A Hybrid Mode Membrane Computing Based Algorithm with Applications for Proton Exchange Membrane Fuel Cells

Abstract

Membrane computing is a branch of natural computing which has been extended to solve various optimization problems. A hybrid mode membrane-computing-based algorithm (HMMCA) is proposed in this paper to solve complex unconstrained optimization problems with continuous variables. The algorithmic framework of HMMCA translates from its distributed cell-like membrane structure and communication rule. A non-deterministic evolutionary programming method and two computational rules are applied to enhance the computational performance. In a numerical simulation, 12 benchmark test functions with different variables are used to verify the algorithmic performance. The test results and comparison with three other algorithms illustrate its effectiveness and superiority. Moreover, a case study on a proton exchange membrane fuel cell (PEMFC) system parameter optimization problem is applied to validate its practicability. The results of the simulation and comparison with seven other algorithms demonstrate its practicability.

1. Introduction

As a branch of natural computing, membrane computing (also called P systems) was presented by Paun in 2000. It is a class of parallel and distributed computing models [1,2]. The computing models of membrane computing (MC) are abstracted from the structures and functions of living cells, the organization and cooperation in tissues and neural nets, etc. The membrane structure, rules and objects are the three essential ingredients of MC (P systems). In a P system, the objects (multi-sets described in mathematics) are processed in a nondeterministic, maximally parallel manner, according to their given local reaction rules in distributed membranes, or compartmentalized by the membranes. Then, transitions of objects operated by rules, or passed between different system configurations of membranes, can be obtained [2,3]. At present, the three main types of P systems are cell-like, tissue-like and neural-like [4]. The research into MC has revealed that, whether cell-like, tissue-like or neural-like, whether with symbol objects or string objects, whether working in the generative or accepting modes, or whether with a certain combination of features, all of these classes of systems are universal [2,3,4], and have reached the computational power of a Turing machine [3]. For example, it has been proved that spiking neural P systems are Turing complete [4], and the computational power of cell-like P systems exploiting some type of “time acceleration” is theoretically capable of trespassing the Turing barrier [5,6]. The theory of MC has been applied to various fields, such as modeling in medicine and biology [7], modeling of metapopulations [8], linguistics in computer science [9], segmentation in computer vision [10], membrane controllers [11], etc.

Inspired by the powerful computational power of MC, more and more optimization algorithms have been studied, based on the theory of using MC for solving complicated optimization problems with a single objective or multiple objectives, constrained or unconstrained, and continuous or discrete variables. Distinguished from other biologically inspired algorithms, membrane-computing-based algorithms are not only analogous to the natural processes in living cells, but also imitate the ideas, parallel-distributed computing models and nondeterministic computational rules of MC. The first “cell-like” membrane algorithms (MAs) were proposed by Nishida to solve the traveling salesman problem [12], which dynamically evolved membranes by creating or destroying them according to certain possibilities, a genetic algorithm and a simulated annealing algorithm, distributed in nested membranes as sub-algorithms. Another “cell-like” algorithm inspired by MC adopted a static nested membrane structure, and three rules were taken from genetic algorithms and inspired by MC [13]. A quantum-inspired evolutionary algorithm based on P systems with a static “cell-like” membrane structure was proposed for solving knapsack problems [14]. The bio-inspired algorithm based on membrane computing (BIAMC), and hybrid method based on membrane computing (HBS), were proposed for solving unconstrained and constrained problems with continuous variables [15,16]. In BIAMC, a static cell-like nested membrane structure and three new rules that simulated the Golgi apparatus of eukaryotic cells were applied. In HBS, an SQP rule cooperated with an improved BIAMC in a variable manner to solve complicated constrained problems [16]. A novel P-system-based optimization algorithm (BIPOA) was proposed for solving parameter estimation problems, where a nested membrane structure with three new rules was applied [17]. The micro-charge field effect P systems optimization algorithm (MFE-POA), as a deeper exploration of the standard P systems optimization algorithm (POA), was proposed to solve the model parameter estimation of two types of PV cell models and multi-cell PV modules [18]. Multi-objective optimization algorithms, based on tissue P systems for optimal control of complex systems, have also been investigated [19,20]. An algorithm, called the membrane computing with harmony search algorithm (MC-HSA), in which an active membrane-dissolving P system was designed and the harmony search algorithm was embedded in the P system, was proposed to quickly select a subset of potential disease-causing genes [21]. A semantic and intelligent focused crawler based on the semantic vector space model (SVSM) and a membrane computing optimization algorithm (MCOA) were proposed for the semantic understanding and intelligent learning abilities. The MCOA method was used to optimize four weighted factors based on the rules of evolution and communication [22]. An optimization numerical spiking neural P system (ONSN P systems or ONSNPS) was proposed to solve continuous constrained optimization problems, in which a guider algorithm is introduced to finish individuals’ crossover and selection, and a random mechanism was used for production function selection to achieve updated parameters [23].

The reported results show that these P systems’ optimization algorithms are readily available and are superior to other compared algorithms in terms of global searching, speed and accuracy. However, novel computational rules derived from MC or living cells are inadequate, and their computational rules from other algorithms carry their own properties, such as rules from GA or DE, etc. Compared with other MC-based algorithms for solving problems with continuous variables, BIAMC works much more effectively, and is distinctive in its algorithmic framework and computational rules, such as 100-dimensional optimization problems. A hybrid mode membrane-computing-based algorithm (HMMCA) is presented to enhance the computational efficiency of BIAMC when the complexity of problems is increased, such as with high dimensions and highly nonlinear and inseparable variables.

