INTERNATIONAL TRANSACTIONS ON ELECTRICAL ENERGY SYSTEMS
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
Published online 20 December 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.1854
A hybrid Univariate Marginal Distribution Algorithm for dynamic
economic dispatch of units considering valve-point effects and
ramp rates
Wei Gu, Yonggang Wu*,† and GuoYong Zhang
School of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan, China
SUMMARY
This paper presents a new approach for dynamic economic dispatch (DED) problem in power system by using a
hybrid Univariate Marginal Distribution Algorithm (HUMDA). The DED problem with valve-point effects and
ramp rate limits is a nonliner constrained optimization problem with non-convex and non-smooth characteristics. In the proposed method, a two-stage adaptive mechanism is devised to control parameters of the Univariate
Marginal Distribution Algorithm in continuous domains (UMDAc) dynamically and lead the algorithm with
better search efficiency; a chaotic local search operator is integrated with UMDAc to effectively avoid premature
convergence. Moreover, a constraint handle according to the two-stage adaptive mechanism is proposed, and the
results show that the strategy can handle constraints effectively. Finally, the efficiency of the proposed method is
validated on two test systems consisting of 5, 10 and 30 thermal units. The results show the superiority of the
proposed method while it is compared with other works in the area. Copyright © 2013 John Wiley & Sons, Ltd.
key words:
dynamic economic dispatch; univariate marginal distribution algorithm; two-stage adaptive
mechanism; chaotic local search operator; constraint handle
1. INTRODUCTION
Dynamic economic dispatch (DED) is a real-time problem in power system operation, which is used to assign the combination of load dispatch of all the units to minimize the total fuel cost while satisfying equality
constraints, inequality constraints and dynamic constraints. As an extension of the static economic dispatch
(SED) problem, DED takes the ramp rate limit as the dynamic constraint which keeps the gradients of the
units within a safe area. Mathematically, the DED problem with the effect of valve-points is a dynamic
non-convex and nonlinear optimization problem, which makes a challenge to find the optimal result.
In the past decades, a lot of optimization methods have been applied to solve DED problem. These optimization methods can be classified into three main categories such as mathematical programming-based
methods, artificial intelligence methods and hybrid methods [1]. The traditional mathematical methods include linear programming (LP) [2], nonlinear programming (NLP) [3], quadratic programming (QP) [4],
dynamic programming (DP) [5,6] and Lagrange relaxation (LR) [7]. However, while applying these mentioned tradition methods into DED problem with a valve-point effects, these methods can hardly achieve
the global optimal solution due to their drawbacks. Large errors would generate during the process of linearization the DED model when applies LP to solve DED problem. For QP and NLP, the objective function
needs to be transformed for the reason that the objective function must be continuous and differentiable,
which would lead inaccuracy to the final solution. Although DP can solve the DED problem without imposing any restrictions, it suffers from the ‘curse of dimensionality’: it may not converge in a possible time
when it is applied in large-scale power systems. When using the Lagrange relaxation method to solve DED
with non-smooth or non-convex cost functions, that can fail to find global optimal solutions [8].
*Correspondence to: Wu, Yonggang, School of Hydropower and Information Engineering, Huazhong University of Science
and Technology, 430074 Wuhan, China.
†
E-mail: [email protected]
Copyright © 2013 John Wiley & Sons, Ltd.
HUMDA FOR DED PROBLEM CONSIDERING VALVE-POINT EFFECTS AND RAMP RATES
375
In addition to the above methods, many modern artificial intelligence method and hybrid methods also
have been applied in DED such as genetic algorithm (GA) [9], particle swarm optimization (PSO) and its
variants [10–14], differential evolution (DE) algorithm and its improvement [15–21], artificial bee colony
algorithm (ABC)[22], simulated annealing (SA) algorithm [23], covariance matrix adapted evolution
strategy algorithm (CMAES)[24] and chaotic search strategy [25,26]. Compared with traditional
algorithms, these algorithms are more suitable for solving constrained nonlinear optimization problems.
However, there are still some drawbacks in these methods. For SA, it is difficult to control parameters,
and the adjustment of parameters is very complicated when it is used to solve DED problems. For GA,
PSO, DE, ABC and CMAES, the premature convergence tends to trap the algorithm into local optimum,
which may reduce their searching ability remarkably, especially when the scale of the problem is large.
The chaotic search strategy, which could lead the algorithm to jump out form local optimum, is commonly
used to combination with other algorithm to improve the performance of the algorithm.
In this respect, hybrid methods such as hybrid evolutionary programming (EP) and SQP method
[27], hybrid PSO and SQP [8], artificial immune systems [28], hybrid artificial immune systems and
sequential quadratic programming (AIS-SQP) [29], hybrid swarm intelligence based harmony search
algorithm (HHS) [30], hybrid approach of Hopfield neural network (HNN) and quadratic programming (QP) [31] combine stochastic heuristics method and deterministic method to try to obtain the
global optimal solution. And the application results in DED problem demonstrate the feasibility of improvement for current optimization algorithm.
In recent years, a new optimization method known as estimation distribution algorithms (EDAs) has
gradually become popular and been applied to solve optimization problems. The concept of EDAs can
be traced back to an early learning algorithm proposed by Ackley in 1987. In this learning algorithm, a
data structure of great significance known as a gene vector has been introduced. EDAs are different
from other evolutionary algorithms. It is modeling from the macro-level of biological evolution. Besides, it can describe the relationship between variables through probability model, and thus can solve
nonlinear and variable coupling optimization problems more effectively [32]. EDA has already applied
in areas like nuclear reactor fuel management optimization [33], protein folding problem [34], software
testing [35] and so on. Some scholars divide EDAs into three kinds of model: No interdependencies,
Pairwise dependencies and Multivariate dependencies [36]. However, when the scale of the problem
becomes very large, the selection of parameters of EDA algorithm will directly influence its global
search capability. At the same time, it also lacks appropriate mechanisms to deal with complex
constraint problems, such as DED problem. To improve the performance of UMDA which has a simple model and easy to implement, this paper presents a hybrid univariate marginal distribution algorithm named as HUMDA and utilize it to solve DED problem. In the proposed method, a two-stage
dynamic parameter control strategy is used to control the mean and variance parameters in order to
preserve the diversity of the population at the beginning of algorithm and improve the local search
capability of the algorithm at the end of the execution. In addition, the chaotic search strategy is adopted
to enhance the precision of solution and search efficiency. To verify the feasibility and effectiveness of
HUMDA, it was used to solve an optimization problem for testing system. Simulation results show that
the proposed algorithm is more effective and robust compared to other existing algorithms.
This paper is organized as follows. Section 2 describes DED problem through modeling. Section 3
briefly introduces the process and mechanism of the HUMDA in details. The HUMDA for DED problem is shown in section 4. Section 5 presents the test system, and section 6 draws the conclusions.
