over days) and $I_{50B,i}$ is a value for the leaf
senescence on day $i$ in
$\frac{\mathrm{\SIUnitSymbolCelsius}\text{\,}\mathrm{d}}{}$ [23]. The yield
for the tomatoes is obtained with $m_{\text{fruit}}=H\\!I\ m_{B,N}$, where
$H\\!I$ is the harvest index ($H\\!I=0.68$ for tomatoes) and $m_{B,N}$ is the
biomass of the plant on the last day.

Following the method employed in [23], we assume that the temperature remains
constant throughout the day. However, we additionally treat the CO2
concentration as a controllable variable. The input vector for this model for
day $i$ is

${\boldsymbol{u}}_{\text{s},i}=\begin{bmatrix}T_{i}&D_{i}&R_{i}&C_{CO2,i}\end{bmatrix}^{\mathsf{T}}$
(3)

where $T_{i}$ is the mean temperature in ℃, $D_{i}$ is the the relative level
of drought between $[0,1]$, $R_{i}$ is the solar radiation in
$\frac{\mathrm{MJ}}{{\mathrm{m}}^{2}\text{\,}\mathrm{d}}$ and $C_{CO2,i}$ is
the CO2 concentration in $\mathrm{p}\mathrm{p}\mathrm{m}$.

###  II-B Reduced TOMGRO Model

The reduced TOMGRO model [15] is a simplified version of the original, widely
used TOMGRO model [8]. Compared to the original model, where the state
dimension is 69, the reduced TOMGRO model contains a state with only a
dimension of 5 and an input dimension of 3. This model was developed under GH
conditions.

Similar to (1), we propose the reduced TOMGRO model describing the state of
the next day $i+1$ as

${\boldsymbol{x}}_{\text{t},i+1}={\boldsymbol{f}}_{\text{t}}({\boldsymbol{x}}_{\text{t},i},{\boldsymbol{u}}_{\text{t},i})\,.$
(4)

The state vector of this model for day $i$ is given as

${\boldsymbol{x}}_{\text{t},i}=\begin{bmatrix}N_{i}&L\\!A\\!I_{i}&W_{i}&W_{f,i}&W_{m,i}\end{bmatrix}^{\mathsf{T}},$
(5)

where $N_{i}$ is the number of main stem nodes, $L\\!A\\!I_{i}$ is the leaf
area index in $\frac{{\mathrm{m}}^{2}}{{\mathrm{m}}^{2}}$ (ratio of leaf area
per ground area), $W_{i}$ is the total plant weight in
$\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}$, $W_{f,i}$ is the fruit dry weight in
$\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}$, and $W_{m,i}$ is the mature fruit dry
weight in $\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}$.

The input vector of this model for day $i$ is given as

${\boldsymbol{u}}_{\text{t},i}=\begin{bmatrix}T_{i}&T_{d,i}&R_{i}&C_{CO_{2},i}\,\end{bmatrix}^{\mathsf{T}},$
(6)

where $T_{i}$ is the average temperature of the whole day in ℃, $T_{d,i}$ is
the average temperature of daytime in ℃, $R_{i}$ is the solar radiation on day
$i$ in $\frac{\mathrm{MJ}}{{\mathrm{m}}^{2}\text{\,}\mathrm{d}}$,
$C_{CO_{2},i}$ is the concentration of CO2 in ppm. From $R_{i}$, we obtain the
photosynthetic photon flux density
$P\\!P\\!F\\!D_{i}=R_{i}/$0.037\text{\,}\frac{\mathrm{MJ}\text{\,}\mathrm{s}}{\mathrm{\SIUnitSymbolMicro
mol}\text{\,}\mathrm{d}}$$ for white light.

As the original model description contains piece-wise defined functions and is
therefore not always differentiable, we use the smoothing function for maximum
operators (cf. [23]), as well as a smoothing function for Heaviside step
function $H(x)\approx H_{\epsilon}(x)=\displaystyle{\frac{1}{1+e^{-\epsilon
x}}}$ to allow for gradient-based optimization approaches. For a big positive
$\epsilon$, the smoothing function converges to the standard Heaviside step
function.

