Ke Yang,Yijun He*,Zifeng Ma

Shanghai Electrochemical Energy Devices Research Center,Department of Chemical Engineering,Shanghai Jiao Tong University,Shanghai 200240,China

1.Introduction

Microbial fuel cells(MFCs)provide a promising technology not only to renewable electricity generation,but also to wastewater treatment[1].MFCs can utilize a variety of materials as carbon sources such as natural organic matter,complex organic waste or renewable biomass.In comparison with the conventional fuel cells,MFCs use living organisms as a biocatalyst and often operate at mild conditions[2–5].Although MFCs have,in recent years,attracted significant attentions from both academia and industry,the intrinsic low biochemical reaction rate of MFCs technically hinders the successful commercial application of MFCs for energy generation or wastewater treatment.On the other hand,the complex bio-electrochemical reactions and transport phenomena are not well understood,which further limits the optimal design and operation ofMFCs.Hence,debottlenecking ofMFCs requires a breakthrough of multidisciplinary cooperation frommicrobiology,electrochemistry,electrical engineering and material science.

Apart from experimental research,mathematical modeling approach provides an alternative promising tool to understand the reaction and transport mechanisms in MFCs.Both lumped and distributed parameter models with different complexities have been developed[6–13].Based on the developed models,itis able to investigate both the design parameters and operating condition effects on the performance of MFCs using sensitivity analysis,which helps to understand the importance ofparameters on system performance[11].Moreover,optimization method can be applied for attaining optimal design and operation of MFCs[13–16].Although the MFCs can simultaneously perform electricity generation and waste treatment,most of the published studies about performance optimization ofMFCs only focus on a single objective such as maximizing power output[13–16].Both experimental and simulation results have shown that the power output cannot be improved without the sacrifice of the maximal waste removal ratio.In addition,obtaining maximal power output often needs degrade the maximal attainable current density.That means these objectives,namely power density,attainable currentdensity and waste removalratio,mightbe conflicting,which implies that there does not exhibit a single solution that simultaneously optimizes each objective.Without a clear understanding of the relationships among those conflicting objectives,it is hard to make reasonable decisions for balancing trade-offs among conflicting objectives.In that case,multi-objective optimization methods are often used to obtain a set of Pareto-optimal solutions.Although it has been recognized that it is important to perform quantitative analysis of these conflicting objectives for obtaining a thorough understanding of their relationships,to our knowledge,there are rare studies about multi-objective optimization and multi-criteria decision making for optimal design and operation in the MFC community.In this study,we will try to establish a systematic multi-objective optimization framework to simultaneously maximize the power density,maximal attainable current density and waste removal ratio.The relationships among these objectives will be thoroughly investigated using the graphical visualization technique,i.e.level diagrams.In addition,the effects of operating conditions on those competing objective will be also investigated.

The remainder of this paper is structured as follows.The mathematical model and multi-objective optimization problems of the twochamber MFCs are first introduced.The detailed implementations of the multi-objective genetic algorithm and the level diagrams method are next proposed.The results and discussion are then presented.Finally,conclusions and future research directions are provided.

2.Two-chamber Microbial Fuel Cells

2.1.Principle and experimental set-up of MFC

MFCs can be broadly divided into two general categories,those that use a mediator and those that are mediator-less.These two types of MFCs are classified based on two transfer electron mechanisms,i.e.indirect electron transfer and direct electron transfer.The former uses a chemical mediator to transfer electron from microorganisms in the cell to the anodic electrode.The oxidized mediators arefirstly reduced by capturing electrons from within the membrane of and release the electrons to the electrode.Modern developed MFCs are often mediator-less,in which the microorganisms typically have electrochemically active redox proteins on their outer membrane that can transfer electrons directly to the anode.There are several electrochemically active microorganisms such asShewanella putrefaciens[17]andRhodoferax ferrireducens[18]which could form a bio film on the anode surface and transfer electrons directly to electrode across the membrane.These microorganisms often have high Coulombic efficiency and can avoid the use of the expensive and toxic mediators.

A typical two-chamber MFC consists of an anodic chamber and a cathodic chamber separated by a cation-exchange membrane(CEM).In the anodic compartment,organic fuel is oxidized by microorganisms to generate carbon dioxide,electrons and protons.Electrons and protons are transferred to the cathodic compartment through an external electric circuit and the CEM,respectively,and then consumed at the cathode through combining with oxygen to form water.

The schematic diagram of the experimental set-up of acetate two-chamber MFC can be referred to the studies reported elsewhere[13,16].The anode and cathode chambers are separated by a Na fion CEM.Acetate fuel and air-saturated water are continuously fed into the anode and cathode chambers respectively.Nitrogen gas is purged to keep the anode chamber anoxic.Each compartment contains two pieces of graphite felt as electrodes,but the graphite felt used for the cathode is coated with platinum powder.The multimeter is used to measure the potential between the anode and cathode,and a personal computer is used to record the potential through a data-acquisition system.Given a specific operating condition,the output power can be regulated through the change of the external resistance.