2. PROBLEM FORMULATIONS
2.1. Non-smooth cost function with valve-point effects
The object of the DED problem is to minimize the total fuel cost over the dispatch periods while satisfying
various constraints. The classic objective function for generating units with T intervals is formulated as follows:
Min
T N
F ¼ ∑ ∑ f ti Pti
(1)
t¼1 i¼1
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
376
W. GU ET AL.
Traditionally, the cost of each generating unit with smooth function can be expressed by
2
f ti Pti ¼ ai þ bi Pti þ ci Pti
(2)
In reality, for more practical and accurate modeling of the problem, the fuel cost function expressed
in Equation (1) needs to be modified because of valve-point effects, which are referred to as the sharp
increase in fuel loss added to the fuel cost curve due to the wire drawing effects when the steam
admission valve starts to open [19]. The valve-point effects can be modeled by adding a sinusoidal
function to the fuel cost function. Hence, the fuel cost function of DED problem with consideration
of valve-point effects can be described as follows:
2 (3)
f ti Pti ¼ ai þ bi Pti þ ci Pti þ ei sin hi Pi; min Pti 2.2. Constraints
The equality and inequality constraints can be described as follows:
2.2.1. Real power balance.
N
∑ Pti ¼ PtD þ PtL
t ¼ 1; 2; …; T
(4)
i¼1
Where PtL is calculated by using the B loss coefficients matrix which can be expressed in the quadratic
form as follows:
N
N
PtL ¼ ∑ ∑ Pti Bij Ptj
(5)
i¼1 j¼1
2.2.2. Real power output limits.
Pi; min ≤Pti ≤Pi; max
i ¼ 1; 2; …; N; t ¼ 1; 2; …; T
2.2.3. Generating unit ramp rate limits:.
( t
Pi Pt1
i ≤URi
(6)
if output increases
Pt1
Pti ≤DRi if output decreases
i
i ¼ 1; 2; …; N; t ¼ 2; 3; ::::; T;
(7)
The notation is summarized as follows:
NOTATION
To formulate the DED problem mathematically, the following notation used in this paper is introduced:
t
period index
T
number of periods
i
generating unit index
N
number of generating units
F
total fuel cost in $
fuel cost of the ith generating unit at interval t in $
fit(Pti)
power output of ith generating unit at interval t in MW
Pti
ai,bi,ci cost coefficients of the ith generating unit
value-point effects coefficients of the ith generating unit
ei,hi
minimum output limit of the ith generating unit in MW
Pi,min
Pi,max
maximum output limit of the ith generating unit in MW
power demand during the tth period in MW
P Dt
power loss during the tth period in MW
PLt
Bij
loss coefficient between the ith unit and jth unit in MW1
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
HUMDA FOR DED PROBLEM CONSIDERING VALVE-POINT EFFECTS AND RAMP RATES
DRi
URi
377
minimum output ramp up rate limit of the ith generating unit in MW/h
maximum output ramp up rate limit of the ith generating unit in MW/h
3. HYBRID UNIVARIATE MARGINAL DISTRIBUTION ALGORITHM
The UMDA is a particular case of EDA where probability distribution pl(xi) is estimated from the
relative marginal frequencies of the i-th variable of the selected individuals in front generation. And
Univariate Marginal Distribution Algorithm in continuous domains (UMDAc) is a kind of EDA in
continuous domains [36]. In UMDAc, the mean value affects the location of the key area of the search,
and the variance affects the search scope. Changes in the search scope can directly affect the convergence of the algorithm and accuracy. In UMDAc, the search scope comes from the selected population; thus, the composition of the population directly impacts the search scope of the next
generation. This kind of mechanism is more likely to make the algorithm become premature.
The formulation of the example of 30-dimensional Rosenbrock function is as follows:
30 2
f 1 ¼ ∑ 100 xiþ1 x2i þ ðxi 1Þ2 ; 30≤ xi ≤30
i¼1
When the population size is 100, the max evolution generation is set to 1000, the theoretical optimal
value is zero and the truncation selection parameter is set to 50%; the convergence of variables and
fitness values changes in evolutionary process shown in Figure 1. The figure shows the convergence
process of the σgj parameter in UMDAc which has fallen to 104 after evolution of 200 generations.
The optimal solution with small search range is difficult to be improved. The best solution’s fitness
is 128.4399. Thus, UMDAc is easily trapped into local optimum when solving complex problems.
To avoid the premature problem existing in UMDAc, a two-stage adaptive mechanism is proposed
to control the parameters of the algorithm. This makes the population obtain stronger search ability
near the optimal solution during the evolutionary process, which is conductive to reserve effective
genes of optimum individuals and to improve search performance. In addition, the HUMDA uses chaotic local search mechanism to make local adjustment on individuals obtained from the global search,
which will help to improve the structure of the population and to prevent the premature problem.
3.1. Two-stage adaptive mechanism
The two-stage adaptive mechanism includes basic search stage and dynamic adaptive search stage. The
parameters calculating method which is related to individuals selected by truncation selection are the
same as UMDAc. However, dynamic adaptive search stage uses a combination of adaptive parameter
control strategy and dynamic search range adjustment strategy to control variance parameter. The
details will be described as follows.
Figure 1. Convergence of variables and fitness values for Rosenbrock function.
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
378
W. GU ET AL.
a. Basic search stage (BSS)
The most important mechanism in the UMDAc is to calculate the variance based on information of
outstanding individuals. The mechanism makes the information of outstanding individuals in the evolutionary process to be retained and to be fully utilized. This kind of mechanism is more likely to make
the algorithm become premature. If the algorithm continues to use the basic search stage to set parameters, that will increase the difficulty to the algorithm to jump out of local optimal solution. Therefore,
a dynamic adaptive search stage is presented to guide the algorithm to jump out of local optimal solution effectively.
b. Dynamic adaptive search stage (DAS)
In dynamic adaptive search stage, the mean parameter is set to the optimal solution. With the variance
of the parameters set, this paper uses a dynamic parameter control strategy that combines an adaptive
control strategy and dynamic search range adjustment strategy. The adaptive parameter control strategy provides an orderly and gradual parameter control strategy which decreases the algorithm search
range during the evolutionary progress. The dynamic search range adjustment strategy improves the
algorithm search efficiency and accuracy of convergence.
(i) Adaptive parameter control strategy (APC)
As mentioned in section 3, σj parameter has a wider search scope when a large value is chosen. On the
contrary, strong local search ability accelerates the convergence speed when σj takes a small value. As
mentioned in the literatures [11,12], a good search strategy we often expect is that the search at the
early evolutionary stage tends to converge to the optimal solution while keeping the search scope being
broad enough. In the latter part of the algorithm, the algorithm is expected to converge soon for
guaranteeing the search near the optimal solution. Based on the above idea, in this paper, we adopt
the adaptive mediation mechanism to adjust the variance of the constructed Gaussian distribution during the evolutionary process so that a wide range of search scope can be obtained at the early evolutionary stage while convergence is guaranteed in the latter part of the algorithm. Thus, to some
extent, we alleviate the premature convergence problem of UMDAc. According to the analysis above,
the proposed method adjusts the σj parameter adaptively and makes it related to the current generation
Gen and the maximize generation Gmax as follows in Equation (8):
σj ¼ σ0j eαGen=G max
σj0
(8)
In Equation (8),
is an initial search range, the value of which is calculated by σj = (uj lj). uj is the
upper limit and lj is the lower limit of the j variable constraint, α is an influence range parameter which
is used to control the rate of change in the process. The adaptive parameter control strategy can adjust
σj adaptively according to the rule which balance between global and local searches by searching globally at the beginning of the evolution while searching locally at the end of execution.
0
(ii) Dynamic search range adjustment strategy (DSRA)
When the search points are near the global optimal solution, getting the global optimal solution is also
in low probability because of the population’s size limit and the complexity of the objective function.
Therefore, a dynamic search range adjustment strategy is proposed to further control search range of
the algorithm. The strategy considers the information in the evolution process and improves the accuracy and efficiency of the algorithm. If the optimal solution is not improved in FSet generations, we
use the dynamic search range adjustment strategy to reduce search range by an order of magnitude because it is difficult to get a better solution in the current search range. If the optimal solution is not improved for a long time, it is considered that the current search range is not suitable. Therefore, the
search range with dynamic adjustment can lead the algorithm to obtain a better solution with higher
accuracy. The strategy combining with the chaotic local search operator will be further introduced
in section 3.2.
