Comparison of different chaotic maps in particle swarm optimization algorithm for long-term cascaded hydroelectric system scheduling

Chaos, Solitons and Fractals 42 (2009) 3169–3176

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Comparison of different chaotic maps in particle swarm optimization algorithm for long-term cascaded hydroelectric system scheduling Yao-Yao He, Jian-Zhong Zhou *, Xiu-Qiao Xiang, Heng Chen, Hui Qin School of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

a r t i c l e

i n f o

Article history: Accepted 9 April 2009

Communicated by: Prof. Ji-Huan He

a b s t r a c t The goal of this paper is to present a novel chaotic particle swarm optimization (CPSO) algorithm and compares the efficiency of three one-dimensional chaotic maps within symmetrical region for long-term cascaded hydroelectric system scheduling. The introduced chaotic maps improve the global optimal capability of CPSO algorithm. Moreover, a piecewise linear interpolation function is employed to transform all constraints into restrict upriver water level for implementing the maximum of objective function. Numerical results and comparisons demonstrate the effect and speed of different algorithms on a practical hydro-system. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction As a kind of complex dynamic behavior of nonlinear systems, chaos has been widely appreciated within the last decade or so [1]. Currently, it has raised enormous interest in different fields of sciences, such as chaos control, synchronization, pattern recognition, optimization theory and so on. In random-based optimization algorithms, the methods using chaotic variables instead of random variables are called chaotic optimization algorithm (COA) [2–7]. Numerical results indicate that COA can more easily escape from local optima for comparing with other stochastic optimization algorithms although it is not mathematically proved yet [8]. However, all of COAs need appropriate initial value due to their sensitivity to the initial conditions. Thus, several scholars try to combine other algorithms with chaotic search to improve the performance of COA. Particle swarm optimization (PSO) is a population-based stochastic evolutionary technique developed in recent years [9– 11]. Due to the simple framework, easy implementation and quick convergence, PSO has attracted much attention in various fields. However, it is easy to be trapped in local optimal point when dealing with some multi-modality functions [12]. Hence, a hybrid approach named CPSO algorithm is introduced [13,14]. Its ability to escape from local optima is better than simple PSO for keeping the balance between global search and local search dynamically. Since 2004, CPSO algorithms have been successfully used in nonlinear optimization [13,14,18], hydro-system scheduling [15], chaos synchronization [16], parameter estimation [17], economic dispatch [19], mechanical engineering design [20] and so on. The aim of long-term scheduling in a cascaded hydro-system is to distribute throughout a long period of time, typically a year, the hydroelectric generation to all hydro plants so as to maximize the total benefit of the hydro generated energy while satisfying various complex constraints. Considering the characteristics of the long-term cascaded hydro-system optimal dispatch in a electric power market environment, this paper presents a novel CPSO algorithm to solve the model of long term scheduling and compares the efficiency of different one-dimensional iterative chaotic maps within symmetrical region. The proposed algorithm introduces three symmetrical region chaotic maps and piecewise linear interpolation function (PLIF)into CPSO algorithm for solving the long term cascaded hydro-system scheduling and is tested on the Three Gorges cascaded hydroelectric system of China.

* Corresponding author. Tel.: +86 27 87543127. E-mail addresses: [email protected] (Y.-Y. He), [email protected] (J.-Z. Zhou). 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.04.019

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The paper is organized as follows. Section 2 is the introduction of three symmetrical region chaotic maps. The model of the long-term cascaded hydro-system scheduling is described in Section 3. Then in Section 4, the novel CPSO algorithm and PLIF is presented. Section 5 shows the numerical results to different maps in a practical hydro-system. At last, the conclusions and future research are outlined in Section 6. 2. Symmetrical region chaotic maps 2.1. Logistic map As a well-known chaotic map, the simplest logistic map was introduced by Robert May in 1976 [21]. It is often cited as an example of how complex behavior can arise from a simple deterministic dynamic system without any stochastic disturbance. This map is written as:

yðnþ1Þ ¼ c  yðnÞ ð1  yðnÞ Þ for 0 < c 6 4; yðnÞ 2 ð0; 1Þ

ð1Þ

in which c is a control parameter and determines whether chaotic variable y stabilizes at a constant value. Setting yðnÞ ¼ ðxðnÞ þ 1Þ=2 when c ¼ 4, the Eq. (1) is changed to xðnþ1Þ ¼ 1  2ðxðnÞ Þ2 . Suppose a control parameter r, then we can define improved logistic map as [22]:

xðnþ1Þ ¼ 1  rðxðnÞ Þ2

for 0 < r 6 2; xðnÞ 2 ð1; 1Þ;

