Comparison of Hybrid PSOGSA & Cuckoo Search (CS) algorithm for Optimization of ETBE
Reactive Distillation
By
Vandana Sakhre*1
1
Madhav Institute of Technology
& Science (MITS),
Gwalior. 474005
Email: vssakhre@gmail.com
Sanjeev Jain1
V.S. Sapkal2,
1
2
Madhav Institute of Technology
SGB Amravati University,
& Science (MITS),
Amravati.444602
Gwalior. 474005
Email: dr_sanjeevjain@yahoo.com Email: vssapkal@gamil.com
Abstract:
Reactive Distillation is highly nonlinear process because of complex chemical interaction and simultaneous
separation of components. Control of process parameters of reactive distillation is a challenging task. The
objective of this contribution is to present a novel approach for optimization of reactive distillation process
parameters. Cuckoo Search (CS) and hybrid combination of Particle Swarm Optimization Gravitational
Search Algorithm (PSOGSA) are newly developed metaheuristic technique for solving optimization
problems. In this paper, we have chosen ETBE Reactive Distillation as a case study. The objective function
for this case is to maximize product purity. After comparison we found that both the algorithm gives best
results, but hybrid PSOGSA gives faster convergence and best solution quality irrespective of number of
iterations.
Key Words: Cuckoo Search, Hybrid Optimization, Metaheuristic, PSOGSA.
1.0 Introduction:
Unlike conventional separate reaction and distillation system, Reactive Distillation is a combination of both
operation. Due to this reason, it comprises of three sections. These three sections are namely non-reactive
rectifying section, reactive section and non-reactive stripping section. The reaction is carried out in reactive
section while separation is carried out in both non-reactive rectifying and stripping sections respectively.
Thus, the system of reactive distillation is affected by type of reacting components, hardware selection,
mode of operation and operating conditions. Reactive Distillation is more advantageous for those reactions
in which reaction equilibrium limits the conversion. During the reaction, the continuous removal of reaction
product causes increased rate of reaction. As reaction and separation are carried out in single unit, it is
benefitted in terms of reduced capital cost [1]. On the other way, it is highly nonlinear process and very
difficult to control process variables. Mathematical Optimization is an applied science which helps
determining the best value of the parameters by minimization or maximization of the given function [2].
Optimization algorithms form the core tools for experimental design, parameter estimation, model
development, and statistical analysis. It also provide tool for model predictive control and real-time
optimization [3]. Particle Swarm Optimization (PSO) is a global stochastic optimization method based on
simulation of social behavior. It exploit the population of potential solution [4]. Nature inspired algorithm
are most powerful algorithm for optimization [5]. New metaheuristic algorithms including Harmony Search
inspired by the improvising process of composing a piece of music, and Firefly which is formulated based
on the flashing behavior of fireflies developed recently [6-7]. Most meta-heuristic algorithms combine rules
and randomness to imitate natural phenomena. These phenomena include the biological evolutionary
processes, such as genetic algorithm (GA) [8-9], evolutionary algorithm [10-11], and differential evolution (DE)
[12]
, animal behavior, such as particle swarm optimization (PSO) [13], Tabu Search (TS) [14] and Ant Colony
Algorithm (ACA) [15], as well as physical annealing processes, such as simulated annealing (SA) [16].
Improved Cuckoo Search (ICS) Algorithm was developed by Ehsan Valian [17]. A new hybrid PSOGSA
developed by Mirjalili, and Hashim (2010) incorporates features of PSO into Gravitational Search
Algorithm (GSA) which is exploitation ability of PSO and exploration ability of GSA [18].
In this research paper, we have implemented hybrid PSOGSA and Cuckoo Search (CS) algorithm for global
optimization of Reactive Distillation (RD). Etherification of Ethyl-Tert-Butyl Ether (ETBE) was taken as
*corresponding author: Email:vssakhre@gmail.com. Phone No.-07512409378/200.
case study. The organization of this paper is as follows: Section 2 describes case study of ETBE reactive
distillation along with problem formulation. Section 3 presents the optimization using heuristic algorithm.
Section 4 describes results and discussion. Section 5 presents conclusion.
