CET 96
DOI: 10.3303/CET2296054
Paper Received: 18 January 2022; Revised: 18 August 2022; Accepted: 14 July 2022
Please cite this article as: Acosta-Pavas J.C., Robles-Rodríguez C.E., Suarez Méndez C.A., Morchain J., Dumas C., Cockx A., Aceves-Lara C.A.,
2022, Dynamic Multi-objective Optimization Applied to Biomethanation Process, Chemical Engineering Transactions, 96, 319-324
DOI:10.3303/CET2296054
CHEMICAL ENGINEERING TRANSACTIONS
VOL. 96, 2022
A publication of
The Italian Association
of Chemical Engineering
Online at www.cetjournal.it
Guest Editors: David Bogle, Flavio Manenti, Piero Salatino
Copyright © 2022, AIDIC Servizi S.r.l.
ISBN 978-88-95608-95-2; ISSN 2283-9216
Dynamic Multi-Objective Optimization Applied to
Biomethanation Process
Juan C. Acosta-Pavasa, Carlos E. Robles-Rodrígueza, Camilo A. Suarez Méndezb,
Jérôme Morchaina*, Claire Dumasa, Arnaud Cockxa, César A. Aceves-Laraa,*
aTBI, Université de Toulouse, CNRS, INRAE, INSA, Toulouse, France
bUniversidad Nacional de Colombia, Medellín, Carrer 65 No. 59a-110, 050034 Medellín, Colombia
aceves@insa-toulouse.fr
Dynamic mathematical models could be beneficial for understanding and simulating processes to achieve an
optimal operation. The optimum, however, could depend on several variables that can be conflicting. In this
regard, Dynamic Multi-objective Optimization (DMO) is necessary for the trade-off between several objectives.
This work aims at proposing a DMO as a control strategy integrating two optimization problems. The objective
is to maximize the yield and productivity of the biomethanation processes by modifying the inlet liquid and
injected gas flowrates. First, multi-objective optimization was applied. Three Pareto optimal points were selected
to develop five cases in dynamic optimization. Case 1 corresponded to experimental data. Cases 2, 3, and 4
considered as objectives the maximum productivity, maximum productivity and yield, and maximum yield,
respectively. Case 5 was performed to assess a switch between objectives. For case 3, the yield decreased to
0.97 times, while the productivity increased 3.26 times with respect to case 1. The injected gas flowrate ranged
from 2.69 to 8.43 𝑚3 𝑑⁄ , and the inlet liquid flowrate reached an approximate value of 7.0×10-3 𝑚3 𝑑⁄ . These
results showed the feasibility and good efficiency of the proposed methodology.
1. Introduction
The use of dynamic models allows to gain a better understanding of different biological processes. One of those
processes is biomethanation, in which organic matter, such as agricultural residues, organic effluents from the
food industry, animal manure, or waste/wastewater residues are transformed through the synergistic work of a
variety of microorganisms into a mixture of CH4, and CO2 (Dar et al., 2021). This process was first modeled by
using the well-known Anaerobic Digestion Model No. 1 (ADM1) (Batstone et al., 2002). This model has been
adapted to solve stiffness problems (Rosen and Jeppsson, 2006), variation of pH (Czatzkowska et al., 2020),
and the inclusion of gas injection to obtain high purity methane (Sun et al., 2021). However, managing the
biomethanation process is still an arduous task due to the multiple molecules and different microorganisms
involved. As a result, obtaining desired objectives, such as high yields, high productivities, low processing times
or low flowrates remains difficult at an industrial scale, especially when you need to optimize several of them
simultaneously. The use of dynamic models plays a crucial role in the design of control strategies, e.g., optimal
control, adaptive control, or model predictive control (MPC) (Morales-Rodelo et al., 2020) to maintain the value
of the variables of interest during the process or to optimize several variables, i.e., a multi-objective optimization
(MO). When we talk about MPC, we refer to optimal controllers, i.e., the control action responds to the
optimization of a criterion (cost function), which is related to the future behavior of the system, determined from
the dynamic model (Camacho and Bordons, 2007). MO involves multiple criteria decision making, i.e., it implies
optimizing problems where there are more than one variable to be optimized simultaneously, and those
variables are usually conflictive (Chang, 2015; Vertovec et al., 2021). In this context, an optimal solution set that
fulfills the desired conditions of the conflicting variables is established and selected as Pareto optimal set. If a
solution point is not dominated by another solution, it is considered a Pareto Optimal Point (POP). Therefore, it
is ideal to have the highest number of Pareto optimal solutions (Deb et al., 2002).
