Quasi-oppositional group search optimization for multi-area dynamic economic dispatch

Electrical Power and Energy Systems 78 (2016) 356–367

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Quasi-oppositional group search optimization for multi-area dynamic economic dispatch M. Basu ⇑ Department of Power Engineering, Jadavpur University, Kolkata 700098, India

a r t i c l e

i n f o

Article history: Received 18 September 2014 Received in revised form 11 November 2015 Accepted 25 November 2015

Keywords: Quasi-oppositional group search optimization Group search optimization Multi-area dynamic economic dispatch Tie line constraints

a b s t r a c t Multi-area dynamic economic dispatch determines the optimal scheduling of online generator outputs and interchange power between areas with predicted load demands over a certain period of time taking into consideration the ramp rate limits of the generators, tie line constraints, and transmission losses. This paper presents quasi-oppositional group search optimization for solving multi-area dynamic economic dispatch problem with multiple fuels and valve-point loading. Group search optimization (GSO) inspired by the animal searching behavior is a biologically realistic algorithm. Quasi-oppositional group search optimization (QOGSO) has been used here to improve the effectiveness and quality of the solution. The proposed QOGSO employs quasi-oppositional based learning (QOBL) for population initialization and also for generation jumping. The QOGSO is tested on two multi-area test systems having valve point loading and mult-fuel option. Results of the proposed QOGSO approach are compared with those obtained from group search optimization (GSO), biogeography-based optimization (BBO), gravitational search algorithm (GSA), differential evolution (DE) and particle swarm optimization (PSO). It is found that the proposed QOGSO based approach is able to provide better solution. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction Multi-area static economic dispatch (MASED) is one of the important optimization problems in power system operation. Generally, the generators are divided into several generation areas interconnected by tie-lines. MASED determines the generation levels of online generators and interchange power between areas for a load demand which is constant for a given interval of time such that total fuel cost in all areas is minimized while satisfying power balance constraints, generating limits constraints, and tieline capacity constraints. The MASED has been the subject of investigation for several decades. Shoults et al. [1] solved economic dispatch problem considering import and export constraints between areas. This study provides a complete formulation of multi-area generation scheduling, and a framework for multi-area studies. Romano et al. [2] presented the Dantzig–Wolfe decomposition principle to the constrained economic dispatch of multi-area systems. Helmick and Shoults [3] solved multi-area economic dispatch with area control error. Wang and Shahidehpour [4] proposed a decomposition approach for solving multi-area generation scheduling with

⇑ Fax: +91 33 23357254. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijepes.2015.11.120 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.

tie-line constraints using expert systems. Network flow models for solving the multi-area economic dispatch problem with transmission constraints have been proposed by Streiffert [5]. Yalcinoz and Short [6] solved multi-area economic dispatch problems by using Hopfield neural network approach. Jayabarathi et al. [7] solved multi-area economic dispatch problems with tie line constraints using evolutionary programming. The direct search method for solving economic dispatch problem considering transmission capacity constraints was presented in Ref. [8]. Manoharan et al. [9] explored the performance of the various evolutionary algorithms on multi-area economic dispatch (MAED) problems. Here, evolutionary algorithms such as the Real-coded Genetic Algorithm (RCGA), particle swarm optimization (PSO), differential evolution (DE) and Covariance Matrix Adapted Evolution Strategy (CMAES) are considered. Sharma et al. [10] have presented a close comparison of classic PSO and DE strategies and their variants for solving the reserve constrained multi-area economic dispatch problem with power balance constraint, upper/lower generation limits, transmission constraints and other practical constraints. In [11], multi-area economic dispatch problem has been solved by using teaching–learning-based optimization algorithm. Multi-area dynamic economic dispatch (MADED) is an extension of multi-area static economic dispatch problem. It schedules the online generator outputs, and interchange power between

357

M. Basu / Electrical Power and Energy Systems 78 (2016) 356–367

areas with the predicted load demands over a certain period of time so as to operate an electric power system most economically. In order to avoid shortening the life of the equipments, plant operators try to keep gradients for temperature and pressure inside the boiler and turbine within safe limits. This mechanical constraint is transformed into a limit on the rate of increase or decrease of the electrical power output. This limit is called ramp rate limit which distinguishes MADED from MASED problem. Thus, the dispatch decision at one time period affects those at later time periods. MADED is the most accurate formulation of multi-area static economic dispatch problem but it is the most difficult to solve because of its large dimensionality. Further, due to increasing competition into the wholesale generation markets, there is a need to understand the incremental cost burden imposed on the system by the ramp rate limits of the generators. Huda et al. [12] developed a hybrid approach of global and local search for constrained optimization problem. In [13], Huda et al. discussed about good convergence on global mathematical optimization approaches. Group search optimization (GSO) is a biologically realistic algorithm which is inspired by the animal (such as lions and wolves) searching behavior. He et al. [14] proposed GSO in 2006, and discussed the effects of designed parameters on the performance of GSO in 2009 [15]. GSO employs a special framework, under which individuals are divided into three classes and evolve separately. This framework is proved to be effective and robust on solving multimodal problems [15]. Shen et al. [16] investigated the performance of GSO and concluded that GSO is an alternative for constrained optimization. Due to its high efficiency, GSO has been applied to solve nonconvex economic dispatch problem [17], distribution network reconfiguration [18], combined heat and power economic dispatch problem [19], etc. The basic concept of opposition-based learning (OBL) [22–24] was originally introduced by Tizhoosh. The main idea behind OBL is for finding a better candidate solution and the simultaneous consideration of an estimate and its corresponding opposite estimate (i.e., guess and opposite guess) which is closer to the global optimum. OBL was first utilized to improve learning and back propagation in neural networks by Ventresca and Tizhoosh [25], and since then, it has been applied to many EAs, such as differential evolution [26], particle swarm optimization [27] and ant colony optimization [28]. In [29] quasi oppositional based differential evolution has been discussed. The utilization of quasi-oppositional based learning (QOBL) improves the effectiveness and quality of the solution. In this paper, QOBL is implemented on group search optimization (GSO). The quasi-oppositional group search optimization QOGSO employs QOBL for population initialization and also for generation jumping. The proposed QOGSO along with basic GSO is applied to solve MADED problem. Here, two types of MADED problems have been considered. These are (A) multi area dynamic economic dispatch with valve point loading, and transmission losses, (B) multi area dynamic economic dispatch with valve point loading multiple fuel sources, and transmission losses. Test results obtained from QOGSO are compared with those obtained from group search optimization (GSO), biogeography-based optimization (BBO), gravitational search algorithm (GSA), differential evolution (DE) and particle swarm optimization (PSO). Problem formulation The objective of MADED is to minimize the total cost of supplying loads to all areas over a certain period of time while satisfying power balance constraints, generating capacity constraints, ramp rate limits of the generators, and tie-line capacity constraints.

