Synthesis/design optimization of SOFC-PEM hybrid system under uncertainty☆

Lingjun Tan,Chen Yang*,Nana Zhou

Key Laboratory of Low-grade Energy Utilization Technologies and Systems,Ministry of Education,Chongqing University,Chongqing 400030,China

Keywords:Solid oxide fuel cell Proton exchange membrane fuel cell Hybrid system Uncertainty Optimization

ABSTRACT Solid oxide fuel cell-proton exchange membrane(SOFC-PEM)hybrid system is being foreseen as a valuable alternative for power generation.As this hybrid system is a conceptual design,many uncertainties involving input values should be considered at the early stage of process optimization.We present in this paper a generalized framework of multi-objective optimization under uncertainty for the synthesis/design optimization of the SOFC-PEM hybrid system.The framework is based on geometric,economic and electrochemical models and focuses on evaluating the effect of uncertainty in operating parameters on three conflicting objectives:electricity efficiency,SOFC current density and capital cost of system.The multi-objective optimization provides solutions in the form of a Pareto surface,with a range of possible synthesis/design solutions and a logical procedure for searching the global optimum solution for decision maker.Comparing the stochastic and deterministic Pareto surfaces of different objectives,we conclude that the objectives are considerably influenced by uncertainties because the two trade-off surfaces are different.

1.Introduction

Energy systems are becoming larger and more complicated,so new configurations and technologies are developed accordingly.Multiple objectives such as cost,process efficiency and pollutant emissions are required in the optimization of energy system,so the optimization becomes more complex due to the conflicting nature of these objectives.In addition,significant uncertainties arise from sources such as scant performance data,use of empirical model,operating parameter fluctuations,inaccuracies in process control,and environment in innovative power systems.Ignorance of these uncertainties may distort the results far from practical conditions.Therefore,uncertainties should be considered in the system design,plant operation,and cost estimates.For fuel cell systems,operating parameters such as cell temperature,system pressure,hydrogen utilization factor, flow rates of fuel and air are subject to control fluctuations and are considered as uncertain.The interactive effects of these parameters affect the electrochemical phenomena,system design and capital cost in the fuel cell power system,and must be understood.

Solid oxide fuel cell-proton exchange membrane(SOFC-PEM)hybrid system is a promising candidate of alternative energy conversion devices for transportation,stationary and power generation applications.In 2002,the conceptual design of the SOFC-PEM hybrid system was proposed by Vollmar et al.[1].They considered that hydrogen and syngas of the SOFC are more valuable than the high-temperature heat converted into chemical energy in the form of CO and hydrogen.The hydrogen-rich gas was used to produce additional power after CO in the syngas was eliminated by a shift reactor.The combined process produced a high-grade chemical energy.Dicks et al.[2]builta simplified model for the hybrid system and used it in a computer simulation to predict and study the performance of the system.The results indicate that the overall efficiency of the SOFC-PEM hybrid system can reach 61%,which is significantly higher than the reformer-PEMFC system and the SOFC-only system.

Yokoo et al.[3-6]developed two configurations of SOFC-PEM system based on the fuel feeding method.In the parallel configuration,the reformed gas is divided into two flow streams for the SOFC and the PEMFC,while in the series configuration the PEMFC is fueled by the SOFC anode exhaust gas.The results show that the electrical efficiency at 190 kW ac output of the SOFC-PEM hybrid system is 5%higher than that of a simple SOFC system and equals that of the SOFCGT system at 1014 kW ac.

Little work has been published on the SOFC-PEM hybrid system optimization under uncertainty.Diwekar and coworkers[7,8]developed a 1472 kW SOFC-PEM fuel cell hybrid system and optimized the operation using a new stochastic multi-objective optimization approach,minimizing number of single objective optimization problems(MINSOOP)[7,9,10]The MINSOOP algorithm based on efficient sampling methods for uncertainty analysis provides explicit trade offs information among objectives and offers a reasonable number of appropriate designs for decision-makers.The computational test indicates that the computation burden of the MINSOOP algorithm is not sensitive to the increased number of decision variables and objectives,and it can converge to the “true”representation of the whole Pareto set quite faster than the traditional multi-objective optimization methods[9].Thus the MINSOOP algorithm can be used to solve large-scale real-world problems with significant computational savings and a better accuracy.In the new system design and optimization stage,designers wish to modify geometric and operational variables for easy manufacture,better performance and less cost.It is necessary to study the relation between objectives and decision variables.However,Diwekar's group mentioned six independent objectives,two uncertainties,and five optimal decision variables,but did not relate the system capital cost to the geometric decision variables[7-10].

