This article was downloaded by: [University Of Pittsburgh]On: 16 April 2013, At: 02:32Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

International Journal of ComputerMathematicsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gcom20

An improved chaos optimizationalgorithm and its application in theeconomic load dispatch problemFang Han a & Qi-Shao Lu aa Science School, Beijing University of Aeronautics andAstronautics, Beijing, 100083, ChinaVersion of record first published: 06 Jun 2008.

To cite this article: Fang Han & Qi-Shao Lu (2008): An improved chaos optimization algorithmand its application in the economic load dispatch problem, International Journal of ComputerMathematics, 85:6, 969-982

To link to this article: http://dx.doi.org/10.1080/00207160701305388

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae, and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.

International Journal of Computer MathematicsVol. 85, No. 6, June 2008, 969–982

An improved chaos optimization algorithm and its applicationin the economic load dispatch problem

FANG HAN* and QI-SHAO LU

Science School, Beijing University of Aeronautics and Astronautics, Beijing 100083, China

(Received 13 December 2006; accepted 29 January 2007)

This paper reforms the Mutative Scale Chaos Optimization Algorithm (MSCOA). Numerical simula-tion demonstrates that the efficiency and performance of the algorithm are improved. An analysis ofthe Improved Mutative Scale Chaos Optimization Algorithm (IMSCOA) is also given. IMSCOA isapplied to examples of economic load dispatch, considering both the valve point effect and transmis-sion loss. The results are compared with those computed by other methods, including MSCOA andanother effective chaos algorithm—CROA. It is shown that IMSCOA provides more optimal resultsthan the other methods, whether or not the problem is large scale. This demonstrates the efficiencyand practicability of IMSCOA in engineering practice.

Keywords: Chaos optimization algorithm; Economic load dispatch; IMSCOA; Valve point effect;Transmission loss

AMS Subject Classification: 46N10; 47N10

1. Introduction

1.1 Chaos optimization algorithm (COA)

Chaotic movement has the properties of ergodicity, intrinsic stochasticity and regularity. It cango through every state in a certain area and every state can be obtained only once. The chaosoptimization algorithm (COA) directly utilizes chaotic variables to search for the optimalsolution. Because of the chaos property, chaos optimization can obtain the global optimalsolution more readily than methods adopted previously. It can more readily escape fromlocal minima than other stochastic algorithms. COA is considered to be a novel stochasticoptimization algorithm and is gradually being applied in engineering practice.

The chaos optimization algorithm can automatically satisfy the interval requirementsof optimization variables. It can deal with optimization problems of continual variables

*Corresponding author. Email: yadiahan@163.com

International Journal of Computer MathematicsISSN 0020-7160 print/ISSN 1029-0265 online © 2008 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/00207160701305388

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

970 F. Han and Q.-S. Lu

as follows:

min f (xi), ai ≤ xi ≤ bi, i = 1, 2, . . . , n, (1)

where ai and bi are, respectively, the lower and upper limits of design variables xi .The chaotic variables used for the carrier can be obtained by choosing Logistic mapping,

that is

xn+1 = μxn(1 − xn), μ ∈ [0, 4], xn ∈ [0, 1], n = 0, 1, 2, . . . , (2)

where μ is a control parameter. The variable xn is in a chaotic situation when μ = 4. Bymaking use of the characteristic of chaos of being sensitive to the initial value, m differentvariables can be obtained by being given m different initial values. In order to avoid settingthe first-order fixed points as the initial values, x0 cannot be 0, 0.25, 0.5, 0.75 or 1.

In addition to Logistic mapping, other chaotic mappings, such as Tent mapping, can alsobe introduced into COA. Researchers are now trying to improve the algorithm’s efficiencyby means of altering the model, configuration, control strategy, search style, etc., and thusvarious chaos optimization algorithms have been proposed. Moreover, these algorithms havegradually been applied in engineering practice.

1.2 The economic load dispatch (ELD) problem

Economic load dispatch in power plants or power systems is the distribution of load among theunits so as to minimize electricity generation costs, while satisfying the operating constraintsof the generating units and power system. It is a typical optimization problem and plays a largerole in the economic dispatch of power systems. However, this problem is high-dimensional,non-convex, discrete and nonlinear, therefore it is very difficult to obtain global optima.