###  II-C DSSAT

The DSSAT toolbox [20] contains widely used and well-established models in
arable agriculture and has been used for decision support processes for a long
time. For tomatoes it uses the CROPGRO model [13]. However, its implementation
is only available as a complete Windows application. It requires manual file
uploads, making it unsuitable for integration into OC processes. Therefore, we
will use this model only as a well-established comparison.

###  II-D Greenhouse Model

The GH model described in [3] is employed for both the SIMPLE crop model [19]
and the reduced TOMGRO model [15]. The GH model comprises the primary GH
compartment, which is linked to three subsystems. These subsystems serve as
the three control inputs for the GH, which are the heat supply through a
heating pipe system, the supply of CO2, and the window ventilation system for
air flow. For a more comprehensive explanation of the GH structure, see [3].

The state-space equation for the GH model describes the state of the GH on an
hourly basis and is given as

${\boldsymbol{x}}_{\text{gh},i+1}={\boldsymbol{f}}_{\text{gh}}({\boldsymbol{x}}_{\text{gh},i},{\boldsymbol{u}}_{\text{gh},i})\,.$
(7)

The state vector of the GH model is

${\boldsymbol{x}}_{\text{gh},i}=\begin{bmatrix}T_{g,i}&T_{s,i}&T_{p,i}&C_{CO_{2},i}&C_{H_{2}O,i}\end{bmatrix}^{\mathsf{T}}\,,$
(8)

which are the greenhouse temperature $T_{g,i}$ in
$\frac{\mathrm{\SIUnitSymbolCelsius}}{}$, soil temperature $T_{s,i}$ in
$\frac{\mathrm{\SIUnitSymbolCelsius}}{}$, pipe temperature $T_{p,i}$ in
$\frac{\mathrm{\SIUnitSymbolCelsius}}{}$, CO2 concentration $C_{CO_{2},i}$ in
$\mathrm{p}\mathrm{p}\mathrm{m}$, and the water vapor concentration
$C_{H_{2}O,i}$ in $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ for the
$i^{\text{th}}$ hour. The three control inputs of the GH

${\boldsymbol{u}}_{\text{gh},i}=\begin{bmatrix}u^{vp}_{q,i}&u^{Ap}_{v,i}&u^{vp}_{CO_{2},i}\end{bmatrix}^{\mathsf{T}}$
(9)

are directly connected to the main GH compartment. The underlying systems of
the control inputs are simplified. The heat and CO2 supply are expressed as a
valve position $u^{vp}_{q,i}$ and $u^{vp}_{CO_{2},i}$ which quantifies the
supply, respectively. The ventilation system is expressed by the aperture of
the windows from the windward and lee side of the greenhouse compartment. Both
windows are considered for simplicity through one control input
$u^{Ap}_{v,i}$.

Note that the the valve positions $u^{vp}_{q,i}$ and $u^{vp}_{CO_{2},i}$ are
between $[0,1]$, where 0 indicates that there is no supply and the valve is
closed. The control input $u^{Ap}_{v,i}$ lies in the interval $[0,2]$ since we
consider both the windward and lee side of the GH compartment. Furthermore,
the GH compartment is subject to environmental influences as it is a partially
open system. Thus, the overall dynamics of the GH model also include six
parameters, which are summarized in the external disturbance vector

$d_{g,i}=\begin{bmatrix}R_{\text{out},i}\ \,T_{\text{out},i}\ \,v_{i}\
\,T_{\text{soil},i}\ \,C_{{H_{2}O\,\text{out},i}}\
\,C_{CO_{2}\,\text{out},i}\end{bmatrix}^{\mathsf{T}}.$ (10)

with the solar radiation $R_{\text{out},i}$ in
$\frac{\mathrm{W}}{{\mathrm{m}}^{2}}$, the temperature outside the greenhouse
$T_{\text{out},i}$ in $\frac{\mathrm{\SIUnitSymbolCelsius}}{}$, wind speed
$v_{i}$ in $\frac{\mathrm{m}}{\mathrm{s}}$, temperature of the subsoil
$T_{\text{soil},i}$ in $\frac{\mathrm{\SIUnitSymbolCelsius}}{}$, the humidity
$C_{{H_{2}O\,\text{out},i}}$ in $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ and
concentration of CO2 $C_{CO_{2}\,\text{out},i}$ in
$\mathrm{p}\mathrm{p}\mathrm{m}$ outside the GH. Just as with the crop models,
smoothing functions are necessary to handle discontinuities. For example, an
approximation of the absolute value is ${|x|\approx\sqrt{x^{2}+\mu}}$ where
$\mu$ is a small positive constant [24]. The derivative of the approximation
exists for all $x\in\mathbb{R}$.

