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Group search optimization for economic fuel scheduling
Electrical Power and Energy Systems 64 (2015) 894–901
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Group search optimization for economic fuel scheduling 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 12 December 2013 Received in revised form 1 August 2014 Accepted 12 August 2014 Available online 17 September 2014 Keywords: Economic fuel scheduling Fuel constraints Load constraints Group search optimization
a b s t r a c t The economical use of fuel available for the generation of power has become a major concern of electric utilities. This paper presents an approach for economic fuel scheduling problem by using group search optimization. This is a minimization technique that includes the standard load constraints as well as the fuel constraints. The generation schedule is compared to that which would result if fuel constraints were ignored. The comparison shows that fuel consumed can be adequately controlled by adjusting the power output of various generating units so that the power system operates within its fuel limitations and within contractual constraints. It has been found that small additional amount of fuel may be required to serve the same power demand but the additional cost of this fuel may well compensate for the penalty that might otherwise be imposed for not maintaining the fuel contract. Numerical results for two test systems have been presented and the test results obtained from group search optimization are compared with those obtained from particle swarm optimization and evolutionary programming. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction Economic dispatch is an important optimization task in power system operation for allocating generation among the committed units. Its objective is to minimize the total generation cost of units, while satisfying the various physical constraints. Economic fuel scheduling which is perhaps more significant than economic dispatch has popped up because of the sudden concern over fuel shortages. Fuel supplies have imposed increased constraints in their fuel supply contracts to the point that utilities have been forced to reschedule generation on the basis of fuel availability. The objective of the fuel scheduling problem is to minimize the total cost by loading units in such a manner that the standard load constraints and fuel constraints are both satisfied. Several papers have been published in this area of fuel scheduling of thermal units [1–6]. The application of linear programming (LP) to solve fuel scheduling problem by dividing the total time period involved into discrete time increments is presented in [6]. The objective is a function of one or more variables from only one time step. Some constraints are made up of variables drawn from one time step whereas others span two or more time steps. The fuel used by a unit may be obtained from different contracts at different prices. Moreover, the worth of the fuel is dependent upon the amount available and the need for it.
⇑ Fax: +91 33 23357254. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijepes.2014.08.016 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
Group search optimization (GSO) is a biologically realistic algorithm. Inspired by the animal (such as lions and wolves) searching behavior, He et al. [7] proposed GSO in 2006, and discussed the effects of designed parameters on the performance of GSO in 2009 [8]. 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 [8]. Shen et al. [9] investigated the performance of GSO and concluded that GSO is an alternative for constrained optimization. Due to its high efficiency, GSO has been applied in many fields. Moreover, some papers also indicate GSO as solutions to some discrete optimization problems, such as optimal design plate structures with discrete variables [10] and optimal design of spatial grid structure [11]. This paper proposes GSO for solving the economic fuel scheduling problem of thermal generating units. The proposed method is validated by applying it to two test systems. The test results obtained by the proposed method are compared with those obtained from particle swarm optimization (PSO) and evolutionary programming (EP). Problem formulation For convenience the entire scheduling period is divided into a number of sub intervals each having a constant load demand. The fuel scheduling problem of N thermal generating units over M time intervals is described mathematically as follows:
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Nomenclature Fim F min ; F max i i FDm Fc Pim Pmin ; Pmax i i PDm
Minimize F c ¼
fuel delivered to thermal unit i in interval m lower and upper fuel delivery limits of thermal unit i fuel delivered in interval m total fuel cost output power of thermal unit i in interval m lower and upper generation limits of thermal unit i load demand in interval m
M X N h n oi X t m ai þ bi P im þ ci P 2im þ di sin ei P min P im i
m¼1 i¼1
ð1Þ
subject to
(i) Power balance constraints N X Pim PDm ¼ 0;
m2M
ð2Þ
i¼1
(ii) Fuel delivery constraints N X F im F Dm ¼ 0;
m2M
ð3Þ
i¼1
(iii) Fuel storage constraints
V im ¼ V iðm1Þ þ F im t m ðai þ bi Pim þ ci P2im Þ;
i 2 N;
m2M
ð4Þ
(iv) Generation limits
Pmin 6 Pim 6 Pmax ; i i
i 2 N;
m2M
ð5Þ
m2M
ð6Þ
(v) Fuel delivery limits
F min 6 F im 6 F max ; i i
i 2 N;
(vi) Fuel storage limits
V min 6 V im 6 V max ; i i
i 2 N; m 2 M
ð7Þ
tm duration of subinterval m Vim fuel storage of thermal unit i in interval m V min ; V max lower and upper fuel storage limits of thermal unit i i i V 0i Initial fuel storage of thermal unit i ai, bi, ci, di, ei cost coefficients of ith thermal unit ai, bi, ci fuel consumption coefficients of thermal unit i
rangers who perform random walk. For convenience of computation, 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. In optimization problems, unknown optima can be taken as open patches randomly distributed in a search space. Group members therefore search for the patches by moving over the search space. It is also assumed that the producer and the scroungers do not differ in their relevant phenotypic characteristics. Therefore, they can switch between the two roles [13]. At each iteration, a group member, which is located in the most promising area and adopts animal scanning to seek the optimal resource and conferred 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. Producer by its vision ability scans the search space for the better states. Vision ability is the ability of testing some points around the producer current position. The producer scans three points around its position in certain distances and head angles.