There are various optimization methods that can satisfy the different operational require-ments in practical use, such as the equal incremental algorithm, dynamic programming [1, 2]and the Lagrange relaxation method. However, the equal incremental algorithm requires theunit input–output curve to be monotonically increasing, while the actual curve generally isnot. As for dynamic programming, the dimensionality problem will become apparent if thereare many units. With the rapid development of artificial intelligence, many other methodshave been put forward to solve this problem, such as the tabu search algorithm [3], the geneticalgorithm (GA) [4], the ant colony algorithm [5], etc. However, these algorithms also fall shortwhen solving the problem. For instance, the genetic algorithm is essentially an unconstrainedoptimization algorithm and how the constraints are dealt with greatly affects the algorithm’sefficiency. Also, as the GA is a stochastic optimization algorithm, it is difficult to ensure theveracity of the optima. Moreover, it requires a fitness function and a form population, andthen selection, crossover and mutation operations have to be carried out to achieve a new pop-ulation to calculate the fitness. This process has to be continued until the object is obtained,therefore the GA is computationally expensive and time-consuming. There is a long historyof 60 years of study of economic load dispatch, but no absolutely exact method has yet beenfound. However, as it can bring tremendous economic benefit, researchers are actively lookingfor better methods for this problem.

1.3 Solving the ELD problem by COA

Some work has been done in the field of economic load dispatch using COA. The chaosoptimization algorithm has been proposed to deal with the economic dispatch problem ofa hydro power plant [6], and the results show that the method is powerful. The MutativeScale Chaos Optimization Algorithm has been applied to the economic operation of power

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

An improved chaos optimization algorithm 971

plants [7], and the results show that the method is effective, but it is time-consuming. Anotherkind of chaos algorithm has been used to solve the economic load dispatch of power systemsand results similar to GA were obtained [8]. Moreover, it can take the valve point effectand transmission loss into account. This paper further develops the research and proposes animproved chaos optimization algorithm, and applies the algorithm to examples of economicload dispatch.

2. Models for the economic load dispatch problem

This section presents the formulation of the economic load dispatch used in this paper.Nonlinear programming with equality and inequality constraints is converted into a newprogramming using a penalty function method.

2.1 Objective function

The economic load dispatch of generation units or power systems is the optimization of unitcommitment to minimize electricity generation costs while satisfying the operating constraintsof the system. It is a problem of nonlinear programming.

In this paper the total coal consumption rate of the units is considered as the objectivefunction. It is given by

min FD =n∑

i=1

Fi(Pi), (3)

where FD is the system’s total coal consumption rate, n is the number of units, Pi is thegeneration by unit i, and Fi(Pi) is the characteristic expression of the coal consumptionof unit i.

2.2 Characteristic curve of the unit’s coal consumption considering the valve point effect

The characteristic curve of unit coal consumption can be approximately expressed as aquadratic function of the active power,

fi(Pi) = aiP2i + biPi + ci, (4)

where ai , bi and ci are coefficients of the coal consumption characteristic of unit i.

Figure 1. Characteristic curve of the unit’s actual coal consumption.

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

972 F. Han and Q.-S. Lu

However, the actual characteristic curve of the coal consumption of the unit is too compli-cated to be expressed by a quadratic function. The actual curve is shown in figure 1. The waveon the curve is caused by the ordinal opening of the steam turbine’s adjusting valve togetherwith the increase in active power. When the previous valve has already completely opened andthe next one is just opening, steam circulation will decrease because of the throttling effectand coal consumption will increase and the curve will rise. This phenomenon is called thevalve point effect [9].

Generally speaking, a smooth quadratic function approximation of the unit coal consump-tion characteristic provides the basis for most economic dispatch. Consequently, the valvepoint effect is often neglected and this makes the dispatch inaccurate. Here, we take the valvepoint effect into account.