The remainder of the paper is organized as follows. HMMCAs are described in Section 2. Section 3 reports the numerical experiment. A case study on optimizing fuel cell model parameters is presented in Section 4. Section 5 concludes this paper.

2. Hybrid Mode Membrane Computing Based Algorithm

2.1. Problem Formulation

Research on non-deterministic versions of MC computing models was expanded several years ago [24]. Inspired by the research and its results, a non-deterministic evolutionary programming method (NEP), a method of nondeterministic computational probability of rules, a nondeterministic abstraction rule and a modified crossover mode of rewriting rule are proposed and applied to enhance the computational performance of the bio-inspired algorithm based on MC.

2.2. Membrane Structure and Operational Rules

The membrane structure and communication rule, non-deterministic evolutionary programming and two modified computational rules of HMMCA are given as follows. The membrane structure is shown in Figure 1.

The surface membrane randomly generates initial objects and a communication object, and contains all inner membranes and outputs results. The computational rules are distributed in parallel element membranes, and the evolved objects are finally sent out or deleted from the quasi-Golgi or parallel element membranes due to the termination condition. Each object comprises a candidate solution vector of problems and other parameters calculated, e.g., an objective function value, function evaluation times, etc.

Without loss of generality, minimization is considered in this paper. The new rules used in the HMMCA are summarized as follows.

• Communication rule: Objects and communication objects are controlled by this rule to transfer in inner membranes. n_o numbered objects kept in membrane O₁ or O₂ are sent into parallel element membranes by this rule. It sends out n_o rewritten objects to membrane O₂ or are deleted. Moreover, it sends a communication object built in a parallel element membrane into the quasi-Golgi according to the activating condition or the next parallel element membrane, or from the quasi-Golgi to a parallel element membrane.
• Crossover mode of rewriting rule:

  $$X_i,X_k\underset{p_{cm}}{{\displaystyle \mapsto }}{X}_i,X_k$$

  (1)

Step 1:

$$\begin{array}{l}{X}_i,X_k\to X_i,X_k\\ where,\{\begin{array}{l}{X}_i,X_k\to X_i{}^{\prime },X_k{}^{\prime }\\ X_i{}^{\prime },X_i,X_k{}^{\prime },X_k\to X_i,X_k\\ X_i{}^{\prime }=\eta \times X_i+(1-\eta )\times X_k,\\ if f(X_i{}^{\prime })\le f(X_i)\mathrm{then}{X}_i=X_i{}^{\prime }\\ else\\ if f(X_i{}^{\prime })\le f(X_{i+1})/f(X_{i-1})\mathrm{then}\\ X_{i+1}/X_{i-1}=X_i{}^{\prime }\\ if f(X_i{}^{\prime })\le f(X_k)\mathrm{then}{X}_k=X_i{}^{\prime }\\ X_k{}^{\prime }=(1-\eta )\times X_i+\eta \times X_k\\ if f(X_k{}^{\prime })\le f(X_i)\mathrm{then}{X}_i=X_k{}^{\prime }\\ else\\ if f(X_k{}^{\prime })\le f(X_{i+1})/f(X_{i-1})\mathrm{then}\\ X_{i+1}/X_{i-1}=X_k{}^{\prime }\\ if f(X_k{}^{\prime })\le f(X_k)\mathrm{then}{X}_k=X_i{}^{\prime }\\ \eta \in [0,1];i=1,\dots ,n_{co};k=n_{co}+1,\dots ,n_o\end{array}\end{array}$$

Step 2:

$$\begin{array}{l}{X}_i,X_{i+1}\to X_i,X_{i+1}\\ where,\{\begin{array}{l}{X}_i,X_{i+1}\to X_i{}^{\prime },X_{i+1}{}^{\prime }\\ X_i{}^{\prime },X_i,X_{i+1}{}^{\prime },X_{i+1}\to X_i,X_{i+1}\\ X_i{}^{\prime }=\eta \times X_i+(1-\eta )\times X_{i+1},\\ X_{i+1}{}^{\prime }=(1-\eta )\times X_i+\eta \times X_{i+1}\\ \eta \in [0,1];i=1,\cdots ,n_o-1;\end{array}\end{array}$$

p₁, p_i and η are random numbers between 0 and 1. p_cm is the probability of crossover mode. i and k are object indices. X_i denotes the solution vector in object i, and $X_i{}^{\prime }$ denotes its rewritten vector. n_o and n_co are the number of objects in a parallel membrane and a communication object.

When the crossover mode operates, the weighted arithmetic mean calculation performs between the top n_co and other n_o − n_co objects. Firstly, generate $n_{co}\times (n_o-n_{co})$ uniformly distributed variables p₁. If and only if p₁ is less than the probability of crossover mode p_cm, object X_i and X_k are replicated and rewritten as described in step 1. When the rewritten $X_i{}^{\prime }$ or $X_k{}^{\prime }$ are less than the initials or the other in top n_co, the rewritten object replaces it. Otherwise, $X_i{}^{\prime }$ and $X_k{}^{\prime }$ are deleted. After the first step, sequentially, n_o rewritten objects are rewritten by the arithmetical crossover between every two neighboring copies of the objects. Similarly, generate n_o − 1 uniformly distributed variable p_i. When p_i is less than p_cm, two neighboring copies of the object X_i and X_i+1 are rewritten. When the rewritten $X_i{}^{\prime }$ or $X_{i+1}{}^{\prime }$ are less than the initials, the rewritten object replaces the initial. Or else, they are deleted.

3.

Nondeterministic abstraction rule (Algorithm 1).

$$X_1,X_2\underset{p_{na}}{{\displaystyle \mapsto }}{X}^{\prime },X_2$$

(2)

The main steps are summarized as follows.

