Thermo-economic multi-objective optimization for a solar-dish Brayton system using NSGA-II and

Thermo-economic multi-objective optimization for a solar-dish Brayton system using NSGA-II and decision

making

Yuqiang Li a ,b ,Shengming Liao a ,?,Gang Liu a

a School of Energy Science and Engineering,Central South University,Changsha,Hunan 410083,China

b

Department of Mechanical Science and Engineering,University of Illinois at Urbana-Champaign,IL 61801,USA

a r t i c l e i n f o Article history:

Received 28April 2014

Received in revised form 3July 2014Accepted 6July 2014

Available online 5August 2014Keywords:

Solar-dish Brayton system Multi-objective optimization NSGA-II

Decision making

a b s t r a c t

A 100kW regenerative Brayton heat engine driven by the hybrid of fossil fuel and solar energy was considered for optimization based on multiple criteria.A thermodynamic model of such hybrid system was developed so that the power output,thermal ef?ciency and dimensionless thermo-economic perfor-mance with the imperfect performance of parabolic dish solar collector,the external irreversibility of Brayton heat engine and conductive thermal bridging loss could be obtained.Evolutionary algorithm based on NSGA-II (Elitist Non-dominated Sorting Genetic Algorithm)was employed to optimize triple-objective and dual-objective functions,where the temperatures of hot reservoir,cold reservoir and work-ing ?uid,the effectiveness of hot-side heat exchanger,cold-side heat exchanger and regenerator were considered as design https://www.360docs.net/doc/9d13508169.html,ing decision makings,including Shannon Entropy,LINMAP and TOPSIS methods,the ?nal optimal solutions were selected from Pareto frontier obtained by NSGA-II.The results show that there exists an appropriate working ?uid temperature to cause optimal solution under each given condition.The comparisons of triple-objective and dual-objective optimization with single-objec-tive optimization indicate that multi-objective optimization can yield the more suitable results due to the lower deviation index from the ideal solution.In the analysis of triple-objective optimization,an expected result is obtained that the optimal values of the power out,ef?ciency and dimensionless thermo-economic performance of solar-dish Brayton system (68.65kW,0.2331and 0.3077)are 22.6%,34.9%and 18.4%respectively less than that of convectional Brayton heat engine.Finally,a range of func-tional relationship between the optimized objectives in Pareto frontier is ?tted to provide more detailed insight into the optimal design of solar-dish Brayton system.

ó2014Elsevier Ltd.All rights reserved.

Introduction

Growing demand for energy and increasing environmental problems have driven the development of solar energy source as a replacement of fossil fuels [1].One of the most cost ef?cient ways of utilizing solar energy is solar thermal power technology includ-ing trough,central tower and dish heat engine systems [2].High ef?ciency,modularity,autonomous and inherent hybrid capability make solar-dish heat engine systems receiving more and more attention.In most contemporary solar-dish heat engine system,Stirling heat engine has been used as the power conversion unit [3].However,Brayton heat engine shows a number of advantages,chie?y including a signi?cant reduction in O&M costs and simpli-?ed hybridization [4].Recent development in the ?eld of high-temperature regenerator will allow the regenerative Brayton heat engine to compete with Stirling heat engine in term of ef?ciency [5].Solar-dish Brayton system has the potential to become one of the least expensive sources of renewable energy power.

Recently,great attention has been drawn to the single-objective optimization of thermodynamic systems driven by solar energy based on various criteria including power output [6],thermal ef?-ciency [7],power density [8],ecological [9],exergetic [10]and thermo-economic performance [11],etc.As the performance of a ?at plate solar air heater is poor due to low heat transfer capability,the system was optimized by Siddhartha [12]to obtain maximum thermal performance.Optimal thermal performance was obtained under different Reynolds number,emissivity of the plate,tilt angle and number of glass plates.Wang et al.[13]optimized a solar-dri-ven Kalina cycle with the net power output as an objective function under the given conditions.Results indicated that an optimal tur-bine inlet pressure can lead to the maximum net power output and maximum system ef?ciency.The optimal performance analy-sis of an internally and externally irreversible solar driven heat

https://www.360docs.net/doc/9d13508169.html,/10.1016/j.ijepes.2014.07.027

0142-0615/ó2014Elsevier Ltd.All rights reserved.