In this work,we propose a more complex and complete configuration for SOFC-PEM hybrid system,which guarantees the dependability of system and makes its application feasible.Geometric,thermodynamic and operation variables are considered and appropriate cost function is developed.The stochastic multi-objective optimizer(MOP)approach,MINSOOP,is used to develop a synthesis/design optimization for SOFC-PEM hybrid system,investigate the effects of operating uncertainties on the objectives by comparing the stochastic and deterministic Pareto surfaces,and determine the optimum values of decision variables.

2.Layout of SOFC–PEM Hybrid System

Fig.1 shows the layout of the proposed SOFC-PEM hybrid system.The fuel is fed to the SOFC.The SOFC anode exhaust gas is partially recycled and joins the feed steam to the reformer installed inside the SOFC stack.The flow rate of the recycled gas is fixed so that the steam-to-carbon ratio(STCR)at the reformer inlet is constant.The exhaust gas remaining in the SOFC anode is supplied to the PEMFC stack,and the carbon monoxide in the exhaust gas is removed as it flows through two water-gas shift reactors and a preferential oxidation reactor.The hydrogen-rich reformate gas from the preferential oxidation reactor is cooled to about 70°C by the condenser before entering the PEMFC stack.The exhaust gas from the condenser is supplied to the anode compartment of the PEMFC stack and hydrogen in the exhaust gas takes part in the electrochemical reactions with humidified air from cathode compartment of the PEMFC stack.The exhaust gas of cathode and anode from the PEMFC stack is then used for cooling the inlet temperature of the two water-gas shifter reactors.The mixer gas and cathode exhaust gas burn in the combustion plenum.All the combustion exhaust gas supplied to the heat exchanger is used to preheat the air and fuel for the SOFC stack.Gross dc outputs of both stacks are converted to gross ac outputs by inverters.Net ac output is determined by subtracting the power consumed in an auxiliary machine from the gross ac output.

In the SOFC-PEM hybrid system,owing to the internal reforming ability of the SOFC,it is possible to produce electrical power and a stream of reformate gas for a PEMFC stack.This eliminates the installation of an upstream reformer before the PEMFC stack and improves the system efficiency accordingly[2].

3.System Model

The model of the entire system is based on following assumptions:one-dimensional flow,steady state,negligible pressure drop,no gas leakage,negligible heat losses toward the environment,negligible kinetic and gravitational terms in the energy and mass balance equations.Thermophysical and transport properties are calculated based on the assumption of ideal gas and Dalton's law.The heat capacity,enthalpy,viscosity and conductivity of each gas are calculated with temperature-dependent function[11].Significant details of models are described below.

3.1.Model of SOFC stack

The SOFC-PEM hybrid system is constructed based on the internal reforming tubular SOFC with an anode re-circulation tubular arrangement[12].The main assumptions are as follows.

(1)In the one-dimensional model,all quantities are in the direction orthogonal to the anode and cathode surfaces.

(2)All reactions are kinetically controlled.The rates are in equilibrium.

(3)Temperature is uniform over the entire cell surface.

(4)No carbon deposits in the SOFC stack.

(5)Pressure changes within the SOFC stack are negligible.

Since our purpose is to study the performance at the system-level rather than the characteristic of a SOFC,the models[13]related to the heat,momentum and charge transport in the fuel cell are not used here.In the SOFC stack model,the stack outlet conditions of components,stack current and voltage are identified as important variables in representing the performance of the SOFC stack.Referred to references[14-16],the SOFC model presented here focuses on the thermodynamic aspect and associated electrochemical processes of the cell operation can determine these key variables accurately.

Fig.1.Layout of SOFC-PEM hybrid power plant.

3.1.1.Internal reforming SOFC stack model

In the SOFC stack,the compressed methane is preheated to keep the reforming and the shift reaction within its anode compartment.A fraction of the water at the anode outlet is recycled to provide steam for reformation of methane.Internal reforming reactions in the SOFC stack are as follows.

Methane steam reformation reaction CH4+H2O→3H2+CO

Water-gas shift reaction CO+H2O→H2+CO2

Electrochemical reaction H2+0.5O2→H2O.