Algorithm 1 Nondeterministic abstraction rule

Initialize p for j = 1: l    if p(j) < pna      for k = 1: l           x = X(1,1: l);           if k = l                a = 0;           Else                a = max(1, $⌊r\times l/2⌋$);                if k + a ≥ l||j + a ≥ l                          a = min(l − k, l − j);                end(if)           end(if)           c = X(2, k: k + a);           x(j: j + a) = c;           if f(x) < f(X(1))                 X(1) updates;            end(if)         end (for)   end(if) end (for)

p_na is the probability of this computational rule. p is a random vector between 0 and 1. j and k are the variables indices. l is the dimension of a solution vector. r is a random number between 0 and 1. a is a random number between 1 and l/2. $\lfloor \rfloor$ is the sign of rounded down.

2.3. Algorithm Description

Definition 1.

 One algorithmic cycle consists of applying the computational rules from parallel membrane 1 to parallel membrane m [16].

The main steps of procedure are summarized as follows (Algorithm 2).

Algorithm 2 Hybrid mode membrane computing

Initialize g, nn, tiv, pop1, cpop   mm = mm + 1   pop2 = pop1   for k = 1:C     for n = 1:Nm      while (FET ≤ fet)       {         g = g + 1        Initialize pcm, pmm, pna, pt.        if n = nn(k)          E = pop2((Nm−1) × No + 1:n × No,:);        else          E = pop1((Nm−1) × No + 1:n × No,:);        end        Crossover mode works on E        Mutation mode and pairing rule works on updated E        pop2 updates        cpop updates        if g > 2              if (~mod(mm,2))                 chg = (wide × 0.0618/(n × 10(k−1))) × ones(Nco,l)                 target indication rule works on cpop                 cpop updates                 transport rule works on cpop                 cpop updates                 nondeterministic abstraction rule works cpop updates              end(if)           end(if)           tiv updates;         }   end (for) end (for)

N_r, N_m, N_o, N_co, C, mm, FET(fet), g, nn, tiv, pop1, pop2 and cpop denote the amount of repeated trials run once, the number of parallel element membranes, the number of objects sending in each parallel membrane, the sum of objects in a communication object, the algorithmic cycles, an activation parameter for the quasi-Golgi membrane, function evaluation times (the setting value used as a termination condition), the current number of N_m × C, the index of parallel element membranes applied nondeterministic evolutionary program, the matrix used for keeping directions of target indication vectors, the initial candidate objects matrix, the updated candidate objects matrix and the communication object matrix, respectively. p_cm, p_mm, p_na and p_t are the probabilities of crossover mode, mutation mode, non-deterministic abstraction, and transition rules. In HMMCA, p_cm, p_mm, p_na and p_t are assigned randomly in each parallel membrane and each algorithmic cycle.

The non-deterministic evolutionary programming method is actualized by the membranes O₁ and O₂, and the communication rule. For each algorithmic cycle, an element membrane is preset randomly to apply evolutionary programming, i.e., besides the communication object, the random preset element membrane receives objects from membrane O₂. Conversely, the remainder element membranes receive objects from membrane O₁. Then, this nondeterministic evolutionary programming is fulfilled after the computing of all algorithmic cycles finishes.

2.4. Computational Complexity

The computational cost of an algorithm is generally to evaluate the CPU time or the times of functions called maximally. For optimization algorithms, the objective functions’ evaluation times (FET) are used. The computational cost of HMMCA under the given parameter settings is the summation of each computational rules’ cost, which is in direct proportion to dimensions of decision variables, probabilities of computational rules, the numbers of algorithmic cycles, parallel element membranes and objects. The computational cost of rewriting rule consumed in the mutation mode is FET ≤ C × N_m × N_o × p_mm × l, and that of the crossover mode is FET ≤ C × N_m × p_cm × N_o × (N_co + 1). In the quasi-Golgi membrane, the computational cost of the non-deterministic abstraction rule is in direct proportion to the square of dimensions of problems and the activating condition of the quasi-Golgi membrane, i.e., FET ≤ 0.5C × N_m × p_na × l². The rules of target direction and transition are all in direct proportion to the number of objects in communication objects, FET ≤ 1.5C × N_m × N_co. Therefore, the worst-case time complexity of HMMCA is O(l²).

3. Performance Evaluation

3.1. Benchmark Function Optimization

To analyze the merits and drawbacks of the HMMCA, we performed a numerical experiment on 12 benchmark test functions with 30 and 100 variables. The names, variable ranges, parameters and optimal solutions of functions are listed in

Table 1. Benchmark functions.

Function	Optimum Solution
f1: Griewangk	xi ∈ [−600, 600], f1*(0, …, 0) = 0
f2: Rosenbrock	xi ∈ [−100, 100], f2*(1, …, 1) = 0
f3: Ackley	xi ∈ [−32.768, 32.768], c1 = 20, c2 = 0.2, c3 = 2π, f3*(0, …, 0) = 0
f4: Weierstrass	xi ∈ [−0.5, 0.5], a = 0.5, b = 3, kmax = 20, f4*(0, …, 0) = 0.
f5: Rastrigin	xi ∈ [−5.12, 512], f5*(0, …, 0) = 0.
f6: Sphere	xi ∈ [−100, 100], f6*(0, …, 0) = 0.
f7: Schwefel 2.26	xi ∈ [−500, 500], f7*(420.968, …, 420.9687) = 0.
f8: Schwefel 1.12	xi ∈ [−100, 100], f8*(0, …, 0) = 0.
f9: Step	xi ∈ [−100, 100], f9*(0, …, 0) = 0.
f10: Schwefel 2.22	xi ∈ [−100, 100], f10*(0, …, 0) = 0
f11: Noncontinuous Rastrigin	xi ∈ [−5.12, 512], f11*(0, …, 0) = 0.
f12: Quatic	xi ∈ [−1.28, 1.28], f12*(0, …, 0) = 0.