Group search optimization Group search optimization (GSO) is a population based optimization algorithm which is inspired by animal searching behavior and group living theory. The framework is based on the producer–scrounger (PS) model which assumes that group members search 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 [8]. 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 ndimensional search space, the ith member of the GSO at the kth searching iteration has a current position X ki 2 Rn and a head angle /ki
¼
ð/ki1 ; . . . ; /kiðn1Þ Þ k
n1
2R
k
n1 Y
n1 Y
Choose scrounger and perform scrounging
Dispersed the rest members to perform ranging Evaluate members
Termination Criterion Satisfied?
q¼1 k
Choose producer and perform producing
NO
cosð/kiq Þ
dij ¼ sinð/kiðj1Þ Þ
Generate and evaluate initial members
. The search direction of the ith mem-
k
ber, Dki ð/ki Þ ¼ ðdi1 ; . . . ; din Þ 2 Rn that can be calculated from /ki via polar to Cartesian coordinate transformation [12]
di1 ¼
Start
cosð/kiq Þ j ¼ 2; . . . ; ðn 1Þ
ð8Þ
q¼j
Yes
k
din ¼ sinð/kiðn1Þ Þ GSO algorithm consists of three kinds of members, i.e., producers and scroungers whose behavior are based on the PS model; and
Terminate Fig. 1. Flowchart of the GSO algorithm.
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M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901
Table 1 Economic generation scheduling for case 1 of test system 1 without considering fuel constraints. Interval
GSO
PSO
EP
Generation (MW)
Cost ($)
Generation (MW)
Cost ($)
Generation (MW)
Cost ($)
1
P1 = 50.2586 P2 = 125.0000 P3 = 175.0000 P4 = 49.7415 P5 = 299.9997
1,057,633
P1 = 50.5020 P2 = 125.0000 P3 = 175.0000 P4 = 49.4980 P5 = 300.0000
1,057,640
P1 = 52.5608 P2 = 125.0000 P3 = 175.0000 P4 = 47.5294 P5 = 299.9098
1,057,655
2
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 125.0000 P5 = 299.9999
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 125.0000 P5 = 300.0000
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 125.0182 P5 = 299.9818
3
P1 = 31.8582 P2 = 120.1521 P3 = 175.0000 P4 = 40.0000 P5 = 282.9895
P1 = 31.6853 P2 = 124.2009 P3 = 175.0000 P4 = 40.0000 P5 = 279.1138
P1 = 30.5297 P2 = 117.5533 P3 = 175.0000 P4 = 40.0000 P5 = 286.9170
1.095
x 10
X z ¼ X kP þ r 1 lmax Dkp ð/k Þ r 2 hmax X r ¼ X kP þ r 1 lmax DkP /k þ 2 r 2 hmax k k k X l ¼ X P þ r 1 lmax DP / 2
6
GSO PSO EP
1.09 1.085
Cost ($)
ð10Þ ð11Þ
where XP is position of the producer, r1 is a normally distributed random number with mean 0 and standard deviation 1 and r2 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. (12).
1.08 1.075 1.07 1.065 1.06 1.055
ð9Þ
0
20
40
60
80
100
120
140
160
180
/kþ1 ¼ /k þ r2 amax
200
Iteration
ð12Þ
1
Fig. 2. Cost convergence obtained from case 1 of test system 1.
where amax e R 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:
At the kth iteration, the producer behaves as follows:
/kþa ¼ /k
(1) The producer scans at zero degree and tests three points toward its position using Eqs. (9)–(11).
ð13Þ
where a e R1 is a constant.
Table 2 Economic generation scheduling for case 2 of test system 1 with initial fuel storage (tons) V 01 ¼ 2000; V 02 ¼ 5000; V 03 ¼ 5000; V 04 ¼ 8000; V 05 ¼ 8000. Interval
GSO
PSO
EP
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
1
P1 = 50.2938 P2 = 125.0000 P3 = 174.9968 P4 = 49.7514 P5 = 299.9579
F1 = 728.5 F2 = 281.6 F3 = 742.1 F4 = 2800.0 F5 = 2447.6
1,057,639
P1 = 49.9704 P2 = 125.0000 P3 = 175.0000 P4 = 50.0296 P5 = 300.0000
F1 = 1000.0 F2 = 558.7 F3 = 2000.0 F4 = 1498.1 F5 = 1943.2
1,057,644
P1 = 49.9074 P2 = 125.0000 P3 = 175.0000 P4 = 50.3265 P5 = 299.7661
F1 = 391.5 F2 = 652.1 F3 = 1916.7 F4 = 3000.0 F5 = 1039.6
1,057,655
2
P1 = 75.0000 P2 = 125.0000 P3 = 174.9957 P4 = 125.0118 P5 = 299.9924
F1 = 291.6 F2 = 848.5 F3 = 1723.5 F4 = 1662.0 F5 = 2474.3
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 125.0000 P5 = 300.0000
F1 = 621.6 F2 = 283.0 F3 = 2000.0 F4 = 2145.8 F5 = 1949.6
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 125.0030 P5 = 299.9970
F1 = 496.2 F2 = 551.4 F3 = 1937.1 F4 = 1791.6 F5 = 2223.7
3
P1 = 31.8957 P2 = 121.0814 P3 = 175.0000 P4 = 40.0762 P5 = 281.9465
F1 = 1000.0 F2 = 713.1 F3 = 1667.6 F4 = 1310.5 F5 = 2308.6
P1 = 31.6587 P2 = 125.0000 P3 = 175.0000 P4 = 40.0000 P5 = 278.3413
F1 = 604.2 F2 = 343.2 F3 = 1343.5 F4 = 2415.2 F5 = 2293.9
P1 = 29.6943 P2 = 118.9272 P3 = 175.0000 P4 = 40.0000 P5 = 286.3785
F1 = 459.9 F2 = 504.4 F3 = 1605.2 F4 = 1618.0 F5 = 2812.6
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M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901 6
6
x 10
1.09
1.105 GSO PSO EP
1.08
1.095
1.075
1.09
1.07
1.085
1.065
1.08
1.06
1.075
1.055
1.07 0
20
40
60
80
100
120
140
160
180
GSO PSO EP
1.1
Cost ($)
Cost ($)
1.085
x 10
200
0
20
40
60
80
100
120
140
160
180
200
Iteration
Iteration
Fig. 4. Cost convergence obtained from case 3 of test system 1.
Fig. 3. Cost convergence obtained from case 2 of test system 1.