?Corresponding author.Tel.:+86073188836936;fax:+86073188879863.

E-mail address:smliao@https://www.360docs.net/doc/9d13508169.html, (S.Liao).

engine was carried out by Yilmaz et al.[14].It was found that the design parameters at maximum power density resulted in higher thermal ef?ciency than those at maximum power output condi-tions.Zhang et al.[15]established an irreversible model of a solar-driven Brayton heat engine.The operating temperature of the solar collector and the temperature ratio in the isobaric process under maximum thermal ef?ciency were obtained.The optimiza-tion of an integrated solar combined cycle system was studied using an exergoeconomic concept,where an attempt was made to minimize investment cost of equipment and cost of exergy destruction[16].Optimization process improved the thermal ef?-ciency and exergy ef?ciency of the system and reduced the rate of fuel cost.

However,the single-objective optimization always brings about unacceptable results with respect to the other objectives when the objectives under consideration con?ict with each other.For many complicated problems,the multi-objective approaches are more accurate and real because it allows decision making to think about the trade-offs between different bene?ts of different objects[17]. In dealing with multi-objective optimization problems,it needs to satisfy a number of different and even contradictory objectives at the same time,which is known as Pareto optimal solutions. For a given Pareto optimal set,the corresponding objective func-tion values in the objective space are called the Pareto frontier. Evolutionary algorithms(EA)are widely applied to solve multi-objective optimization in recent years[18–22],because it has been proved to be more ef?cient than the classic optimization methods such as weighted sum methods,goal programming and min–max methods.

A number of research works on multi-objective optimization of the solar-driven thermodynamic systems have been investigated by researchers[23–30],because of the limitation of single objec-tive optimization.However,few works have involved the multi-objective optimization of solar-dish Brayton system.In this study, multi-objective optimization is conducted to achieve the optimal power output,thermal ef?ciency and dimensionless thermo-economic performance for a solar-dish Brayton system.The EA method based on the elitist non-dominated sorting genetic algorithm(NSGA-II)is employed to obtain the Pareto optimal set and Pareto frontier in objective space[31,32].The?nal optimal solutions from available solutions located at the Pareto frontier are selected by the decision makings including Shannon Entropy, LINMAP and TOPSIS methods.

Thermodynamic analysis of the solar-dish Brayton system Solar-dish Brayton system description

The schematic diagram for a100kW regenerative Brayton heat engine driven by the hybrid of fossil fuel and solar energy is shown in Fig.1.During the day time,the parabolic dish concentrator con-trolled by a sun tracker dramatically focuses sun light on the focal point where the receiver is located.The receiver absorbs energy re?ected by the concentrator and transfers it to the Brayton heat engine’s working?uid.The Brayton heat engine produces power by the controlled burning of fuel.In the solar-dish Brayton system, solar energy is used to replace(or supplement)the fuel.The result-ing hot working?uid expands rapidly and is used to produce power.In the gas turbine,the burning is continuous and the expan-sion working?uid is used to drive the turbine and alternator.The waste heat exhausted from the gas turbine is used to preheat working?uid from the compressor in the regenerator,which is a key to improve system ef?ciency.

Thermodynamic model of solar-dish Brayton system

A thermodynamic model is established to calculate the power output,thermal ef?ciency and dimensionless thermo-economic performance of solar-dish Brayton system based on the following assumptions:

The solar-dish Brayton system operates at a steady state.

The heat source and heat sink of Brayton heat engine have in?-nite thermal capacitance rates.