Two different types of MADED problems have been considered here. Multi area dynamic economic dispatch with valve point loading and transmission losses The objective function F c , total cost of committed generators of all areas over T number of intervals in the scheduled horizon considering the valve-point effect may be written as

Fc ¼

Mi T X N X X   F ijt Pijt t¼1 i¼1 j¼1

¼

X

t ¼ 1T

Mi N X X aij

þ bij Pijt þ cij P2ijt

i¼1 j¼1

 n  o   þ dij  sin eij  Pmin  Pijt  ij

ð1Þ

  where F ijt Pijt is the cost function of j th generator in area i at time t. aij ; bij , cij ; dij and eij are the cost coefficients of j th generator in area i; N is the number of areas, M i is the number of committed generators in area i; P ijt is the real power output of j th generator in area i at time t. The MADED problem minimizes F c subject to the following constraints. Real power balance constraint Mi X X Pijt ¼ P Dit þ PLit þ T ikt j¼1

i 2 N;

t2T

ð2Þ

k;k–i

The transmission loss P Lit of area i at time t may be expressed by using B-coefficients as

PLit ¼

Mi X Mi Mi X X P ijt Bilj P ilt þ B0ij Pijt þ B00i l¼1 j¼1

t2T

ð3Þ

j¼1

where P Dit real power demand of area i at time t; T ikt is the tie line real power transfer from area i to area k at time t. T ikt is positive when power flows from area i to area k, and T ik is negative when power flows from area k to area i. Real power generation capacity constraints The real power generated by each generator should be within max its lower limit Pmin , so that ij , and upper limit P ij

Pmin 6 P ijt 6 P max ij ij ;

i 2 N;

j 2 Mi ;

t2T

ð4Þ

Generator ramp rate limits constraints The ramp rate limits of each generator should be within its ramp-up rate limit URij , and ramp-down rate limit DRij , so that

Pijt  P ijðt1Þ 6 URij Pijðt1Þ  P ijt 6 DRij

i 2 N; i 2 N;

j 2 Mi ; j 2 Mi ;

t ¼ 2; 3; . . . ; T t ¼ 2; 3; . . . ; T

ð5Þ

Tie line capacity constraints The tie line real power transfer T ikt from area i to area k at time t should not exceed the tie line transfer capacity for security consideration.

T max 6 T ikt 6 T max ik ik T max ik

where is the power flow limit from area i to area k and the power flow limit from area k to area i.

ð6Þ T max ik

is

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M. Basu / Electrical Power and Energy Systems 78 (2016) 356–367

Multi area dynamic economic dispatch with valve point loading multiple fuel sources and transmission losses Practically generators are supplied with multi-fuel sources. Each generator should be represented with several piecewise quadratic functions superimposed sine terms reflecting the effect of fuel type changes, and the generator must identify the most economical fuel to burn. The fuel cost function of the j th generator in area i with N F fuel types at time t considering valve-point loading is expressed as

  F ijt Pijt ¼ aijm þ bijm Pijt þ cijm P2ijt  n  o   þ dijm  sin eijm  P min  ijm  P ijt

ð7Þ

max if Pmin ijm 6 P ijt 6 P ijm for fuel type m and m ¼ 1; 2; . . . ; N F . The objective function F c is given by

Mi T X N X X   Fc ¼ F ijt Pijt

either for ‘finding’ or for joining opportunities. Based on this framework, concepts from animal scanning mechanisms are employed metaphorically for designing an optimum searching strategy in order to solve continuous optimization problems [15]. The population of the GSO algorithm is called a group and each individual in the population is called a member. During each iteration, a member is defined by its position and head angle. In n-dimensional search space, the i th member of the GSO at the k th searching iteration has a current position X ki 2 Rn and a head   angle /ki ¼ /ki1 ; . . . ; /kiðn1Þ 2 Rn1 . The search direction of the i th     k k member, Dki /ki ¼ di1 ; . . . ; din 2 Rn that can be calculated from /ki via polar to Cartesian coordinate transformation [20]. k

di1 ¼ ð8Þ

The objective function F c is to be minimized subject to the constraints given in (2), (4), (5) and (6).

k din

t¼1 i¼1 j¼1

Determination of generation level of slack generator M i committed generators in area i at time t deliver their power output subject to the power balance constraint (2), the respective generation capacity constraints (4), the respective generator ramp rate limits constraints (5), and tie line capacity constraints (6). Assuming the power loading of first (Mi  1Þ generators are known, the power level of the Mi th generator (i.e. the slack generator) is given by

X

T ikt 

M i 1 X

k;k–i

ð9Þ

Pijt

j¼1

The transmission loss PLit is a function of all generator outputs including the slack generator and it is given by

PLit ¼

M i 1M i 1 X X

Pijt Bilj Pilt þ 2PiMi t

l¼1 j¼1

þ

M i 1 X

M i 1 X

!

BiMi j Pijt

  cos /kiq

q¼1 k dij

PiMi t ¼ PDit þ P Lit þ

n1 Y

n1  Y   ¼ sin /kiðj1Þ cos /kiq j ¼ 2; . . . ; ðn  1Þ

¼ sin



ð12Þ

q¼j

/kiðn1Þ



GSO algorithm consists of three kinds of members, i.e., producers, scroungers and rangers who perform random walk. It is assumed that there is only one producer at each iteration and the remaining members are scroungers and rangers. Here, simplest joining policy is used where all scroungers will join the resource found by the producer. The producer and the scroungers do not differ in their relevant phenotypic characteristics and they can switch between the two roles [21]. At each iteration, a group member, located in the most promising area, adopts animal scanning to seek the optimal resource and the best fitness value is chosen as the producer. A number of group members except the producer are randomly selected as scroungers and then the rest of members are rangers. Scroungers perform area copying to join the resource found by the producer and do local searching around it. Rangers employ ranging behavior by random walk in the searching space to increase the chance of GSO to escape local optima.

þ BiMi Mi P2iMi t

j¼1

B0ij Pijt þ B0iMi PiMi t þ B00i

ð10Þ

Start

j¼1

Expanding and rearranging, Eq. (9) becomes M i 1 X

BiMi Mi P2iMi t þ 2

Generate and evaluate initial members

! BiMi j P ijt þ B0iMi  1 P iMi t

Choose producer and perform producing

j¼1

þ PDit þ

X k;k–i

T ikt þ

M i 1M i 1 X X

M i 1 X

j¼1 l¼1

j¼1

P ijt Bilj P ilt þ

B0ij P ijt 

M i 1 X

! Pijt þ B00i

Choose scrounger and perform scrounging

j¼1

Dispersed the rest members to perform ranging

¼0 ð11Þ

Evaluate members

The loading of the slack generator (i.e. M i th) can then be found by solving Eq. (11) using standard algebraic method. Description of quasi-oppositional group search optimization

Termination Criterion Satisfied?

A brief description of group search optimization Group search optimization (GSO) is a population based optimization algorithm inspired by animal searching behavior and group living theory. The framework is based on the producerscrounger (PS) model which assumes that group members search

Yes Terminate Fig. 1. Flowchart of the GSO algorithm.