Parameter setting During each iteration, some of group members are selected as scroungers. The scroungers will keep searching for opportunities to join the resources found by the producer. At the kth iteration, the area copying behavior of the ith scrounger can be modeled as a random walk toward the producer using Eq. (14)
X kþ1 ¼ X ki þ r 3 ðX kP X ki Þ i
ð14Þ
where X ki is the position of the ith scrounger at the kth iteration. r3 is a uniform random number in the range of (0, 1). Operator ‘‘’’ is the Hadamard product, which calculates the entry wise product of the two vectors. At each iteration, some of group members are selected as rangers. Rangers are dispersed from their positions and randomly walk at search space. At the kth iteration, a ranger generates a random head angle /i using Eq. (14), and then it chooses a random distance using Eq. (15) and move to the new point using Eq. (16).
li ¼ br1 lmax
ð15Þ
¼ X ki þ li Dki ð/kþ1 Þ X kþ1 i
ð16Þ
where b is a constant and r1 is a normally distributed random number with mean 0 and standard deviation 1.
The initial population of GSO is generated uniformly at random in the search space. The initial head angle /0 of each individual pffiffiffiffiffiffiffiffiffiffiffiis ffi set to be (P/4, . . ., P/4). The constant a is given by roundð n þ 1Þ 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
qX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ðU i Li Þ2 i¼1
ð17Þ
where Li and Ui are the lower and upper bounds for the ith dimension. The most important control parameter that affects the search performance 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. Simulation results The proposed method has been applied to two test systems. In order to show the effectiveness of the proposed GSO approach, these two test systems are solved by using particle swarm optimi-
Table 3 Economic generation scheduling for case 3 of test system 1 with initial fuel storage (tons) V 01 ¼ 2000; V 02 ¼ 5000; V 03 ¼ 5000; V 04 ¼ 8000; V 05 ¼ 500. Interval
GSO
PSO
EP
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
1
P1 = 74.6392 P2 = 124.6857 P3 = 174.9533 P4 = 112.0461 P5 = 213.6754
F1 = 0 F2 = 1000.0 F3 = 0 F4 = 3000.0 F5 = 3000.0
1,070,340
P1 = 60.7327 P2 = 124.8970 P3 = 174.9973 P4 = 102.6182 P5 = 236.7549
F1 = 596.0 F2 = 890.7 F3 = 1684.8 F4 = 836.9 F5 = 2991.6
1,071,476
P1 = 75.0000 P2 = 121.8625 P3 = 175.0000 P4 = 123.5708 P5 = 204.5667
F1 = 264.2 F2 = 631.7 F3 = 1580.8 F4 = 1544.8 F5 = 2978.5
1,071,174
2
P1 = 75.0000 P2 = 124.9925 P3 = 174.8302 P4 = 144.6305 P5 = 280.5466
F1 = 985.4 F2 = 1000.0 F3 = 2000.0 F4 = 16.9 F5 = 2997.5
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 166.6609 P5 = 258.3391
F1 = 14.5 F2 = 506.2 F3 = 1443.7 F4 = 2035.7 F5 = 3000.0
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 140.5695 P5 = 284.4305
F1 = 542.7 F2 = 810.9 F3 = 1724.7 F4 = 926.9 F5 = 2994.8
3
P1 = 55.0912 P2 = 124.2366 P3 = 175.0000 P4 = 65.7868 P5 = 229.8853
F1 = 1000.0 F2 = 1000.0 F3 = 2000.0 F4 = 0 F5 = 3000.0
P1 = 55.9316 P2 = 125.0000 P3 = 175.0000 P4 = 63.8360 P5 = 230.2324
F1 = 665.0 F2 = 549.1 F3 = 1669.7 F4 = 1116.2 F5 = 3000.0
P1 = 57.5784 P2 = 123.2328 P3 = 175.0000 P4 = 61.6197 P5 = 232.5691
F1 = 827.3 F2 = 692.0 F3 = 832.6 F4 = 1656.5 F5 = 2991.7
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M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901
Table 4 Economic generation scheduling for case 1 of test system 2 without considering fuel constraints. Interval
GSO
PSO
EP
Generation (MW)
Cost ($)
Generation (MW)
Cost ($)
Generation (MW)
Cost ($)
1
P1 = 50.4239 P2 = 125.0000 P3 = 174.9994 P4 = 48.0101 P5 = 300.0000 P6 = 50.8705 P7 = 124.9943 P8 = 174.9596 P9 = 50.7613 P10 = 299.9810
2,115,313
P1 = 50.2034 P2 = 125.0000 P3 = 175.0000 P4 = 49.9442 P5 = 300.0000 P6 = 49.6952 P7 = 125.0000 P8 = 175.0000 P9 = 50.1572 P10 = 300.0000
2,115,350
P1 = 51.6145 P2 = 124.8992 P3 = 174.5670 P4 = 53.2483 P5 = 300.0000 P6 = 50.6754 P7 = 125.0000 P8 = 175.0000 P9 = 46.1097 P10 = 298.8860
2,115,573
2
P1 = 74.9165 P2 = 124.9975 P3 = 174.9939 P4 = 125.3525 P5 = 300.0000 P6 = 74.9245 P7 = 124.9785 P8 = 175.0000 P9 = 124.9377 P10 = 299.8988
P1 = 74.9996 P2 = 125.0000 P3 = 175.0000 P4 = 128.9239 P5 = 300.0000 P6 = 74.9439 P7 = 124.9775 P8 = 175.0000 P9 = 121.1551 P10 = 300.0000
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 129.2501 P5 = 300.0000 P6 = 72.4004 P7 = 125.0000 P8 = 175.0000 P9 = 123.4143 P10 = 299.9352
3
P1 = 32.5830 P2 = 124.3235 P3 = 174.5465 P4 = 40.0123 P5 = 288.7465 P6 = 30.3859 P7 = 118.7845 P8 = 175.0000 P9 = 40.0000 P10 = 275.6178
P1 = 30.9489 P2 = 125.0000 P3 = 175.0000 P4 = 40.0000 P5 = 300.0000 P6 = 30.9489 P7 = 125.0000 P8 = 175.0000 P9 = 40.0000 P10 = 258.1022
P1 = 25.5707 P2 = 118.5388 P3 = 171.9169 P4 = 40.0144 P5 = 300.0000 P6 = 28.5973 P7 = 118.9537 P8 = 173.2292 P9 = 40.0000 P10 = 283.1789
zation (PSO) and evolutionary programming (EP). All the algorithms i.e. GSO, PSO and EP used in this paper for solving economic fuel scheduling problem are implemented by using MATLAB 7.0 on a PC (Pentium-IV, 80 GB, 3.0 GHz).
