Multi-objective thermodynamic optimization of combined Brayton and inverse
Brayton cycles using genetic algorithms

S.M. Besarati

a

, K. Atashkari

a,*

, A. Jamali

a

, A. Hajiloo

a

, N. Nariman-zadeh

a,b

a

Department of Mechanical Engineering, University of Guilan, PO Box 3756, Rasht, Iran

b

Intelligent-based Experimental Mechanics Center of Excellence, School of Mechanical Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran

a r t i c l e

i n f o

Article history:
Received 14 August 2008
Received in revised form 27 April 2009
Accepted 22 September 2009
Available online 23 October 2009

Keywords:
Brayton cycle
Inversed Brayton cycle
Multi-objective optimization
Genetic algorithms

a b s t r a c t

This paper presents a simultaneous optimization study of two outputs performance of a previously pro-
posed combined Brayton and inverse Brayton cycles. It has been carried out by varying the upper cycle
pressure ratio, the expansion pressure of the bottom cycle and using variable, above atmospheric, bottom
cycle inlet pressure. Multi-objective genetic algorithms are used for Pareto approach optimization of the
cycle outputs. The two important conflicting thermodynamic objectives that have been considered in this
work are net specific work (w

s

) and thermal efficiency ð

g

th

Þ. It is shown that some interesting features

among optimal objective functions and decision variables involved in the Baryton and inverse Brayton
cycles can be discovered consequently.

Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The world energy demand has increased steadily and is expected

to increase continuously in the future. The limited sources of fossil
fuels and stringent environmental restrictions on thermal pollution
have necessitated a greater effort in developing more efficient en-
ergy systems. Thermodynamic power cycles are the basis for the
operation of heat engines, which supply most of the world’s electric
power. Gas turbines reject exhaust gases to the atmosphere that
contain a considerable heat content which could be further used
to augment power production or for other purposes.

A development in the search for higher thermal efficiency of con-

ventional power plant has been the introduction of combined cycle.
The gas turbine combined cycle has been extensively used in power
generation. Many techniques have been used to improve the effi-
ciency of the combined cycle. Most of these techniques improve
the efficiency of the gas cycle or steam cycle in order to enhance
the efficiency of the combined cycle. The efficiency of the steam cy-
cle can be increased by applying reheating and by reducing the irre-
versibility of the steam generation

[1]

. Steam and gas turbine

combined cycles use the exhaust heat from the gas turbine to in-
crease the power plant output and boost overall efficiency in order
of 55%

[2]

. These efficient combined cycles are constructed with

gas turbine as topping cycle and steam cycle as the bottom one.

As an alternative form of bottom cycle, an inverse Brayton has

been recently proposed using different configurations for both
power production and cogeneration applications. In this manner,
the expansion process extends to below atmospheric conditions
to generate more power output. Various cycle types have been pre-
sented in available literature using different bottom cycle arrange-
ments and different cycle expansion pressures. In an investigation
Frost et al.

[3]

proposed a hybrid gas turbine cycle (Braysson cycle)

based on the conventional Brayton cycle for high-temperature heat
addition process while adoption the Ericsson cycle for the low tem-
perature heat rejection process, and perform first law analysis
based on energy balance. The predicted thermal efficiency was
54% which is similar to that currently obtainable from steam and
gas combined cycles. In another study Fuji et al.

[4]

investigated

a bottom cycle constructed from an expander followed by an inter-
cooled compression process (mirror arrangement). They limited
the bottom cycle expansion pressure to 0.25 bar to avoid rapid in-
crease in gas flow axial velocity. Bianchi et al.

[5]

proposed an in-

verted Brayton cycle in which the processes were expansion,
cooling at constant pressure and compression to atmospheric pres-
sure followed by a heat recovery heat exchanger. Agnew et al.

[6]

studied a simple inverse Brayton cycle arrangement that consisted
of an expansion and cooling at constant pressure followed by a
recompression process. The main difference between this study
and the work mentioned in

[4]

is that the inlet pressure to the bot-

tom cycle is not set to atmospheric pressure, but is allowed to vary
to maximize the total system output. A recent study examined the
developed configurations of the proposed simple Brayton and

0196-8904/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:

10.1016/j.enconman.2009.09.015

*

Corresponding author. Tel.: [phone]; fax: [phone].
E-mail address:

[e-mail]

(K. Atashkari).