ð2Þ

where the distribution of y with different c and x with different r are depicted in Fig. 1. 2.2. Chebyshev map Chebyshev map is a common symmetrical region chaotic map, and finds wide application as for example neural network, digital communication and security problems [23–25]. At the same time, it is also a polynomial map with degree k is defined as:

xðnþ1Þ ¼ cosðk  arccosðxðnÞ ÞÞ for k > 0; xðnÞ 2 ½1; 1: ð0Þ

ð1Þ

2

ð2Þ

ð3Þ ðnþ1Þ

ðnÞ

ðn1Þ

The recurrent formulas are x ¼ 1; x ¼ x; x ¼ 2x  1;    ; x ¼ 2x  x  x . Especially, the distinction between Chebyshev map and improved logistic map is only a minus sign while k = 2 and r = 2. Thus, we will only consider degree 3–10 of Chebyshev map from 3 to 10 in Section 5. 2.3. ICMIC map The iterative chaotic map with infinite collapses (ICMIC) within symmetrical region ½1; Þ [ ð0; 1 is proposed in [26]. It has infinite fixed points in comparison with above-mentioned finite collapses one-dimensional maps[27]. The ICMIC map is described by following equation:

 a  for a > 0; xðnÞ 2 ½1; 0Þ [ ð0; 1; xðnÞ

ð4Þ

Logistic map

1

Improved logistic map

1

0.9

0.8

0.8

0.6

0.7

0.4

0.6

0.2

0.5

x

y

xðnþ1Þ ¼ sin

0

0.4

−0.2

0.3

−0.4

0.2

−0.6

0.1

−0.8

0

−1 0

1

2

c

3

4

0

0.5

1

r

Fig. 1. The distribution of y with different c and x with different r.

1.5

2

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Chebyshev map 0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

x

1

x

1

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1 0

5

10

−1 0

ICMIC map

1

k

2

3

4

a

Fig. 2. The distribution of x with different k and a.

where the distribution of x with different k and a are depicted in Fig. 2. 3. Model of long-term scheduling in a cascaded hydro-system 3.1. Objective function Considering the flood and dry power price in an electric market environment, the object of long-term cascaded hydro-system scheduling is to maximize the total generation benefit in a long period under lots of complex constrained conditions. The model of objective function is

max

N X T X i¼1

Ai Q i ðtÞHi ðtÞCðtÞ

ð5Þ

t¼1

where N is the number of hydro plants; T is the length of term; Ai is the power generation coefficient of hydro plant i; Q i ðtÞ; Hi ðtÞ are the water discharge and net head of hydro plant i at time t, respectively; CðtÞ is the power price at time t. 3.2. Constrained conditions 1. Water dynamic balance equation with travel time



V 1 ðt þ 1Þ ¼ V 1 ðtÞ þ q1 ðtÞ  Q 1 ðtÞ  S1 ðtÞ V i ðt þ 1Þ ¼ V i ðtÞ þ qi ðtÞ þ Q i ðt  sÞ  Q i ðtÞ  Si ðtÞ

ð6Þ

2. Hydro plant power generation limits

Pi;min 6 Pi ðtÞ 6 P i;max

ð7Þ

3. Reservoir storage volumes upriver water level limits

Z ti;min 6 Z i ðtÞ 6 Z ti;max

ð8Þ

4. Hydro plant discharge limits

Q i;min 6 Q i ðtÞ 6 Q i;max

ð9Þ

5. Initial and terminal water level volumes

Z i ð0Þ ¼ Z 0i

Z i ðTÞ ¼ Z Ti

ð10Þ

In the equations above, s is water travel time; V i ðtÞ; qi ðtÞ; Si ðtÞ are reservoir storage volume, natural inflow into reservoir and water spillage of hydro plant i at time t, respectively; P i;min ; Pi;max ; P i ðtÞ are minimum, maximum power generation of hydro plant i and power generation of hydro plant i at time t; Z ti;min ; Z ti;max ; Z i ðtÞ are minimum, maximum upriver water level

