=Paper=
{{Paper
|id=Vol-1498/HAICTA_2015_paper19
|storemode=property
|title=Use of Branch and Bound Algorithms for Greenhouse Climate Control
|pdfUrl=https://ceur-ws.org/Vol-1498/HAICTA_2015_paper19.pdf
|volume=Vol-1498
|dblpUrl=https://dblp.org/rec/conf/haicta/DimokasK15
}}
==Use of Branch and Bound Algorithms for Greenhouse Climate Control==
https://ceur-ws.org/Vol-1498/HAICTA_2015_paper19.pdf
Use of Branch and Bound Algorithms for Greenhouse
Climate Control
George Dimokas1, Constantinos Kittas2
1
Department of Agriculture Crop Production and Rural Environment, University of Thessaly,
Greece, e-mail: gedimokas@gmail.com
2
Department of Agriculture Crop Production and Rural Environment, University of Thessaly,
Greece
Abstract. Optimization of greenhouse climate management during winter
period is an issue that intensely preoccupied researchers over the last two
decades as it is directly linked to energy saving, products quality, and
reduction of chemical inputs. Goal of this project was the use of Branch and
Bound algorithms in order to obtain the optimize greenhouse climate control.
For this purpose a biophysical simulator were used and experiments were
carried out in the farm of the University of Thessaly in the region of Volos
(Greece), during the autumn and winter periods of 2005 and 2007. Branch and
Bound algorithms used for two different climate scenarios and the results
showed the difference between the classical greenhouse climate control and the
control according B & B optimization technique. Finally results showed the
contribution of optimization technique to increase tomato production and to
reduce energy consumption.
Keywords: Branch and Bound Algorithms, Climate Control, Greenhouse,
tomato production, optimization.
1 Introduction
Most of the greenhouse climate control problems show large number of possible
solutions, which are therefore entered the need for finding the best "route" or optimal
solution (Dimokas 2009). The use of advanced optimization models, may contribute
to the variation of the classical greenhouse climate management during winter period
which usually consists in the management of heating by specifying two desired
thermostat setting one temperature, for the night and one for day period, based
mainly on the producer experience (Tap et al. 1993). The desired values of the
temperature setting (set- points) are depending on the type and the age of crop.
Nevertheless, experimental work showed that the growth and development of many
vegetable and horticultural species appears to respond more to an average daily
temperature than on accurate temperature evolution during the day (Heuvelink 1989,
Vogelezang et al. 2000).
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The heating of the greenhouse by the sum of the temperatures (Integrated
Temperature Control) during the day has already been applied in several floriculture
(Rijsdijk and Vogelezang 2000) and horticultural species (De Konning 1988),
allowing an energy saving of about 10 - 20% (Bailey and Seginer 1989).
Many experimental projects are referred to optimization of greenhouse climate.
The optimal control is one of the processes studied further (van Henten 1994,
Tchamitchian and Tantau 1996). According to this method, a model function of the
optimization system is used, consisting of differential first degree equations and an
algebraic criterion that measures the result quality of this operation.
The principle of Bellman (Bellman 1957), allows to solve the problem with
dynamic programming, while the principle of Pontryagin (Pontryagin et al. 1962)
uses the Lagrange multipliers to convert the power system model and the algebraic
criterion to one function for minimization. These experimental works have not yet
reached to commercial systems. Goal of this project was the use of Branch and
Bound algorithms in order to obtain the optimize greenhouse climate control for
energy saving.
2 Material & Methods
Branch and bound (BB or B&B) is an algorithm design paradigm for discrete and
combinatorial optimization problems, as well as general real valued problems. A
branch-and-bound algorithm consists of a systematic enumeration of candidate
solutions by means of state space search: the set of candidate solutions is thought of
as forming a rooted tree (Figure 1) with the full set at the root. The algorithm
explores branches of this tree, which represent subsets of the solution set. Before
enumerating the candidate solutions of a branch, the branch is checked against upper
and lower estimated bounds on the optimal solution, and is discarded if it cannot
produce a better solution than the best one found so far by the algorithm.