Energy Conversion and Management 51 (2010) 212–217

Contents lists available at

ScienceDirect

Energy Conversion and Management

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n c o n m a n

inverse Brayton cycle by introducing regeneration and reheat. It
was demonstrated that the system with regeneration attained
the best efficiency, but with the smallest work output

[7]

. Due to

two main performance parameters of combined Brayton and in-
verse Brayton cycles and developed configurations a multi-objec-
tive optimization investigation is necessary to be carried out to
determine the best possible of outputs.

Optimization in engineering design has always been of great

importance and interest particularly in solving complex real-world
design problems. Basically, the optimization process is defined as
finding a set of values for a vector of design variables so that it leads
to an optimum value of an objective or cost function. In such single-
objective optimization problems, there may or may not exist some
constraint functions on the design variables and they are respec-
tively referred to as constrained or unconstrained optimization
problems. There are many calculus-based methods including gradi-
ent approaches to search for mostly local optimum solutions and
these are well documented in

[8]

. However, some basic difficulties

in the gradient methods such as their strong dependence on the ini-
tial guess can cause them to find a local optimum rather than a glo-
bal one. This has led to other heuristic optimization methods,
particularly genetic algorithms (GAs) being used extensively during
the last decade. Such nature-inspired evolutionary algorithms

[9]

differ from other traditional calculus based techniques. The main
difference is that GAs work with a population of candidate solu-
tions, not a single point in search space. This helps significantly to
avoid being trapped in local optima as long as the diversity of the
population is well preserved. In multi-objective optimization prob-
lems, there are several objective of cost functions (a vector of objec-
tives) to be optimized simultaneously. These objectives often
conflict with each other so that as one objective function improves,
another deteriorates. Therefore, there is no single optimal solution
that is best with respect to all the objective functions. Instead, there
is a set of optimal solutions, well known as Pareto optimal solutions

[10]

, which distinguish significantly the inherent natures between

single-objective and multi-objective optimization problems. The
concept of a Pareto front in the space of objective functions in mul-
ti-objective optimization problems stand for a set of solutions that
are non-dominated to each other but are superior to the rest of
solutions in the search space. Evidently, changing the vector of de-
sign variables in such Pareto optimal solutions consisting of these
non-dominated solutions would not lead to the improvement of
all objectives simultaneously. Consequently, such change leads to
a deterioration of at least one objective to an inferior one. Thus,
each solution of the Pareto set includes at least one objective infe-
rior to that of another solution in that Pareto set, although both are
superior to others in the rest of search space. The inherent parallel-
ism in evolutionary algorithms makes them suitably eligible for
solving multi-objective optimization problems.

In thermal systems, like many real world engineering design

problems, there are many complex optimization design problems

which can also be multi-objective in nature. The objectives in ther-
mal systems are usually conflicting and non-commensurable, and
thus Pareto solutions provide more insights into the competing
objectives. Recently, there has been a growing interest in evolu-
tionary Pareto optimization in thermal systems. A thermo-eco-
nomic analysis has been performed by Toffolo and Lazzaretto

[11]

in which two exergic and economic issues in a cogeneration

power plant have been considered as conflicting objectives. A mon-
etary multi-objective optimization of a combined cycle power sys-
tem has been studied by Roosen et al.

[12]

.

In this paper a simultaneous optimization of net specific work

and thermal efficiency based on the system proposed by Agnew
et al.

[6]

will be presented. Most of the papers in literature present

either a study of performance analysis of combined Brayton and in-
verse Brayton cycles or a separate single optimization investigation
of power or efficiency in terms of changes in one of inputs vari-
ables. The results of this study provides more choices for optimal
inputs. The two optimal variables, namely upper cycle pressure
(P

R

) and expansion pressure of inverse Brayton cycle (P

5

), are found

using a Pareto approach to multi-objective optimization. Two con-
flicting outputs (net specific work and thermal efficiency) are se-
lected as objective functions for optimization. In this way,
diversity preserving algorithm called

e

– elimination diversity

algorithm is used to enhance the performance of NSGA-II in terms
of diversity of population and Pareto fronts.

2. System description

The detail simulation and performance analysis of the com-

bined Brayton and inverse Brayton cycles has been given in

[6]

.

