Energy and Power Engineering, 2013, 5, 294-299
doi:10.4236/epe.2013.54B058 Published Online July 2013 (http://www.scirp.org/journal/epe) 295
Modeling and Solution of Economic Dispatch Problem
for GTCC Units
Yongmei Wang1*, Hua Xiao2*
1College of Information Engineering, Zhengzhou University, Zhengzhou, China
2College of Electric Power, South China University of Technology, Guangzhou, China
Received February, 2013
ABSTRACT
Economic dispatch problem lies at the kernel among different issues in GTCC units’ operation, which is about mini-
mizing the fuel consumption for a period of operation so as to accomplish optimal load dispatch among units. This pa-
per has analyzed the load dispatch model of gas turbine combined-cycle (GTCC) units and utilizes a quantum genetic
algorithm to optimize the solution of the mode l. The performance of gas turbin e combined-cycle units v aries with many
factors and this directly leads to variation of model parameters. To solve the dispatch problem, variable constraints are
adopted to correct the parameters influenced by ambient conditions. In the simulation, comparison of dispatch models
for GTCC units considering and not considering the influence of ambient conditions indicates that it is necessary to
adopt variable constraints for the dispatch model of GTCC units. To optimize the solution of the model, a Quantum
Genetic Algorithm is used considering its advantages in searching performance. QGA combines the quantum theory
with evolutionary theory of genetic algorithm. It is a new kind of intelligence algorithm which has been successfully
employed in optimization problems. Utilizing quantum code, quantum gate and so on, QGA shows flexibility, high
convergent rate, and global optimal capacity and so on. Simulations were performed by building up models and opti-
mizing the solu tions of the models by QGA. QGA shows better effect than equal micro incremental method used in the
previous literatu re. The operational economy is proved by the results obtained by QGA. It can be concluded that QGA
is quite effective in optimizing economic dispatch problem of GTCC units.
Keywords: GTCC Units; Economic Dispatch; Variable Constraints; Quantum Genetic Algorithm
1. Introduction
Due to the availability of natural gas and the advantages
[1,2] of GTCC units, GTCC plants continue to gain strength
in power industry. In this situation, economic operation
of the GTCC units becomes much important. Economic
dispatch problem lies at the kernel among different issues
in GTCC units’ operation [3, 4].
The economic dispatch problem is about minimizing
the fuel cost for a period of operation so as to accomplish
optimal load dispatch among units and in return satisfy-
ing the total load demand and operation constraints.
Modeling of the load dispatch problem is necessary for
the calculation of the problem. In practical, performance
of a GTCC unit varies with many factors [5-7]. And this
will lead to variation of load dispatch model. Environ-
mental factors are uncontrollable and always changing
with time and location. So the influence of environmental
conditions on establishing the economic dispatch model
is analyzed.
Mathematically, the problem of economic dispatch is a
complex nonlinear problem containing integer and con-
tinuous variables. Many efforts [8-10] have been made to
solve the problem, through various mathematical pro-
gramming and optimization techniques. But these meth-
ods all have certain limitations such as requiring the
formulation in continuous differentiable form, high
computation time, failing to provide g lobal optimal solu-
tion and so on.
With the development of computer technology and ar-
tificial intelligence, modern intelligent algorith ms [11-14]
show great advantages in solving economic dispatch
problem. Quantum genetic algorithm (QGA) is a search-
ing probabilistic method which combines the quantum
theory with evolutionary theory of genetic algorithm. It
utilizes the methods of quantum technology to improve
the diversity of genetic algorithms’ coding so that the
algorithm will have faster calculation speed and stronger
global optimization ability. Nowadays, quantum genetic
algorithm has been widely applied to combination opti-
mization problems [15] and the optimization ability has
*These authors contributed equally to this work.