Although several models[17-19]assume a simplified approach of thermodynamic equilibrium,more correct models for the four reactions are based on a kinetic/equilibrium approach.Specific literature[14,15]on the internal reforming in the presence of typical mixture of Ni and yttrium stabilized zirconia SOFC materials suggest an area-based methane reformation model for the calculation of methane reaction rates rCH4.The reaction express is shown as

where k0is the pre-exponential factor,Eais the activation energy,pCH4is the partial pressure of methane,and TSOis the outlet temperature of SOFC stack.

Actual methane conversion XCH4is defined as the ratio of the molar flow rate of CH4reacted zCH4and the inlet methane nCH4,in.Once the geometric parameters of the SOFC are fixed,XCH4and zCH4can be determined by using the plus flow design equation[20]:

where nSOis the number of SOFC tubes,LSOis the length of SOFC,ASOis the active area,and ρBis the bulk density of catalyst.

Comparing to the methane reformation reactions,the water-gas shift reaction is faster and reaches completion very quickly,so it is commonly considered to be atequilibrium[14].Its equilibrium constant Kshiftis calculated from the molar rate of each gas component at the anode side of the tubular SOFC[17]:

Kshiftis also calculated according to the relation between the equilibrium constant and the outlet temperature of SOFC stack[15,19].

In the electrochemical reaction,the hydrogen utilization factor of SOFC Uf,SOallows the control of the molar flow rate of H2reacted zH2:

The flow rate ofair nair,inis evaluated by the SOFC stack temperature,with the energy balance on the control surface of the fuel cell.For the tubular SOFC,the energy balance is formulated as

where ΔHkrepresents enthalpy changes in steam reforming,shift reactions and electrochemical reaction.

The SOFC current density iSOis derived from the stoichiometric balance on the anode semi-reaction,assuming all cells have the same behavior:

3.1.2.SOFC electrochemical model

The simulation of electrochemical phenomena and parameters are based on the model described in[21,22].The SOFC voltage VSOis affected by activation,concentration and ohmic overpotential

The reversible cell voltage of SOFC is given by[21]

The concentration overpotential ηconcis calculated by

where

The activation overpotential ηactis calculated using the following equation

The activation overpotential depends principally on the exchange current density of the anode and cathode[22]

According to Ohm's law,the ohmic overpotential ηohmcan be expressed in terms of the properties of electrolyte[21]

where Leis the thickness of electrolyte measured in micrometers.The required parameters in the SOFC stack model are shown in Table 1.

3.2.Model of PEMFC stack

A steady state electrochemical model is used for the PEMFC stack.All quantities vary in the direction orthogonal to the anode and cathodesurfaces.The membrane electrolyte is assumed to be saturated with water so that its conductivity is only a function of temperature.Thus the current density of PEMFC strongly depends on the amount of reactants,hydrogen utilization factor and geometric characteristics

Table 1Parameters used in the SOFC stack model

The voltage of single PEMFC,VPEM,is regarded as a function of current density,temperature,pressure,chemical composition and geometric/material characteristics,and can be calculated according to the difference between the reversible potential and all overvoltages

Particularly,the open circuit reversible voltage is calculated on the basis of the Nernst equation,given by

The active overvoltage is derived from the Tafel equation,expressed as[21]

In the PEMFC,the anode i0,afor hydrogen oxidation is very high compared to the cathode i0,cfor oxygen reduction,so the anode contribution to this overvoltage is neglected.

The concentration overvoltage used in the model is given by[23]

The ohmic overvoltage includes the electronic Relecand ionic Rionicresistances contributed to the fuel cell resistance,written as[24,25]

Relecis assumed to be a constant at the operating temperature of PEMFC.Rionicis determined by the following empirical equation[25]

where rMis the membrane-specific resistance for the flow of hydrated protons(Ω·cm),Leis the thickness of membrane,and APEMis the active cell area.

It is difficult to describe the term rMphenomenologically,so the following semi-empirical expression is employed

Usually,adjustable fitting parameter λ is assigned a value between 10 and 20[25].In this case λ takes the value of 14.6.Other fixed parameters used in the PEMFC model are given in Table 2.