2.2.Two-chamber MFC model

Although there are different modeling approaches available in published literature concerning the MFCs,most reported models consisting of partial differential equations[7–11]have inherent complexity and heavy computational cost,which may not meetthe requirements of optimization and control in a real-time way and consequently may not be readily implemented by the majority ofthe MFC community.Recentdeveloped ordinary differential equations based lumped parameter models[12,13],which can be able to adequately capture MFC behavior atvarious operating conditions,could significantly decrease the computational cost and provide an alternative solution for meeting the requirements of optimal design,operation and control of MFCs.It is worth noting that although simplified models neglect spatial nonuniformities and cannot describe all the detailed phenomena occurring in the MFC,itmay be the only practicaland effective way for performing real-time optimization and control.Hence,in this study,the MFC model developed by Zenget al.[13]is used to perform multi-objective optimization.In this model,both the anodic and cathodic compartments are treated as a continuously stirred tank reactor(CSTR).It should be mentioned that although the CSTR model neglects the bio film formation on the anode side,both fitting and validation results have illustrated the effectiveness of this CSTR model,which is ready for further model analysis and optimization.The mathematical description of a two-chamber MFC CSTR model developed by Zenget al.[13]is briefly shown as follows.

For acetate fueled two-chamber MFC,the oxidation of acetate and the reduction of dissolved oxygen are shown in Eqs.(1)and(2)at the anode and cathode,respectively.

The rates of electrochemical reaction at the anode and cathode are formulated in Eqs.(3)and(4)using Butler–Volmer type expressions respectively.

The mass balances of the four components in the anode chamber,namely acetate,dissolved carbon dioxide,hydrogen ion and biomass,are shown in Eqs.(5)–(8)respectively.

where subscript‘a(chǎn)’denotes the anode.

The mass balances of the three components in the cathode chamber,namely dissolved oxygen,hydroxyl and cation M+,are shown in Eqs.(9)–(11)respectively.

where subscript ‘c’denotes the cathode;NMis the flux of M+ions transported from the anode to cathodeviathe CEM and is shown in Eq.(12).

The charge balances at the anode and cathode are given by Eqs.(13)and(14)respectively.

where ηaand ηcdenote the overpotential of anode and cathode respectively;Fis Faraday's constant andicellis the current density.

The cellvoltageUcellis calculated as Eq.(15)by neglecting the ohmic drops in the current-collectors and electric connections.

In practical,the output power is regulated through the change of external resistanceRex,and the cell current is calculated withIcell=Ucell/Rex.Note that the relationship between the current density and the cell current isicell=Icell/Am,whereAmis the area of membrane.Thus,the current density can be calculated as Eq.(16)by means of external resistance.

By replacing the current density in Eqs.(12)–(14)with the external resistance using Eq.(16),the MFC model becomes more practical and could avoid determining the maximal current density for attaining meaningful system output.For solving the steady-state MFC model,ther1andr2can be firstly determined when using the current density as the input;the concentrations of seven state variables can be then calculated using Eqs.(5)–(11); finally,the overpotential of anode and cathode can be analytically computed using Eqs.(3)–(4)respectively.However,ifan inappropriate currentdensity is selected,the systemoutput would be illogical,e.g.the cell voltage mightbecome negative.If the external resistance is used,unreasonable system output can be effectively avoided.However,it is worth noting that the steady-state MFC model needs to be solved using a nonlinear optimization method because of the existence of coupling between external resistance and overpotential.

The description and nominalvalues ofthe above parameters and operating conditions are summarized in Table 1 obtained from Zenget al.[13].Ithas been recognized thatthe performance ofMFCs is strongly dependent on operating conditions,such as pH,temperature,organic loading rate,feed rate and shear stress.Detailed analyses of the influences of operating conditions on the MFC performance can be referred to the reported review[2].In this study,four operating conditions,namely flow rate(Qa)and concentration(CiAnC)of acetate to anode,flow rate feeding to cathode(Qc)and external resistance(Rex)are selected as decision variables.

2.3.Multi-objective optimization problem of MFCs

A general mathematical formulation of multi-objective operational optimization problem for MFCs is as follows:

where x and y denote the vectors ofcontrolvariables and state variables respectively;xminand xmaxare the lower and upper bounds of control variables respectively;Hrepresents the steady-state MFC modelobtained from Eqs.(3)–(16);F is theM-dimensionalobjective function vector,fm(x,y)indicates themth objective function.

To evaluate the MFC performance,three objectives,namely power density,attainable current density and waste removal ratio,are used.

Table 1Nominal values of parameters and operating conditions for MFC model[13]

The power densityPDcellis calculated as

Given specific values ofQa,CiAnCandQc,the currentdensity increases with the decreasing of the external resistance,and the maximal attainable current densityicell,maxcan be calculated using the minimal external resistanceRex,min.The larger value oficell,maxindicates the better current regulation ability of MFCs.