“*” means the global optimal solution.

. Meanwhile, BIAMC, the evolution strategy with covariance matrix adaptation (CMA-ES) and the genetic algorithm (GAs) were executed for comparison and analyses. CMA-ES, proposed by Nikolaus Hansen and Andreas OsterMeier in 2001, is a powerful evolutionary algorithm using the evolutionary strategy and adaptive covariance matrix method, and has been used in hundreds of non-linear, non-convex continuous, etc. optimization problems and applications [25].

3.2. Experiment Setup

The parameter settings of the HMMCA derived from the previous research and trial and method were N_r = 100; N_m = 10; C = 30/15/15; the stop condition, FET = 30,000/200,000/500,000; N_o = 7; N_co = 2; p_cm ≥ 0.8; p_mm ≥ 0.8; p_t ≤ 0.3; p_na ≥ 0.8; the increment used in mutation mode, $(r\times wid{e}_j)/(ml\times {10}^c)$, where wide_j = x_j^u − x_j^l, x_j^u and x_j^l are the upper limit and lower limit, respectively, ml and c are the current numbers of the parallel element membrane and algorithmic cycle, respectively, the magnitude of target indication vector, $0.0618wid{e}_j/(ml\times {10}^{c-1})$; the activating condition of quasi-Golgi, the multiplication of the current value of the algorithmic cycle by that of the parallel element membrane is exactly divisible by 2. To implement nondeterministic computing and keep a better computational performance, the upper or lower limits of p_cm, p_mm, p_t and p_na are given, that is, in each element membrane, they are randomly set within the upper or lower limits.

Most of the parameter settings of BIAMC followed previous research on BIAMC combined with trial and error: N_r = 100; N_m = 10; C = 30/15/15; FET = 30,000/200,000/500,000; N_o = 7; N_co = 2; p_cm = 1; p_mm = 1; p_t = 0.3; p_a = 1; the increment used in mutation mode was $(r\times wid{e}_j)/(ml\times {10}^c)$; the magnitude of target indication vector was $0.0618wid{e}_j/(ml\times {10}^{c-1})$; the activating condition of quasi-Golgi was that the multiplication of the current value of algorithmic cycle by that of parallel element membrane was exactly divisible by 2.

The MATLAB code of CMA-ES was downloaded from reference [25]. Most of the parameters of CMA-ES were set to default as described in the CMA-ES algorithm. The changed parameters were derived from the trial and error method: the amount of repeated trials, N_r = 100; sigma = (x^u− x^l)/3; Options: MaxFunEvals = 30,000/200,000/500,000; MaxIter = Inf; TolHistFun = 1/Inf; TolUpX = Inf; TolFun = 1/Inf; TolX = 1/Inf; Restarts = 2; StopFunEvas = 30,000/200,000/500,000; StopOnWarnings = no; StopOnStagnation = off; StopOn-EqualFunctionValues = Inf; StopFitness = −Inf.

The ga function in the MATLAB optimization toolbox was applied because of its popularization. In the same way, the parameters of ga were set: the amount of repeated trials, N_r = 100, ‘PopulationSize’ = 50, ‘MutationFcn’ = mutation-adaptfeasible, ‘CrossoverFraction’ = 0.9, ‘MigrationFraction’ = 0.2, ‘SelectionFcn’ = selectionstochunif, ‘CrossoverFcn’ = crossoverheuristic/crossoverintermediate (used for f₁, f₄, f₉, f₁₁), ‘Generations’ = 600/5000/10,000, ‘StallGenLimit’ = 600/4000/10,000, ‘Display’ = off. Other optional parameters were default.

All the experiments in this paper ran on the same PC with Win 7 64-bit (Inter Core^TM i7-2600 processor with 3.40 GHz and 8 GB RAM). The listed data were obtained by executing each algorithm once when MATLAB 2009b was open or when the test functions were changed.

3.3. Result and Discussion

For a compact representation, most test result of functions with 30 and 100 variables are listed in

Table 2. Result of functions f₁–f₁₂ with 30 dimensions calculated by HMMCA, BIAMC, CMA-ES, GAs.