319
This work aims at proposing a Dynamic Multi-objective Optimization (DMO) to find the trade-off between the
maxima of yield and productivity of the biomethanation process through a Pareto optimal set. Afterwards, a POP
is selected and used as the optimal reference trajectory. Then, a dynamic optimization is formulated in terms of
a MPC to modify the inlet liquid and injected gas flowrates to achieve the optimal values of yield and productivity
obtained from the POP.
2. Biomethanation Model Extension Proposal
The biomethanation process can be divided in four phases: hydrolysis, acidogenesis, acetogenesis and
methanogenesis (Dar et al., 2021). In the first phase, the fermentative bacteria excrete enzymes that hydrolyse
complex organic polymers (carbohydrates, proteins, and lipids) into soluble monomers, such as
monosaccharides, amino acids, and long chain fatty acids. In the second phase, these monomers are
transformed into volatile fatty acids (VFA), such as acetate, propionate, and butyrate. In the third phase, all the
VFA are transformed into acetate, H2, and CO2. The fourth phase involves the conversion of these components
by methanogenic archaea into biogas, i.e., a mixture of CH4, and CO2. Finally, this process is extended to
biological methanation which includes methane production by the biological activity of methanogenic bacteria
converting injected H2, and CO.The model was based on experimental data from the literature (Sun et al., 2021).
The entire experiment was carried out in a bioreactor with a working volume (𝑉𝑙𝑖𝑞) of 3.75×10-2 𝑚3 and hydraulic
retention time (𝐻𝑅𝑇) of 20 days operating at 37°C for 207 days. The organic loading rate (𝑂𝐿𝑅) was 0.53
𝑘𝑔𝐶𝑂𝐷 𝑚
3⁄ ∙ 𝑑 of glucose with an inlet liquid flowrate (𝑞𝑙𝑖𝑞,𝑖𝑛) of 1.9×10-3 𝑚3 𝑑⁄ (Chemical Oxygen Demand COD:
amount of oxygen needed to degrade the organic matter into CO2 and H2O). The gas injection was carried out
in five stages, in which the injected gas flowrate (𝑞𝑔𝑎𝑠,𝑖𝑛) and the gas loading rate (𝐺𝐿𝑅) were varied in time.
These values and the injected concentration of component 𝑖 in gas phase (𝑆𝑖,𝑖𝑛
𝑔
) are reported in Table 1.
Table 1: Experimental conditions from literature data without DMO (Sun et al., 2021).
Stage Time (Day)
Injected gas flowrate
(𝒎𝟑/𝒅)
Gas loading rate×10-2
(𝒎𝟑/𝒅)