The waste removal ratio WRR based on the inlet and outlet acetate concentration is defined as

WRR provides a quantitative criterion of waste treatment ability.If the outlet acetate concentration is equal to 0,it implies that the overall waste is ideally removed and the value ofWRR is equalto 1;ifthe outlet acetate concentration is equal toCiAnC, it means that there is no waste is removed and the value of WRR is equal to 0.Note that the value of WRR lies in the range[0,1].The larger value of WRR indicates the better waste treatment ability of MFCs.

Optimal operation of MFCs should balance the trade-offs among the following maximization criteria:power density,maximal attainable current density and waste removal ratio.It is worth noting that these objectives are generally conflicting,and consequently there exists no single solution that simultaneously optimizes all objective functions.Thatmeans none ofthe objectives can be improved withoutthe sacrifice of the other objectives.Therefore,it is desirable to use the multiobjective optimization method to generate a set of Pareto-optimal solutions for revealing the relationship among these conflict objectives.In this study,we will thoroughly investigate three bi-objective optimization problems,namely maximization of power density and waste removal ratio,maximization of power density and maximal attainable current density and maximization of waste removal ratio and maximal attainable current density,and one three-objective optimization problem,namely simultaneous maximization of power density,waste removal ratio and maximal attainable current density.

3.Methodology

3.1.Multi-objective optimization method

Multi-objective optimization(MOO)is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints.MOO has been widely applied in many fields such as engineering and economics,in which optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives.For a nontrivial MOO problem,there usually does not exhibit a single solution that simultaneously optimizes each objective and there exists a set of Pareto-optimal solutions.A solution is called nondominated or Pareto-optimal,if none of the objective functions can be improved in value without degrading some of the other objective values.The region defined by the objectives for all Pareto-optimal solutions is often called the Pareto front.The basic goal of MOO methods is to find a uniformly distributed set of Pareto-optimal solutions,which could approximate discretely the Pareto front.

The MOO methods can be broadly decomposed into two categories:scalarization approaches and Pareto approaches.The former solves the MOOproblems by translating itback to a single(or a series ofsingle)objective scalarproblem.The formation ofaggregate objective function requires prescribinga prioripreferences or weights between objectives.Weighted sum approach,goal programming and lexicographic approach are three common types ofscalarization methods.In comparison with the scalarization approach,the Pareto approach could obtain a set of Pareto-optimal solutions in one run showing a more efficient way for solving MOO problems.Multi-objective genetic algorithms,adaptive weighted sum method and normal boundary intersection are three types of Pareto approach.In practice,it can be very difficult to precisely and accurately set up the preferences or weights,even for someone familiar with the problem domain.Therefore,the best way to solve the MOO problems is through the generation of a set of Pareto-optimal solutions using Pareto approaches,which could provide a wide range of decision options to the practitioners.

Genetic algorithm(GA)is a kind of population-based optimization methods based on the principles of the evolutionviagenetics operation and natural selection and has been widely applied in engineering optimization[19].In a GA,a population of solutions to optimization problems evolves toward better solutions using techniques inspired by natural evolution,such as selection,mutation,and crossover.Non-dominated sorting genetic algorithm(NSGA)[20]and its enhanced variants,i.e.elitist NSGA-II[21]and controlled elitist NSGA-II[22],are an outstanding type of Pareto approach for solving MOO problems.The controlled elitist NSGA-II algorithm not only favors individuals with better fitness value,but also favors individuals that can help increase the diversity of the population even if they have a lower fitness value.The characteristics of trade-off between exploration and exploitation in controlled elitist NSGA-II could play an important role in maintaining the diversity of population for convergence to an optimal Pareto front.The details of controlled elitist NSGA-II algorithm can be referred to Deb and Goel's study[22]and the implementation procedure of NSGA-II is briefly described below.

Step 1:Initially create a random parent populationP0of sizeNum,and assign each solution a fitness equal to its non-domination level obtained by fast non-dominated sorting method.

Step 2:Use binary tournament selection,recombination and mutation operators to create a child populationQ0of sizeNum.

Step 3:Sett=0.

Step 4:Form a combined populationRt=Pt∪Qtof size 2Num,and each solution ofRtis assigned a fitness equal to its non-domination level.

Step 5:Generate the new parentpopulationPt+1ofsizeNum,which uses a geometric distribution to maintain the number of individuals in each front.

Step 6:Generate a new child population of sizeNumfrom parent populationPt+1by using binary tournament selection,recombination and mutation operators.

Step 7:Sett=t+1;iftis less than maximum generation number,go back to Step 4;otherwise,exit and output the optimization results.