6
2.19
GSO PSO EP
2.18 2.17
Test system 1
2.16
Cost ($)
This test system considers five coal-burning generating units which remain on line for a 3-week period. All the generator data containing coefficients of cost and coal consumption, fuel delivery limits, fuel storage limits, load demand and fuel delivered during the scheduling period are given in Tables A.1 and A.2 in Appendix A. Here, three cases are considered. In case 1, economic generation scheduling is done without considering fuel constraints. In case 2, economic generation scheduling is done considering fuel constraints when all the units have sufficient coal. Case 3 is purposely structured to show the interaction of the fuel deliveries and economic dispatch of the generating units when there is fuel shortage at unit 5. The problem is solved by using GSO. Here, the population size (NP) and the maximum iteration number (Nmax) have been selected as 100, and 200 respectively for the test system under consideration. The same problem is solved by using PSO and EP. In case of PSO, parameters are selected as NP = 100, Nmax = 200, wmax = 0.9, wmin = 0.4, c1 = 0.5 and c2 = 0.5. The population size (NP), scaling factor (F) and the maximum iteration number (Nmax) have been selected as 100, 0.25 and 200 respectively in case of EP. Table 1 shows the economic generation scheduling of case 1 without considering fuel constraints. Fig. 2 depicts cost convergence obtained from case 1. Table 2 summarizes the economic generation scheduling of case 2 considering fuel constraints when all the units have sufficient coal. Cost convergence obtained from case
x 10
2.15 2.14 2.13 2.12 2.11
0
20
40
60
80
100
120
140
160
180
200
Iteration Fig. 5. Cost convergence obtained from case 1 of test system 2.
2 is shown in Fig. 3. The dispatch solution for case 3 considering fuel constraints when there is fuel shortage at unit 5 is shown in Table 3. Fig. 4 depicts cost convergence obtained from case 3. It is seen from Tables 1–3 that minimum fuel cost obtained from GSO is less than that obtained from PSO and EP. It is seen from Tables 1 and 2 that minimum fuel cost obtained from economic generation scheduling without considering fuel constraints and economic generation scheduling considering fuel constraints when all the units have sufficient coal is same in case of EP and quite close to each other in case of GSO and PSO. It is seen from Table 3 that minimum fuel cost obtained from case 3 is more compared to case 1 and case 2.
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M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901
Table 5 Economic generation scheduling for case 2 of test system 2 with initial fuel storage (tons) V 01 ¼ 2000; V 02 ¼ 5000; V 03 ¼ 5000; V 04 ¼ 8000; V 05 ¼ 8000; V 06 ¼ 2000; V 07 ¼ 5000; V 08 ¼ 5000; V 09 ¼ 8000; V 010 ¼ 8000. Interval
GSO
PSO
EP
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
1
P1 = 55.4326 P2 = 124.9783 P3 = 175.0000 P4 = 47.7339 P5 = 299.9991 P6 = 47.9198 P7 = 124.9998 P8 = 175.0000 P9 = 48.9865 P10 = 299.9501
F1 = 432.7 F2 = 817.8 F3 = 1412.8 F4 = 30,000 F5 = 2999.0 F6 = 493.2 F7 = 873.5 F8 = 1962.0 F9 = 0 F10 = 2009.0
2,115,341
P1 = 52.6067 P2 = 125.0000 P3 = 175.0000 P4 = 54.8588 P5 = 300.0000 P6 = 52.5346 P7 = 125.0000 P8 = 175.0000 P9 = 40.0000 P10 = 299.9999
F1 = 1000.0 F2 = 999.3 F3 = 896.9 F4 = 2887.2 F5 = 3000.0 F6 = 622.2 F7 = 1000.0 F8 = 983.8 F9 = 1749.2 F10 = 861.4
2,115,402
P1 = 53.0949 P2 = 125.0000 P3 = 175.0000 P4 = 52.8567 P5 = 300.0000 P6 = 52.9295 P7 = 125.0000 P8 = 175.0000 P9 = 41.8547 P10 = 299.2641
F1 = 997.3 F2 = 580.9 F3 = 1884.7 F4 = 2853.8 F5 = 2066.3 F6 = 848.2 F7 = 368.2 F8 = 1780.6 F9 = 993.9 F10 = 1626.2
2,115,672
2
P1 = 74.8972 P2 = 124.9988 P3 = 174.9942 P4 = 125.2755 P5 = 300.0000 P6 = 75.0000 P7 = 125.0000 P8 = 175.0000 P9 = 124.9284 P10 = 299.9059
F1 = 907.9 F2 = 5.7 F3 = 1874.1 F4 = 2999.4 F5 = 3000.0 F6 = 0 F7 = 260.2 F8 = 1981.1 F9 = 911.5 F10 = 2059.9
P1 = 75.0000 P2 = 125.0000 P3 = 174.9999 P4 = 124.8507 P5 = 300.0000 P6 = 75.0000 P7 = 125.0000 P8 = 174.9998 P9 = 125.1496 P10 = 300.0000
F1 = 685.1 F2 = 513.6 F3 = 1807.4 F4 = 2490.6 F5 = 1199.4 F6 = 944.6 F7 = 76.4 F8 = 1553.9 F9 = 2069.2 F10 = 2659.8