###  II-E Integration of the models

In a comparison of the OC results of the two harvesting models, the respective
state-space model is used in combination with the GH model. Since the GH model
operates on an hourly basis while the crop models are evaluated once a day,
evaluating the combined model is not a simple task. Consequently, the crop
model receives the mean value of the GH conditions as input, and the combined
state vector comprises the state of the tomato model along with the hourly
state of the greenhouse, i.e.,

$\displaystyle{\boldsymbol{x}}_{i}=\begin{bmatrix}{\boldsymbol{x}}_{\text{s/t},i}^{\top}&{\boldsymbol{x}}_{\text{gh},24i}^{\top}&{\boldsymbol{x}}_{\text{gh},24i+1}^{\top}&..&{\boldsymbol{x}}_{\text{gh},24i+23}^{\top}\end{bmatrix}^{\top},$
(11)

where ${\boldsymbol{x}}_{\text{s/t},i}$ is either the state resulting from the
SIMPLE model ${\boldsymbol{x}}_{\text{s},i}$ or the state based on the TOMGRO
model ${\boldsymbol{x}}_{\text{t},i}$.

In order to account for and optimize the performance of the entire system
comprising the tomato crop models within a greenhouse setting, we amalgamate
them into a single model

$\displaystyle{\boldsymbol{x}}_{i+1}={\boldsymbol{f}}({\boldsymbol{x}}_{i},{\boldsymbol{u}}_{\text{gh},i})\,,$
(12)

where the components of ${\boldsymbol{x}}_{i+1}$ are computed by

$\displaystyle{\boldsymbol{x}}_{\text{s/t},i+1}$
$\displaystyle={\boldsymbol{f}}_{\text{s/t}}\left({\boldsymbol{x}}_{\text{s/t},i},{\boldsymbol{g}}_{\text{s/t}}\left({\textstyle\frac{1}{24}\sum_{j=0}^{23}{\boldsymbol{x}}_{\text{gh},i+j}}\right)\right)$
(13) $\displaystyle{\boldsymbol{x}}_{\text{gh},i+j+1}$
$\displaystyle={\boldsymbol{f}}_{\text{gh}}({\boldsymbol{x}}_{\text{gh},i+j},{\boldsymbol{u}}_{\text{gh},i})\quad\forall
j\in[0,23].$ (14)

and arranged in the same form as in (11). The function
${\boldsymbol{g}}_{\text{s/t}}\left(\cdot\right)$ maps the mean states of the
GH to the inputs for either the SIMPLE or the TOMGRO model. In the following
section, the combined model (12) of tomato growth and GH is used in an optimal
control problem (OCP).

##  III Optimal Control for Plant Growth

In the following, the framework for the OC of plant growth is presented. The
goal is to find a sequence of input values for the greenhouse (9) resulting in
a maximum economic yield. The inputs of the greenhouse over the optimization
period $N$ is combined to the input sequence vector
${{\boldsymbol{U}}=\left[{\boldsymbol{u}}_{\text{gh},0}^{\top},{\boldsymbol{u}}_{\text{gh},1}^{\top},...,{\boldsymbol{u}}_{\text{gh},N-1}^{\top}\right]^{\top}}$.

The optimization selects one from all possible input sequences
${\boldsymbol{U}}$ that minimizes the cost function

$\displaystyle
J({\boldsymbol{U}},{\boldsymbol{x}}_{N})=\sum_{i=0}^{N-1}l({\boldsymbol{u}}_{\text{gh},i})-V({\boldsymbol{x}}_{N}),$
(15)