Nomenclature

q obtained energy,W

I direct solar?ux intensity,W/m2

A area,m2

f dimensionless objective functions

h heat transfer coef?cient,W mà2Kà1

T temperature,K

e emissivity factor

R concentrator ratio

_Q heat transfer rate,W

_C heat capacity rate,W Kà1

N number of transfer units

C i annual investment cost,ncu yearà1

C f annual fuel cost,nuc yearà1

P power output

F0dimensionless thermo-economic performance

ED Euclidian distance

SE Shannon Entropy

a investment cost parameter,ncu yearà1mà2

b fuel consumption cost parameter,ncu yearà1Wà1

k a/b,economical parameter,W mà2

d deviation index

X assessment parameter of Shannon Entropy decision making Y assessment parameter of TOPSIS decision making Greek

g thermal ef?ciency

d Stefan’s constant,W mà2Kà4

e heat exchanger effectiveness

n percentage of the internal conductance of heat engine with respect to the thermal capacity rate of working

?uid

Subscript

c parabolic dish solar collector

app aperture

rec receiver

B Brayton heat engine

wf working?uid

H heat resource

L heat sink

R regenerator

0ambient condition,optics

1–6process states of Brayton cycle

168Y.Li et al./Electrical Power and Energy Systems64(2015)167–175

The Brayton heat engine is an endoreversible regenerative gas-turbine power cycle.

The imperfect performance of parabolic dish solar collector,the external irreversibility of Brayton heat engine and conductive thermal bridging loss are considered.

Fig.2shows the T–S diagram of solar-dish Brayton system based on the assumptions,which consists of four processes includ-ing two constant pressure(1–2and3–4)and two isentropic pro-cesses(2–3and4–1).The working?uid at state1is heated by the regenerator and then the hot isothermal reservoir at tempera-ture T H to state2.After this process,the working?uid passes through the turbine to state3for producing power.After state3, the working?uid releases heat to the regenerator and then to the cold isothermal reservoir at temperature T L to state4.Finally the working?uid undergoes the compression process4–1in the compressor and return to state1.

The obtained energy of the parabolic dish collector

Given the conduction,convection and radiation losses,the actual useful obtained energy of the parabolic dish collector is given by[3,33]q

c

?IA app g0àA rec?h ceT HàT0Tte c deT4HàT40

where I is the direct solar?ux intensity;A app

tor aperture area and the absorber area,respectively;

lector optical ef?ciency;h c is the conduction/convection transfer coef?cient;T H and T0are the average

and the ambient temperature,respectively;

tor of the collector;and d is the Stefan’s constant.

Thermal ef?ciency g c of the parabolic

obtained as[3,33]

g

c

?

q

c

IA app

?g0à

1

IR c

h ceT HàT0Tte c deT4

H

à

h

where R c is the collector concentrating ratio.

The conductive thermal bridging losses from

heat sink

Using Bejan’s linear-mode[34],the rate

from the hot reservoir at temperature T H to

temperature T L is given by

_Q

HL

?_C BeT HàT LT?_C wf neT HàT LT

where_C B is the internal conductance of Brayton heat engine;C wf is the heat capacity rate of the working?uid;and n denotes the per-centage of the internal conductance with respect to the heat capac-ity rate of the working?uid(n?_C B=_C wf).

The absorbed heat from the heat source and the released heat to the heat sink

The rates at which heat is supplied from heat source to working ?uid(_Q H),heat is rejected from the working?uid to the heat sink (_Q L)and heat is exchanged in the regenerator(_Q R)are obtained by [35,36]

_Q

H

?_C wf e HeT HàT5T?_C wfeT2àT5Te4T

_Q

L

?_C wf e LeT6àT LT?_C wfeT6àT4Te5T

_Q

R

?_C wf e ReT3àT1T?_C wfeT5àT1T?_C wfeT3àT6Te6T

where the effectiveness of hot-side heat exchanger e H,cold-side heat exchanger e L and regenerative heat-exchanger e R are de?ned as:

e H?1àeàN He7Te L?1àeàN Le8T

Fig.1.Schematic diagram of a solar-dish Brayton system. Fig.2.T–S diagram of a solar-dish Brayton system.

e R?N R

N Rt1

e9T

where the number of transfer units N H,N L,N R are based on the min-imum heat conductance rates as N H?h H A H=_C wf,N L?h L A L=_C wf, N R?h R A R=_C wf.

