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Modeling of ammonia conversion rate in ammonia synthesis based on a hybrid algorithm and least squares support vector regression.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2012; 7: 150–158
Published online 7 October 2010 in Wiley Online Library
(wileyonlinelibrary.com) DOI:10.1002/apj.517
Research article
Modeling of ammonia conversion rate in ammonia
synthesis based on a hybrid algorithm and least squares
support vector regression
Wei Xu, Lingbo Zhang and Xingsheng Gu*
Research Institute of Automation, East China University of Science and Technology, Shanghai, China
Received 10 March 2010; Revised 18 August 2010; Accepted 19 August 2010
ABSTRACT: In ammonia synthesis production, the ammonia conversion rate reflects how well the synthesis proceeds.
In this paper, a model, which characterizes the relationship between operational variables and ammonia conversion
rate, is established using least squares support vector regression (LSSVR). A hybrid algorithm of particle swarm
optimization and differential evolution (HPSODE) is proposed to identify the hyperparameters of LSSVR, i.e. the
regulation parameter and the width of the kernel function. HPSODE is first tested through benchmark functions and
the performance is evaluated with traditional particle swarm optimization (PSO), differential evolution (DE), and a
hybrid particle swarm optimization with differential evolution operator (DEPSO). It is then applied to the modeling of
ammonia synthesis process. Results using other modeling methods [back propagation neural network (BPNN), LSSVR,
PSO–LSSVR, and DE–LSSVR] are presented for comparison purpose. The proposed HPSODE–LSSVR modeling
shows good feasibility of the algorithm and reliability of global convergence.  2010 Curtin University of Technology
and John Wiley & Sons, Ltd.
KEYWORDS: ammonia synthesis; ammonia conversion rate; particle swarm optimization; differential evolution; least
squares support vector regression; operational parameter
INTRODUCTION
Ammonia synthesis system is an important chemical
process, in which the ammonia synthesis reactor is the
key device. It is necessary to study the reactor so as
to know the effects of operational variables upon the
reactor performance. Mathematical models for simulation and optimization purpose have been established
by many researchers.[1 – 5] Based on the mathematical
models, optimization of ammonia synthesis reactors was
promoted. Patnaik et al .[6] applied Powell’s direct optimization method for both parameter identification and
steady-state optimization of a tubular reactor used in
ammonia synthesis. Reddy and Husain[7] formulated the
synthesis loop of an ammonia plant. Effects of operational parameters on ammonia production rate, fractional hydrogen conversion, and gross profitability were
studied and the optimum value of H2 /N2 ratio in the
recycle gas was found. Mansson and Andresen[8] used
a Temkin–Pyzhev rate equation to achieve an approximate temperature profile of the ammonia synthesis
*Correspondence to: Xingsheng Gu, Research Institute of Automation, East China University of Science and Technology, Shanghai,
China, 200237. E-mail: xsgu@ecust.edu.cn
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Curtin University is a trademark of Curtin University of Technology
converter and calculated the optimal temperature profile along the reactor. A similar optimization study of
an ammonia synthesis reactor with three adiabatic beds
has also been conducted by Elnashaie et al .[9] . Babu
et al .[10] presented the simulation of an autothermal
ammonia synthesis reactor using Numerical Algorithms
Group (NAG) subroutine (D02EJF) in MATLAB, and
combined the quasi-Newton method for optimization
of the length of reactor for different top temperatures.
Sadeghi and Kavianiboroujeni[11] utilized two models
to evaluate the behavior of an industrial ammonia synthesis reactor, and genetic algorithm, thereafter, was
employed to optimize the reactor performance in varying its quench flows.
Particle swarm optimization (PSO), which is a new
intelligent algorithm, was proposed by Eberhart and
Kennedy[12,13] in 1995. It was motivated by the social
behavior of bird flocking and fish schooling. As a swarm
intelligence algorithm, PSO provides reliable performance and satisfactory results on a lot of problems.
