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Solving combined heat and power economic dispatch problem using real coded genetic algorithm with improved Mühlenbein mutation
Accepted Manuscript Title: Solving combined heat and power economic dispatch problem using real coded genetic algorithm with improved mühlenbein mutation Author: A. Haghrah, M. Nazari-Heris, B. Mohammadi-ivatloo PII: DOI: Reference:
S1359-4311(16)00027-2 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.12.136 ATE 7550
To appear in:
Applied Thermal Engineering
Received date: Accepted date:
9-10-2015 29-12-2015
Please cite this article as: A. Haghrah, M. Nazari-Heris, B. Mohammadi-ivatloo, Solving combined heat and power economic dispatch problem using real coded genetic algorithm with improved mühlenbein mutation, Applied Thermal Engineering (2016), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.12.136. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Solving combined heat and power economic dispatch problem using real coded genetic algorithm with improved Mühlenbein mutation A. Haghraha, M. Nazari-Herisa, B. Mohammadi-ivatlooa,* a
Department of Electrical Engineering, University of Tabriz, Tabriz, Iran
*Corresponding author: 109, ECE Department, University of Tabriz, 29 Bahman Blvd., Tabriz, Iran Tel: +98-41-33393744, Fax: +98-41-33300829 (Attention: Mohammadi) Email addresses: [email protected] (A. Haghrah), [email protected] (M. Nazari-Heris), [email protected] (B. Mohammadi-ivatloo) Highlights
An improved Muhlenbein mutation is proposed for GA algorithm
Proposed algorithm is evaluated using different benchmark functions.
The proposed algorithm has shown better convergence and constraint handling capability.
Proposed algorithm found lower cost for CHPED problem in comparison with other algorithms.
Abstract The combined heat and power economic dispatch (CHPED) is a complicated optimization problem which determines the production of heat and power units to obtain the minimum production costs of the system, satisfying the heat and power demands and considering operational constraints. This paper presents a real coded genetic algorithm with improved Mühlenbein mutation (RCGA-IMM) for solving CHPED optimization task. Mühlenbein mutation is implemented on basic RCGA for speeding up the convergence and improving the optimization problem results. To evaluate the performance features, the proposed RCGA-IMM procedure is employed on six benchmark functions. The effect of valve-point and transmission losses are considered in cost function and four test systems are presented to demonstrate the effectiveness and superiority of the proposed method. In all test cases the obtained solutions utilizing RCGA-IMM optimization method are feasible and in most instances express a marked improvement over the provided results by recent works in this area. Keywords: Combined heat and power (CHP), economic dispatch, real code genetic algorithm
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(RCGA), non-convex optimization problem. Nomenclature Total production cost.
C N
p
Number of conventional thermal units.
Nc
Number of co-generation units.
N
h
Number of heat only units.
i
Index utilized to indicate conventional thermal units. j
Index utilized to indicate co-generation units.
k
Index utilized to indicate heat only units. p
Pi
c
Power production of jth co-generation unit.
Pj
c
H
j
h
H
Power production of ith thermal unit.
k
Heat production of jth co-generation unit. Heat production of kth heat only unit.
i
Cost coefficient of ith thermal unit.
i
Cost coefficient of ith thermal unit.
i
Cost coefficient of ith thermal unit. j
Cost coefficient of jth co-generation unit.
bj
Cost coefficient of jth co-generation unit.
cj
Cost coefficient of jth co-generation unit.
a
d
j
Cost coefficient of jth co-generation unit.
ej f
Cost coefficient of jth co-generation unit.
j
Cost coefficient of jth co-generation unit.
ak
Cost coefficient of kth heat only unit.
bk
Cost coefficient of kth heat only unit.
ck
Cost coefficient of kth heat only unit.
Pd
Electric power demand of system.
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Transmission loss.
Ploss H
Thermal power demand of system.
d pmin
Pi
pmax
Pi
Minimum power output of the ith thermal unit in MW. Maximum power output of the ith thermal unit in MW.
cmin
Minimum power output of the jth co-generation unit in MW.
cmax
Maximum power output of the jth co-generation unit in MW.
Pj
Pj H H
cmin j
cmax j
hmin
H
k
H
k
hmax
Minimum heat output of the jth co-generation unit in MWth. Maximum heat output of the jth co-generation unit in MWth. Minimum heat output of the kth heat only unit in MWth. Maximum heat output of the kth heat only unit in MWth.
i
Valve-point effect cost coefficient.
i
Valve-point effects cost coefficient.
B
Loss coefficient matrix.
1. Introduction The most efficient combined cycle generation plants generate electric power at an efficiencies of between 50-60%. Heat is the most wasted energy in the conversion of fossil fuels into electricity. Combined heat and power (co-generation) recovers the heat wasted during such conversion. CHP production unit, not only achieves energy efficiency as much as 90% [1], but also serves an important impress for reducing greenhouse gas emission around 13-18%, which is considered as an environmental advantage [2]. Due to its energy saving and environmental advantages, CHP systems are considered as the main alternative for conventional systems [3, 4]. CHP economic dispatch involves the utilization optimizing of the heat and power units with minimum cost of generation to meet the heat and power demands considering operational constraints [2]. Mutual dependency of multiple demand (heat and power) and heat-power capacity of the co-generation units present complexity for solving the optimization problem [5]. CHPED problem will be more complicated while considering several constraints which consist of valve-point loading, transmission losses, prohibited operation zones of conventional thermal generators. For solving CHPED problem, which has attracted attention of researchers in recent
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years, in prior approaches, non-linear optimization algorithm such as dual and quadratic programming [6], and gradient decent methods, such as Lagrangian relaxation [7] have been employed. However non-convex fuel cost function of the generating units were not considered for solving the problem. In [8], differential evolution with Gaussian mutation (DEGM) is introduced for solving CHPED problem considering valve-point loading and prohibited operating zones of conventional thermal generators. Implementation of Gaussian mutation to DE optimization method resulted to better search efficiency and providing the global optimal solution with high probability. The performance of Lagrangian relaxation in solution of CHPED problem is improved in [9] by utilization of surrogate subgradient multiplier updating procedure. An optimization method based on benders decomposition (BD) has been employed in [10] for solving the CHPED problem, where non-convex feasible operation region of co-generation units has been taken into account. The CHPED problem is solved by proposing a hybrid optimization tool based on harmony search (HS) and genetic algorithm (GA) in [11]. The authors recommended to utilize HSGA which encompass the advantages of adaption and parallelism of GA and inferior individuals identification of HS, in order to obtain the global optimum with high probability. The authors utilized time varying acceleration coefficients PSO (TVAC-PSO) in [12] for the solution of CHPED problem, considering valve-point loading, system losses and capacity limits. This paper introduced a new large test system, considering valve-point loading and the proposed algorithm which has capability to be applied in large systems, obtains the optimal feasible solution. Self adaptive real-coded genetic algorithm has been employed for solving the CHPED problem in [13], considering the optimization problem with equality and inequality constraints. Simulated binary crossover (SBX) is applied for achieving self adaptation. In [14] HS algorithm as a new optimization technique has been implemented for obtaining the optimal solution of the CHPED problem. Optimal solution of CHPED problem by applying invasive weed optimization (IWO) procedure is presented in [15]. A solution for CHPED problem in large scale power systems has been introduced in [16] by proposing an improved group search optimization procedure (IGSO) but the obtained results for system 4 are not feasible in which the minimum obtained cost for this test instance is 58049.019 $. In this paper a novel real coded genetic algorithm (RCGA) with an upgraded mutation process is employed for solving CHPED optimization problem. Valve point effects and system
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transmission losses are taken into account for the solution of the problem. Benchmark test cases and test systems have been utilized to prove the effectiveness of the proposed method. The proposed RCGA-IMM has the capability for dealing with CHPED problem considering valve-point loading effect and transmission losses. The obtained solutions for generation of system units by implementing the proposed RCGA-IMM show feasibility and better solution in terms of total cost, compared with reported studies in this area. The rest of this paper is organized as follows: Section 2 represents the mathematical formulation of the CHPED problem, in which valve point effects and transmission losses are taken into account. Section 3 provides the brief description and basic aspects of GA and a detailed description of the proposed RCGA-IMM. Section 4 expresses the implementation of the proposed procedure to four test instances and provides a comparison of the obtained optimal results with the recent researches in the area of CHPED problem. The paper conclusions are presented in Section 5. 2. Formulation of the CHPED Problem The CHPED is stated to obtain the minimum operation cost of heat and power units, satisfying the heat and power demands. The objective function of the CHPED problem considering conventional thermal units, combined heat and power units and heat-only units is formulated as (1): N
m in
N
p
C
( Pi ) p
i
i =1
N
c
C
c
j
c
( Pj , H j )
j =1
h
C
k
(H
h k
(1)
) ($/ h )
k =1
In which C is the total production cost.
N
p
,
Nc
and
N
h
are the respective number of
conventional thermal units, co-generation units, and heat-only units. The heat and power output of the unit are defined by H and P, respectively. i, j and k are utilized for indicatiing the above mentioned units. The production cost of different unit types can be stated as follows: C i ( Pi ) = i ( Pi ) i Pi p
p
2
p
i
($ / h )
(2)
C j ( Pj , H j ) = a j ( Pj ) b j Pj c j d j ( H j ) c
e jH
c
c j
c
2
f j H j P j ($/ h ) c
c
c
c
2
(3)
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Ck (H
p
Pi
h k
) bk H
h
2
) = ak (H
k
Where
C i Pi
p
c k ($ / h )
h k
(4)
is the respective fuel cost of conventional thermal unit i for producing
MW for 1 hour period. The cost function of conventional thermal units are modelled by
utilization of quadratic function approximation (2) [14, 17, 18]. the co-generation unit
and
j
a
,
j
bj
,
cj
,
d
j
,
ej
and
f
j
C
j
P
c j
,H
c j
is utilized to define
are the cost coefficients of this
unit. The cost function of the co-generation unit is convex in both power output output
c
H
P
c
and heat
, which can be observed from (3).
The cost of heat-only unit k is defined by MWth heat.
ak
,
bk
, and
Ck H
h k
which is considered for producing
H
h
are the cost coefficients of kth heat-only unit.
ck
In order to obtain the optimal solution of the objective function (1), the following constraints should be taken into account: • Power production and demand balance N
N
p
Pi
p
i =1
c
P
c
(5)
= Pd
j
j =1
• Heat production and demand balance N
N
c
H
c j
j =1
h
H
h
= H
k
(6)
d
k =1
• Capacity limits of conventional units Pi
p m in
Pi
p
Pi
pm ax
i = 1,
,N
(7)
p
• Capacity limits of CHP units c m in
Pj
( H j ) Pj Pj c
c
c m in j
( Pj ) H c
j = 1, , N c
Where
cmin
Pj
c
(H j )
(8)
j = 1, , N c
H
cm ax
c j
H
cm ax j
c
( Pj )
(9) c
(H j )
and
cmax
Pj
c
(H j )
which are functions of generated heat
H
c j
represent
minimum and maximum power limits of jth CHP unit respectively. Heat generation limits are identified by
H
cmin j
c
( Pj )
and
H
cmax j
c
( Pj )
which are functions of generated power
c
Pj
. It should
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be mentioned that there are dependency between limitations of the CHP units power production and unit heat production plus limitations of the heat production and unit power production. • Production limits of heat-only units H
H
h m in k
H
h k
hm ax
k = 1,
k
(10)
,Nh
2.1. Valve point impact consideration Most of the reported studies have implemented quadratic and cubic cost function [19, 17]. When steam admission valve starts to open, because of the wire drawing impacts, a ripple is created in the production cost. A sinusoid term has been added to the production cost of the generation units for modeling this impact [20, 21]. Valve-point effects is utilized to express this ripple in the production cost, which is taken into account in the proposed work, making the optimization problem non-convex and non-differentiable. The fuel cost function with the consideration of valve-point effects can be stated as: C i ( Pi ) = i ( Pi ) i Pi p
p
| i sin ( i ( Pi
i
In which
2
pmin
and
p
i
Pi )) | p
i
(11)
are the valve-point effects cost coefficients. The unit fuel cost by
consideration of valve-point effects is shown in Fig. 2 for i = 300
i = 0.035
,
,
Pi
pmin
= 0
and
Pi
pmax
= 680
i = 0.00028
,
i = 8.10
,
i = 550
,
.
2.2. Transmission loss consideration Transmission loss is a function of power production of all units. Two approaches have been introduced for calculating transmission loss including load flow approach [22] and Krons loss formula which is known as B-matrix coefficient loss procedure [23]. Krons loss formula is utilized in proposed work. The transmission loss
Ploss
utilizing B-coefficient formula can be
represented as follows: N
Plo s s =
N
p
N
p
Pi B im Pm p
p
i =1 m =1
N
c
N
p
N
c
P
i
i =1
p
c
B ij P j
j =1
c
P
c j
B
c
jn
Pn
(12)
j =1 n =1
By consideration of transmission loss, power production and demand balance expressed in (5) needs to be modified as follows:
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N
N
p
Pi
p
c
P
i =1
c j
= Pd Plo s s
(13)
j =1
3. Real coded genetic algorithm based on Mühlenbein mutation Genetic algorithm is a meta-heuristic method based on modelling natural selection which is efficiently utilized for solving different optimization problems. There are some positive features of GA with respect to other optimization techniques which attract researchers attention: (a) possibility decrement of local minimum trapping, (b) minimum computations of going from current state to another, (c) derivations or other auxiliary functions are not required [24, 25]. The main steps introduced in GA method are as follows: 1. The problem and algorithm parameters initiation. 2. Initial population generation 3. A new generation improvisation. 4. New generation evaluation. 5. Checking the termination criterion. 3.1. Mühlenbein mutation Let
be the parameter to be mutated and
ci
'
be the resulting parameter by Mühlenbein
ci
mutation: c i' = c i r a n g e i .