F	A	30					100
		Mean	st.dev	avg.t	sr	t	Mean	st.dev	avg.t	sr	t
f1	H	4.30 × 10−67	1.63 × 10−66	15.83	1	=	2.87 × 10−21	7.35 × 10−21	37.99	1	=
	B	1.73 × 10−3	8.68 × 10−3	19.57	0.95	≈	1.30 × 10−3	1.30 × 10−2	26.19	0.99	≈
	C	1.28 × 10−3	3.16 × 10−3	47.22	0.85	+	4.93 × 10−4	2.24 × 10−3	223.48	0.95	+
	G	3.76 × 10−1	3.54 × 10−1	21.35	0	+	7.24 × 101	1.17 × 101	85.92	0	+
f2	H	4.99 × 100	2.98 × 100	18.87	0.79	=	1.26 × 101	3.28 × 101	37.86	0.85	=
	B	4.10 × 100	1.07 × 101	17.30	0.07	+	1.88 × 101	5.11 × 101	33.32	0.33	+
	C	5.98 × 10−1	1.43 × 100	52.17	0.84	−	1.54 × 100	2.30 × 100	236.22	0.1	+
	G	1.79 × 102	3.83 × 102	19.01	0	+	5.50 × 102	1.07 × 103	79.69	0	+
f3	H	6.57 × 10−15	3.31 × 10−15	15.23	1	=	2.98 × 10−11	6.98 × 10−11	30.34	1	=
	B	7.15 × 10−2	5.03 × 10−1	19.24	0.98	≈	1.90 × 10−9	2.84 × 10−9	19.95	1	+
	C	4.76 × 10−1	1.38 × 100	19.93	0.89	+	4.20 × 100	4.62 × 100	205.05	0.19	+
	G	4.38 × 10−1	1.08 × 100	19.19	0	+	1.11 × 101	1.78 × 100	75.96	0	+
f4	H	7.11 × 10−16	5.85 × 10−15	134.34	1	=	1.45 × 10−9	3.96 × 10−9	1014.80	0.94	=
	B	2.85 × 10−6	7.35 × 10−6	139.17	0.43	+	1.93 × 10−6	4.96 × 10−6	1055.84	0.46	+
	C	1.34 × 100	2.03 × 100	133.00	0.65	+	2.49 × 101	4.44 × 100	1371.89	0	+
	G	6.08 × 100	2.02 × 100	140.49	0	+	5.29 × 101	4.95 × 100	1347.69	0	+
f5	H	2.99 × 10−1	2.98 × 100	18.87	0.99	=	0	0.00 × 100	38.84	1	=
	B	0.00 × 100	0.00 × 100	15.98	1	≈	0	0.00 × 100	33.77	1	=
	C	5.93 × 101	2.67 × 101	46.32	0	+	4.43 × 102	6.16 × 101	222.14	0	+
	G	9.95 × 10−3	9.95 × 10−2	18.84	0.99	≈	6.99 × 100	3.59 × 100	70.35	0	+
f6	H	9.26 × 10−65	4.15 × 10−64	18.35	1	=	3.57 × 10−19	1.08 × 10−18	31.68	1	=
	B	2.26 × 10−58	5.09 × 10−58	15.34	1	+	5.84 × 10−16	1.28 × 10−15	27.55	1	+
	C	2.72 × 10−110	1.7 × 10−109	47.27	1	−	1.01 × 10−82	8.34 × 10−82	203.93	1	−
	G	1.31 × 10−18	2.16 × 10−18	18.87	1	+	4.52 × 10−15	2.60 × 10−15	63.97	1	+
f7	H	7.11 × 101	5.00 × 102	16.34	0.98	=	9.55 × 10−11	2.59 × 10−12	34.25	1	=
	B	−3.22 × 10−12	8.12 × 10−13	20.69	1	≈	9.76 × 10−11	1.23 × 10−11	24.64	1	≈
	C	5.55 × 103	7.02 × 102	21.33	0	+	1.92 × 104	1.40 × 103	177.56	0	+
	G	1.82 × 10−14	4.08 × 10−13	19.44	1	≈	3.01 × 10−11	1.04 × 10−11	65.84	1	+
f8	H	3.30 × 10−63	1.43 × 10−62	28.90	1	=	2.41 × 10−15	2.15 × 10−14	122.62	1	=
	B	2.77 × 103	1.20 × 103	27.60	0	+	4.72 × 104	1.47 × 104	138.54	0	+
	C	4.71 × 10−79	2.24 × 10−78	75.20	1	−	6.20 × 10−296	0	421.73	1	−
	G	2.33 × 101	2.29 × 101	29.96	0	+	1.39 × 104	6.77 × 103	160.57	0	+
f9	H	0	0	18.39	1	=	0	0.00 × 100	31.91	1	=
	B	0	0	15.56	1	=	0	0	30.24	1	=
	C	0	0	3.07	1	=	0	0	11.84	1	=
	G	1.48 × 103	4.77 × 102	19.21	0	+	1.11 × 104	1.78 × 103	73.85	0	+
f10	H	1.04 × 10−32	2.22 × 10−32	18.54	1	=	2.01 × 10−9	3.76 × 10−9	32.24	0.92	=
	B	3.32 × 10−29	5.95 × 10−29	15.70	1	+	1.28 × 10−7	2.08 × 10−7	27.19	0.47	+
	C	4.12 × 10−71	4.12 × 10−70	51.31	1	−	1.69 × 10−3	1.11 × 10−2	212.15	0.07	+
	G	1.76 × 101	3.92 × 101	18.94	0	+	1.17 × 103	6.97 × 102	63.12	0	+
f11	H	3.00 × 10−1	3.00 × 100	18.18	0.99	=	2.73 × 10−14	1.04 × 10−13	46.32	1	=
	B	1.80 × 100	7.16 × 100	13.88	0.94	≈	5.00 × 10−14	2.47 × 10−13	35.30	1	≈
	C	2.02 × 101	4.71 × 100	19.64	0	+	1.24 × 102	2.58 × 101	124.60	0	+
	G	2.97 × 101	6.91 × 101	29.70	0	+	2.39 × 102	5.27 × 101	97.94	0	+
f12	H	2.90 × 10−4	2.48 × 10−4	15.76	0	=	4.86 × 10−4	2.09 × 10−4	36.87	0	=
	B	9.22 × 10−4	8.14 × 10−4	16.87	0	+	4.11 × 10−3	1.88 × 10−3	25.16	0	+
	C	2.10 × 10−1	8.17 × 10−2	48.57	0	+	4.47 × 10−1	9.03 × 10−2	224.94	0	+
	G	1.66 × 10−1	5.94 × 10−2	26.17	0	+	5.00 × 10−1	9.54 × 10−2	72.63	0	+

. In Table 2, A denotes algorithms; mean refers to averages of 100 groups of results; st.dev refers to the standard deviation of 100 groups of result; avg.t. denotes the averages of CPU time consumed in 100 repeated trials; sr is the success ratio of the times that results approached optimum value (i.e., |f–f*| ≤ 1 × 10⁻⁸) to 100; t refers to the result of a t-test between HMMCA and the other algorithms. Before, the optional parameters of selection and crossover in GAs and the setting of Restart in CMA-ES were fixed by trial and error while considering sr and st.dev.