3.2.Graphical visualization of Pareto front

Once obtaining a set of Pareto-optimal solutions using NSGA-II,the next important step is to select one,or more,solutions inside the Pareto-optimal solutions.Without given sufficient preferences,visualization techniques could play a significant key role in helping decision-makers make proper decisions.Scatter diagrams and parallel coordinates are two common types of visualization techniques for multidimensional visualization of Pareto front.For the scatter diagrams,the complexity of the representation increases significantly with the dimension.The parallel coordinates,on the other hand,shows a compact representation through plotting a multidimensional point in a two-dimensional graph,but it would lose clarity and be difficult to perform analysis for large number of Pareto-optimal solutions.Level diagrams(LD)method,developed by Blascoet al.[23],is a new graphical visualization of both the Pareto front and set,and can be easily combined with preferences for helping the decisionmaker.The LD method firstly classifies the Pareto front points by means of their proximity to ideal points measured with a specific norm of normalized objectives and then synchronizes the objective and parameter diagrams.A description of the LD method is briefly described as follows.

Let xi∈ X?? X denote theith Pareto-optimal solution,where X?is the obtained finite set of Pareto-optimal solutions using NSGA-II,andfm(xi) ∈ F(X?)denote themth objective value in Pareto front with respect to theith Pareto-optimal solution.The maximum and minimum values of themth objective on the approximated Pareto front are given by Eqs.(20)and(21),respectively.

A smaller value of‖F(xiàn)（x）‖p(p=1,2,∞)indicates a better solution.Based on the above normalization and norm computation procedures,every Pareto-optimal solution is classified and the points of the Pareto front are sorted in ascending order of the value of‖F(xiàn)（x）‖p.In the level diagrams,each objective or decision variable has its own graphical representation.The verticalaxis on allgraphs corresponds the levelvalue of‖F(xiàn)（x）‖p,which means that all graphs are synchronized with respect to this axis.The horizontal axis corresponds to values of the objectives,or decision variables.Note that level diagrams draw a point at the same level for all graphs in objective and decision variable space,which is useful for making proper decisions.Further detailed information about the LD method can be referred to the study of Blascoet al.[23].It should be noted that although the classical scatter diagrams are sufficient for a bi-objective problem,the level diagrams can provide more information about the distance to the ideal point,which is useful for decision-maker.In this study,the LD method is only carried out for the three-objective optimization problem.

4.Results and Discussion

The above described controlled elitist NSGA-II and level diagrams are applied to multi-objective operational optimization of twochamber MFC.Three bi-objective optimization and one threeobjective optimization problem are investigated.Both controlled elitist NSGA-II algorithm and LD method are implemented in MATLAB,and all computations are carried out on a PC with 2.60 GHz processor and 8 GB of RAM.The parameter values of controlled elitist NSGA-II algorithm are shown as follows:number of generations is 500,population size is 500,cross fraction is 0.8,uniform mutation rate is 0.01,and Pareto fraction is 0.25.The lower bounds ofQa,CiAnC,QcandRexare 0.5×10-5m3·h-1,1 mol·m-3,0.5 × 10-3m3·h-1and 10 Ω,respectively;the upper bounds ofQa,CiAnC,QcandRexare 3 × 10-5m3·h-1,2 mol·m-3,2 × 10-3m3·h-1and 10000 Ω,respectively.

4.1.Maximizing power density and waste removal ratio

The Pareto front obtained by maximizing power density and waste removal ratio without considering the criterion of attainable current density is presented in Fig.1.The Pareto front is smooth and well distributed over a wide range.It is found that as the power density increases,the waste removal ratio would decrease,indicating that a conflict exists between power density and waste removal ratio.It also means that improvement of power density would result in deterioration of waste removal ratio,and vice versa.From Fig.1,the maximum power density of 3.82 W·m-2corresponds to the minimum waste removalratio of0.67,while the maximumwaste removalratio of0.97 corresponds to the minimum power density of 0.64 W·m-2.It can be observed that at high power density region,waste removal ratio can be significantly improved with a relatively small sacrifice in power density.For example,increasing of waste removal ratio from 0.67 to 0.80 only leads to decreasing of power density from 3.82 to 3.50 W·m-2.While at high waste removal ratio region,achieving a relatively small improvement of waste removal ratio needs a much higher sacrifice in power density.For example,increasing of waste removal ratio from 0.90 to 0.97 would result in a significant decreasing of power density from 2.52 to 0.64 W·m-2.If a MFC is designed to only generate electricity,the Pareto point with maximum power density of 3.82 W·m-2becomes a best option.On the other hand,if a MFC is designed to wastewater treatment,the Pareto point with maximum waste removal ratio of 0.97 is a best option.However,practical MFC operation often avoids the extreme objective values and calls for a trade-off between power density and waste removalratio.Moreover,although the obtained Pareto front provides a wide range of competing options for optimal operation from a multi-objective perspective,additional preferences should be provided for assisting in final decision-making.The maximal attainable currentdensity(icell,max)againstpowerdensity and waste removal ratio corresponding to the Pareto front in Fig.1 is also plotted in Fig.2.Itis observed that(1)the maximum oficell,maxis 8.87 A·m-2;and(2)the maximal attainable current density increases with the increasing of power density,but with the decreasing of waste removal ratio.It is implied that without considering the criterion of maximization oficell,max,maximal attainable current density and power density seem to be not conflicting,where improving power density would also result in increasing attainable current density,whereas maximal attainable currentdensity and waste removalratio are conflicting,where waste removal ratio is improved with the sacrifice of attainable current density.However,itshould be noted thatiftreating attainable currentdensity as optimization objective,the exhibition of a clear conflict between power density and attainable current density would be illustrated and the maximum oficell,maxcan be significantly improved.