2.0
ETBE Reactive Distillation: Case Study
In this research paper, etherification of ETBE by reactive distillation is chosen as case study. ETBE is one
of the most important fuel oxygenating agent for gasoline. ETBE can be synthesized by exothermic
reversible reaction between Isobutylene (IB) and ethanol [24], but the availability of IB is limited. Therefore,
alternative routes to synthesize ETBE are under substantial consideration. By far the most important
substitute of Iso-Butylene is Tertiary Butyl Alcohol (TBA) which is a by-product of propylene oxide
production in ARCO process. ETBE can be synthesized by direct reaction of TBA and Ethanol in the
presence of acidic ion exchange catalyst according to following reactions:
TBA + EtOH <------> ETBE + H2O
This main reaction is also accompanied by two side reactions and major side reaction is dehydration of
TBA into IB and water.
TBA <---> IB + H2O
IB + EtOH <-----> ETBE
The reactive distillation process can be used as an efficient process to achieve separations of equilibrium
limited reactions. The difference in reactivity can be exploited advantageously.
2.1 Modeling of Catalytic Packed Reactive Distillation:
Modeling of catalytic packed RD was done using first principle model. Two reactants (A and B)
producing two products (C and D) as per the following chemistry:
A+BC+D
The schematic diagram of the reactive distillation column is shown in fig 1. Two reactants with feed flow
rates of F1 and F2 fed at top and bottom segment of the reactive section respectively. The product is collected
at the top of the column while byproduct is collected at the bottom.
Fig 1: Schematic of Reactive Distillation
Some assumptions are made as follows:
1. The process is ideal in Vapor-Liquid Equilibrium
2. Saturated Liquid Feed and Reflux flow rate
3. The products have constant relative volatilities
4. Heat of reaction and vaporization and saturated liquid feed and reflux are fixed
5. Constant liquid hold up in reactive zone, reboiler and condenser
6. Reactive zone to be a single stage
7. Negligible vapor holdup.
Rectifying and stripping trays:
𝑑(𝑥𝑛,𝑗 𝑀𝑛 )
= 𝐿𝑛+1 𝑥𝑛+1,𝑗 + 𝑉𝑛−1 𝑦𝑛−1,𝑗 − 𝐿𝑛 𝑥𝑛,𝑗 − 𝑉𝑛 𝑦𝑛,𝑗
𝑑𝑡
Reactive trays:
𝑑(𝑥𝑛,𝑗 𝑀𝑛 )
= 𝐿𝑛+1 𝑥𝑛+1,𝑗 + 𝑉𝑛−1 𝑦𝑛−1,𝑗 − 𝐿𝑛 𝑥𝑛,𝑗 − 𝑉𝑛 𝑦𝑛,𝑗 + 𝑅𝑛,𝑗
𝑑𝑡
Feed trays:
𝑑(𝑥𝑛,𝑗 𝑀𝑛 )
= 𝐿𝑛+1 𝑥𝑛+1,𝑗 + 𝑉𝑛−1 𝑦𝑛−1,𝑗 − 𝐿𝑛 𝑥𝑛,𝑗 − 𝑉𝑛 𝑦𝑛,𝑗 + 𝑅𝑛,𝑗 + 𝐹𝑛 𝑧𝑛,𝑗
𝑑𝑡
Column base:
𝑑(𝑥𝐵,𝑗 𝑀𝐵 )
= 𝐿1 𝑥1,𝑗 − 𝐵𝑥𝐵,𝑗 − 𝑉𝑆 𝑦𝐵,𝑗
𝑑𝑡
Column pressure:
𝑁𝐶
𝑆
𝑃 = ∑ 𝑥𝑛,𝑗 𝑃𝑗(𝑇𝑛)
𝑗=1
Equilibrium:
𝑦𝑛,𝑗 =
𝑃𝑗𝑆
𝑃
𝑥𝑛,𝑗
2.2 Problem Formulation for ETBE RD:
The condition in the reactive distillation column are suboptimal because it is a combined reactor-separator
unit. The in-situ separation leads to complex vapor liquid equilibrium and fast conversion. The aim of
reactive distillation to combine reaction and separation together to achieve higher conversion for
equilibrium limited reactions. In this case study, we use simple mass balance equations for maximization
of distillate composition:
F=B+D
F*XF=B*XB+D*XD
Using above equation and calculating in terms of distillate D we get
D/F = (X𝑓 − X 𝑤 )/(X𝑑 − X 𝑤 )
Rearranging above equation to get Xd in terms of D/F and let us denote this ratio by Z then equation
reduced to:
X𝑓 − X 𝑤
X𝑑 = (
) + X𝑤
Z
Let the objective is to maximize distillate purity then our Objective function will be
X𝑓 − X 𝑤
Max Xd = (
) + X𝑤
z
Subjected to constraints
0<=Xw<=1
0<=z<=1
Where
Xw=f (T, L, R); T is temperature, L is liquid flow rate, R is reflux ratio
z=f (T, D, F); D is distillate flow rate, F is feed flow rate
Xf if fraction reacting component in the feed.