Where
range
i
(14)
a i < c i < bi
represents the mutation range which is normally set to
is chosen with a probability of
0.1( b i a i ) .
The sign
and
0.5
15
=
k
2
k
(15)
k=0
k 0 , 1
is randomly generated with
Values on the interval c i
range
i
P (
, c i range
probability of generating a neighbourhood of proximity is produced with a precision of
k
i
= 1) =
1
.
16
are generated utilizing this operator, with the being very high. The minimum possible
ci
range
i
.2
15
[26].
3.2. Improved Mühlenbein mutation Improvement in the Mühlenbein mutation method is done by replacing with formula below:
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b
=
k2
k
(16)
k = a 1
In which a = 1
C u r r e n t I te r a tio n
b = a
(18)
2
k 0 , 1
and
(17)
M a x I te r a tio n
is randomly generated with
P (
k
= 1) =
k a 2
. The minimum possible
2
proximity in improved version of this mutation is produced with a precision of 1
range
i
.2
(
1
2
)
.
and 2 in the formula above are algorithm parameters and must be determined by the
implementer. 3.3. Implementation of RCGA-IMM for CHPED problem Steps of implementing RCGA-IMM for CHPED problem and some of its intricacies are discussed in this section. 3.3.1. Generating initial population For generating initial population, the upper and lower limitations of all of the power and heat production of generation units except a power only one and a heat only one should be taken into account. The production of the excepted units will be determined considering the power and heat demand equality constraints. In order to calculate values of these two units, already allocated power and heat must be calculated. Excluded heat unit generation can be calculated by using (6) as below: N
H
h
= H
b
d
N
c
H
j
p
b
h
(19)
k
is the index of excluded heat unit. The remaining power requirement to be supported
by
Pb
1.
Ploss
2.
Pb = Pd Ploss
, can be calculated using (13) and steps described below:
3.
Ploss
,temp
= 0
p
4. if
H
k =1,k b
j =1
Where
h
c
, temp
= Ploss
Pd Ploss
, temp
N
Pi p
p
i = 1, i b
N
c
j =1
c
Pj
, current
, current
N
p
i =1
Pi p
N
c
j =1
Pj c
then finish process else go to 2
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Where
Ploss
is a temporary variable and
, temp
Ploss
, current
is loss calculated for current state of the
individual. This process for satisfying power and heat equality constraints are also applied after crossover and mutation operators. It must be noticed that in the case of combined heat and power units, which have feasibility region, upper and lower limits of generated heat and power are defined virtually as below:
= c
m ax P max H
c
=
m ax P H F R
max
H
P FR
min P
c
=
min
P
H FR
min H
c
=
max
H
(20)
P FR
which
FR
stands for
Feasible
Region
.
Initial population generated by this method may have infeasible individuals which will be directed to feasible region by means of additional penalty in fitness function. 3.3.2. Fitness assignment and constraint handling Fitness function plays an essential role in directing individuals to feasible and optimal subspaces. So it is important how to assign fitness to individuals as a function of the total cost and penalties. Penalty function in linear form, quadratic form and also constant only penalty are used for inequality, equality and feasible region constraints, respectively as below: k1 ( X i X i ) k 2 P e n a lty L in ( X i ) = m in k1 ( X i X i ) k 2 m ax
P e n a lty Q u a d ( X i ) = k 1 ( X i X i
d e s ir e d
0 P e n a lty C o n s ta n t ( X ) = k
m ax
if X i > X
i
if X i < X
) k2 | X i X i 2
m in i
, .
d e s ir e d
| k3
if X is in th e fe a s ib le r e g io n , if X is n o t in th e fe a s ib le r e g io n .
3.3.3. Crossover Weighted averaging of both parents parameters is used as the crossover operator in the proposed algorithm. Corresponding weights are generated randomly in such a way that parent
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with better fitness is more likely to get higher weights. If the individual obtained from crossover operator is more appropriate than its infirm parent (parent with weaker fitness), it will be replaced in new generation. The prerequisite of applying crossover operator is selection of two individuals as parents. Selection method may affect the final solution of optimization process, so it is important how it is implemented in the algorithm. In the proposed algorithm a sequential selection is used for choosing parents, in which crossover operator is applied to ith index and population, where 1 i <
N
pop
N
pop
N
pop
ith index in the
is the population size.
(21)
2
As in generating initial population, crossover is not applied to two units that will satisfy the shortage in power and heat demand equality constraint. Calculating excluded heat and power units generation are as discussed previously. After finishing crossover application on population in every generation, population is sorted considering fitness value. 3.3.4. Mutation As the success factor of proposed algorithm, Mühlenbein mutation in an improved form which is described formerly is implemented on conventional GA. Mutation operator is applied to all parameters of individuals except two excepted units defined before generating initial population. These two units will satisfy power and heat demand equalities by means of aforementioned calculation method. In the proposed algorithm, mutation operator is only applied to a pre-determined top percent of individuals from the aspect of fitness. Also population is sorted after ending application of mutation operator. 3.3.5. Pseudo-code of the algorithm Pseudo-code of the algorithm is presented here and also the flowchart of the algorithm is provided in Fig. 3 for clarification. For
generation
for
=1
i =1
to
to
Iteration
population
num size
do do
2
o ffs p r in g c r o s s o v e r ( p o p u la tio n [ i ] , p o p u la tio n [ p o p u la tio n s iz e i ] )
if
fitn e s s o f o ffs p r in g is b e tte r th a n p o p u la tio n [ p o p u la tio n s iz e i ]
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then replace it by offspring end if end for sort population by fitness For
j =1
to
determined
mutation
num
do
Select an individual from determined top percent of population and mutate it if
m u ta te d in d iv id u a l is fitte r th a n p o p u la tio n [ p o p u la tio n s iz e ]
then
replace it by mutated individual end if end for sort population by fitness end for 4. Case studies The proposed RCGA-IMM is implemented on six benchmark functions and four test systems. A comparison between proposed optimization method and conventional GA is done in terms of convergence characteristics. It should be noted that the steps of the optimization process and parameters for conventional GA and proposed RCGA-IMM are all similar, except mutation process. Algorithm parameters used for test systems are represented in Table 1. The data provided in this paper is achieved by independently running algorithm for 100 times. It should be mentioned that the obtained results for CHPED are rounded up to four decimal digits. 4.1. Benchmark functions Six benchmark functions are studied in this section in order to evaluate the performance of the proposed RCGA-IMM algorithm. Data of benchmark functions are adopted from [27] and are shown in Table 2. Proposed RCGA-IMM is applied to mentioned benchmark functions and mean and standard deviation of the results are presented in Table 3. 4.2. Test system I The first system tested, contains a power-only unit and a heat-only unit which has been taken from [7]. The linear cost functions and capacity limits of power-only unit (unit 1) and heat-only unit (unit 4) are shown in Eqs. (22) and (23), respectively.
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C 1 ( P1 ) = 5 0 P1
0 P1 1 5 0 M W
(22)
0 H 4 2 6 9 5 .2 M W th
C 4 ( H 4 ) = 2 3 .4 H 4
(23)
The system power and heat demand are 200 MW and 115 MWth, respectively. The parameters of cost functions of CHP units are given in Table 4. Figs. 1 and 4 shows the heat-power feasible operation regions of the cogeneration units. The performance of the proposed RCGA-IMM method on CHPED optimization problem is validated by comparing the obtained results with eighteen references. From Table 5, it is obvious that by implementing RCGA-IMM procedure, total cost provided is 9257.075 which is equal with the results of recent studies. 4.3. Test system II This test contains a conventional power unit, three cogeneration units and a heat-only units, is proposed by [2]. Equations (24) and (25) represents the cost functions of power-only unit (unit 1) and heat-only unit (unit 5), respectively. C 1 ( P1 ) = 0 .0 0 0 1 1 5 P1 0 .0 0 1 7 2 P1 7 .6 9 9 7 P1 2 5 4 .8 8 6 3 3
35 P1 135 MW C 5 ( H 5 ) = 0 .0 3 8 H
0 H 5 60 MWth
2
(24) 2 5
2 .0 1 0 9 H
5
950
(25)
Three different load profiles (LPs) are considered for solving the CHPED problem. The power and heat demand for first load profile (LP1) are 300 MW and 150 MWth, respectively. For second load profile (LP2), the power and heat demand are 250 MW and 175 MWth and third load profile (LP3) is considered the power and heat demand 160 MW and 220 MWth, respectively. The provided results for test system 2 considering three different load profiles are shown in Table 6, comparing the obtained results with the results of recent applied methods. The convergence characteristics of RCGA-IMM and conventional GA for LP1 are presented in Fig. 5. The obtained optimal results for LP1 have been compared with the results of HS [2], GA [2], EDHS [14], CPSO [12], TVAC-PSO [12], FA [28], IWO [15] and BD [10]. As it can be observed in Table VI, total cost obtained for LP1 is 13660.5322 $/h and the obtained results are feasible. The total cost reported for EDHS is 13613 $/h in which the obtained results are not feasible, since the power output of unit 4 is out of feasible region. A comparison of the obtained results for LP2 is done with respect to HS [2], GA [2], EDHS [14], CPSO [12], TVAC-PSO [12],
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FA [28], IWO [15] and BD [10]. The results provided for LP2 are feasible and the total cost obtained for this profile is 12104.8682 $/h. The prepared optimal results for LP3 have been compared with the results of HS [2], GA [2], EDHS [14], CPSO [12], TVAC-PSO [12] and BD [10]. The obtained results for LP2 and LP3 utilizing EDHS [14] are not feasible too. The total cost obtained for LP3 using RCGA-IMM is 11758.6349 $/h. 4.4. Test system III A test system which consists of 7 units including 4 power-only units (units 1-4), 2 CHP units (units 5 and 6) and a heat-only unit (unit 7), considering valve-point effects and transmission losses is taken in to account in this section. Unit data has been taken from [29]. Table 7 contains the cost function parameters of this test instance with the feasible operation region coordinates of CHP units. B-matrix is utilized to show the coefficients of the network loss. 49 14 15 2B = 15 20 25
14
15
15
20
45
16
20
18
16
39
10
12
20
10
40
14
18
12
14
35
19
15
11
17
25 19 15 10 11 17 39
7
(26)
The unit of the mentioned matrix elements are 1/MW. Table 8 compares the prepared optimal results with the results of PSO [18], EP [30], DE [20], RCGA [29], BCO [29], CPSO [12], TVAC-PSO [12] and KH [31]. Total cost provided utilizing RCGA-IMM method is 10094.0552 $/h lower than the total cost of compared methods in which the optimum result is related to TVAC-PSO [12] which is 10100.3164 $/h. The convergence characteristics of the proposed method in comparison with conventional GA for this test system are depicted in Fig. 6. 4.5. Test system IV This test system consists of 13 power only units, 6 CHP units, and 5 heat-only units. Data for each unit of this test instance considered has been adopted from [12], which are shown in Table 9. The power and heat demands of the system are 2350 MW and 1250 MWth, respectively. Table 10 includes the optimal solution of this case utilizing RCGA-IMM, which is compared with results prepared utilizing CPSO [12], TVAC-PSO [12], GSO[16], IGSO[16], OTLBO[32] and GWO[33]. Fig. 7 presents the convergence characteristics of the proposed method in comparison with conventional GA for this test system.
Page 14 of 29
5. Conclusion In this paper RCGA-IMM as a meta-heuristic method is proposed for the solution of CHPED optimization problem. Six benchmark functions and four test instances are utilized to show the efficiency of this method and a marked improvement and feasibility in providing the optimal solution in all cases. The proposed RCGA-IMM shows a great capability for handling different constraints including valve-point loading, system transmission losses, capacity limits, and heat-power dependency. The highlights of the proposed method can be categorized into two pivotal fields: superiority of performance and ease of implementation. Better convergence, capability of handling several constraints in non-convex and complex search spaces and the most important, achieving feasible solutions with lesser cost function values is observed by comprehensive simulation results. Outcomes of the simulation results, indicate the superiority of the proposed method compared to primitive and recently developed methods. The other advantage of the proposed algorithm is the ease of implementation as it is developed based on the conventional and well known algorithm, GA. References [1] Alipour M, Mohammadi-Ivatloo B, Zare K. Stochastic risk-constrained short-term scheduling of industrial cogeneration systems in the presence of demand response programs. Applied Energy 2014;136:393–404. [2] Vasebi A, Fesanghary M, Bathaee S. Combined heat and power economic dispatch by harmony search algorithm. International Journal of Electrical Power & Energy Systems 2007;29(10):713–9. [3] Dong L, Liu H, Riffat S. Development of small-scale and micro-scale biomass-fuelled chp systems–a literature review. Applied thermal engineering 2009;29(11):2119–26. [4] Wang H, Lahdelma R,Wang X, JiaoW, Zhu C, Zou P. Analysis of the location for peak heating in chp based combined district heating systems. Applied Thermal Engineering 2015. [5] Alipour M, Zare K, Mohammadi-Ivatloo B. Short-term scheduling of combined heat and power generation units in the presence of demand response programs. Energy 2014;71:289–301. [6] Rooijers FJ, van Amerongen RA. Static economic dispatch for co-generation systems. Power Systems, IEEE Transactions on 1994;9(3):1392–8. [7] Guo T, Henwood MI, van Ooijen M. An algorithm for combined heat and power economic dispatch. Power Systems, IEEE Transactions on 1996;11(4):1778–84.
