Solving Dynamic Economic Dispatch With Modified PSO Algorithm ...

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Solving Dynamic Economic Dispatch With Modified PSO Algorithm Considering Valve Point Effects And Ramp Rate Limits M. Hamed, B. Mahdad, K. Srairi and N. Mancer Department of Electrical Engineering, Biskra University, Biskra, 07000 Algeria email: [email protected]

Abstract: The practical dynamic economic dispatch (DED),

methods enable to obtain the optimal solution and these

with consideration of valve-point effects, and ramp up, ramp

method leads to suboptimal solution [6].

down generators constraints considered as a complicated non-linear constrained optimization problem. In this paper,

New techniques are being used in the last years to tackle

a new variant swarm optimization based time varying

the (DED) problem in a more efficient and quality

acceleration (PSO-TVAC) proposed to solve this problem.

convergence. A lot of works are studied and reported in

This algorithm has been compared and found to be superior

literature, recently evolutionary algorithm is applied by G.

compared to the results of classical (PSO) and (GA) method

Ching et al. to solve dynamic economic dispatch with

in term of solution quality and convergence characteristic.

energy saving and emission reduction [7]. Ivatloo et al. proposed time varying acceleration coefficients IPSO for

Key words: Dynamic economic dispatch, Non-smooth cost function, PSO-TVAC, GA, Valve point effect, Ramp rate limits.

solving dynamic economic dispatch with non-smooth cost function [6.] A group search optimizer with multiple producers

1. INTRODUCTION

algorithm is treated by C.X. Guo et al. to solve the

Dynamic economic dispatch (DED) is one of the

dynamic economic emission dispatch problem [8].The

important power system optimization problems which is a

artificial immune system algorithm is proposed by S.

non-linear

optimization

Hemamalini et al. to solve the dynamic economic

problem. (DED) is a method to dispatch the generating

dispatch for units with valve point effect [2]. A hybrid

units to the predicted load demands over a certain period

algorithm approach based on sequential combination of

of time at minimum operating cost while satisfying

(GA) and active power optimization using Newton’s

equality and inequality constraints[1-2]. There were many

second order approach is presented by T. Nadeem Malik

methods applied to solve the dynamic economic dispatch,

et al to solve economic dispatch problem with valve point

such as dynamic programming [3], linear programming

effect [9]. The Hopfield neural network method is applied

[4], Lagrange relaxation method [5], but the nonlinearity

too by A.Y. Abdelaziz et al to solve this problem [10].

and discontinuity of the search space makes all these

Mahdad and Srairi [18] proposed a combined method

and

complicated

dynamic

1


based GA-DE-PS to solve practical economic dispatch

electrical network test system using 5 and 10 unit test

considering power losses and valve point effects.

system.

It can be seen that recently the meta-heuristic optimization methods have been significantly used in (ED) and considered as an alternative to the classical methods, primarily due to theirs nice feature of population-based search. Particle swarm optimization is

2. Problem Formulations 2.1 Objective Function The objective function of (DED) problem is to minimize the total production cost over the operation period, which can be written as :

such a technique. We adopt PSO to handle the complexity and nonlinearity of the problem [11-12]. PSO has several key

advantages

over

other

existing

robustness [11-13]. PSO is easy to implement in computer using

basic

mathematical

and

ng

(1)

t =1 i =1

optimization

techniques in terms of simplicity, convergence speed, and

simulations

T

minTC = ∑∑ Cit ( Pit ) Where Cit is the unit i production cost at time t; number of generation units and

logic

operations, since its working mechanism only involves two fundamental updating rules. PSO also has fewer operators to adjust in the implementation, and it can be flexibly combined with other optimization techniques to

ng is the

Pit is the power output of

it unit at time t. T is the total number of hours in the operation period. The cost function is nonlinear characteristic

which represented by the following

formula: ng

F ( pit ) = ∑ ai + bi pit + ci pit 2

build hybrid algorithms [11-14-15].