Initial concentration gas
phase ×10-3 (𝒌𝒈𝑪𝑶𝑫 𝒎𝟑⁄ )
H2 CO
First 32 days 1-32 - - - -
I 33-64 0.09 0.75 20.83 38.69
II 65-101 1.44 0.75 1.30 2.42
III 102-135 2.88 1.50 1.30 2.41
IV 136-171 2.88 3.75 3.26 6.04
V 172-207 5.76 3.75 1.63 3.02
An extension of the Anaerobic Digestion Model No. 1 (ADM1_ME) to consider the injection of H2 and CO to
improver CH4 production was proposed in a previous work (Acosta-Pavas et al., 2022). Here, it is rewriten as,
𝑑𝑆𝑔𝑎𝑠,𝑖
𝑑𝑡
=
𝑞𝑔𝑎𝑠,𝑖𝑛
𝑉𝑔𝑎𝑠
𝑆𝑖,𝑖𝑛
𝑔
+𝑁𝑖 (
𝑉𝑙𝑖𝑞
𝑉𝑔𝑎𝑠
) −
𝑞𝑔𝑎𝑠
𝑉𝑔𝑎𝑠
𝑆𝑔𝑎𝑠,𝑖 (1)
𝑑𝑆𝑙𝑖𝑞,𝑗
𝑑𝑡
=
𝑞𝑙𝑖𝑞,𝑖𝑛
𝑉𝑙𝑖𝑞
(𝑆𝑗,𝑖𝑛
𝑙 −𝑆𝑙𝑖𝑞,𝑗) + ∑𝛽𝑗,𝑘µ𝑘𝑋𝑘
𝑘
− 𝑁𝑖 (2)
𝑑𝑋𝑘
𝑑𝑡
=
𝑞𝑙𝑖𝑞,𝑖𝑛
𝑉𝑙𝑖𝑞
(𝑋𝑘,𝑖𝑛−𝑋𝑘)+ 𝑌𝑘,𝑗µ𝑘𝑋𝑘 − 𝐾𝑘,𝑑𝑒𝑐 𝑋𝑘 (3)
where sub-index 𝑖 indicates the components H2, CH4, and CO in the gas phase; sub-index 𝑗 indicates the
components glucose, butyrate, propionate, acetate, H2, CH4, and CO in the liquid phase; and sub-index 𝑘
indicates the biomass that degraded glucose, butyrate, propionate, acetate, H2, and CO in liquid phase. 𝑉𝑔𝑎𝑠 is
the molar fraction volume, 𝑆𝑗,𝑖𝑛
𝑙 the inlet concentration of component 𝑗 in liquid phase, 𝑞𝑔𝑎𝑠 the outlet gas flowrate,
𝛽𝑗,𝑘 the stoichiometric coefficients, 𝑋𝑘,𝑖𝑛 the inlet concentration of biomass 𝑘, µ𝑘 and 𝐾𝑘,𝑑𝑒𝑐 the growth rate and
decay constant of biomass k, 𝑌𝑘,𝑗 the yield of biomass k, and 𝑁𝑖 the mass transfer rate of component 𝑖.
3. Dynamic Multi-objective Optimization Construction as Control Strategy
3.1 Multi-objective Optimization
There are several variables that can be optimized in biological processes, yields, productivities, process times,
etc. Most of these variables are often conflicting. Thus, it is necessary to find a trade-off between them. This is
called a MO problem. In this case, multiple optimal solutions that satisfy the desired conditions of both variables
can be found. This is known as the Pareto optimal set. In general, a MO can be formulated as follows,
320
min
𝑥,𝑢,𝑝,𝑡
{𝐽1(𝑥,𝑢,𝑝),…,𝐽𝑚(𝑥,𝑢,𝑝)} s.𝑡.
{
𝑑𝑥 𝑑𝑡⁄ = 𝑓(𝑥,𝑢,𝑝,𝑡) 𝑡 ∈ [0,𝑡𝑓]
𝑓𝑖(𝑥,𝑢,𝑝,𝑡) ≤ 0 𝑖 = 1,2,…,𝑛
𝑔𝑖(𝑥,𝑢,𝑝,𝑡) = 0 𝑖 = 1,2,…,𝑘
𝑢𝐿 ≤ 𝑢 ≤ 𝑢𝑈
(4)
where 𝐽1,…,𝐽𝑚 are the 𝑚 objective functions, 𝑥 the state variables, 𝑓𝑖 and 𝑔𝑖 indicate inequality and equality
constraints on the variable states, 𝑢 and 𝑝 denote the control variables and parameters, and 𝑢𝐿,𝑢𝑈correspond
to the lower and upper bounds of the control variables (Tsiantis et al., 2018).