Table 2 Parameters used in the PEMFC model

3.3.Model of heat exchangers

The plate- fin type compact heat exchangers with a single-pass and cross- flow arrangement are used in the SOFC-PEM hybrid system.In the particular models for these heat exchangers[26],the length,volumes of hot and cold sides,and the height are considered as synthesis/design decision variables.The effectiveness-NTU method is used to correlate the geometric models of heat exchanges with the thermodynamic properties at any operation moment.The geometric parameters for the selected plate- fin heat exchangers as well as the thermal conductivity of fin(code name “11.1”)material are given in Table 3.

Table 3 Parameters used in the plate- fin heat exchanger model

3.4.Model of water–gas shift reactors

The high-temperature water-gas shift(HTS)reactors and the lowtemperature water-gas shift(LTS)reactors are modeled considering the water-gas shift reaction in an adiabatic catalyst bed,

The equations for HTS and LTS reactors are given by Georgopoulos[27].

The geometric variables(i.e.length L and diameter D),inlet molar flow rate of CO nCO,i,actual conversion of carbon monoxide XCO,and steam-to-carbon ratio STCR are correlated in the models of shift reactors.The geometries of the HTS and LTS reactors(length L and diameter D)as well as the actual conversion of carbon monoxide XCOare considered as synthesis/design decision variables.The temperature and composition of the reformate gas at the outlets of both HTS and the LTS reactors are evaluated.The constants used in the equation are listed in Table 4.

Table 4 Parameters used for kinetic modeling of HTS and LTS reactors

3.5.Model of preferential oxidation reactor

Preferential oxidation reactor(PROX)is modeled simply.As air is added to the reformate gas,two reactions take place in this reactor:

Compared to CO abundant O2must be provided,i.e.the ratio of O2/CO should be above 0.5.It is assumed that the reaction between CO and O2is complete in 99%and remaining O2completely reacts with H2.

3.6.Model of the compressor,combustor and mixer

A centrifugal compressor is selected,which can be operated at high efficiency over a wide range of flow rates by changing flow rate and pressure.In the compressor model,the heat flow and kinetic energy of gas are neglected.The gas is considered as an ideal gas,so it is assumed that the specific heat is constant at a constant pressure.Taking into account the inefficiencies of the compression process,the temperature at the end of the compression can be found from the following equation

where k is the ratio of specific heats(for air,k=1.4;for fuel gas,k=1.3),and εACis the compressor efficiency,which is assigned a value of 90%in this work.

The compressor power is expressed as

Mixers are modeled on the basis of energy and mass balance.The combustor reactions are assumed to complete within the catalytic burner.The outletchemical composition and energy rates are determined by the energy and mass balance.

3.7.Cost model

The life cycle cost of the SOFC-PEM hybrid system is composed of operating cost and capital cost of each sub-system and component.The operating cost is associated with the consumption of natural gas throughout the lifetime of the hybrid system.The capital cost consists of purchase cost,maintenance cost,and annual amortization cost representing the income tax,depreciation and property insurance.When the synthesis/design decision variables are setatany possible operating point,the estimated capital costs are the total costs of all components calculated by corresponding cost models.

The capital cost of each component is based on a production volume of 1482 units per year.Actually,the number of units produced per year could be changed significantly.The effect of a change in the production volume on the unitcostis taken into account by applying an appropriate scale factor α to the components capital cost[28].

where Cunitis the single unit purchase cost based on the production volume Nunit,CNbaseis the baseline manufacturing cost based on the production volume Nbase(Nbase=500000).Kim[28]suggests that α=?0.362 based on a production volume of 500000 units.Other economic assumptions made in the evaluation of the system are presented in Table 5.

Table 5 Economic assumptions included in the cost model of SOFC-PEM hybrid system

3.7.1.Cost model of PEMFC stack

To describe the effect of size(i.e.number of cells NPEMand cell active area APEM)on the optimization of the system cost,the PEMFC purchase cost is divided into the purchase cost of stack components(i.e.membrane,bipolar plates,and electrodes)and the assembly cost.The specific functions are as follows[27].

Membrane electrode assembly and purchase cost:

The costs of endplates,collectors,insulators,and tie bolts are much lower than those of other major components,so they are considered constant:Cextramaterial=248.96.

PEMFC stack assembly cost:Cassembly=30.8+0.3465NPEM.

PEMFC stack purchase cost:Cstack=CMEA+Cbipolarplates+Ccoolerplates.

Auxiliary equipment purchase cost:Caux=0.1Cstack.

