A steadystate mathematical model is built in order to represent plant behavior under stationary operating conditions. A novel modeling using LSSVR based on Cultural Differential Evolution with Ant Search is proposed. LSSVM is adopted to establish the model of the net value of ammonia. The modeling method has fast convergence speed and good global adaptability for identification of the ammonia synthesis process. The LSSVR model was established using the abovementioned method. Simulation results verify the validity of the method.
Ammonia is one of the important chemicals that has innumerable uses in a wide range of areas, that is, explosive materials, pharmaceuticals, polymers, acids and coolers, particularly in synthetic fertilizers. It is produced worldwide on a large scale with capacities extending to about 159 million tons at 2010. Generally, the average energy consumption of ammonia production per ton is 1900 KG of standard coal in China, which is much higher than the advanced standard of 1570 KG around the world. At the same time, the haze and particulate matter 2.5 has been serious exceeded in big cities in China at recent years, and one of the important reasons is the emission of coal chemical factories. Thus, an economic potential exists in energy consumption of the ammonia synthesis as prices of energy rise and reduce the ammonia synthesis pollution to protect the environment. Ammonia synthesis process has the characteristics of nonlinearity, strong coupling, large timedelay and great inertia load, and so forth. Steadystate operationoptimization can be a reliable technique for output improvement and energy reduction without changing any devices.
The optimization of ammonia synthesis process highly relies on the accurate system model. To establish an appropriate mathematical model of ammonia synthesis process is a principal problem of operation optimization. It has received considerable attention since last century. Heterogeneous simulation models imitating different types of ammonia synthesis reactors have been developed for design, optimization and control [
The above study indicated that both the productive capacity and the stability of the ammonia reactor are influenced by the cold quench and the feed temperature significantly. Babu and Angira [
In order to achieve the required accuracy of the model, some researches focus on the novel modeling methods combining some heuristic methods such as ANN (Artificial Neural Network), LSSVM (Least Squares Support Vector Machine) with Evolutionary Algorithm, for example, genetic algorithm, ant colony optimization (ACO), particle swarm optimization (PSO), differential evolution (DE), and so forth. DE is one of the most popular algorithms for this problem and has been applied in many fields. Sacco and Hendersonb [
To describe the relationship between net value of ammonia in ammonia synthesis reactor and the key operational parameters, least squares support vector machine is employed to build the structure of the relationship model, in which a novel algorithm called CDEAS is proposed to identify the parameters. The experiment results showed that the proposed CDEASLSSVM optimizing model is very effective of being used to obtain the optimal operational parameters of ammonia synthesis converter.
The remaining of the paper is organized as follows. Section
A normal ammonia production flow chart includes the synthesis gas production, purification, gas compression, and ammonia synthesis. Ammonia synthesis loop is one of the most critical units in the entire process. The system has been realized by LuHua Inc., a medium fertilizers factory of YanKuang Group, China.
Figure
Ammonia synthesis system.
The reaction is limited by the unfavorable position of the chemical equilibrium and by the low activity of the promoted iron catalysts with high pressure and temperature [
Figure
The ammonia synthesis unit.