3.2 Dynamic Optimization as a Model Predictive Control
MPC is one of the most widely used control methods in the industry (Morales-Rodelo et al., 2020). A MPC is an
advanced control strategy that solves an optimal control problem at every sampling time. The control uses an
explicit model to predict the outputs of the system at a future time by calculating the future control sequences to
minimize a cost function (Giraldo et al., 2022). The dynamic optimization (control problem) determines the future
control value that minimize a specified performance index, i.e., the input variables, that minimizes the following
objective function,
𝑚𝑖𝑛
𝑢
(∑ |𝑦∗ − 𝑦(𝑡 + 𝑗|𝑡)|2
𝑡+𝐻𝑝
𝑗=𝑡
+ ∑ 𝑊𝑢
𝑡+𝐻𝑐
𝑗=𝑡
∆𝑢(𝑡 + 𝑗|𝑡)2) 𝑠. 𝑡.{
𝑑𝑥 𝑑𝑡⁄ = 𝑓(𝑥,𝑢,𝑝,𝑡)
𝑓𝑖(𝑥,𝑢,𝑝,𝑡) ≤ 0 𝑖 = 1,2,…,𝑛
𝑔𝑖(𝑥,𝑢,𝑝,𝑡) = 0 𝑖 = 1,2,…,𝑘
𝑢𝐿 ≤ 𝑢 ≤ 𝑢𝑈
(5)
where 𝑢 is the vector of the control variables, 𝐻𝑝 and 𝐻𝑐 the prediction and control horizons, 𝑦(𝑡 + 𝑗|𝑡) the output
prediction calculated at time instant 𝑡 + 𝑗 using the information available at time instant 𝑡, 𝑦∗ a reference
trajectory which enables to reach the set point. These variables are determined by the MO, ∆𝑢(𝑡 + 𝑗|𝑡) is the
control move at time instant 𝑡 + 𝑗 calculated using information available at time instant 𝑡.
A DMO strategy is proposed to determine the optimal values of the objective functions. This strategy entails five
steps. Step 1 - Model definition: proposition of the dynamic model representative of the biological process.
Step 2 - Definition of the multi-objective control problem: definition of the objective functions 𝐽1,…,𝐽𝑚 to be
maximized/minimized by the MO optimization, the vector of the control variables 𝑢, the constraints 𝑓𝑖 and 𝑔𝑖,
and the bounds 𝑢𝐿 and 𝑢𝑈of the control variables in the MO optimization. Step 3 - Selection of the POP:
determination of the Pareto optimal set 𝐽1∗,…,𝐽𝑚∗ and select the POP to be used as the reference trajectory in
the dynamic optimization. Step 4 - Definition of the MPC problem defining a single weighted objective
based on the POP: formulation of an objective function considering the previously identified POP in terms of a
MPC problem. Indication of the initial guess values 𝑢0, the constraints 𝑓𝑖 and 𝑔𝑖, and the bounds 𝑢𝐿 and 𝑢𝑈 of
the control variables in the dynamic optimization. Step 5 - Determination of control and optimized variables:
execution of the dynamic optimization and determination of the optimal values of the control and optimized
variables at each time.
4. Case Study: Dynamic Multi-objective Optimization in Biomethanation Process
The main objective was to optimize yield (𝑌) and productivity (𝑃) and to achieve larger values with respect to
the ones obtained in the literature (data without DMO). Two control variables were proposed 𝑞𝑔𝑎𝑠,𝑖𝑛 and 𝑞𝑙𝑖𝑞,𝑖𝑛.
The yield 𝑌 was defined as the CH4 outlet flowrate (𝑞𝑔𝑎𝑠,𝐶𝐻4) produced over the total COD kilograms added per
day, while productivity 𝑃 was the ratio between 𝑞𝑔𝑎𝑠,𝐶𝐻4 and 𝑉𝑙𝑖𝑞, expressed as,
𝑌 =
𝑞𝑔𝑎𝑠,𝐶𝐻4
𝑆𝑠𝑢,𝑖𝑛
𝑙 𝑞𝑙𝑖𝑞,𝑖𝑛 + 𝑆𝐻2,𝑖𝑛
𝑔
𝑞𝑔𝑎𝑠,𝑖𝑛 + 𝑆𝐶𝑂,𝑖𝑛
𝑔
𝑞𝑔𝑎𝑠,𝑖𝑛
(6)
𝑃 =
𝑞𝑔𝑎𝑠,𝐶𝐻4
𝑉𝑙𝑖𝑞
(7)
4.1 Multi-objective Optimization
In this study, the simulations were run using an Intel® Core i7 8665U 2.11 GHz, 16 GB RAM computer. To obtain
the Pareto optimal set for each stage, the paretosearch function from MATLAB® was used. The MO was
proposed as,
𝑚𝑎𝑥
{𝑞𝑔𝑎𝑠,𝑖𝑛 𝑞𝑙𝑖𝑞,𝑖𝑛}
(𝑌,𝑃) s.𝑡.