NO

M. Basu / Electrical Power and Energy Systems 78 (2016) 356–367

359

Start Generate and evaluate initial members and its quasi-opposite members If fitness value of quasi-opposite member is less than the fitness value of initial member replace the initial member with its quasi-opposite member

Choose producer and perform producing Choose scrounger and perform scrounging Dispersed the rest members to perform ranging Generate quasi-opposite members of rest members Evaluate rest members and its quasi-opposite members If fitness value of quasi-opposite member is less than the fitness value of rest member replace the rest member with its quasi-opposite member NO Termination Criterion Satisfied? Yes Stop Fig. 2. Flowchart of the QOGSO algorithm.

Producer by its vision ability scans the search space for the better states. The producer scans three points around its position in certain distances and head angles. At the k th iteration, the producer behaves as follows: (1) The producer scans at zero degree and tests three points toward its position using Eqs. (13)–(15).



X z ¼ X kP þ r 1 lmax Dkp /k



ð13Þ

  r2 hmax X r ¼ X kP þ r 1 lmax DkP /k þ 2

ð14Þ

  r2 hmax X l ¼ X kP þ r 1 lmax DkP /k  2

ð15Þ

where X P is position of the producer, r1 is a normally distributed random number with mean 0 and standard deviation 1 and r 2 is a uniformly distributed random number in the range of (0, 1), lmax is maximum pursuit distance and hmax is maximum pursuit angle. (2) The producer will then find the best point. If the best point has a better value in comparison with its current position, producer will fly to that point. If not, it will stay in its current position and turn its head using Eq. (16).

/kþ1 ¼ /k þ r 2 amax

ð16Þ

where amax 2 R1 is the maximum turning angle. (3) If the producer cannot find a better area after a iterations, it will turn its head back to zero degree as follows:

/kþa ¼ /k where a 2 R1 is a constant.

ð17Þ

During each iteration, some of group members are selected as scroungers which will keep searching for opportunities to join the resources found by the producer. At the k th iteration, the area copying behavior of the i th scrounger can be modeled as a random walk toward the producer using Eq. (18)

  X ikþ1 ¼ X ki þ r3  X kP  X ki

ð18Þ

where X ki is the position of the i th scrounger at the k th iteration. r3 is a uniform random number in the range of (0, 1). Operator ‘‘” is the Hadamard product or the Schur product, which calculates the entry wise product of the two vectors. At each iteration, some of group members are selected as rangers which are dispersed from their positions and randomly walk at search space. At the k th iteration, a ranger generates a random head angle /i using Eq. (13), and then it chooses a random distance using Eq. (19) and move to the new point using Eq. (20).

li ¼ br 1 lmax

ð19Þ

  X ikþ1 ¼ X ki þ li Dki /kþ1

ð20Þ

where b is a constant and is a normally distributed random number with mean 0 and standard deviation 1. Parameter setting The initial head angle /0 of each individual is set to be pffiffiffiffiffiffiffiffiffiffiffiffi ðP=4; . . . ; P=4Þ. Constant a is given by round n þ 1 where n is the dimension of the search space. The maximum pursuit angle hmax is P=a2 . The maximum turning angle amax is set to be hmax =2. The maximum pursuit distance lmax is calculated from the following equation.

lmax ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Xn 2 ð U  L Þ i i i¼1

ð21Þ

360

M. Basu / Electrical Power and Energy Systems 78 (2016) 356–367

Table 1 Hourly generation (MW) schedule and tie line power flow (MW) of test system 1. Hour

P 1;1

P 1;2

P 1;3

P 1;4

P 2;1

P 2;2

P 2;3

P 3;1

P 3;2

P 3;3

T 21

T 31

T 32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

151.9654 123.0205 116.7819 172.4256 141.0505 205.8323 180.0439 220.0213 243.4103 241.1833 236.7180 237.6618 242.3153 189.7903 215.5978 240.5637 178.1432 239.7972 236.0135 238.2420 237.5060 187.0103 204.1365 227.2505

124.6050 173.7695 183.4086 219.0842 221.0308 164.5992 201.5083 220.3388 217.8412 224.3407 217.6517 220.0598 220.7074 218.7838 217.8548 228.7539 196.9275 219.0577 221.6522 210.3471 225.9685 153.4868 167.5467 169.1498

431.1583 423.4586 460.9913 456.5057 454.4104 452.9249 467.8767 479.1940 473.1436 499.8268 488.7048 499.6641 477.8423 447.6376 492.1062 477.0851 403.9753 480.9397 484.4515 496.0830 448.2970 426.1637 362.1525 423.6382

136.6290 137.9884 137.8966 137.8085 213.6647 235.8349 237.0443 244.1658 242.0164 248.3323 244.5677 255.4538 234.0885 245.3751 250.2129 243.6284 236.6408 239.7321 250.3466 244.0318 247.3907 231.8021 231.1306 166.3406

241.0157 188.1391 213.1749 147.5643 189.7431 129.6897 211.8108 188.1240 127.3611 150.3922 174.1417 216.0765 241.1609 207.1975 209.9092 246.3903 234.0053 209.1668 241.1168 265.7972 227.1119 173.7376 202.4311 130.8515

113.6095 86.9521 121.5149 180.2868 180.1529 146.8614 155.9006 188.9144 194.5388 197.6194 232.3406 245.3797 240.6720 180.7674 150.6425 175.3689 224.9507 235.4299 241.7476 170.0561 142.0219 183.2789 150.5332 114.1032

185.3611 190.8286 182.5713 176.7140 216.8155 278.0775 265.7254 284.6178 313.7361 340.5806 327.7478 323.8249 293.0689 316.9433 273.7308 264.3422 275.5949 283.6093 352.5413 329.6839 310.7305 278.8587 222.7623 196.0281

81.9190 109.2906 156.4317 146.3886 133.3509 137.2696 157.2233 96.0937 149.8076 180.4709 233.2815 223.4723 189.5086 141.8611 165.8550 108.1536 139.6727 181.6953 168.0616 220.5168 178.1498 241.3442 210.9762 156.7057

113.6908 168.7050 136.9054 188.9114 170.3571 226.4232 210.7816 238.8742 229.7268 230.7993 235.9198 217.8706 228.9535 221.1531 238.9739 226.2654 196.0545 216.0095 237.7583 232.7251 205.0602 200.8017 147.5344 119.9816

118.7816 138.7080 172.6999 177.5584 214.6263 260.9590 265.2815 278.4907 341.4482 324.0368 345.1420 348.6533 317.2358 365.6402 289.5364 260.0910 250.5956 266.7046 273.8000 337.9717 311.4820 238.8110 199.2112 202.4475

92.6716 25.7404 94.1994 97.1578 74.3258 46.6429 19.6004 56.2680 37.8614 94.7806 95.7509 92.7544 95.4860 71.1610 25.2742 5.0146 71.0128 74.7147 98.1702 93.3813 97.4209 82.2046 38.3092 18.4999