Following is a brief description for the application of multi-objec-
tive optimization technique, which needs the objective functions
to be related to the input variables (design variables). The system
which is divided into two subsystems is schematically shown in

Fig. 1

. The upper cycle is a Brayton cycle and is used as a gas gen-

erator to power the bottom cycle. Moreover, it is assumed that the
turbine in upper cycle is designed solely to power the compressor
and the bottom cycle is responsible for producing the output
power of complete system. The bottom subsystem works based
on the inverse Brayton cycle and consists of three processes,
namely expansion, cooling at constant pressure and compression.
In the second cycle, the gas turbine exhaust gas is further expanded
beyond the atmospheric condition resulting increases in the power
and efficiency. Applying simple thermodynamic relations for the
system components the outputs can be evaluated as below.

The work required for compression in the upper cycle is:

w

c

1

¼ ðh

2

h

1

Þ=

g

m

ð1Þ

The amount of work produced by turbine in the upper cycle is

used to drive the compressor. It can be determined as:

Nomenclature

h

specific enthalpy (kJ/kg)

P

pressure (bar)

P

R

upper cycle pressure ratio

P

5

bottom cycle expansion pressure

q

in

system heat input (kJ/kg)

w

work (kJ/kg)

t

temperature (°C)

g

th

system thermal efficiency

g

m

mechanical efficiency of turbomachines

Subscripts
a

ambient

c

1

compressor of upper cycle

c

2

compressor of bottom cycle

s

specific

t

1

turbine of upper cycle

t

2

turbine of bottom cycle

S.M. Besarati et al. / Energy Conversion and Management 51 (2010) 212–217

213

w

t

1

¼ ðh

3

h

4

Þ

g

m

ð2Þ

Similarly, the amount of work produced in bottom cycle can be

described as:

w

t

2

¼ ðh

4

h

5

Þ

g

m

ð3Þ

The amount of heat added to the system is equal to the differ-

ence between inlet and outlet enthalpy of the combustion
chamber:

q

in

¼ h

3

h

2

ð4Þ

The work required for driving the second compressor can be

calculated as:

w

c

2

¼ ðh

7

h

6

Þ=

g

m

ð5Þ

System specific work is determined as:

w

s

¼ w

t

2

w

c

2

ð6Þ

The thermal efficiency can be evaluated as:

g

th

¼ w

s

=

q

in

ð7Þ

In the present work, the temperature and pressure of the intake

air

are

assumed

to

be

constant

at

t

1

¼ t

a

¼ 15

C

and

P

1

¼ P

a

¼ 1:013 bar, respectively. The turbine inlet temperature

of the upper cycle is chosen to be constant as, t

3

¼ 1300

C. More-

over, the temperature of the expanded gas before entering the
compressor of the bottom cycle is fixed as, t

6

¼ 65

C, whereas

the pressure at the exhaust of the bottom compressor is chosen
as, P

8

¼ 1:04 bar. In addition, the isentropic efficiencies of the com-

pressors and turbines are taken as 0.85 and 0.9, respectively while
the mechanical efficiency of turbo machines is assumed 99%.

3. Multi-objective Pareto optimization

Multi-objective optimization which is also called multi-criteria

optimization or vector optimization has been defined as finding a
vector of decision variables satisfying constraints to give optimal
values to all objective functions. In general, it can be mathemati-
cally defined as:

Find the vector X

¼ ½x

1

;

x

2

; . . . ;

x

n

T

to optimize

FðXÞ ¼ ½f

1

ðXÞ; f

2

ðXÞ; . . . ; f

k

ðXÞ

T

ð8Þ

subject to m inequality constraints

l

i

ðXÞ 6 0;

i ¼ 1 to m

ð9Þ

and p equality constraints

h

j

ðXÞ ¼ 0;

j ¼ 1 to p

ð10Þ

where X

2 R

n

is the vector of decision or design variables, and

FðXÞ 2 R

k

is the vector of objective functions. Without loss of gen-

erality, it is assumed that all objective functions are to be mini-
mized. Such multi-objective minimization based on the Pareto
approach can be conducted using some definitions:

3.1. Definition of Pareto dominance

A vector U ¼ ½u

1

;

u

2

; . . . ;

u

k

2 R

k

dominates to vector V ¼

½

v

1

;

v

2

; . . . ;

v

k

2 R

k

(denoted

by

U V)

if

and

only

if

8

i 2 1; 2; . . . ; kg

f

, u

i

6

v

i

^ 9 j 2 1; 2; . . . ; kg

f

:

u

j

<

v

j

It means

that there is at least one u

j

which is smaller than

v

j

whilst the rest

u’s are either smaller or equal to corresponding

v

’s.