Both annual amortization and maintenance cost are given as a percentage of the purchase cost,which is also applied to SOFC and FPS.

3.7.2.Cost model of SOFC stack

The SOFC stack purchase cost includes its purchase cost and that of auxiliary equipment.The cost functions are constructed in combination with the PEMFC capital cost model,which can correlate the SOFC stack costs with proper geometric variables,i.e.cell active area ASOand total number of cells NSO.The specific functions are given as follows[29].

Stack purchase cost:Cass=(0.02687ASO+0.88)NSO.

Endplates,collectors,insulators,and tie bolts:Cextramaterial=248.96.

Stack assembly cost:Cassembly=30.8+0.3465NSO.

Stack purchase cost:Cstack=Cass+Cextrameterial+Cassembly.

The capital costs for auxiliary devices such as combustor,mixers,and by-pass valves are 10%of the stack cost.

3.7.3.Cost model of fuel processing sub-system

The fuel processing sub-system(FPS)consists of HTS,LTS,PROX,and heat exchangers.The FPS capital cost model relates the cost to proper geometric variables,e.g.,volume V(πLD2),system power produced WSOFC-PEM,heat transfer area AHX(LhLcH)and compressor duty WFC.The specific functions are given as follows[27].

3.7.4.Capital cost of the system

The capital cost of the SOFC-PEM hybrid system is defined as the sum of those for fuel and all components based on the economic assumption in Table 5.

where Tlifeis the life time,toperis the operating hours per year,cfuelis the natural gas price,obtained from the Energy Information Administration(2012),and the average residential price of 0.118USD·m-3[30].

3.8.Model validation

In order to validate the SOFC and PEMFC models,the polarization curves generated by the code are compared with experimental data.Fig.2 shows that the simulation results by SOFC model agree well with the experimental data[31].However,the data available cannot deal with methane-fuelled SOFCs,so the internal reformation aspect of the SOFC cannot be validated here.Fig.3 shows that the PEMFC model agrees well with the experimental data[32]except at relatively high current density(8000-10000 A·m?2).The simplification to the ohmic overpotential might be the main reason of the derivation.Since the PEMFC mostly operates over a current density range from 2000 A·m?2to 6000 A·m?2,the model can be used to predict the PEMFC performance over a large range of current density.

Fig.2.Validation of the SOFC model using the experiment data[31].

Fig.3.Validation of the PEMFC model using the experiment data[32].

4.Multi-objective Optimization under an Uncertainty Framework

The multi-objective optimization strategy is applied to the synthesis/design optimization of a steady-state 220 kWSOFC-PEMhybrid system under a full load condition.Three objectives are to be optimized simultaneously with some constraints.The multi-objective optimization formulation is described as the following problems:

?

According to the knowledge and previous modeling experiences,26 key decision variables are selected and identified important to the hybrid system performance and cost.These decision variables are defined in Table 6,with 20 geometric variables and 6 operational variables.The ranges and the initial design variables are determined by heuristics and experiences with this process.

Table 6 Values of synthesis/design(S/D)decision variables

Fig.4 presents an efficient multi-objective optimization under uncertainty framework,named MINSOOP[7-10],for the system synthesis/design optimization,in which the Hammersley sequence sampling(HSS)technique is employed due to its good uniformity properties and great computational efficiency[33-36].The framework includes four levels:(1)stochastic simulator,(2)system model,(3)multi objective optimizer,and(4)nonlinear programming(NLP)optimizer.

Level1 Stochastic simulator.In order to quantify uncertainties and their effect on the final synthesis/design of the system,the inputs and outputs should be treated probabilistically.Thus the stochastic simulator generates probability information such as the probability distribution of the uncertain parameters.This work focuses on evaluating the uncertainties of operating parameters uibecause of their control fluctuations and unpredictability.Operating parameters with uncertainties are described in the form of normal probability distributions quantified in terms of mean and variance values.Table 7 shows the distribution,mean and variance values of the operating parameters.Once probability distributions are assigned to the uncertain parameters,the next step is to sample the probability distribution of each uncertain parameter.The HSS method[35]is used in this framework.In the method,a quasi-random number generator based on the Hammersley points is used to uniformly sample a unit hypercube and generate a set of N samples from the distributions of the m uncertain parameters,as depicted in Fig.4.As mentioned above,the stochastic simulator considers 4 uncertain parameters,i.e.,m=4.Each sample,j,represents a combination of the uncertain parameters values{uji,i=1,…,m}.These samples are then propagated through the system model.