Because of such properties as globally exploring ability
and convergence accuracy, PSO has drawn much attention from researchers and scholars around the world.
However, PSO cannot escape from the premature convergence completely, and is easily stuck to the local
Asia-Pacific Journal of Chemical Engineering
MODELING OF AMMONIA CONVERSION RATE IN AMMONIA SYNTHESIS
minima. In order to improve its performance, a large
number of developments have been carried out. Shi
and Eberhart[14,15] found a significant improvement on
PSO with the linearly decreasing inertia weight over
the generations. Instead of inertia weight, Clerc and
Kennedy[16] introduced a constriction factor to enhance
the convergence of basic PSO. Hybridizing PSO with
other methods is also an interesting research trend.
Differential evolution (DE), which was developed
by Price and Storn,[17] is a population-based algorithm
like PSO. Among DEs, the advantages are the simple
structure, ease of use, speed, and robustness. Already,
DE has been successfully applied for solving several
complex problems[18 – 21] and attracted growing concerns in recent years. Consequently, some authors considered the integration of PSO and DE. In Refs [22]
and[23], the authors proposed a hybrid DE and PSO,
i.e. DEPSO, thus eliminating the particles from falling
into local minima. This algorithm was implemented
based on the PSO strategy in every odd iteration
and differential operation in every even iteration. Tim
Hendtlass[24] presented a swarm differential evolution
algorithm (SDEA), in which individuals following the
PSO strategy continuously were moved to discrete
points, each better than the last. Ben Niu and Li Li[25]
presented a novel algorithm, in which PSO and DE were
executed in parallel to enhance frequent information
sharing between populations. In Ref [26], a differential
vector operator was imported into the velocity updating
scheme of PSO.
In 1995, Cortes and Vapnik[27] put forward the
support vector machine (SVM). In contrast to some
traditional neural networks which are based on the
principle of empirical risk minimization, SVM was
inspired by the statistical learning theory based on
the principle of structural risk minimization. It comprises support vector machine for classification (SVC)
and support vector machine for regression (SVR).
SVM has strong capabilities of small sample learning and generalization, yet is time consuming and
demands huge space. To overcome these shortcomings, least squares SVM (LSSVM) was developed
by Suykens et al .[28,29] in 1999. Furthermore, least
squares SVR (LSSVR) has been successfully used
for systems modeling, especially for nonlinear systems
modeling.[30 – 34]
In this paper, a model of ammonia conversion rate
in ammonia synthesis is established using LSSVR
for the description of the relationship between the
operational variables and ammonia conversion rate.
Model hyperparameters are identified by a hybrid
algorithm of PSO and DE (HPSODE), which contains the steps of population decomposition, PSO and
DE computation in parallel, and population recomposition. HPSODE is first tested by five benchmarks
in comparison with the performance from traditional
PSO, DE, and DEPSO. The HPSODE-LSSVR is then
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
applied to the modeling of ammonia synthesis process. Simulations have also been run using other
modeling methods [back propagation neural network
(BPNN), LSSVR, PSO–LSSVR, and DE–LSSVR] for
comparison purpose. The results show better generalization performance for our proposed method. This
HPSODE–LSSVR modeling is the first step toward
the operational optimization for providing optimum
operational parameters so as to increase ammonia
production.
MODELING USING HPSODE–LSSVR
LSSVR is a modeling method based on statistical
learning theory. The performance of LSSVR depends
on its hyperparameters, i.e. the regulation parameter
and the width of the kernel function, so the main
issue for users trying to apply LSSVR is how to set
these parameters. In this section, the hybrid algorithm
HPSODE is proposed to offer a proper setting of
the parameters for ensuring the good generalization of
LSSVR.