85.7713 40.6096 48.5809 77.4443 33.9366 14.2954 71.3972 0.4640 52.2667 10.7135 83.9566 87.5164 71.3491 92.6696 52.1006 41.0384 80.5030 32.7023 62.1685 88.4417 9.0705 76.9478 46.0013 11.9978

20.8722 56.4270 47.2355 91.6811 20.7862 50.8664 25.2536 3.9828 34.1867 65.1631 45.2875 5.0707 6.8105 0.5906 16.7721 62.4530 77.0811 9.8838 57.4796 16.5883 52.6174 25.8247 11.5860 15.9435

Table 2 Comparison of performance for test system 1. Techniques

QOGSO

GSO

GSA

BBO

DE

PSO

Best cost ($) Average cost ($) Worst cost ($) CPU time (s)

12976.90 12983.56 12992.38 75.0243

13013.66 13021.20 13031.93 73.3451

13121.05 13134.32 13149.53 75.7481

13081.08 13092.74 13106.53 75.9517

13042.28 13050.04 13062.47 72.5739

13134.05 13151.32 13170.27 72.7598

where Li and U i are the lower and upper bounds for the i th dimension. The most important control parameter of GSO is the percentage of scroungers and rangers. In this paper scroungers are taken as 70% and rangers are taken as 30% of the total population. Fig. 1 shows the flowchart of GSO algorithm.

Opposition-based learning Opposition-based learning (OBL) was developed by Tizhoosh [22] to improve candidate solution by considering current population as well as its opposite population at the same time. 1.6

x 10

6

4

2.75

QOGSO GSO BBO GSA DE PSO

1.55

x 10

QOGSO GSO BBO GSA DE PSO

2.7

2.65

Cost ($)

1.5

Cost ($)

Evolutionary algorithms start with some random initial population and try to improve them toward some optimal solution. The process of searching terminates when some predefined criteria are satisfied. The process can be improved by starting with a closer i.e. fitter solution by simultaneously checking the opposite solution. By doing this, the fitter one (guess or opposite guess) may be chosen as an initial solution. According to the theory of probability, 50% of the time, a guess is further from the solution than its opposite guess. Therefore, process starts with the closer of the two guesses. The same approach can be applied not only to the initial solution but also continuously to each solution in the current population.

1.45

1.4

2.6

1.35 2.55

1.3

1.25

2.5

0

10

20

30

40

50

60

70

80

Iteration Fig. 3. Cost convergence characteristic of test system 1.

90

100

0

20

40

60

80

100

120

140

160

Iteration Fig. 4. Convergence characteristic of test system 2.

180

200

361

M. Basu / Electrical Power and Energy Systems 78 (2016) 356–367 Table 3 Hourly generation (MW) schedule of area 1 of test system 2. Hour

P 1;1

P 1;2

P 1;3

P 1;4

P 1;5

P 1;6

P 1;7

P 1;8

P 1;9

P 1;10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

88.4636 76.7347 63.7560 86.2055 76.3305 39.3490 53.4397 83.4918 79.1198 40.1372 68.1841 75.1645 106.2483 73.4426 105.1328 113.8199 112.4286 91.4739 94.7939 103.4216 110.6976 96.9659 105.4970 80.7003

87.3840 75.2708 45.4042 68.4834 78.4936 83.7204 103.5357 83.4315 44.6076 64.3616 88.2680 111.5937 108.4138 100.9707 92.1666 72.7821 73.5532 73.1554 86.9869 108.5738 68.9016 46.3520 38.8058 72.0946

111.1301 97.3402 72.0272 60.6169 62.6612 94.8833 99.8395 109.7822 78.7469 98.9323 100.3638 91.5070 97.3133 60.1421 60.7236 97.2628 111.8402 97.0064 93.7368 84.6496 68.9557 62.1101 60.0237 97.4073

179.7492 132.1167 175.5017 129.5891 172.6526 173.9746 161.7045 102.3338 149.3089 179.7462 170.6071 129.8170 110.2012 137.5764 91.8077 82.0652 129.5445 179.6476 186.4989 179.7672 173.5678 176.0481 139.2862 130.8657

77.3767 88.6841 87.6243 86.5063 66.9036 87.7963 83.6027 63.3027 72.7967 87.8173 92.2923 92.4836 90.0659 89.6496 88.7219 85.0242 58.3099 86.9530 80.5796 60.7423 57.2687 87.2365 94.5440 85.2062

115.5094 103.0649 129.2717 103.2343 105.7729 105.3798 97.6376 110.4057 139.9988 135.3263 138.8349 123.7135 139.9962 107.3453 86.5718 123.8836 125.6504 75.6675 122.3203 131.7974 130.7547 102.2845 138.8269 105.6634

267.5881 295.6046 297.6697 260.4232 258.9234 276.6705 259.6081 298.8342 293.8760 259.6775 255.8386 263.1618 290.0555 256.5871 219.9425 271.9719 287.5847 267.5148 296.0587 221.4935 184.7771 200.1399 239.7649 195.4630

137.0415 166.1334 157.8605 157.3979 178.0475 209.7974 233.2328 292.2617 265.9128 284.6052 236.0436 265.0735 242.6208 203.8177 212.9918 137.6417 207.9194 284.1401 209.8250 191.6821 209.3131 269.6152 216.8056 142.4874

209.8138 158.7786 186.2633 256.0324 216.0185 196.7651 135.0099 183.3587 219.8074 149.8070 205.0928 255.1359 272.7364 197.5704 239.4212 190.3093 209.7899 284.2273 212.9659 186.2137 190.0520 187.4677 254.4585 211.6883

130.0001 130.0000 130.0000 130.0000 130.0001 204.0619 156.6112 204.7996 204.7940 130.0004 130.0001 164.3164 130.0002 130.0001 130.0001 130.0000 204.7978 130.0003 204.7991 130.0002 130.0000 130.0000 204.7464 204.7999

Table 4 Hourly generation (MW) schedule of area 2 of test system 2. Hour

P 2;1

P 2;2

P 2;3

P 2;4

P 2;5

P 2;6

P 2;7

P 2;8

P 2;9

P 2;10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

319.5527 258.6217 252.0273 291.3191 307.7656 319.3828 296.4986 275.5431 261.0402 331.3319 260.2242 327.5875 244.7221 313.9282 241.6956 205.2598 164.4869 169.9356 242.8495 335.1175 315.1779 317.7162 330.1095 305.8049

169.1925 152.8667 168.8012 105.5888 169.3073 267.3388 318.7387 232.6531 216.3230 260.2235 354.5811 263.1410 212.3853 167.8806 151.3065 228.7152 224.6076 318.2346 223.6412 248.1236 243.4743 331.3150 266.7141 198.8237

192.0997 205.5656 214.1128 291.2733 346.2025 407.2059 394.2793 479.4559 395.6397 454.3162 479.5366 473.0090 436.7973 390.8429 469.8882 449.8652 392.2347 392.6218 440.2423 419.4759 352.5607 391.1846 399.1320 307.4867