Evolutionary Algorithms, which are inspired by the evolution of species, have been adopted to solve a wide range of optimization problems successfully in different fields. The primary advantage of Evolutionary Algorithms is that they just require the objective function values, while properties such as differentiability and continuity are not necessary [
Differential evolution, proposed by Storn and Price, is a fast and simple population based stochastic search technique [
As an effective and powerful random optimization method, DE has been successfully used to solve real world problems in diverse fields both unconstrained and constrained optimization problems.
As we mentioned in Section
In this paper, based on the theory of Cultural Algorithm and Ant Colony Optimization (ACO), an improved Cultural Differential Algorithm incorporation with Ant Colony Search is presented. In order to accelerate searching out the global solution, the Ant Colony Search is used to search the optimal combination of
The framework of CDEAS algorithm.
The population space is divided into two parts: subpopulation 1 and subpopulation 2. The two subpopulations contain equal number of the individuals.
In subpopulation 1, the individual is set as ant at each generation.
Then, the pheromone trail
In order to prevent the ants from being limited to one ant path and improve the possibility of choosing other paths considerably, the probability of each ant chooses
Relationship between pheromone and ant paths of
Figure
In subpopulation 2, the individual is set as ant at each generation. Mutation strategies which are listed at (
Then, the pheromone trail
In order to prevent the ants from being limited to one ant path and improve the possibility of choosing other paths considerably, the probability of each ant choosing
Figure
Relationship between and ant paths of mutation strategy.
In our approach, the belief space is divided into two knowledge sources, situational knowledge and normative knowledge.
Situational knowledge consists of the global best exemplar
The normative knowledge contains the intervals that decide the individuals of population space where to move.
Acceptance function controls the amount of good individuals which impact on the update of belief space [
In the CDEAS, situational knowledge and normative knowledge are involved to influence each individual in the population space, and then population space is updated.
The individuals in population space are updated in the following equation:
After
The procedure of CDEAS is proposed as follows.
Initialize the population spaces and the belief spaces; the population space is divided into subpopulation 1 and subpopulation 2.
Evaluate each individual’s fitness.
To find the proper
According to acceptance function, choose good individuals from subpopulation 1 and subpopulation 2, and then update the normative knowledge and situational knowledge.
Adopt the normative knowledge and situational knowledge to influence each individual in population space through the influence functions, and generate two corresponding subpopulations.
Select individuals from subpopulation 1 and subpopulation 2, and update the belief spaces including the two knowledge sources for the next generation.
If the algorithm reaches the given times, exchange the knowledge of
If the stop criteria are achieved, terminate the iteration; otherwise, go back to Step
The proposed CDEAS algorithm is compared with original DE algorithm. To get the average performance of the CDEAS algorithm 30 runs on each problem instance were performed and the solution quality was averaged. The parameters of CDEAS and original DE algorithm are set as follows: the maximum evolution generation is 2000; the size of the population is 50; for original DE algorithm
To illustrate the effectiveness and performance of CDEAS algorithm for optimization problems, a set of 18 representative benchmark functions which were listed in the appendix were employed to evaluate them in comparison with original DE. The test problems are heterogeneous, nonlinear, and numerical benchmark functions and the global optimum for
The comparison results of the CDEAS algorithm and original DE algorithm.
Original DE  CDEAS  