Fig. 1. System layout

[6]

.

214

S.M. Besarati et al. / Energy Conversion and Management 51 (2010) 212–217

3.2. Definition of Pareto optimality

A point X

2

X

(

X

is a feasible region in R

n

satisfying Eqs. (8)

and (9)) is said to be Pareto optimal (minimal) with respect to all
X 2

X

if and only if FðX

Þ FðXÞ. Alternatively, it can be readily re-

stated

as

8

i 2 1; 2; . . . ; kg

f

,

8

X 2

X

fX

g

f

i

ðX

Þ 6 f

i

ðXÞ ^ 9j 2

f1; 2; . . . ; kg :

f

j

ðX

Þ < f

j

ðXÞ. It means that the solution X

is said

to be Pareto optimal (minimal) if no other solution can be found
to dominate X

using the definition of Pareto dominance.

3.3. Definition of Pareto set

For a given multi-objective problem, a Pareto set P

is a set in

the decision variable space consisting of all the Pareto optimal vec-
tors, P

¼ fX 2

X

j 9

=

X

0

2

X

FðX

0

Þ FðXÞg In other words, there is

no other X

0

in

X

that dominates any X 2 P

3.4. Definition of Pareto front

For a given multi-objective problem, the Pareto front PT

is a

set of vectors of objective functions which are obtained using the
vectors of decision variables in the Pareto set P

, that is,

PT

¼ fFðXÞ ¼ ðf

1

ðXÞ; f

2

ðXÞ; . . . ; f

k

ðXÞÞ : X 2 P

. Therefore, the Pare-

to fron PT

is a set of the vectors of objective functions mapped

from P

.

Evolutionary algorithms have been widely used for multi-

objective optimization because of their natural properties suited
for these types of problems. This is mostly because of their par-
allel or population-based search approach. Therefore, most diffi-
culties and deficiencies within the classical methods in solving
multi-objective optimization problems are eliminated. For exam-
ple, there is no need for either several runs to find the Pareto
front or quantification of the importance of each objective using
numerical weights. It is very important in evolutionary algo-
rithms that the genetic diversity within the population be pre-
served sufficiently. This main issue in multi-objective problems
has been addressed by much related research work

[13]

. Conse-

quently, the premature convergence of multi-objective evolution-
ary algorithms is prevented and the solutions are directed and
distributed along the true Pareto front if such genetic diversity
is well provided. The Pareto-based approach of NSGA-II

[14]

has been recently used in a wide range of engineering multi-
objective problems because of its simple yet efficient non-domi-
nance ranking procedure in yielding different levels of Pareto
frontiers. However, the crowding approach in such a state-of-
the-art multi-objective evolutionary algorithm works efficiently
for two-objective optimization problems as a diversity-preserving
operator which is not the case for problems with more than two
objective functions. The reason is that the sorting procedure of
individuals based on each objective in this algorithm will cause
different enclosing hyper-boxes. It must be noted that, in a
two-objective Pareto optimization, if the solutions of a Pareto
front are sorted in a decreasing order of importance to one objec-
tive, these solutions are then automatically ordered in an
increasing order of importance to the second objective. Thus,
the hyper-boxes surrounding an individual solution remain un-
changed in the objective-wise sorting procedure of the crowding
distance of NSGA-II in the two-objective Pareto optimization
problem. However, in multi-objective Pareto optimization prob-
lem with more than two objectives, such sorting procedure of
individuals based on each objective in this algorithm will cause
different enclosing hyper-boxes. Thus, the overall crowding dis-
tance of an individual computed in this way may not exactly re-
flect the true measure of diversity or crowding property for the
multi-objective Pareto optimization problems with more than
two objectives.

In this work, a new method called

e

– elimination diversity algo-

rithm is deployed to modify NSGA-II so that it can safely be used
for any number of objective functions (particularly for more than
two objectives) in multi-objective optimization problems.