i =1

The mechanism of PSO facilitates a better convergence

+ ei + sin( fi ( pit

performance than some other optimization procedures like

min

(2)

− pit ))

computationally

ai ,bi ,ci ,ei , and f Cost generators coefficients. The

expensive evolutionary operations such as crossover and

objective function of the (DED) problem should be

mutation [11]. Unlike the traditional optimization

minimized subject to following equality and inequality

algorithms, PSO is a derivative-free algorithm and thus it

constraints:

is especially effective in dealing with complex and

A.

genetic

algorithms,

which

have

The equality constraints are

nonlinear problems. PSO is more robust to deal with such ng

problems, since it is less sensitive to the nature of the

∑p i =1

objective function in terms of convexity and continuity [16], and the inner working of PSO helps to escape local minima. The robustness of PSO can also be reflected by its less sensitivity to the optimizer parameters as well as

it

= pd ( t )

t = 1.2.3.......T

(3)

,

B.

Inequality constraints:

p mini ,t ≤ pi ,t ≤ p maxi ,t

Which: i=1:ng ,

(4)

the initial solutions to start its iteration process [11]. In this paper, a novel Time Varying Acceleration particle

pi min , pi max are the maximum and the minimum of unit’s production.

swarm optimization (PSO-TVAC) algorithm is proposed to deal with the dynamic economic dispatch problem without considering losses. The effectiveness of the

3. Particle swarm optimization with time varying acceleration coefficients PSO-TVAC

proposed approach is demonstrated on a practical

In this new algorithm based (PSO) the cognitive and social factors are not constant but they are function of

2


generation (time) to explore most all the positions space

Step 1: Initialized of the Population.

research due eliminate the local minima , overcome the

Step 2: Evaluation of the mechanism search of PSO-

non linearity of the equation (2), so it represent then an

TVAC:

challenge advantage. The position and velocity of the ith

1-Velocity equation

particle are modeled by the following equations

2-Position equation

v (t + 1) = w * v (t ) + α1 * rand1 * ( Pi − x (t ))  + α 2 * rand 2 * ( Pb − X (t ))   x (t + 1) = x (t ) + v (t + 1) 

Step 3: if iter< itermax return to the Step 2. (5)

α1 =((c1f −c1i ).iter / itermax ) +c1i  α2 =((c2 f −c2i ).iter / itermax ) +c2i

(6)

Step 4: stock the best cost and their optimal unit generation.

5. Simulation results: Test system 1

Where: x (t ) is the Particle initial position, v (t ) is

First we are applied the proposed approach at 5 unit

Particle initial velocity, v (t + 1) is new particle

test system, the cost coefficients, generators limits and

velocity, x (t + 1) : A new particle position, Pi : Best

load demand in each hour are taken from [6] and

w : Inertia

power generation obtained without considering ramp

iter Iteration number, itermax : Maximum

rate limits. Table. 3 shows the optimal power

0.4 ≤ w ≤ 0.9 , α 1 , α 2 , w are

generation obtained considering ramp rate limits.

local solution, Pb : Best global solution, factor,

depicted in appendix. Table. 1 shows the optimal

iteration number, and

respectively cognitive, Social, and inertia factors.

C1i , C1 f , C2i , C2 f initial and final values of cognitive

Table. 2, table.4 represent a comparison between GA, PSO and the proposed algorithm in each case respectively. 4

and social factors [17].

5.66

x 10

5.65

5.1 Algorithms parameters

GA PSO PSO-TVAC

5.64

5.63

selection and the maximum of iteration are set respectively 16; 50%, 100.The crossover and the mutation set respectively 0.5; 0.15.

C Ost($/h)

1. GA: binary genetic algorithm, the population size;

5.62

5.61

5.6

2. PSO: standard (PSO), the population size; and the

5.59

maximum of iteration are set respectively 16; 100. 5.58

The

coefficients

of

the

equation

(5)

are

constants α1 = α 2 = w = 1 3. PSO-TVAC: the population size; and the maximum of iteration are set respectively 16; 100.