{
𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 2 − 4
𝑌 ≤ 0.39 𝑚𝐶𝐻4
3 /𝑘𝑔𝐶𝑂𝐷𝑎𝑑𝑑𝑒𝑑
1 × 10−3 ≤ 𝑞𝑔𝑎𝑠,𝑖𝑛 ≤ 10 𝑚
3/𝑑
1 × 10−3 ≤ 𝑞𝑙𝑖𝑞,𝑖𝑛 ≤ 0.1 𝑚
3/𝑑
(8)
321
The constrains are reported in equation (8) where equation 1-3 correspond to the model dynamics (ADM1_ME),
and 0.39 𝑚𝐶𝐻43 /𝑘𝑔𝐶𝑂𝐷𝑎𝑑𝑑𝑒𝑑 represents the theoretical cumulative CH4 volume at 37°C or 0.35
𝑚𝐶𝐻4
3 /𝑘𝑔𝐶𝑂𝐷𝑎𝑑𝑑𝑒𝑑 at standard temperature and pressure conditions (Filer et al., 2019). The MO was performed
for stages I-V. 60 Pareto points were computed for each stage. In which 20, 22, 20, 28, and 23 iterations were
executed in 14.65, 19.01, 23.15, 38.29, 56.33 min, respectively. Figure 1A displays the Pareto optimal sets for
each stage. In stage I and IV, the Pareto optimal set is far from the experimental value, indicating that
optimization can perform a representative change in both optimum variables. For the other stages, the
experimental point is in the proximity of the Pareto optimal set, indicating that the experiment was performed to
maximize yield. The POP selection was performed by normalizing the Pareto optimal set [0,1] and determining
the maximum Euclidean length (𝑑𝑚𝑎𝑥) from the origin on the normalized coordinates. Other POP were selected
to maximize the yield and productivity. The first POP considered maximization of productivity, the second one
the maximization of the Euclidean length, and the third one maximization of yield (orange, yellow, and purple
squares in Figure 1A). Table 2 summarizes the selected POP for yield and productivity maximization in each
stage.
Table 2: Multi-objective optimization results.
Stage
POP
for maximum productivity
POP
for maximum Euclidean length
POP
for maximum yield
Yield
(𝐦𝐂𝐇𝟒
𝟑 𝐤𝐠𝐂𝐎𝐃𝒂𝒅𝒅𝒆𝒅⁄ )
Productivity
(𝒎𝑪𝑯𝟒
𝟑 𝒎𝒓𝒆𝒂𝒄𝒕𝒐𝒓
𝟑⁄ ∙ 𝒅)
Yield
(𝐦𝐂𝐇𝟒
𝟑 𝐤𝐠𝐂𝐎𝐃𝒂𝒅𝒅𝒆𝒅⁄ )
Productivity
(𝐦𝐂𝐇𝟒
𝟑 𝐦𝐫𝐞𝐚𝐜𝐭𝐨𝐫
𝟑⁄ ∙ 𝐝)
Yield
(𝐦𝐂𝐇𝟒
𝟑 𝐤𝐠𝐂𝐎𝐃𝒂𝒅𝒅𝒆𝒅⁄ )
Productivity
(𝐦𝐂𝐇𝟒
𝟑 𝐦𝐫𝐞𝐚𝐜𝐭𝐨𝐫
𝟑⁄ ∙ 𝐝)
I 0.3155 0.8011 0.3341 0.7080 0.3443 0.2984
II 0.3156 0.8012 0.3340 0.7107 0.3443 0.2984
III 0.3159 0.8471 0.3330 0.7599 0.3417 0.4042
IV 0.3149 0.9847 0.3308 0.9042 0.3371 0.5940
V 0.3167 0.9844 0.3313 0.8943 0.3371 0.5940
4.2 Dynamic Multi-objective Optimization
To perform the dynamic optimization, the patternsearch function from MATLAB® was used. The dynamic
optimization problem was proposed as,
𝑚𝑖𝑛
{𝑞𝑔𝑎𝑠,𝑖𝑛 𝑞𝑙𝑖𝑞,𝑖𝑛}
( ∑ (
|𝑌∗ − 𝑌(𝑡)|
𝑌∗
)
2
𝑡+𝐻𝑝
𝑗=𝑡
+ (
|𝑃∗ − 𝑃(𝑡)|
𝑃∗
)
2
+ ∑ 𝑊𝑢,1 ∆𝑞(𝑡)𝑔𝑎𝑠,𝑖𝑛
2 + 𝑊𝑢,2 ∆𝑞(𝑡)𝑙𝑖𝑞,𝑖𝑛
2
𝑡+𝐻𝑐
𝑗=𝑡
) 𝑠. 𝑡.