Page 15 of 29
[8] Jena C, Basu M, Panigrahi C. Differential evolution with gaussian mutation for combined heat and power economic dispatch. Soft Computing 2014;1–8. [9] Sashirekha A, Pasupuleti J, Moin N, Tan C. Combined heat and power (chp) economic dispatch solved using lagrangian relaxation with surrogate subgradient multiplier updates. International Journal of Electrical Power & Energy Systems 2013;44(1):421–30. [10] Abdolmohammadi HR, Kazemi A. A benders decomposition approach for a combined heat and power economic dispatch. Energy Conversion and Management 2013;71:21–31. [11] Huang SH, Lin PC. A harmony-genetic based heuristic approach toward economic dispatching combined heat and power. International Journal of Electrical Power & Energy Systems 2013;53:482–7. [12] Mohammadi-Ivatloo B, Moradi-Dalvand M, Rabiee A. Combined heat and power economic dispatch problem solution using particle swarm optimization with time varying acceleration coefficients. Electric Power Systems Research 2013;95:9–18. [13] Subbaraj P, Rengaraj R, Salivahanan S. Enhancement of combined heat and power economic dispatch using self adaptive real-coded genetic algorithm. Applied Energy 2009;86(6):915–21. [14] Khorram E, Jaberipour M. Harmony search algorithm for solving combined heat and power economic dispatch problems. Energy Conversion and Management 2011;52(2):1550–4. [15] Jayabarathi T, Yazdani A, Ramesh V, Raghunathan T. Combined heat and power economic dispatch problem using the invasive weed optimization algorithm. Frontiers in Energy 2014;8(1):25–30. [16] Hagh MT, Teimourzadeh S, Alipour M, Aliasghary P. Improved group search optimization method for solving chped in large scale power systems. Energy Conversion and Management 2014;80:446–56. [17] Song Y, Chou C, Stonham T. Combined heat and power economic dispatch by improved ant colony search algorithm. Electric Power Systems Research 1999;52(2):115–21. [18] Wang L, Singh C. Stochastic combined heat and power dispatch based on multi-objective particle swarm optimization. International Journal of Electrical Power & Energy Systems 2008;30(3):226–34. [19] Su CT, Chiang CL. An incorporated algorithm for combined heat and power economic dispatch. Electric Power Systems Research 2004;69(2):187–95.
Page 16 of 29
[20] Basu M. Combined heat and power economic dispatch by using differential evolution. Electric Power Components and Systems 2010;38(8):996–1004. [21] Mohammadi-Ivatloo B, Rabiee A, Soroudi A. Nonconvex dynamic economic power dispatch problems solution using hybrid immune-genetic algorithm. IEEE Systems Journal 2013;7(4):777–85. [22] Abdelaziz A, Kamh M, Mekhamer S, Badr M. A hybrid hnn-qp approach for dynamic economic dispatch problem. Electric Power Systems Research 2008;78(10):1784–8. [23] Victoire TAA, Jeyakumar AE. Reserve constrained dynamic dispatch of units with valve-point effects. Power Systems, IEEE Transactions on 2005;20(3):1273–82. [24] Lee KY, El-Sharkawi MA. Modern heuristic optimization techniques: theory and applications to power systems; vol. 39. John Wiley & Sons; 2008. [25] Haghrah A, Mohammadi-Ivatloo B, Seyedmonir S. Real coded genetic algorithm approach with random transfer vectors-based mutation for short-term hydro–thermal scheduling. IET Generation, Transmission & Distribution 2014;9(1):75–89. [26] Herrera F, Lozano M, Verdegay JL. Tackling real-coded genetic algorithms: Operators and tools for behavioural analysis. Artificial intelligence review 1998;12(4):265–319. [27] He S, Wu QH, Saunders J. Group search optimizer: an optimization algorithm inspired by animal searching behavior. Evolutionary Computation, IEEE Transactions on 2009;13(5):973–90. [28] Yazdani A, Jayabarathi T, Ramesh V, Raghunathan T. Combined heat and power economic dispatch problem using firefly algorithm. Frontiers in Energy 2013;7(2):133–9. [29] Basu M. Bee colony optimization for combined heat and power economic dispatch. Expert Systems with Applications 2011;38(11):13527–31. [30] Wong KP, Algie C. Evolutionary programming approach for combined heat and power dispatch. Electric Power Systems Research 2002;61(3):227–32. [31] Adhvaryyu PK, Chattopadhyay PK, Bhattacharjya A. Application of bio-inspired krill herd algorithm to combined heat and power economic dispatch. In: Innovative Smart Grid Technologies-Asia (ISGT Asia), 2014 IEEE. IEEE; 2014, p. 338–43. [32] Roy PK, Paul C, Sultana S. Oppositional teaching learning based optimization approach for combined heat and power dispatch. International Journal of Electrical Power & Energy Systems 2014;57:392–403.
Page 17 of 29
[33] Jayakumar N, Subramanian S, Ganesan S, Elanchezhian E. Grey wolf optimization for combined heat and power dispatch with cogeneration systems. International Journal of Electrical Power & Energy Systems 2016;74:252–64. [34] Song Y, Xuan Q. Combined heat and power economic dispatch using genetic algorithm based penalty function method. Electric machines and power systems 1998;26(4):363–72. [35] Ramesh V, Jayabarathi T, Shrivastava N, Baska A. A novel selective particle swarm optimization approach for combined heat and power economic dispatch. Electric Power Components and Systems 2009;37(11):1231–40. [36] Rao PN. Combined heat and power economic dispatch: A direct solution. Electric Power Components and Systems 2006;34(9):1043–56. Figure Captions • Fig. 1: The heat-power feasible regions for a combined heat and power unit (CHP unit 3 in case I). • Fig. 2: Illustration of unit fuel cost considering valve-point effects. • Fig. 3: Flowchart of the algorithm. • Fig. 4: The heat-power feasible regions for a combined heat and power unit (CHP unit 2 in case I). • Fig. 5: Convergence characteristics of the proposed algorithm in comparison with conventional GA for test system 2 load profile 1. • Fig. 6: Convergence characteristics of the proposed algorithm in comparison with conventional GA for test system 3. • Fig. 7: Convergence characteristics of the proposed algorithm in comparison with conventional GA for test system 4. Tables Table 1: Algorithm parameters used for test systems. Test
Case
system
Population
Determined
Iteration
Determined
size
mutation
number
top percent
1
2
number 1
2000
4000
1000
0.5
35
8
2
1
2000
4000
1000
0.5
20
8
2
2
2000
4000
1000
0.5
20
8
Page 18 of 29
2
3
2000
5000
1000
0.5
20
8
3
2000
4000
1000
0.5
20
8
4
3000
4000
1000
0.5
20
8
Table 2: Benchmark functions data. Benchmark functions
n
Search space
Global minimum
n
f1 ( x ) =
i =1
f2 (x) =
i =1
f3 ( x) =
(100( x i 1 x i ) ( x i 1)) 2
n
( x i 10 cos (2 x i ) 10) 2
n i =1
j =1
2
xj
f6 (x) =
| x i
1 4000
2
4
i
2
2
i
f 4 ( x ) = 4 x 1 2.1 x 1
f 5 ( x ) = max
2
1 3
x1 x1 x 2 4 x 2 4 x 2 6
2
4
|,1 i n
n i =1
( x i 100)
2
x i 100 cos 1 i =1 i n
30
30, 30
30
5 .1 2 , 5 .1 2
30
100, 100
2
5, 5
30
100, 100
n
0
30
600, 600
n
0
n
0
n
n
0
0
1.0316285
n
Table 3: Comparison of different algorithm mean and standard deviation for benchmark functions.[27] Method GA
f1
f2
f3
f4
f5
338.5616
0.6509
9749.9145
1.0298
7.9610
1.0038
Std.
361.497
0.3594
2594.9593
3 .1 3 1 4 1 0
1.5063
6 .7 5 4 5 1 0
Me
37.3582
20.7863
1 .1 9 7 9 1 0
3
1.0160
0.4123
0.2323
Std.
32.1436
5.9400
2 .1 1 0 9 1 0
3
1 .2 7 8 6 1 0
0.2500
0.4434
Me
49.8359
1.0179
5.7829
1.031628
0.1078
3 .0 7 9 2 1 0
2
Std.
30.1771
0.9509
3.6813
0
3 .9 9 8 1 1 0 3 .0 8 6 7 1 0
2
Me
5.06
4 .6 1 0
1.03
0.3
Me
f6
an
PSO
3
2
an
GSO
2
an
FEP
2
1 .6 1 0
2
2
1 .6 1 0
2
Page 19 of 29
an 1 .4 1 0
2
4 .9 1 0
89.0
5 .0 1 0
2
1.03
13.61
23.1
6 .6 1 0
2
4 .9 1 0
33.28
0.16
1 .4 1 0
3
1.0316
Std.
43.13
0.33
5 .3 1 0
4
6 .0 1 0
Me
6.69
70.82
1 .3 1 0
Std.
14.45
21.49
8 .5 1 0
RCGA-I
Me
5 .9 0 8 7 1 0 1 .2 6 2 2 1 0
8
31
14
MM
an 9
31
13
CEP
Std.
5.87
1 .2 1 0
Me
6.17
Std. Me
2
4
0.5
2 .2 1 0
2
2.0
8 .6 1 0
2
1.2
0.12
an
FES
4
5 .5 1 0
3
3 .7 1 0
2
6 .5 1 0
4
5 .0 1 0
2
an
CES
7
1.0316
4
0.35
0.38
0.42
0.77
an
Std.
1 .4 8 7 0 1 0 7 .6 6 3 9 1 0
5
6 .0 1 0
2 .9 5 6 8 1 0
1 .7 1 3 7 1 0
7
1.03162845
1 .5 6 2 1 1 0
351 .3 6 1 9 1 0 41 .3 3 1 0 1 0 3
15
4
2 .0 6 9 3 1 0 3 .1 2 1 9 1 0
3
Table 4: Cost function parameters of the CHP units of cases I and II. Unit
a
b
c
d
e
f
Feasible region coordinates
P
c
,H
c
Case I 2
0.0345
14.5
2650
0.03
4.2
0.031
[98.8,0], [81,104.8], [215,180], [247,0]
3
0.0435
36
1250
0.027
0.6
0.011
[44,0], [44,15.9], [40,75], [110.2,135.6], [125.8,32.4], [125.8,0]
Case II 2
0.0435
36
1250
0.027
0.6
0.011
[44,0],
Page 20 of 29
[44,15.9], [40,75], [110.2,135.6], [125.8,32.4], [125.8,0] 3
0.1035
34.5
2650
0.025
2.203
0.051
[20,0], [10,40], [45,55], [60,0]
4
0.072
20
1565
0.02
2.34
0.04
[35,0], [35,20], [90,45], [90,25], [105,0]
Table 5: Comparison of simulation results for case I. TPa
THb
TCc
0.37
200.01
115
9452.2
75.03
0
200.05
115
9265.1
39.99
75
0
200
114.99
9257.09
40.01
39.99
75
0
200
114.99
9257.07
160
40
40
75
0
200
115
9257.07
0
200
0
0
115
0
200
115
8606.07
GA_PF[34]
0
159.23
40.77
39.94
75.06
0
200
115
9267.28
SPSO*[35]
0
159.706
39.909
40
75
0
199.616
115
9278.17
5
7
Method
P1
P2
P3
H
IACS[17]
0.08
150.93
49
48.84
65.79
PSO[18]
0.05
159.43
40.57
39.97
IGA[19]
0
160
40
SARGA[13]
0
159.99
HS[2]
0
EDHS*[14]
2
H
3
H
4
2
CPSO[12]
0.00
160.00
40.00
40.00
75.00
0.00
200.00
115.00
9257.08
TVAC-PSO
0
160
40
40
75
0
200
115
9257.07
LR[7]
0
160
40
40
75
0
200
115
9257.07
ACSA[17]
0.08
150.93
49
48.84
65.79
0.37
200.00
115.00
9452.20
DM[36]
0
160
40
40
75
0
200
115
9257.07
LRSS[9]
0
160
40
40
75
0
200
115
9257.07
EP[30]
0.00
160.00
40.00
40.00
75.00
0.00
200.00
115.00
9257.10
FA[28]
0.001
159.998
40.00
40.00
75.00
0.00
200.00
115.00
9257.10
4
6
0.000
159.999
40.00
40.00
75.00
0.00
200.00
115.00
9257.08
[12]
IWO[15]
Page 21 of 29
2
8
BD[10]
0.00
160.00
40.00
40.00
75.00
0.00
200.00
115.00
9257.07
RCGA-IM
0.0000
160.0000
40.0000
40.0000
75.0000
0.0000
200.0000
115.0000
9257.0750
M
* Not feasible. a Total power (MW). b Total heat (MWth). c Total cost ($). Table 6: Comparison of simulation results for case II. Lo
TPa
THb
TCc
38.7
300.0
150.0
13723.