Least squares support vector regression
Assume that training samples are denoted as [(xk , yk ),
k = 1, . . . , m]. The purpose of applying LSSVR is to
find out a nonlinear mapping φ(•) which maps the data
space to a high dimension feature space and construct
an optimal linear regression function. Using equality
constraints instead of inequality, the regression problem
could be equivalent to the optimization problem as
below:
m
1 2
1 T
ξk
min J (ω, ξ ) = ω ω + C
2
2
(1)
k =1
s.t. yk = ωT φ(xk ) + b + ξk
(2)
where J (ω, ξ ) is structural risk, C is regulation parameter, and ξk is the error between the target output and
estimated value of sample k . ω and b are the weight
and bias, respectively.
The Lagrange function for this problem is established
as follows:
L(ω, b, ξ, α) =
m
1 2
1 T
ω ω+ C
ξk
2
2
k =1
−
m
αk [ωT φ(xk ) + b + ξk − yk ]
(3)
k =1
where αk is the Lagrange coefficient. Based on the optimality conditions ∂L/∂ω = 0, ∂L/∂b = 0, ∂L/∂α = 0,
Asia-Pac. J. Chem. Eng. 2012; 7: 150–158
DOI: 10.1002/apj
151
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W. XU, L. ZHANG AND X. GU
Asia-Pacific Journal of Chemical Engineering
and ∂L/∂ξ = 0, the optimal solution could be calculated
according to Eqn (4).
0
IT
b
0
=
(4)
−1
I K +C I
α
y
where I is a unit matrix and K (xk , x ) = φ(xk )T φ(x ) is
the Mercer kernel function.
Therefore, the resulting LSSVM model is given by
f (x ) =
m
αk K (x , xk ) + b
(5)
k =1
In Eqn (5), the suitable kernel function should be
chosen. There are several kinds of kernel function,
such as polynomial, hyperbolic tangent, and radial
basis function (RBF). Some literatures proved that RBF
kernel function has strong generalization, so we adopt
RBF in this paper, which corresponds to
K (x , xk ) = exp − x − xk 2 2σ 2
(6)
where σ is the width of the kernel function.
A hybrid algorithm of PSO and DE
The proposed hybrid algorithm, called HPSODE, includes four major steps: initialization, population
decomposition, intelligent computations (PSO computation and DE computation), and population recomposition. This is explained in the following.
Initialization
In HPSODE, N individuals are generated randomly in
the bounds to form an initial population. In addition,
the parameters of PSO and DE, such as acceleration
constants, mutation factor, and crossover factor, etc,
should be set by users according to conventional
settings.
Population decomposition
After being evaluated by the fitness function, all individuals are sorted according to the fitness values. The population is decomposed into PSO subpopulation and DE
subpopulation. The two subpopulations possess unequal
number of particles or individuals. The objective of
introducing DE computation is to maintain the diversity of the population in the evolution but never to
overpower PSO, so the DE subpopulation is assigned
no more than 50% of the total individuals under study.
This percentage pd , which the individuals in the DE
subpopulation account for, should be given before the
search. Each of the particles composing the PSO subpopulation is chosen from the top half, best-performing
part, by the selection rate ps , or from the lower half
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
part. The rest of the individuals in the population are
assigned to the DE subpopulation. It is hoped that most
of the better performing individuals are distributed into
the PSO subpopulation, so ps is set to be more than 0.5.
PSO computation
The PSO subpopulation is regarded as a swarm consisting of many quasi-elite particles. By using those
good-performing quasi-elites, new particles in the next
iteration will achieve better performance than those
generated by ordinary individuals; meanwhile, the badperforming quasi-elites in the swarm could effectively
get rid of premature convergence to local optima. In
this paper, instead of using traditional PSO, we adopt an
improved algorithm of HPSO–TVAC.[35] The updating
of velocity and position are formulated as follows:
t
vidt+1 = c1 r1 (pidt − xidt ) + c2 r2 (pgd
− xidt )
(7)
xidt+1 = xidt + vidt+1
(8)
where i = 1, 2, . . . , N , and N is the size of the swarm.