while the input and state sequences are a solution to the system model (12)
that consists of the greenhouse (7) and the tomato model, either the SIMPLE
model (1) or the TOMGRO model (4). It is evaluated over a given growth period
of $N$ days. The cost function (15) represents the negative economic yield,
thus the terminal cost is used with a negative sign, similar to [23]. The
stage cost $l({\boldsymbol{u}}_{\text{gh},i})$ represents the energy cost used
for the growing process and is given as
$l({\boldsymbol{u}}_{\text{gh},i})={\boldsymbol{r}}^{\mathsf{T}}{\boldsymbol{u}}_{\text{gh},i}$,
where ${\boldsymbol{r}}\in\mathbb{R}^{3}$ is a weight vector. The terminal
cost $V({\boldsymbol{x}}_{N})$ gives the yield at the harvest on day $N$. The
price evolves linearly, leading to a linear cost term, i.e.
$V({\boldsymbol{x}}_{N})={\boldsymbol{q}}^{\mathsf{T}}{\boldsymbol{x}}_{N}$,
with the weight ${\boldsymbol{q}}\in\mathbb{R}^{3}$. The temperature,
ventilation and CO2 can be controlled in a GH within lower and upper bounds.
Therefore, we add constraints ${\boldsymbol{u}}_{\text{gh},i}\in\mathcal{U}$
to the OCP, yielding

$\displaystyle{\boldsymbol{U}}^{*}=$
$\displaystyle\operatorname*{\arg\\!\min}_{{\boldsymbol{U}}}J({\boldsymbol{U}},{\boldsymbol{x}}_{N})$
(16a) s.t.
$\displaystyle{\boldsymbol{x}}_{i+1}={\boldsymbol{f}}({\boldsymbol{x}}_{k},{\boldsymbol{u}}_{\text{gh},i})$
$\displaystyle\forall i\in[0,N-1]$ (16b)
$\displaystyle{\boldsymbol{u}}_{i}\in\mathcal{U}\quad$ $\displaystyle\forall
i\in[0,N-1]$ (16c)
$\displaystyle{\boldsymbol{x}}_{0}={\boldsymbol{x}}_{\text{init}},$ (16d)

where ${\boldsymbol{U}}^{*}$ is the sequence of optimal inputs of the GH. The
states on day $i=0$ are given as initial states
${\boldsymbol{x}}_{\text{init}}$. The optimal input sequence
${\boldsymbol{U}}^{*}$ results in a state sequence that yields the smallest
value of the cost function (15). The OCP is implemented in Python and solved
with CasADi [25].

##  IV Comparison of the crop models

After the defintion of the models, we now compare these models regarding their
structure and their validity compared to the DSSAT model and a GH data set.
Additionally, we compare the findings of the OC study.

###  IV-A Structure

As already described in Sec. II-E, the two tomato models follow a similar
state-space representation, which are integrated with the GH. Furthermore,
their inputs, denoted as ${\boldsymbol{u}}_{\text{s},i}$ and
${\boldsymbol{u}}_{\text{t},i}$, have a similar structure and contain
variables such as daily average temperature, solar radiation, and CO2.
However, there are some differences between the two models in terms of inputs.
For instance, the SIMPLE model includes an input for a drought factor $D$,
which is set to 0 in a GH environment under the assumption that we guarantee
sufficient water supply for each day. Another key difference is that TOMGRO
follows a more complex model approach, e.g., it has a state with five elements
(5), whereas the state of the SIMPLE model only considers three elements (1).
While the TOMGRO model state is readily accessed and measurable through simple
techniques such as counting stem nodes, assessing leaf area, and weighing
different parts of the plant, the state of the SIMPLE model is not as easily
observable. It requires expert knowledge along with analytical methods.
Nonetheless, both models share a common state, the dry weight of the fruit
($m_{\text{fruit}}$ for the SIMPLE model and $W_{f}$ for the TOMGRO model),
which is of interest in this study as it helps in obtaining the optimal yield.
The results are given in fresh weight to compare it with experimental data.

###  IV-B Validation of Models

For the validation of both models, we use as real-world ground truth a data
set consisting of GH climate data compiled from different experiments [22].
This data set, which we compare to both models, is generated from the
experiments of five different multi-disciplinary teams. As an example and for
brevity, we only show the comparison for one set (The Automators team) for our
validation. We use the environmental conditions of the experiment, including
temperature, solar radiation, and CO2 concentration, for each day.

In Fig. 1, we observe that both models present a similar behavior for around
$160$ days of cultivation in comparison to the experimental data. However, the
TOMGRO model fits the data better and yields about
$2\text{\,}\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}$ more biomass than the
experimental data, while the SIMPLE model results in an underestimate of
$4\text{\,}\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}$, under the same conditions.