180.8644 258.1577 200.4789 127.4698 136.1873 214.7656 125.9865 170.8023 216.6600 311.7359 305.8236 272.5429 314.0148 247.0504 215.8729 127.8088 214.1248 247.1254 229.4211 251.7060 188.3088 264.9650 186.5932 205.8047

303.9307 242.8469 309.4928 298.5681 234.3827 201.6170 297.7686 250.9398 290.3459 299.8793 306.3032 303.6616 299.7421 304.9924 281.4339 252.0869 219.9851 205.7612 215.3865 237.4158 315.2384 302.9253 216.9606 270.2435

283.4864 306.7322 396.6708 304.7907 312.2731 255.3090 352.0021 397.2367 335.6656 359.5719 404.5153 483.0706 418.4910 482.5111 407.5941 405.7018 365.4907 394.0088 447.7739 416.8867 329.0596 303.9267 302.7847 217.4631

404.9554 400.4697 394.6988 424.8350 328.2806 308.0100 339.9227 381.4898 477.6804 498.8991 489.1338 489.0734 411.6652 370.7963 328.2844 271.4711 306.5547 399.5428 321.9497 253.9041 316.8116 271.2461 226.3632 310.4889

399.5364 350.3388 299.6818 229.7258 309.5509 220.4341 314.1902 309.7765 367.9267 408.9926 428.3856 424.3955 489.2871 452.0763 388.5547 436.6439 464.4608 489.2799 402.7907 399.2569 352.9582 384.6522 340.9986 348.3206

394.5316 331.8446 308.6523 329.8818 318.7944 351.2780 413.3907 341.0968 269.3307 360.3879 402.0987 331.4826 312.4203 272.8741 337.9892 345.0724 245.9902 331.7601 421.5437 436.4329 501.5446 403.7470 426.0120 420.6700

242.0005 331.8826 421.5200 421.5198 511.2791 421.5613 421.5196 511.2799 511.2793 423.6705 421.5196 421.5195 421.5196 421.5196 511.2797 421.5198 511.2786 511.2796 421.5200 422.0003 421.5197 331.7598 328.2989 242.0002

Quasi-opposition-based learning Quasi-opposition-based learning (QOBL) was introduced by Rahnamayan et al. [29] to improve candidate solution by considering current population as well as its quasi-opposite population at the same time. The process can be improved by starting with a closer i.e. fitter solution by simultaneously checking the quasi-opposite solution. By doing this, the fitter one (guess or quasi-opposite guess) may be chosen as an initial solution. The process starts with the closer of the two guesses. The same approach can be applied not only to the initial solution but also continuously to each solution in the current population. It is proved that a quasi-opposite number is usually closer than a random number to the solution. It is also proved that a quasi-opposite number is usually closer than an

opposite number to the solution [29]. The idea of QOBL technique is used in population initialization and generation jumping. Due to incorporation of QOBL technique in group search optimization (GSO), quasi-opposition group search optimization (QOGSO) provides better quality solution and less iteration cycles compared to group search optimization (GSO), biogeography-based optimization (BBO), gravitational search algorithm (GSA), differential evolution (DE) and particle swarm optimization (PSO). Definition of opposite point and quasi-opposite point Let X ¼ ðx1 ; x2 ; . . . ; xn Þ be a point in n- dimensional space where xi 2 ½lbi ; ubi  and i 2 1; 2; . . . ; n. The opposite point X o ¼ ðxo1 ; xo2 ; . . . ; xon Þ is completely defined by its components as in [22].

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M. Basu / Electrical Power and Energy Systems 78 (2016) 356–367

Table 5 Hourly generation (MW) schedule of area 3 of test system 2. Hour

P 3;1

P 3;2

P 3;3

P 3;4

P 3;5

P 3;6

P 3;7

P 3;8

P 3;9

P 3;10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

265.3835 309.5852 357.4984 341.3779 369.1194 424.1325 343.9443 407.0786 441.1262 436.6166 433.5726 352.8269 338.5697 299.3116 344.7102 344.7916 261.6559 358.5049 420.2066 520.0152 434.0637 347.0317 254.8793 301.1753

310.9382 322.6578 257.2414 337.3115 404.1251 360.6514 343.8092 299.9144 377.3548 434.6854 434.1909 446.5072 523.2942 480.3446 433.8422 434.8213 424.7327 518.2992 427.7120 504.1634 444.0595 360.6392 276.2630 311.0168

256.8649 292.9499 345.0390 258.5771 258.4382 343.7979 430.2935 345.5764 344.0610 363.6066 411.4973 434.9107 433.5250 350.4854 425.6033 470.2601 371.4781 347.2247 366.5645 343.7612 431.8146 360.2138 301.1109 318.7364

299.6052 293.9513 268.7450 262.0616 331.9290 423.7539 508.8916 504.4064 534.3711 483.6099 537.5603 464.2393 411.3091 422.5667 387.5290 401.3899 317.5084 345.0287 400.8638 500.7838 455.1130 388.4992 306.8105 265.0831

311.5936 372.7580 345.0835 421.0251 344.6649 254.5237 263.5343 353.1114 433.5868 382.6909 481.0608 538.5826 496.3330 435.4386 384.8799 357.5819 266.5461 286.7561 346.8678 299.3702 394.0549 315.6041 343.7724 321.4261

254.5532 343.7388 279.1190 301.4092 369.0830 350.2589 417.4687 343.7784 348.6274 437.9927 433.5327 527.6630 464.2806 369.7681 369.2250 304.2255 256.7659 351.0715 442.4350 433.5386 417.3576 334.0969 373.7646 335.0264

10.2311 10.0007 10.5851 30.2261 13.5656 11.0109 36.5293 31.5652 28.4136 16.4759 18.8396 24.3303 34.2013 10.0880 15.1834 11.3742 10.0209 15.0490 30.9970 41.8013 14.1125 12.6044 21.4731 10.0018

16.8716 28.9089 12.7314 16.4766 32.7484 14.3124 20.1603 22.8320 31.6367 40.9021 38.0863 45.8506 27.0507 10.0678 14.8622 11.0726 10.0032 18.5143 24.3124 27.5636 11.4801 10.1390 24.7737 22.3799

10.0851 10.0099 10.0038 10.0228 12.5412 14.7450 10.1242 24.9059 25.5039 19.6556 39.1601 28.6185 19.9605 32.4212 22.5710 11.9026 10.0992 15.0054 29.2934 27.6184 35.8668 10.0010 16.6486 10.0913

91.8839 87.8001 95.4859 87.8008 89.7474 87.8066 87.8510 96.9843 96.9999 95.5933 89.1774 96.9997 88.4286 87.8351 88.2561 93.0709 87.8014 96.7717 96.8004 88.8341 88.1178 87.8000 92.7240 88.1218

Table 6 Hourly generation (MW) schedule of area 4 of test system 2. Hour

P 4;1

P 4;2

P 4;3

P 4;4

P 4;5

P 4;6

P 4;7

P 4;8

P 4;9

P 4;10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

60.4203 71.9162 81.1821 60.0860 109.7669 97.3053 87.0352 13.6716 102.5236 109.8668 104.0260 159.7341 152.6027 144.4167 138.8124 162.2532 156.9475 110.7737 165.5969 163.3594 150.7556 91.8809 110.2713 120.2325