S (Α)
S
S1 S1 S2 S3 S4 (Β)
S4
S2
S3
S1
S31
S21 S21 S22 S 31 S32 (C)
S22 S32
S3
Fig. 1. Display in the form of a tree Branch & Bound method.
The method of branching and bounding (Branch & Bound) has found application
in solving various and important optimization problems, eg, in integer programming,
nonlinear problems, programming problems, plant sitting problems (Dimokas 2009).
137
2.1 Optimization Method for Greenhouse Climate Control
Optimization of the biophysical simulator with the use of Branch & Bound
method uses a space selection strategy in order to be investigated in accordance with
the algorithms (1), (2) below. The algorithm (1), is responsible for setting the (Vh)
ventilation (system controller) inside the greenhouse
RH − pRT 1 (1)
Vh = 0.1 ⋅ (1 + ( )) + pb1−Ti
( Abs ⋅ ( RH − pRT )) + 0.5 (
pc1
)
e⋅
where pb1, pc1, pRΤ, the values of the variables that have to be optimized in order to
give the best solution. At the same time the algorithm (1), uses the biophysical
simulator and more specifically the results obtained for the relative humidity (RH)
and the air temperature inside the greenhouse (Ti), during the process of
optimization.
The algorithm (2) below is responsible for determining the operation of the (Vt)
heating system (system controller) within the greenhouse.
RH − pRT 1 (2)
Vt = 0.1 ⋅ (1 + ( )) + pb 2 − DTi
( Abs ⋅ ( RH − pRT )) + 0.5 (
pc 2
)
e⋅
where pb2, pc2, pRΤ, the values of the variables that have to be optimized in order to
give the best solution. At the same time the algorithm (2), uses the biophysical
simulator and more specifically the results obtained for the relative humidity (RH)
and the air temperature difference inside and outside the greenhouse (DTi), during
the process of optimization.
The problem to be solved is to minimize the objective function J to a range of
possible solutions, S:
min J = ∑ ( Fcon) − ∑ ( D.W .F ) (3)
where ∑ ( Fcon) is the total energy gives the heating system inside the greenhouse,
while respectively ∑ ( D.W .F ) the resulting dry weight of mature fruit.
The limiting function used to optimize the biophysical simulator using Branch &
Bound method shown below:
min J (one) > max J (other ) (4)
Function (4) reject some subset of possible solutions by further exploring inside,
to find the possible solution when the above condition is true, that the optimal
minimal solution that offers the particular subset is greater than the largest value of
another subset.
138
2.2 Climate and Biological Measurements
The measurements that used to optimize the model were data for tomato
production and development simultaneously with the greenhouse climatic data,
during the autumn and winter periods of 2005 and 2007. For the experimental
periods were used also calculated values by the modified TOMGRO (Dimokas et. al.,
2009), for: i) plant development and the number of leaves, fruits, flowers, ii) biomass
and fruit production. The aim was to identify differences between the optimization
method, and the climate control during the experimental period.
3 Results
This section presents results of treatment followed during the experimental
measurements and optimum proposed in accordance with the branch & bound
method. The results are giving detailed greenhouse climatic conditions and all
features related to the development and production of tomato plants. From these were
selected and are presented in the following sections the results concerning: a. number
of plants node, b. shoot dry weight, c. leaves dry weight, d. whole plant dry weight,
e. simulation curve of windows opening, f. air temperature inside the greenhouse, g.
temperature of the greenhouse cover. Values that are used as input variables for the
number of nodes, dry weight of leaves, stems and fruits are the same that used of the
modified TOMGRO.
3.1 Results of the first simulated period
Initially a variation was observed in greenhouse air - cover temperature and
presented in Figures 2 and 3. The optimum calculated values fall short of measured
values and this is due to the diversification arising from the way of windows
opening. Control of when and how much, windows are opening through the use of
the algorithm (1) are indicated in Figure 5. Time variation of window opening rate
leads to both reduce the temperature and reduce the humidity inside emissions (data
not shown).