3.2. Chaotic local search operator
Recently, some successful applications of evolutionary algorithm combined with chaotic sequences in
optimization problems have been reported in literatures [21,25,26,37]. Chaotic local search strategy
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
HUMDA FOR DED PROBLEM CONSIDERING VALVE-POINT EFFECTS AND RAMP RATES
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increases the algorithm exploitation capability in the search space and enhances its convergence.
Because chaotic local search strategy has characteristics like ergodicity, randomness and irregularity,
it can effectively prevent the algorithm from falling into the local optimum.
In this paper, a chaotic local search operator is adopted to enhance the precision of solution and
search efficiency. Using a simplest dynamic system to evidence chaotic behavior is the iterator named
the logistic map, which can be described as follows in Equation (9):
(9)
¼ 4cxki 1 cxki ; i ¼ 1; 2…; D
cxkþ1
i
In the above equation, k represents the iteration number, cxki is the ith chaotic variable. cxki is distributed between [0,1] under the condition that the initial cx0i ∈(0,1) and cx0i ∉f0:25; 0:5; 0:75g.
The chaotic search steps are as follows:
Step: 1 set EDSetG to be the best individual in the current evaluation set, and f g is the fitness value
of the corresponding individual.
Step: 2 initialize the chaos vector CX0 = (cx00, cx01,….,cx0D) through randomization.
Step: 3 produce the next iteration variable according to the logistic equation (9).
Step: 4 Use formula (10) for chaotic local search
XCk = (xck0, xck1,…,xckD) is generated by combining EDSetG = (xbest k0, xbest k2,…xbest kD) and CXk =
(cx k0, cx k1,…, cx kD). Linearly, given as follows:
xcki ¼ EDSetG ± DFσi cxki ; i ¼ 1; 2…; D
(10)
In the above equation, XCk is the value of the individual after kth chaos local search, σj is the parameter
which was mentioned in the adaptive parameter control strategy to control the search range. DF is the
dynamic search range adjustment parameter which is used to further improve the accuracy and efficiency
of the chaotic local search, which is described as follows in Equation (11):
DF
F < FSet
DF ¼
(11)
0:1DF
F > Fset
In Equation (11), F is the optimal solution which has not been improved in F generations. When F is
greater than Fset, the DF parameter is reduced to one tenth of the original which is an order of magnitude. After the searching range is dynamically reduced, the optimal solution which is not improved in
the previous stages will obtain a better solution with higher possibility.
The range of local search within a feasible solution should try to satisfy the constraints limit during
the local search. When the local search is executing, two variables are adjusted at the same time but in
the opposite directions. This strategy maintains the local search within the current equality constraints.
Step: 5 calculate the fitness value of XCk. If XCk has better fitness value than that of EDSetG, then it
will be taken as EDSetG, and the corresponding fitness value is taken as f g.
Step: 6 if k reaches to kmax, the chaotic local search ends; otherwise, k = k + 1, and go to Step 3
3.3. HUMDA steps
The general steps of HUMDA are shown as below. In the UMDAc, the Gaussian function is
constructed as the generating function of the next-generation population through evaluation of the selected individuals. The mean and variance of the selected population are calculated and set as the mean
parameter and variance parameter of the Gaussian function, respectively. In HUMDA, we use the characteristics of Gaussian distribution and the two-stage adaptive mechanism to make the algorithm
quickly converge the global optimal solution. Besides, the use of chaotic local search strategy and
adaptive mechanism can prevent premature effectively.
In order to ensure that in the early stages of the algorithm, the information of excellent individuals
can be fully utilized. The basic search strategy is used to set the parameters before T generations, and
the dynamic adaptive search strategy is used after T generations. The steps are as follows:
Step: 1 Initialize population PopSet, set Gen = 0;
Step: 2 Calculate the fitness of individuals, use truncation selection to choose best M individuals;
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
380
W. GU ET AL.
Step: 3 Calculate the mean and variance parameters of Gaussian distribution. If the generation
number is larger than T, use dynamic adaptive search stage to calculate parameters; otherwise, use basic search stage.
Step: 4 Use the chaotic local search operator in the dynamic adaptive search stage. If the value of F is
larger than Fset, the dynamic search range adjustment strategy is used to reset the search range.
Step: 5 Use the Gaussian distribution function to generate the next generation population.
Step: 6 Judge whether the algorithm reaches the specified generation, if yes, then the algorithm
ends; otherwise, Gen = Gen + 1, go to step 2.
4. IMPLEMENTATION OF HUMDA FOR DED PROBLEM
In this section, the implementation procedure of the HUMDA to solve DED problem with value-point
effects is described in detail.
Step 1. Initialize the population. The initial solution of nonlinear problems is often generated
randomly. The initial population is generated randomly within inequality constraints, making the individual generated be a feasible solution. A solution is initialized as following
Equation (12):
Pj;t ¼ Pj; min þ Pj; max Pj; min rand ð0; 1Þ
(12)
j ¼ 1; 2; …; N; t ¼ 1; 2; …; T
In Equation (12), rand (0,1) is a random generated number between 0 and 1, which obeys uniform
distribution. Pj,min, Pi,max are the minimum and maximum output limit of the ith generating unit.
Step 2. Calculate the fitness of individuals.
If the individual is a feasible solution, the fitness value is calculated by Equation (3) directly. Otherwise, in one case when the individual violates the inequality constraints, the variable which violates
the constraint is assigned to the inequality of boundary value. A heuristic strategy was proposed for
solving inequality and equality constraints in literature [37]. Equation (13) is utilized to calculate the
feasible horizon of each generating unit at t interval, and Equation (14) is utilized to adjust the outputs
of each generating unit at t interval to the feasible horizon.
8
8
Pj; min if t ¼ 1
>
<
>
>
> Pt
¼
>
j; min
>
>
: max Pt1 DRj ; Pj; min
<
others
j
8
j ¼ 1; 2; …; N; t ¼ 1; 2; ::; T
(13)
>
Pj; max if t ¼ 1
<
>
>
>
>
> Ptj; max ¼
>
: min Pt1 þ URj ; Pj; max
:
others
j
8 t
Pj;t < Ptj; min
P
if
>
< j; min
Pj;t ¼ Pj;t if Ptj; min < Pj;t < Ptj; max
(14)
>
: t
t
Pj;t > Pj; max
Pj; max if
In the other case when the individual violates the equality constraints, a penalty function method is
usually mentioned to punish the individual so that the infeasible solution is more likely to be removed
during the evolutionary process. The penalty function method follows several key principles [18,19].
(1) The fitness value of the individuals of the feasible solution is better than that of the non-feasible
solution
(2) Between two feasible solutions, the one having the better objective function value is preferred
(3) Between two infeasible solutions, the one having the smaller constraint violation is preferred.
During the basic search stage, only handle inequality constraints with heuristic strategy are used to
reduce the effect of inequality constraints and handle equality constraints with penalty function. When
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
HUMDA FOR DED PROBLEM CONSIDERING VALVE-POINT EFFECTS AND RAMP RATES
381
the algorithm turns into the dynamic adaptive search stage, the heuristic strategy is used to handle both
inequality and equality constraints. While handling equality with heuristic strategy, feasible solutions
are easy to obtain. But it is hard to escape from these feasible solutions because it is easy to violate the
equality constraints when searching for a better solution. Therefore, the heuristic strategy is provided
to handle equality constraints only in dynamic adaptive search stage to keep the diversity of the
population during the early part of the algorithm.