Comparing the data in Table 2, the computational performance of HMMCA is not satisfactory in terms of robustness because not all sr are 1; nor is that the case for the other algorithms. However, HMMCA keeps its high computational performance in terms of speediness, accuracy and robustness in solving 30- and 100-dimensional functions considering its values of sr, avg.t and st.dev. HMMCA is mostly superior to BIAMC in robustness and accuracy considering sr, avg.t, st.dev., and the t test result. The avg.t of HMMCA is higher than that of BIAMC, although they share the same stop condition. This was partly caused by much more space in memory occupied and more time spent in the nondeterministic computation for high dimensional problems. CMA-ES is outstanding at approaching optimal solutions, due to its covariance matrix adaptation with evolution strategy, considering its test results for f₂, f₆, f₈, f₉, f₁₀ with 30D and 100D, although this computational performance is not so robust. The avg.t of CMA-ES increased more compared with other algorithms. Except for f₆ and f₇, GAs can not approach optimal solutions, and the dimensions of problems caused it more serious trouble than the other algorithms.

To exhibit the behavior of four algorithms in solving problems with different dimensions, the mean values of f₁, f₂, f₃ and f₁₀ with 10 variables together with 30D and 100D are plotted in Figure 2.

It is shown from the four sub-graphs in Figure 2 that the mean values tested on the same functions with different dimensions deviate from the optimum on the different dimension. The change of mean curves calculated by four test algorithms reveals their dissimilar computational performance, such as in the sub-graph of f₁ and f₂. The global searching performance of HMMCA is superior to the others, according to the data in Figure 2. Meanwhile, the computational performance of HMMCA is the most robust and highest efficiency considering the results of sr, avg.t, t-test, etc. in Table 2.

4. Industrial Applications

A fuel cell is an electrochemical device that converts fuel directly and continuously to electricity and heat by using an oxidant (typically oxygen in air), and its by-products are water, very small amounts of nitrogen dioxide and other emissions (depending on the fuel source) [26,27,28]. Because of its quick start-up, low temperature and high power density, etc., the Polymer Electrolyte Membrane fuel cell (PEMFC) is widely researched and applied in power distribution systems, e.g., transportation and industries [17,26,27,28].

Fuel cells generally have three adjacent segments, i.e., the anode, the cathode and the electrolyte that allows charges to move between two poles (shown in Figure 3). A proton-conducting membrane covers two electrodes where two chemical reactions occur (described in Equations (3)–(5)). Only protons (H⁺) can pass through the membrane, which are transported from the hydrogen electrode to the oxygen electrode.

The electrochemical reactions occurring in PEMFC are as follows [28]:

Anode reaction:

$${2\mathrm{H}}_2\to 4{\mathrm{H}}^{+}+4e^{-}$$

(3)

Cathode reaction:

$$4{\mathrm{H}}^{+}+4e^{-}+{\mathrm{O}}_2\to 2{\mathrm{H}}_2\mathrm{O}$$

(4)

Overall cell electrochemical reaction:

$$2{\mathrm{H}}_2+{\mathrm{O}}_2\to 2{\mathrm{H}}_2\mathrm{O}$$

(5)

The electricity produced by unit fuel cells is about 0.7 volts [26], so cells can be placed in series or parallel circuits to increase the voltage and current output to supply hundreds of kilowatts. Moreover, the performance of fuel cells can be improved by pressurizing the air to higher pressures. These characters have been reflected in the following mathematical model of PEMFC.

4.1. A. PEMFC Stack Model

In this case study, a widely used semi-empirical model of PEMFC for predicting i–v character is applied as a parameter optimization problem [17,28,29]. The voltage produced by PEMFC can not attain its theoretical maximum E_Nernst (thermodynamic cell potential), which is defined via the Nernst equation in expanded form [17,27,28,29], and results from resistance. There are three main loss terms in this model: The activation overvoltage V_act is caused by activating the anode and the cathode; ohmic loss V_ohmic is the resistance to electrons transferred through the collecting plates and carbon electrodes, and protons transferred through the solid membrane; concentration overvoltage V_con is caused by the drop in the concentration of oxygen and hydrogen. The fuel cell electrochemical model [17,27,28,29,30,31,32,33] is given as follows.

The stack voltage V_stack connected by n unit cell:

V_stack = nV_FC

(6)

The output voltage of unit fuel cell V_FC:

V_FC = E_Nernst − V_act − V_ohmic − V_con

(7)

The thermodynamic cell potential of fuel cell E_Nernst:

$$\begin{array}{ll}{E}_{\mathrm{Nernst}}& =1.229-8.5\times {10}^{-4}(T-298.15)+4.3085\times {10}^{-5}\\ & T(\ln (P_{{\mathrm{H}}_2})+0.5\ln (P_{{\mathrm{O}}_2}))\end{array}$$

(8)

Three voltage loss terms:

$$V_{\mathrm{act}}=-[{\zeta }_1+{\zeta }_{2}T+{\zeta }_{3}T\ln (C_{{\mathrm{O}}_2})+{\zeta }_{4}T\ln (i)]$$