Rearranging Eqs.(4)–(6)gives

T2?e H T Hte1àe HTT5e10TT4?e L T Lte1àe LTT6e11TT5?e1àe RTT1te R T3e12TT6?e1àe RTT3te R mT1e13TSince processes(4–1)and(2–3)are isentropic compression and expansion,the second law of thermodynamics requires that

T1T3?T2T4e14T

After substituting Eqs.(10)–(13)into Eq.(14),it can be obtained that

a1T2

3

ta2T3ta3?0e15T

where

a1?e1àe HTe1àe LTe1àe RTe R

a2?a4T1ta5

a3?a1T2

1

ta6T1ta7

a4?e1àe HTe1àe LTe2Rte1àe HTe1àe LTe1àe RT2à1

a5?e He1àe LTe1àe RTT Hte1àe HTe L e R T L

a6?e He1àe LTe R T Hte1àe HTe Le1àe RTT L

a7?e H e L T H T L

Then T2,T3,T4,T5and T6can all be expressed as a function of T1, T H,T L and e R,e L,e H:

T3?a8?àa2àea2

2à4a1a3T1=2

h i

=2a1e16TT2?e H T Hte1àe RT?e1àe RTT1te R a8 e17T

T4?e L T Lte1àe RT?e1àe RTa8te R T1 e18TT5?e1àe RTT1te R a8e19TT6?e1àe RTa8te R T1e20TThe total heat absorbed rate_Q HT from the hot reservoir and heat released rate_Q LT to cold reservoir are obtained using the following expressions:

_Q

HT

?_Q Ht_Q HL?_C wf e HeT HàT5Ttn_C wfeT HàT LTe21T

_Q

LT

?_Q Lt_Q HL?_C wf e LeT6àT LTtn_C wfeT HàT LTe22TSubstituting from Eqs.(19),(20)into Eqs.(21),(22)gives

_Q

HT

?_C wf e H?T Hàe1àe RTT1àe R a8 t_C wf neT HàT LTe23T

_Q LT ?_C wf e L?e1àe RTa8te R T1àT L t_C wf neT HàT LTe24T

Power output,thermal ef?ciency and dimensionless thermal-economic

performance

According to the?rst law of thermodynamics,the power output

P and the thermal ef?ciency g B of Brayton heat engine are given by:

P?_Q HTà_Q LT?_C wf e H?T Hàe1àe RTT1àe R a8 à_C wf e L?e1àe RTa8

te R T1àT L e25T

g

B

?1à

_Q

LT

_Q

HT

?

e H?T Hàe1àe RTT1àe R a8 àe L?e1àe RTa8te R T1àT L

e H?T Hàe1àe RTT1àe R a8 tneT HàT LT

e26T

Therefore,the thermal ef?ciency of solar-dish Brayton system is

obtained by

g

m

?g B g c?g0à

1

IR c

h ceT HàT0Tte c deT4

H

àT4

T

h i

?

e H?T Hàe1àe RTT1àe R a8 àe L?e1àe RTa8te R T1àT L

e H?T Hàe1àe RTT1àe R a8 tneT HàT LTe27T

Given the thermo-economic performance of solar-dish Brayton

system,the power output per total cost accounting for both invest-

ment and fuel consumption cost is also considered as the optimiza-

tion objective,which is de?ned as follows[11]:

F?

