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

код для вставкиСкачатьASIA-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 151 152 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 155 156 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). REFERENCES [1] D. Annable. Chem. Eng. Sci., 1952; 1, 145–154. [2] R.F. Baddour, P.L.T. Brian, B.A. Logeais, J.P. Eymery. Chem. Eng. Sci., 1965; 20, 281–292. [3] M.J. Shah. Ind. Eng. Chem., 1967; 59, 72–83. [4] L.D. Gaines. Chem. Eng. Sci., 1979; 34, 37–50. [5] S.S.E.H. Elnashaie, A.T. Mahfouz, S.S. Elshishini. Chem. Eng. Process., 1988; 23, 165–177. [6] L.M. Patnaik, N. Viswanadham, I.G. Sarma. In Proceedings of the IEEE Conference on Decision and Control including the 16th Symposium on Adaptive Processes and A Special Symposium on Fuzzy Set Theory and Applications, v16, New Orleans, IEEE Press: Piscataway, NJ, 1977; pp.185–190. [7] K.V. Reddy, A. Husain. Ind. Eng. Chem. 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