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 126.8406 P5 = 300.0000 P6 = 75.0000 P7 = 125.0000 P8 = 174.5579 P9 = 124.1440 P10 = 299.4576
F1 = 159.1 F2 = 741.9 F3 = 1212.5 F4 = 2279.2 F5 = 2804.7 F6 = 42.3 F7 = 699.4 F8 = 970.3 F9 = 2817.8 F10 = 2272.8
3
P1 = 32.2989 P2 = 119.1383 P3 = 175.0000 P4 = 40.0000 P5 = 281.8965 P6 = 32.5553 P7 = 124.2220 P8 = 174.9636 P9 = 40.0000 P10 = 279.9255
F1 = 606.1 F2 = 1000.0 F3 = 1548.2 F4 = 271.8 F5 = 2294.2 F6 = 949.2 F7 = 406.0 F8 = 1911.9 F9 = 3000.0 F10 = 2012.6
P1 = 31.0925 P2 = 117.2705 P3 = 174.9753 P4 = 40.0000 P5 = 300.0000 P6 = 31.3586 P7 = 125.0000 P8 = 174.6601 P9 = 40.0000 P10 = 265.6430
F1 = 665.4 F2 = 22.8 F3 = 1906.8 F4 = 1726.3 F5 = 2917.0 F6 = 886.4 F7 = 69.8 F8 = 2000.0 F9 = 2918.9 F10 = 886.5
P1 = 21.1081 P2 = 122.6257 P3 = 174.4971 P4 = 40.0000 P5 = 288.0505 P6 = 33.5099 P7 = 118.4827 P8 = 175.0000 P9 = 40.0000 P10 = 286.7259
F1 = 874.0 F2 = 951.4 F3 = 1943.7 F4 = 2296.3 F5 = 2280.7 F6 = 645.0 F7 = 225.6 F8 = 2000.0 F9 = 658.7 F10 = 2124.5
Test system 2
6
2.19
GSO PSO EP
2.18 2.17 2.16
Cost ($)
This test system considers 10 coal-burning generating units which remain on line for a 3-week period. The data of test system 1 is doubled to obtain a 10 unit system. Load demand and fuel delivered during the scheduling period are also doubled. Here, three cases are considered. In case 1, economic generation scheduling is done without considering fuel constraints. In case 2, economic generation scheduling is done considering fuel constraints when all the units have sufficient coal. Case 3 is purposely structured to show the interaction of the fuel deliveries and economic dispatch of the generating units when there is fuel shortage at unit 2. The problem is solved by using GSO. Here, the population size (NP) and the maximum iteration number (Nmax) have been selected as 100, and 200 respectively for the test system under consideration. The same problem is solved by using PSO and EP. In case of PSO, parameters are selected as NP = 100, Nmax = 200, wmax = 0.9, wmin = 0.4, c1 = 0.5 and c2 = 0.5. The population size (NP), scaling factor (F) and the maximum iteration number (Nmax) have been selected as 100, 0.25 and 200 respectively in case of EP. Table 4 summarizes the economic generation scheduling of case 1 without considering fuel constraints. Fig. 5 shows cost convergence obtained from case 1. Table 5 shows the economic generation scheduling of case 2 considering fuel constraints when all the units have sufficient coal. Cost convergence obtained from case 2 is depicted in Fig. 6. The dispatch solution for case 3 considering fuel constraints when there is fuel shortage at unit 2 is shown in Table 6. Fig. 7 shows cost convergence obtained from case 3.
x 10
2.15 2.14 2.13 2.12 2.11
0
20
40
60
80
100
120
140
160
180
200
Iteration Fig. 6. Cost convergence obtained from case 2 of test system 2.
It is seen from Tables 4–6 that minimum fuel cost obtained from GSO is less than that obtained from PSO and EP. It is seen from Tables 4 and 5 that minimum fuel cost obtained from economic generation scheduling without considering fuel constraints and economic generation scheduling considering fuel constraints when all the units have sufficient coal is quite close to each other. It is seen from Table 6 that minimum fuel cost obtained from case 3 is more compared to case 1 and case 2.
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M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901
Table 6 Economic generation scheduling for case 3 of test system 2 V 01 ¼ 2000; V 02 ¼ 500; V 03 ¼ 5000; V 04 ¼ 8000; V 05 ¼ 8000; V 06 ¼ 2000; V 07 ¼ 5000; V 08 ¼ 5000; V 09 ¼ 8000; V 010 ¼ 8000. Interval
GSO
with
PSO
initial
fuel
storage
(tons)
EP
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
1
P1 = 64.3155 P2 = 80.8564 P3 = 175.0000 P4 = 72.0319 P5 = 299.8533 P6 = 52.7862 P7 = 125.0000 P8 = 175.0000 P9 = 60.0350 P10 = 295.1217
F1 = 0 F2 = 996.7 F3 = 2000.0 F4 = 2943.5 F5 = 2997.6 F6 = 0 F7 = 0 F8 = 1999.7 F9 = 2451.8 F10 = 610.6
2,129,740
P1 = 66.1757 P2 = 69.0337 P3 = 171.8788 P4 = 42.8439 P5 = 300.0000 P6 = 58.6313 P7 = 121.2165 P8 = 175.0000 P9 = 95.4076 P10 = 299.8126
F1 = 391.2 F2 = 922.0 F3 = 2000.0 F4 = 2118.4 F5 = 2010.4 F6 = 998.3 F7 = 30.2 F8 = 1632.3 F9 = 3000.0 F10 = 897.2