$0$$25$$50$$75$$100$$125$$150$$0$$4$$8$$12$$16$Time since planting in
d$\displaystyle m_{\mathbf{fruit}},W_{\mathbf{f}}$ in
$\displaystyle\mathrm{\frac{kg}{m^{2}}}$ExperimentSIMPLETOMGRO Figure 1:
Comparison of the tomato fresh weight computed with the SIMPLE and TOMGRO
model with experimental data.

Furthermore, we verify the accuracy of both models by subjecting them to
validation in the DSSAT environment [20], which is widely used for crop growth
modeling. For simulation of tomato growth and weather data, we utilize the
DSSAT model. Fig. 2 illustrates that the SIMPLE and TOMGRO models produce
nearly identical trajectories, whereas the DSSAT simulation behaves
differently starting from day 60. However, all three models generate a similar
yield of approximately $13\text{\,}\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}$.

$0$$20$$40$$60$$80$$100$$120$$0$$3$$6$$9$$12$Time since planting
[d]$\displaystyle m_{\mathbf{fruit}},W_{\mathbf{f}}$ in
$\displaystyle{\mathrm{\frac{kg}{m^{2}}}}$DSSATSIMPLETOMGRO Figure 2:
Comparison of the SIMPLE and TOMGRO with the DSSAT model.

Hence, the models demonstrate their ability to simulate real-life scenarios.
In the following, the suitability of the models for use in OC is investigated
to find optimal inputs for the greenhouse.

###  IV-C Comparing of Optimal Control Results

In this section, we present the results of applying the OC framework, as
described in Section III, to the two crop models. The cost function (15)
weights are chosen such that the cost represents the negative value of the
economic yield, given that the fresh weight of the tomato is sold for
$2\text{\,}\frac{\mathrm{\mbox{€}}}{\mathrm{kg}}$. The cost for CO2 is assumed
to be $0.15\text{\,}\frac{\mathrm{\mbox{€}}}{\mathrm{kg}}$, while ventilation
is considered to be neutral in terms of cost. Heating is assumed to cost
$0.02\text{\,}\frac{\mathrm{\mbox{€}}}{\mathrm{kW}\text{\,}\mathrm{h}}$. The
results include the optimal values of the crop model states, greenhouse
states, and control inputs. Fig. 3 and Fig. 4 show the optimal biomass
evolution of the SIMPLE and TOMGRO model, respectively. Their optimal results
can be found in Tab. I.

$0$$20$$40$$60$$80$$100$$0$$6$$12$$18$Time since planting in
d$\displaystyle{\mathrm{m_{\mathrm{B}}}}$ in
$\displaystyle{\mathrm{\frac{kg}{m^{2}}}}$$m_{\mathrm{B}}$$0$$20$$40$$60$$80$$100$$0$$800$$1{,}600$$2{,}400$$\displaystyle{\mathrm{\tau}},{\mathrm{I_{50B}}}$
in ∘Cd${\mathrm{\tau}}$${\mathrm{I_{50B}}}$ Figure 3: Optimized states of the
SIMPLE model. $0$$25$$50$$\displaystyle
N$$N$$0$$20$$40$$60$$80$$100$$0$$10$$20$Time since planting in d$\displaystyle
W_{f},W$ in
$\displaystyle\mathrm{\frac{kg}{m^{2}}}$$W_{f}$$W$$0$$20$$40$$60$$80$$100$$0$$2.5$$5$$\displaystyle
LAI$$LAI$$0$$20$$40$$60$$80$$100$$0$$1.5$$3$$\displaystyle W_{m}$ in
$\displaystyle\mathrm{\frac{kg}{m^{2}}}$$W_{m}$ Figure 4: Optimized states of
the TOMGRO model.

Fig. 5 displays the OC inputs of the greenhouse. The optimal value for
$u^{Ap}_{v,i}$ is 0 throughout for both crop models since ventilation leads to
a decrease in the CO2 concentration.

$0$$20$$40$$60$$80$$100$$0$$0.2$$0.4$$0.6$Time since planting in
d$v^{\mathrm{vp}}_{CO_{2}}$$v^{\mathrm{vp}}_{q}$ Figure 5: Optimal inputs of
the GH with the use of different tomato models in the optimization: Solid
lines: SIMPLE model; Dashed lines: TOMGRO model.