Step 3. Calculate the mean and variance parameters of Gaussian distribution. If the generation
number is larger than T, use dynamic adaptive search stage to calculate parameters; otherwise, use basic search stage.
Step 4. Use the chaotic local search operator during the dynamic adaptive search stage and adopt a
local search for the selected area. If the value of F is larger than Fset, use dynamic search
range adjustment strategy to reset the search range.
Step 5. Use the Gaussian distribution function to generate the next generation population.
Step 6. Judge whether the algorithm reaches the specified generation. If so, the algorithm ends;
otherwise, Gen = Gen + 1, go to step 2.
The flow chart of HUMDA is described in Figure 2.
Figure 2. HUMDA flow for DED problem.
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
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W. GU ET AL.
5. NUMERICAL SIMULATIONS
In this section, the proposed HUMDA algorithm is applied on two benchmark DED test systems with
different number of generating units, i.e. 5 and 10 units. The proposed HUMDA were implemented by
using C++ builder 6.0 on an AMD3000+ and 1 GB of RAM personal computer.
The selection of proper parameters for algorithm plays a significant role in both quality of solution
and the convergence speed of the algorithm. Considering the minimum cost for different set of population sizes, the optimal value for the population size is set to 200 for all test cases. Besides, the proper
maximum iteration number of different test system, which is carried out by numerous studies, is mentioned in corresponding test cases. The parameters in basic search stage and dynamic adaptive search
stage which is reported above are set as follows: the truncation selection parameter is set to 50%, α
parameter is set to 8.0, parameter T is set to 100, parameter F is set to 150 and parameter k in chaotic
local search is set to 20.
5.1. Test system 1
The first test system is a 5-unit system with one case studied: in this system, generator capacity limits,
ramp-rate constraints, cost coefficients, load demand and transmission losses which are taken from
[38]. The details are given in Tables I and II. In addition, the B matrix of the transmission loss coefficients is given in the Appendix A. The proper maximum iteration number is set to 500.
Table I. Units’ characteristics in 5-unit test system.
Unit
1
2
3
4
5
ai
bi
ci
ei
hi
Pi,min
Pi,max
URi
DRi
0.008
0.003
0.0012
0.001
0.0015
2
1.8
2.1
2
1.8
25
60
100
120
40
100
140
160
180
200
0.042
0.04
0.038
0.037
0.035
10
20
30
40
50
75
125
175
250
300
30
30
40
50
50
30
30
40
50
50
Table II. Load demand for 24 h in 5-unit test system.
Hour
Demand (MW)
Hour
Demand (MW)
1
2
3
4
5
6
7
8
9
10
11
12
410
13
704
435
14
690
475
15
654
530
16
580
558
17
558
608
18
608
626
19
654
654
20
704
690
21
680
704
22
605
720
23
527
740
24
463
Table III. Comparison of optimization results with different methods for Case 1.
Method
SA[23]
AIS[28]
GA[22]
PSO[22]
ABC[22]
MLS[38]
IPS[39]
APSO[11]
TVAC-IPSO[40]
CMAES[24]
Original UMDAc
Proposed
HUMDA
Minimum cost ($)
Average cost ($)
Maximum cost ($)
Simulation time
(min)
47 356
44 385.43
44 862.42
44 253.24
44 045.83
49 216.81
46 530
44 678
43 136.561
43 526
51 717.973
43 128.920
NA
44 758.8363
44 921.76
45 657.06
44 064.73
NA
NA
NA
43 185.664
43 915
51 889.698
43 437.73
NA
45 553.7707
45 893.95
46 402.52
44 218.64
NA
NA
NA
43 302.233
44 191
51 995.876
43 750.823
5.86
4
3.3242
3.5506
3.2901
0.024
4.53
NA
1.1
0.27
0.26
1.28
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
HUMDA FOR DED PROBLEM CONSIDERING VALVE-POINT EFFECTS AND RAMP RATES
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5.1.1. Case 1. The DED problem of test system 1 is solved by using the proposed algorithm. Table I
shows the obtained results. We compared it with other algorithms such as simulated annealing (SA)
algorithm [23], particle swarm optimization (PSO) algorithm [22], artificial bee colony (ABC)
algorithm [22], genetic algorithm (GA) [22], adaptive PSO (APSO) algorithm [11], artificial immune
system (AIS) [28], Maclaurin series (MLS) [38], improved pattern search algorithm (IPS) [39], covariance matrix adapted evolution strategy algorithm (CMAES) [24], time varying acceleration coefficients iteration particle swarm optimization (TVAC-IPSO) method [40] and continuous univariate
marginal distribution algorithm (UMDAc) in Table III. Results of the proposed algorithm are in bold.
Table IV shows that HUMDA gives the best solutions in terms of minimum from 20 trial runs, which
Figure 3. Convergence process of the minimum cost for Case 1.
Table IV. Optimal solution of 5-unit using HUMDA considering transmission losses.
Hour
P1
P2
P3
P4
P5
Loss(MW)
Cost ($)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Total
20.60791
10.00007
10.65103
10.09596
10.00005
10.00003
10.00002
13.73614
43.73612
64.01069
74.99998
74.99996
64.01072
49.61948
31.68924
10
10.00001
10.00007
21.31258
51.31254
39.3529
10.00002
10
10.00007
98.53989
98.53984
98.53982
98.53981
87.58161
98.53979
72.45058
97.50712
99.13059
98.53984
99.9352
124.7112
98.53983
98.53984
98.53983
97.04813
87.58156
98.53983
96.99118
99.4745
98.53984
98.53982
97.39956
72.22412
30
65.92281
105.9228
112.6735
112.6734
112.6735
112.6735
112.6721
117.9524
112.6735
116.743
112.6735
112.6735
112.6735
112.6735
109.9495
112.6735
112.6735
105.6181
124.3969
112.6735
111.3367
71.33674
31.33674
124.9079
124.9079
124.9079
124.9079
124.9079
165.218
209.8158
209.8158
209.8158
209.8158
209.8158
209.8158
209.8158
209.8158
190.7221
140.7221
124.9079
165.2179
209.8157
209.8158
209.8158
163.4814
124.9079
124.9079
139.7598
139.7598
139.7598
189.7598
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
229.5196
3.815523
4.130417
4.781323
5.976974
6.682536
7.950862
8.459525
9.250806
10.15452
10.55946
11.01358
11.71998
10.55946
10.16827
9.14422
7.239319
6.682536
7.950862
9.257164
10.51937
9.9016
7.877562
6.163731
4.988364
195.3259
1249.575
1418.866
1399.227
1682.557
1615.299
1853.472
1840.604
1807.088
2009.902
1996.595
2039.629
2180.023
1996.595
1977.661
1980.592
1731.838
1615.299
1853.472
1881.855
2085.87
1944.597
1853.099
1649.448
1465.756
43 128.920
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
384
W. GU ET AL.
shows its ability in finding better solutions for DED problems. In order to test the convergence of proposed algorithm, a comparison between proposed HUMDA and original UMDAc which is mentioned
in [36], is examined and the result is shown in Figure 3 (The solid line represents HUMDA method and
the dotted line represents UMDAs method). This figure implies that the proposed HUMDA outperforms the original UMDAc in the optimality of the objective function. Minimum costs of using
HUMDA and UMDAc are $43 128.920 and $44 211.563, respectively.