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and upriver water level of hydro plant i at time t; Q i;min ; Q i;max ; Q i ðtÞ are minimum, maximum water discharge of hydro plant i and water discharge of hydro plant i at time t. 4. A novel CPSO algorithm for cascaded hydro-system scheduling 4.1. Simple PSO PSO algorithm, first introduced by Kenny and Eberhart [9], is one of evolutionary computation technique that includes several individuals. As a population-based optimization technique, all individuals called particle change theirs positions with time. In a PSO system, particles fly around in the problem search space to look for the optimal solution. During flight, each particle adjusts its position according to its own ”experience” as well as the experience of neighboring particles. Let x and v denote a particle position and its corresponding flight velocity in the d-dimensional search space, respectively. v i ¼ ðv i1 ; v i2 ; . . . ; v id Þ stands for the velocity of the i-th particle. xi ¼ ðxi1 ; xi2 ; . . . ; xid Þ indicates the position of the i-th particle of population, and pi ¼ ðpi1 ; pi2 ; . . . ; pid Þ represents the best position of the i-th particle. The global best position among all the particles is represented as pg ¼ ðpg1 ; pg2 ; . . . ; pgd Þ. The updates of velocity and position of the i-th particle can be shown according to the following simple PSO formulas:

(

v tþ1 ¼ xt  v tij þ c1  r 1  ðptij  xtij Þ þ c2  r 2  ðptgj  xtij Þ ij tþ1 xij ¼ xtij þ v tij ; i ¼ 1; 2; . . . ; M; j ¼ 1; 2; . . . ; d; t ¼ 0; 1; . . . ; gen:

ð11Þ

where M denotes the number of particles in population; gen is the number of iterative generation; c1 ; c2 are positive constants referred as acceleration constants; r 1 ; r 2 are uniformly distributed random numbers in the range of [0,1]; x is called the inertia factor described by following equation:

x ¼ xmax 

xmax  xmin gen

t

ð12Þ

where x decreases linearly from xmax to xmin in a run. 4.2. CPSO algorithm using symmetrical region chaotic maps CPSO algorithms mainly utilize the ergodicity and pseudo-randomness of chaotic sequences instead of stochastic sequences rather than the unique properties of chaos so as to implement the global optimization. However, almost all CPSO algorithms choose the chaotic map distributed in (0,1) to carry out exploration. This paper tries to apply chaotic maps within symmetrical region referred by Section 2 and compares their capability in complex engineering problems. The procedure of novel approach can be summarized as follows: Step 1 Step 2 Step 3

Set the iteration counter t ¼ 0 and initialize a population including M particles with random position in the ddimensional space and the velocity of particles as a random value in the range of 20% search space. Evaluate the value of objective function for all particles F i , update the individual maximal fitness F pi the global maximal fitness F pg , the individual optimal position pi and the global optimal position pg . Identify each particle whether inactive or not by following equation

DF i ¼ ðF pi  F i Þ=F i

Step 4 Step 5 Step 6

Step 7

if DF i < d is satisfied for T c times in succession then renew the velocity and position; else go to step 4 directly, where the d is a predefined critical constant according to the precision requirement and the T c is a predefined count constant. Calculate the inertia weight factor x using Eq. (12). , then v ij ¼ v max ; if v ij < v min , then Update the velocity and position of all particles according to Eq. (11). If v ij > v max j j j max max min min v ij ¼ v min ; if x > x , then x ¼ x ; if x < x , then x ¼ x . ij ij ij ij j j j j j Initialize the chaotic variable cxð0Þ ¼ r and ½1; 1 except several fixed points in conformity with different maps ð0Þ ð0Þ within symmetrical region. Furthermore, set l ¼ 1; n ¼ 0, then use medial variable zj and let zj ¼ pg . The maximum number of chaotic iteration is C max . In order to improve the computing efficiency, employ chaotic search with little probability in the former phase of evolvement due to quick convergence of PSO, and use chaotic search in all probability in the latter phase of evolvement for easily trapping in local optimal point. The probability to progress chaotic search may denote as follows:

P t ¼ 1  1=ð1 þ lnðtÞÞ ðnÞ jzj j

Step 8 Step 9

ð13Þ

ð14Þ

if < Pt , then start step 8; otherwise go to step 12. Begin chaotic search using chaotic maps within symmetrical region The chaotic variables, which are proportional to inertia factor x, alter each component nearby the optimal position by following equation

Y.-Y. He et al. / Chaos, Solitons and Fractals 42 (2009) 3169–3176 ðnþ1Þ

zj

ðnþ1Þ

if zj Step Step Step Step

10 11 12 13

ðnÞ

ðnþ1Þ

¼ zj þ xðtÞ  cxj

ðnþ1Þ

> xmax , then zj j

3173

ð15Þ ðnþ1Þ

¼ xmax ; if zj j

ðnþ1Þ

< xmin , then zj j

¼ xmin . j

ðnþ1Þ zj

Let pgj ¼ and compute the function value F, if F > F pg , then F pg ¼ F. If n < C max , then n = n + 1, and go back to step 7; otherwise go to step 12. If l < d, then l = l + 1, and go back to step 7; otherwise go to step 13. Update the iteration counter by t = t + 1 and loop to step 2 until t = gen.

