HU Chunping (胡春平) and YAN Xuefeng (颜学峰)



An Immune Self-adaptive Differential Evolution Algorithm with Application to Estimate Kinetic Parameters for Homogeneous Mercury Oxidation*

HU Chunping (胡春平) and YAN Xuefeng (颜学峰)**

Automation Institute, East China University of Science and Technology, Shanghai 200237, China

A new version of differential evolution (DE) algorithm, in which immune concepts and methods are applied to determine the parameter setting, named immune self-adaptive differential evolution (ISDE), is proposed to improve the performance of the DE algorithm. During the actual operation, ISDE seeks the optimal parameters arising from the evolutionary process, which enable ISDE to alter the algorithm for different optimization problems and improve the performance of ISDE by the control parameters’ self-adaptation. The performance of the proposed method is studied with the use of nine benchmark problems and compared with original DE algorithm and other well-known self-adaptive DE algorithms. The experiments conducted show that the ISDE clearly outperforms the other DE algorithms in all benchmark functions. Furthermore, ISDE is applied to develop the kinetic model for homogeneous mercury (Hg) oxidation in flue gas, and satisfactory results are obtained.

differential evolution, immune system, evolutionary computation, parameter estimation

1 Introduction

Differential evolution (DE) [1] is a new generation evolutionary algorithm (EA) and has been successfully applied to solve a wide range of optimization problems. Differential evolution is stochastic, population- based, and direct search algorithm for globally optimizing functions with real valued parameters. In the first international IEEE Competition on evolutionary optimization, DE proved to be one of the fastest EAs. Storn and Price [1] have compared DE with adaptive simulated annealing [2], the annealed Nelder and Mead approach [3], the breeder genetic algorithm [4], the EA with soft genetic operators [5], and the method of stochastic differential equations [6]. In most instances, DE outperformed all of the above minimizations approaches. Vesterstrom and Thomsen [7] applied 34 widely used benchmark problems to evaluate the performance of DE, particle swarm optimization (PSO) [8], and EA. Their study shows that DE generally outperforms the other algorithms. Krink. [9] introduced three search approaches [genetic algorithm (GA) [10], PSO, and DE] to develop bank rating systems, respectively, and turned out that DE is clearly and consistently superior compared with GA and PSO both in respect to precision and reliability.

In recent years, researchers have developed various strategies for adjusting control parameters dynamically. Abbass [15] proposed a self-adaptive Pareto DE (SPDE). In SPDE, the mutation rate is sampled for each individual from a Gaussian distribution,(0, 1). The crossover rate is first initialized for each individual from a uniform distribution,(0, 1). Then, CR is adapted as

where

In self-adaptation parameter control, the idea of an evolutionary search can be used to implement the self-adaptation of search parameters [19]. In other words, the concept of coevolution can be used to adapt the control parameters. Coevolution method is an effective approach to decompose complex structure and achieve better performance. Several applications of coevolution method, which have been proven to be useful, were described in the literatures [20-22]. In this article, the immune concepts and methods are applied to determine the parameter setting of DE. Further, the proposed method, named immune self-adaptive differential evolution (ISDE), is compared with the versions of DE proposed by Price and Storn [23], the SPDE proposed by Abbass [15], and the self-adaptive DE (SDE) proposed by Omran[16].

For illustration, ISDE was applied to develop the kinetic model for homogeneous mercury (Hg) oxidation in flue gas. Homogeneous mercury oxidation in flue gas is a highly nonlinear reaction with reference to optimal operating conditions with many equality and inequality constraints. The kinetic model involves five reactions. Two of these reactions are reversible and three are irreversible. The preexponential factor and activation energy values in the rate constant term for each reaction need to be determined. Then, ISDE was used to determine the kinetic parameters for homogeneous mercury oxidation in flue gas with the data obtained in a laboratory scale apparatus, reported by Agarwal. [24, 25], and the kinetic model with good precision for homogeneous mercury oxidation in flue gas was developed.