4. Multi-objective thermodynamic optimization of combined
Brayton and inverse Brayton cycles

Some predicted results of thermal efficiency and specific work

output produced from the simulation procedure, for three selected
bottom cycle expansion pressure at various upper cycle pressure
ratio, are shown graphically in

Figs. 2 and 3

. As obviously shown

in these figures, there is a conflict between the values of two out-
puts of the system. It clearly indicates that improving cycle effi-
ciency deteriorates the specific work output and vice versa. The
results obtained in this paper show close agreements with those
given with the use of a commercial software in

[6]

. As one can de-

duce to find simultaneous optimum values for ðw

s

Þ and ð

g

th

Þ; using

above procedure, with respect to varying P

R

and P

5

is a very diffi-

cult and time consuming work.

In order to investigate the optimal thermodynamic behavior of

the system, specific output work and thermal efficiency are consid-
ered as objective functions. The design variable vector is
P ¼ ½P

R

;

P

5

, which has to be optimally determined based on the

multi-objective Pareto approach. Evidently, it is expected that both
objective functions to be maximized.

The evolutionary process of optimum selection of the design

variables vector to obtain the Pareto front of those objective func-
tions is accomplished with a population size of 70 with crossover
probability, P

c

;

and mutation probability, P

m

;

of 0.8 and 0.02,

respectively, using the modified NSGA-II

[15]

. The range of varia-

tions for P

R

and P

5

are assumed to be 8–25 bar and 0.2–0.8 bar,

respectively. Consequently, a total number of 31 non-dominated
optimum design points have been obtained, as shown in

Fig. 4

in

the plane of the specific work output and efficiency. It can be ob-
served from the Pareto front of the figure improving one objective
will cause deterioration of the other objective. However, optimum
design point A demonstrates a trade-off design point in terms of
the two selected objective functions. The values assigned to this
point are P = [13, 0.34 bar]. Moreover, design points B and C repre-
sent the highest efficiency and highest specific output work respec-
tively. The values of the objective functions in these optimal points
are very close to those obtained by single-objective optimization.
These agreements confirm the accuracy of multi-objective optimi-
zation. The values assigned to points B and C are P = [25, 0.49 bar]
and P = [10, 0.31 bar], respectively. As might be expected, the max-
imum efficiency is obtained at the highest possible value of P

R

.

Fig. 2. Variation of system specific work versus upper compressor pressure ratio.

S.M. Besarati et al. / Energy Conversion and Management 51 (2010) 212–217

215

The variations of the corresponding design variables with re-

spect to the same horizontal axis (efficiency) are shown in

Fig. 5

.

The most interesting feature of the figure is that the values of both
design variables corresponding to the optimal values of thermal
efficiency are maximized while the efficiency reaches to its highest
value.

Fig. 6

depicts the variations of the design variables with re-

spect to specific output work. As may be observed, the values of

both design variables corresponding to the optimal values of spe-
cific work are minimized while the work approaches its maximum
value.

The values of corresponding design variables to 31 non-domi-

nated optimum outputs are shown in

Fig. 7

. As can be observed,

while P

R

varied considerably, there are little differences in the val-

ues of P

5

. It means that for a wide range of variations in upper cycle

pressure ratio, the expansion pressure of inverse Brayton cycle
should be adjusted to few values to achieve the optimal perfor-
mance of the combine cycles. The range of variations for P

R

and

their corresponding P

5

at which the cycle operates optimally are

summarized in

Table 1

. The trade off point design (A) is also shown

in

Fig. 7

.

5. Conclusion

The work presented in this paper provided a multi-objective GA

(non-dominated sorting genetic algorithm, NSGA II) with a new
diversity preserving mechanism to obtain Pareto based optimiza-
tion of the performance of combined Brayton and inverse Brayton
cycles. Applying the first law of thermodynamic two objective
functions, namely, specific output work and thermal efficiency
were determined in terms of two design variables (upper cycle
pressure ratio and bottom cycle inlet pressure). Simultaneous opti-
mization of the two outputs revealed some interesting features
among optimal objective functions and decision variables involved
in the thermodynamic cycle of the proposed system that would
have not been obtained without the use of a multi-objective opti-
mization approach. It was also demonstrated that two extreme
points in the Pareto included those of single-objective optimization
results, therefore, provided more choices for optimal design
variables.

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Fig. 4. Pareto front of non-dominated efficiency–specific output work.

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Table 1
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