P

C itC f

e28T

where C i and C f refer to annual investment and fuel consumption

costs,respectively.The investment cost of the system is assumed

to be proportional to the size of the system.The size of the system

can be taken proportional to the total heat transfer area.Thus the

annual investment cost of the system can be given as:

C i?aeA HtA LTe29T

where a is equal to the capital recovery factor times investment cost

per unit heat transfer area;the hot-side exchanger area A H and cold-

side exchanger area A L are obtained as

A H?à

_C

wf

lne1àe HT

h H

e30T

A L?à

_C

wf

lne1àe LT

h L

e31T

The annual fuel consumption cost is proportional to the heat

rate input,i.e.

C f?b_Q HTe32T

where the coef?cient b is equal to the annual operation hours times

price per unit heat input.After substituting Eqs.(25)and(29)–(32)

into Eq.(28),the dimensionless thermo-economic performance is

given by

F0?

e H?T Hàe1àe RTT1àe R a8 àe L?e1àe RTa8te R T1àT L

e H?T Hàe1àe RTT1àe R a8 àk lne1àe HT

h H

tlne1àe LT

h L

h ie33T

where k=a/b is the economical parameter.

Multi-objective optimization with evolution algorithm based

on NSGA-II

Multi-objective optimization formulation

In multi-objective optimization,there does not typically exist a

feasible solution that maximizes or minimizes all objective func-

tions simultaneously.Therefore,attention is paid to Pareto optimal

solutions which are not dominated by other solutions and cannot

170Y.Li et al./Electrical Power and Energy Systems64(2015)167–175

be improved with respect to any objective

least one objective.The set of all

referred to as the Pareto optimal set,and the

tive function values are called the Pareto

tier is bounded by an ideal objective

objective vector z nadir,which de?ne upper

the objective function values of Pareto

tively[17].All mentioned above have been

illustrates the Pareto frontier for the

functions f1and f2.

Visualizing the Pareto frontier via

NSGA-II

In the present investigation,the Pareto

responding Pareto frontier are achieved

algorithm based on elitist non-dominated

(NSGA-II)proposed by Deb[37].Shi and Reitz

formance of different schemes for

algorithms and concluded that NSGA-II was

ods because it carried out an elite-preserving

cit diversity-preserving mechanism

depicted in Fig.4.In an evolutionary cycle

parent population is?rst generated and is

teria:(i)rank and(ii)crowding distance.

by a tournament selection are stored in an

pool,where crossover and mutation

ate the child populations.Next,the old set of solutions and newly

created solutions are merged to create a larger population,and?t-ness values are assigned to all individuals by the non-dominated sorting.Finally,individuals with better?tness are selected by elitist sorting and these become the parent individuals.These steps are repeated until the maximum generation number is reached.When NSGA-II terminates,non-dominated solutions of the?nal popula-tion are the approximate Pareto frontier of multi-objective space. Decision making in multi-objective optimization

In the case of multi-objective problems,informing the decision making is needed to select the?nal optimal solution from the Par-eto frontier.A number of decision makings can be applied[39].In this study,most recognized and common type of decision makings,which is a useful tool to acquire weights of alternatives[40]. Consider P ij in decision matrix f ij with n alternatives and m objectives,the element of this matrix for j th objective is obtained by:

P ij?

f ij

P n

i?1

f ij

;i?1;...;n;j?1;...;me34TShannon Entropy is calculated as below

SE j?àM

X n

i?1

P ij ln P ij;where M?

1

lnenT

e35T

Next step is to calculate deviation degree(d j)which shows that to what extent j th objective has useful information for decision making.Deviation degree is given by

d j?1àSE je36T

Then the weight of j th objective can be obtained based on fol-lowing equation:

w j?

d j

P m

j?1

d j

e37T

A comprehensive assessment parameter is expressed as:

X i?P ij?w je38TThe solution with maximum X i is selected as a?nal desired optimal solution in Shannon Entropy decision making.Therefore, the index of?nal optimal solution i?nal is

i final?i2maxeX iTe39TLINMAP decision making

The Euclidian distance of every solution on the Pareto frontier from the ideal solution marked by ED i+is de?ned as:

ED it?