2 Differential evolution algorithm

The procedure of executing DE can be described in the following:

(7) Repeat steps 2-6 as long as the number of generations is smaller than the allowable maximum numbermand the best individual is not obtained.

The mutation strategy described above is known as DE/rand/1, meaning that the vector to be perturbed is randomly chosen, and that the perturbation consists of one weighted difference vector. DE/rand/1 is the most successful and the most widely used strategy [14].Other useful strategies are:

“DE/rand-to-best/1”:

“DE/best/2”:

“DE/rand/2”:

3 Immune Self-adaptive Differential Evolution

In self-adaptation parameter control, the idea of an evolutionary search can be used to implement the self-adaptation of search parameters. The parameters to be adapted are coded into the chromosomes that undergo mutation and recombination. Better values for these encoded parameters are supposed to result in better individuals that in turn are more likely to survive and produce offspring and hence propagate better parameter values [19]. In other words, self-adaptation is a strategy in which the idea of an evolutionary search was used to choose the optimal parameters.

Immune system, a highly evolved biological system with learning, memory, and pattern recognition capabilities [26], has been successfully integrated into many other evolution algorithms [27-29]. In our work, the immune concepts and methods are applied to determine the parameter setting of DE. To be exact, the aim of leading immune concepts and methods into DE is theoretically to use the previous state information of search for seeking the optimal parameters,and CR. During the actual operation, ISDE seeks the optimal parameters arising from the evolutionary process, which enable ISDE to alter the algorithm for different optimization problems and improve the performance of ISDE by the control parameters’ self-adaptation. In ISDE, the first initial antibodies are randomly generated within the feasible range. The two parameters of each individual are initialized from a normal distribution within the feasible range. The affinity values of the antibodies are calculated. Then, depending on the affinity values, the parameters are replaced by antibody with a certain probability defined previously. In each generation, a percentage of antibodies in the antibody population are replaced by created new antibodies. Thus, the coevolution method is established. Differential evolution is used to perform evolution search in spaces of solutions, and immune system is used to perform evolution search in spaces of control parameters. The solutions and control parameters evolve interactively and selfadaptively, and both the satisfactory solutions and suitable control parameters can be obtained simultaneously.

The procedure of executing ISDE can be described in the following:

(1) Initialization operation

(3) Mutation operation

(4) Crossover operation

(5) Evaluation operation

(6) Create new antibodies

(7) Update antibodies

(8) Generate the parameters for next generation

whereis the parameter that controls the probability between different antibodies.

(10) Repeat steps 2-9 as long as the number of generations is smaller than the allowable maximum numbermand the best individual is not obtained.

4 Benchmark function

Nine benchmark functions were used in our experimental studies. These benchmark functions were divided into three classes: functions with single optima, many local minima, and a few local minima. The benchmark functions are given in Table 1.stands for the dimension of the function,0denotes their ranges, andminis a function value of the global optimum. A more detailed description of each function is given in Yao. [30], Krink. [31], and Salman[32].

5 Experimental results

Maximal number of evaluations: 50000;

The results reported in this section are average results of 30 independent runs.

Table 1 Benchmark function

5.1 No-noisy benchmark functions

Table 3 summarizes the results obtained by applying the different approaches to the multimodal benchmark functions. The results show that the ISDE significantly outperformed (or at least equal to) the other methods in all the multimodal functions.

From above experiments, it can be turned out that ISDE is clearly superior compared with the original DE strategies, SPDE, and SDE in all benchmark functions.