Therefore, for simplicity, ventilation is not additionally marked in the
graph. We conclude that in the optimal solution, the greenhouse ventilation is
not used, indicating that the optimization aims to maintain a stable
environment throughout the cultivation period. The optimal values of
$u^{vp}_{q}$ and $u^{vp}_{CO_{2}}$ of the greenhouse show similar results.

The optimal states of the greenhouse are shown in Fig. 6. We obtain similar
results for both crop models, particularly for the $T_{s}$, $T_{p}$, and
$C_{H_{2}O}$. The optimized temperature $T_{g}$ for the SIMPLE model is around
$27\text{\,}\frac{\mathrm{\SIUnitSymbolCelsius}}{}$ during the cultivation
period, which is close to the optimal temperature for this tomato variety
($26\text{\,}\frac{\mathrm{\SIUnitSymbolCelsius}}{}$) according to [19]. The
optimal solution of $C_{CO_{2}}$ concentration differs between the models,
with the concentration resulting from the SIMPLE model saturating at
$700\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$, since further increase makes no
difference in the simplified model.

$0$$0.25$$0.5$$0.75$$1$$\displaystyle C_{co2}$ in
$\displaystyle{\mathrm{\frac{kg}{m^{3}}}}$$C_{co2}$$0$$20$$40$$60$$80$$100$$20$$30$$\displaystyle
T_{\mathrm{g}},T_{\mathrm{s}}$ in
${}^{\circ}C$$T_{\mathrm{g}}$$T_{\mathrm{s}}$$0$$20$$40$$60$$80$$100$$100$$200$$300$Time
since planting in d$\displaystyle T_{\mathrm{p}}$ in
${}^{\circ}C$$T_{\mathrm{p}}$$0$$20$$40$$60$$80$$100$$0$$5$$10$$15$$20$$\displaystyle
C_{h2o}$ in $\displaystyle{\mathrm{\frac{g}{m^{3}}}}$$C_{h20}$ Figure 6:
Optimal sequence of states of the GH: Solid lines: SIMPLE model; Dashed lines:
TOMGRO model. $0$$20$$40$$60$$80$$100$$0$$5$$10$$15$$20$Time since planting
[d]$\displaystyle{\mathrm{m_{\mathrm{B}}}}\
\left[{\mathrm{\frac{kg}{m^{2}}}}\right]$OT-SIMPLEOT-TOMGROOT-DSSATOS-
SIMPLEOS-TOMGROOS-DSSAT Figure 7:  Comparison of the optimized inputs applied
on each model. OT: Inputs generated with OC based on TOMGRO, OS: Inputs
generated with OC based on SIMPLE. I.e., OT-SIMPLE is the biomass predicted by
the SIMPLE given the optimized inputs generated by the TOMGRO.

##  V Discussion

Overall, all three models achieve similar yields under standard weather and GH
conditions, even though only one of the models, the reduced TOMGRO, has
initially been designed for GH applications. The existing DSSAT implementation
rendered it inaccessible for immediate use in the current implementation of
our OC approaches. The remaining two models have different structures and
states, with TOMGRO being more intuitive to measure. All models share similar
inputs, but DSSAT allows for more degrees of freedom, while SIMPLE allows for
the least.

Based on the optimization results, the crop models have different
sensitivities to environmental variables. We applied the OC trajectories of
the GH to both models and found that while the resulting biomass depends on
the chosen trajectory, the differences between the models are not substantial,
even when the GH is optimized with the other model, shown in Fig. 7. Comparing
to DSSAT, its prediction lies in between the other two models, leaving
uncertainty as to which prediction is closer to reality. These findings
emphasize the importance of thorough model calibration, as accurate validation
does not necessarily lead to similar optimal GH environment trajectories for
different crop models.