d represents the d th dimension of the problem space. c1
and c2 are two positive acceleration constants. r1 and r2
are two uniformly distributed random coefficients in the
range of [0, 1]. p t i and p t g represent the personal best
experience and the global best experience in the swarm
in iteration t, respectively. According to Eqn (7), the
updating of velocity ignores the influence of current
velocity. This improvement is suitable for the particles
in the PSO subpopulation. That is because those particles which are produced by DE computation in the
previous iteration do not possess the characteristics of
velocity. Based on this improved PSO algorithm, there
is no need to focus on where current particles come
from. c1 and c2 vary when the iteration proceeds, formulated as Eqn (9) and (10).
iter
+ c1i
MaxIt
iter
+ c2i
c2 = (c2f − c2i )
MaxIt
c1 = (c1f − c1i )
(9)
(10)
where c1i and c2i are the initial values of c1 and c2 ,
respectively; c1f and c2f are the final values. iter is the
current iteration number and MaxIt is the maximum
number of allowable iterations.
The pseudocode of updating particles in the PSO
subpopulation is described as follows:
For particle i
For d th dimension
calculate the new velocity as Eqn (7)
vid = c1 r1 (pid − xid ) + c2 r2 (pgd − xid )
If (vid = 0)
If rand1 <0.5
vid = rand2 × Vrein
Else
Asia-Pac. J. Chem. Eng. 2012; 7: 150–158
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
MODELING OF AMMONIA CONVERSION RATE IN AMMONIA SYNTHESIS
vid = −rand3 × Vrein
End If
End If
update the position as Eqn (8)
Increase d
update the personal best experience
Increase i
In the pseudocode, rand1, rand2, and rand3 are
separately generated random numbers uniformly distributed in the range of [0, 1]. Vrein is the reinitialization velocity which decreases linearly along with the
evolution.
DE computation
As aforementioned descriptions, the performance of the
individuals in the DE subpopulation is in general not
better than those particles in the PSO subpopulation.
These individuals are employed to utilize the differential
information and implement the genetic operations of DE
so as to yield better results. In the DE subpopulation,
the inferior individuals in the majority participate in
the evolution for making full use of the environmental
knowledge and increasing the diversity of the whole
population. The involvement of the superior ones would
prevent the subpopulation from being confined to the
areas where inferior individuals locate, and control
the progress made previously from being ruined. If
the generated individual is adopted as a particle in
the PSO subpopulation in the next iteration, its own
personal best experience so far will represent the current
position.
Population recomposition
After completing the updating, both the PSO and DE
subpopulations are combined to recompose a new population for the next iteration. The global best experience
till now is calculated, and the learning process would
stop when the terminal criterions are satisfied.
The pseudocode of HPSODE is presented as below:
Begin
iter = 0;
Initialize the population;
Repeat
Evaluate the fitness of individuals in the
population;
Decompose the population into PSO and DE
subpopulations;
Implement the PSO computation in the PSO
subpopulation;
Implement the DE computation in the DE
subpopulation;
Combine the updated PSO subpopulation and
DE subpopulation to recompose a new
population;
iter = iter +1;
Until termination condition achieved;
End
TEST OF HPSODE USING BENCHMARK
FUNCTIONS
To verify the performance of HPSODE, five benchmark
functions are used. The functions and their parameters
are summarized in Table 1.
Both the Sphere and the Rosenbrock function are
unimodal, each of which has only one peak value in
the range. Sphere is a simple function and it is easy
to reach the optimum. For the Rosenbrock function, the
global minimum locates in a long, narrow, and parabolashaped area. The feature results in that the minimum
would rarely be found. Therefore, the Rosenbrock
function is widely employed to make the evaluation
on the performance of algorithms. The rest of the
benchmarks are multimodal functions with numerous
local minima. It is a tough task to explore the global
minimum of each of the functions. Therefore, all these
benchmarks are competent to verify the capability
of a variety of algorithms, including the proposed
HPSODE.
HPSODE would be compared with three other algorithms, i.e. traditional PSO,[14] traditional DE,[17] and
Table 1. Benchmark functions and their parameters.