3.0
Optimization using Heuristic Algorithm:
Optimization problems are concerned with finding the values for one or several decision variables that meet
the objective without violating the constraint. Depending on the objective function, optimization problems
might have multiple solutions some of which might be local optima. For solving non-linear objective
function, metaheuristic algorithm are commonly used, which uses pattern matrix to give random solution.
In Particle Swarm Optimization (PSO), swarm is referred by pattern matrix and each pattern is corresponds
to artificial particle. A pattern is considered as an artificial nest in the Cuckoo Search (CS) algorithm.
Gravitational Search Algorithm (GSA) uses two solution strategies, exploration and exploitation. The
exploration process succeeds in enabling the algorithm to reach the best local solutions within the search
space, the exploitation process expresses the ability to reach the global optimum solution which is likely to
exist around the local solutions obtained. Hybrid PSOGSA combines the ability of global search in
PSOGSA with the local search capability of GSA.
3.1 Optimization using Hybrid PSOGSA:
Particle Swarm has two primary operators-Velocity update and Position update. During each generation
each particle is accelerated toward the particles previous best position and the global best position. At each
iteration a new velocity value for each particle is calculated based on its current velocity, the distance from
its previous best position, and the distance from the global best position. The new velocity value is then
used to calculate the next position of the particle in the search space. This process is then iterated a set
number of times [19]. The algorithm of GSA is based on Law of Gravity. The gravitation is the tendency of
masses to accelerate toward each other. The GSA starts with a set of agents, selected at random or based
on some criteria, with certain positions and masses representing possible solutions to a problem, and iterates
by changing the positions based on some values like fitness function, velocity and acceleration that gets
updated [20].
The PSOGSA algorithm incorporates some features of particle swarm optimization algorithm into
gravitational search algorithm i.e. exploitation ability of PSO with ability of exploration in GSA to unify
their strength. Random initialization of agents in the search space is attracted towards the agent having a
good solution. The agents near the optimal solution moves more slowly and assures the exploitation step of
algorithm. The position and velocity are updated until it reaches to the stopping criterion [21].
At a specific time t we define the force acting on mass i from mass j as following:
𝑀𝑝𝑖 (𝑡) × 𝑀𝑎𝑗 (𝑡) 𝑑
𝐹𝑖𝑗𝑑 (𝑡) = 𝐺(𝑡)
(𝑥𝑗 (𝑡) − 𝑥𝑖𝑑 (𝑡))
𝑅𝑖𝑗 (𝑡) + 𝜀
Where Maj is the active gravitational mass of agent j, Mpi is the passive gravitational mass of agent i, G(t)
is gravitational constant at time t, ε is a small constant, and Rij(t) is the Euclidian distance between two
agents i and j:
𝑅𝑖𝑗 (𝑡) = ||𝑋𝑖 (𝑡), 𝑋𝑗 (𝑡)||2
In case of Reactive Distillation Column, the active force for consideration may be taken as either of flow
rates of both the feeds, reaction rate constant, vapor phase tray efficiency, Temperature and pressure of the
column. The reboiler heat duty and Reflux Ratio can be fixed as constraints.