TABLE I: Overall Comparison of the three tomato models.

|  | Developed for
---
GHs

| Valid under
---
GH conditions

| Usable for
---
OC

| Optimal
---
harvest in $\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}$

| Economic
---
yield in $\frac{\mathrm{\mbox{€}}}{{\mathrm{m}}^{2}}$

| Optimal harvest
---
based on DSSAT
in $\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}$

| Economic yield
---
based on DSSAT
in $\frac{\mathrm{\mbox{€}}}{{\mathrm{m}}^{2}}$

|  Computational
---
cost in $\frac{\mathrm{s}}{}$

SIMPLE |  |  |  | 16.73 | 16.16 | 17.19 | 16.78 | 14
| Reduced
---
TOMGRO
|  |  | 19.76 | 21.88 | 17.43 | 18.02 | 19
DSSAT |  |  |  | - | - | - | - | -

The results are presented in Table I, where it is shown that the SIMPLE model
has a computation time 1.35 times faster than the TOMGRO model on a standard
laptop, given an identical parallelized implementation in Python. This outcome
holds noteworthy implications for closed-loop control systems that depend on
updating constrained optimization using measured states. While the current GH
model only permits hourly updates, adapting to a finer granularity would be
advantageous for resource optimization, making the differences in computation
time more consequential. Furthermore, the current configuration does not
account for spatially distributed control approaches that consider spatial
discrepancies in the GH and crop states, which would lead to a substantial
increase in computational resource and real-time capability requirements. In
order to compensate for the explained model parameter uncertainties and to
enable a fair comparison, we compare the achievable harvest and yield using
the DSSAT model predictions, using the inputs generated in the optimization
with both SIMPLE and TOMGRO (see columns 6 and 7). Although both models
achieve similar final biomass, the TOMGRO model produces slightly better
economic yield (7 % higher). However, in a closed-loop regime, small model
inaccuracies are less important due to constant feedback. In summary, both
models have their strengths and weaknesses, but the SIMPLE model seems more
attractive for closed-loop applications since it can be easily adapted to
other crops. Nevertheless, obtaining the states poses difficulties that must
be overcome to use the SIMPLE model in a closed-loop.

##  VI Conclusion

In this study, we compared three different models for the growth of a tomato
plant, namely the SIMPLE, reduced TOMGRO and the DSSAT model. All crop models
can achieve similar accurate yields under standard weather and GH conditions.
While the DSSAT model turned out to be inaccessible for optimal control
approaches, the remaining two models, TOMGRO and SIMPLE, were compared in a
combination with the same GH model with respect to the usability in an OC
approach. The optimization results revealed that the models have different
sensitivities to environmental variables, emphasizing the importance of
careful model calibration. The SIMPLE model had a faster computation time,
making it more attractive for closed-loop control systems, but the TOMGRO
model achieved an overall higher yield, leaving uncertainty about which model
is closer to reality. Despite their strengths and weaknesses, both models are
relevant for closed-loop applications. The SIMPLE model is particularly useful
due to its adaptability to other crops. However, challenges associated with
obtaining states need to be overcome for the use in a closed-loop control
system which will be left for future work.