295 Y. M. WANG, H. XIAO
been approved. Therefore QGA is adopted to solve the
economic dispatch problem for GTCC units. Simulations
were performed by building up models using data from
previous literature [16] and optimizing the solutions of
the models by QGA. The results show th at QGA is quite
effective in solving the optimization of economic dis-
patch problem of GTCC units.
2. Economic Dispatch Model of GTCC Units
2.1. Basic Model of Economic Dispatch Problem
The economic dispatch prob lem of GTCC units is to find
the optimum combination of units that minimizes the
total gas consumption while satisfying the total load de-
mand and operation constraints. In order to analyze the
problem through a mathematical model, the total gas
consumption of all the GTCC units is described as a
function of units’ power outputs. And to optimize the
solution of the load dispatch problem is to get the value
of each unit’s power output while th e total gas consump-
tion function achieves a minimum. Thus the formulation
of a GTCC units load dispatch problem with operation
constraints can be described as follows.
1
1
min max
max
1
min( )
..
n
iii
i
n
ii D
i
iii
n
ii D
i
F
UfP
stU PP
PPP
UPP R

















(1)
where F is the objective function corresponding to the
total gas consumption (in ); n is the total number
of GTCC units.
3
m/h
()
ii
f
P
hP is the gas consumption for the
ith unit (in ); i is the power output of unit i (in
MW); n is the number of units in the system; i repre-
sents ith unit’s running state and it can only be 0 or 1
which represents stop or running.
3
m/
U
D
P
P is the system’s total demand (in MW); mini and maxi are the lower
and upper bounds of power outputs for the ith unit (in
MW); R is the spinning reserve and generally
P
R

0.07
D
P is adopted.
For gas consumption function ii ()
f
P, binomial ex-
pression is usually adopted to fit its characteristic curve
as follow.
2
()
iiiiiii
f
PaPbPc (2)
where are the gas consumption characteristic
coefficients of ith unit.
,,
iii
abc
There are some simplified hypotheses of the model:
All the n units can be arranged to start and stop; fuel
consumption for start or stop is not considered; line
losses are not considered; the fuel property of natural gas
is constant; and the total load demand keeps constant
within a certain interval.
2.2. Economic Dispatch Model of GTCC Units
Adopting Variable Constraints
In the model above, minmax are influenced by the
performance of a GTCC unit. And the performance of a
unit varies with many factors, such as environmental
conditions, condenser pressure, inlet and exhaust losses,
fuel properties, and so on. Considering environmental
factors are always changing with time and location, the
influence of environmental conditions is mainly studied
to show how the performance of GTCC units varies with
environmental conditions.
,
ii
PP
According to Ref [17,18], for the influence of ambient
temperature, generally speaking, every 1℃depression in
inlet air temperature results in about 0.45% power in-
crease of GTCC and very slight variation in heat con-
sumption rate [19]. For the influence of barometric pres-
sure, the output and the barometric pressure generally
decrease proportionately, but the heat rate and other cy-
cle parameters are not affected. However, for the plant
already installed, the variation of this variable is so subtle
that can be neglected. For the influence of humid air, the
variation is to be too small to be consid ered.
To calculate the value of mini and maxi, correction
for ambient temperature needs to be known. The tem-
perature correcti o n fa ctor can be d ef ined as fol low.
P P
0
t
P
P
  (3)
where t is the output power of a GTCC unit under full
load operation when the ambient temperature is t. 0 is
the output power of a GTCC unit under ISO condition
(15℃). Thus, when the ambient temperature is t, the up-
per bound for power output of the ith unit can be
calculated as follow.
PP
maxi
P
max 0i
PP


(4)
Considering the stability and economy of GTCC units,
min max
60%
i
Pi P

is usually adopted.
3. Optimization of Economic Dispatch
Problem for GTCC Units Based on
Quantum Geneti c Algorith m
3.1. Quantum Genetic Algorithm
AjitNarayanan and MarkMoore proposed the concept of
Quantum Genetic Algorithm, which is a searching prob-
abilistic method combining the quantum theory and the
genetic algorithm. The algorithm is based on some prin-
ciples of quantum theory such as interference, superposi-
tion of states. It adopts quantum bits code to create
chromosomes and achieves evolutionary search through

Y. M. WANG, H. XIAO 296
updating of quantum gates. QCA has the advantages of
small population size, stro ng ability of globally o pti miza-
tion, fast convergent speed and so on.
3.2. Quantum Code
QGA uses vector representation to express quantum state,
of which the smallest unit o f information for rep resenting
individuals is quantum bit. Using probability amplitude
of quantum bit to represent the codes of chromosome,
quantum bit coded chromosome probabilistically repre-
sents several states in the search space. The state of a
quantum bit can be represented by a superposition of two
quantum states as follow [20].
||0|1