Gravitational and inertia masses are simply calculated by the fitness evaluation. We update the gravitational
and inertial masses by the following equations:
𝑀𝑎𝑖 = 𝑀𝑝𝑖 = 𝑀𝑗𝑖 = 𝑀𝑖 , 𝑖 = 1, 2, … … 𝑁
𝑓𝑖𝑡𝑖 (𝑡) − 𝑤𝑜𝑟𝑠𝑡(𝑡)
𝑚𝑖 (𝑡) =
,
𝑏𝑒𝑠𝑡(𝑡) − 𝑤𝑜𝑟𝑠𝑡(𝑡)
𝑚𝑖 (𝑡)
𝑀𝑖 (𝑡) = 𝑁
,
∑𝑗−1 𝑚𝑗 (𝑡)
Where fiti(t) represent the fitness value of the agent i at time t, and, worst(t) and best(t) are given for a
minimization.
The velocity of an agent is calculated as𝑣𝑖𝑑 (1) = 𝑤. 𝑣𝑖𝑑 (𝑡)𝑐1′ × 𝑟𝑎𝑛𝑑 × 𝑎𝑐𝑖𝑑 (𝑡) + 𝑐2′ × 𝑟𝑎𝑛𝑑 × (𝑔𝑏𝑒𝑠𝑡 − 𝑥𝑖𝑑 (𝑡))
Where 𝑣𝑖𝑑 (𝑡) is the velocity of agent i at iteration t in dimension 𝑑, 𝑐𝑗′ is a weighting factor, w is a weighting
function, rand is a random number between 0 and 1, 𝑎𝑐𝑖𝑑 (𝑡) is the acceleration of ith agent at iteration t in
dimension d and gbest is the best solution found so far.
At each, iteration the position of an agent is calculated as:
𝑥𝑖𝑑 (𝑡 + 1) + 𝑥𝑖𝑑 (𝑡) + 𝑣𝑖𝑑 (𝑡 + 1)
Where 𝑣𝑖𝑑 (𝑡 + 1) is the velocity of next agent and 𝑥𝑖𝑑 is the position of ith agent in dth dimension at iteration
t.
3.2 Optimization using Cuckoo Search Algorithm:
Cuckoo search (CS) is developed by Xin-she Yang and Suash Deb in 2009. The algorithm is inspired by
the reproduction strategy of cuckoos. At the most basic level, cuckoos lay their eggs in the nests of other
host birds, which may be of different species. Some cuckoos have evolved in such a way that female
parasitic cuckoos can imitate the colors and patterns of the eggs of a few chosen host species. Some host
birds can engage direct conflict with the intruding cuckoos. Cuckoo search idealized such breeding
behavior, and thus can be applied for various optimization problems.
Each egg in a nest represents a solution, and a cuckoo egg represents a new solution. The aim is to use the
new and potentially better solutions (cuckoos) to replace a not-so-good solution in the nests.
CS is based on three idealized rules:
1. Each cuckoo lays one egg at a time, and dumps its egg in a randomly chosen nest.
2. The fraction of best nests with high quality of eggs will carry over to the next generation.
3. The number of nests is fixed and there is a probability that a host can discover an alien egg. If this
happens, the host can either discard the egg or the nest and this result in building a new nest in a new
location.
Each egg in a nest represents a solution, and a cuckoo egg represents a new solution. The aim is to employ
the new and potentially better solutions (cuckoos) to replace not-so-good solutions in the nests. The
algorithm can be extended to more complicated cases in which each nest has multiple eggs representing a
set of solutions [22-23].
When generating new solutions xi (t + 1) for the ith cuckoo, the following Levy flight is performed:
𝑋𝑖 (𝑡 + 1) = 𝑋𝑖 (𝑡 ) + 𝛼 𝐿é𝑣𝑦 (𝜆)
where α > 0 is the step size which should be related to the scales of the problem of interest. The product
means entry-wise multiplications. L´evy flights essentially provide a random walk while their random steps
are drawn from L´evy distribution for large steps where α > 0 is the step size which should be related to the
scales of the problem of interest.
4.0 Results and Discussion:
Intelligent optimization algorithms under stochastic optimization such as Cuckoo Search (CS) and
PSOGSA, which is developed recently, was implemented in our case of ETBE Reactive Distillation. The
MATLAB code is written for both the algorithms and program was run for specified objective function.