2,130,525
P1 = 67.2790 P2 = 48.4807 P3 = 175.0000 P4 = 85.4257 P5 = 300.0000 P6 = 58.4337 P7 = 121.7879 P8 = 175.0000 P9 = 73.6433 P10 = 294.9496
F1 = 414.4 F2 = 921.0 F3 = 1504.3 F4 = 2219.5 F5 = 1479.6 F6 = 638.2 F7 = 866.1 F8 = 1894.6 F9 = 1336.5 F10 = 2725.7
2,130,572
2
P1 = 75.0000 P2 = 76.1953 P3 = 175.0000 P4 = 123.3936 P5 = 299.9087 P6 = 75.0000 P7 = 125.0000 P8 = 175.0000 P9 = 175.8587 P10 = 299.6437
F1 = 1000.0 F2 = 1000.0 F3 = 2000.0 F4 = 3000.0 F5 = 2167.2 F6 = 1000.0 F7 = 0 F8 = 0 F9 = 3000.0 F10 = 832.8
P1 = 74.9798 P2 = 84.0597 P3 = 175.0000 P4 = 141.9937 P5 = 300.0000 P6 = 75.0000 P7 = 125.0000 P8 = 175.0000 P9 = 148.9846 P10 = 299.9823
F1 = 0 F2 = 1000.0 F3 = 1991.9 F4 = 2714.7 F5 = 2199.2 F6 = 969.1 F7 = 992.2 F8 = 1335.5 F9 = 1987.4 F10 = 810.0
P1 = 75.0000 P2 = 91.7479 P3 = 175.0000 P4 = 132.8599 P5 = 300.0000 P6 = 73.8834 P7 = 124.9989 P8 = 174.8171 P9 = 152.5742 P10 = 299.1186
F1 = 618.9 F2 = 855.6 F3 = 1360.8 F4 = 1878.7 F5 = 2712.4 F6 = 514.9 F7 = 670.3 F8 = 645.8 F9 = 2398.2 F10 = 2344.5
3
P1 = 45.9335 P2 = 60.1562 P3 = 173.3245 P4 = 43.7764 P5 = 299.9995 P6 = 44.6254 P7 = 121.8623 P8 = 174.8939 P9 = 40.0043 P10 = 295.4239
F1 = 1000.0 F2 = 1000.0 F3 = 1999.8 F4 = 417.5 F5 = 2999.9 F6 = 999.9 F7 = 942.6 F8 = 2000.0 F9 = 3.2 F10 = 2637.0
P1 = 40.8748 P2 = 58.5879 P3 = 175.0000 P4 = 40.0000 P5 = 300.0000 P6 = 63.1483 P7 = 124.1625 P8 = 175.0000 P9 = 40.0000 P10 = 283.2265
F1 = 673.2 F2 = 989.8 F3 = 2000.0 F4 = 87.3 F5 = 2906.3 F6 = 49.2 F7 = 1000.0 F8 = 1043.2 F9 = 2449.8 F10 = 2801.1
P1 = 47.4505 P2 = 57.8164 P3 = 175.0000 P4 = 40.0000 P5 = 300.0000 P6 = 45.5320 P7 = 125.0000 P8 = 172.6829 P9 = 40.0000 P10 = 296.5182
F1 = 831.2 F2 = 982.3 F3 = 1800.5 F4 = 1883.0 F5 = 218.5 F6 = 799.2 F7 = 840.3 F8 = 1223.9 F9 = 2581.5 F10 = 2839.6
6
2.22
x 10
Conclusion
2.21
GSO PSO EP
2.2
This paper examines the usefulness of the group search optimization for solving economic fuel dispatch. The results show that fuel consumption can be adequately controlled to satisfy constraints imposed by suppliers using the proposed method. Sum additional amount of fuel is usually required in this method to serve the same power demand, but the cost of this fuel is generally much less than the penalty that could be imposed for violating the fuel system constraints. The test results are compared with those obtained from particle swarm optimization and evolutionary programming. It has been observed from the comparison that the proposed group search optimization has the ability to converge to a better quality solution than particle swarm optimization and evolutionary programming.
Cost ($)
2.19 2.18 2.17 2.16 2.15 2.14 2.13 2.12
0
20
40
60
80
100
120
140
160
180
200
Iteration
Appendix A See Tables A.1 and A.2.
Fig. 7. Cost convergence obtained from case 3 of test system 2.
Table A.1 Generator characteristics. Unit
1 2 3 4 5
P min i MW
P max i
ai
bi
ci
di
ei
ai
bi
ci
$/h
$/MW h
$/(MW)2 h
$/h
rad/MW
ton/h
ton/MW h
ton/(MW)2 h
F min i ton
F max i
MW
ton
V min i ton
ton
V max i
20 20 30 40 50
75 125 175 250 300
25 60 100 120 40
2.0 1.8 2.1 2.2 1.8
0.0080 0.0030 0.0012 0.0040 0.0015
10 20 30 40 50
0.012 0.010 0.009 0.008 0.007
0.83612 2.00669 3.34448 4.01338 1.33779
0.066889 0.060200 0.070230 0.073578 0.060200
0.00026756 0.00010033 0.00004013 0.00013378 0.00005017
0 0 0 0 0
1000 1000 2000 3000 3000
0 0 0 0 0
10,000 10,000 20,000 30,000 30,000
M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901 Table A.2 Load demand and fuel delivered during scheduling period. Interval
Duration (h)
Load demand PD (MW)
Fuel delivered FD (ton)
1 2 3
168 168 168
700 800 650
7000 7000 7000
References [1] Trefny FJ, Lee KY. Economic fuel dispatch. IEEE Trans Power Apparatus Syst PAS-100 1981;7:3468–77. [2] Vemuri S, Kumar ABR, Hackett DF, Eisenhauer JT, Lugtu R. Fuel resource scheduling. Part-I – overview of an energy management problem. IEEE Trans Power Apparatus Syst PAS-103 1984;7:1542–8. [3] Kumar ABR, Vemuri S. Fuel resource scheduling, Part-II – constrained economic dispatch. IEEE Trans Power Apparatus Syst PAS-103 1984;7:1549–55. [4] Kumar ABR, Vemuri S, Gibbs LA, Hackett DF, Eisenhauer JT. Fuel resource scheduling, Part-III – the short-term problem. IEEE Trans Power Apparatus Syst PAS-103 1984;7:1556–61.