?????????????????????????????????????

X n

j?1

ef ijàf ideal

j

T2

r

e40T

Fig.3.Pareto frontier of a multi-objective optimization.Fig.4.Flow chart of NSGA-II algorithm.

Y.

where f ideal

j is the ideal solution of j th objective

optimization.In the LINMAP decision minimum distance from ideal point is optimal solution[41],hence,i?nal is

i final?i2mineED itT

TOPSIS decision making

Besides the ideal solution,the nadir TOPSIS decision making.Therefore,beside the of each solution from ideal solution ED i+,the of each solution from the nadir solution mented as a criterion for the selection of?nal

ED ià?

??????????????????????????????????????X n

j?1

ef ijàf nadir

j

T2 r

A new assessment parameter is de?ned as

Y i?

ED ià

ED iàtED it

e43T

In TOPSIS decision making,a solution with maximum Y i is selected as a desired?nal solution[42],therefore,i?nal is

i final?i2maxeY iTe44TResults and discussion

Multi-objective functions,including triple-objective(P–g m–F0) and dual-objective(P–g m,P–F0and g m–F0),are optimized by imple-menting NSGA-II and decision makings.In this regard,six design variables with corresponding constraints expressed in Eqs.(45)–

(50)have been considered as follows:

Temperature at hot reservoir;700K6T H61000Ke45T

Temperature at cold reservoir;400K6T L6500Ke46T

Temperature of the working fluid at state1of Brayton cycle;

T L

Effectiveness of regenerator;0:56e R60:8e50TAll the other operating parameters will remain constant as g0=0.85,I=1000W mà2,R c=1300,e c=0.9,d=5.67?10à8 W mà2Kà4,h c=20W mà2Kà1,T0=300K,k=4,n=0.02,h H= h L=2000W mà2Kà1,and_C wf?1050W Kà1.

Fig.5shows the Pareto frontier for triple-objective(P,g m and F0) optimization obtained using NSGA-II algorithm with the speci?ed options in Table1.Three?nal optimal solutions selected by Shannon Entropy,LINMAP and TOPSIS decision makings are also indicated separately.The ideal solution is in coordination(69.72, 0.2352and0.3114)which are P,g m and F0respectively.The corresponding nadir solution is in coordination(65.64,0.2255 and0.2968).With respect to the20–30%thermal ef?ciency of solar-dish Bratyton system,optimal solution regions are appropriate according to the previous works[4,15].To assist the optimal design of the system,the?tted curve(F0=0.1018–0.0185ln P+0.0198g m+13.4277g m2à52.1478g m3+71.2245g m4) derived from Pareto frontier is obtained.

Table2lists the?nal optimal results of triple-objective optimi-zation and single-objective optimization in detail.It can be observed that the?nal optimal solutions always occur in the max-imum temperature at hot reservoir,the minimum temperature at cold reservoir and the maximum effectiveness of cold-side heat exchanger,hot-side heat exchanger and regenerator within the de?ned ranges.The differences between the optimal solutions depend on the temperature of the working?uid in Brayton cycle. In order to explore the reasonable status of various solutions obtained in this study,the deviation index of each solution from the ideal solution is presented as[25].

d?

?????????????????????????????????????

P n

j?1

f jàf ideal

j

2

r

?????????????????????????????????????

P n

j?1

f jàf ideal

j

2

r

t

?????????????????????????????????????

P n

j?1

f jàf nadir

j

2

re51T

The last column of Table2represents the deviation indexes for the results in each optimization approach.As is clear,the deviation indexes(0.3682,0.2614and0.2652)for the triple-objective opti-mization are less than those for maximum P,maximum g m and maximum F0which are0.2848,0.4071and0.5645,respectively. Therefore,the?nal optimal solution selected by LINMAP decision making is the most preferred.