{
𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 2 − 4
𝑌 ≤ 0.39 𝑚𝐶𝐻4
3 /𝑘𝑔𝐶𝑂𝐷𝑎𝑑𝑑𝑒𝑑
1 × 10−3 ≤ 𝑞𝑔𝑎𝑠,𝑖𝑛 ≤ 10 𝑚
3/𝑑
1 × 10−3 ≤ 𝑞𝑙𝑖𝑞,𝑖𝑛 ≤ 0.1 𝑚
3/𝑑
(9)
where 𝑌(𝑡) and 𝑃(𝑡) were calculated by using (6) and (7). 𝑌∗, and 𝑃∗ are the POP values for yield and
productivity computed by the MO, ∆𝑞(𝑡)𝑔𝑎𝑠,𝑖𝑛
2 and ∆𝑞(𝑡)𝑙𝑖𝑞,𝑖𝑛
2 represent the differences between the injected gas
and inlet liquid flowrates before and after each step in the dynamic optimization. 𝑊𝑢,1 and 𝑊𝑢,2 are the
parameters that weight the importance of the control effort for each input of the optimization.
Five cases were studied to assess the dynamic optimization. Case 1: ADM1_ME without DMO (experimental
value). Case 2: ADM1_ME with DMO (POP for maximum productivity). Case 3: ADM1_ME with DMO (POP for
maximum Euclidean length). Case 4: ADM1_ME with DMO (POP for maximum yield). Case 5: ADM1_ME with
DMO switching between the maximum productivity (stages I, V), maximum Euclidean length (stages II, III) and
maximum yield (stage IV). In all cases, the initial guess (𝑢0) was 1 × 10−3 𝑚3/𝑑 for both control variables. The
lower and upper bounds of the objective variables 𝑌 and 𝑃, and the constrains were the same presented in the
MO. 𝐻𝑝 and 𝐻𝑐 were considered equal with values corresponding to the time of each stage (see Table 1).
For cases 2, 3, 4 and 5 the simulation times were 2.12, 3.35, 3.37, and 2.89 min, respectively, in which the
weights Wu,1 and Wu,2 were manually adjusted to values of 1 × 10−6. Figure 4 shows the dynamical behavior of
optimum and control variables. Regarding case 2 in stage V, the productivity was maximized from 0.419 without
DMO to 0.982 𝑚𝐶𝐻43 𝑚𝑟𝑒𝑎𝑐𝑡𝑜𝑟3⁄ · 𝑑 with a slightly decrease of yield from 0.334 without DMO to 0.316
𝑚𝐶𝐻4
3 𝑘𝑔𝐶𝑂𝐷𝑎𝑑𝑑𝑒𝑑⁄ with DMO. The inlet liquid flowrate increased from 1.9×10-3 𝑚3/𝑑 without DMO to 8.4×10-3
𝑚3/𝑑 with DMO and remained constant for all stages. The injected gas flowrate increased from 5.76 𝑚3/𝑑
without DMO to 10 𝑚3/𝑑 with DMO in stage V.
In case 3, the yield increased from 0.318 to 0.333 𝑚𝐶𝐻43 𝑚𝑟𝑒𝑎𝑐𝑡𝑜𝑟3⁄ · 𝑑 in stage I, but decreased with respect to
case 1 in stage V. The productivity increased from 0.419 without DMO to 0.982 in stage V. The inlet liquid
flowrate increased from 1.9×10-3 𝑚3/𝑑 without DMO to 7.0×10-3 𝑚3/𝑑 with DMO in stage V. The injected gas
flowrate increased from 5.76 𝑚3/𝑑 without DMO to 8.40 𝑚3/𝑑 with DMO in stage V. In case 4, the yield achieved
a value of 0.343 𝑚𝐶𝐻43 𝑚𝑟𝑒𝑎𝑐𝑡𝑜𝑟3⁄ · 𝑑 in stage I, but decreased slightly to 0.337 𝑚𝐶𝐻43 𝑚𝑟𝑒𝑎𝑐𝑡𝑜𝑟3⁄ · 𝑑 in stage V. The
productivity reached a value of 0.59 𝑚𝐶𝐻43 𝑚𝑟𝑒𝑎𝑐𝑡𝑜𝑟3⁄ · 𝑑 in stage V. The inlet liquid flowrate increased up to
322
3.69×10-3 𝑚3/𝑑 in stage V. The injected gas flowrate increased from 5.76 𝑚3/𝑑 in case 1 stage V to 8.01 𝑚3/𝑑
in stage V.