5.57

10

20

30

40

50

60

70

80

90

100

generatipon

Fig.1 Convergence characteristic of GA; PSO and PSOTVAC for 5 unit test system without considering ramp rate limits.

The algorithm of optimization is based on the following steps:

3


Test system 2 To verify the robustness of the proposed approach the

Table. 5 shows the optimal power generation obtained

algorithm has been applied on a large scale network 10

in ten execution solution of dynamic economic

unit test system. All generators data are taken from [6].

dispatch without considering ramp rate limits. Table.7 shows

2400

the

optimal

power

generation

obtained

considering ramp rate limits. Table. 6 and Table. 8 2350

illustrate a comparison between GA, PSO and the

PSO PSO-TVAC GA

proposed algorithm in each case respectively.

C Ost($/h)

2300

Fig. 1 show the characteristic convergence of the three algorithms for solving the dynamic economic dispatch

2250

of 5 units test system without considering ramp rate 2200

limits for active power demand equal 710 MW. Fig 2 shows the optimal solution calculated by the three

2150

algorithms for 10 unit test system considering ramp 2100

10

20

30

40

50

60

70

80

90

100

Generation

Fig.2 Convergence characteristic of GA, PSO and PSOTVAC for 10 units test system considering ramp rate limits

rate limits when the active power demand set 2131MW. The best cost found by the proposed new variant (PSO-TVAC) is better in

Table 1. Optimal solution of 5 unit considering valve point effect without Ramp rate limits based PSO-TVAC

two cases (5 and 10 units) than the optimum value found using PSO and GA in term of solution quality and execution time. Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Pg1 10 30.571 10 40.665 10 35.991 60.359 10 39.451 53.452 10 75 53.45 12.991 10 10 10 35.991 10 53.452 29.45 44.211 37.665 58.572

Pg2 20 20.002 87.66 20 81.118 20 98.54 91.993 98.54 98.539 95.794 112.99 98.54 125 91.992 20 85.752 20 91.993 98.54 98.541 98.54 20 20

Pg3 30 30 112.67 30 112.68 112.67 112.67 112.67 112.67 112.67 174.87 112.67 112.67 112.67 112.67 110.66 112.67 112.67 112.67 112.67 112.67 112.67 30 30

Pg4 120.48 124.91 124.91 209.82 124.79 209.82 124.91 209.82 209.82 209.82 209.82 209.82 209.82 209.82 209.82 209.82 209.81 209.82 209.82 209.82 209.81 209.82 209.82 124.91

Pg5 229.52 229.52 139.76 229.52 229.41 229.52 229.52 229.52 229.52 229.52 229.52 229.52 229.52 229.52 229.52 229.52 139.76 229.52 229.52 229.52 229.52 139.76 229.52 229.52

Cost($) 1244.1 1348.5 1403.8 1592.7 1632 1760.9 1796.3 1800.7 1945 1985.7 2053.2 2105.8 1985.7 1986.4 1800.7 1618.1 1607.1 1760.9 1800.7 1985.7 1897.9 1751 1580.5 1437.6

4


Table 2. Comparison of optimization results: 5 units with valve point effect. Total cost

GA

PSO

PSOTVAC

Min

42905

43640

41881

Mean

43027

43962

42054

Max

43256

44478

42190

Time (s)

2.494

2.191

2.120

Table 3. Optimal solution of 5 unit considering valve point effect with Ramp rate limits based PSO-TVAC Time

Pg1

Pg2

Pg3

Pg4

Pg5

Cost($)

