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Particle Swarm Optimization Based on Cultural Algorithm

for Short-term Optimal Operation of Cascade Hydropower Stations

Wei Xie, Chang-ming Ji, Xin-wu LiResearch Institute of water Resources and Hydro-Electric Engineering,

Renewable Energy School North China Electric Power University

Beijing, China

[email protected]

AbstractDifficulties likely exist in optimal reservoir operation

along with discreteness and nonlinear constraints. Therefore,

Particle Swarm Optimization based on Cultural Algorithm

(PSO-CA) is presented in this paper for overcoming these

defects. In this article the evolutionary mechanism of Particle

Swarm Optimization Algorithm (PSO) is guided by Cultural

Algorithm (CA). PSO-CA uses PSO in population space and

guides the evolution by shape knowledge and standardizationknowledge in belief space. An example is also used to show that

the PSO-CA algorithm has a better applied prospect for its

high reliability and fast operation speed in global optimization.

Keywords-optimal operation of reservoir; particle swarm

optimization; cultural algorithm;

I. INTRODUCTIONThe short-term optimal operation of cascade hydropower

stations is typical nonlinear and multi-dimensionaloptimization problem with unequal constraints in nature.Because of its dominating characteristics such asmultivariable, highly-complexity and multi-constraint [1], it

is more complicated and difficult to be solved rather thanother nonlinear optimal problems. Systematic methods andalgorithms have been quickly developed exemplified bydynamic programming (DP), large system decompositioncoordination model (LSDC), network flow method [2],genetic algorithm (GA) [3] etc., for their easy realization andsimple programme. However, they werent versatile to settlereservoir regulation, and some defects were exposed in theirapplication, such as dimension disaster and overlongcalculating time of the DP, high complexity and lowconvergence of LSDC caused by coordination factor, areeasily plunged into local optimum of GA caused by lowlimits constructing of object function and constraints.

Particle swarm optimization algorithm [4] (PSO) has

been widely applied to the nonlinear and nondifferentiableproblems of power system [5, 6] as a stochastic globaloptimization technology for its low dependence of objectfunction and constraint conditions, easy use and rapidconvergence. However, some questions still exist. Forexample, PSO algorithm is easy to fall into local optimumand lead to precocity phenomenon as its not a globalconvergence algorithm [7]. As previous research has notgreat perfection on reservoirs operation, Particle swarmoptimization based on cultural algorithm (PSO-CA) isadopted to overcome the non-high precision in the paper,

which may realize searching optimal point on continuousspace by multipoint optimization. Furthermore, aiming toavoid the precocity in the application of cascade hydropowerstations and how to settle them with shape andstandardization knowledge in belief space is discussed in thispaper.

II. ANALYSIS ON PSO-CAALGORITHMA. Principle of PSO

The PSO algorithm is a new optimization algorithm,which derived from researching of bird and fish flockspreying. The algorithm modifies the individual behaviorstrategies by the information of flock and their ownexperiences given by evaluating the fitness of solution, andfinally finds global optimal solution [8].

The main idea of PSO algorithm can be described asfollows: firstly, a swarm of particles is randomly initializedwith a position and velocity. The position which decides theparticle is flying direction can be expressed as an n-dimensional vectorXi= (xi,1, xi,2 xi,n), and the velocity can

be also expressed as Vi= (vi,1, vi,2...vi,n). Each particle has afitness value (Yi) which can be attained when put vector intoobjective function. And each particle do self-renewal followstwo peaks. One is the optimal solution of a particle itself inflying named individual optimal solution Pi

k = (pki,1,pki,2p

ki,n). The other is the optimal solution among all the

particle flying named global optimal solution Pgk

= (pkg,1,

pkg,2p

kg,n)., where kis iteration number and k[1,K], and

K is the biggest generation number in k. During eachiteration, particle will regulate its velocity and then renew itsposition. The movement equations are as follows:

][][ 22111 k

i

k

g

k

i

k

i

k

i

k

i XPrcXPrcVwV ++=+ (1)

11 +++=

k

i

k

i

k

i VXX (2)

Where w is inertia weight coefficient and can alter theparticles searching capacity by the way of adjusting itsvalue. The value ofw is linear degression in the interval [0.1,0.9] along with the increase of iteration number in this paper;r1 and r2 are stochastic numbers between [0, 1]; c1 and c2 aretwo constant values, normally c1=c2=2.