The valve point effect can be expressed as

Ei = |gi sin(hi(Pi − Pi min))|, (5)

where gi and hi are constants.Therefore, the actual characteristic curve of the unit’s coal consumption can be expressed as

Fi(Pi) = fi(Pi) + Ei. (6)

2.3 Constraints considering transmission loss

The unit operating constraint is

Pi min < Pi < Pi max, (7)

where Pi min and Pi max are, respectively, the minimum and maximum generation of unit i.The power balance constraint is

n∑i=1

Pi = PLD, (8)

where PLD is the total load demand.In most cases, transmission loss is neglected in the power balance constraint. However, it

sometimes represents a relatively larger part of the total load (this can be seen from example 1in section 5). If the transmission loss is considered, the power balance constraint should bemodified as

n∑i=1

Pi = PLD + PS, (9)

where PS is the total transmission loss.The transmission loss can generally be obtained by power flow computation or the B coef-

ficient method. If we use the B coefficient method, then the relationship between transmissionloss and the generation of units is

PS = PTBGGP + PTBG0 + B00, (10)

where P is the active power vector of n dimensions. BGG, BG0 and B00 are called B coefficientsand can be measured and modified once every few seconds.

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

An improved chaos optimization algorithm 973

2.4 Optimal objective function

Considering the above constraints, the mathematical model of the optimal dispatch for thetotal load among units can be expressed as

min FD =n∑

i=1

Fi(Pi) =n∑

i=1

[fi(Pi) + Ei], (11)

subject to

n∑i=1

Pi − PLD = 0

(n∑

i=1

Pi − PLD − PS = 0

),

Pi min < Pi < Pi max, i = 1, 2, . . . , n. (12)

If we make unit generation a design variable, the minimum and maximum generation of theunit can be satisfied spontaneously using the chaos optimization algorithm. Equal constraintscan be denoted by a penalty function. The final optimization objective function is (13) or (14).

When transmission loss is neglected,

f =n∑

i=1

fi(Pi) +n∑

i=1

Ei + k

∣∣∣∣∣n∑

i=1

Pi − PLD

∣∣∣∣∣β

, i = 1, 2, . . . , n. (13)

When transmission loss is considered,

f =n∑

i=1

fi(Pi) +n∑

i=1

Ei + k

∣∣∣∣∣n∑

i=1

Pi − PLD − PS

∣∣∣∣∣β

, i = 1, 2, . . . , n, (14)

where k is a penalty factor, and is a given constant satisfying β ≥ 1.

3. The improved chaos optimization algorithm

3.1 Basic idea of MSCOA

The idea of a mutative scale comes from the DFP algorithm, which is a quasi-Newtonmethod [10]. The DFP algorithm, which is based on the original scale matrix, continuallytransforms the scale of the matrix to accelerate the speed of the optima search. Zhang et al. [11]introduced the idea of a mutative scale to the chaos optimization algorithm and proposed theMutative Scale Chaos OptimizationAlgorithm (MSCOA). It considers the value interval of thedesign variables as the scale, continually transforms the scale to accelerate the optimizationprocess and greatly improves the efficiency of COA.

3.2 Three improvements made to MSCOA

This paper reports improvements to three aspects: the choice of the initial value, the searchconfiguration and the convergence criterion. The improved method will be called the ImprovedMSCOA, or IMSCOA.

First is the choice of the initial states. In practice, chaotic variables cannot go through everystate in a certain time. The choice of the initial states will greatly affect the results of the chaos

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

974 F. Han and Q.-S. Lu

optimization algorithm. MSCOA takes different values in the interval [0, 1] as its initial states(fixed points are excluded). It requires a selection and comparison many times to find possibleoptima, and, accordingly, other parameters also need to be adjusted, but it is still difficult toguarantee the accuracy of the optimization results. Random numbers are taken as the initialchaotic variables in this paper. The method is: if there are n independent design variables, thenuse a random number generator to produce n chaotic variables in the interval (0, 0.5). Thereason for limiting the scale of the random number is that formula (2), in which μ = 4, hassymmetry properties about 0.5. If two initial states xi and xj satisfy the equation xi = 1 − xj ,the same sequence will be obtained during the iteration process and the chaos search will fail.

If random numbers are used as initial states, the results computed by the program will bedifferent each time. Therefore, a paratactic search configuration is adopted: after all of thesearch steps have been performed, choose another group of initial states and repeat the searchsteps. After looping several times, compare these groups of results and find the most optimalone as the final optimal solution.