Table V. Units’ characteristics in 10-unit test system.
Unit
1
2
3
4
5
6
7
8
9
10
ai
bi
ci
ei
hi
Pi,min
Pi,max
URi
DRi
0.00043
0.00063
0.00039
0.0007
0.00079
0.00056
0.00211
0.0048
0.10908
0.00951
21.6
21.05
20.81
23.9
21.62
17.87
16.51
23.23
19.58
22.54
958.2
1313.6
604.97
471.6
480.29
601.75
502.7
639.4
455.6
692.4
450
600
320
260
280
310
300
340
270
390
0.041
0.036
0.028
0.052
0.063
0.048
0.086
0.082
0.098
0.094
150
135
73
60
73
57
20
47
20
55
470
460
340
300
243
160
130
120
80
55
80
80
80
50
50
50
30
30
30
30
80
80
80
50
50
50
30
30
30
30
Table VI. Load demand for 24 h in 10-unit test system.
Hour
Demand
(MW)
Hour
Demand
(MW)
1
2
3
4
5
6
7
8
9
10
11
12
1036
1110
1258
1406
1480
1628
1702
1776
1924
2072
2146
2220
13
2072
14
1924
15
1776
16
1554
17
1480
18
1628
19
1776
20
2072
21
1924
22
1628
23
1332
24
1184
Table VII. Comparison of optimization results with different methods for Case 2.
Method
DE[16]
HDE[19]
IDE[15]
MDE[18]
CS-DE[20]
CDE[21]
EP-SQP[27]
MHEP-SQP[27]
PSO-SQP[7]
ABC[22]
CMAES[24]
AIS[28]
HHS[30]
AIS-SQP[29]
DGPSO[10]
IPSO[12]
ICPSO[37]
HQPSO[14]
CDE[25]
TVAC-IPSO[40]
Original UMDAc
Proposed
HUMDA
Minimum cost ($)
Average cost ($)
Maximum cost ($)
Simulation time (min)
1 019 786
1 031 077
1 026 269
1 031 612
1 023 432
1 019 123
1 031 746
1 028 924
1 027 334
1 021 576
1 023 740
1 021 980
1 019 091
1 029 900
1 028 835
1 023 807
1 019 072
1 031 559
1 019 123
1 018 217.224
1 033 242.374
1 018 894.022
NA
NA
NA
1 033 630
1 026 475
1 020 870
1 035 748
1 031 179
1 028 546
1 022 686
1 026 307
1 023 156
NA
NA
1 030 183
1 026 863
1 020 027
1 033 837
1 020 870
1 018 965.355
1 037 811.247
1 020 074.678
NA
NA
NA
NA
1 027 634
1 023 115
NA
NA
1 033 986
1 024 316
1 032 939
1 024 973
NA
NA
NA
NA
NA
1 036 681
1 023 115
1 020 417.821
1 039 242.374
1 020 952.182
11.25
NA
NA
12.50
0.24
0.32
20.51
21.23
16.37
2.6029
0.63
19.01
12.233
NA
15.39
0.06
0.467
NA
0.32
2.8
0.85
1.65
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
P1
150
226.6285
303.234
379.8663
456.4986
456.4998
456.478
456.497
456.4909
456.5215
456.4973
456.5141
456.4973
456.4878
379.8551
303.2749
226.6039
303.282
379.8701
456.4906
379.8749
303.2475
226.6244
226.6121
Hour
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Total
135
135
215
222.267
222.2821
302.2821
309.5043
316.797
396.7931
460
459.3332
460
396.8057
396.8117
396.7951
316.8182
309.5237
309.5658
389.5658
460
396.7848
316.7848
236.7848
222.2679
P2
193.9981
191.4065
182.8963
196.9766
194.3964
262.4462
279.4133
297.958
297.3922
302.1611
297.027
325.0154
314.8879
296.8669
283.742
318.1531
289.5079
308.9411
300.4427
312.5586
297.3883
292.0238
212.8956
179.344
P3
60.00486
60.01206
60
60
60.0008
60
60.0021
110.0021
130.8181
180.818
230.6416
241.2782
234.055
184.055
180.3886
130.3886
119.1762
120.4312
120.4343
170.4343
180.8335
130.8335
120.4504
70.45042
P4
122.9087
122.8766
122.8306
172.7701
172.7306
172.7244
222.5668
183.1489
222.6
222.5975
222.6
222.5984
222.6115
172.7308
122.8571
73.00509
122.8574
172.7728
172.74
222.6037
222.604
172.7416
122.8817
73
P5
122.4905
122.4885
122.4526
122.5162
122.4933
122.432
122.4442
160
160
160
160
159.9997
122.5382
122.453
122.4686
122.4509
122.4443
123.0079
123.0202
160
156.5827
122.4502
122.4554
122.4244
P6
129.5945
129.5882
129.5834
129.6042
129.5949
129.6147
129.5922
129.5971
129.5902
129.5898
129.5903
129.5928
129.606
129.5942
129.5879
129.5955
129.5859
129.682
129.5892
129.5997
129.6175
129.6088
129.5816
129.5852
P7
47
47
47.0009
47
47
47
47
47
55.3121
85.3121
115.3211
120
120
90
85.3072
85.3153
85.2926
85.3174
85.3351
85.3098
85.3107
85.3115
85.3244
85.3169
P8
Table VIII. Optimal solution of 10-unit using HUMDA without transmission losses.
20
20
20
20
20
20
20
20
20
20
20
49.998
20
20
20.0008
20
20.0077
20
20
20
20
20
20
20
P9
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
P10
28 239.07
29 828.49
33 142.49
36 291.95
37 896.53
41 373.1
42 711.81
44 882.68
48 037.45
51 578.08
53 705.85
55 513.23
51 479.11
47 896.38
44 596.03
39 967.15
37 976.85
41 219.76
44 511.98
51 768.66
47 877.47
41 479.03
35 280.07
31 640.8
1 018 894.02
Cost ($)
HUMDA FOR DED PROBLEM CONSIDERING VALVE-POINT EFFECTS AND RAMP RATES
Copyright © 2013 John Wiley & Sons, Ltd.
385
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
386
W. GU ET AL.
5.2. Test system 2
The second test system is a 10-unit system with three cases being studied. In the first case (Case 2),
generator capacity is limited, and ramp-rate constraint and value-point effects are considered. In the
second case (Case 3), transmission losses are also considered, and in the last case (Case 4), a 30-unit
test system by tripling the number of units in Case 2 is used to validate the effectiveness of the
HUMDA. The data in this system are from the literature [40]. The characteristics of the units and load
demand are summarized in Table V and Table VI. The proper maximum iteration number is set to 500
for Case 2, Case 3 and set to 1000 for Case 4.
5.2.1. Case 2. Table VII shows a comparison between the results obtained by the HUMDA and the
other methods which have been reported in the literature. It can be inferred from Table VII that the solution obtained by the HUMDA method is better than most of the other methods. The best solution is
obtained as $1 018 894.022. Accordingly, the detailed results of the best solution are shown in Table
VIII. Table VIII shows that the best results attained by HUMDA satisfy both ramp-rate limits and load
balance constraints. The experiments are run for 20 times independently by using the proposed
HUMDA. The calculated standard deviation is $483.8954, which proves the HUMDA method converges to a small variation range of total cost. We use Figures 4 and 5 to clarify the quality of the solution and the robustness of HUMDA method. Figure 4 (the solid line represents HUMDA method and
Figure 4. Convergence process of the minimum cost for Case 2.