Fig.1.Pareto front obtained from the simultaneous maximization of power density and waste removal ratio.

To quantify the decision variables'role in realizing the competing objective values shown in Fig.1,Fig.3 plots the decision variables against power density and waste removal ratio.Fig.3(A)shows a clear increase in flow rate of fuel feed to anode(Qa)with increasing power density and decreasing waste removal ratio.The possible reason is that increasing ofQaresults in an increase of available carbon source,which favors to improve power output.However,it should be noted that the maximum value ofQais 1.1 × 10-5m3·h-1corresponding to maximal power density of 3.82 W·m-2in Fig.1,which only lies in the medium region in the search space.It is,thus,implied that further increasing ofQawill not improve power output.On the other hand,increasing ofQawould decrease the residence time of reactants,which might result in reducing waste removal capability.Fig.3(B)shows that almost the largest available concentration of fuel(CiAnC) is chosen uniformly for the entire Pareto.It indicates that increasing concentration of fuel is greatly helpful to increase both power density and waste removal ratio.Fig.3(C)indicates that the values of flow rate of feeding to cathode(Qc)are scattering and have no clear trends compared to the smooth variation of the Pareto front they generated(Fig.1).A two-chamber MFC is often operated with an excessive oxygen supply in the cathode for maximizing power output and stimulating waste removal.Based on our simulation results,it is found that if fixing a relatively highQcvalue of 2.0×10-3,the obtained Pareto front is almost the same as thatin Fig.1.Itis,thus,implied that maintaining a technical feasible high enough flow rate of feeding to cathode(Qc)could be helpful to maximize power output and waste removal.However,it is worth noting that if a MFC has been operated at an anode reaction limiting region,a furtherincrease ofQcwould be meaninglessand cannotimprove power output and waste removal.From Fig.3(D),it can be observed that(1)at the low power density region of[0.60,1.44]W·m-2and high waste removal ratio region of[0.95,0.97],external resistance(Rex)approximately increases linearly from 70 to 170 Ω with the increasing of power density and the decreasing of waste removal rate;and(2)at the power density region of[1.44,3.82]W·m-2and waste removal ratio region of[0.64,0.95],the values of internal resistance(Rex)almost uniformly scatter at the region of[170,190]Ω.In general,the external resistance is optimally manipulated to regulate the current for tracking the maximal power point.Fig.4 plots the currentdensity(icell)againstexternalresistance corresponding to the Pareto frontin Fig.1.When the externalresistance ismanipulated from70 to 170 Ω,the current density of MFC almost remains constant at 4.2 A·m-2.When the external resistance varies from 170 to 190 Ω,the values of current density scatter at the region of 4.2-6.6 A·m-2.Moreover,the currentdensity againstpowerdensity and waste removal ratio is also plotted in Fig.5.Itis observed that increasing of power density mostly requires the increasing of current density,whereas increasing of waste removal ratio is mostly attributed to the decreasing of current density.

Fig.2.Maximal attainable current density versus power density()and waste removal ratio(○)corresponding to the Pareto front in Fig.1.

Fig.3.Decision variables versus power density()and waste removal ratio(○)corresponding to the Pareto front in Fig.1.

Fig.4.Externalresistance versus currentdensity corresponding to the Pareto front in Fig.1.

Fig.5.Current density versus power density()and waste removal ratio(○)corresponding to the Pareto front in Fig.1.

4.2.Maximizing power density and maximal attainable current density

Fig.6 shows the Pareto frontby maximizing power density and attainable current density without considering the criterion of waste removal ratio.It is observed that power density decreases with the increasing of the maximal attainable current density.The maximum power density of 3.78 W·m-2corresponds to the minimum maximal attainable current density of 9.96 A·m-2,while the maximum maximal attainable current density of 14.88 A·m-2corresponds to the minimum power density of 2.28 W·m-2.In comparison with Fig.2,it is obviously found that maximalattainable currentdensity can be significantly improved when explicitly considering it as an optimization objective.Moreover,it is observed that at high maximal attainable current density,achieving a relatively small improvement of maximal attainable current density needs a much higher sacrifice in power density.For example,increasing of maximal attainable currentdensity from14.49 to 14.88 A·m-2would resultin a significant decreasing of power density from 2.80 to 2.28 W·m-2.Fig.7 plots the waste removal ratio against power density and maximal attainable current density corresponding to the Pareto front in Fig.6.It is found that waste removal ratio increases with the increasing of power density,whereas it increases with the decreasing of maximal attainable current density.The maximum waste removal ratio is only 0.67,which approximately corresponds to the minimum waste removal ratio in Fig.1.This means withoutexplicitly considering waste removalratio as optimization objective,itfailed to obtain a relatively high waste removalratio.Hence,it is important to explicitly introduce waste removal ratio as an optimization objective if a MFC is designed to simultaneous electricity generation and waste treatment.