A new convergence criterion is then defined. It is not sufficient to make the convergencecriterion as ‘iterating for N times’. If N is too large, much time would be lost, and moreiteration steps will not contribute to a more optimal result. However, if it is too small, it isdifficult to find the optimal result. Moreover, N needs to be chosen as different values fordifferent optimization problems. Hence, this paper defines a new convergence criterion asfollows: during the subtle search process, after a search for N1(N2) times, save the optimalobjective function value f ∗ as f ∗

old; continue the search for N1(N2) times, if the current optimalobjective function value f ∗ and f ∗

old satisfy |f ∗ − f ∗old| > ε (ε is a small positive number),

then save the smaller value as f ∗old; continue the search steps until |f ∗ − f ∗

old| < ε is satisfied,then save the current optimal value f ∗ and, accordingly, the optimal design variables. Thus, weonly need to set the iteration steps to a reasonable number, say 600, and the program will definethe appropriate number of iteration steps automatically. This reduces manual adjustment andensures the effectiveness of the optimal solutions.

3.3 The improved algorithm

The improved algorithm is presented as follows.

Step 1 Initialize the parameters. Let m = 0, where m is a paratactic search indicator. Let x∗be the current optimal chaotic variable and f ∗ the current optimal solution. Set theinitial value of f ∗ to a large number.

Step 2 Choose the initial states. Initialize xi with n different values by means of a randomnumber generator, where i = 1, 2, . . . , n. Let xk

i = xi , k = 0, r = 0, ari = ai and

bri = bi , where k is the iterative indicator of the chaotic variables and r is the subtle

search indicator.

Step 3 Design variables are obtained at intervals utilizing the first carrier wave of

mxi = ari + (br

i − ari )x

ki . (15)

Formula (15) can make design variables mxi change within the variance ranges of theoptimization variables.

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

An improved chaos optimization algorithm 975

Start searching as follows.

if f (mxi) < f ∗, then x∗i = xk

i , mx∗i = mxi, f

∗ = f (mxi), k = k + 1,

xki = 4xk

i (1 − xki ).

Continue the above steps. After iterating and searching N1 times, save the currentoptimal objective value f ∗ as f ∗

old and optimal solution mx∗i as mx∗

i_old; then iterate andsearch another N1 times. If |f ∗ − f ∗

old| > ε (ε is a small positive number), then save thesmaller value as f ∗

old and its corresponding optimal solutions as mx∗i_old. Continue with

the above search steps until |f ∗ − f ∗old| < ε, and save the current optimal objective

value f ∗ and optimal solution mx∗i .

Step 4 Transform the scale using the following formulae:

ar+1i = mx∗

i − γ ∗ (bri − ar

i ),

br+1i = mx∗

i + γ ∗ (bri − ar

i ),(16)

where γ ∈ (0, 0.5). The transformation may transcend the original interval, thereforemodification should be performed as follows:

if ar+1i < ar

i , then ar+1i = ar

i ; if br+1i > br

i , then br+1i = br

i .

Then mx∗i should be reverted using the formula

x∗i = mx∗

i − ar+1i

br+1i − ar+1

i

. (17)

Use x∗i and xk

i to formulate new chaotic variables according to

yi = (1 − α)x∗i + αxk

i , 0 < α < 1. (18)

Perform a subtle search with the new chaotic variables around the optimum and theiterative calculation process is similar to Step 3. Corresponding to N1 in Step 3, theiteration number in the repeated search process will be N2, which is generally smallerthan N1. When the desired precision is satisfied, end the iteration and save the currentoptimal value f ∗, optimal solution mx∗

i and their corresponding chaotic variables x∗i .

Step 5 Reduce α utilizing α = α − h, where h is the step length. It can be a small number,such as 0.01. Repeat Step 4.

Step 6 Repeat Step 5 until α is less than a fixed number, such as 0.02, then output and savethe optimal design variables and optimal objective function value.

Step 7 Let m = m + 1, then return to Step 2. Loop until m = M . By comparison, denote thesmallest of the M objective function values as the most optimal, f ∗, and select thecorresponding optimal design variables mx∗

i as the final optimal values.

End the search process.

4. Numerical simulation and analysis

In this section, IMSCOA is tested using several optimization functions and an analysis of thealgorithm is presented. In order to demonstrate the improved algorithm’s efficiency, MSCOAis also tested. MATLAB is adopted as the computer language in this paper. The optimizationrate of the algorithms is obtained by calculating each function 20 times. The calculation timewas tested on a PIV computer running at 2.50 GHz with 256 MB of RAM.