  (5)
where |

,|0> and |1> represent respectively the states
of a quantum bit;

and

are two complex numbers
represent the probability a mplitudes of th e corresponding
quantum state. They should satisfy the following equa-
tion.
22
1   (6)
where 2

is the probability to have the value 0 and
2

is the probability to have the value 1.
The quantum bit can contain information of 0 and 1
simultaneously. Therefore a chromosome containing m
quantum bits can represent states. 2m
3.3. Quantum Gate
Quantum gate is the actuator of evolution operation.
Quantum rotation gate strategy is adopted to achieve
population evolution as follow.
cos sin '
'sin cos








 

 
 


(7)
where


,

 and


', '

 are the probability ampli-
tudes before and after quantum rotation updating.

is
the rotation angle and its value adopts the methods in
paper [2 1].
3.4. The Procedure of the Optimization by QGA
1) Initialize a population Q(t) of n chromosomes ran-
domly and each chromosome contains m quantum bits.
Each pair of quantum bit probability amplitudes i

and
i

, i = 1,2, ..., m, are initialized with 1/2, which
means the same probability to emerge at the beginning of
the algorithm.
2) Observe each chromosome in Q(t) so that a set of
binary solutions 23
1n are obtained.
Each bit of (){ ,,...}
tt tt
Ptpppp
j
t
p, j=1,2, ..., n, is formed as follow. Gener-
ate a number r randomly with uniform distribution in
range [0, 1]. If r > 2

or 2

, the result of the obser-
vation will be 1, otherwise, it will 0.
3) Measure the fitness of all the solutions observed
above. For the load dispatch problem in this paper, the
objective function can be described as follow.
2
11
[()](
nn
iiiii D
ii
)
g
UfPC UPP

 

(8)
where C is a penalty factor which usually takes a large
value. In the function above, it can lead to small value of
the function if a solution does not satisfy the constraints,
so the solution will be deleted. Record the best solution
and its fitness.
4) Judge whether the convergence condition is satis-
fied. If not, the algorithm will use the quantum rotation
gate to update the population Q(t) so that a new popula-
tion Q(t+1) is obtained.
5) The algorithm will go back to step (2) and the proc-
ess will be repeated iteratively until the convergence
condition is satisfied.
4. Simulation
In paper [16] the author uses equal micro incremental
method to optimize the load dispatch of 3×390 MW gas
turbine generation units. In this paper, the gas consump-
tion characteristic equations from paper [16] are used for
the calculation of optimal load dispatch by QGA.
According to paper [16] the characteristic coefficient
of gas consumption function and bound constraints are
shown in Table 1.
Take total load demand =491 MW on July 25, 2007 as
an example. The food consistence function is described
as follow.
33
2
11
[()(
iiiii D
ii
gUfPCUPP

 

)] (9)
The units in operation are just 1# and 3#, so
13
UU1

, 2 U0

. Take 100 for the penalty factor C.
2
111 1
2
333 3
( )0.058427117.2911904
( )0.1268175.69517860
fPP P
fPP P




 (10)
In this QGA, the size of the population is set to 40; the
maximum iteration number is 200. Through the optimi-
zation of QGA, when the load of 1# unit is 234.34 MW
and 2# unit is 255.94 MW, the maximum value of fitness
function reaches-87229.83. The total gas consumption is
85605.81m3N/h.
Table 1. Gas consumption character coefficients and limit
of load.
unit ai b
i c i [pmin , pma x]
1
2
3
0.058427
0.12583
0.12681
117.29
75.695
71.863
11904
18274
17860
[234, 390]
[234, 390]
[234, 390]