Fig.8 plots the decision variables against power density and maximal attainable current density corresponding to the Pareto front in Fig.6.From Fig.8(A),it is observed that power density decreases with the increasing ofQa,whereas maximal attainable current density increases with the increasing ofQa.It is also found that the operating region ofQais(1.3-2.6)× 10-5m3·h-1,which is significantly higher than that in Fig.3(A).Fig.8(B)shows that almost the largest availableCiAnCof 2 mol·m-3is chosen for the entire Pareto.It is thus indicated that achieving higher maximal attainable current density requires a substantial increase ofQa,but increasing ofQacannot further improve the maximal attainable current density without increasingCiAnC. Fig.8(C)shows that almostQcscatters at a narrow region of(1.95-2.0)× 10-3m3·h-1for the entire Pareto.Fig.8(D)shows that at the low power density region of 2.28-3.20 W·m-2,almostRexis chosen in a narrow region of 76-82 Ω,whereas at the high power density region of 3.20-3.78 W·m-2,power density approximately linearly increases with the increasing ofRex.Note that the plot of maximal attainable current density againstRexis meaningless,because maximal attainable current density is calculated at the minimal value ofRexof 10 Ω.Fig.9 plots the current density(icell)againstRexcorresponding to the Pareto front in Fig.6.WhenRexis manipulated from 76 to 90 Ω,the values oficellscatter at the region of 7.7-9.2 A·m-2.WhenRexvaries d from 90 to 142Ω,icellshows an approximately lineardecreasing trend with the increasing ofRex.Moreover,the current density against power density and maximal attainable current density is plotted in Fig.10.It is observed that at the high power density region of 3.20-3.78 W·m-2,power density decreases with the increasing oficell,whereas at the low power density region of 2.28-3.20 W·m-2,power density increases with the increasing oficell.

Fig.6.Pareto front obtained from the simultaneous maximization of power density and maximal attainable current density.

Fig.7.Waste removal ratio versus power density()and maximal attainable current density(○)corresponding to the Pareto front in Fig.6.

4.3.Maximizing waste removal ratio and maximal attainable current density

Fig.11 shows the Pareto front by maximizing waste removal ratio and attainable current density without considering the criterion of power density.It is observed that waste removal ratio decreases with the increasing of the maximalattainable currentdensity.The maximum waste removal ratio of 0.98 corresponds to the minimum maximal attainable current density of 4.20 A·m-2,while the maximum maximal attainable currentdensity of14.88 A·m-2corresponds to the minimum waste removalratio of0.67.Atthe high maximalattainable currentdensity region of 12.96-14.88 A·m-2,a relatively small improvement oficell,maxwould result in a significant deterioration of waste removal ratio.Fig.12 plots the power density against waste removal ratio and maximal attainable current density corresponding to the Pareto front in Fig.11.It is observed that power density approximately linearly increases with the increasing of maximal attainable current density,whereas it decreases with the increasing of waste removal ratio.It is also found that the maximum power density is only 1.16 W·m-2without explicitly considering it as an optimization objective.It is worth noting that the indication of maximal attainable current density is considered to be of importance only when it is combined with the power density for evaluating the MFC performance in electrical applications.This study only attempts to illustrate the relationship between maximal attainable current density and waste removal ratio,practical optimization of maximal attainable current density should be performed simultaneously with power density.

Fig.9.Externalresistance versus currentdensity corresponding to the Pareto front in Fig.6.

Fig.10.Current density versus power density()and maximal attainable current density(○)corresponding to the Pareto front in Fig.6.

Fig.11.Pareto frontobtained from the simultaneous maximization of waste removalratio and maximal attainable current density.

Fig.12.Power density versus waste removal ratio()and maximal attainable current density(○)corresponding to the Pareto front in Fig.11.