Table 7 Distribution,mean and variance of operating parameters

Level 2 System model.Decision variables and uncertainties(samples){uji,i=1,…,m}are used as input variables for the deterministic system model simulation.The results of objectives{f1f2… fN}′and constraints{g1g2… gN}′corresponding to the N samples are used to construct the probabilistic information(i.e.means,variances,and/or probability distribution)of objective functions and constraints.Then these system responses are sent to the multi-objective optimizer.

Fig.4.Multi-objective optimization under uncertainty framework.

Level 3 Transformation of multi-objective problem to a series of singleobjective problems by MOP.The basic strategy is to select one objective fkand minimize it while other objectives fk?1are turned into inequality constraints with parametric right-hand sides,εk.The Pareto set is derived by solving N singleobjective optimization problems created by the combination of different values of right-hand side,{εjk:j=1,…,N}.The next step is to find the potential ranges of εjk.The minimum value of εjkis the individual optimal solution.The maximum value of εjkis the maximum value for the selected objective when other objectives are minimized individually.In this way,the potential ranges of εjkfor all objectives are obtained and listed in Table 8.

Table 8 Potential ranges of εi for all objectives:a deterministic payoff table

Level 4 NLP optimizer.For each single-objective optimization problem,the NLP optimizer determines new values for the decision variables and the inner NLP optimization cycle repeats until it finds the optimal values of objective functions and the decision variables forming the Pareto optimal set.The successive quadratic programming method is used for this framework as it needs fewer functions and gradient evaluations for the highly nonlinear constrained optimization and does not need feasible points at intermediate iterations[37,38].Unlike the multiobjective optimization under an uncertainty framework,the deterministic multi-objective optimization only includes multi-objective optimizer and NLP optimizer.The procedure is described as follows.Once the multi-objective optimizer receives inputs,it formulates single objective NLP optimization problems and delivers them to the NLP optimizer.The optimal values of the objective functions and the decision variables are determined by the NLP optimizer and then return to the multi-objective optimizer.This step is repeated about 100 times until a good approximation of the Pareto optimal set of solutions is obtained.

5.Results and Discussion

5.1.Deterministic MOP results

As mentioned in Section 4,the capital cost objective along with the SOFC current density and electricity efficiency is employed in the multi-objective optimization framework to obtain approximate Pareto set of solution.This includes 100 optimal solutions.Decision makers can search these 100 optimal designs and find the most appropriate design to fit their implicit value function.

As seen in Fig.5,the capital cost decreases rapidly at low SOFC current densities(＜3000 A·m?2),while it decrease slowly in the region of higher SOFC current density(＞3000 A·m?2).This is because the SOFC capital cost dominates the capital cost,which is mainly dependent on the number of cells and SOFC active area.When the SOFC current density is low(＜3000 A·m?2),the number of cells and SOFC active area decrease significantly,reducing the capital cost greatly.However,the number of cells and SOFC active area decrease slowly at high SOFC current density,so the capital cost is affected less.Fig.5 shows that the maximum electricity efficiency of the hybrid system can reach 68%.This is acceptable because 99.99%of methane is converted to hydrogen before being delivered to the PEMFC stack and much more hydrogen is used effectively for the generation of the electricity in the SOFC-PEM hybrid system.

Fig.5.Deterministic Pareto surfaces for electricity efficiency,SOFC current density and capital cost.

Specifically,as shown in Fig.5,there are four identical high cost circles(＞850000USD)in the lower cost region(＜850000USD)where the SOFC current density is between 3000 and 4500 A·m?2.For a decision maker,designs in these four high cost circles are not good candidates compared with the designs at the lower capital cost and the same SOFC current density.To get the detailed information of the Pareto surface,we found that designs in the middle-bottom region of the plot can be accomplished by the designs from the top circle because they have the same cost,higher electricity efficiency and similar SOFC current density,and can also be accomplished by the designs from the right circle because they have the same cost,higher SOFC current density and similar electricity.

For a complete trade-off representation,we normalize the objective values of the 100 optimal designs and obtain values between 0 and 1.A better design is obtained as the value of the normalized objective function is reduced.