Function
Sphere
Rosenbrock
Ackley
Rastrigin
Griewank
f1 (x ) =
f2 (x ) =
n
i =1
n
xi2
(100 × (xi +1 − xi2 )2 + (1 − xi )2 )
n
n
1
2
f3 (x ) = 20 + e − 20 × exp −0.2 × n
xi − exp n1
cos(2π xi )
i =1
i =1
n 2
xi − 10 × cos(2π xi ) + 10
f4 (x ) =
i =1
n
n
xi + 1
1 x 2 − cos √
f5 (x ) = 4000
i
i
i =1
i =1
Min.
Range
Goal
0
[−100,100]
0.01
0
[−30,30]
100
0
[−30,30]
10−5
0
[−5.12,5.12]
100
0
[−60,60]
0.1
i =1
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2012; 7: 150–158
DOI: 10.1002/apj
153
W. XU, L. ZHANG AND X. GU
Asia-Pacific Journal of Chemical Engineering
Table 2. Parameter setting for algorithms.
20
Algorithms
N
w
c1
c2
F
CR
PSO[14]
DE[17]
DEPSO[22]
HPSODE
40
40
40
40
0.9–0.4
–
0.4
–
2.0
–
2.0
2.5–0.5
2.0
–
2.0
0.5–2.5
–
0.8
–
0.8
–
0.4
–
0.4
DEPSO.[22] The swarm size N = 40. Each of these
algorithms is iterated for 2000 generations on 30
dimensions. All the experiments are run 50 times. The
parameters of all algorithms are illustrated in Table 2.
Table 3 shows the performance of these algorithms.
Best, Worst, Mean, Std and Success represent the best
minimum, worst minimum, mean minimum in 50 runs,
standard deviation and success rate of having reached
the goal, respectively. The average convergence curves
of algorithms on five benchmark functions are shown
in Figs 1–5.
It can be seen from the results that the Sphere function
was easily optimized by all algorithms, and HPSODE
behaved better than others. For the Rosenbrock function, HPSODE could completely reach the goal in the
whole experiments, and was always excellent in the
evolution. In the test of the Achley function, HPSODE
exceeded other algorithms after about 700 iterations and
PSO
DE
DEPSO
HPSODE
10
0
Fitness (log)
154
-10
-20
-30
-40
-50
0
200 400 600 800 1000 1200 1400 1600 1800 2000
Iteration
Figure 1. The convergence curves on Sphere. This figure
is available in colour online at www.apjChemEng.com.
was successful in reaching the goal in each of 50 trials, although not very well at the beginning. For the
Rastrigin and Griewank functions, the performances of
HPSODE were better than other algorithms as shown
in Figs 4 and 5. It is therefore concluded that HPSODE
exhibits its strong capability on those typical optimizations, especially on the difficult high-dimensional
problems.
Table 3. Results of benchmark functions using four algorithms.