Fig.13 plots the decision variables against waste removal ratio and maximal attainable current density corresponding to the Pareto front in Fig.11.It is observed from Fig.13(A)that waste removal ratio decreases with the increasing ofQa,whereas maximal attainable current density increases with the increasing ofQa.The operating region ofQais(0.5-2.6) × 10-5m3·h-1,which is much wider than that both in Figs.3(A)and 8(A).From Fig.13(B),it is observed that almost the largest availableCiAnCof 2 mol·m-3is chosen for the entire Pareto,which is similar with Figs.3(B)and 8(B).Fig.13(C)and(D)shows that almost the values ofQcscatter at the region of 1.3-2.0 m-3·h-1and almost the values ofRexscatter at a narrow region of 11.1-11.5 Ω.From Fig.14,it is found that the values oficellcan vary from 4.2 to 14.9 A·m-2with a small variation inRex.The large variation inicellis mainly attributed to the large variation inQa.Fig.15 shows the plot oficellagainst maximal attainable current density and waste removal ratio corresponding to the Pareto front in Fig.11.It is observed that waste removal ratio shows a decreasing trend with the increasing oficell.The linear relationship with an approximate slope of 1 between maximal attainable current density andicellis attributed to the fact that almost the values of Pareto optimalRexapproach its minimum value of 10 Ω,at which the maximal attainable current density is calculated.Moreover,from Figs.12 and 15,it is found that power density approximately increases with the increasing oficell.

4.4.Maximizing power density,waste removal ratio and maximal attainable current density

Fig.13.Decision variables versus waste removal ratio()and maximal attainable current density(○)corresponding to the Pareto front in Fig.11.

The Pareto front obtained by simultaneously maximizing power density,waste removal ratio and maximal attainable current density is shown in Fig.16.It is obtained that(1)the maximum power density of 3.82 W·m-2corresponds to the maximal attainable current density of 8.86 A·m-2and the waste removal ratio of 0.67;(2)the maximum waste removal ratio of 0.97 corresponds to the power density of 0.39 W·m-2and the maximal attainable current density of 4.21 A·m-2;and(3)the maximum maximal attainable current density of 14.88 A·m-2corresponds to the power density of 2.28 W·m-2and the waste removalratio of0.35.Moreover,the minima ofpower density,waste removal ratio and maximal attainable current density are 0.39 W·m-2,0.35 and 4.10 A·m-2,respectively.It is thus implied that achieving the highest waste removal ratio would result in the minimum power output,whereas a highest maximal attainable current density is achieved with the lowest waste removal ratio.

Fig.17 shows a 2-Dvisualization ofthe Pareto frontby the projection ofthe 3-DPareto frontin Fig.16.Itis observed from Fig.17(A)that(1)at the high waste removal ratio region of 0.80-0.97,it decreases with the increasing of power density;(2)at the low waste removal region of 0.35-0.55,it increases with the increasing of power density;and(3)at the medium waste removal region of 0.55-0.80,the relationship between waste removal ratio and power density shows scattering characteristic.Moreover,it is observed from Fig.17(A)that the low power density region of 0.39-2.30 W·m-2is mostly obtained at the high waste removal ratio region and the high power density region of 3.0-3.82 W·m-2is obtained at the medium waste removal ratio region.From Fig.17(B),it can be seen that(1)at the low maximal attainable current density region of 4.10-9 A·m-2,power density approximately increases with the increasing of maximal attainable current density;(2)at the high maximal attainable current density region of 12-15 A·m-2,increasing of power density would result in a decreasing of maximal attainable current density;and(3)at the medium maximal attainable current density region of 9-12 A·m-2,the changes in power density with respect to maximal attainable current density show scattering characteristics.It is seen from Fig.17(C)that at both the low and high waste removal ratio regions,waste removal ratio decreases with the increasing of maximal attainable current density,whereas at the medium waste removal ratio region,waste removal ratio shows a decreasing scattering characteristic with respect to maximal attainable current density.

According to the Pareto front in Fig.16,the LD method is applied to aid in graphical visualization and optimal decision making.Fig.18 shows the 2-norm level diagram representation of the Pareto front in Fig.16.It is found that the lowest level value is always over 0.52.That means the Pareto front is relatively far from the ideal point.It should be noted that in the LD method,all points have been normalized and the ideal point is thus of[0,0,0]for three-objective maximization problems.Moreover,note that the points situated at the lower levels correspond to the zones of the Pareto front nearer to the ideal point.It is observed from Fig.18 that there exhibits several points at the lower levels,which means that there are several points relatively close to the ideal point.The nearest points to the ideal are approximately at 2.84-3.25 W·m-2for power density,0.69-0.78 for waste removal ratio and 10.82-12.19 A·m-2for maximal attainable current density.These points provide a set of promising candidates for final optimal decision making.However,it should be noted that the LD method with 2-norm treats each objective equally,and level computation is based on the distance from the ideal point.The final decision would be significantly affected if each criterion is given a proper preference or constraint.If power density needs to be above 3.50 W·m-2,it can be seen from Fig.18 that the 2-norm level should be at the range of 0.57-0.74.At this level,the waste removal ratio and maximal attainable current density are situated at the ranges of 0.57-0.73 and 8.73-11.81 A·m-2,respectively.If waste removal ratio needs to be above 0.95,it is found that the 2-norm level should be above 1.20,which means that these points are far from the ideal point.At this level,the power density and maximal attainable current density are situated at the ranges of 0.39-1.09 W·m-2and 4.10-5.88 A·m-2,respectively.That means achieving extreme high waste removal ratio needs significantly sacrifice power density and maximal attainable current density.Moreover,if maximal attainable current density needs to be above 14.0 A·m-2,the 2-norm level should be at the range of 0.74-1.10,which means that these points are relatively far from the ideal point.At this level,the power density and waste removal ratio are situated at the ranges of 2.28-3.03 W·m-2and 0.35-0.55,respectively.It also implies that achieving the highest maximal attainable current density would result in the lowest waste removal ratio.