2009 Fifth International Conference on Natural Computation

978-0-7695-3736-8/09 $25.00 2009 IEEE

DOI 10.1109/ICNC.2009.182

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978-0-7695-3736-8/09 $25.00 2009 IEEE

DOI 10.1109/ICNC.2009.182

289


978-0-7695-3736-8/09 $25.00 2009 IEEE

DOI 10.1109/ICNC.2009.182

289


978-0-7695-3736-8/09 $25.00 2009 IEEE

DOI 10.1109/ICNC.2009.182

289


978-0-7695-3736-8/09 $25.00 2009 IEEE

DOI 10.1109/ICNC.2009.182

289


978-0-7695-3736-8/09 $25.00 2009 IEEE

DOI 10.1109/ICNC.2009.182

289


978-0-7695-3736-8/09 $25.00 2009 IEEE

DOI 10.1109/ICNC.2009.182

289


978-0-7695-3736-8/09 $25.00 2009 IEEE

DOI 10.1109/ICNC.2009.182

289


978-0-7695-3736-8/09 $25.00 2009 IEEE

DOI 10.1109/ICNC.2009.182

289


978-0-7695-3736-8/09 $25.00 2009 IEEE

DOI 10.1109/ICNC.2009.182

289


978-0-7695-3736-8/09 $25.00 2009 IEEE

DOI 10.1109/ICNC.2009.182

289


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B. Principle of CAIn human society, culture can be viewed as a vehicle for

the storage of information that is potentially accessible to allthe members of the society, and used in guiding theirproblem solving activities. CA, developed by RobertG.Reynlods in 1994 [9], is used to model the evolution of thecultural component of an evolutionary computational systemover time as it accumulates experience in solving a given setof problems. Culture guides the populations evolution witha certain speed which transcends that only by genes. Theknowledge, collected from identifying patterns in thepopulations problem solving experiences, is used toinfluence the generation of candidate solutions, either topromote more instances of desirable candidates or to reducethe number of less desirable candidates in the population.The framework of CA is depicted in Figure 1.

Different from other evolutionary algorithms, CAcontains two evolutionary spaces. They are population spacethat evolutionary computation methods have and belief spacecomposed by individuals knowledge and experience.Evolution can be viewed as an optimization and as a learning

process through a supporting communication mechanismthat between the two components. Individual Experiencesselected from the population space are used to generateproblem solving knowledge that resides in the belief space.The belief space stores and manipulates the knowledge andin turn influences the evolution of the population component.In this way, the population space and the belief space interactwith each other and support each other.

Acceptance ()

Influence ()

Belief Space

Update ()

Population Space

Evolution ()

Objective ()

Figure 1. Cultural Algorithm framework

C. The PSO-CA AlgorithmCA provides an efficient algorithm framework and each

kind of species evolutionary algorithm can provide thespecies for population space, such as GA, PSO, EC, etc.. Thedefinition of belief space usually adopts shape knowledge(SK) and standardization knowledge (NK), namely structure. In this paper, species are generated by PSO inpopulation space and the evolution of species is leaded by structure in belief space. Although PSO has virtuesadvanced above, it has shortcomings on lower searchprecision and tending to partial optimization. SO shapeknowledge is used to improve the capabilities of researchand standardization knowledge to enhance the computationalefficiencies of the algorithm.

1) Definition and Update of Belief Space: Shapeknowledge has n+1 elements (n is the size of species), the

last element is bg which consists of the history optimal

position (X), the corresponding fitness value (Y) and

continuous not update iteration number (nStaCount) of

species. The othern elements described as b1~bn, and bi is

composed of the history optimal position (Xi), thecorresponding fitness value (Yi) and continuous not update

iteration number (nStaCount) of the ith particle. Data

structure of shape knowledge shown as Figure 2:

1x 2x mx y nStaCount

1b 2b 3b nb gb

Figure 2. Data structure of shape knowledge

When initializing shape knowledge, bi consists of initialposition, the corresponding fitness value of the ith particle

and nStaCount is equal to 0; bg also should use initialposition, the corresponding fitness value of the particlewhich has the largest fitness value in the initial particleswarm and nStaCount also equals 0. Shape knowledge isupdated by Acceptance () function during each iteration.When the current fitness value of a particle is higher thanthat which stores in shape knowledge, replaces the latter withthe former. At the same time, the corresponding nStaCount isequal to 0 otherwise add 1. So does bg.

Standardization knowledge represented asN= means the range of decision variable (Xi) and m is thenumber of that Niexpresses as . AndI= [li, ui] ={Xi| liXiui} which indicates the range ofXi.Li is the fitnessvalue which can be obtained when li is put into the objective

function and its initial value is +. The meaning ofUi is asthe same as Li. Then standardization knowledge is renewedaccording to the probability conditions in (3) under theassumption that the lower limit of the jth particle is affectedby the ith particle and the upper limit is affected by the dthparticle.