901
[5] Asgarpoor Sohrab. Comparison of linear, nonlinear, and network flow programming techniques in fuel scheduling. Elect Power Syst Res 1994;30(3):169–74. [6] Wood AJ, Wollenberg BF. Power generation, operation and control. New York: Wiley; 1996. [7] He S, Wu QH, Saunders JR. A novel group search optimizer inspired by animal behavioral ecology. In: IEEE congress on evolutionary computation, CAS; 2006. [8] He S, Wu QH, Saunders JR. Group search optimizer: an optimization algorithm inspired by animal searching behavior. IEEE Trans Evol Comput 2009;13(5):973–90. [9] Shen H, Zhu Y, Zou W, Zhu Z. Group search optimizer algorithm for constrained optimization. Comput Sci Environ Eng EcoInf 2011. [10] Zhong G, Liu F. Optimal design of plate structures with discrete variables by group search optimizer. Adv Sci Lett 2011;4. [11] Yuan B, Ren FM, Zhong GQ, Zhou J. Optimal design of spatial grid structure using group search optimization. Adv Mater Res 2011;243. [12] Mustard D. Numerical integration over the n-dimensional spherical shell. Math Comput 1964;18(88):578–89. [13] Giraldeau LA, Lefebvre L. Exchangeable producer and scrounger roles in a captive flock of feral pigeons – a case for the skill pool effect. Animal Behav 1986;34(3):797–803.
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Group search optimization for economic fuel scheduling 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 12 December 2013 Received in revised form 1 August 2014 Accepted 12 August 2014 Available online 17 September 2014 Keywords: Economic fuel scheduling Fuel constraints Load constraints Group search optimization
a b s t r a c t The economical use of fuel available for the generation of power has become a major concern of electric utilities. This paper presents an approach for economic fuel scheduling problem by using group search optimization. This is a minimization technique that includes the standard load constraints as well as the fuel constraints. The generation schedule is compared to that which would result if fuel constraints were ignored. The comparison shows that fuel consumed can be adequately controlled by adjusting the power output of various generating units so that the power system operates within its fuel limitations and within contractual constraints. It has been found that small additional amount of fuel may be required to serve the same power demand but the additional cost of this fuel may well compensate for the penalty that might otherwise be imposed for not maintaining the fuel contract. Numerical results for two test systems have been presented and the test results obtained from group search optimization are compared with those obtained from particle swarm optimization and evolutionary programming. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction Economic dispatch is an important optimization task in power system operation for allocating generation among the committed units. Its objective is to minimize the total generation cost of units, while satisfying the various physical constraints. Economic fuel scheduling which is perhaps more significant than economic dispatch has popped up because of the sudden concern over fuel shortages. Fuel supplies have imposed increased constraints in their fuel supply contracts to the point that utilities have been forced to reschedule generation on the basis of fuel availability. The objective of the fuel scheduling problem is to minimize the total cost by loading units in such a manner that the standard load constraints and fuel constraints are both satisfied. Several papers have been published in this area of fuel scheduling of thermal units [1–6]. The application of linear programming (LP) to solve fuel scheduling problem by dividing the total time period involved into discrete time increments is presented in [6]. The objective is a function of one or more variables from only one time step. Some constraints are made up of variables drawn from one time step whereas others span two or more time steps. The fuel used by a unit may be obtained from different contracts at different prices. Moreover, the worth of the fuel is dependent upon the amount available and the need for it.
⇑ Fax: +91 33 23357254. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijepes.2014.08.016 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
Group search optimization (GSO) is a biologically realistic algorithm. Inspired by the animal (such as lions and wolves) searching behavior, He et al. [7] proposed GSO in 2006, and discussed the effects of designed parameters on the performance of GSO in 2009 [8]. 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 [8]. Shen et al. [9] investigated the performance of GSO and concluded that GSO is an alternative for constrained optimization. Due to its high efficiency, GSO has been applied in many fields. Moreover, some papers also indicate GSO as solutions to some discrete optimization problems, such as optimal design plate structures with discrete variables [10] and optimal design of spatial grid structure [11]. This paper proposes GSO for solving the economic fuel scheduling problem of thermal generating units. The proposed method is validated by applying it to two test systems. The test results obtained by the proposed method are compared with those obtained from particle swarm optimization (PSO) and evolutionary programming (EP). Problem formulation For convenience the entire scheduling period is divided into a number of sub intervals each having a constant load demand. The fuel scheduling problem of N thermal generating units over M time intervals is described mathematically as follows:
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M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901
Nomenclature Fim F min ; F max i i FDm Fc Pim Pmin ; Pmax i i PDm
Minimize F c ¼
fuel delivered to thermal unit i in interval m lower and upper fuel delivery limits of thermal unit i fuel delivered in interval m total fuel cost output power of thermal unit i in interval m lower and upper generation limits of thermal unit i load demand in interval m
M X N h n oi X t m ai þ bi P im þ ci P 2im þ di sin ei P min P im i
m¼1 i¼1
ð1Þ
subject to
(i) Power balance constraints N X Pim PDm ¼ 0;
m2M
ð2Þ
i¼1
(ii) Fuel delivery constraints N X F im F Dm ¼ 0;
m2M
ð3Þ
i¼1
(iii) Fuel storage constraints
V im ¼ V iðm1Þ þ F im t m ðai þ bi Pim þ ci P2im Þ;
i 2 N;
m2M
ð4Þ
(iv) Generation limits
Pmin 6 Pim 6 Pmax ; i i
i 2 N;
m2M
ð5Þ
m2M
ð6Þ
(v) Fuel delivery limits
F min 6 F im 6 F max ; i i
i 2 N;
(vi) Fuel storage limits
V min 6 V im 6 V max ; i i
i 2 N; m 2 M
ð7Þ
tm duration of subinterval m Vim fuel storage of thermal unit i in interval m V min ; V max lower and upper fuel storage limits of thermal unit i i i V 0i Initial fuel storage of thermal unit i ai, bi, ci, di, ei cost coefficients of ith thermal unit ai, bi, ci fuel consumption coefficients of thermal unit i
rangers who perform random walk. For convenience of computation, 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. In optimization problems, unknown optima can be taken as open patches randomly distributed in a search space. Group members therefore search for the patches by moving over the search space. It is also assumed that the producer and the scroungers do not differ in their relevant phenotypic characteristics. Therefore, they can switch between the two roles [13]. At each iteration, a group member, which is located in the most promising area and adopts animal scanning to seek the optimal resource and conferred 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. Producer by its vision ability scans the search space for the better states. Vision ability is the ability of testing some points around the producer current position. The producer scans three points around its position in certain distances and head angles.
