(444e) Studying Various Optimal Control Problems in Biodiesel Production in a Batch Reactor Under Uncertainty

Abstract

Optimal
control problems encountered in biodiesel production can be formulated using
various performance indices like maximum concentration, minimum time, and
maximum profit.  The problems involve
determining optimal temperature profile so as to maximize these performance
indices. This paper presents these three formulations and analyzes the
solutions in biodiesel production.  We
also present the maximum profit problem where the variability and uncertainties
in the feed composition of soybean are considered.  

Introduction

Optimal control problems are
defined in the time domain and their solution requires establishing a
performance index for the system. 
Because of the dynamic nature, optimal control problems are much more
difficult to solve compared to normal optimization.  In this paper, we are proposing an alternative
approach that avoids the use of large-scales NLP solvers. Moreover, we study
the three optimal control problems on batch reactors: maximum concentration problem
(MCP) of methyl ester, the minimum time problem (MTP), and the maximum profit
problem (MPP) for biodiesel production. The MCP and MTP are solved using the
maximum principle, and the approach is based on the Steepest Ascent of Hamiltonian.  On the other hand, the MPP is solved using an
algorithm that combines the maximum principle and NLP techniques.   This algorithm is an efficient approach
which avoids the solution of the two-point boundary value problem that results
in the pure maximum principle or in the solution of the partial differential
equations for the pure dynamic programming formulation. Biodiesel is one of the
most well-known examples for alternative energy and is also environmentally
friendly emission profile [1].  Here we
present the optimal control problem where optimal temperature profile is
derived using optimal control theory.

Optimal
control problems become more challenging when variability in any parameter or
variable is included.  In biodiesel
production, there are inherent uncertainties that have a significant impact on
the process.  One of the most influential
uncertainties in this process is the feed composition since the percentage and
type of triglycerides in biodiesel composition varies considerable [2].  This uncertainty
can be modeled using probabilistic techniques, and can be propagated using
stochastic modeling iterative procedures [3]. 
Therefore, at the end of this paper we propose a stochastic
maximum profit problem (SMPP) that regards the variability in the feed
composition to observe how this uncertainty affects the process economic in the
batch reactor.      

Optimal Control Problems

Table 1 summarizes the
optimization problems presented in this paper.   

Table
1 Optimal Control problems

The kinetic model presented here
is based on [4].

F1=dCTGdt=-k1CTGCA+k2CDGCE
                                                                                      (1)

F2 =dCDGdt=k1CTGCA-k2CDGCE-k3CDGCA+k4CMGCE
                                                 (2)

F3 =dCMGdt=k3CDGCA-k4CMGCE-k5CMGCA+k6CGLCE
                                                (3)

F4 =dCEdt=k1CTGCA-k2CDGCE+k3CDGCA-k4CMGCE+k5CMGCA-k6CGLCE           (4)
                          

F5=dCAdt=-dCE dt
                                                                                                                   (5)

F6=dCGLdt=k5CMGCA-k6CGLCE
  
                                                                                      (6)

where C_TG, C_DG,
C_MG, C_E, C_A, and C_GL are the state
variables and represent: triglycerides, diglycerides, monoglycerides, biodiesel,
methanol, and glycerol, respectively. 
The initial conditions are: C_i (t₀) = [C_TG;
0; 0; 0; C_A; 0]   [mol/L] and   k_i is the reaction constant.

The objective function for the MPP
is represented by Eq.7 [5]

maxJ''=MEPr-BoCot+ts
                                                                                                             (7)

where M_E is the amount
if product (kg), P_r is the sales value of the product ($/kg), B_o
is the amount of feed F (kg), C_o is the cost of feed ($/kg), t is
batch time (minutes) and t_s is the setup time for each batch.
However, this equation can be converted as:

maxJ''= max MEPr-BoCot+ts
                                                                                                     (8)

Table 2 shows the information for
profit function calculation.  The amount
of feed involves the quantity of methanol and triglycerides at the beginning of
the reaction while the amount of product is the final concentration of methyl
ester which is maximized by finding a temperature profile as a control
variable. 

Table
2 Information for maximum profit problem

For the MCP and MTP, maximum
principle is used to solve the optimal control problem.  On the other hand, a combination of maximum
principle and nonlinear optimization (NLP) technique based on SQP algorithm is
used to solve the MPP.