00
000
200
000
2000
39.8
0.00
29.6
299.9
149.9
13779.
400
100
00
400
300
900
5000
133.7
84.0
37.7
0.00
28.1
300.0
149.9
13613.
749
688
626
657
00
118
000
401
0000
40.7
19.2
105.0
64.4
26.4
0.00
59.1
300.0
150.0
13692.
000
309
728
000
003
119
00
955
037
076
5212
TVAC-P
135.0
41.4
18.5
105.0
73.3
37.4
0.00
39.2
300.0
150.0
13672.
SO[12]
000
019
981
000
562
295
00
143
000
000
8892
FA[28]
134.7
40.0
20.2
105.0
75.0
27.8
0.00
47.1
299.9
149.9
13683.
4
0
5
0
0
7
2
9
9
22
134.7
40.0
20.8
104.4
75.0
37.6
37.4
300.0
150.0
13683.
3
0
6
1
0
0
0
65
135.0
40.7
19.2
105.0
73.5
36.7
0.00
39.6
300.0
150.0
13672.
000
687
313
000
957
759
00
284
000
000
83
Method
P1
P2
P3
P4
H
LP HS[2]
134.7
48.2
16.2
100.8
81.0
23.9
6.29
1
400
000
300
500
900
200
135.0
70.8
10.8
83.28
80.5
000
100
400
00
EDHS*[
135.0
18.1
13.0
14]
000
563
CPSO[1
135.0
2]
2
H
3
H
4
H
5
ad
GA[2]
IWO[15]
BD[10]
RCGA-I
0.00
135.0000 40.7680 19.2320 105.0000 7 3.5960 36.7760 0.0000
39.6280 300.0000 150.0000 13660.5322
LP HS[2]
134.6
52.9
10.1
52.23
85.6
39.7
4.18
45.4
250.0
175.0
12284.
2
700
900
100
00
900
300
00
000
000
000
4500
119.2
45.1
15.8
69.89
78.9
22.6
18.4
54.9
250.0
174.9
12327.
200
200
200
00
400
300
000
900
500
600
3700
MM
GA[2]
Page 22 of 29
EDHS*[
135.0
0.11
0.00
114.8
85.8
56.3
0.00
32.8
250.0
174.9
11836.
14]
000
12
00
888
178
198
00
135
000
511
0000
CPSO[1
135.0
40.3
10.0
64.60
70.9
39.9
4.07
60.0
250.0
175.0
12132.
2]
000
446
506
60
318
918
73
000
012
009
8579
TVAC-P
135.0
40.0
10.0
64.94
74.8
39.8
16.1
44.1
250.0
175.0
12117.
SO[12]
000
118
391
91
263
443
867
428
000
000
3895
FA[28]
134.8
40.0
10.0
65.18
75.0
40.0
16.9
43.0
249.9
174.9
12119.
1
0
0
0
0
7
2
9
9
86
134.5
40.0
10.9
75.0
38.9
8.81
52.2
250.0
175.0
12134.
9
0
4
0
8
1
0
0
33
135.0
40.0
10.0
65.00
75.0
40.0
14.4
45.5
250.0
175.0
12116.
000
000
000
00
000
000
029
971
000
000
60
IWO[15]
BD[10]
RCGA-I
64.47
135.0000 40.0000 10.0000 65.0000 75.0000 40.0000 14.0595 45.9405 250.0000 175.0000 12104.8682
MM LP HS[2]
41.41
66.6
10.5
41.39
97.7
40.2
22.8
59.2
160.0
220.0
11810.
3
00
100
900
00
300
300
300
100
000
000
8800
37.98
76.3
10.4
35.03
106.
38.3
15.8
59.9
159.8
220.1
11837.
00
900
100
00
0000
700
400
700
100
800
4000
EDHS*[
135.0
0.00
0.00
25.00
87.2
58.1
40.1
34.3
160.0
219.9
93181.
14]
000
00
00
00
560
586
823
703
000
672
0000
CPSO[1
35.59
57.3
10.0
57.05
89.9
40.0
30.0
60.0
160.0
220.0
11781.
2]
72
554
070
87
767
025
232
000
183
024
3690
TVAC-P
42.14
64.6
10.0
43.22
96.2
40.0
23.7
60.0
160.0
220.0
11758.
SO[12]
33
271
001
95
593
001
404
000
000
000
0625
BD[10]
42.14
64.6
10.0
43.22
96.2
40.0
23.7
60.0
160.0
220.0
11758.
54
296
000
50
614
000
386
000
000
000
06
RCGA-I
42.16
64.6
10.0
43.18
96.2
40.0
23.7
60.0
160.0
220.0
11758.
MM
60
523
000
17
810
000
190
000
000
000
6349
GA[2]
* Not feasible. a Total power (MW). b TOtal heat (MWth). c Total cost ($). Table 7: Cost function parameters of test system III.
Page 23 of 29
1
0.008
2
25
100
0.042
10
75
2
0.003
1.8
60
140
0.04
20
125
3
0.0012
2.1
100
160
0.038
30
175
4
0.001
2
120
180
0.037
40
250
Unit
P
min
P
max
Power only units
a
b
c
d
e
f
Feasible region coordinates
P
c
,H
c
CHP units 5
0.0345
14.5
2650
0.03
4.2
0.031
[98.8,0], [81,104.8], [215,180], [247,0]
6
0.0435
36
1250
0.027
0.6
0.11
[44,0], [44,15.9], [40,75], [110.2,135.6], [125.8,32.4], [125.8,0]
a
b
c
H
0.038
2.0109
950
0
hmin
H
hmax
Heat only units 7
2695.20
Table 8: Comparison of the proposed algorithm with previous methods for case III. Outp
PSO[1
ut
8]
P1
18.462
EP[30]
61.361
DE[20
RCGA[
BCO[2 CPSO[1
TVAC-PSO
]
29]
9]
2]
[12]
44.211
74.6834
43.945
75
47.3383
KH[31]
RCGA-I MM
46.3835
45.6614
Page 24 of 29
6 P2
P3
P4
P5
8
124.26
95.120
98.538
02
5
3
112.77
99.942
112.69
94
7
209.81
7 97.9578
98.588
112.380
98.5398
104.122
8
0
167.230
112.93
30
112.6735
64.3729
112.6735
13
8
2
208.73
209.77
124.907
209.77
250
209.81582
246.185
209.8158
58
19
41
9
19
98.814
98.8
98.821
98.8008
98.8
93.2701
92.3718
98.9736
93.9960
98.5398
3
3
7 44.010
44
44
44.0001
44
40.1585
40.0000
40.7401
40.0000
57.923
18.071
12.537
58.0965
12.097
32.5655
37.8467
0.0000
28.2842
6
3
9
32.760
77.554
78.348
72.6738
74.9999
66.7100
75.0000
3
8
1
59.316
54.373
59.113
1
9
9
Total
608.14
607.95
pow
27
150
P6
7 H
H
H
5
6
7
4 32.4116
78.023 6
59.4919
59.879
44.7606
37.1532
83.2900
46.7158
608.03
607.580
608.03
600.808
600.7392
600.777
600.6865
61
72
8
84
6
150
149.99
150
150
150.000
7
er Total heat Total
99 10613
10390
10317
150
150.000
0 10667
10317
cost
150.0000
0
10325.3
10100.3164
339
10111.1
10094.0552
501
Table 9: Cost function parameters of test system IV.
1
0.00028
8.1
550
300
0.035
0
680
2
0.00056
8.1
309
200
0.042
0
360
3
0.00056
8.1
309
200
0.042
0
360
4
0.00324
7.74
240
150
0.063
60
180
5
0.00324
7.74
240
150
0.063
60
180
Unit
P
min
P
max
Power only units
Page 25 of 29
6
0.00324
7.74
240
150
0.063
60
180
7
0.00324
7.74
240
150
0.063
60
180
8
0.00324
7.74
240
150
0.063
60
180
9
0.00324
7.74
240
150
0.063
60
180
10
0.00284
8.6
126
100
0.084
40
120
11
0.00284
8.6
126
100
0.084
40
120
12
0.00284
8.6
126
100
0.084
55
120
13
0.00284
8.6
126
100
0.084
55
120
a
b
c
d
e
f
Feasible region coordinates
P
c
,H
c
CHP units 14
0.0345
14.5
2650
0.03
4.2
0.031
[98.8,0], [81,104.8], [215,180], [247,0]
15
0.0435
36
1250
0.027
0.6
0.11
[44,0], [44,15.9], [40,75], [110.2,135.6], [125.8,32.4], [125.8,0]
16
0.0345
14.5
2650
0.03
4.2
0.031
[98.8,0], [81,104.8], [215,180], [247,0]
17
0.0435
36
1250
0.027
0.6
0.11
[44,0], [44,15.9], [40,75], [110.2,135.6],
Page 26 of 29
[125.8,32.4], [125.8,0] 18
0.1035
34.5
2650
0.025
2.203
0.051
[20,0], [10,40], [45,55], [60,0]
19
0.072
20
1565
0.02
2.34
0.04
[35,0], [35,20], [90,45], [90,25], [105,0]
a
b
c
H
20
0.038
2.0109
950
0
2695.20
21
0.038
2.0109
950
0
60
22
0.038
2.0109
950
0
60
23
0.052
3.0651
480
0
120
24
0.052
3.0651
480
0
120
hmin
H
hmax
Heat only units
Table 10: Comparison of the proposed algorithm with previous methods for case IV. Output
CPSO[12] TVAC-PSO[1
GSO[16]
2]
IGSO*[16 OTLBO*[3
GWO*[3
RCGA-IM
]
2]
3]
M
P1
680
538.5587
627.7455
628.152
538.5656
538.8440
448.8000
P2
0
224.4608
76.2285
299.4778
299.2123
299.3423
299.9568
P3
0
224.4608
299.5794
154.5535
299.1220
299.3423
299.2108
P4
180
109.8666
159.4386
60.846
109.9920
109.9653
109.8694
P5
180
109.8666
61.2378
103.8538
109.9545
109.9653
109.8679
P6
180
109.8666
60
110.0552
110.4042
109.9653
159.7353
P7
180
109.8666
157.1503
159.0773
109.8045
109.9653
109.8684
P8
180
109.8666
107.2654
109.8258
109.6862
109.9653
60.6545
Page 27 of 29
P9
180
109.8666
110.1816
159.992
109.8992
109.9653
159.7354
P10
50.5304
77.5210
113.9894
41.103
77.3992
77.6223
75.8146
P11
50.5304
77.5210
79.7755
77.7055
77.8364
77.6223
40.1672
P12
55
120
91.1668
94.9768
55.2225
55.0000
92.6079
P13
55
120
115.6511
55.7143
55.0861
55.0000
92.4056
P14
117.4854
88.3514
84.3133
83.9536
81.7524
83.4650
83.0376
P15
45.9281
40.5611
40
40
41.7615
40.0000
40.0071
P16
117.4854
88.3514
81.1796
85.7133
82.2730
82.7732
81.4577
P17
45.9281
40.5611
40
40
40.5599
40.0000
41.6937
P18
10.0013
10.0245
10
10
10.0002
10.0000
10.0042
P19
42.1109
40.4288
35.0970
35
31.4679
31.4568
35.1058
H 14
125.2754
108.9256
106.6588
106.4569
105.2219
106.0991
105.9431
H 15
80.1175
75.4844
74.9980
74.998
76.5205
75.0000
75.0059
H 16
125.2754
108.9256
104.9002
107.4073
105.5142
105.7890
105.0550
H 17
80.1174
75.484
74.9980
74.998
75.4833
75.0000
76.4619
H 18
40.0005
40.0104
40
40
39.9999
40.0000
40.0007
H 19
23.2322
22.4676
19.7385
20
18.3944
18.3782
20.0477
H
20
415.9815
458.7020
469.3368
466.2575
468.9043
469.7337
467.4871
H
21
60
60
60
60
59.9994
60.0000
59.9999
H
22
60
60
60
60
59.9999
60.0000
59.9997
H
23
120
120
119.6511
120
119.9854
120.0000
119.9991
H
24
120
120
119.7176
119.8823
119.9768
120.0000
119.9998
Mean
59853.47
58498.3106
58295.92
58156.51
57883.2105
-
58066.6354
cost ($)
8
43
92
Maximu
60076.69
58318.87
58219.14
57913.7731
-
58301.9013
m cost
03
92
13
58359.552
($)
Page 28 of 29
Minimu
59736.26
m cost
35
58122.7460
58225.74
58049.01
50
97
57856.2676
57846.84
57927.6919
($)
* Not feasible.