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

976 F. Han and Q.-S. Lu

4.1 Test functions and optimization results

Various optimization functions are used for testing the improved algorithm’s efficiency. Welist some of them as follows.

F1 : max f (x1, x2) = 0.5 −sin(

√x2

1 + x22 )2 − 0.5

[1 + 0.001(x21 + x2

2 )]2,

x1 ∈ [−100, 100], x2 ∈ [−100, 100],F2 : min f (x1, x2) = −21.5 − x1 sin(4πx1) − x2 sin(20πx2),

x1 ∈ [−3, 12.1], x2 ∈ [4.1, 5.8],

F3 : min f =2∏

k=1

{5∑

i=1

i cos[(i + 1)xk + i]}

, xi ∈ [−10, 10],

F4 : max f = 0.002 +25∑

j=1

1

j + ∑2i=1(xi − aij )6

, xi ∈ [−65536, 65536],

where (aij )=(−32 −16 0 16 32 −32 −16 · · · 0 16 32

−32 −32 −32 −32 −32 −16 −16 · · · 32 32 32

).

Three-dimensional figures of these functions are shown in figure 2. A comparison of theoptimization efficiency of IMSCOA and MSCOA is shown in table 1.

Figure 2. Three-dimensional representation of optimization functions F1–F4.

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

An

improved

chaosoptim

izationalgorithm

977

Table 1. Comparison of the optimization efficiency of IMSCOA and MSCOA.

Optimization OptimizationFunction Optimum Algorithm results Time (s) rate

F1 x1 = 0, x2 = 0, MSCOA x1 = −0.4501 ∗ 10−5, x2 = 0.0324 ∗ 10−8, f ∗ = 1 1.140 Unstable

f ∗ = 1 IMSCOA x1 = −0.9059 ∗ 10−8, x2 = 0.2528 ∗ 10−8, f ∗ = 1 5.609 90%F2 Reference minimum MSCOA x1 = 11.6255, x2 = 5.7250, f ∗ = −38.8503 1.140 Unstable

x1 = 11.6255, IMSCOA x1 = 12.1255, x2 = 5.7250, f ∗ = −39.3503 1.172 100%

x2 = 5.7250,

f ∗ = −38.8503

F3 Reference minimum MSCOA f ∗ = −186.7309088310239 0.109 100%

f ∗ = −186.730908831 IMSCOA f ∗ = −186.7309088310239 0.250 100%

F4 Reference minimum MSCOA x1 = −31.978335, x2 − 31.978338, f ∗ = 1.0020 1.157 Unstable

x1 = −31.978335, IMSCOA x1 = −31.978334, x2 = −31.978345, f ∗ = 1.0020 12.515 100%

x2 = −31.978335,f ∗ = 1.002

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

978 F. Han and Q.-S. Lu

4.2 Analysis of IMSCOA

IMSCOA adopts random numbers as the initial states of the chaotic variables and reducesthe fussiness of manual evaluation and parameter adjustment. The introduction of a paratacticsearch configuration can produce stable results of high precision, as the most optimal value ofthe multi-orbit calculation results is made the final optimum value. However, the introductionof a paratactic search comes with the cost of increased computational burden. However, fromanother aspect, a mutative scale accelerates the convergence speed of the algorithm and thiscan make up for the lack of optimization speed of the paratactic configuration. The new con-vergence criterion is helpful as the algorithm determines the iteration steps automatically andguarantees the current optimal solution to a certain degree. It can be seen from the numericalsimulation that the optimization efficiency of IMSCOA is high and it can generally find theoptimal solution 100% of the time. Even for some difficult functions, changing some param-eters of the algorithm will improve the optimization efficiency. The time cost of the algorithmis reasonable and can be decreased by adjusting the parameters. MSCOA takes less time, butneeds to frequently change the initial states and, accordingly, adjustment of the parameters canbe laborious, debasing the algorithm’s efficiency. Comparing one with the other, IMSCOA ismore effective and practical than MSCOA.