The number of agents for each is selected as 50. The performance of PSOGSA is shown in fig 2 to fig 4 for
50, 100 and 200 iterations respectively. In each run the maximum product purity obtained is 99.5% which
shows best performance of hybrid PSOGSA.
Fig 2: Performance of PSOGSA using 50 Iterations
Fig 3: Performance of PSOGSA using 100 Iterations
Fig 4: Performance of PSOGSA using 200 Iterations
The performance of CS is shown in fig 5 to fig 7 using 50, 100 and 200 Iterations respectively. It can
observe from the fig below that the maximum product purity obtained for each iterations are 99.4%, 99.5%
and again 99.5% respectively. As we go on increasing the number of iterations we observed the same purity.
The maximum iterations was set as 1000.
Fig 5: Performance of CS using 50 Iterations
Fig 6: Performance of CS using 100 Iterations
Fig 7: Performance of CS using 200 Iterations
The comparison of hybrid PSOGSA and CS was also performed. This comparison is shown in fig 8 and fig
9 respectively for 50 and 100 iterations. We can clearly observe from the fig below that hybrid PSOGSA
give best performance. The results of PSOGSA and CS are tabulated in the Table 1 below for various gbest,
fbest and %Xd values. From the table also we can conclude that PSOGSA gives better performance.
Fig 8: Comparison of PSOGSA & CS using 50 Iterations
Fig 9: Comparison of PSOGSA and CS using 100 Iterations
No of Agents
50
50
50
50
50
Table 1: Results of PSOGSA and CS
No of Iterations
Hybrid PSOGSA
gbest
%Xd
50
100
99.5
100
100
99.5
200
100
99.5
500
100
99.5
1000
100
99.5
Cuckoo Search (CS)
fbest
%Xd
98.2205
98.8787
96.2850
96.9499
98.4794
99.3051
98.2570
99.1461
98.7083
99.2065
5.0 Conclusion:
Optimization using newly developed heuristic algorithm such as hybrid PSOGSA and Cuckoo Search (CS)
was implemented in case of ETBE reactive distillation for optimization of process parameters. One of the
most important process parameter is distillate purity. Maximization of distillate purity was selected as
objective function. The MATLAB code is written for algorithms. The maximum number of agents selected
was 50. The maximum number of iteration was set as 1000. After the run for 50, 100, 200, 500 and 1000
iterations we observed maximum distillate purity as 99.5 % in case of hybrid PSOGSA and 99.2% in case
of CS. After comparison we found that both hybrid PSOGSA and CS performance gives best results, but
hybrid PSOGSA gives constant results irrespective of number of iterations.
Nomenclature:
F
B
D
Xf
XB
XD
𝑣𝑗
𝑅𝑛,𝑗
𝑀𝑛
𝑘𝐹𝑛
𝑘𝐵𝑛
𝑥𝑛,𝑗
𝑉𝑛
𝐿𝑛
𝜆
∆𝐻𝑣
𝑁𝑇
𝑦𝑛,𝑗
𝑅𝑅
𝐹𝑛
𝑧𝑛,𝑗
𝑃𝑗𝑆
𝑇𝑛
P
feed flow rate
bottom flow rate
distillate flow rate
mole fraction of more volatile component in feed
mole fraction of more volatile component in bottom
mole fraction of more volatile component in distillate
: Stoichiometric coefficient of component j
: Net reaction rate of component j on nth tray
: Liquid holdup on nth tray
: Forward specific reaction rate on nth tray
: Backward specific reaction rate on nth tray
: Liquid composition of component j on nth tray
: Vapor flow rate on nth tray
: Liquid flow rate on nth tray
: Heat of reaction
: Heat of vaporization
: Total number of trays
: Vapor composition of component j on nth tray
: Reflux ratio
: Input feed flow rate on nth tray
: Feed composition of component j on nth tray
: Pure vapour pressure of component j
: Temperature at nth tray
: Total pressure
Acknowledgement:
The author is gratefully acknowledge the financial assistance provided by All India Council of Technical
Education (AICTE) New Delhi, under Research Promotion Scheme (RPS) in 2012.
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