Page 29 of 29
S1359-4311(16)00027-2 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.12.136 ATE 7550
To appear in:
Applied Thermal Engineering
Received date: Accepted date:
9-10-2015 29-12-2015
Please cite this article as: A. Haghrah, M. Nazari-Heris, B. Mohammadi-ivatloo, Solving combined heat and power economic dispatch problem using real coded genetic algorithm with improved mühlenbein mutation, Applied Thermal Engineering (2016), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.12.136. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Solving combined heat and power economic dispatch problem using real coded genetic algorithm with improved Mühlenbein mutation A. Haghraha, M. Nazari-Herisa, B. Mohammadi-ivatlooa,* a
Department of Electrical Engineering, University of Tabriz, Tabriz, Iran
*Corresponding author: 109, ECE Department, University of Tabriz, 29 Bahman Blvd., Tabriz, Iran Tel: +98-41-33393744, Fax: +98-41-33300829 (Attention: Mohammadi) Email addresses: [email protected] (A. Haghrah), [email protected] (M. Nazari-Heris), [email protected] (B. Mohammadi-ivatloo) Highlights
An improved Muhlenbein mutation is proposed for GA algorithm
Proposed algorithm is evaluated using different benchmark functions.
The proposed algorithm has shown better convergence and constraint handling capability.
Proposed algorithm found lower cost for CHPED problem in comparison with other algorithms.
Abstract The combined heat and power economic dispatch (CHPED) is a complicated optimization problem which determines the production of heat and power units to obtain the minimum production costs of the system, satisfying the heat and power demands and considering operational constraints. This paper presents a real coded genetic algorithm with improved Mühlenbein mutation (RCGA-IMM) for solving CHPED optimization task. Mühlenbein mutation is implemented on basic RCGA for speeding up the convergence and improving the optimization problem results. To evaluate the performance features, the proposed RCGA-IMM procedure is employed on six benchmark functions. The effect of valve-point and transmission losses are considered in cost function and four test systems are presented to demonstrate the effectiveness and superiority of the proposed method. In all test cases the obtained solutions utilizing RCGA-IMM optimization method are feasible and in most instances express a marked improvement over the provided results by recent works in this area. Keywords: Combined heat and power (CHP), economic dispatch, real code genetic algorithm
Page 1 of 29
(RCGA), non-convex optimization problem. Nomenclature Total production cost.
C N
p
Number of conventional thermal units.
Nc
Number of co-generation units.
N
h
Number of heat only units.
i
Index utilized to indicate conventional thermal units. j
Index utilized to indicate co-generation units.
k
Index utilized to indicate heat only units. p
Pi
c
Power production of jth co-generation unit.
Pj
c
H
j
h
H
Power production of ith thermal unit.
k
Heat production of jth co-generation unit. Heat production of kth heat only unit.
i
Cost coefficient of ith thermal unit.
i
Cost coefficient of ith thermal unit.
i
Cost coefficient of ith thermal unit. j
Cost coefficient of jth co-generation unit.
bj
Cost coefficient of jth co-generation unit.
cj
Cost coefficient of jth co-generation unit.
a
d
j
Cost coefficient of jth co-generation unit.
ej f
Cost coefficient of jth co-generation unit.
j
Cost coefficient of jth co-generation unit.
ak
Cost coefficient of kth heat only unit.
bk
Cost coefficient of kth heat only unit.
ck
Cost coefficient of kth heat only unit.
Pd
Electric power demand of system.
Page 2 of 29
Transmission loss.
Ploss H
Thermal power demand of system.
d pmin
Pi
pmax
Pi
Minimum power output of the ith thermal unit in MW. Maximum power output of the ith thermal unit in MW.
cmin
Minimum power output of the jth co-generation unit in MW.
cmax
Maximum power output of the jth co-generation unit in MW.
Pj
Pj H H
cmin j
cmax j
hmin
H
k
H
k
hmax
Minimum heat output of the jth co-generation unit in MWth. Maximum heat output of the jth co-generation unit in MWth. Minimum heat output of the kth heat only unit in MWth. Maximum heat output of the kth heat only unit in MWth.
i
Valve-point effect cost coefficient.
i
Valve-point effects cost coefficient.
B
Loss coefficient matrix.
1. Introduction The most efficient combined cycle generation plants generate electric power at an efficiencies of between 50-60%. Heat is the most wasted energy in the conversion of fossil fuels into electricity. Combined heat and power (co-generation) recovers the heat wasted during such conversion. CHP production unit, not only achieves energy efficiency as much as 90% [1], but also serves an important impress for reducing greenhouse gas emission around 13-18%, which is considered as an environmental advantage [2]. Due to its energy saving and environmental advantages, CHP systems are considered as the main alternative for conventional systems [3, 4]. CHP economic dispatch involves the utilization optimizing of the heat and power units with minimum cost of generation to meet the heat and power demands considering operational constraints [2]. Mutual dependency of multiple demand (heat and power) and heat-power capacity of the co-generation units present complexity for solving the optimization problem [5]. CHPED problem will be more complicated while considering several constraints which consist of valve-point loading, transmission losses, prohibited operation zones of conventional thermal generators. For solving CHPED problem, which has attracted attention of researchers in recent
Page 3 of 29
years, in prior approaches, non-linear optimization algorithm such as dual and quadratic programming [6], and gradient decent methods, such as Lagrangian relaxation [7] have been employed. However non-convex fuel cost function of the generating units were not considered for solving the problem. In [8], differential evolution with Gaussian mutation (DEGM) is introduced for solving CHPED problem considering valve-point loading and prohibited operating zones of conventional thermal generators. Implementation of Gaussian mutation to DE optimization method resulted to better search efficiency and providing the global optimal solution with high probability. The performance of Lagrangian relaxation in solution of CHPED problem is improved in [9] by utilization of surrogate subgradient multiplier updating procedure. An optimization method based on benders decomposition (BD) has been employed in [10] for solving the CHPED problem, where non-convex feasible operation region of co-generation units has been taken into account. The CHPED problem is solved by proposing a hybrid optimization tool based on harmony search (HS) and genetic algorithm (GA) in [11]. The authors recommended to utilize HSGA which encompass the advantages of adaption and parallelism of GA and inferior individuals identification of HS, in order to obtain the global optimum with high probability. The authors utilized time varying acceleration coefficients PSO (TVAC-PSO) in [12] for the solution of CHPED problem, considering valve-point loading, system losses and capacity limits. This paper introduced a new large test system, considering valve-point loading and the proposed algorithm which has capability to be applied in large systems, obtains the optimal feasible solution. Self adaptive real-coded genetic algorithm has been employed for solving the CHPED problem in [13], considering the optimization problem with equality and inequality constraints. Simulated binary crossover (SBX) is applied for achieving self adaptation. In [14] HS algorithm as a new optimization technique has been implemented for obtaining the optimal solution of the CHPED problem. Optimal solution of CHPED problem by applying invasive weed optimization (IWO) procedure is presented in [15]. A solution for CHPED problem in large scale power systems has been introduced in [16] by proposing an improved group search optimization procedure (IGSO) but the obtained results for system 4 are not feasible in which the minimum obtained cost for this test instance is 58049.019 $. In this paper a novel real coded genetic algorithm (RCGA) with an upgraded mutation process is employed for solving CHPED optimization problem. Valve point effects and system
Page 4 of 29
transmission losses are taken into account for the solution of the problem. Benchmark test cases and test systems have been utilized to prove the effectiveness of the proposed method. The proposed RCGA-IMM has the capability for dealing with CHPED problem considering valve-point loading effect and transmission losses. The obtained solutions for generation of system units by implementing the proposed RCGA-IMM show feasibility and better solution in terms of total cost, compared with reported studies in this area. The rest of this paper is organized as follows: Section 2 represents the mathematical formulation of the CHPED problem, in which valve point effects and transmission losses are taken into account. Section 3 provides the brief description and basic aspects of GA and a detailed description of the proposed RCGA-IMM. Section 4 expresses the implementation of the proposed procedure to four test instances and provides a comparison of the obtained optimal results with the recent researches in the area of CHPED problem. The paper conclusions are presented in Section 5. 2. Formulation of the CHPED Problem The CHPED is stated to obtain the minimum operation cost of heat and power units, satisfying the heat and power demands. The objective function of the CHPED problem considering conventional thermal units, combined heat and power units and heat-only units is formulated as (1): N
m in
N
p
C
( Pi ) p
i
i =1
N
c
C
c
j
c
( Pj , H j )
j =1
h
C
k
(H
h k
(1)
) ($/ h )
k =1
In which C is the total production cost.
N
p
,
Nc
and
N
h
are the respective number of
conventional thermal units, co-generation units, and heat-only units. The heat and power output of the unit are defined by H and P, respectively. i, j and k are utilized for indicatiing the above mentioned units. The production cost of different unit types can be stated as follows: C i ( Pi ) = i ( Pi ) i Pi p
p
2
p
i
($ / h )
(2)
C j ( Pj , H j ) = a j ( Pj ) b j Pj c j d j ( H j ) c
e jH
c
c j
c
2
f j H j P j ($/ h ) c
c
c
c
2
(3)
Page 5 of 29
Ck (H
p
Pi
h k
) bk H
h
2
) = ak (H
k
Where
C i Pi
p
c k ($ / h )
h k
(4)
is the respective fuel cost of conventional thermal unit i for producing
MW for 1 hour period. The cost function of conventional thermal units are modelled by
utilization of quadratic function approximation (2) [14, 17, 18]. the co-generation unit
and
j
a
,
j
bj
,
cj
,
d
j
,
ej
and
f
j
C
j
P
c j
,H
c j
is utilized to define
are the cost coefficients of this
unit. The cost function of the co-generation unit is convex in both power output output
c
H
P
c
and heat
, which can be observed from (3).
The cost of heat-only unit k is defined by MWth heat.
ak
,
bk
, and
Ck H
h k
which is considered for producing
H
h
are the cost coefficients of kth heat-only unit.
ck
In order to obtain the optimal solution of the objective function (1), the following constraints should be taken into account: • Power production and demand balance N
N
p
Pi
p
i =1
c
P
c
(5)
= Pd
j
j =1
• Heat production and demand balance N
N
c
H
c j
j =1
h
H
h
= H
k
(6)
d
k =1
• Capacity limits of conventional units Pi
p m in
Pi
p
Pi
pm ax
i = 1,
,N
(7)
p
• Capacity limits of CHP units c m in
Pj
( H j ) Pj Pj c
c
c m in j
( Pj ) H c
j = 1, , N c
Where
cmin
Pj
c
(H j )
(8)
j = 1, , N c
H
cm ax
c j
H
cm ax j
c
( Pj )
(9) c
(H j )
and
cmax
Pj
c
(H j )
which are functions of generated heat
H
c j
represent
minimum and maximum power limits of jth CHP unit respectively. Heat generation limits are identified by
H
cmin j
c
( Pj )
and
H
cmax j
c
( Pj )
which are functions of generated power
c
Pj
. It should
Page 6 of 29
be mentioned that there are dependency between limitations of the CHP units power production and unit heat production plus limitations of the heat production and unit power production. • Production limits of heat-only units H
H
h m in k
H
h k
hm ax
k = 1,
k
(10)
,Nh
2.1. Valve point impact consideration Most of the reported studies have implemented quadratic and cubic cost function [19, 17]. When steam admission valve starts to open, because of the wire drawing impacts, a ripple is created in the production cost. A sinusoid term has been added to the production cost of the generation units for modeling this impact [20, 21]. Valve-point effects is utilized to express this ripple in the production cost, which is taken into account in the proposed work, making the optimization problem non-convex and non-differentiable. The fuel cost function with the consideration of valve-point effects can be stated as: C i ( Pi ) = i ( Pi ) i Pi p
p
| i sin ( i ( Pi
i
In which
2
pmin
and
p
i
Pi )) | p
i
(11)
are the valve-point effects cost coefficients. The unit fuel cost by
consideration of valve-point effects is shown in Fig. 2 for i = 300
i = 0.035
,
,
Pi
pmin
= 0
and
Pi
pmax
= 680
i = 0.00028
,
i = 8.10
,
i = 550
,
.
2.2. Transmission loss consideration Transmission loss is a function of power production of all units. Two approaches have been introduced for calculating transmission loss including load flow approach [22] and Krons loss formula which is known as B-matrix coefficient loss procedure [23]. Krons loss formula is utilized in proposed work. The transmission loss
Ploss
utilizing B-coefficient formula can be
represented as follows: N
Plo s s =
N
p
N
p
Pi B im Pm p
p
i =1 m =1
N
c
N
p
N
c
P
i
i =1
p
c
B ij P j
j =1
c
P
c j
B
c
jn
Pn
(12)
j =1 n =1
By consideration of transmission loss, power production and demand balance expressed in (5) needs to be modified as follows:
Page 7 of 29
N
N
p
Pi
p
c
P
i =1
c j
= Pd Plo s s
(13)
j =1
3. Real coded genetic algorithm based on Mühlenbein mutation Genetic algorithm is a meta-heuristic method based on modelling natural selection which is efficiently utilized for solving different optimization problems. There are some positive features of GA with respect to other optimization techniques which attract researchers attention: (a) possibility decrement of local minimum trapping, (b) minimum computations of going from current state to another, (c) derivations or other auxiliary functions are not required [24, 25]. The main steps introduced in GA method are as follows: 1. The problem and algorithm parameters initiation. 2. Initial population generation 3. A new generation improvisation. 4. New generation evaluation. 5. Checking the termination criterion. 3.1. Mühlenbein mutation Let
be the parameter to be mutated and
ci
'
be the resulting parameter by Mühlenbein
ci
mutation: c i' = c i r a n g e i .