(9)

$$V_{\mathrm{ohmic}}=i(R_{\mathrm{M}}+R_{\mathrm{C}})$$

(10)

$$V_{\mathrm{con}}=-b\ln (1-\frac{J}{J_{\max }})$$

(11)

$$\begin{array}{ll}{R}_{\mathrm{M}}& =\frac{{\rho }_{\mathrm{M}}l}{A}=\frac{l}{A}\times \\ & \frac{181.6\left[1+0.03(\frac{i}{A})+0.062{(\frac{T}{303})}^2{(\frac{i}{A})}^{2.5}\right]}{[\lambda -0.634-3(\frac{i}{A})]\exp \left[4.18(\frac{T-303}{T})\right]}\end{array}$$

(12)

$$\begin{array}{ll}{P}_{{\mathrm{H}}_2}& =0.5R{H}_a\times P_{O_2}\times \\ & \left[{\left(\exp (\frac{1.635(i/A)}{T^{1.334}})\times \frac{R{H}_a\times P_{{\mathrm{H}}_2\mathrm{O}}}{P_a}\right)}^{-1}-1\right]\end{array}$$

(13)

$$\begin{array}{l}{\log }_{10}(P_{{\mathrm{H}}_2\mathrm{O}})=2.95\times {10}^{-2}(T-273.15)-9.18\times {10}^{-5}\\ {(T-273.15)}^2+1.44\times {10}^{-7}{(T-273.15)}^3-2.18\end{array}$$

(14)

$$P_{{\mathrm{O}}_2}=P_c-P{\mathrm{H}}_{\mathrm{c}}\times P_{{\mathrm{H}}_2\mathrm{O}}-P_{{\mathrm{N}}_2}\times \exp (\frac{0.291(i/A)}{T^{0.832}})$$

(15)

$$P_{{\mathrm{N}}_2}=\frac{0.79}{0.21}{P}_{{\mathrm{O}}_2}$$

(16)

$$C_{{\mathrm{O}}_2}=\frac{P_{{\mathrm{O}}_2}}{5.08\times {10}^6\exp (\frac{-498}{T})}$$

(17)

where T is the absolute temperature (K). $P_{{\mathrm{H}}_2}$ and $P_{{\mathrm{O}}_2}$ refer to the partial pressure (atm) of hydrogen and oxygen. ζ₁, ζ₂, ζ₃, ζ₄, R_C (Ω) and b (V) are six parameters of the cell model. R_C is the resistance to transfer of protons through the membranes, and is usually considered as a constant. R_M and R_C denote the equivalent membrane resistance inside and the equivalent contact resistance to the outside (Ω·cm²). b depends on the cell and its operation state. $C_{{\mathrm{O}}_2}$ is the concentration of oxygen in the catalytic interface of the cathode (mol·cm⁻³). i is the cell operating current (A). J and J_max are the actual current density and the maximum current density, $J=\frac{i}{A}$. l is the thickness of the PEM (cm). A is the cell active area (cm²). ${\rho }_{\mathrm{M}}$ is the membrane’s specific resistivity (Ω·cm). λ is an adjustable parameter. $P_{{\mathrm{N}}_2}$ and $P_{{\mathrm{H}}_2\mathrm{O}}$ represent nitrogen partial pressure at the cathode gas flow channel and the saturation pressure of water vapor, respectively. P_a and P_c are the anode and cathode gas flow channel (atm), respectively. RH_a and RH_c are the relative humidity of vapor in the anode and cathode, respectively.

The parameters’ definition of the semi-empirical mathematical model is essential for approaching the actual performance of PEMFC. Minimizing errors between the output of the model and the actual performance is used as the objective function of this parameter optimization problem.

$$\underset{X}{\min }f(X)={\displaystyle \sum _{j=1}^N{(V_{\mathrm{stackm}}-V_{\mathrm{stack}}(X))}^2}$$

(18)

where X is the parameter vector. V_stackm is the experimental data of output voltage. V_stack is the model output voltage. N is the number of experimental data.

4.2. Experimental Data

A set of experimental data was adopted in which four polarization curves for a 250 W fuel cell were recorded in different process conditions accordingly which were measured at (3/5 bar, 353.15 K), (1/1 bar, 343.15 K), (2.5/3 bar, 343.15 K) and (1.5/1.5 bar, 343.15 K) [17,28,29]. The experimental data used in this case study were taken from four polarization curves, with 15 i–v pairs in each group. Other parameters and the operation conditions are shown in

Table 3. Stack parameters and the range of operation conditions [17,28,29].

Stack Parameters		Range of Operation Conditions
Number of cells in series	N	24
Inlet Anode pressure	Pa (bar)	3.0–1.0
Cell’s effective active area	A (cm)	27
Inlet Cathode pressure	Pc (bar)	5.0–1.0
Nafion 115:5	mil (μm)	127
Stack temperature	T (K)	353.15–343.15
Maximum current density	Imax (A cm−2)	0.86
Relative humidity in anode	Rha	1
Rated power	(W)	250
Relative humidity in cathode	RHc	1

. Two groups of data measured at (3/5 bar, 353.15 K) and (1/1 bar, 343.15 K) were used for estimation, and the remaining two groups of data were used for validation. The upper and lower bounds of parameters are given in

Table 4. Value range of parameters [17,32,33].

Parameters	ζ1	ζ2	ζ3	ζ4	RC	B	λ
Upper bound	−0.80	6.00 × 10−3	1.00 × 10−4	−8.00 × 10−5	9.90 × 10−4	0.60	24
Lower bound	−1.20	8.00 × 10−4	3.50 × 10−5	−3.00 × 10−4	8.00 × 10−5	0.01	10

.

4.3. Optimization and Simulation Results

We applied HMMCA, BIAMC, GAs and CMA-ES to solve this parameter optimization problem.