4.3. The implement of objective function In order to solve the long-term cascaded hydro-system optimal dispatch with complex constrained conditions, we try to transform all constraints into restrict upriver water level. For this purpose, the calculation of objective function is summarized as follows: Step 1

Consult the known table of reservoir storage volumes V i ðtÞ about upriver water level Z i ðtÞ, then calculate the reservoir storage volumes using following PLIF:

V i ðtÞ ¼

n X

v j lj ðZi ðtÞÞ

ð16Þ

j¼1

8 > < ðZ i ðtÞ  zj1 Þ=ðzj  zj1 Þ; zj1 6 Z i ðtÞ 6 zj ðj–0Þ lj ðZ i ðtÞÞ ¼ ðZ i ðtÞ  zjþ1 Þ=ðzj  zjþ1 Þ; zj 6 Z i ðtÞ 6 zjþ1 ðj–n Þ > : 0 Z i ðtÞ 2 ½a; zj1  [ ½zjþ1 ; b

Step 2 Step 3 Step 4

ð17Þ

in which lj is the basis function of PLIF; ðzj ; v j Þ; ðj ¼ 0;    ; n Þ is relevant value in the table of reservoir storage volumes about upriver water level; a; b are the boundary values of upriver water level. Calculate the hydro plant discharge Q i ðtÞ according to the Eq. (6) and foregoing results. Consult the known table of downriver water level Z xi ðtÞ about hydro plant discharge, then compute the downriver water level using new variables Z xi ðtÞ; Q i ðtÞ instead of the variables V i ðtÞ; Z i ðtÞ in PLIF Eqs. (16), (17). Endue the Eq. (5) with the net head Hi ðtÞ ¼ Z i ðtÞ  Z xi ðtÞ to solve objective function.In above-mentioned process, we adopt boundary value when the variable go beyond the boundary. So all constraints can be transformed into restrict upriver water level.

5. Numerical experiments 5.1. The parameters of model The model is applied in the Three Gorges cascaded hydro-system of China including Three Gorges hydro power plant and Gezhou dam hydro power plant. Assume that the flood and dry power price is employed in this area. The length of term is one year, and can been subdivided into 12 stages of 1 month each. According to the natural inflow into reservoir, assume that 1st–4th, 12th month are dry water period with 0.45 RMB per KW h; 5th, 11th month are normal water period with 0.3 RMB per KW h; 6th–10th month are flood water period with 0.225 RMB per KW h. Moreover, Table 1 lists the actual main parameters in the Three Gorges cascaded hydro-system. 5.2. Numerical results and comparisons In our experiment, the population size is set to 30. The inertia weight factor x linearly decreases from xmax ¼ 1:2 to xmin ¼ 0:2 using c1 ¼ c2 ¼ 2. The maximum generation of PSO iteration is set to 1000 with d ¼ 0:001; T c ¼ 5, and the maximum iterative number of chaotic search is 100. For the purpose of comparison, we call the CPSO algorithm using Eq. Table 1 The parameters of the three Gorges cascaded hydro-system. Parameters

Three Gorges

Gezhou dam

Hydro plant discharge range (m3/s) Downriver water level range (m) Power generation coefficient Upriver water level range (m) Power generation range (MW) Initial water level (m)

[1580,98800] [63,71.8] 8.5 [145,175] [4990,18200] 175

[3200,86000] [38,58.63] 8.4 [63,66.5] [946,2715] 64.5

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(1) CPSO-L algorithm (initial value of chaotic variables generate randomly in (0,1)). Three CPSO algorithms using Eqs. (2)–(4) within symmetrical region are called CPSO-IL, CPSO-C, CPSO-ICMIC algorithms, respectively.

Table 2 Contrastable generate electricity benefit of the Three Gorges cascaded hydro-system of China in a period of flood year. Optimization algorithm Simple PSO CPSO-L(c = 4) CPSO-IL(r = 2) CPSO-C(k = 3) CPSO-C(k = 4) CPSO-C(k = 5) CPSO-C(k = 6) CPSO-C(k = 7) CPSO-C(k = 8) CPSO-C(k = 9) CPSO-C(k = 10) CPSO-ICMIC(a = 2)

Mean time (s)

Max

Min

Mean

Std

16.370 197.157 45.990 58.856 68.062 62.796 66.335 57.951 62.384 63.304 64.065 54.813

(Billion RMB) 39.884 40.449 41.427 41.476 41.594 41.758 41.910 41.673 41.719 41.613 41.330 41.113