Where
range
i
(14)
a i < c i < bi
represents the mutation range which is normally set to
is chosen with a probability of
0.1( b i a i ) .
The sign
and
0.5
15
=
k
2
k
(15)
k=0
k 0 , 1
is randomly generated with
Values on the interval c i
range
i
P (
, c i range
probability of generating a neighbourhood of proximity is produced with a precision of
k
i
= 1) =
1
.
16
are generated utilizing this operator, with the being very high. The minimum possible
ci
range
i
.2
15
[26].
3.2. Improved Mühlenbein mutation Improvement in the Mühlenbein mutation method is done by replacing with formula below:
Page 8 of 29
b
=
k2
k
(16)
k = a 1
In which a = 1
C u r r e n t I te r a tio n
b = a
(18)
2
k 0 , 1
and
(17)
M a x I te r a tio n
is randomly generated with
P (
k
= 1) =
k a 2
. The minimum possible
2
proximity in improved version of this mutation is produced with a precision of 1
range
i
.2
(
1
2
)
.
and 2 in the formula above are algorithm parameters and must be determined by the
implementer. 3.3. Implementation of RCGA-IMM for CHPED problem Steps of implementing RCGA-IMM for CHPED problem and some of its intricacies are discussed in this section. 3.3.1. Generating initial population For generating initial population, the upper and lower limitations of all of the power and heat production of generation units except a power only one and a heat only one should be taken into account. The production of the excepted units will be determined considering the power and heat demand equality constraints. In order to calculate values of these two units, already allocated power and heat must be calculated. Excluded heat unit generation can be calculated by using (6) as below: N
H
h
= H
b
d
N
c
H
j
p
b
h
(19)
k
is the index of excluded heat unit. The remaining power requirement to be supported
by
Pb
1.
Ploss
2.
Pb = Pd Ploss
, can be calculated using (13) and steps described below:
3.
Ploss
,temp
= 0
p
4. if
H
k =1,k b
j =1
Where
h
c
, temp
= Ploss
Pd Ploss
, temp
N
Pi p
p
i = 1, i b
N
c
j =1
c
Pj
, current
, current
N
p
i =1
Pi p
N
c
j =1
Pj c
then finish process else go to 2
Page 9 of 29
Where
Ploss
is a temporary variable and
, temp
Ploss
, current
is loss calculated for current state of the
individual. This process for satisfying power and heat equality constraints are also applied after crossover and mutation operators. It must be noticed that in the case of combined heat and power units, which have feasibility region, upper and lower limits of generated heat and power are defined virtually as below:
= c
m ax P max H
c
=
m ax P H F R
max
H
P FR
min P
c
=
min
P
H FR
min H
c
=
max
H
(20)
P FR
which
FR
stands for
Feasible
Region
.
Initial population generated by this method may have infeasible individuals which will be directed to feasible region by means of additional penalty in fitness function. 3.3.2. Fitness assignment and constraint handling Fitness function plays an essential role in directing individuals to feasible and optimal subspaces. So it is important how to assign fitness to individuals as a function of the total cost and penalties. Penalty function in linear form, quadratic form and also constant only penalty are used for inequality, equality and feasible region constraints, respectively as below: k1 ( X i X i ) k 2 P e n a lty L in ( X i ) = m in k1 ( X i X i ) k 2 m ax
P e n a lty Q u a d ( X i ) = k 1 ( X i X i
d e s ir e d
0 P e n a lty C o n s ta n t ( X ) = k
m ax
if X i > X
i
if X i < X
) k2 | X i X i 2
m in i
, .
d e s ir e d
| k3
if X is in th e fe a s ib le r e g io n , if X is n o t in th e fe a s ib le r e g io n .
3.3.3. Crossover Weighted averaging of both parents parameters is used as the crossover operator in the proposed algorithm. Corresponding weights are generated randomly in such a way that parent
Page 10 of 29
with better fitness is more likely to get higher weights. If the individual obtained from crossover operator is more appropriate than its infirm parent (parent with weaker fitness), it will be replaced in new generation. The prerequisite of applying crossover operator is selection of two individuals as parents. Selection method may affect the final solution of optimization process, so it is important how it is implemented in the algorithm. In the proposed algorithm a sequential selection is used for choosing parents, in which crossover operator is applied to ith index and population, where 1 i <
N
pop
N
pop
N
pop
ith index in the
is the population size.
(21)
2
As in generating initial population, crossover is not applied to two units that will satisfy the shortage in power and heat demand equality constraint. Calculating excluded heat and power units generation are as discussed previously. After finishing crossover application on population in every generation, population is sorted considering fitness value. 3.3.4. Mutation As the success factor of proposed algorithm, Mühlenbein mutation in an improved form which is described formerly is implemented on conventional GA. Mutation operator is applied to all parameters of individuals except two excepted units defined before generating initial population. These two units will satisfy power and heat demand equalities by means of aforementioned calculation method. In the proposed algorithm, mutation operator is only applied to a pre-determined top percent of individuals from the aspect of fitness. Also population is sorted after ending application of mutation operator. 3.3.5. Pseudo-code of the algorithm Pseudo-code of the algorithm is presented here and also the flowchart of the algorithm is provided in Fig. 3 for clarification. For
generation
for
=1
i =1
to
to
Iteration
population
num size
do do
2
o ffs p r in g c r o s s o v e r ( p o p u la tio n [ i ] , p o p u la tio n [ p o p u la tio n s iz e i ] )
if
fitn e s s o f o ffs p r in g is b e tte r th a n p o p u la tio n [ p o p u la tio n s iz e i ]
Page 11 of 29
then replace it by offspring end if end for sort population by fitness For
j =1
to
determined
mutation
num
do
Select an individual from determined top percent of population and mutate it if
m u ta te d in d iv id u a l is fitte r th a n p o p u la tio n [ p o p u la tio n s iz e ]
then
replace it by mutated individual end if end for sort population by fitness end for 4. Case studies The proposed RCGA-IMM is implemented on six benchmark functions and four test systems. A comparison between proposed optimization method and conventional GA is done in terms of convergence characteristics. It should be noted that the steps of the optimization process and parameters for conventional GA and proposed RCGA-IMM are all similar, except mutation process. Algorithm parameters used for test systems are represented in Table 1. The data provided in this paper is achieved by independently running algorithm for 100 times. It should be mentioned that the obtained results for CHPED are rounded up to four decimal digits. 4.1. Benchmark functions Six benchmark functions are studied in this section in order to evaluate the performance of the proposed RCGA-IMM algorithm. Data of benchmark functions are adopted from [27] and are shown in Table 2. Proposed RCGA-IMM is applied to mentioned benchmark functions and mean and standard deviation of the results are presented in Table 3. 4.2. Test system I The first system tested, contains a power-only unit and a heat-only unit which has been taken from [7]. The linear cost functions and capacity limits of power-only unit (unit 1) and heat-only unit (unit 4) are shown in Eqs. (22) and (23), respectively.
Page 12 of 29
C 1 ( P1 ) = 5 0 P1
0 P1 1 5 0 M W
(22)
0 H 4 2 6 9 5 .2 M W th
C 4 ( H 4 ) = 2 3 .4 H 4
(23)
The system power and heat demand are 200 MW and 115 MWth, respectively. The parameters of cost functions of CHP units are given in Table 4. Figs. 1 and 4 shows the heat-power feasible operation regions of the cogeneration units. The performance of the proposed RCGA-IMM method on CHPED optimization problem is validated by comparing the obtained results with eighteen references. From Table 5, it is obvious that by implementing RCGA-IMM procedure, total cost provided is 9257.075 which is equal with the results of recent studies. 4.3. Test system II This test contains a conventional power unit, three cogeneration units and a heat-only units, is proposed by [2]. Equations (24) and (25) represents the cost functions of power-only unit (unit 1) and heat-only unit (unit 5), respectively. C 1 ( P1 ) = 0 .0 0 0 1 1 5 P1 0 .0 0 1 7 2 P1 7 .6 9 9 7 P1 2 5 4 .8 8 6 3 3
35 P1 135 MW C 5 ( H 5 ) = 0 .0 3 8 H
0 H 5 60 MWth
2
(24) 2 5
2 .0 1 0 9 H
5
950
(25)
Three different load profiles (LPs) are considered for solving the CHPED problem. The power and heat demand for first load profile (LP1) are 300 MW and 150 MWth, respectively. For second load profile (LP2), the power and heat demand are 250 MW and 175 MWth and third load profile (LP3) is considered the power and heat demand 160 MW and 220 MWth, respectively. The provided results for test system 2 considering three different load profiles are shown in Table 6, comparing the obtained results with the results of recent applied methods. The convergence characteristics of RCGA-IMM and conventional GA for LP1 are presented in Fig. 5. The obtained optimal results for LP1 have been compared with the results of HS [2], GA [2], EDHS [14], CPSO [12], TVAC-PSO [12], FA [28], IWO [15] and BD [10]. As it can be observed in Table VI, total cost obtained for LP1 is 13660.5322 $/h and the obtained results are feasible. The total cost reported for EDHS is 13613 $/h in which the obtained results are not feasible, since the power output of unit 4 is out of feasible region. A comparison of the obtained results for LP2 is done with respect to HS [2], GA [2], EDHS [14], CPSO [12], TVAC-PSO [12],
Page 13 of 29
FA [28], IWO [15] and BD [10]. The results provided for LP2 are feasible and the total cost obtained for this profile is 12104.8682 $/h. The prepared optimal results for LP3 have been compared with the results of HS [2], GA [2], EDHS [14], CPSO [12], TVAC-PSO [12] and BD [10]. The obtained results for LP2 and LP3 utilizing EDHS [14] are not feasible too. The total cost obtained for LP3 using RCGA-IMM is 11758.6349 $/h. 4.4. Test system III A test system which consists of 7 units including 4 power-only units (units 1-4), 2 CHP units (units 5 and 6) and a heat-only unit (unit 7), considering valve-point effects and transmission losses is taken in to account in this section. Unit data has been taken from [29]. Table 7 contains the cost function parameters of this test instance with the feasible operation region coordinates of CHP units. B-matrix is utilized to show the coefficients of the network loss. 49 14 15 2B = 15 20 25
14
15
15
20
45
16
20
18
16
39
10
12
20
10
40
14
18
12
14
35
19
15
11
17
25 19 15 10 11 17 39
7
(26)
The unit of the mentioned matrix elements are 1/MW. Table 8 compares the prepared optimal results with the results of PSO [18], EP [30], DE [20], RCGA [29], BCO [29], CPSO [12], TVAC-PSO [12] and KH [31]. Total cost provided utilizing RCGA-IMM method is 10094.0552 $/h lower than the total cost of compared methods in which the optimum result is related to TVAC-PSO [12] which is 10100.3164 $/h. The convergence characteristics of the proposed method in comparison with conventional GA for this test system are depicted in Fig. 6. 4.5. Test system IV This test system consists of 13 power only units, 6 CHP units, and 5 heat-only units. Data for each unit of this test instance considered has been adopted from [12], which are shown in Table 9. The power and heat demands of the system are 2350 MW and 1250 MWth, respectively. Table 10 includes the optimal solution of this case utilizing RCGA-IMM, which is compared with results prepared utilizing CPSO [12], TVAC-PSO [12], GSO[16], IGSO[16], OTLBO[32] and GWO[33]. Fig. 7 presents the convergence characteristics of the proposed method in comparison with conventional GA for this test system.
Page 14 of 29
5. Conclusion In this paper RCGA-IMM as a meta-heuristic method is proposed for the solution of CHPED optimization problem. Six benchmark functions and four test instances are utilized to show the efficiency of this method and a marked improvement and feasibility in providing the optimal solution in all cases. The proposed RCGA-IMM shows a great capability for handling different constraints including valve-point loading, system transmission losses, capacity limits, and heat-power dependency. The highlights of the proposed method can be categorized into two pivotal fields: superiority of performance and ease of implementation. Better convergence, capability of handling several constraints in non-convex and complex search spaces and the most important, achieving feasible solutions with lesser cost function values is observed by comprehensive simulation results. Outcomes of the simulation results, indicate the superiority of the proposed method compared to primitive and recently developed methods. The other advantage of the proposed algorithm is the ease of implementation as it is developed based on the conventional and well known algorithm, GA. References [1] Alipour M, Mohammadi-Ivatloo B, Zare K. Stochastic risk-constrained short-term scheduling of industrial cogeneration systems in the presence of demand response programs. Applied Energy 2014;136:393–404. [2] Vasebi A, Fesanghary M, Bathaee S. Combined heat and power economic dispatch by harmony search algorithm. International Journal of Electrical Power & Energy Systems 2007;29(10):713–9. [3] Dong L, Liu H, Riffat S. Development of small-scale and micro-scale biomass-fuelled chp systems–a literature review. Applied thermal engineering 2009;29(11):2119–26. [4] Wang H, Lahdelma R,Wang X, JiaoW, Zhu C, Zou P. Analysis of the location for peak heating in chp based combined district heating systems. Applied Thermal Engineering 2015. [5] Alipour M, Zare K, Mohammadi-Ivatloo B. Short-term scheduling of combined heat and power generation units in the presence of demand response programs. Energy 2014;71:289–301. [6] Rooijers FJ, van Amerongen RA. Static economic dispatch for co-generation systems. Power Systems, IEEE Transactions on 1994;9(3):1392–8. [7] Guo T, Henwood MI, van Ooijen M. An algorithm for combined heat and power economic dispatch. Power Systems, IEEE Transactions on 1996;11(4):1778–84.