Figures 4 (a-d) are showing the variation of calculated values using the modified
TOMGRO, and the optimum values obtained after the use of branch & bound
method for the number of nodes, dry weight of stem, leaves and whole plant
correspondingly. In Figure 4 (a) observed that the climate change does not change
the number of plant nodes formed. Correspondingly there is no change at the number
of leaves and the number of produced flowers (data not shown). Variation is
observed in Figure 4 (b) showing the dry weight of the shoot. The calculated optimal
values are above those calculated by the modified TOMGRO. Reverse change is
observed in the dry weight of leaves Figure 4 (c), wherein the optimum values are
below those estimated by the modified TOMGRO.
Smaller differences observed in Figure 4 (d) illustrating the total output of the
plant biomass. The differences are due to the diversification of air temperature
139
values, Figure 2. Reduction of air temperature leads to a small hysteresis of growth
and development of tomato plants. The reduction in biomass production when the
plants are in the initial stage of development helps to create more robust plants and
there is a regular practice for the producers that mainly use the chemical composition
of the nutrient solution and the irrigation dose to achieve it.
3.2 Results of the second simulated period
Figures 8 (a-d) are showing the variation of calculated values by using the
modified TOMGRO and best values obtained after the use of the branch & bound
method on the number of nodes and dry weight of the shoot, leaf and whole plant
respectively for the second experimental period. In Figure 8 (a) is observed that the
best calculated values relating to the formation of nodes, are slightly below than
those created by the modified TOMGRO. The growth rate resulting from the
optimum climate management is less than the one followed during the experimental
period. Simultaneously a reduction is presented for the values of shoot dry weight,
leaves dry weight and whole plant biomass. The reduction is almost 15.5% for shoot
dry weight, 18.5% for leaves dry weight and 17.4% for whole plant dry weight. The
decrease results from lower average value for the air temperature, kept inside the
greenhouse (Figure 6). It is observed that during the experimental period air
temperature was maintained above the fixed price of 15 oC. A similar differentiation
was observed in cover temperature as shown in Figure 7.
The gain from the use of Branch and Bound method was observed and that was
the reduction of heating system cost that was calculated 19.72% compared to the
initial treatment. The reduction in production costs resulting from the reduction of
the heating system cost may lead to a reduction in growth and development of
production, but may be a target for the producers. The decrease that caused to plants
growth and development can be balanced by increasing the temperature inside the
greenhouse to a period prior or after the reduction. This will give to the producers
smaller costs for the climate management and simultaneously energy saving.
4 Discussion & Conclusions
The results presented above were according two different climate scenarios within
greenhouses, with simultaneous display of the changes caused in the development
and production of the tomato crop. The periods used were selected of producer’s
interest and was at the start of the growing season and the second in the medium.
The study of the results for the first scenario, it is established that the reduction of
air temperature contributes to the reduction of both dry weight of leaves and whole
plant. The decrease in air temperature was due to the different treatment of the
windows operating system (system controller and B & B method).
140
40
35
Air Temperature (o C)
30
25
20
15
10
5
0
9/10/2007 10/10/2007 11/10/2007 12/10/2007 13/10/2007 14/10/2007
Days
Fig. 2. Variation of measured (–) and optimally calculated (-) values, for air temperature (oC)
during the first simulated period.
50
45
Cover Temperature (o C)
40
35
30
25
20
15
10
5
0
9/10/2007 10/10/2007 11/10/2007 12/10/2007 13/10/2007 14/10/2007
Days
Fig. 3. Variation of measured (–) and optimally calculated (–) values, for cover temperature
(oC) during the first simulated period.
The practice followed in accordance with the use of branch & bound method leads
to an opening of a greenhouse window for longer period than that followed in the
experimental procedure. However, the reduction of biomass produced when the
plants are in the initial stage of development helps to create more robust plants in the
future.