In order to analyze the performance difference between hybrid and conventional systems,the same Brayton heat engine without solar?eld is optimized based on the indicated triple-objective. Without the effect of solar?eld on the Brayton cycle,the maximum effectiveness of hot-side heat exchanger,cold-side heat exchanger and regenerator can reach to0.8,0.8and0.9,respectively.Based on the increase of fuel consumption rate and the decrease of invest-ment cost because of no solar share,the economical parameter k=2.5is applied in the optimization.Fig.6presents the Pareto frontier and the?nal optimal result selected by LINMAP decision making which shows the minimum deviation index from ideal solution in triple-objective optimization.It can be seen that

5.Pareto frontier and optimal solutions for triple-objective(P–g m optimization.

Table1

Speci?ed NSGA-II options for multi-objective optimization in this

study.

Speci?ed options for AGA Value

Population250

Generations500

Pool size200

Tour size2

Distribution index for crossover20

Distribution index for mutation20

172Y.Li

the optimal solutions locate in 76.04kW 6P 693.02kW,0.32656g B 60.3681and 0.34156F 060.3916.Table 3compares the corresponding optimal results of hybrid and conventional sys-tems,where the optimal values of E H ,E L ,E R ,T H and T L are not included because they are always approximately equal to their maximum or minimum limits as shown in Table 2.It is found that the optimal power out,ef?ciency and dimensionless thermo-eco-nomic performance of solar-dish Brayton system decrease 22.6%,34.9%and 18.4%respectively when compare with convec-tional Brayton heat engine,which is expected outcome according the previous studies [43,44].

Figs.7–9present the Pareto frontiers for the dual-objective optimization (P –g m ,P –F 0and g m –F 0),where the ?nal optimal solu-tions selected by decision makings are shown too.As can be seen Figs.7–9,the thermal ef?ciency and dimensionless thermo-eco-nomic performance decrease with the rising power output,while the dimensionless thermo-economic increases with the ef?ciency.Likewise,three curves are ?tted to the Pareto frontiers obtained in the dual objective optimization,which are g m =(0.2731à0.0283P 0.5)/(1à0.1192P 0.5),F 0=(0.3131à0.0045P )/(1à0.0142P )and F 0=1.4727g m à0.0354,respectively.

Tables 4–6compare the ?nal optimal results for dual-objective (P –g m ,P –F 0and g m –F 0)optimizations based on Shannon Entropy,LINMAP and TOPSIS decision makings and for single-objective

optimizations based on maximum P ,g m and F 0.From the last vol-ume of Tables 4–6,it is found that the deviation indexes for the dual-objective optimization are less than the corresponding values in single-objective optimization,which is consistent with the result in triple-objective optimization.Hence,it implies that the solutions for multi-objective optimization are more desirable

Table 2

Comparison between optimal solutions for triple-objective (P –g m –F 0)and single-objective optimizations.Fig.6.Pareto frontier and optimal solution for convectional Brayton heat engine.

Table 3

Comparison of optimal results in convectional Brayton heat engine and solar-dish Brayton system based on the NSGA-II and LINMAP decision making.Systems

Dependent variables Objectives T 1(K)

T 2(K)T 3(K)T 4(K)T 5(K)T 6(K)_Q R (kW)_Q HT (kW)_Q LT (kW)P (kW)g

F 0Brayton heat engine 556.38943.93737.86434.91719.71574.52171.49247.98159.1988.790.3581(g B )0.3773Solar-dish Brayton

586.87

904.57

718.38

466.09

692.08

613.17

110.47

235.68

167.04

68.65

0.2331(g m )

0.3077

Pareto frontier and optimal solutions for dual-objective (P –g m )optimization.

Pareto frontier and optimal solutions for dual-objective (P –F 0)optimization.

Y.Li et al./Electrical Power and Energy Systems 64(2015)167–175

173

results.In addition,the ?nal optimal solution selected by Shannon Entropy decision making presents the minimum deviation index,which is different from the case in triple-objective optimization.It should be noted that the application of decision makings has not general rule as other decision makings may yield better optimal solutions in other cases.