Page 15 of 29
[8] Jena C, Basu M, Panigrahi C. Differential evolution with gaussian mutation for combined heat and power economic dispatch. Soft Computing 2014;1–8. [9] Sashirekha A, Pasupuleti J, Moin N, Tan C. Combined heat and power (chp) economic dispatch solved using lagrangian relaxation with surrogate subgradient multiplier updates. International Journal of Electrical Power & Energy Systems 2013;44(1):421–30. [10] Abdolmohammadi HR, Kazemi A. A benders decomposition approach for a combined heat and power economic dispatch. Energy Conversion and Management 2013;71:21–31. [11] Huang SH, Lin PC. A harmony-genetic based heuristic approach toward economic dispatching combined heat and power. International Journal of Electrical Power & Energy Systems 2013;53:482–7. [12] Mohammadi-Ivatloo B, Moradi-Dalvand M, Rabiee A. Combined heat and power economic dispatch problem solution using particle swarm optimization with time varying acceleration coefficients. Electric Power Systems Research 2013;95:9–18. [13] Subbaraj P, Rengaraj R, Salivahanan S. Enhancement of combined heat and power economic dispatch using self adaptive real-coded genetic algorithm. Applied Energy 2009;86(6):915–21. [14] Khorram E, Jaberipour M. Harmony search algorithm for solving combined heat and power economic dispatch problems. Energy Conversion and Management 2011;52(2):1550–4. [15] Jayabarathi T, Yazdani A, Ramesh V, Raghunathan T. Combined heat and power economic dispatch problem using the invasive weed optimization algorithm. Frontiers in Energy 2014;8(1):25–30. [16] Hagh MT, Teimourzadeh S, Alipour M, Aliasghary P. Improved group search optimization method for solving chped in large scale power systems. Energy Conversion and Management 2014;80:446–56. [17] Song Y, Chou C, Stonham T. Combined heat and power economic dispatch by improved ant colony search algorithm. Electric Power Systems Research 1999;52(2):115–21. [18] Wang L, Singh C. Stochastic combined heat and power dispatch based on multi-objective particle swarm optimization. International Journal of Electrical Power & Energy Systems 2008;30(3):226–34. [19] Su CT, Chiang CL. An incorporated algorithm for combined heat and power economic dispatch. Electric Power Systems Research 2004;69(2):187–95.
Page 16 of 29
[20] Basu M. Combined heat and power economic dispatch by using differential evolution. Electric Power Components and Systems 2010;38(8):996–1004. [21] Mohammadi-Ivatloo B, Rabiee A, Soroudi A. Nonconvex dynamic economic power dispatch problems solution using hybrid immune-genetic algorithm. IEEE Systems Journal 2013;7(4):777–85. [22] Abdelaziz A, Kamh M, Mekhamer S, Badr M. A hybrid hnn-qp approach for dynamic economic dispatch problem. Electric Power Systems Research 2008;78(10):1784–8. [23] Victoire TAA, Jeyakumar AE. Reserve constrained dynamic dispatch of units with valve-point effects. Power Systems, IEEE Transactions on 2005;20(3):1273–82. [24] Lee KY, El-Sharkawi MA. Modern heuristic optimization techniques: theory and applications to power systems; vol. 39. John Wiley & Sons; 2008. [25] Haghrah A, Mohammadi-Ivatloo B, Seyedmonir S. Real coded genetic algorithm approach with random transfer vectors-based mutation for short-term hydro–thermal scheduling. IET Generation, Transmission & Distribution 2014;9(1):75–89. [26] Herrera F, Lozano M, Verdegay JL. Tackling real-coded genetic algorithms: Operators and tools for behavioural analysis. Artificial intelligence review 1998;12(4):265–319. [27] He S, Wu QH, Saunders J. Group search optimizer: an optimization algorithm inspired by animal searching behavior. Evolutionary Computation, IEEE Transactions on 2009;13(5):973–90. [28] Yazdani A, Jayabarathi T, Ramesh V, Raghunathan T. Combined heat and power economic dispatch problem using firefly algorithm. Frontiers in Energy 2013;7(2):133–9. [29] Basu M. Bee colony optimization for combined heat and power economic dispatch. Expert Systems with Applications 2011;38(11):13527–31. [30] Wong KP, Algie C. Evolutionary programming approach for combined heat and power dispatch. Electric Power Systems Research 2002;61(3):227–32. [31] Adhvaryyu PK, Chattopadhyay PK, Bhattacharjya A. Application of bio-inspired krill herd algorithm to combined heat and power economic dispatch. In: Innovative Smart Grid Technologies-Asia (ISGT Asia), 2014 IEEE. IEEE; 2014, p. 338–43. [32] Roy PK, Paul C, Sultana S. Oppositional teaching learning based optimization approach for combined heat and power dispatch. International Journal of Electrical Power & Energy Systems 2014;57:392–403.
Page 17 of 29
[33] Jayakumar N, Subramanian S, Ganesan S, Elanchezhian E. Grey wolf optimization for combined heat and power dispatch with cogeneration systems. International Journal of Electrical Power & Energy Systems 2016;74:252–64. [34] Song Y, Xuan Q. Combined heat and power economic dispatch using genetic algorithm based penalty function method. Electric machines and power systems 1998;26(4):363–72. [35] Ramesh V, Jayabarathi T, Shrivastava N, Baska A. A novel selective particle swarm optimization approach for combined heat and power economic dispatch. Electric Power Components and Systems 2009;37(11):1231–40. [36] Rao PN. Combined heat and power economic dispatch: A direct solution. Electric Power Components and Systems 2006;34(9):1043–56. Figure Captions • Fig. 1: The heat-power feasible regions for a combined heat and power unit (CHP unit 3 in case I). • Fig. 2: Illustration of unit fuel cost considering valve-point effects. • Fig. 3: Flowchart of the algorithm. • Fig. 4: The heat-power feasible regions for a combined heat and power unit (CHP unit 2 in case I). • Fig. 5: Convergence characteristics of the proposed algorithm in comparison with conventional GA for test system 2 load profile 1. • Fig. 6: Convergence characteristics of the proposed algorithm in comparison with conventional GA for test system 3. • Fig. 7: Convergence characteristics of the proposed algorithm in comparison with conventional GA for test system 4. Tables Table 1: Algorithm parameters used for test systems. Test
Case
system
Population
Determined
Iteration
Determined
size
mutation
number
top percent
1
2
number 1
2000
4000
1000
0.5
35
8
2
1
2000
4000
1000
0.5
20
8
2
2
2000
4000
1000
0.5
20
8
Page 18 of 29
2
3
2000
5000
1000
0.5
20
8
3
2000
4000
1000
0.5
20
8
4
3000
4000
1000
0.5
20
8
Table 2: Benchmark functions data. Benchmark functions
n
Search space
Global minimum
n
f1 ( x ) =
i =1
f2 (x) =
i =1
f3 ( x) =
(100( x i 1 x i ) ( x i 1)) 2
n
( x i 10 cos (2 x i ) 10) 2
n i =1
j =1
2
xj
f6 (x) =
| x i
1 4000
2
4
i
2
2
i
f 4 ( x ) = 4 x 1 2.1 x 1
f 5 ( x ) = max
2
1 3
x1 x1 x 2 4 x 2 4 x 2 6
2
4
|,1 i n
n i =1
( x i 100)
2
x i 100 cos 1 i =1 i n
30
30, 30
30
5 .1 2 , 5 .1 2
30
100, 100
2
5, 5
30
100, 100
n
0
30
600, 600
n
0
n
0
n
n
0
0
1.0316285
n
Table 3: Comparison of different algorithm mean and standard deviation for benchmark functions.[27] Method GA
f1
f2
f3
f4
f5
338.5616
0.6509
9749.9145
1.0298
7.9610
1.0038
Std.
361.497
0.3594
2594.9593
3 .1 3 1 4 1 0
1.5063
6 .7 5 4 5 1 0
Me
37.3582
20.7863
1 .1 9 7 9 1 0
3
1.0160
0.4123
0.2323
Std.
32.1436
5.9400
2 .1 1 0 9 1 0
3
1 .2 7 8 6 1 0
0.2500
0.4434
Me
49.8359
1.0179
5.7829
1.031628
0.1078
3 .0 7 9 2 1 0
2
Std.
30.1771
0.9509
3.6813
0
3 .9 9 8 1 1 0 3 .0 8 6 7 1 0
2
Me
5.06
4 .6 1 0
1.03
0.3
Me
f6
an
PSO
3
2
an
GSO
2
an
FEP
2
1 .6 1 0
2
2
1 .6 1 0
2
Page 19 of 29
an 1 .4 1 0
2
4 .9 1 0
89.0
5 .0 1 0
2
1.03
13.61
23.1
6 .6 1 0
2
4 .9 1 0
33.28
0.16
1 .4 1 0
3
1.0316
Std.
43.13
0.33
5 .3 1 0
4
6 .0 1 0
Me
6.69
70.82
1 .3 1 0
Std.
14.45
21.49
8 .5 1 0
RCGA-I
Me
5 .9 0 8 7 1 0 1 .2 6 2 2 1 0
8
31
14
MM
an 9
31
13
CEP
Std.
5.87
1 .2 1 0
Me
6.17
Std. Me
2
4
0.5
2 .2 1 0
2
2.0
8 .6 1 0
2
1.2
0.12
an
FES
4
5 .5 1 0
3
3 .7 1 0
2
6 .5 1 0
4
5 .0 1 0
2
an
CES
7
1.0316
4
0.35
0.38
0.42
0.77
an
Std.
1 .4 8 7 0 1 0 7 .6 6 3 9 1 0
5
6 .0 1 0
2 .9 5 6 8 1 0
1 .7 1 3 7 1 0
7
1.03162845
1 .5 6 2 1 1 0
351 .3 6 1 9 1 0 41 .3 3 1 0 1 0 3
15
4
2 .0 6 9 3 1 0 3 .1 2 1 9 1 0
3
Table 4: Cost function parameters of the CHP units of cases I and II. Unit
a
b
c
d
e
f
Feasible region coordinates
P
c
,H
c
Case I 2
0.0345
14.5
2650
0.03
4.2
0.031
[98.8,0], [81,104.8], [215,180], [247,0]
3
0.0435
36
1250
0.027
0.6
0.011
[44,0], [44,15.9], [40,75], [110.2,135.6], [125.8,32.4], [125.8,0]
Case II 2
0.0435
36
1250
0.027
0.6
0.011
[44,0],
Page 20 of 29
[44,15.9], [40,75], [110.2,135.6], [125.8,32.4], [125.8,0] 3
0.1035
34.5
2650
0.025
2.203
0.051
[20,0], [10,40], [45,55], [60,0]
4
0.072
20
1565
0.02
2.34
0.04
[35,0], [35,20], [90,45], [90,25], [105,0]
Table 5: Comparison of simulation results for case I. TPa
THb
TCc
0.37
200.01
115
9452.2
75.03
0
200.05
115
9265.1
39.99
75
0
200
114.99
9257.09
40.01
39.99
75
0
200
114.99
9257.07
160
40
40
75
0
200
115
9257.07
0
200
0
0
115
0
200
115
8606.07
GA_PF[34]
0
159.23
40.77
39.94
75.06
0
200
115
9267.28
SPSO*[35]
0
159.706
39.909
40
75
0
199.616
115
9278.17
5
7
Method
P1
P2
P3
H
IACS[17]
0.08
150.93
49
48.84
65.79
PSO[18]
0.05
159.43
40.57
39.97
IGA[19]
0
160
40
SARGA[13]
0
159.99
HS[2]
0
EDHS*[14]
2
H
3
H
4
2
CPSO[12]
0.00
160.00
40.00
40.00
75.00
0.00
200.00
115.00
9257.08
TVAC-PSO
0
160
40
40
75
0
200
115
9257.07
LR[7]
0
160
40
40
75
0
200
115
9257.07
ACSA[17]
0.08
150.93
49
48.84
65.79
0.37
200.00
115.00
9452.20
DM[36]
0
160
40
40
75
0
200
115
9257.07
LRSS[9]
0
160
40
40
75
0
200
115
9257.07
EP[30]
0.00
160.00
40.00
40.00
75.00
0.00
200.00
115.00
9257.10
FA[28]
0.001
159.998
40.00
40.00
75.00
0.00
200.00
115.00
9257.10
4
6
0.000
159.999
40.00
40.00
75.00
0.00
200.00
115.00
9257.08
[12]
IWO[15]
Page 21 of 29
2
8
BD[10]
0.00
160.00
40.00
40.00
75.00
0.00
200.00
115.00
9257.07
RCGA-IM
0.0000
160.0000
40.0000
40.0000
75.0000
0.0000
200.0000
115.0000
9257.0750
M
* Not feasible. a Total power (MW). b Total heat (MWth). c Total cost ($). Table 6: Comparison of simulation results for case II. Lo
TPa
THb
TCc
38.7
300.0
150.0
13723.