Function
Items
Sphere (f1 )
Best
Worst
Mean
Std
Success
Best
Worst
Mean
Std
Success
Best
Worst
Mean
Std
Success
Best
Worst
Mean
Std
Success
Best
Worst
Mean
Std
Success
Rosenbrock (f2 )
–
–
–
–
Ackley (f3 )
–
–
–
–
Rastrigin (f4 )
–
–
–
–
Griewank (f5 )
–
–
–
–
PSO[14]
DE[17]
DEPSO[22]
HPSODE
2.7939 × 10−13
7.3313 × 10−10
6.8196 × 10−11
1.4063 × 10−10
1.00
5.3584
252.0733
63.5232
57.1822
0.90
2.0669 × 10−7
3.7673 × 10−5
4.6643 × 10−6
7.2921 × 10−6
0.90
26.8639
63.6773
38.2722
8.8286
1.00
3.0753 × 10−14
6.3899 × 10−2
1.4512 × 10−2
1.6385 × 10−2
1.00
2.3812 × 10−7
1.3364 × 10−6
5.9194 × 10−7
2.3226 × 10−7
1.00
26.8450
105.9900
40.6881
21.0129
0.98
1.2191 × 10−4
2.7380 × 10−4
1.9480 × 10−4
4.0766 × 10−5
0
115.1400
144.1100
129.1416
6.7774
1.00
8.1262 × 10−6
1.7724 × 10−1
1.8033 × 10−2
3.9425 × 10−2
1.00
1.7036 × 10−20
7.1637 × 10−16
5.5050 × 10−17
1.2074 × 10−16
1.00
4.1653
221.2061
48.9169
41.6557
0.94
9.0662 × 10−11
1.3404
7.6720 × 10−2
3.0442 × 10−1
0.94
20.8941
75.6167
45.8836
13.8085
1.00
0
6.1404 × 10−2
1.2845 × 10−2
1.3697 × 10−2
1.00
1.2905 × 10−21
3.2091 × 10−18
5.5115 × 10−19
7.8365 × 10−19
1.00
1.1142
81.4919
34.8060
25.8507
1.00
3.6259 × 10−11
8.6319 × 10−10
3.3556 × 10−10
2.2994 × 10−10
1.00
4.9748
19.8990
11.7802
2.9671
1.00
0
5.9050 × 10−2
7.6732 × 10−3
1.3080 × 10−2
1.00
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2012; 7: 150–158
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
MODELING OF AMMONIA CONVERSION RATE IN AMMONIA SYNTHESIS
20
6.5
PSO
DE
DEPSO
HPSODE
18
16
5.5
5
Fitness(log)
Fitness (log)
14
12
10
4
3.5
6
3
4
2.5
2
0
200 400 600 800 1000 1200 1400 1600 1800 2000
Iteration
The convergence curves on Rosenbrock. This
figure is available in colour online at www.apjChemEng.com.
Figure 2.
200 400 600 800 1000 1200 1400 1600 1800 2000
Iteration
2
1
-5
0
Fitness(log)
0
-10
-15
PSO
DE
DEPSO
HPSODE
-20
-25
0
Figure 4. The convergence curves on Rastrigin. This figure
is available in colour online at www.apjChemEng.com.
5
Fitness (log)
4.5
8
2
PSO
DE
DEPSO
HPSODE
6
0
200 400 600 800 1000 1200 1400 1600 1800 2000
Iteration
Figure 3. The convergence curves on Ackley. This figure is
available in colour online at www.apjChemEng.com.
APPLICATION OF HPSODE–LSSVR
TO AMMONIA SYNTHESIS PROCESS
The HPSODE–LSSVR has been applied to structuring the connection between operational variables and
ammonia conversion rate in a fertilizer plant in Shandong province. The process is briefly described below.
The ammonia synthesis system shown in Fig. 6 is
usually applied in small and medium fertilizers in
China. The synthesis gas mixed with the gas from
the circulator outlet flows through the condenser and
ammonia cooler, and is cooled to separate a part of
the liquid ammonia. And then the gas is recycled
to the annular space between the catalyst basket and
outer shell in the ammonia converter. As the ammonia
synthesis reaction is exothermic, a large quantity of
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
PSO
DE
DEPSO
HPSODE
-1
-2
-3
-4
-5
0
200 400 600 800 1000 1200 1400 1600 1800 2000
Iteration
Figure 5. The convergence curves on Griewank. This figure
is available in colour online at www.apjChemEng.com.
heat is generated, which is transferred to the gas
following through the annular space. After that, the gas,
which contains hydrogen, nitrogen, ammonia, methane,
and argon, is preheated by the counter-current flowing
product gas. Thereafter, ammonia is produced based
on the reaction between hydrogen and nitrogen in the
presence of the catalyst. The product gas passes through
the waste heat boiler and the preheater where it is cooled
to a low temperature. Through further cooling, liquid
ammonia is separated from the gas in the ammonia
separator. A small stream of purge gas is necessary to
maintain inerts (methane and argon) to a low level.