Sphere function 

Best 


Worst 


Mean 


Std. 


Success rate (%)  100 

Times (s) 

14.6017 
Shifted sphere function 

Best  0 

Worst 


Mean 


Std. 


Success rate (%)  100 

Times (s) 

18.1117 
Schwefel’s Problem 1.2 

Best 


Worst 


Mean 


Std. 


Success rate (%)  100 

Times (s) 

24.1178 
Shifted Schwefel’s Problem 1.2 

Best  0  0 
Worst 


Mean 


Std. 


Success rate (%)  100 

Times (s) 

27.7058 
Rosenbrock’s function 

Best  13.0060 

Worst  166.1159 

Mean  70.9399 

Std.  40.0052 

Success rate (%)  86.67 

Times (s) 

16.7233 
Schwefel’s Problem 1.2 with noise in fitness 

Best 


Worst 


Mean 


Std. 


Success rate (%)  100 

Times (s) 

24.2426 
Shifted Schwefel’s Problem 1.2 with noise in fitness 

Best 


Worst 


Mean 


Std. 


Success rate (%)  100 

Times (s) 

28.5638 
Ackley’s function 

Best 


Worst 

1.3404 
Mean 

0.1763 
Std. 

0.4068 
Success rate (%) 

83.33 
Times (s) 

20.9353 
Shifted Ackley’s function 

Best 


Worst 


Mean 

0.0620 
Std. 

0.2362 
Success rate (%) 

93.33 
Times (s) 

21.6841 
Griewank’s function 

Best  0 

Worst  0.0367 

Mean 

0.0054 
Std. 

0.0076 
Success rate (%) 

56.67 
Times (s) 

20.7793 
Shifted Griewank’s function 

Best 

0 
Worst 

0.0343 
Mean 

0.0060 
Std  0.0089 

Success rate (%) 

76.67 
Times (s) 

22.8541 
Rastrigin’s function 

Best  8.1540 

Worst  35.5878 

Mean  20.3594 

Std.  6.3072 

Success rate (%)  3.33 

Times (s) 

22.3237 
Shifted Rastrigin’s function 

Best  5.9725 

Worst  36.9923 

Mean  19.4719 

Std.  8.9164 

Success rate (%)  16.67 

Times (s) 

23.8838 
Noncontiguous Rastrigin’s function 

Best  20.7617 

Worst  29.9112 

Mean  25.4556 

Std.  2.9078 

Success rate (%)  0 

Times (s) 

25.5374 
Shifted noncontiguous Rastrigin’s function 

Best  0 

Worst  16 

Mean  6.7666 

Std.  3.4509 

Success rate (%)  40 

Times (s) 

25.9430 
Schwefel’s function 

Best 

236.8770 
Worst 

1362.0521 
Mean 

676.4166 
Std. 

324.2317 
Success rate (%) 

40 
Times (s) 

19.0009 
Schwefel’s Problem 2.21 

Best 

0.3254 
Worst 

4.7086 
Mean 

1.9849 
Std. 

1.16418 
Success rate (%) 

23.33 
Times (s) 

19.2505 
Schwefel’s Problem 2.22 

Best 


Worst 


Mean 


Std. 


Success rate (%)  100 

Times (s) 

20.8573 
From simulation results of Table
The convergence figures of CDEAS comparing with original DE for 18 instances are listed as Figure
Convergence figure of CDEAS comparing with original DE for
From Figure
All these comparisons of CDEAS with original DE algorithm have shown that CDEAS is a competitive algorithm to solve all the unimodal function problems and most of the multimodal function optimization problems listed above. As shown in the descriptions and all the illustrations before, CDEAS is efficacious on those typical function optimizations.
There are some process variables which have the greatest influence on the net value of ammina, such as system pressure, recycle gas flow rate, feed composition (H/N ratio), ammonia and inert gas cencetration in the gas of reactor inlet, hot spot temperatures, and so forth. The relations between the process variables are coupling and the operational variables interact with each other.
The inlet ammonia concentration is an important process variable which is beneficial to operationoptimization but the device of online catharometer is very expensive. According to the mechanism and soft sensor model, a IIOBP model was built to get the more accurate value of the inlet ammonia concentration [
From the analysis discussed above, some important variables have significant effects on the net value of ammonia. By discussion with experienced engineers and taking into consideration a priori knowledge about the process, the system pressure, recycle gas flow rate, the H/N ratio, hotspot temperatures in the catalyst bed, and ammonia and methane concentration in the recycle gas are identified as the key auxiliary variables to model net value of ammonia which is listed in Table
Auxiliary variables of model of net value of Ammonia.
List  Symbols  Name  Unit 

1 

H/N ratio  % 
2 

Methane concentration in recycled synthesis gas at the reactor inlet  Mole ratio 
3  A_{NH3}  Ammonia concentration in recycled synthesis gas at the reactor inlet  Mole ratio 
4 