Figure 1: (A) Pareto optimal sets. (B) Yield, productivity, injected gas and inlet liquid flowrates in the DMO.
Table 3: Yield and productivity ratio with DMO.
Stage
Case 1 Case 2 Case 3 Case 4 Case 5
Yield (𝒎𝑪𝑯𝟒
𝟑 𝒌𝒈𝑪𝑶𝑫𝒂𝒅𝒅𝒆𝒅⁄ )
Value Value Ratio Value Ratio Value Ratio Value Ratio
I 0.318 0.315 0.99 0.333 1.05 0.343 1.08 0.315 0.99
II 0.342 0.315 0.92 0.333 0.97 0.344 1.00 0.333 0.97
III 0.339 0.316 0.93 0.332 0.98 0.341 1.01 0.332 0.98
IV 0.331 0.315 0.95 0.330 1.00 0.336 1.02 0.336 1.02
V 0.334 0.316 0.95 0.331 0.99 0.337 1.01 0.316 0.95
Productivity (𝒎𝑪𝑯𝟒
𝟑 𝒎𝒓𝒆𝒂𝒄𝒕𝒐𝒓
𝟑 ∙ 𝒅⁄ )
I 0.217 0.807 3.72 0.708 3.26 0.298 1.37 0.798 3.67
II 0.234 0.800 3.42 0.710 3.04 0.298 1.28 0.711 3.04
III 0.28 0.846 3.02 0.76 2.71 0.404 1.44 0.760 2.71
IV 0.415 0.985 2.37 0.904 2.18 0.594 1.43 0.594 1.43
V 0.419 0.982 2.34 0.894 2.13 0.594 1.42 0.982 2.34
In case 5, the yield, productivity, and the inlet liquid flowrate followed the behavior of case 2 in stage I, the
behavior of case 3 in stages II and III, the behavior of case 4 in stage IV, and again the behavior of case 2 in
stage V. However, the injected gas flowrate differs for all cases and stages.
Table 4 reports the obtained values for yield and productivity as well as a ratio of their respective values with
respect to case 1(experimental data). Values larger than 1 show that the DMO is better than the experimental
data. For case 2, the yield was 0.99 times higher than the yield for case 1 in stage I and 0.95 times in stage V.
On the other hand, the productivity increases 3.72 times and decreases to 2.34 times from stages I to V,
respectively.
Concerning case 3, the yield ratio varied between 1.05 and 0.99 times while the productivity ratio changed
between 3.26 and 2.13 times concerning without DMO (case 1) for stages I and V, respectively. For case 4, the
yield was 1.08 times higher than the yield for case 1 in stage I and 1.01 times in stage V. On the other hand,
the productivity increased between 1.28 and 1.44 for the different stages.
For case 2, the inlet liquid flowrate increased slightly, ranging from 1.9×10-3 𝑚3/𝑑 (case 1) to a value up to
8.4×10-3 𝑚3/𝑑 in stage V. However, the injected gas flowrate needed was higher ranging from the values
reported in Table 1 (case 1) to values between 2.69 𝑚3/𝑑 and 10.00 𝑚3/𝑑 between stages I and V. In case 3,
to maintain the maximum yield and productivity, the DMO suggested keeping the inlet liquid flowrate at 7.0×10-
3 𝑚3/𝑑 during all stages while changing the injected gas flowrate between 2.73 and 8.44 𝑚3/𝑑 from stage I to
V. For case 4, an inlet liquid flowrate between 2.56×10-3 𝑚3/𝑑 and 3.70×10-3 𝑚3/𝑑 from stage I to V was used
to maintain the maximum yield. The injected gas flowrate ranged between 2.69 and 8.07 𝑚3/𝑑 from stage I to
323
V. Case 5 followed the same behavior for the inlet liquid flowrate, whereas the injected gas flowrate behavior
was different for all cases.