The converter is the key unit in the ammonia synthesis loop. The operations in the converter directly
influence the consumption of feed gas and power as well
Asia-Pac. J. Chem. Eng. 2012; 7: 150–158
DOI: 10.1002/apj
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W. XU, L. ZHANG AND X. GU
Asia-Pacific Journal of Chemical Engineering
Condenser
T1
T2
T3
T4
Ammonia
converter
Ammonia
cooler
Purge
gas
Ammonia
Ammonia
Waste heat
boiler
Preheater
Water
cooler
Ammonia
separator
Synthesis
gas
Circulator
Figure 6. Ammonia synthesis system.
as the working of other devices. Currently, the average energy consumption per ton of ammonia product in
China is about 1900 kg of standard coal, which is much
higher than the advanced standard of about 1570 kg
around the world. Therefore, it has much potential to
economize on energy. Without changing the devices,
operational optimization is a reliable technique in the
ammonia synthesis loop for production improvement
and energy reduction, which in turn achieves capital and
operational cost savings. For the purpose of operational
optimization, priority must be given to the establishment of the relationship description between operational
parameters and product quality. The product quality
used here is the ammonia conversion rate, which is the
ratio of the mixture of nitrogen and hydrogen that has
been converted to ammonia to that in the feed gas. In
the ammonia synthesis system, the parameters, namely,
inerts concentration in the recycle gas, ammonia concentration at the inlet of the catalyst bed, the ratio of
hydrogen to nitrogen in the recycle gas, and the hotspot temperatures in the catalyst bed have pronounced
effects on the ammonia conversion rate.
In a real-world fertilizer plant, the actual concentration of inerts exceeds the maximum measurable value
of the online analyzer; thus the displayed value is unreliable. What is worse, the analyzer which calculates
the inlet ammonia concentration was broken. Therefore,
the following operations are performed to minimize
the effects of inerts concentration and inlet ammonia
concentration on the ammonia conversion rate: (1) the
flowrate of purge gas is set to be constant so as to avoid
large variation of inerts concentration; (2) the temperature of the gas from the ammonia cooler outlet is kept
within a small range for ensuring that the inlet ammonia concentration is invariable. Through discussion with
experienced engineers, five of the operational parameters, which are the hydrogen concentration in the recycle
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
gas and four hot-spot temperatures in the four-stage catalyst bed, are considered to be the key variables to the
ammonia conversion rate.
Samples of five operational variables are collected
from distributed control system (DCS) while those of
ammonia conversion rate are obtained through manual
analysis. After being screened for incompleteness and
evident inaccuracies, 281 groups of modeling data are
gathered totally, in which 201 groups are training data
for identifying the hyperparameters of the LSSVR and
the remaining are to verify the generalization as testing
data.
RESULTS AND DISCUSSION
The sample data are exposed to the HPSODE–LSSVR.
As shown in Fig. 7, the training results fit the practical
data well. After training, the regulation parameter and
the width of the kernel function are identified. And then,
the testing results are illustrated in Fig. 8 as a good prediction performance of the HPSODE–LSSVR on testing data. The results indicate that the HPSODE–LSSVR
can predict the ammonia conversion rate accurately and
provide good generalization capability.
BPNN, LSSVR, PSO-based LSSVR (PSO–LSSVR)
and DE-based LSSVR (DE–LSSVR) are also employed
to describe the relationship, respectively. BPNN is a
5–10–1 three-layer network using BP algorithm, and
LSSVR adopts grid search and crossvalidation to gain
the regulation parameter and the width of the kernel
function. The parameter settings of PSO and DE used in
PSO–LSSVR and DE–LSSVR are the same as those in
the benchmark tests. The comparisons of the five modeling methods are showed in Table 4. In comparison with
the four other methods on testing data, the relative error
(RE) when using the HPSODE–LSSVR is reduced by
23.0, 32.1, 14.3, and 8.7%, respectively; the absolute
Asia-Pac. J. Chem. Eng. 2012; 7: 150–158
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
MODELING OF AMMONIA CONVERSION RATE IN AMMONIA SYNTHESIS
by 33.1, 47.7, 30.5, and 21.0%, respectively. The outcomes indicate that the proposed HPSODE–LSSVR is
superior to the other methods and the model calculated
values have better agreement with the plant data.