As seen in Fig.6,three different groups are identified for three types of designs,which are max electricity efficiency(Group I),max SOFC current density(Group II)and min capital cost(Group III).Designs in Group I show electricity efficiency from 0.66 to 0.68,with the corresponding costs from 938460USD to 973455USD and SOFC current density from 1798 to 2019 A·m?2.Designs in Group II present high SOFC current density of 3300 to 3500 A·m?2,with capital cost ranging from 826421USD to 859290USD and electricity efficiency from 0.54 to 0.56.Designs in Group III have low capital cost from 800000USD to 820000USD,with electricity efficiency ranging from 0.484 to 0.532 and SOFC current density from 2557 to 2842 A·m?2.

Fig.6.Normalized objectives of the 100 designs for SOFC-PEM hybrid fuel cell system.▲CDSOFC;●COST;▼EFF.

For a decision maker,if he/she wants to see the potential ability of the hybrid system for electricity efficiency and is willing to sacrifice slightly in SOFC current density and capital cost,the optimal designs in Group I are better candidates.If the decision maker wants to obtain a higher SOFC current density and lower capital cost with a sacrifice in efficiency,the designs in Groups II and III are better choices.

5.2.Stochastic MOP results

Tables 8 and 9 are the pay-off tables for deterministic case and stochastic case,respectively,presenting considerable differences.In the stochastic case,the bound of SOFC current density is lower,the bound of electricity efficiency is higher,and the capital cost shifts toward higher levels.Thus the deterministic Pareto surface underestimates the capital cost for a majority of the designs.

Table 9 The potential ranges of εi for all objectives:A stochastic payoff table

Fig.7 presents a complete stochastic Pareto surface for the three objective functions with the uncertainty factor of operating parameter considered.The effects of the operating uncertainties on the objectives are significant because the Pareto surfaces are different considerably in the contour shape and levels from the deterministic Pareto surfaces.Many regions in stochastic Pareto surface are covered by the capital cost of 800000-1000000USD,as compared to a range of 750000-850000USD in deterministic Pareto surface.Both Pareto surfaces present general contour shape in the lower SOFC current density region.However,in the higher SOFC current density region,the four cost circles in the deterministic Pareto surface disappear in the stochastic Pareto surface.

Fig.7.Stochastic Pareto surface for electricity efficiency,SOFC current density and capital cost.

Fig.8 shows the similar qualitative designs using normalized objectives corresponding to the trade-off surfaces in Fig.7.We also identify three groups for three types of design similar to the deterministic case and tabulate their corresponding objectives in Table 10.The objectives in the same design group have different ranges in the deterministic and stochastic cases.For example,in Group I,the ranges of SOFC current density and the capital cost for the stochastic case increase compared to those for the deterministic case.In Group II and Group III,three objectives have almost the same range but different values,since the relative effect of uncertainties on the three types of design results in considerable design changes due to the nonlinear nature of the performance and cost model.Moreover,the number of designs for all the groups increases when uncertainties are incorporated into the operating parameters,therefore,more options are provided to designers considering the effect of uncertainties.

Fig.8.Normalized objectives for the design of SOFC-PEM hybrid fuel cell power plant:a stochastic case.▲CDSOFC;●COST;?EFF.

6.Conclusions

In this work,detailed geometric,economic and electrochemical models are developed and implemented for a SOFC-PEM hybrid system.The approach of multi-objective optimization considering uncertainties is presented and applied to the synthesis/design optimization of a 220 kW SOFC-PEM hybrid system.100 optimal designs are provided by the multi-objective optimization to form an approximation for the Pareto surface between conflicting objectives,e.g.electricity efficiency,SOFC current density and capital cost of system.Then,decision makers can backtrack and find the values of the decision variables from the Pareto surface to fit their target performance.Moreover,normalizing these objectives can deepen the understanding of the characteristic of system performance.In the optimization of the system,the uncertainty of operating parameters resulted from control fluctuations and unpredictability is also evaluated,which is often overlooked by traditional synthesis/design optimization.Comparing the stochastic and deterministic trade-off surfaces of different objectives,it is found that the trade off surfaces are different and the uncertainty of operating parameters has considerable effects on the objectives.

Table 10 Comparison between deterministic and stochastic objective function values for three type of designs

This work provides an effective way to deal with the hybrid fuel cell system synthesis/design optimization under uncertainty.This paradigm can be applied to the development of any other energy systems.