5. Application of IMSCOA in economic load dispatch

In this section, the proposed algorithm IMSCOA is applied to several examples of economicload dispatch and the results are compared with those computed by other algorithms, suchas MSCOA. In addition, another effective chaos algorithm (CROA) [12] is applied to theseexamples. By comparing the results, some conclusions can be made.

5.1 Example 1: a small-scale ELD problem

Problem Distribute a total load of 500 MW in a power system of three units and sixgeneratrices and optimize the units’ commitment. The parameters of each unit are shownin table 2. The B coefficients [13] used for the power system are as follows:

BGG =⎛⎝ 0.676 × 10−3 0.953 × 10−4 −0.507 × 10−4

0.953 × 10−4 0.521 × 10−3 0.901 × 10−4

−0.507 × 10−4 0.901 × 10−4 0.294 × 10−3

⎞⎠ ,

BG0 =⎛⎝ −7.66 × 10−2

−0.342 × 10−2

1.89 × 10−2

⎞⎠ , B00 = 4.0357.

Table 2. Parameters of the units in the power system.

Cost parameters of the fuel Limits of units

Unit ai ($/MW2·h) bi ($/MW·h) ci ($) gi ($) hi Pi max Pi min

1 0.00156 7.92 561 300 0.0315 600 1002 0.00194 7.85 310 200 0.0425 400 1003 0.00482 7.97 78 150 0.0635 200 50

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

An improved chaos optimization algorithm 979

Table 3. Feasible solutions found by IMSCOA and a comparison of the results of the different algorithms.

Algorithm P1 (MW) P2 (MW) P3 (MW)∑

Pi (MW) Objective value time (s)

Algorithm in [8] 299.41 100.70 99.90 500.0100 5121.47CROA 199.7331 248.9746 51.2923 500.0000 5106.84 0.922MSCOA 299.4737 100.5674 99.9595 500.0007 5120.62 0.546IMSCOA 200.2041 249.7814 50.0145 500.0000 5095.83 4.203IMSCOA 199.7658 250.1471 50.0879 500.0008 5095.64 4.265IMSCOA 199.9253 250.0763 49.9992 500.0009 5095.72 4.078

Table 4. Feasible solutions found by IMSCOA and a comparison of the results of the different algorithms.

P1 P2 P3∑

Pi PS ObjectiveAlgorithm (MW) (MW) (MW) (MW) (MW) value time (s)

Algorithm in [8] 299.46 172.00 198.84 570.30 70.24 5735.93CROA 299.47 171.43 99.87 570.76 71.10 5735.81 1.297MSCOA 199.73 174.79 177.10 551.62 52.24 5736.11 0.750IMSCOA 299.46 171.49 99.86 570.81 71.11 5735.84 8.828IMSCOA 299.46 171.48 99.82 570.76 71.11 5735.87 8.344IMSCOA 299.47 171.54 99.84 570.85 71.13 5735.85 8.750

A kind of chaos optimization algorithm has been used to solve this problem [8]. This paperuses IMSCOA to compute the problem thrice and compares the results with those computedby CROA and MSCOA. Two cases are considered, and the test results are listed in tables 3and 4, respectively.

Case 1 The valve point effect is considered, whereas transmission loss is neglected. Inthis case, the parameters of the penalty function are penalty factor k = 100 and β = 1. Theparameters of IMSCOA are m = 10, γ = 0.35, α = 0.21, N1 = 600 and N2 = 100. The testresults are listed in table 3.

Case 2 The valve point effect and transmission loss are both considered. In this case, theparameters of the penalty function are penalty factor k = 100 and β = 1. The parametersof IMSCOA are m = 10, γ = 0.35, α = 0.21, N1 = 600 and N2 = 100. The test results arelisted in table 4.

It can be seen that the results calculated by IMSCOA are better than those provided by theother algorithms; not only are they more stable, but also more optimal, although the methodis more time-consuming.

For the above two cases, IMSCOA can provide many stable feasible solutions as referencefor dispatch officers. In this example, the transmission loss exceeds the unit’s total active powerby 10%, and cannot be neglected, therefore it is necessary to consider both the transmissionloss and the valve point effect for accurate solutions.

5.2 Example 2: a large-scale ELD problem

Problem Optimize the commitment among 40 units for a total load of 10 500 MW with thevalve point effect considered. The data are from [14].