39.714 39.775 39.714 40.145 39.658 39.992 39.074 39.914 39.401 39.390 39.583 39.708

39.871 40.005 40.732 40.764 40.667 40.710 40.577 40.672 40.611 40.791 40.748 40.569

0.03613 0.12669 0.41101 0.36422 0.43898 0.47680 0.64098 0.49496 0.47717 0.55040 0.40208 0.34795

Table 3 Contrastable generate electricity benefit of the Three Gorges cascaded hydro-system of China in a period of dry year. Optimization algorithm

Mean time (s)

Max

Min

Mean

Std

Simple PSO CPSO-L(c = 4) CPSO-IL(r = 2) CPSO-C(k = 3) CPSO-C(k = 4) CPSO-C(k = 5) CPSO-C(k = 6) CPSO-C(k = 7) CPSO-C(k = 8) CPSO-C(k = 9) CPSO-C(k = 10) CPSO-ICMIC(a = 2)

16.504 210.772 45.972 62.724 58.436 61.569 67.976 62.986 61.932 66.274 61.882 47.321

(Billion RMB) 22.033 22.207 23.774 23.425 23.545 23.863 23.296 23.343 23.673 24.223 23.323 23.372

21.736 21.549 21.802 21.391 21.877 21.808 21.912 21.897 21.907 21.557 21.881 21.582

21.982 21.969 22.443 22.450 22.530 22.620 22.443 22.410 22.387 22.459 22.410 22.564

0.08515 0.12776 0.40705 0.40379 0.48575 0.54519 0.33324 0.33167 0.35873 0.49847 0.37575 0.42197

10

4.2

x 10

4.15

Simple PSO CPSO−L(c=4) CPSO−C(k=6)

Best fitness

4.1 4.05 4 3.95 3.9 3.85 3.8

0

200

400

600 Iteration

800

Fig. 3. Comparison of the algorithms in a flood year.

1000

Y.-Y. He et al. / Chaos, Solitons and Fractals 42 (2009) 3169–3176

3175

10

2.45

x 10

2.4

Simple PSO CPSO−L(c=4) CPSO−C(k=9)

Best fitness

2.35 2.3 2.25 2.2 2.15 2.1 2.05 0

200

400

600

800

1000

Iteration Fig. 4. Comparison of the algorithms in a dry year.

For eliminating the discrepancy of stochastic search, each algorithm will run 30 times independently. Tables 2 and 3 show the numerical results by applying simple PSO; CPSO-L(c = 4), CPSO-IL(r = 2), CPSO-C(k = 3-10), CPSO-ICMIC(a = 2) algorithms, including the mean time of 30 runs, maximum (best value), minimum, mean value, standard deviation (Std). From Table 2, it can be seen that the maximum is obtained by CPSO-C(k = 6) as compared with the others in a period of flood year. Table 3 show the maximum obtained by CPSO-C(k = 9) in a period of dry year. Moreover, the results of two tables display three CPSO algorithms using chaotic maps within symmetrical region outperform simple PSO and CPSO-L algorithms although the simple PSO algorithm employs the fastest speed and minimal Std. Comparison of mean values show the discrepancy of them are unconspicuous. But the CPSO algorithm using improved logistic map is the fastest among algorithms using chaotic maps within symmetrical region. Due to the unconspicuous discrepancy of CPSO algorithms using chaotic maps within symmetrical region, we depict the best results to compare effect with PSO and CPSO-L algorithms in Figs. 3 and 4. Apparently, the proposed new algorithm achieves a better performance regarding computation effect. Furthermore, a comparison between Figs. 3 and 4 shows the increased generate electricity benefit of the Three Gorges cascaded hydro-system in a dry year is better than the benefit in a flood year. 6. Conclusions and future research Contraposing the model of long-term scheduling in a cascaded hydro-system, this paper introduces a novel CPSO algorithm using chaotic map within symmetrical region, and compares the effect and speed of improved logistic map, Chebyshev map, ICMIC map. By employing three symmetrical region chaotic maps, the numerical results demonstrate similar calculation effects and discover the improved logistic map is the fastest. At the same time, it is easy to detect that all CPSO algorithms using chaotic map within symmetrical region are more effective than PSO and CPSO-L algorithms. The future research is to consider a CPSO algorithm combined with other methods for solving multi objective scheduling in a cascaded hydro-system. Acknowledgements This paper is supported by Special Research Foundation for the Public Welfare Industry of the Ministry of Science and Technology and the Ministry of Water Resources (No. 200701008), National 973 Program of China (No.2007CB714107), and the National Natural Science Foundation (No.50579022). References [1] [2] [3] [4]

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