Figure 5. Distribution of the best solution of each trial.
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
HUMDA FOR DED PROBLEM CONSIDERING VALVE-POINT EFFECTS AND RAMP RATES
387
the dotted line represents UMDAs method) shows the comparison of the results between the HUMDA
and the original UMDAc during the evolutionary process. In addition, Figure 5 shows that the best
total cost varies in a small range within 20 trial numbers by using the HUMDA method. Therefore,
the proposed HUMDA method has good performance in quality of the solution and convergence
features.
In order to verify the fact that the proposed strategy can greatly improve UMDA algorithm performance, the HUMDA without dynamic adaptive search stage (DAS) and the HUMDA without chaotic
local search (CLS) operator are provided to be compared with HUMDA and original UMDAc method.
In Table IX, it shows that although CLS operator can significantly improve UMDA, it can get a better
solution when both DSA and CLS strategy are used in UMDA. Therefore, the HUMDA improves the
convergence performance greatly compared with UMDAc.
5.2.2. Case 3. The transmission losses are taken in case 3. The B matrix of the transmission loss coefficients is given in the literature [40], and the details are shown in Appendix A. Table X shows the
comparison of the best, the average and the worst solution obtained by the HUMDA method with other
methods reported in the literature. The best result obtained by the proposed method is $1 040 512.224
which is much better than the mentioned methods in Table X, and the detail of the result is given in
Table XI. A comparison between the HUMDA and original UMDA during the evolutionary process
is shown in Figure 6 (the solid line represents HUMDA method and the dotted line represents UMDAs
method).
5.2.3. Case 4. In this case, the dimension of the problem is increased and the result is compared with
that from other methods reported in the literature. The transmission losses are neglected in this case. In
Table XII, the best result obtained by HUMDA is $3 056 191.9855. It can be seen that the proposed
HUMDA can provide lower best total cost than other peer methods.
Table IX. Effects of the proposed strategies on the optimal solution of case.
Case 1
Method
HUMDA
HUMDA
without DSA
HUMDA
without CLS
Original
UMDAc
Case 2
Minimum
cost ($)
Average
cost ($)
Maximum
cost ($)
Minimum
cost ($)
Average
cost ($)
Maximum
cost ($)
43 128.950
43 732.365
43 437.73
43 987.257
43 750.823
44 256.406
1 018 894.022
1 021 220.197
1 020 074.678
1 022 702.523
1 020 952.182
1 023 252.632
46 985.256
47 080.474
49 289.369
1 029 342.269
1 030 524.563
1 031 034.306
51 717.973
51 889.698
51 995.876
1 033 242.374
1 037 811.247
1 039 242.374
Table X. Comparison of optimization results with different methods for Case 3.
Method
EP[27]
EP-SQP[27]
MHEP-SQP[27]
GA[22]
PSO[22]
IPSO[12]
ABC[22]
AIS[28]
TVAC-IPSO[40]
Original UMDAc
Proposed
HUMDA
Minimum cost ($)
Average cost ($)
Maximum cost ($)
Simulation time (min)
1 054 685
1 052 668
1 050 054
1 052 251
1 048 410
1 046 275
1 043 381
1 045 715
1 041 066.196
1 047 923.236
1 040 512.224
1 057 323
1 053 771
1 052 349
1 058 041
1 052 092
1 048 145
1 044 963
1 047 050
1 042 118.472
1 048 457.258
1 041 356.056
NA
NA
NA
1 062 511
1 057 170
NA
1 046 805
1 048 431
1 043 625.977
1 051 339.874
1 042 724.611
47.23
27.53
24.33
3.4436
4.0933
NA
3.4083
23.22
3.25
1.96
2.38
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
P1
150
150
226.6242
303.2484
303.2484
379.8726
379.8726
456.4968
456.4968
456.4968
456.4968
456.4969
456.4968
379.8726
303.2484
226.6242
226.6242
303.2484
379.8726
456.4968
456.4969
379.8726
303.2484
226.6242
Hour
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Total
222.2665
222.2665
222.2665
222.2665
222.2665
302.2665
309.5329
309.5329
389.5329
460
460
460
396.7994
396.7994
396.7994
316.7994
309.5329
389.5329
396.7994
460
396.7994
316.7994
236.7994
222.2665
P2
83.26266
157.8819
182.7811
208.7962
284.1827
297.3995
301.5195
293.0708
331.2354
338.1317
330.8883
340
297.3995
306.562
294.8149
318.5231
299.0956
313.6442
297.0111
338.132
301.3283
247.6171
185.1997
205.851
P3
60
60
60
60.17606
60.00001
60
110
119.8773
120.4152
170.4152
220.4152
270.4152
241.2457
241.2457
234.6495
184.6495
134.655
120.4152
120.415
170.415
180.8305
130.8305
92.27757
60
P4
122.8665
122.8665
172.7331
222.5997
222.5996
172.7331
222.5997
222.5996
222.5996
222.5997
222.5996
235.006
222.5997
222.5997
172.7331
122.8666
122.8666
122.8666
172.7331
222.5997
222.5996
172.7331
122.8665
73
P5
122.6119
122.4498
122.4498
122.4594
122.4498
157.6272
122.7374
122.4498
160
160
160
160
151.0019
124.5283
122.4498
122.4527
123.3083
122.5271
159.6219
160
122.4669
122.4499
122.4499
122.4508
P6
129.5904
129.5904
129.5904
129.5904
129.5904
129.5904
129.5904
129.5904
129.5904
129.5904
129.5904
129.6316
129.5904
129.5904
129.5904
129.5904
129.5904
129.5905
129.5904
129.5904
129.5905
129.5904
129.5904
129.5905
P7
85.31212
85.31211
85.3121
85.31211
85.31211
85.31211
85.31211
85.31211
85.3121
85.3121
115.3121
120
120
90
85.31211
85.31211
85.31211
85.31211
85.31211
85.31211
85.31211
85.31211
85.3121
85.31211
P8
20
20
20
20
20
20
20
20
20
50
52.05707
52.05707
52.05707
22.05707
20
20
20
20
20
50
20
20
20
20
P9
Table XI. Optimal solution of 10-unit using HUMDA considering transmission losses.
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
55
P10
14.91012
15.36733
18.75729
23.44871
24.64965
31.80137
34.16473
37.92991
46.18262
55.546
56.35963
58.60686
50.19054
44.25525
38.59763
27.81802
25.98511
34.13702
40.35568
55.546
46.42423
32.20508
20.74408
16.09497
850.0778
Loss(MW)
28 708.24
30 393.65
33 472.43
36 965.65
38 461.38
42 015.18
43 635.44
45 243.6
49 126.98
53 336.12
55 404.11
57 348.9
52 733.49
49 099.71
45 538.32
40 619.82
38 682.83
42 117.44
45 369.53
53 336.12
48 752.49
42 489.57
35 727.52
31 933.69
1 040 512.224
Cost ($)
388
W. GU ET AL.
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
HUMDA FOR DED PROBLEM CONSIDERING VALVE-POINT EFFECTS AND RAMP RATES
389
Figure 6. Convergence process of the minimum cost for Case 3.
Table XII. Comparison of optimization results with different methods for Case 4.