Fig.14.External resistance versus current density corresponding to the Pareto front in Fig.11.

Fig.15.Current density versus waste removal ratio()and maximal attainable current density(○)corresponding to the Pareto front in Fig.11.

Fig.16.Pareto front obtained from the simultaneous maximization of power density,waste removal ratio and maximal attainable current density.

Fig.19 shows the 2-norm level diagrams representation of the decision variables corresponding to the Pareto front in Fig.16.It is observed that(1)the values ofQaare situated at the range of(0.5-2.56)× 10-5m3·h-1;(2)the values ofQcare almost over 1.5× 10-5m3·h-1;(3)the values ofCiAnCmostly reach its highest limit of 2 mol·m-3,which is in accordance with the three bi-objective optimization results;(4)the values ofRexare below 200 Ω and most ofRexare manipulated at the range of 50-80Ω,and(5)the values oficellare regulated atthe range of 4.07-10.56 A·m-2.Moreover,it is found that the ranges of operating variables corresponding to the lower level are approximately atQa∈ [1.4,1.7]× 10-5m3·h-1,Qc∈ [1.5,1.8]× 10-3m3·h-1,CiAnC≈ 2 mol·m-3,Rex∈ [50,70]Ω andicell∈[9.44,10.56]A·m-2.

Note that Blascoet al.have recommended representing the same Pareto front with different norms to see the difference[23].However,it is found that for this three objective optimization problem,both 1-norm and in finity norm representations do not illustrate much new information by comparing to 2-norm representation,and we neglect to depict these two representations.Moreover,although the level diagrams provide a promising graphical analysis tool for high-dimensional Pareto front,more sophisticated preferences need to be introduced for final decision making.

Fig.17.2-D visualization of Pareto front by the projection of a 3-D Pareto front in Fig.16:(A)power density versus waste removal ratio;(B)power density versus maximal attainable current density;(C)waste removal ratio versus maximal attainable current density.

5.Conclusions

Multi-objective steady-state optimization of MFC was successfully carried out to understand the complex relationship between power density,waste removal ration and maximal attainable current density.Three bi-objective and one three-objective optimization problems were thoroughly investigated and the LD method was applied to assist in graphical analysis and decision making for three-objective optimization problem.The main findings of the present study can be summarized as follows:

Fig.18.2-norm level diagram representation of the Pareto front in Fig.16.

(1)Results ofthree bi-objective optimization problems indicated that power density,waste removal ratio and maximal attainable current density are mutually conflicting.The maximum values of power density,waste removal ratio and maximal attainable current density are 3.82 W·m-2,0.97 and 14.88 A·m-2,respectively.The effects offour operating conditions,namelyQa,Qc,CiAnCandRex,on MFC performance illustrated that(a)the flow rate of feeding to cathode(Qc)can be usually chosen to supply excessive oxygen;(b)the flow rate of fuel feed to anode(Qa)plays significant and different effects on objectives in different bi-objective problems and should be properly determined for balancing tradeoffs between conflicting objectives;(c)the acetate concentration(CiAnC)often reaches its highest limit of 2 mol·m-3for all problems;and(d)the externalresistance(Rex)should be properly manipulated below 200 Ω for different problems.

(2)The three-objective optimization results demonstrated that these three objectives are conflicting in nature.Graphicalanalysis ofPareto front using level diagrams illustrated that the nearest points to the ideal point are at 2.84-3.25 W·m-2for power density,0.69-0.78 for waste removal ratio and 10.82-12.19 A·m-2for maximal attainable current density.Moreover,it was found that(a)achieving the high power density range of 3.50-3.82 W·m-2would result in the medium waste removal ratio range of 0.57-0.73 and maximal attainable current density range of 8.73-11.81 A·m-2;(b)achieving the high waste removal ratio above 0.95 would lead to low power density below 1.09 W·m-2and maximal attainable current density below 5.88 A·m-2;and(c)achieving the high maximal attainable current density above 14.0 A·m-2would resultin low waste removal ratio below 0.55.

Hence,the aforementioned findings indicate that the integrated methodology of multi-objective genetic algorithm and level diagram providesa promising approach to trade-offconflicting objectives foroptimal operation of MFC.However,the present study only investigates the steady-state operational optimization of MFC.For improving the dynamic operational performance,future work should focus on multiobjective dynamic optimization of MFC.

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