1

10

20

30

120.48

229.52

1244.1

2

30.573

20

30

124.91

229.52

1348.5

3

20.573

20

30

174.91

229.52

1579.4

4

10.665

50

30

209.82

229.52

1617.9

5

10

78.665

30

209.82

229.52

1645.8

6

40

98.665

30

209.82

229.52

1760.4

7

18.125

98.54

70

209.82

229.52

1892.3

8

10

94.665

110

209.82

229.52

1802.3

9

39.451

98.54

112.67

209.82

229.52

1945

10

53.451

98.54

112.67

209.82

229.52

1985.7

11

69.452

98.54

112.67

209.82

229.52

1996.7

12

75

112.99

112.67

209.82

229.52

2105.8

13

53.451

98.54

112.67

209.82

229.52

1985.7

14

39.451

98.54

112.67

209.82

229.52

1945

15

10

91.991

112.67

209.82

229.52

1800.7

16

10

61.991

72.673

205.82

229.52

1919.5

17

10

75.991

32.673

209.82

229.52

1671.7

18

10

98.54

60.125

209.82

229.52

1804.4

19

16

98.54

100.12

209.82

229.52

1862

20

46

105.99

112.67

209.82

229.52

2027

21

29.45

98.541

112.67

209.82

229.52

1897.9

22

10

82.991

72.674

209.82

229.52

1892

23

10

52.991

32.674

201.82

229.52

1680.4

24

10

22.991

30

170.49

229.52

1530.6

Table 4. Comparison of optimization results: 5 units with valve point effect and ramp rate limits Total cost

GA

PSO

PSOTVAC

Min

43708

44525

42941

Mean

44207

44960

44260

Max

44693

45296

44821

Time (s)

24.555

21.467

19.725

5


Table 5. Optimal solution of 10 units considering valve point effect without ramp rate limits based PSO-TVAC time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Pg1 150 150 150 216.99 150 150.39 266.12 315.95 420.73 464.94 469.7 465.26 466 454.34 436.98 150 150 158.91 430.94 469.07 432.35 150 150 165.83

Pg2 135 135 135 135 285.4 433.41 456.07 460 460 460 460 460 460 460 460 352.32 411.34 460 460 460 460 423.95 298.09 135

Pg3 206 280.15 337.91 340 340 340 340 340 340 340 340 340 340 340 333.83 340 340 338.11 340 340 340 339.72 340 340

Pg4 60 60 60 60 60 60 60 60 60 152.68 151.23 300 137.6 60 60 60 60 60 60 147.09 60 60 60 60

Pg5 73 73 163.48 243 232.6 232.59 167.81 188.05 231.28 242.52 240.07 243 243 197.66 73.185 240.14 109 198.98 73.063 243 219.65 242.33 73 73

Pg6 160 160 159.61 160 160 159.6 160 160 160 159.86 160 160 160 160 160 159.54 157.66 160 160 159.92 160 160 158.91 158.54

Pg7 130 129.85 130 129.01 130 130 130 130 130 130 130 129.74 130 130 130 130 130 130 130 129.96 130 130 130 129.64

Pg8 47 47 47 47 47 47 47 47 47 47 120 47 60.399 47 47 47 47 47 47 47.962 47 47 47 47

Pg9 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

Pg10 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55

Cost($) 28120 29602 32794 36100 37621 40767 42406 44025 47257 50695 52512 54358 50742 47271 44026 39181 37537 40788 44036 50712 47270 40780 34339 31165

Table 6. Comparison of optimization results: 10 units with valve point effect Total cost Min Mean Max Time (s)

GA 1006400 1006600 1006800 2.6622

PSO 1005500 1006000 1006300 1.9937

PSOTVAC 1004100 1004400 1004900 2.0423

Table 7. Optimal solution of 10 units considering valve point effect with ramp rate limits based PSO-TVAC time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Pg1 150 150 226.63 303.25 379.87 456.5 456.49 456.5 456.49 456.5 456.5 470 456.5 456.5 379.87 303.25 226.62 303.25 379.88 459.88 456.5 379.87 303.25 226.62

Pg2 135 135 215 295 375 396.8 396.8 396.8 460 460 460 460 396.8 396.8 396.8 316.8 309.53 389.53 396.81 460 396.81 316.81 236.82 222.27