The parameter settings of the HMMCA were: the amount of repeated trials, N_r = 100; the number of parallel membranes, N_m = 10; the number of algorithm cycle for one trial, C = 35; the stop condition, FET = 30,000; the number of objects sending in each parallel membrane, N_o = 7; the sum of objects in a communication object, N_co = 2; p_cm ≥ 0.8; p_mm ≥ 0.8; p_t ≤ 0.3; p_na ≥ 0.8; the increment used in mutation mode, $(r\times wid{e}_j)/(ml\times c)$. The magnitude of target indication vector, $wid{e}_j\times 0.618/(ml\times c)$. The activating condition of the quasi-Golgi was that the multiplication of the current value of algorithmic cycle by that of membranes does divide exactly by 2.

The parameter settings of BIAMC were: N_r = 100; N_m = 10; C = 35; FET = 30,000; N_o = 7; N_co = 2; p_mm = 1; p_cm = 1; p_t = 0.3; p_a = 1. The increment used in mutation mode was $(r\times wid{e}_j)/(ml\times c)$. The magnitude of target indication vector was $wid{e}_j\times 0.618/(ml\times c)$. The activating condition of the quasi-Golgi was that the multiplication of the current value of algorithmic cycle by that of membranes was exactly divisible by 2. Most of the parameter settings were the same as the subsection of function examination, except the parameters for mutation mode and the target indication rule were changed for a faster searching.

The main optional parameter settings of ga were: the amount of repeated trials, N_r = 100, ‘PopulationSize’ = 50, ‘MutationFcn’ = mutationadaptfeasible, ‘CrossoverFraction’ = 0.9, ‘MigrationFraction’ = 0.2, ‘SelectionFcn’ = selectionstochunif, ‘CrossoverFcn’ = crossoverheuristic/crossoverintermediate, ‘Generations’ = 600, ‘StallGen-Limit’ = 600, ‘Display’ = off. Other optional parameters were default.

The changed parameter settings were: the amount of repeated trials, N_r = 100; sigma = min((x^u − x^l)/3); Options: MaxFunEvals = 30,000; MaxIter = Inf; TolHistFun = 1/Inf; TolUpX = Inf; TolFun = 1/Inf; TolX = 1/Inf; Restarts = 2; StopFunEvas = 30,000; StopOnWarnings = no; StopOnStagnation = off; StopOnEqualFunctionValues = Inf; StopFitness = -Inf; etc.

All the results in

Table 5. The results calculated by HMMCA, BIAMC, GA and CMA-ES.

A	Min	Mean	max	avg.t(s)	st.dev
HMMCA	5.4881	5.8091	6.2466	6.28	1.61 × 10−1
BIAMC	5.5105	5.8171	6.3642	6.36	1.76 × 10−1
CMA-ES	5.4639	5.4639	5.4640	15.85	1.00 × 10−5
GA	5.6624	6.2618	8.3809	9.12	4.31 × 10−1

were obtained by running them once in MATLAB 2009b.

Meanwhile, the parameters calculated by four algorithms in [17,28,29] were introduced directly and their square deviations on the estimation data were used for comparison. In 100 groups of parameters calculated by each algorithm, the group of parameters that minimized the square deviation on the estimation data (SD1) are listed in

Table 6. Comparison with results from [17,28,29].

A	ζ1	ζ2 × 10−3	ζ3 × 10−5	ζ4 × 10−4	RC × 10−4	b	λ	SD1
HMMCA	−0.9113	2.8731	7.4531	−1.1482	1.9944	0.0301	12.0815	5.4881
BIAMC	−1.1257	3.2957	6.1573	−1.1207	1.6779	0.0273	11.2850	5.5105
GA	−0.9415	3.0181	7.5145	−1.1196	6.7769	0.0346	13.9963	5.6624
CMA-ES	−1.2000	3.4962	6.0781	−1.1618	0.8000	0.0299	11.9888	5.4639
BIPOA	−0.8016	2.6673	8.1288	−1.2713	0.8000	0.0324	13.5158	5.6053
RGA	−1.1568	3.4243	6.4161	−1.1544	1.4504	0.0343	12.8989	5.5676
HGA	−0.944957	3.01801	7.4010	−1.8800	1.0000	0.02914489	23.0000	11.3400
SGA	−0.94731	3.0641	7.7134	−1.9390	2.7197	0.023981	19.7650	13.1686

. The evaluation of computational cost used in the four algorithms in [17,28,29] was different from this paper. According to the data shown in those references, even if the objective function was evaluated once for a candidate solution in one generation, the minimum FET was 20,000.

It is shown that CMA-ES is superior to HMMCA at solving this seven-dimensional parameter optimization problem considering the explored best parameters with the smallest SD1. However, HMMCA is consistently excellent compared with other performances, such as avg.t, st.dev, mean and the minimal value of objective function listed in Table 5. It can be seen that all the best parameters in Table 6 are close neighbors of the parameters calculated by the CMA-ES algorithm. Perhaps it will be helpful to explore the best parameters to apply other termination conditions or parameter settings for HMMCA and BIAMC.

5. Conclusions

To goal of precisely solving as many types of complicated optimization problems as possible with lowest computational complexity is pursued by all optimization algorithms. Distributed structures, nondeterministic and evolutionary mechanism are generally used for most optimization algorithms. The hybrid mode membrane computing algorithm (HMMCA) was proposed in this paper as a modified algorithm of the bio-inspired algorithm based on membrane computing (BIAMC), which contained a new netted membrane structure together with a communication rule, a nondeterministic computation mechanism of evolutionary programming used for the communication rule, rewriting rule, target indication rule and abstraction rule. The test results on benchmark functions with 30D and 100D, and a seven-dimensional parameter optimization problem of PEMFC revealed its higher efficiency, global searching and robustness of computational performance in general, compared with the test results of BIAMC, CMA-ES and GAs.