In the SMPP, the
objective function is subject to fluctuations due to the uncertainty arising in
the feedstock content.  In a previous
work of our group [6], we showed the uncertainty characterization and the
stochastic simulation for the feed stock composition of soybean oil.  The interest of these types of problems is to
determine the expected value of the maximum profit.   Then, the objective function for the SMPP is
shown in Eq. 9.

maxJ''=E max MEPr-BoCot+ts
                                                                                       (9)

Result and discussion

 Figure 1 shows the concentration profiles
obtained for the three optimal control problems and two base cases (base case
1: 315K and 2: 323K).  To start with, consider
the concentration profile of methyl ester for the MCP.  Here, we are comparing the concentration
values at constant temperature with the values calculated at optimal temperature
profile. It can be seen that at 100 minutes, the concentration of biodiesel using
optimal control reaches its maximum value, 0.7944mol/L; while at constant
temperature, the maximum concentration is 0.7324mol/L and 0.7829mol/L,
respectively.  This change represents an
increase of 8.46 % (base case 1) and 1.47% (base case 2) on the concentration
of methyl ester.  The increment for the
second base case is not significant since the constant profile at 323K belongs
to the constant optimal profiles reported in the literature [7].  Moreover, if we fix the concentration at
0.7324mol/L (concentration reached in base case1), the reaction time needed is
30.5 minutes which represents 69.5% less than it was at the beginning (100
minutes) after using optimal control. 
Compare with base case 2, the reduction on time represents 46% of the
original time.  This improvement does not
affect the behavior of the other components because after 50 minutes their
concentrations remain constant.  For the
MTP, after fixing the concentration of methyl ester to 0.7324mol/L the minimum time
reached is 30.6 minutes when optimal control is applied, while for base case 1
and 2 their minimum time is reached at 100 and 54 minutes, respectively.  Although, the optimal control profiles shown
in Figure 2 for the two optimal control problems are significantly different,
their results are similar showing that this problem have multiple solutions.

Figure 1
Concentration Profiles.

Figure 2
Optimal Temperature Profiles

For
the case of MCP, it can be seen that at 50 of minimum time there is an increase
of Biodiesel concentration of 25.56% (case 1) and 8.50% (case 2).  Moreover, if we compute the profit in the MCP
and MTP using Eq. 9 and compare these values with the profit found in the MPP,
there is an increment of 45.32% and 355.58%, respectively.

Table 3 Comparison of the optimal
control problems

Table 4 shows results of these
problems regarding uncertainty.  As shown,
in stochastic case there is an improvement of 6.68% compared to the
deterministic case and very significant improvement compare with the two base
cases.  In other words, the SMPP gives 9.99$/hr
more than MPP (deterministic) and 148.163$/hr and 76.951$/hr more than base
case 1 and 2, respectively.

Table 4 Comparison deterministic
and stochastic cases.

Conclusions

The
problems presented in this article involved determining optimal temperature
profile so as to maximize or minimize three performance indices: concentration,
time, and profit for biodiesel production. MCP and MTP were solved using maximum
principle and steepest ascent of the Hamiltonian.  The solution of these two problems results in
similar equations for maximum principle. 
In both cases reaction time was around 30.5 to 30.6 minutes and concentration
of biodiesel was the same.  While in MPP,
the technique used was based on combining the maximum principle and NLP
techniques.  Applying optimal control
under uncertainty it resulted in better time to produce the same amount of
biodiesel which improved the profit value of the problem.

References

1.       
Zhang, Y.; Dube, M.A.;
McLean, D.D.; Kates, M. Biodiesel production from waste cooking oil: 1. Process
design and technological assessment. Bioresour. Technol. 89 (2003) 1. 

2.       
Linstromberg WW. Organic chemistry. MA:
DC Heath and Co. Lexington. (1970) 129.

3.       
Diwekar U, Rubin ES. Stochastic modelling
of chemical processes. Comput. Chem. Eng. 15 (1991) 105

4.       
Noureddini H, Zhu D. Kinetic of
transesterification of soybean oil. J Am Oil Chem Soc.  74 (1997) 1457.

5.       
Kerkhof, L H. J.; Vissers, J. M.  On the profit of optimum control in batch
distillation.  Chem. Eng. Sci. 33 (1978)
961.

6.       
Benavides, P.; Diwekar, U. 
Optimal Control of Biodiesel Production in a Batch Reactor Part II:
Stochastic Control.  Fuel.
94 (2012) 218.

7.       
Leung,
D, Guo, Y.  Transesterification of neat and used frying
oil: optimization for biodiesel
production.  Fuel
Process Technol.  87 (2006)
883.