Pg3 205.24 229.37 190.77 185.2 179.78 179.01 200.64 186.83 251.58 331.58 340 340 329.28 303.18 300.9 297.4 317.15 288.52 302.11 340 337.82 257.82 196.29 182.21

Pg4 60 60 60 110 60 60 110 120.52 120.29 121.01 171.01 221.01 180.83 130.83 80.83 60 60 60 60 110 60 60 60 60

Pg5 73 122.87 122.87 73 73 122.93 122.87 172.87 222.87 222.6 233.9 243 222.6 222.6 172.73 172.73 122.94 172.94 222.94 243 222.56 172.73 122.88 73

Pg6 122.45 122.45 122.45 149.65 122.45 122.45 125 122.46 122.45 160 160 160 160 122.79 123.57 142.52 122.45 122.45 123.95 160 160 150.46 122.45 160

Pg7 130 130 129.96 129.59 129.59 130 129.88 130 130 130 129.59 130 130 130 130 130 130 130 130 130 130 130 130 129.59

Pg8 85.312 85.312 115.31 85.312 85.312 85.313 85.313 85.311 85.312 115.31 120 120 90 85.314 85.312 55.312 85.312 85.312 85.316 85.312 85.312 85.312 85.312 55.312

Pg9 20 20 20 20 20 20 20 49.706 20 20 20 20.992 50.992 20.992 50.992 20.992 50.992 20.992 20 28.812 20.002 20 20 20

Pg10 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55

Cost($) 28410 30134 33535 36977 38307 41132 43076 44673 48370 51939 53890 56084 51666 47814 44713 39888 38113 41238 44266 52352 47863 41664 35043 31766

6


Table 8. Comparison of optimization results: 10 units with valve point effect and ramp rate limits Total cost Min Mean Max Time (s)

GA 1029100 1030100 1031600 4.0966

PSO 1027900 1030700 1032800 2.691

6. Conclusion

Data of generators coefficients for 10 unit test system

In this paper, the non-convex dynamic ED problem with valve-point effects and ramp rate limits was solved using new variant based PSO called (PSO-TVAC). To validate the proposed new variant, 5 units and 10 units with practical generator units were considered. Compared with the standard previous approaches such as: GA and PSO, the results showed the effectiveness of the PSOTVAC algorithm in terms of high-quality solution

ng

a

b

c

e

1 2 3 4 5 6 7 8 9 10

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

450 600 320 260 280 310 300 340 270 380

ng

f

p min

p max

UR

DR

1 2 3 4 5 6 7 8 9 10

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

convergence and good computation efficiency.

. Appendix Data of generators coefficients for 5 units test system

ng

a

b

c

e

1 2 3 4 5

0.0080 0.0030 0.0012 0.0010 0.0015

2.0000 1.8000 2.1000 2.0000 1.8000

25.0000 60.0000 100.0000 120.0000 40.0000

100.0000 140.0000 160.0000 180.0000 200.0000

ng 1 2 3 4 5

f 0.0420 0.0400 0.0380 0.0370 0.0350

p min 10 20 30 40 50

p max 75 125 175 250 300

UR

DR

30 30 40 50 50

30 30 40 50 50

Hourly demand for 10 unit test system hour 1 2 3 4 5 6 7 8 9 10 11 12

Hourly demand for 5 unit test system hour 1 2 3 4 5 6 7 8 9 10 11 12

PSOTVAC 1022900 1025400 1027600 2.632

load 410 435 475 530 558 608 626 654 690 704 720 740

Hour 13 14 15 16 17 18 19 20 21 22 23 24

load 704 690 654 580 558 608 654 704 680 605 527 463

load 1036 1110 1258 1406 1480 1628 1702 1776 1924 2072 2146 2220

hour 13 14 15 16 17 18 19 20 21 22 23 24

load 2072 1924 1776 1554 1480 1628 1776 2072 1924 1628 1332 1184

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