00
000
200
000
2000
39.8
0.00
29.6
299.9
149.9
13779.
400
100
00
400
300
900
5000
133.7
84.0
37.7
0.00
28.1
300.0
149.9
13613.
749
688
626
657
00
118
000
401
0000
40.7
19.2
105.0
64.4
26.4
0.00
59.1
300.0
150.0
13692.
000
309
728
000
003
119
00
955
037
076
5212
TVAC-P
135.0
41.4
18.5
105.0
73.3
37.4
0.00
39.2
300.0
150.0
13672.
SO[12]
000
019
981
000
562
295
00
143
000
000
8892
FA[28]
134.7
40.0
20.2
105.0
75.0
27.8
0.00
47.1
299.9
149.9
13683.
4
0
5
0
0
7
2
9
9
22
134.7
40.0
20.8
104.4
75.0
37.6
37.4
300.0
150.0
13683.
3
0
6
1
0
0
0
65
135.0
40.7
19.2
105.0
73.5
36.7
0.00
39.6
300.0
150.0
13672.
000
687
313
000
957
759
00
284
000
000
83
Method
P1
P2
P3
P4
H
LP HS[2]
134.7
48.2
16.2
100.8
81.0
23.9
6.29
1
400
000
300
500
900
200
135.0
70.8
10.8
83.28
80.5
000
100
400
00
EDHS*[
135.0
18.1
13.0
14]
000
563
CPSO[1
135.0
2]
2
H
3
H
4
H
5
ad
GA[2]
IWO[15]
BD[10]
RCGA-I
0.00
135.0000 40.7680 19.2320 105.0000 7 3.5960 36.7760 0.0000
39.6280 300.0000 150.0000 13660.5322
LP HS[2]
134.6
52.9
10.1
52.23
85.6
39.7
4.18
45.4
250.0
175.0
12284.
2
700
900
100
00
900
300
00
000
000
000
4500
119.2
45.1
15.8
69.89
78.9
22.6
18.4
54.9
250.0
174.9
12327.
200
200
200
00
400
300
000
900
500
600
3700
MM
GA[2]
Page 22 of 29
EDHS*[
135.0
0.11
0.00
114.8
85.8
56.3
0.00
32.8
250.0
174.9
11836.
14]
000
12
00
888
178
198
00
135
000
511
0000
CPSO[1
135.0
40.3
10.0
64.60
70.9
39.9
4.07
60.0
250.0
175.0
12132.
2]
000
446
506
60
318
918
73
000
012
009
8579
TVAC-P
135.0
40.0
10.0
64.94
74.8
39.8
16.1
44.1
250.0
175.0
12117.
SO[12]
000
118
391
91
263
443
867
428
000
000
3895
FA[28]
134.8
40.0
10.0
65.18
75.0
40.0
16.9
43.0
249.9
174.9
12119.
1
0
0
0
0
7
2
9
9
86
134.5
40.0
10.9
75.0
38.9
8.81
52.2
250.0
175.0
12134.
9
0
4
0
8
1
0
0
33
135.0
40.0
10.0
65.00
75.0
40.0
14.4
45.5
250.0
175.0
12116.
000
000
000
00
000
000
029
971
000
000
60
IWO[15]
BD[10]
RCGA-I
64.47
135.0000 40.0000 10.0000 65.0000 75.0000 40.0000 14.0595 45.9405 250.0000 175.0000 12104.8682
MM LP HS[2]
41.41
66.6
10.5
41.39
97.7
40.2
22.8
59.2
160.0
220.0
11810.
3
00
100
900
00
300
300
300
100
000
000
8800
37.98
76.3
10.4
35.03
106.
38.3
15.8
59.9
159.8
220.1
11837.
00
900
100
00
0000
700
400
700
100
800
4000
EDHS*[
135.0
0.00
0.00
25.00
87.2
58.1
40.1
34.3
160.0
219.9
93181.
14]
000
00
00
00
560
586
823
703
000
672
0000
CPSO[1
35.59
57.3
10.0
57.05
89.9
40.0
30.0
60.0
160.0
220.0
11781.
2]
72
554
070
87
767
025
232
000
183
024
3690
TVAC-P
42.14
64.6
10.0
43.22
96.2
40.0
23.7
60.0
160.0
220.0
11758.
SO[12]
33
271
001
95
593
001
404
000
000
000
0625
BD[10]
42.14
64.6
10.0
43.22
96.2
40.0
23.7
60.0
160.0
220.0
11758.
54
296
000
50
614
000
386
000
000
000
06
RCGA-I
42.16
64.6
10.0
43.18
96.2
40.0
23.7
60.0
160.0
220.0
11758.
MM
60
523
000
17
810
000
190
000
000
000
6349
GA[2]
* Not feasible. a Total power (MW). b TOtal heat (MWth). c Total cost ($). Table 7: Cost function parameters of test system III.
Page 23 of 29
1
0.008
2
25
100
0.042
10
75
2
0.003
1.8
60
140
0.04
20
125
3
0.0012
2.1
100
160
0.038
30
175
4
0.001
2
120
180
0.037
40
250
Unit
P
min
P
max
Power only units
a
b
c
d
e
f
Feasible region coordinates
P
c
,H
c
CHP units 5
0.0345
14.5
2650
0.03
4.2
0.031
[98.8,0], [81,104.8], [215,180], [247,0]
6
0.0435
36
1250
0.027
0.6
0.11
[44,0], [44,15.9], [40,75], [110.2,135.6], [125.8,32.4], [125.8,0]
a
b
c
H
0.038
2.0109
950
0
hmin
H
hmax
Heat only units 7
2695.20
Table 8: Comparison of the proposed algorithm with previous methods for case III. Outp
PSO[1
ut
8]
P1
18.462
EP[30]
61.361
DE[20
RCGA[
BCO[2 CPSO[1
TVAC-PSO
]
29]
9]
2]
[12]
44.211
74.6834
43.945
75
47.3383
KH[31]
RCGA-I MM
46.3835
45.6614
Page 24 of 29
6 P2
P3
P4
P5
8
124.26
95.120
98.538
02
5
3
112.77
99.942
112.69
94
7
209.81
7 97.9578
98.588
112.380
98.5398
104.122
8
0
167.230
112.93
30
112.6735
64.3729
112.6735
13
8
2
208.73
209.77
124.907
209.77
250
209.81582
246.185
209.8158
58
19
41
9
19
98.814
98.8
98.821
98.8008
98.8
93.2701
92.3718
98.9736
93.9960
98.5398
3
3
7 44.010
44
44
44.0001
44
40.1585
40.0000
40.7401
40.0000
57.923
18.071
12.537
58.0965
12.097
32.5655
37.8467
0.0000
28.2842
6
3
9
32.760
77.554
78.348
72.6738
74.9999
66.7100
75.0000
3
8
1
59.316
54.373
59.113
1
9
9
Total
608.14
607.95
pow
27
150
P6
7 H
H
H
5
6
7
4 32.4116
78.023 6
59.4919
59.879
44.7606
37.1532
83.2900
46.7158
608.03
607.580
608.03
600.808
600.7392
600.777
600.6865
61
72
8
84
6
150
149.99
150
150
150.000
7
er Total heat Total
99 10613
10390
10317
150
150.000
0 10667
10317
cost
150.0000
0
10325.3
10100.3164
339
10111.1
10094.0552
501
Table 9: Cost function parameters of test system IV.
1
0.00028
8.1
550
300
0.035
0
680
2
0.00056
8.1
309
200
0.042
0
360
3
0.00056
8.1
309
200
0.042
0
360
4
0.00324
7.74
240
150
0.063
60
180
5
0.00324
7.74
240
150
0.063
60
180
Unit
P
min
P
max
Power only units
Page 25 of 29
6
0.00324
7.74
240
150
0.063
60
180
7
0.00324
7.74
240
150
0.063
60
180
8
0.00324
7.74
240
150
0.063
60
180
9
0.00324
7.74
240
150
0.063
60
180
10
0.00284
8.6
126
100
0.084
40
120
11
0.00284
8.6
126
100
0.084
40
120
12
0.00284
8.6
126
100
0.084
55
120
13
0.00284
8.6
126
100
0.084
55
120
a
b
c
d
e
f
Feasible region coordinates
P
c
,H
c
CHP units 14
0.0345
14.5
2650
0.03
4.2
0.031
[98.8,0], [81,104.8], [215,180], [247,0]
15
0.0435
36
1250
0.027
0.6
0.11
[44,0], [44,15.9], [40,75], [110.2,135.6], [125.8,32.4], [125.8,0]
16
0.0345
14.5
2650
0.03
4.2
0.031
[98.8,0], [81,104.8], [215,180], [247,0]
17
0.0435
36
1250
0.027
0.6
0.11
[44,0], [44,15.9], [40,75], [110.2,135.6],
Page 26 of 29
[125.8,32.4], [125.8,0] 18
0.1035
34.5
2650
0.025
2.203
0.051
[20,0], [10,40], [45,55], [60,0]
19
0.072
20
1565
0.02
2.34
0.04
[35,0], [35,20], [90,45], [90,25], [105,0]
a
b
c
H
20
0.038
2.0109
950
0
2695.20
21
0.038
2.0109
950
0
60
22
0.038
2.0109
950
0
60
23
0.052
3.0651
480
0
120
24
0.052
3.0651
480
0
120
hmin
H
hmax
Heat only units
Table 10: Comparison of the proposed algorithm with previous methods for case IV. Output
CPSO[12] TVAC-PSO[1
GSO[16]
2]
IGSO*[16 OTLBO*[3
GWO*[3
RCGA-IM
]
2]
3]
M
P1
680
538.5587
627.7455
628.152
538.5656
538.8440
448.8000
P2
0
224.4608
76.2285
299.4778
299.2123
299.3423
299.9568
P3
0
224.4608
299.5794
154.5535
299.1220
299.3423
299.2108
P4
180
109.8666
159.4386
60.846
109.9920
109.9653
109.8694
P5
180
109.8666
61.2378
103.8538
109.9545
109.9653
109.8679
P6
180
109.8666
60
110.0552
110.4042
109.9653
159.7353
P7
180
109.8666
157.1503
159.0773
109.8045
109.9653
109.8684
P8
180
109.8666
107.2654
109.8258
109.6862
109.9653
60.6545
Page 27 of 29
P9
180
109.8666
110.1816
159.992
109.8992
109.9653
159.7354
P10
50.5304
77.5210
113.9894
41.103
77.3992
77.6223
75.8146
P11
50.5304
77.5210
79.7755
77.7055
77.8364
77.6223
40.1672
P12
55
120
91.1668
94.9768
55.2225
55.0000
92.6079
P13
55
120
115.6511
55.7143
55.0861
55.0000
92.4056
P14
117.4854
88.3514
84.3133
83.9536
81.7524
83.4650
83.0376
P15
45.9281
40.5611
40
40
41.7615
40.0000
40.0071
P16
117.4854
88.3514
81.1796
85.7133
82.2730
82.7732
81.4577
P17
45.9281
40.5611
40
40
40.5599
40.0000
41.6937
P18
10.0013
10.0245
10
10
10.0002
10.0000
10.0042
P19
42.1109
40.4288
35.0970
35
31.4679
31.4568
35.1058
H 14
125.2754
108.9256
106.6588
106.4569
105.2219
106.0991
105.9431
H 15
80.1175
75.4844
74.9980
74.998
76.5205
75.0000
75.0059
H 16
125.2754
108.9256
104.9002
107.4073
105.5142
105.7890
105.0550
H 17
80.1174
75.484
74.9980
74.998
75.4833
75.0000
76.4619
H 18
40.0005
40.0104
40
40
39.9999
40.0000
40.0007
H 19
23.2322
22.4676
19.7385
20
18.3944
18.3782
20.0477
H
20
415.9815
458.7020
469.3368
466.2575
468.9043
469.7337
467.4871
H
21
60
60
60
60
59.9994
60.0000
59.9999
H
22
60
60
60
60
59.9999
60.0000
59.9997
H
23
120
120
119.6511
120
119.9854
120.0000
119.9991
H
24
120
120
119.7176
119.8823
119.9768
120.0000
119.9998
Mean
59853.47
58498.3106
58295.92
58156.51
57883.2105
-
58066.6354
cost ($)
8
43
92
Maximu
60076.69
58318.87
58219.14
57913.7731
-
58301.9013
m cost
03
92
13
58359.552
($)
Page 28 of 29
Minimu
59736.26
m cost
35
58122.7460
58225.74
58049.01
50
97
57856.2676
57846.84
57927.6919
($)
* Not feasible.
Page 29 of 29