75.9621 79.1235 122.5592 62.5984 81.5804 113.7931 168.7280 159.7501 170.3277 189.2222 188.6689 187.5204 131.0368 159.7688 107.1353 112.2685 109.7773 110.0341 163.6937 150.7821 163.9496 143.7498 127.1986 71.2279

77.0458 67.2690 72.3974 97.6922 151.7681 167.3757 156.5423 98.1934 120.1045 147.7137 180.9367 189.4559 144.5513 94.0254 122.3843 111.9545 120.6569 75.9195 133.0276 75.6270 106.9562 70.0362 69.6032 65.3497

105.7053 95.4045 105.5201 96.9599 143.3996 93.4731 152.2228 129.0179 127.9653 175.5552 152.3443 113.0663 151.5373 170.6183 156.0163 116.3115 134.0688 92.4790 152.3709 164.6976 138.6323 108.8871 114.8685 104.8389

94.6604 91.4975 101.1740 160.1303 107.5738 90.1953 150.1465 193.7074 199.4383 193.5012 166.9235 195.0848 164.8895 163.8813 141.1741 189.1383 147.9208 172.0028 198.5241 143.3301 95.4857 92.3887 91.8514 124.6687

90.2693 106.7538 92.1881 90.0653 105.7075 140.2618 105.2488 134.1500 152.0036 164.7359 154.1517 176.0960 146.0808 141.2507 156.8993 189.8544 148.7641 142.6365 197.9790 142.3272 104.7267 135.4090 91.2935 90.5793

30.7300 33.3204 56.8073 54.2557 60.4566 59.6282 51.0370 77.2415 56.0888 57.0019 86.4943 102.8288 63.8002 94.7345 80.4573 98.1084 93.3520 63.3883 99.3070 86.9447 59.6558 55.8588 34.2559 25.6780

25.2462 47.7586 84.4996 58.3278 35.0507 36.5761 65.7930 55.0180 89.1445 82.7427 87.4058 107.1197 88.9888 76.1050 62.8835 102.3100 81.8206 63.1007 100.6161 61.4838 72.9307 56.2726 57.8358 39.6855

35.7431 31.5406 30.6242 60.3743 58.9065 90.6705 87.3613 57.0616 85.3422 102.8921 85.4444 97.8354 61.5841 61.2591 60.9167 89.1258 67.9277 90.5276 109.8686 89.0583 84.4054 89.4699 57.1136 58.0036

50.0002 50.0009 50.0003 139.7599 50.0003 96.4268 139.7598 139.7599 229.5195 229.5197 319.2795 319.2791 319.2794 409.0390 319.2793 319.2793 229.5197 229.5761 319.2796 319.2790 229.5193 139.7599 139.7621 50.1946

xoi ¼ lbi þ ubi  xi

ð22Þ

  The quasi-opposite point X qo ¼ xqo1 ; xqo2 ; . . . ; xqon is completely defined by its components as in (23).

xqoi



  lbi þ lui ; ðlbi þ lui  xi Þ ¼ rand 2

ð23Þ

Quasi-opposition based optimization Let X ¼ ðx1 ; x2 ; . . . ; xn Þ be a point in n-dimensional space i.e. a candidate solution. Assume f ¼ ðÞ is a fitness function which is used to measure the candidate’s fitness. According to the definition   of the quasi-opposite point, X qo ¼ xqo1 ; xqo2 ; . . . ; xqon is the   quasi-opposite of X ¼ ðx1 ; x2 ; . . . ; xn Þ. Now, if f X qo < f ðX Þ (for a minimization problem), then point X can be replaced with X qo ; otherwise, the process is continued with X. Hence, the point and

its quasi-opposite point are evaluated simultaneously in order to continue with the fitter one. Quasi-oppositional group search optimization In the present work, the concept of the quasi-opposition-based learning [29] is incorporated in group search optimization (GSO). The original GSO is chosen as a parent algorithm and the quasiopposition-based ideas are embedded in GSO. Fig. 2 shows the flowchart of quasi-opposition group search optimization (QOGSO) algorithm. Simulation results To verify the effectiveness and performance of the proposed QOGSO and GSO two multi-area test systems have been

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M. Basu / Electrical Power and Energy Systems 78 (2016) 356–367 Table 7 Hourly tie line power flow (MW) of test system 2. Hour

T 21

T 31

T 32

T 41

T 42

T 43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

107.1998 32.2097 50.9805 55.2117 125.5595 189.6148 17.6897 66.6722 18.6289 24.8407 57.5811 156.2110 31.8601 4.7774 14.6027 86.5044 43.8350 74.0370 101.4132 41.3151 14.0971 97.4873 185.9817 175.4237

173.7182 70.3376 188.6599 115.9300 171.3816 99.6162 41.9025 71.7368 120.5420 9.7659 188.4201 87.9983 68.9683 161.7502 32.9688 149.7388 182.8915 92.0081 114.5084 87.8292 13.3353 191.6916 184.4484 51.7940

192.0820 41.0623 0.0891 26.4069 8.8729 13.3683 50.7433 115.6153 100.2710 129.3253 97.0798 51.2146 49.8627 101.1465 85.3273 76.9256 147.3169 110.2064 139.2009 164.4786 176.4420 119.2885 176.7156 177.1395

107.9385 184.6807 144.1993 100.8475 68.6373 26.6673 81.8795 123.0929 77.0482 49.8858 149.3643 167.7423 139.0434 119.1207 100.9517 188.4825 103.9920 123.9412 4.6565 177.0356 35.9500 83.2155 8.8790 66.1085

98.0681 148.5984 79.2453 9.7770 110.7100 59.1491 72.7566 144.5612 93.2085 91.8934 99.3770 93.5199 184.9527 199.4521 163.6310 198.2763 169.4676 10.5803 65.0673 57.7168 61.3987 62.1378 64.0330 70.5906

163.4107 28.6359 43.1032 75.6256 171.2168 84.9776 69.7523 154.7255 16.4524 198.9304 108.6623 108.7417 33.0583 122.7238 9.1793 153.1550 34.5798 5.1600 193.2396 165.7580 15.2660 176.0093 146.4841 145.8923

Table 8 Comparison of performance for test system 2. Techniques

QOGSO

GSO

GSA

BBO

DE

PSO

Best cost ($) Average cost ($) Worst cost ($) CPU time (s)