Method
Minimum cost ($/24 h)
EP[27]
EP-SQP[27]
MHEP-SQP[27]
DGPSO[10]
IPSO[12]
ICPSO[37]
CDE[25]
Proposed HUMDA
3 164 531
3 169 093
3 151 445
3 148 992
3 090 570
3 064 497
3 083 930
3 056 191 9855
5.3. DISCUSSION
According to the above cases, the performance of HUMDA is much better than most of the mentioned
methods, except for several superior methods such as TVAC-IPSO and ICPSO. In case 1, the minimum
cost obtained by the HUMDA method is better than the TVAC-IPSO method, but the TVAC-IPSO
method does better in average cost and maximum cost. In case 2, the performance of HUMDA is worse
than TVAC-IPSO, but better than ICPSO in the minimum cost. In case 3, the HUMDA method can get
better results than other methods. In case 4, when the dimension increases, the result obtained by HUMDA
is better than ICPSO. Compared with other methods, it seems that the proposed HUMDA has better
performance for solving high dimensional problems such as DED problem, especially when the problem
considers the transmission loss coefficients. Finally, the HUMDA method provides a superior method in
better convergence and robustness to solving high dimensional problems with complex constraints.
6. CONCLUSIONS
This paper introduces a new approach to solve dynamic economic dispatch problem in power system with
value-point effects, ramp-rate constraints and transmission losses. To improve the performance of the
original UMDAc, a two-stage adaptive mechanism is devised to control parameters. Moreover, a chaotic
local search operator is integrated to avoid premature convergence effectively. In addition, the combination between the proposed two-stage adaptive mechanism and the heuristic strategies can handle the
equality and inequality constraints efficiently. The simulation results in two test systems show that the results obtained by HUMDA method are better than most of other methods in recent literatures. Besides,
HUMDA has outstanding ability in finding better solutions quickly and effectively when we compare
the convergence performance of the HUMDA method with original UMDAc. In summary, HUMDA
method is good at solving high dimensional problems with complex constraints such as DED problems.
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
390
W. GU ET AL.
7. LIST OF ABBREVIATIONS
DED
HUMDA
UMDAc
SED
LP
NLP
QP
DP
LR
GA
PSO
DE
ABC
SA
CMAES
EP
AIS-SQP
HHS
HNN
EDAs
BSS
DAS
APC
DSRA
CLS
APSO
AIS
MLS
IPS
CMAES
TVAC-IPSO
Dynamic economic dispatch
hybrid Univariate Marginal Distribution Algorithm
Univariate Marginal Distribution Algorithm in continuous domains
static economic dispatch problem
linear programming
non-linear programming
quadratic programming
dynamic programming
Lagrange relaxation
Genetic algorithm
particle swarm optimization
differential evolution
artificial bee colony algorithm
simulated annealing
covariance matrix adapted evolution strategy algorithm
evolutionary programming
hybrid artificial immune systems and sequential quadratic programming
hybrid swarm intelligence based harmony search algorithm
hybrid approach of Hopfield neural network
estimation distribution algorithms
Basic search stage
Dynamic adaptive search stage
Adaptive parameter control strategy
Dynamic search range adjustment strategy
Chaotic local search
adaptive PSO
artificial immune system
Maclaurin Series
improved pattern search algorithm
covariance matrix adapted evolution strategy algorithm
time varying acceleration coefficients iteration particle swarm optimization
REFERENCES
1. X Xia, AM Elaiw. Optimal dynamic economic dispatch generation: A review. Electric Power Systems Research,
2010, 80(8):975–986. DOI: 10.1016/j.epsr.2009.12.012.
2. Jabr R, Coonick A, Cory B A homogeneous linear programming algorithm for the security constrained economic
dispatch problem. IEEE Transactions on Power Systems 2000; 15(3):930–937. DOI: 10.1109/59.871715.
3. Chen CL. Non-convex economic dispatch: A direct search approach. Energy Conversion and Management. 2007;
48(1):219–225. DOI: 10.1016/j.enconman.2006.04.010.
4. Pathom A, Hiroyuki K, Eiichi T. A hybrid EP and SQP for dynamic economic dispatch with nonsmooth fuel cost
function. IEEE Transactions on Power Systems 2002; 17(2):411–416. DOI: 10.1109/TPWRS.2002.1007911.
5. Ross, D, Kim, S. Dynamic economic dispatch of generation. IEEE Transactions on Power Systems 1980; 99(6):
2060–2068.
6. Balamurugan, R, Subramanian, S. Hybrid integer coded differential evolution-dynamic programming approach for
economic load dispatch with multiple fuel options. Energy Conversion and Management 2008; 49(4):608–614.
DOI: 10.1016/j.enconman.2007.07.039.
7. Hindi K, Ghani M. Dynamic economic dispatch for large-scale power systems: a Lagrangian relaxation approach.
International Journal of Electric Power Systems Research 1991; 13(1):1272–1282. DOI: 10.1016/0142-0615(91)
90018-Q
8. Victoire TAA, Jeyakumar AE. Reserve constrained dynamic dispatch of units with valve-point effects. IEEE Transactions on Power Systems 2005; 20(3):1273–1282. DOI: 10.1109/TPWRS.2005.851958.
9. Amjady, N, Nasiri-Rad, H. Solution of non-convex and non-smooth economic dispatch by a new adaptive real code
genetic algorithm. Expert Systems with Applications 2010; 37(7):5239–5245. DOI: 10.1016/j.eswa.2009.12.084.
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
HUMDA FOR DED PROBLEM CONSIDERING VALVE-POINT EFFECTS AND RAMP RATES
391
10. Victoire T, Jeyakumar A. Deterministically guided PSO for dynamic dispatch considering valve-point effect Electric
Power Systems Research 2005; 73:313–22. DOI: 10.1016/j.epsr.2004.07.005.
11. Panigrahi B, Ravikumar PV, Sanjoy D. Adaptive particle swarm optimization approach for static and dynamic
economic load dispatch. Energy Conversion and Management 2008; 49:1407–15. DOI: 10.1016/j.enconman.
2007.12.023.
12. Yuan X, Su A, Yuan Y, Nie H, Wang L. An improved PSO for dynamic load dispatch of generators with valve-point
effects. Energy 2009; 34:67–74. DOI: 10.1016/j.energy.2008.09.010.
13. Safari A, Shayeghi H. Iteration particle swarm optimization procedure for economic load dispatch with generator
constraints. Expert Systems with Applications 2011; 38:6043–8. DOI: 10.1016/j.eswa.2010.11.015.
14. Chakraborty S, Senjyu T, Yona A, et al. Solving economic load dispatch problem with value-point effects using a
hybrid quantum mechanics inspired particle swarm optimization. IET Generation, Transmission & Distribution
2011; 5(10):1042–1052. DOI: 10.1049/iet-gtd.2011.0038.
15. Balamurugan R, Subramanian S. An improved differential evolution based dynamic economic dispatch with
nonsmooth fuel cost function. Journal of Electrical Systems 2007; 3:151–61.
16. B Balamurugan, R Subramanian, Differential evolution-based dynamic economic dispatch of generating units
with valve-point effects. Electric Power Components and Systems 2008; 36(8):828–843. DOI: 10.1080/
15325000801911427.
17. Nomana N, Iba H. Differential evolution for economic load dispatch problems. Electric Power Systems Research
2008; 78:1322–31. DOI: 10.1016/j.epsr.2007.11.007.
18. Yuan X, Wang L, Yuan Y, Zhang Y, Cao B, Yang B. A modified differential evolution approach for dynamic
economic dispatch with valve point effects. Energy Conversion and Management 2008; 49:3447–53. DOI:
10.1016/j.enconman.2008.08.016.
19. Yuan X, Wang L, Zhang Y, Yuan Y. A hybrid differential evolution method for dynamic economic dispatch with
valve-point effects. Expert Systems with Applications 2009; 36:4042–8. DOI: 10.1016/j.eswa.2008.03.006.