5.2 Noisy benchmark functions

In this subsection, the effect of noise on the performance of ISDE is investigated. The noisy versions of the benchmark functions are defined as:

Table 4 and Table 5 summarize the results obtained for the noisy problems for the unimodal and multimodal functions, respectively. Table 4 and 5 show that the ISDE was less prone to noise than other DE strategies for all benchmark functions. The ISDE retained its position as the best performer when applied to all benchmark functions even in the presence of noise. The only exception is the noisy Rastrigin’s function where SDE outperformed the ISDE. However, even for the noisy Rastrigin’s function where ISDE’s average is worse than SDE, it is not significantly worse. In addition, the improvement is even more significant for the noisy Ackley’s function, where all strategies were trapped in a local optimum. Hence, compared with the other tested strategies, the ISDE seems to be less badly affected by noise. This is a significant improvement over the conventional DE, which is not a good approach to achieve results with high accuracy for noisy functions [31].

Table 2 Mean and standard deviation (±SD) of the unimodal function optimization results (The data about DE/rand/1, DE/best/1, DE/rand-to-best/1, DE/rand/2, DE/best/2, SPDE, and SDE were reported by Salman et al. [32])

Table 3 Mean and standard deviation (±SD) of the multimodal function optimization results (The data about DE/rand/1, DE/best/1, DE/rand-to-best/1, DE/rand/2, DE/best/2, SPDE, and SDE were reported by Salman et al. [32])

Table 4 Mean and standard deviation (±SD) of the noisy unimodal function optimization results (The data about DE/rand/1, DE/best/1, DE/rand-to-best/1, DE/rand/2, DE/best/2, SPDE, and SDE were reported by Salman et al. [32])

Table 5 Mean and standard deviation (±SD) of the noisy multimodal function optimization results (The data about DE/rand/1, DE/best/1, DE/rand-to-best/1, DE/rand/2, DE/best/2, SPDE, and SDE were reported by Salman et al. [32])

5.3 Effect of α, β, and

6 Application

Mercury emissions from coal-fired power plants are highly dependent upon mercury speciation [33]. Mercury in the flue gas is most commonly classified in three forms: elemental mercury (Hg0), oxidized mercury (Hg2+), and particulate bound mercury (HgP). The particulate bound mercury is usually trapped by ash collection devices within power plants, such as electrostatic precipitators, mechanical hoppers, or bag houses. Elemental mercury is relatively inert and difficult to capture because of its nonreactivity. It is also volatile at high temperatures and insoluble in water. In contrast, oxidized mercury is very water soluble and has an affinity for adsorbing onto particulate matter such as fly ash or on metal surfaces in the duct. As a result of these physical and chemical properties of Hg0and Hg2+, the removal of mercury is enhanced when elemental Hg is converted to its oxidized form [34].

In order to better understand the reaction mechanism that takes place in the gas phase, a model needs to be developed where the percentage of mercury oxidized can be predicted based on the concentrations of these flue gas components. The reaction mechanism, proposed by Agarwal and Stenger [34], is a five-reactionsystem, where two reactions are reversible and three reactions are irreversible. These reactions are listed below:

where the term (s, g) denotes that the species could be in solid (s), gas (g) or both phases.

The reaction rate equations can be written as follows:

Table 6 Influence of α on the performance of ISDE

Table 7 Influence of β on the performance of ISDE

Table 8 Influence of  on the performance of ISDE

Table 9 Optimal parameters and objective function values corresponding to reported data [34] and ISDE

7 Conclusions

This article presents an efficient self-adaptive DE algorithm for global optimization. In the proposed method, the immune concepts and methods are applied to determine the parameter setting, which utilizes the previous state information of search for seeking the optimal parameters,and CR. In the benchmark tests, both noisy and nonoisy functions, the results show that the performance of ISDE is outstanding in comparison with the original DE strategies, SPDE and SDE tested. Among the tested strategies, the ISDE can rightfully be regarded as an excellent first choice when faced with a new optimization problem to solve. Thus, the algorithm was subsequently used to estimate the kinetic parameters for homogeneous mercury oxidation in flue gas. The results marked a noticeable improvement over previously reported solutions.

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the National Natural Science Foundation of China (20506003, 20776042) and the National High-Tech Research and Development Program of China (2007AA04Z164).

** To whom correspondence should be addressed. E-mail: yan_xuefeng@hotmail.com