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2) Influence on Population Space by the Belief Space:To change the default, adjust the template as follows.

Population space obtains the information from belief space

through Influence () function and these influences include

the following two aspects:Firstly, influence of shape knowledge. It is clear that the

particle will be the changed one when nStaCount is greaterthan staMax, where staMax is a given value. When thequantity of the variable particles reaches , in that case, arandomized crossover and mutation will be carried out tothose ones, where is a supposed ratio. And thecorresponding information in the belief space also should berenewed at one time. If the given individuals of a and bshould be changed, the laws are given as:

k

a

k

b

k

b

k

b

k

a

k

a

XsXsX

XsXsX

+=

+=

+

+

)1(

)1(

1

1

(4)

Wheres is the parameter that obtained by the experience ofexperiment.

Secondly, influence of standardization knowledge.Different particle has different position and velocity and theinfluence for the next flying is also not the same. The nextvelocity is determined by the particles current position, so(1) can be replaced by the (5).

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IV. PSO-CAAPPLICATIONS ON SHORT-TERM OPTIMALOPERATION OF CASCADE HYDROPOWERSTATIONS

Reservoir optimal operation is a continuous and dynamicproblem with multi-time intervals and unequal constraint innature. Reservoir operating rule is signified as powerdischarge sequence in this paper. But in the programmingprocess the power discharge sequence may usuallytransformed into water level change sequence in each storagetime interval. The paper unified PSO-CA theory andreservoir operation together, reservoir water levelscorresponding to particles, amount of scheduling period todimension of searching space, generation object function tofitness function. In that way, one particle can be indicated asmn matrix (Xmn) where m is the number of hydropowerstations and n is the quantity of time interval. Thus thecomponent xit(i=1,2,,m; t=1,2,,n)of the matrix meansthe water level in tth time interval of the ith station. The onlyproblem to be solved is to dispose restriction condition bypunishment function, which shows as the follow [10]:

== =j

j

jj

k

k

kk xgRxhRExF1

2

1

2 )](,0max[)]([)( (11)

Where F(x) is fitness value function and E is the objectivefunction of reservoir optimal operation; hk(x) expressesequation restriction andgi(x) indicates inequation constraint;Rj andRkare weight coefficient.

Reservoir optimal operation calculation steps based onPSO-CA algorithm are represented as the followings:

Step 1: Initialization particle swarm. To shortensearching time of the algorithm, the initial particles and theircorresponding velocities are randomly generated inconstraint.

Step 2: Ascertaining the parameters: constant c1 and c2,

inertia weight coefficient w and the largest number ofevolutionary iterationsK.Step 3: Initialization the corresponding information of

shape knowledge and standardization knowledge accordingto the initial swarm in belief space.

Step 4: Evaluation fitness value of each particle inpopulation space.

Step 5: Updating the belief space by means of Update ()function which gains the messages through Acceptance ()function.

Step 6: Producing a new generation of groups byEvolution () function based on the guidance of belief space.

Step 7: Judgment. Estimating whether the program meetsthe convergence condition or reaches the maximum iterationnumber or not. If it can satisfy the condition, then withdrawit, otherwise return to the step 4.

To understand the process of the program better pleaselook at Figure 3.

V. EXPERIMENT SETUP AND RESULT ANALYSISAn experiment is designed mainly to verify the validity

and reliability of PSO-CA algorithm. This algorithm hasbeen used in two hydropower stations. One (expresses as A)

of the cascade hydropower stations is annual regulatingmode while the other (shows as B) is daily. Here, the deadwater level of A is 172m and B is 69m, the maximumrestriction water levels are not higher than 210m and 90mrespectively. The output of A is in the range of 55~1220MWand B is 20~300MW. The outflow of A is in the range of70~1200m

3/s and B is 20~860 m

3/s. Note that the scheduling

period is 24h and the time interval is 1h. A is 60km awayfrom B thus we set the water flow time i is 2h. Here, theprocess of inflow and the initial water level of the twohydropower stations respectively 185m and 80m have beenknown.

Producinganewg

enerationofgroups

Initialization particle swarm

Initialization belief space

Beliefspace

Calculation the fitness value

of each particle

Updating particle swarm

Produce a new generation of

particle

Whether satisfied the

condition of change?

Crossover and mutation

Whether to meet the

end condition?