System pressure  Mpa 
5 

Recycle gas flow rate  Nm^{3}/h 
6 

Quench gas flows of axial layer  Nm^{3/}h 
7 

Cold quench gas flows of 1st radial layers  Nm^{3}/h 
8 

Quench gas flows of 2nd radial layers  Nm^{3}/h 
9 

Hot quench gas flows of 1st radial layers  Nm^{3}/h 
10 

Hotspot temperatures of axial bed  °C 
11 

Hotspot temperatures of radial bed I  °C 
12 

Hotspot temperatures of radial bed II  °C 
13 

Outlet gas temperature of evaporator  °C 
LSSVM is an alternate formulation of SVM, which is proposed by Suykens. The einsensitive loss function is replaced by a squared loss function, which constructs the Lagrange function by solving the problem linear Karush_Kuhn_Tucker (KKT)
There are several kinds of kernel functions, such as hyperbolic tangent, polynomial, and Gaussian radial basis function (RBF) which are commonly used. Literatures have proved that RBF kernel function has strong generalization, so in this study RBF kernel was used:
As we can see from (
Grid search is a commonly used method to select the parameters of LSSVM, but it is timeconsuming and inefficient. CDEAS algorithm has strong search capabilities, and the algorithm is simple and easy to implement. Therefore, this paper proposes the CDEAS algorithm to calculate the best parameters
Operational parameters such as
The extreme values are eliminated from the data using the
BPNN, LSSVM, and DELSSVM are also used to model the net value of ammonia, respectively. BPNN is a 13151 threelayer network with backpropagation algorithm. LSSVM gains the
The comparisons of training error and testing error of LSSVM.
Method  Type of error  RE*  MAE*  MSE* 

BPNN  Training error 



Testing error  0.008085 

 


LSSVM  Training error  0.002231 


Testing error  0.005328 





DELSSVM  Training error  0.002739 


Testing error  0.005252 





CDEASLSSVM  Training error  0.002830 


Testing error 



Global optimum, search ranges, and initialization ranges of the test functions.

Dimension  Global optimum 

Search range  Target 


30  0  0 




0 

 

0  0 

 


0 

 

1  1 

100  

0  0 

 


0 

 

0  0 

 


0 

0.1  

0  0 

0.001  


0 

0.01  

0  0 

10  


0 

10  

0  0 

10  


0 

5  

418.9829  0 

500  

0  0 

1  

0  0 


The analyzed results, training results, and testing results of BPNN, LSSVM, DELSSVM, and CDEASLSSVM.
Despite the fact that the training error using BPNN is smaller than that using CDEASLSSVM, which is because BPNN is overfitting to the training data, the mean square error (MSE) on training data using CDEASLSSVM is reduced by 25.6% and 23.2% compared with LSSVM and DELSSVM, respectively. In comparison with the other models (BPNN, LSSVM, and DELSSVM), testing error using CDEASLSSVM model is reduced by 14.1% and 11.2%, respectively. The results indicate that the proposed CDEASLSSVM model has a good tracking precision performance and guides production better.
In this paper, an optimizing model which describes the relationship between net value of ammonia and key operational parameters in ammonia synthesis has been proposed. Some representative benchmark functions were employed to evaluate the performance of a novel algorithm CDEAS. The obtained results show that CDEAS algorithm is efficacious for solving most of the optimization problems comparisons with original DE. Least squares support vector machine is used to build the model while CDEAS algorithm is employed to identify the parameters of LSSVM. The simulation results indicated that CDEASLSSVM is superior to other models (BPNN, LSSVM, and DELSSVM) and meets the requirements of ammonia synthesis process. The CDEASLSSVM optimizing model makes it a promising candidate for obtaining the optimal operational parameters of ammonia synthesis process and meets the maximum benefit of ammonia synthesis production.
(1) Sphere function
(2) Shifted sphere function
(3) Schwefel’s Problem 1.2
(4) Shifted Schwefel’s Problem 1.2
(5) Rosenbrock’s function
(6) Schwefel’s Problem 1.2 with noise in fitness
(7) Shifted Schwefel’s Problem 1.2 with noise in fitness
(8) Ackley’s function
(9) Shifted Ackley’s function
(10) Griewank’s function
(11) Shifted Griewank’s function
(12) Rastrigin’s function
(13) Shifted Rastrigin’s function
(14) Noncontiguous Rastrigin’s function
(15) Shifted noncontiguous Rastrigin’s function
(16) Schwefel’s function
(17) Schwefel’s Problem 2.21
(18) Schwefel’s Problem 2.22
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the anonymous reviewers for giving us helpful suggestions. This work is supported by National Natural Science Foundation of China (Grant nos. 61174040 and 61104178) and Fundamental Research Funds for the Central Universities, Shanghai Commission of Science and Technology (Grant no. 12JC1403400).