2,511,795 2,511,812 2,511,833 98.0368

2,512,155 2,512,174 2,512,197 96.4507

2,512,381 2,512,397 2,512,418 98.4478

2,512,314 2,512,329 2,512,347 98.4478

2,512,208 2,512,230 2,512,256 96.1436

2,512,423 2,512,445 2,512,461 96.1577

considered. The results obtained from proposed QOGSO method are compared with those obtained from group search optimization (GSO), biogeography-based optimization (BBO), gravitational search algorithm (GSA), differential evolution (DE) and particle swarm optimization (PSO). All the algorithms i.e. QOGSO, GSO, BBO, GSA, DE, and PSO used in this paper for solving multi-area dynamic economic dispatch (MADED) problem are implemented by using MATLAB 7.0 on a PC (Pentium-IV, 80 GB, 3.0 GHz). Test system 1 This system comprises ten generators with valve-point loading and multi-fuel sources having three fuel options. Transmission loss is considered here. The demand of the system is divided into 24 intervals. Unit data has been modified from [31] and can be found in Tables A.1.1 and A.1.2 in Appendix A. The B-coefficients are also given in Appendix A. The ten generators are divided into three areas. Area 1 consists of the first four units; area 2 includes the next three units and area 3 includes the last three units. The load demand in area 1 is assumed as 50% of the total demand. The load demand in area 2 is assumed as 25% and in area 3 is taken as 25% of the total demand. The power flow limit from area 1 to area 2 or from area 2 to area 1 is 100 MW. The power flow limit from area 1 to area 3 or from area 3 to area 1 is 100 MW. Also the power flow limit from area 2 to area 3 or from area 3 to area 2 is 100 MW. The problem is solved by using QOGSO and GSO. Here, the population size (N P Þ and maximum number of iterations (N max Þ have been selected 100 and 100 respectively for this test system under consideration. In order to validate the proposed QOGSO based approach, the same test system is also solved by using BBO, GSA, DE and PSO.

In BBO, habitat size, habitat modification probability, immigration probability bounds per gene, step size for numerical integration, maximum immigration emigration rate for each island, mutation probability and maximum number of iterations are taken as 100, 1, [0, 1], 1, 1, 0.005 and 100 respectively. In GSA, starting value of gravitational constant G0 , number of agents, initial value of K 0 , constant a and maximum iteration number are taken as 0.16, 100, 100, 20 and 100 respectively. The population size, scaling factor and crossover constant have been selected as 100, 1.0 and 1.0 respectively in DE. The parameters are taken as N P ¼ 100; wmax ¼ 0:9; wmin ¼ 0:4; c1 ¼ 0:3 and c2 ¼ 0:3 in PSO. Maximum number of iterations has been selected 100 for DE and PSO. Hourly generation schedule and tie line power flow corresponding to the lowest minimum cost obtained from QOGSO have been presented in Table 1. The best, average and worst cost and average CPU time among 100 runs of solutions obtained from proposed QOGSO, GSO, BBO, GSA, DE and PSO are summarized in Table 2. The cost convergence characteristic obtained from QOGSO, GSO, BBO, GSA, DE and PSO is shown in Fig. 3. It is seen from Fig. 3 that cost converges more quickly in case of QOGSO than all other methods. It is observed from Table 2 that the cost found by using QOGSO is the lowest among all other methods. Test system 2 This system comprises forty generators with valve-point loading. The unit data has been modified from [30] and can be found in Tables A.2.1–A.2.3 in Appendix B. Transmission loss is neglected here. Forty generators are divided into four areas. Area 1 includes first ten generators and 15% of the total load demand. Area 2 has

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M. Basu / Electrical Power and Energy Systems 78 (2016) 356–367

Table A.1.1 Generator characteristics of test system 1. Unit

1

2

3

4

5

6

7

8

9

10

Generation Min (MW) Max (MW) F1 F2 F3 50

50

196 250 1 2

2

114

3

157 230 1

100 332 388 500 1 3 2

50

1

100

50

2

138 200 265 2 3

1

338 407 490 2 3

138 200 265 1 3

100 331 391 500 1 2 3

50 138 200 265 1 2 3

100 213 370 440 3 1 3

100 362 407 490 1 3 2

Fuel type

Cost coefficients

URi (MW/h)

ai ($/h)

bi ($/MW h)

ci ($/(MW)2h)

di ($/h)

ei (rad/MW)

1

0.2697e2

0.3975e0

0.2176e2

0.2697e1

0.3975e1

2

0.2113e2

0.3059e0

0.1861e2

0.2113e1

0.3059e1

1

0.1184e3

0.1269e1

0.4194e2

0.1184e0

0.1269e2

2 3

0.1865e1 0.1365e2

0.3988e1 0.1980e0

0.1138e2 0.1620e2

0.1865e2 0.1365e1

0.3988e0 0.1980e1

1

0.3979e2

0.3116e0

0.1457e2

0.3979e1

0.3116e1

2 3

0.5914e2 0.2875e1

0.4864e0 0.3389e1

0.1176e4 0.8035e3

0.5914e1 0.2876e2

0.4864e1 0.3389e0

1

0.1983e1

0.3114e1

0.1049e2

0.1983e2

0.3114e0

2 3

0.5285e2 0.2668e3

0.6348e0 0.2338e1

0.2758e2 0.5935e2

0.5285e1 0.2668e0

0.6348e1 0.2338e2

1

0.1392e2

0.8733e1

0.1066e2

0.1392e1

0.8733e0

2 3

0.9976e2 0.5399e2

0.5206e0 0.4462e0

0.1597e2 0.1498e3

0.9976e1 0.5399e1

0.5206e1 0.4462e1

1

0.5285e2

0.6348e0

0.2758e2

0.5285e1

0.6348e1

2 3

0.1983e1 0.2668e3

0.3114e1 0.2338e1

0.1049e2 0.5935e2

0.1983e2 0.2668e0

0.3114e0 0.2338e2

1

0.1893e2

0.1325e0

0.1107e2

0.1893e1

0.1325e1

2 3

0.4377e2 0.4335e2

0.2267e0 0.3559e0

0.1165e2 0.2454e3

0.4377e1 0.4335e1

0.2267e1 0.3559e1

1

0.1983e1

0.3114e1

0.1049e2

0.1983e2

0.3114e0

2 3

0.5285e2 0.2668e3

0.6348e0 0.2338e1

0.2758e2 0.5935e2

0.5285e1 0.2668e0

0.6348e1 0.2338e2

1

0.8853e2

0.5675e0

0.1554e2

0.8853e1

0.5675e1

2 3

0.1530e2 0.1423e2

0.4514e1 0.1817e1

0.7033e2 0.6121e3

0.1423e1 0.1423e1

0.1817e0 0.1817e0

1

0.1397e2

0.9938e1

0.1102e2

0.1397e1

0.9938e0

2 3

0.6113e2 0.4671e2

0.5084e0 0.2024e0

0.4164e4 0.1137e2

0.6113e1 0.4671e1

0.5084e1 0.2024e1

second ten generators and 40% of the total load demand. Area 3 consists of third ten generators and 30% of the total load demand. Area four includes last ten generators and 15% of the total load demand. The problem is solved by using QOGSO and GSO. Here, the population size (N P Þ and maximum number of iterations (N max Þ have been selected 200 and 200 respectively for this test system under consideration. In order to validate the proposed QOGSO based approach, the same test system is also solved by using BBO, GSA, DE and PSO. In BBO, habitat size, habitat modification probability, immigration probability bounds per gene, step size for numerical integration, maximum immigration emigration rate for each island, mutation probability and maximum number of iterations are taken as 200, 1, [0, 1], 1, 1, 0.005 and 200 respectively. In GSA, starting value of gravitational constant G0 , number of agents, initial value of K 0 , constant a and maximum iteration number are taken as 0.16, 200, 200, 20 and 200 respectively.