End

Begin

Yes

No

Yes

No

Initialization belief space

Updating particle swarm

Guide the evolution

Figure 3. Algorithm process of PSO-CA

Setting parameters: the number of particle swarm is 25,c1=c2=2, StaMax=5, =0.3, nStaCount=10, K=500. In theexperiment the problem has been solved with PSO-CA underthe condition of Visual C++6.0. The results are given in tableI.

It is obvious that better result can be gotten by PSO-CAalgorithm compared with PSO algorithm. The growth rate of

output of A and B, respectively, is 2.7% and 5.42%. In otherwords, the average growth rate of the cascade hydropowerstations is 3.36%.

Figure 4 illustrates the convergence of algorithm duringthe iterative process. It is clear that PSO quickly falls intolocal optimum although its convergence speed is faster thanthat of PSO-CA. Nevertheless the belief space stores andmanipulates the knowledge of species and in turn influencesthe evolution of the population. In that way, PSO-CA can

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jump out the local optimum and eventually reach the globaloptimum.

In terms of the relative diversity which represented asFigure 5, it is higher of PSO-CA than PSO before theiteration is end. This means that the relative diversity ofPSO-CA keeps the high level before finding the optimalvalue through the crossover and mutation of population.

TABLE I. RESULTS OF OPERATION OPTIMIZATION

Hydro-

power

station

Total

water

inflow

per day

/km3

Total water

consumption

per day

/km3

Output

by

PSO-

CA

/MWh

Output

by

PSO

/MWh

Output

growth

rate

/%

A 54324.0 72948.4 18099.0 17623.0 2.70

B 72948.4 71380.3 5990.2 5682.2 5.42

Total 127272.4 144328.6 24089.2 23305.2 3.36

Figure 4. Convergence comparison

Figure 5. Comparison of the relative diversity

VI. CONCLUSIONThere is a variety of mechanisms which guide

evolutionary search within population space. The goal of thispaper is to use a CA to acquire knowledge which isnecessary to control the evolution. In belief space, saving theinformation of outstanding individuals and ensuring the

diversity of particles by shape knowledge to prevent thealgorithm into local optimization; guiding the evolution ofparticle swarm with different selection by standardizationknowledge to increase the speed of convergence.

The experiment has verified that PSO-CA is a veryimportant technique for solving the discreteness andnonlinear problems. It is testified that PSO-CA is a new andeffective optimal method and better solution can be gained.So going to investigate the performance of PSO-CA intracking the optimum on more complicated problems such aslong-term optimal operation of cascade hydropower stationsis suggested.

REFERENCES

[1] Oliveira R, Loucks DP. Operating rules for multireservoir systems.Water Resources Research, 1997, 33(4), pp839-852.

[2] Chang Jianxia, Huang Qiang, Wang Yimin. Optimal operation ofhydropower station reservoir by using an improved genetic algorithm.Journal of hydroelectric engineering, 2001, (3), pp85-90.

[3] Howson H R, Sancho NCF.A new algorithm for the solution ofmultistate dynamic programming problems. Math program, 1975,8(1), pp104-116.

[4] Kennedy J , Eberhart R . Particle Swarm Optimization. Proceedingsof IEEE conference on Neural Networks. Perth, Australia, 1995, 4,

pp1942-1948.

[5] Wang Ni, Wang Shao-bo, Xie Jiangcang. Modified Particle SwarmOptimization And Its Application on Optimal Operation ofHydropower station. Third International Conference on NaturalComputation. Haikou, China: IEEE, 2007, pp549-553.

[6] Huang Qiang, Zhang Hong-bo, Chen Xiao-nan, Lv Yu-jie.Application of Particle Swarm Optimization algorithm to ReservoirOperation.Third International Conference on Natural Computation.Haikou, China: IEEE, 2007, pp595-600.

[7] Van den BerghF. Analysis of Particle Swarm Optimizers. SouthAfrica: University of Pretoria, 2001.

[8] Zeng Jianchao, Jie Jing, Cui Zhihua. Particle SwarmAlgorithm.Beijing: Science publishing company.2004.

[9] Robert G. Reynolds. An introduction to cultural algorithms.Proceeding of the Third Annual Conference on EvolutionaryProgramming. New Jersey: World Scientific, 1994, pp131-139.

[10] Zhang shuang-hu, Huang Qiang, Sun Ting-rong. Study on the optimaloperation of hydropower station based on parallel recombinationsimulated annealing algorithms. Journal of hydroelectric engineering,2004,23(4), pp16-19.

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