Conclusions

A thermodynamic model is developed to determine power output,thermal ef?ciency and dimensionless thermo-economic performance of a solar-dish Brayton system.Multi-objective optimizations,including triple-objective (P –g m –F 0)optimization and dual-objective (P –g m ,P –F 0and g m –F 0)optimization,are carried out using evolutionary algorithm based on NSGA-II,in which the temperatures of hot reservoir,cold reservoir and the working ?uid of Brayton cycle,as well as the effectiveness of hot-side heat exchanger,cold-side heat exchanger and regenerator are consid-ered as design variables.

The ?nal optimal solutions located at the Pareto frontier obtained by NSGA-II are selected by three decision makings based on Shannon Entropy,LINMAP and TOPSIS methods,respectively.The results show that an optimal working ?uid can be achieved in a given case.By comparing the deviation index of each solution from the ideal solution,it is indicated that the multi-objective opti-mization can lead to a more desirable design than the single-objec-tive optimization.In the analysis related to the performance difference between hybrid and convectional systems,it is found that the optimal power out,ef?ciency and dimensionless thermo-economic performance of solar-dish Brayton system decrease by 22.6%,34.9%and 18.4%respectively when compare with that of the same Brayton heat engine without solar ?eld.For triple-objective and dual-objective optimizations,the optimal solutions with the least deviation index are yielded by different decision makings,which illustrates that the application of

decision

9.Pareto frontier and optimal solutions for dual-objective (g optimization.

Table 4

Comparison between optimal solutions for dual-objective (P –g m )and single-objective optimizations.Optimization algorithms

Decision makings

Design variables Objectives Deviation index

E H

E L E R T H (K)T L (K)T 1(K)P (kW)g m

d NSGA-II

Shannon Entropy 0.690.690.8999.99400.03591.2368.990.23210.1768LINMAP 0.690.70.79999.97400.01590.4168.940.23230.1913TOPSIS

0.70.690.8999.97400.09590.8268.970.23220.1840Maximum P 0.690.690.8999.83400.19604.9271.420.22030.2278Maximize g m

0.69

0.69

0.79

999.78

400.11

564.21

68.06

0.2376

0.4065

Table 5

Comparison between optimal solutions for dual-objective (P –F 0)and single-objective optimizations.Optimization algorithms

Decision makings

Design variables Objectives Deviation index E H

E L E R T H (K)T L (K)T 1(K)P (kW)F

d NSGA-II

Shannon Entropy 0.690.690.8999.98400.01590.1568.910.30670.1978LINMAP 0.690.70.8999.96400.09588.9368.820.30710.2199TOPSIS

0.70.70.791000400.08589.3468.850.30690.2125Maximum P 0.690.690.8999.83400.19604.9271.420.29320.2278Maximize F 0

0.69

0.69

0.79

999.78

400.11

564.21

67.42

0.3144

0.5636

Table 6

Comparison between optimal solutions for dual-objective (g m –F 0)and single-objective optimizations.Optimization algorithms

Decision makings

Design variables Objectives

Deviation index

E H

E L E R T H (K)T L (K)T 1(K)g m

F

d NSGA-II

Shannon Entropy 0.69

0.69

0.8

999.99

400.01

565.59

0.2352

0.3114

LINMAP TOPSIS

Maximum g m 0.70.690.79999.84400.15567.970.23760.31240.1169Maximum F 00.690.690.79999.78400.11564.210.23720.31440.1465

Energy Systems 64(2015)167–175

makings has not general rule and depends on speci?c conditions.In addition,the curves?tted to Pareto frontiers for multi-objective optimization are obtained for better insight into the optimal design of solar-dish Brayton system.

Acknowledgments

This work was partially supported by the Fundamental Research Funds for the Central Universities(Grant No. 2012zzts016)and the Key Laboratory of Regenerative Energy Electric-Technology of Hunan Province(Grant No.2012ZNDL009). References

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