297 Y. M. WANG, H. XIAO
5. Results and Discussions
Using the foregoing QGA method to optimize the eco-
nomic load dispatch problem of GTCC units, the load
dispatch results obtained by QGA on October 26, 2007
are compared with those by other methods, as shown in
Table 2.
According to the results above, QGA could achieve
savings of total gas consumption compared to the equal
micro incremental method and AGC instruction. For in-
stance, on October 26, 2007 , when the total load demand
is 500 MW, the load dispatch results of unit 1# and 3#
are 236.03 MW and 263.25 MW. The gas consumption is
42842.31 m3N/h and 43609.51 m3N/h respectively. The
total gas consumption is 86451.83 m3N/h, which has a
reduction of 2055.50 m3N/h compared to equal micro
incremental method and 1327.151 m3N/h compared to
AGC instruction. It indicates that using QGA to optimize
the load dispatch problem of gas turbine units can im-
prove the economic efficiency of the units.
When considering the influence of ambient conditions,
according to related literature, it can be assumed that
. If the ambient temperature is 25℃, then the
maximum limit of unit i is max 372.45
i
PMW1,3i

,
and the minimum limit of unit i is minmax 60%
ii
PP
223.47
M
W . Then the load dispatch results
are shown in Table 3.
1, 3i
In Tables 2 and 3 the results of load dispatch optimi-
zation are compared. It has shown that there are big dif-
ferences in the results considering the influence of am-
bient condition or not. For example, when the total load
demand is 491 MW, the load dispatch results of the
model with variable constraints are 223.50 MW and
266.80 MW while the results not considering the influ-
ence of ambient temperature are 234.34 MW and 255.94
MW. In fact, considering the influence of ambient condi-
tions can help to make full use of GTCC units’ output
performance in this situation. The process for QGA to
optimize the solution of load dispatch model is shown in
Figure 1.
6. Conclusions
In this paper, the economic dispatch problem of gas tur-
bine combined-cycle units is proposed and the quantum
genetic algorithm is employed on the optimization of the
problem.
050100 150 200
-1.15
-1.1
-1.05
-1
-0.95
-0.9
-0.85 x 10
5
Number of iteration
Best f itness of each generation
Figure 1. Iterative process of QGA.
Table 2. Load dispatch results of 1#,3# by different methods On October 26,2007.
Load
MW
P1
(QGA)
MW
P3
(QGA)
MW
1# gas consumption
(QGA)
m3 N /h
3# gas consumption
(QGA)
m3 N /h
Total gas consumption
(QGA)
m3 N /h
Equal micro
increme ntal meth od
m3 N /h
AGC
m3 N /h
491 234.34 255.94 42598.68 43007.13 85605.81 87114.16 88041.39
500 236.03 263.25 42842.31 43609.51 86451.83 88507.33 87778.98
600 287.24 312.00 50415.61 50846.51 101262.12 103229.10 103538.45
651 324.82 325.41 56166.31 54630.19 110796.50 111009.30 111040.91
701 355.25 344.95 60945.12 58478.96 119424.08 118737.80 119029.83
Table 3. Load dispatch results of 1#,3# considering the influence of ambient temperature.
Load
MW
P1
(QGA)
MW
P3
(QGA)
MW
1# gas consumption
(QGA)
m3 N /h
3# gas consumption
(QGA)
m3 N /h
Total gas consumption
(QGA)
m3 N /h
Equal micro
increme ntal meth od
m3 N /h
AGC
m3 N /h
491 223.50 266.80 41412.29 47426.59 83985.02 87114.16 88041.39
500 224.56 274.72 41558.06 48565.27 84757.70 88507.33 87778.98
600 287.97 311.28 50730.95 54028.17 101366.38 103229.10 103538.45
651 326.05 324.18 56697.07 56036.98 110974.87 111009.30 111040.91
701 356.62 343.58 61736.62 59135.23 119621.05 118737.80 119029.83

Y. M. WANG, H. XIAO 298
On the basis of analyzing the influence of th e ambient
conditions on the performance of GTCC units, tempera-
ture correction factor is used to improve the economic
dispatch model.
The optimization results by QGA are compared with
those by equal micro incremental method. Almost all the
results obtained by QGA in the simulations are better
than other methods used in the literature. With QGA
method the gas consumptions are significantly reduced.
The searching processes show that QGA has good char-
acteristics of globally optimization, fast convergent speed
robustness for initial values and so on.
The comparison of dispatch models for GTCC units
considering and not considering the influence of ambient
conditions indicates that it is so necessary to adopt vari-
able constraints for the dispatch model of GTCC units.
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