The literature [14] uses different algorithms to solve this problem. Here we use IMSCOA,CROA and MSCOA. A comparison of the methods can be seen in table 5 (all the algorithmsare computed 20 times).

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

980 F. Han and Q.-S. Lu

Table 5. Comparison of statistical test results.

Algorithm Average value Maximum Minimum Average time (s)

CEP 126 810.00 128 080.00 125 420.00 160.25PSO 126 848.67 126 850.01 126 848.66 167.80CEP−PSO 124 145.60 124 900.00 123 670.00 196.45CROA 128 178.68 129 740.72 125 453.69 5.62MSCOA 139 162.52 153 300.70 127 201.99 1.75IMSCOA 124 234.50 126 760.02 121 410.50 12.97

The parameters of the penalty function are penalty factor k = 1000 and β = 1. The parame-ters for IMSCOA are m = 20, γ = 0.25, α = 0.21, N1 = 1500 and N2 = 100. The test resultsare listed in table 4.

It can be seen that IMSCOA has advantages with respect to calculation precision and timetaken compared with the algorithms reported in [14]. This demonstrates that IMSCOA iseffective in solving large-scale problems with multi-variables. CROA does not provide veryaccurate solutions, but is less time-consuming, therefore its optimization potential could beexplored more. MSCOA consumes the least time, but its precision is not sufficiently high.

5.3 Example 3: an ELD problem in a practical system

Problem There are five units in a large power plant. The units’ coal consumption modificationcurves can be expressed as

y1 = −1.614 × 10−7x31 + 4.39 × 10−4x2

1 − 3.265 × 10−1x1 + 3.84 × 102,

y2 = 0.0003x22 − 0.3651x2 + 404.0631,

y3 = −2.7719 × 10−7x33 + 5.6722 × 10−4x2

3 − 4.0829 × 10−1x3 + 4.1188 × 102,

y4 = 3.2044 × 10−7x34 − 2.1458 × 10−4x2

4 − 9.8175 × 10−2x4 + 3.7007 × 102,

y5 = 7.9055 × 10−7x35 − 7.0732 × 10−4x2

5 + 5.1926 × 10−2x5 + 3.6118 × 102,

where xi is the load (in MW) and yi is the coal consumption (in g/KW·h). The curves arestandard and some factors have been eliminated that affect coal consumption, such as watertemperature, vacuum deviation, valve leakage, etc. The rated load of each unit is 600 MW andthe safe operation range is 40–100% of the rated load.

Table 6. Optimal dispatch results computed by IMSCOA.

Total Total coalload x1 x2 x3 x4 x5 consumption(MW) (MW) (MW) (MW) (MW) (MW) (g/KW·h) t̄ (s)

2374.63 350.4952 544.0614 499.0359 450.6330 530.4044 1544.98 16.562335.1748 535.2386 418.2626 526.8689 559.0851 1545.37350.4952 544.0614 499.0359 450.6330 530.4044 1544.98

2469.94 374.1041 556.9930 513.2002 530.9798 494.6628 1537.78 15.625374.1041 556.9930 513.2002 530.9798 494.6628 1537.78396.1672 512.9677 450.0004 598.2448 512.5600 1538.56

2718.88 418.8551 601.9837 583.0215 575.3587 539.6609 1528.15 15.640384.4118 570.9630 602.3887 589.8756 571.2409 1529.43450.4324 586.1229 566.3803 548.6060 567.3384 1528.48

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

An improved chaos optimization algorithm 981

We computed the load dispatch condition at three time points on 1 September 2005. TheIMSCOA parameters are as follows: k = 1000, β = 1, m = 10, γ = 0.35, α = 0.21, N1 =600 and N2 = 500. The optimization process for the optimal load dispatch was performedthree times. The results are listed in table 6.

It can be seen that the total coal consumption calculated using IMSCOA is much lessthan the plant’s actual coal consumption (the actual coal consumption of the power plant was1625.22, 1619.00 and 1597.15 on the three occasions measured) and about 4–5% energy couldbe saved (however, more constraints should be considered, therefore this value would actuallybe smaller). Moreover, the results computed by the proposed algorithm are stable and the timerequired could be reduced by modifying some of the parameters. The results suggest that thealgorithm would be very promising for use in real-time load dispatch.