## References

  * [1] T. Searchinger, R. Waite, C. Hanson, J. Ranganathan, P. Dumas, E. Matthews, and C. Klirs.  Creating a sustainable food future: A menu of solutions to feed nearly 10 billion people by 2050.  WRI, 2019.
  * [2] R. Shamshiri, F. Kalantari, K.C. Ting, K.R. Thorp, I.A. Hameed, C. Weltzien, D. Ahmad, and Z.M. Shad.  Advances in greenhouse automation and controlled environment agriculture: A transition to plant factories and urban agriculture.  Int. J. Agric. Biol. Eng., 11:1–22, 2018.
  * [3] G. Van Straten, G. van Willigenburg, E. van Henten, and R. van Ooteghem.  Optimal control of greenhouse cultivation.  CRC press, 2010.
  * [4] A. Ramírez-Arias, F. Rodríguez, J.L. Guzmán, and M. Berenguel.  Multiobjective hierarchical control architecture for greenhouse crop growth.  Automatica, 48(3):490–498, 2012.
  * [5] D. Lin, L. Zhang, and X. Xia.  Hierarchical model predictive control of venlo-type greenhouse climate for improving energy efficiency and reducing operating cost.  J. Clean. Prod., 264:121513, 2020.
  * [6] Gerrit Van Straten.  Optimal greenhouse cultivation control: Quo vadis?  IFAC Proc. Vol., 46(4):11–16, 2013.
  * [7] N. Engler and M. Krarti.  Review of energy efficiency in controlled environment agriculture.  Renew. Sust. Energ. Rev., 141:110786, 2021.
  * [8] J.W. Jones, E. Dayan, L.H. Allen, H. Van Keulen, and H. Challa.  A dynamic tomato growth and yield model (TOMGRO).  Trans. ASAE, 34(2):663–0672, 1991.
  * [9] E. Heuvelink.  Evaluation of a dynamic simulation model for tomato crop growth and development.  Annals of Botany, 83(4):413–422, 1999.
  * [10] I. Seginer, C. Gary, and M. Tchamitchian.  Optimal temperature regimes for a greenhouse crop with a carbohydrate pool: A modelling study.  Scientia Horticulturae, 60(1):55–80, 1994.
  * [11] L.F.M. Marcelis, A. Elings, P.H.B. De Visser, and E. Heuvelink.  Simulating growth and development of tomato crop.  In International Symposium on Tomato in the Tropics 821, pages 101–110, 2008.
  * [12] A. Martínez-Ruíz, I.L. López-Cruz, A. Ruiz-García, J. Pineda-Pineda, and J.V. Prado-Hernández.  Hortsyst: A dynamic model to predict growth, nitrogen uptake, and transpiration of greenhouse tomatoes.  Chilean journal of agricultural research, 2019.
  * [13] K.J. Boote, M. R. Rybak, J.M.S. Scholberg, and J.W. Jones.  Improving the cropgro-tomato model for predicting growth and yield response to temperature.  Hortscience, 47:1038–1049, 2012.
  * [14] A. Ramirez, F. Rodriguez, M. Berenguel, and E. Heuvelink.  Calibration and validation of complex and simplified tomato growth models for control purposes in the southeast of spain.  In Acta Hortic. 654, pages 147–154, 2003.
  * [15] J.W. Jones, A. Kenig, and C.E. Vallejos.  Reduced state–variable tomato growth model.  Transactions of the ASAE, 42(1):255, 1999.
  * [16] F. Tap.  Economics-based optimal control of greenhouse tomato crop production.  PhD thesis, Wageningen University and Research, 2000.
  * [17] R.J.C. Van Ooteghem.  Optimal control design for a solar greenhouse.  PhD thesis, Wageningen University and Research, 2007.
  * [18] W.J.P. Kuijpers, M.J.G van de Molengraft, S. van Mourik, A. van’t Ooster, S. Hemming, and E.J. van Henten.  Model selection with a common structure: Tomato crop growth models.  Biosystems Engineering, 187:247–257, 2019.
  * [19] C. Zhao, B. Liu, L. Xiao, G. Hoogenboom, K.J. Boote, B.T. Kassie, W. Pavan, V. Shelia, W.S. Kim, I.M. Hernandez-Ochoa, et al.  A SIMPLE crop model.  Eur. J. Agron., 104:97–106, 2019.
  * [20] J.W. Jones, G. Hoogenboom, C.H. Porter, K.J. Boote, W.D. Batchelor, L.A. Hunt, P.W. Wilkens, U. Singh, A.J. Gijsman, and J.T. Ritchie.  The DSSAT cropping system model.  Eur. J. Agron., 18(3-4):235–265, 2003.
  * [21] S. Hemming, F. de Zwart, A. Elings, A. Petropoulou, and I. Righini.  Cherry tomato production in intelligent greenhouses—sensors and ai for control of climate, irrigation, crop yield, and quality.  Sensors, 20(22):6430, 2020.
  * [22] S. Hemming, H.F. de Zwart, A. Elings, A. Petropoulou, and I. Righini.  Autonomous greenhouse challenge, second edition (2019), 2020.
  * [23] A. Daniels, M. Fink, M. Leibold, and D. Wollherr.  Optimal control for indoor vertical farms based on crop growth (accepted).  IFAC Proceedings Volumes, 2023.
  * [24] C. Ramirez, R. Sanchez, V. Kreinovich, and M. Argaez.  x2+ $\mu$ is the most computationally efficient smooth approximation to $|x|$ : a proof.  Journal of Uncertain Systems, 8, 2014.
  * [25] J.A.E. Andersson, J. Gillis, G. Horn, J.B. Rawlings, and M. Diehl.  CasADi – A software framework for nonlinear optimization and optimal control.  Math. Program. Comput., 11(1):1–36, 2019.

  *[GHs]: greenhouse
  *[GH]: greenhouse
  *[OC]: optimal control
  *[OCP]: optimal control problem