Conversion Rate of Ammonia(%)
21.5
21
20.5
CONCLUSIONS
20
19.5
Analyzed Results
Training Results
of HPSODE-LSSVR
19
18.5
0
20
40
60
80 100 120 140 160 180 200
Sample Number
The analyzed results and training results of
HPSODE–LSSVR. This figure is available in colour online
at www.apjChemEng.com.
Figure 7.
Conversion Rate of Ammonia (%)
21.5
21
20.5
20
19.5
19
Analyzed Results
Testing Results
of HPSODE-LSSVR
0
10
20
30
40
50
Sample Number
60
70
80
Modeling for the relationship between the key operational parameters and ammonia conversion rate in
ammonia synthesis has been proposed. LSSVR is
employed to set up the model framework while a
hybrid algorithm HPSODE is used to identify the hyperparameters (the regulation parameter and the width
of the kernel function) of LSSVR. The hybrid algorithm divides the population into two subpopulations,
which separately implement the computations based on
an improved PSO algorithm and DE algorithm. The
improved PSO algorithm ignores the influence of current velocity on velocity updating, which ensures that
individuals can be distributed to any one of the subpopulations regardless of which subpopulation they come
from. Subsequently, the subpopulations are recomposed
to form a new population for the next iteration. The proposed HPSODE algorithm was verified for the optimization of benchmark functions. The HPSODE–LSSVR
method has been used to describe the effects of some
key operational parameters on ammonia conversion rate
in a real-world ammonia synthesis process. It is indicated that HPSODE–LSSVR performed better than four
other modeling methods and can yield good generalization in ammonia synthesis process. Due to the
availability, the HPSODE–LSSVR method will lay the
foundation for obtaining the optimal operational parameters, thus fulfilling the maximum benefit of ammonia
synthesis production in our future work.
The analyzed results and testing results of
HPSODE–LSSVR. This figure is available in colour online
at www.apjChemEng.com.
Acknowledgements
error (ABE) is reduced by 23.0, 31.6, 14.3, and 8.6%,
respectively; the mean square error (MSE) is reduced
We are very grateful to the editor and anonymous
reviewers for their valuable comments and suggestions
to help improve our paper. This work is supported
Figure 8.
Table 4. The comparisons of five modeling methods.
Training error
Modeling
methods
BPNN
LSSVR
PSO–LSSVR
DE–LSSVR
HPSODE–LSSVR
RE
ABE
−3
1.7759 × 10
3.5007 × 10−3
3.1474 × 10−3
2.9176 × 10−3
2.6233 × 10−3
0.0370
0.0718
0.0649
0.0602
0.0543
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Testing error
MSE
0.0514
0.1097
0.0907
0.0812
0.0724
RE
−3
4.6049 × 10
5.2154 × 10−3
4.1367 × 10−3
3.8802 × 10−3
3.5437 × 10−3
ABE
MSE
0.0953
0.1073
0.0856
0.0803
0.0734
0.1336
0.1708
0.1286
0.1131
0.0894
Asia-Pac. J. Chem. Eng. 2012; 7: 150–158
DOI: 10.1002/apj
157
158
W. XU, L. ZHANG AND X. GU
by National High Technology Research and Development Program of China (863 Program) (Grant No.
2009AA04Z141), National Natural Science Foundation
of China (Grant No. 60774078), Shanghai Commission
of Science and Technology (Grant No. 08JC1408200),
and Shanghai Leading Academic Discipline Project
(Grant No. B504).
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