6. Conclusion

An improved chaos optimization algorithm has been applied to examples of the economic loaddispatch problem. The results show that, regardless of the scale of the problem, and regardlessof how many nonlinear constraints or units there are, the proposed algorithm can obtain resultsthat are more optimal than those obtained by other algorithms, including MSCOA and CROA.CROA is also an effective algorithm but, compared with IMSCOA, it is less practical in actualuse. The application of IMSCOA in economic load dispatch makes economic sense as energyis scarce. IMSCOA is highly efficient and can be applied to many power system problems,such as economic operation, optimal power flow, system identification and optimal control.The application of the chaos optimization algorithm in power systems will bring enormouseconomic benefit. Moreover, the proposed algorithm can be applied in other engineering fields.

However, there are more complicated problems in economic dispatch that need to be solvedusing the chaos optimization algorithm. For example, there are more complicated constraints inthe unit commitment problem, such as unit minimum starting up/down time constraints, ramprate limits and network operation constraints. How to solve these problems using IMSCOA isthe subject of future research.

Acknowledgements

The authors gratefully acknowledge the contributions of Shuangxin Wang for her work onthe original version of this document and Wuchao Qi for his help and suggestions during theprocess of modification. The hard work of the reviewers is also greatly appreciated.

References[1] Wang, J.Q., Zhang, B.M. and Zhao, Z.C., 1996, Optimal unit commitment based on dynamic comparison method.

Proceedings of the CSEE, 16, 254–256.[2] Chen, H.Y. and Wang, X.F., 1999, A survey of optimization-based methods for unit commitment. Automation

of Electric Power Systems, 23, 51–56.[3] Wu, J.H., Wu, Y.W. and Xiong, X.Y. et al., 2003, Random tabu search with simulated annealing approach for

the unit commitment problem. Power System Technology, 27, 35–38.[4] Cai, X.G. and Chu, Z., 2003, Unit commitment based on genetic algorithms. Power System Technology, 27,

36–39.[5] Hou, Y.H., Xiong, X.Y. and Wu, Y.W. et al., 2003, Economic dispatch of power systems based on generalized

ant colony optimization method. Proceedings of the CSEE, 23, 59–64.[6] Jiang, C.W., Quan, X.Z. and Zhang, Y.C., 1999, A chaotic optimization method for economical operation of

hydro power plants. Journal of Huangzhong University of Science and Technology, 27, 39–40.[7] Xu, H.P., Zhu,Y. and Zhang, T. et al., 2000, Application of mutative scale chaos optimization algorithm in power

plant units economic dispatch. Journal of Harbin Institute of Technology, 32, 55–58.

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13

982 F. Han and Q.-S. Lu

[8] Tang, W. and Li, D.P., 2000, Chaotic optimization for economic dispatch of power systems. Proceedings of theCSEE, 20, 36–40.

[9] Luo, J.S. and Zhang, C., 1990, In: H. Wu (Ed.) Optimal Operation of Power Systems (Huazhong University ofScience and Technology Press).

[10] Lou, S.H., Wu, Y.W. and Xiong, X.Y., 2005, Mutative scale chaos optimization algorithm for reactive poweroptimization of power system. Power System Technology, 29, 20–24.

[11] Zhang, T., Wang, H.W. and Wang, Z.C., 1999, Mutative scale chaos optimization algorithm and its application.Control and Decision, 14, 285–288.

[12] Chen, Z.F., Shi, H.Y. and An, Y.J. et al., 2003, Globally convergent approach based on chaotic theory forunderwater robot motor optimization. Paper presented at the International Conference on Robotics, IntelligentSystems and Signal Processing, Vol. 2, pp. 996–1001.

[13] Yuan, Z.Q., Hou, Z.J. and Jiang, C.W., 2003, Economic dispatch and optimal power flow based on chaoticoptimization. Proceedings of IEEE, 2313–2317.

[14] Nidul, S. and Purkayastha, B., 2004, PSO embedded evolutionary programming technique for non-convexeconomic load dispatch. IEEE PES on Power Systems Conference and Exposition, 10, 66–71.

Dow

nloa

ded

by [

Uni

vers

ity O

f Pi

ttsbu

rgh]

at 0

2:32

16

Apr

il 20

13