141
10 1
(c)
Dry weight of leaves (g plant-1 )
)
-1t
(a) 0,8
n
a
l
p 9 0,6
o
(n
se
d
o 0,4
n
fo 8
er 0,2
b
m
u
N 0
7 0 1 2 3 4 5
0 1 2 3 4 5 Days
Days
1
) 1,4
0,6 -t
n
)
(b)
la 1,2
p (d)
-1t 0,5 (g 1
t
an
l n
p 0,4 la
g( p 0,8
fo
s t
m
e 0,3 h 0,6
st ig
e
f w 0,4
o
t
h 0,2 ry
ige d
l 0,2
w ta
yr 0,1 o
T 0
D
0 1 2 3 4 5
0
Days
0 1 2 3 4 5
Days
Fig. 4 (a), (b), (c), (d). Variation of calculated values according modified TOMGRO ( ) and
the optimal values according B & B algorithms ( ), growth and biomass production of tomato
plants, during the first simulated period.
100%
Percentage of Window Opening (%)
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Air Temperature (o C)
Fig. 5. Percentage (%) of window opening (-), during the first simulated period.
By observing the results of the second scenario it was found that the reduction of
production costs by 19.72% led to a corresponding reduction of the biomass
production by 15.5% of shoot dry weight, 18.5% of leaves dry weight and 17.4% of
whole plant dry weight. However, the reduction caused in plants growth and
development can be balanced with an increase in air temperature inside the
greenhouse to a period prior to the reduction.
25
20
Air Temperature (o C)
15
10
5
0
28/11/2007 29/11/2007 30/11/2007 1/12/2007 2/12/2007 3/12/2007 4/12/2007
Days
Fig. 6. Variation of measured (–) and optimally calculated (-) values, for air temperature (oC)
during the second simulated period.
25
Cover Temperature (o C)
20
15
10
5
0
28/11/2007 29/11/2007 30/11/2007 1/12/2007 2/12/2007 3/12/2007 4/12/2007
Days
Fig. 7. Variation of measured (–) and optimally calculated (-) values, for cover temperature
(oC) during the second simulated period.
Managing greenhouse climate as mentioned is a daily activity for the producers,
which despite its frequency poses many problems. Further analysis of possible
climate scenarios will help to create strategies that will shape the conditions for
reducing production costs and improving the climate inside the greenhouse units.
143
15,0 5,0
(c)
Dry weight of leaves (g plant-1 )
(a)
Number of nodes (no plant-1 )
4,0
14,0
3,0
13,0
2,0
12,0
1,0
11,0 0,0
0 1 2 3 4 5 0 1 2 3 4 5
Days Days
3,0 7,0
Total dry weight of plant (g plant-1 )
(b) (d)
Dry weight of stmes (g plant-1 )
2,5 6,0
5,0
2,0
4,0
1,5
3,0
1,0
2,0
0,5
1,0
0,0 0,0
0 1 2 3 4 5 0 1 2 3 4 5
Days Days
Fig. 8 (a), (b), (c), (d). Variation of calculated values according modified TOMGRO ( ) and
the optimal values according B & B algorithms ( ), growth and biomass production of tomato
plants, during the second simulated period.
100% 100%
Percentage of Heating System Opening (%)
90% 90%
Percentage of Window Opening (%)
80% 80%
70% 70%
60% 60%
50% 50%
40% 40%
30% 30%
20% 20%
10% 10%
0% 0%
0 5 10 15 20 25 30
Air Temperature (o C)
Fig. 9. Percentage (%) of window (-) and heating system (–) opening during the second
simulated period.
Acknowledgments. This paper is part of the 03ED526 research project, implemented
within the framework of the “Reinforcement Program of Human Research
144
Manpower” (PENED) and co-financed by National and Community Funds (25%
from the Greek Ministry of Development-General Secretariat of Research and
Technology and 75% from E.U.-European Social Fund).
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