DRi (MW/h)

80

80

80

80

100

100

80

80

100

100

80

80

100

100

80

80

100

100

100

100

Table A.1.2 Total load demands of test system 1. Hour

P D (MW)

Hour

P D (MW)

Hour

P D (MW)

1 2 3 4 5 6 7 8

1678 1720 1858 1980 2109 2212 2323 2407

9 10 11 12 13 14 15 16

2500 2602 2700 2750 2650 2500 2471 2438

17 18 19 20 21 22 23 24

2309 2540 2671 2706 2500 2290 2078 1887

In DE, the population size, scaling factor and crossover constant have been selected as 200, 1.0 and 1.0 respectively. In case of PSO, the parameters are taken as N P ¼ 200; wmax ¼ 0:9; wmin ¼ 0:4; c1 ¼ 0:3 and c2 ¼ 0:3. Maximum number of iterations has been selected 200 for DE and PSO.

Table A.2.1 Generator characteristics of test system 2. P max (MW) i

(MW) P min i

ai ($/h)

bi ($/MW h)

ci ($/(MW)2 h)

di ($/h)

ei (rad/MW)

URi (MW/h)

DRi (MW/h)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

36 36 60 80 47 68 110 135 135 130 94 94 125 125 125 125 220 220 242 242 254 254 254 254 254 254 10 10 10 47 60 60 60 90 90 90 25 25 25 50

114 114 120 190 97 140 300 300 300 300 375 375 500 500 500 500 500 500 550 550 550 550 550 550 550 550 150 150 150 97 190 190 190 200 200 200 110 110 110 550

94.705 94.705 309.540 369.030 148.890 222.330 287.710 391.980 455.760 722.820 635.200 654.690 913.400 1760.400 1760.400 1760.400 647.850 649.690 647.830 647.810 785.960 785.960 794.530 794.530 801.320 801.320 1055.100 1055.100 1055.100 148.890 222.920 222.920 222.920 107.870 116.580 116.580 307.450 307.450 307.450 647.830

6.73 6.73 7.07 8.18 5.35 8.05 8.03 6.99 6.60 12.90 12.90 12.80 12.50 8.84 8.84 8.84 7.97 7.95 7.97 7.97 6.63 6.63 6.66 6.66 7.10 7.10 3.33 3.33 3.33 5.35 6.43 6.43 6.43 8.95 8.62 8.62 5.88 5.88 5.88 7.97

0.00690 0.00690 0.02028 0.00942 0.01140 0.01142 0.00357 0.00492 0.00573 0.00605 0.00515 0.00569 0.00421 0.00752 0.00752 0.00752 0.00313 0.00313 0.00313 0.00313 0.00298 0.00298 0.00284 0.00284 0.00277 0.00277 0.52124 0.52124 0.52124 0.01140 0.00160 0.00160 0.00160 0.00010 0.00010 0.00010 0.01610 0.01610 0.01610 0.00313

100 100 100 150 120 100 200 200 200 200 200 200 300 300 300 300 300 300 300 300 300 300 300 300 300 300 120 120 120 120 150 150 150 200 200 200 80 80 80 300

0.084 0.084 0.084 0.063 0.077 0.084 0.042 0.042 0.042 0.042 0.042 0.042 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.077 0.077 0.077 0.077 0.063 0.063 0.063 0.042 0.042 0.042 0.098 0.098 0.098 0.035

40 40 40 60 30 50 80 80 80 80 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 50 50 50 30 60 60 60 60 60 60 40 40 40 100

40 40 40 60 30 50 80 80 80 80 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 50 50 50 30 60 60 60 60 60 60 40 40 40 100

M. Basu / Electrical Power and Energy Systems 78 (2016) 356–367

Unit

365

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M. Basu / Electrical Power and Energy Systems 78 (2016) 356–367

Hourly generation schedules of area 1, area 2, area 3 and area 4 of this test system obtained from QOGSO are summarized in Tables 3–6 respectively. Hourly tie line power flows of this test system obtained from QOGSO are given in Table 7. The best, average and worst cost and average CPU time among 100 runs of solutions obtained from proposed QOGSO, GSO, BBO, GSA, DE and PSO are summarized in Table 8. The cost convergence characteristic obtained from QOGSO, GSO, BBO, GSA, DE and PSO is depicted in Fig. 4. It is seen from Fig. 4 that cost converges more quickly in case of QOGSO than all other methods. It is observed from Table 8 that the cost obtained from QOGSO is the lowest among all other methods. Conclusion A novel approach based on quasi-oppositional group search optimization has been presented to solve multi-area dynamic economic dispatch problem. The effectiveness of the proposed method is illustrated by using two test systems. The results have been compared with those obtained from group search optimization, biogeography-based optimization, gravitational search algorithm, differential evolution and particle swarm optimization. It has been observed from the comparison that the proposed quasioppositional group search optimization has the ability to converge to a better quality solution with superior computational efficiency.

2

8:60

6 B2 ¼ 4 0:80 0:37

0:80 9:08 4:90

0:37

3

7 4:90 5  105 8:24

B02 ¼ ½ 21:6 66:35 50:34   105 B002 ¼ 0:045 2

3 1:70 0:96 0:56 6 7 B3 ¼ 4 0:96 4:93 0:30 5  105 0:56 0:30 5:99 B03 ¼ ½ 32:16 46:35 35:03   105 B003 ¼ 0:055:

Appendix B See Tables A.2.1–A.2.3. References

Appendix A See Tables A.1.1 and A.1.2. The transmission loss formula coefficients of test system 1 are:

2

8:70

0:43

4:61

0:36

0:36

0:22

2:00

5:30

3

6 0:43 8:30 0:97 0:22 7 7 6 5 B1 ¼ 6 7  10 4 4:61 0:97 9:00 2:00 5

B01 ¼ ½ 39:08 12:97 70:47 5:91   105 B001 ¼ 0:056

Table A.2.2 Tie line power flow (MW) limits of test system 2. T ik

T max ik

T max ik

T 21 T 31 T 32 T 41 T 42 T 43

200 200 200 200 200 200

200 200 200 200 200 200

Table A.2.3 Total load demands of test system 2. Hour

P D (MW)

Hour

P D (MW)

Hour

P D (MW)

1 2 3 4 5 6 7 8

6768 6910 7090 7110 7450 7710 8285 8470

9 10 11 12 13 14 15 16

8885 9304 9780 9970 9410 8795 8494 8380

17 18 19 20 21 22 23 24

7938 8532 9182 9003 8594 7872 7423 6887

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