5. Conclusions
In this work, a DMO strategy for the biomethanation process based on the dynamic model ADM1_ME was
presented. Optimizations for two objectives were performed: maximization of yield and productivity by modifying
the inlet liquid and injected gas flowrates. The proposed strategy showed the conflicting behavior of both
objectives. Five cases studies were compared, where it was observed that maximizing productivity lowers yield
ratio, and vice versa. Case 5 reported a switching strategy between objectives, which allows to demonstrate the
robustness of the process and the well accounted adaptations of the input variables in simulations. Additionally,
it was demonstrated that both input variables have a role in DMO. For instance, the injected gas flowrate made
a higher effort with respect to the inlet liquid flowrate. This was observed in case 5, where the behavior of the
injected gas flowrate differed in all cases. These results show the feasibility of the proposed methodology and
its use for multiple control objectives.
Acknowledgments
This research was supported by the Ministerio de Ciencias, Tecnología e Innovación (Minciencias) through the
Scholarship Program No. 860.
References
Acosta-Pavas, J. C., Morchain, J., Dumas, C., Ngu, V., Cockx, A., & Aceves-Lara, C. A. (2022). Towards
Anaerobic Digestion (ADM No.1) Model’s Extensions and Reductions with In-situ Gas Injection for
Biomethane Production, Accepted. 10th Vienna International Conference on Mathematical Modelling
(MATHMOD), 6.
Batstone, D. J., Keller, J., Angelidaki, I., Kalyuzhnyi, S. V., Pavlostathis, S. G., Rozzi, A., Sanders, W. T. M.,
Siegrist, H., & Vavilin, V. A. (2002). The IWA Anaerobic Digestion Model No 1 (ADM1). Water Science and
Technology, 45(10), 65–73.
Camacho, E. F., & Bordons, C. (2007). Model Predictive control. Springer London. https://doi.org/10.1007/978-
0-85729-398-5
Chang, K.-H. (2015). Multiobjective Optimization and Advanced Topics. In e-Design (pp. 1105–1173).
Czatzkowska, M., Harnisz, M., Korzeniewska, E., & Koniuszewska, I. (2020). Inhibitors of the methane
fermentation process with particular emphasis on the microbiological aspect: A review. Energy Science and
Engineering, 8(5), 1880–1897.
Dar, R. A., Parmar, M., Dar, E. A., Sani, R. K., & Phutela, U. G. (2021). Biomethanation of agricultural residues:
Potential, limitations and possible solutions. Renewable and Sustainable Energy Reviews, 135(August
2020), 110217.
Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm:
NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2), 182–197.
Filer, J., Ding, H. H., & Chang, S. (2019). Biochemical Methane Potential (BMP) Assay Method for Anaerobic
Digestion Research. Water, 11(5), 921.
Giraldo, S. A. C., Melo, P. A., & Secchi, A. R. (2022). Tuning of Model Predictive Controllers Based on Hybrid
Optimization. Processes, 10(2), 351.
Morales-Rodelo, K., Francisco, M., Alvarez, H., Vega, P., & Revollar, S. (2020). Collaborative control applied to
bsm1 for wastewater treatment plants. Processes, 8(11), 1–22.
Rosen, C., & Jeppsson, U. (2006). Aspects on ADM1 Implementation within the BSM2 Framework. In Technical
report.
Sun, H., Yang, Z., Zhao, Q., Kurbonova, M., Zhang, R., Liu, G., & Wang, W. (2021). Modification and extension
of anaerobic digestion model No.1 (ADM1) for syngas biomethanation simulation: From lab-scale to pilot-
scale. Chemical Engineering Journal, 403(1), 126177.
Tsiantis, N., Balsa-Canto, E., & Banga, J. R. (2018). Optimality and identification of dynamic models in systems
biology: An inverse optimal control framework. Bioinformatics, 34(14), 2433–2440.
Vertovec, N., Ober-Blöbaum, S., & Margellos, K. (2021). Multi-objective minimum time optimal control for low-
thrust trajectory design.
324
141acostapavas.pdf
Dynamic Multi-Objective Optimization Applied to Biomethanation Process