20. Dakuo H, Dong G, Wang F, Mao Z. Optimization of dynamic economic dispatch with valve-point effect using
chaotic sequence based differential evolution algorithms. Energy Conversion and Management 2011; 52:1026–32.
DOI: 10.1016/j.enconman.2010.08.031.
21. Lu Y, Zhou J, Qin H, Wang Y, Zhang Y. Chaotic differential evolution methods for dynamic economic dispatch
with valve-point effects. Engineering Applications of Artificial Intelligence 2011, 24:378–87. DOI: 10.1016/j.
engappai.2010.10.014.
22. Hemamalini S, Simon SP. Dynamic economic dispatch using artificial bee colony algorithm for units with valvepoint effect. European Transactions on Electrical Power 2011; 21:70–81. DOI: 10.1002/etep.413.
23. Panigrahi CK, PK Chattopadhyay, Chakrabarti RN, et al. Simulated annealing technique for dynamic economic
dispatch. Electric Power Components and Systems 2006; 34(5):577–586. DOI: 10.1080/15325000500360843.
24. Manoharan P, Kannan P, Baskar S, Willjuice IM, Dhananjeyan V. Covariance matrix adapted evolution strategy
algorithm-based solution to dynamic economic dispatch problems. Engineering Optimization 2009; 41:635–57.
DOI: 10.1080/03052150902738768.
25. Youlin L, J Zhou, H Qin, Y Wang, Y Zhang. Chaotic differential evolution methods for dynamic economic dispatch
with valve-point effects. Engineering Application of Artificial Intelligence 2011; 24:378–387. DOI: 10.1016/j.
engappai.2010.10.014.
26. Cai J, Li Q, Li L, Peng H, Yang Y. A fuzzy adaptive chaotic ant swarm optimization for economic dispatch.
Electrical Power and Energy Systems 2012; 34:154–160. DOI: 10.1016/j.ijepes.2011.09.020.
27. Victoire T, Jeyakumar A. A modified hybrid EP-SQP approach for dynamic dispatch with valve-point effect. International Journal of Electrical Power and Energy Systems 2005; 27:594–601. DOI: 10.1016/j.ijepes.2005.06.006.
28. Hemamalini S, Simon SP. Dynamic economic dispatch using artificial immune system for units with valve-point
effect. International Journal of Electrical Power and Energy Systems 2011; 33:868–74. DOI: 10.1016/j.
ijepes.2010.12.017.
29. Basu M. Hybridization of artificial immune systems and sequential quadratic programming for dynamic economic
dispatch. Electric Power Components and Systems 2009; 37:1036–45. DOI: 10.1080/15325000902918941.
30. Pandi VR, Panigrahi BK. Dynamic economic load dispatch using hybrid swarm intelligence based harmony search
algorithm. Expert Systems with Applications 2011; 38:8509–14. DOI: 10.1016/j.eswa.2011.01.050.
31. Abdelaziz A, Kamh M, Mekhamer S, Badr M. A hybrid HNN-QP approach for dynamic economic dispatch problem.
Electric Power Systems Research 2008; 78:1784–8. DOI: 10.1016/j.epsr.2008.03.011.
32. Larrañaga P, Lozano JA. Estimation of Distribution Algorithms:A New Tool for Evolutionary Computation, Kluwer
Academic Publishers: Boston, 2002.
33. Jiang S, Ziver AK, Carter JN, et al.. Estimation of distribution algorithms for nuclear reactor fuel management
optimization. Annals of Nuclear Energy. 2006; 33(11–12):1039–57. DOI: 10.1016/j.anucene.2006.03.012.
34. Santana R, Larrañaga P, Lozano JA. Protein folding in simplified models with estimation of distribution algorithms.
IEEE transactions on Evolutionary Computation 2008; 12(4):418–438. DOI: 10.1109/TEVC.2007.906095.
35. Sagarna R, Lozano JA. Scatter Search in software testing, comparison and collaboration with Estimation of Distribution
Algorithms. European Journal of Operational Research 2006; 169(2):392–412. DOI: 10.1016/j.ejor.2004.08.006.
36. Larrañaga P, Etxeberria R, Lozano JA, Peña JM. Optimization by learning and simulation of Bayesian and Gaussian
networks. University of the Basque Country Technical Report EHU-KZAA-IK-4/99. http://www.sc.ehu.es/
ccwbayes/postscript/kzaa-ik-04-99.ps, 1999.
Copyright © 2013 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep
392
W. GU ET AL.
37. Wang Y, Zhou J, Qin H, Lu Y. Improved chaotic particle swarm optimization algorithm for dynamic economic
dispatch problem with valve-point effects. Energy Conversion and Management 2010; 51(12):2893–900. DOI:
10.1016/j.enconman.2010.06.029.
38. Hemamalini S, Simon SP. Dynamic economic dispatch using Maclaurin series based lagrangian method. Energy
Conversion and Management 2010; 51(11):2212–9. DOI: 10.1016/j.enconman.2010.03.015.
39. Alsumait J, Qasem M, Sykulski J, Al-Othman A. An improved pattern search based algorithm to solve the dynamic
economic dispatch problem with valve-point effect. Energy Conversion and Management 2010; 51(10):2062–7.
DOI: 10.1016/j.enconman.2010.02.039.
40. Mohammadi-ivatloo B, Rabiee A, Ehsan M. Time-varying acceleration coefficients IPSO for solving dynamic
economic dispatch with non-smooth cost function. Energy Conversion and Management 2012; 56:175–183. DOI:
10.1016/j.enconman.2011.12.004.
APPENDIX A
The B-matrix coefficients of 5-unit test system are as follows.
2
0:000049
6 0:000014
6
6
Bij ¼ 6
6 0:000015
6
4 0:000015
0:000020
0:000015 0:000020
3
0:000014
0:000015
0:000045
0:000016
0:000016
0:000039
0:000020
0:000010
0:000020 0:000018 7
7
7
0:000010 0:000012 7
7
7
0:000040 0:000014 5
0:000018
0:000012
0:000014 0:000035
The B-matrix coefficients of 10-unit test system are as follows.
2
8:7
6 0:43
6
6
6 4:61
6
6
6 0:36
6
6 0:32
6
Bij ¼ 6
6 0:66
6
6 0:96
6
6
6 1:6
6
6
4 0:8
0:1
4:61
0:36
0:32
0:66
0:96
1:6
0:8
8:3
0:97
0:22
0:75
0:28
5:04
1:7
0:54
0:97
9
2
0:63
3
1:7
4:3
3:1
0:22
2
5:3
0:47
2:62
1:96
2:1
0:67
0:75
0:63
0:47
8:6
0:8
0:37
0:72
0:9
0:28
3
2:62
0:8
11:8
4:9
0:3
3
5:04
1:7
1:96
0:37
4:9
8:24
0:9
5:9
1:7
4:3
2:1
0:72
0:3
0:9
1:2
0:96
0:54
3:1
0:67
0:9
3
5:9
0:96
0:93
7:2 7
7
7
2 7
7
7
1:8 7
7
0:69 7
7
7
3 7
7
0:6 7
7
7
0:56 7
7
7
0:3 5
7:2
2
1:8
0:69
3
0:6
0:56
0:3
0:99
Copyright © 2013 John Wiley & Sons, Ltd.
0:1
3
0:43
Int. Trans. Electr. Energ. Syst. 2015; 25:374–392
DOI: 10.1002/etep