Group search optimization Group search optimization (GSO) is a population based optimization algorithm which is inspired by animal searching behavior and group living theory. The framework is based on the producer–scrounger (PS) model which assumes that group members search 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 [8]. 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 ndimensional search space, the ith member of the GSO at the kth searching iteration has a current position X ki 2 Rn and a head angle /ki
¼
ð/ki1 ; . . . ; /kiðn1Þ Þ k
n1
2R
k
n1 Y
n1 Y
Choose scrounger and perform scrounging
Dispersed the rest members to perform ranging Evaluate members
Termination Criterion Satisfied?
q¼1 k
Choose producer and perform producing
NO
cosð/kiq Þ
dij ¼ sinð/kiðj1Þ Þ
Generate and evaluate initial members
. The search direction of the ith mem-
k
ber, Dki ð/ki Þ ¼ ðdi1 ; . . . ; din Þ 2 Rn that can be calculated from /ki via polar to Cartesian coordinate transformation [12]
di1 ¼
Start
cosð/kiq Þ j ¼ 2; . . . ; ðn 1Þ
ð8Þ
q¼j
Yes
k
din ¼ sinð/kiðn1Þ Þ GSO algorithm consists of three kinds of members, i.e., producers and scroungers whose behavior are based on the PS model; and
Terminate Fig. 1. Flowchart of the GSO algorithm.
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M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901
Table 1 Economic generation scheduling for case 1 of test system 1 without considering fuel constraints. Interval
GSO
PSO
EP
Generation (MW)
Cost ($)
Generation (MW)
Cost ($)
Generation (MW)
Cost ($)
1
P1 = 50.2586 P2 = 125.0000 P3 = 175.0000 P4 = 49.7415 P5 = 299.9997
1,057,633
P1 = 50.5020 P2 = 125.0000 P3 = 175.0000 P4 = 49.4980 P5 = 300.0000
1,057,640
P1 = 52.5608 P2 = 125.0000 P3 = 175.0000 P4 = 47.5294 P5 = 299.9098
1,057,655
2
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 125.0000 P5 = 299.9999
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 125.0000 P5 = 300.0000
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 125.0182 P5 = 299.9818
3
P1 = 31.8582 P2 = 120.1521 P3 = 175.0000 P4 = 40.0000 P5 = 282.9895
P1 = 31.6853 P2 = 124.2009 P3 = 175.0000 P4 = 40.0000 P5 = 279.1138
P1 = 30.5297 P2 = 117.5533 P3 = 175.0000 P4 = 40.0000 P5 = 286.9170
1.095
x 10
X z ¼ X kP þ r 1 lmax Dkp ð/k Þ r 2 hmax X r ¼ X kP þ r 1 lmax DkP /k þ 2 r 2 hmax k k k X l ¼ X P þ r 1 lmax DP / 2
6
GSO PSO EP
1.09 1.085
Cost ($)
ð10Þ ð11Þ
where XP is position of the producer, r1 is a normally distributed random number with mean 0 and standard deviation 1 and r2 is a uniformly distributed random number in the range of (0, 1), lmax is maximum pursuit distance and hmax is maximum pursuit angle. (2) The producer will then find the best point. If the best point has a better value in comparison with its current position, producer will fly to that point. If not, it will stay in its current position and turn its head using Eq. (12).
1.08 1.075 1.07 1.065 1.06 1.055
ð9Þ
0
20
40
60
80
100
120
140
160
180
/kþ1 ¼ /k þ r2 amax
200
Iteration
ð12Þ
1
Fig. 2. Cost convergence obtained from case 1 of test system 1.
where amax e R is the maximum turning angle. (3) If the producer cannot find a better area after a iterations, it will turn its head back to zero degree as follows:
At the kth iteration, the producer behaves as follows:
/kþa ¼ /k
(1) The producer scans at zero degree and tests three points toward its position using Eqs. (9)–(11).
ð13Þ
where a e R1 is a constant.
Table 2 Economic generation scheduling for case 2 of test system 1 with initial fuel storage (tons) V 01 ¼ 2000; V 02 ¼ 5000; V 03 ¼ 5000; V 04 ¼ 8000; V 05 ¼ 8000. Interval
GSO
PSO
EP
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
1
P1 = 50.2938 P2 = 125.0000 P3 = 174.9968 P4 = 49.7514 P5 = 299.9579
F1 = 728.5 F2 = 281.6 F3 = 742.1 F4 = 2800.0 F5 = 2447.6
1,057,639
P1 = 49.9704 P2 = 125.0000 P3 = 175.0000 P4 = 50.0296 P5 = 300.0000
F1 = 1000.0 F2 = 558.7 F3 = 2000.0 F4 = 1498.1 F5 = 1943.2
1,057,644
P1 = 49.9074 P2 = 125.0000 P3 = 175.0000 P4 = 50.3265 P5 = 299.7661
F1 = 391.5 F2 = 652.1 F3 = 1916.7 F4 = 3000.0 F5 = 1039.6
1,057,655
2
P1 = 75.0000 P2 = 125.0000 P3 = 174.9957 P4 = 125.0118 P5 = 299.9924
F1 = 291.6 F2 = 848.5 F3 = 1723.5 F4 = 1662.0 F5 = 2474.3
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 125.0000 P5 = 300.0000
F1 = 621.6 F2 = 283.0 F3 = 2000.0 F4 = 2145.8 F5 = 1949.6
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 125.0030 P5 = 299.9970
F1 = 496.2 F2 = 551.4 F3 = 1937.1 F4 = 1791.6 F5 = 2223.7
3
P1 = 31.8957 P2 = 121.0814 P3 = 175.0000 P4 = 40.0762 P5 = 281.9465
F1 = 1000.0 F2 = 713.1 F3 = 1667.6 F4 = 1310.5 F5 = 2308.6
P1 = 31.6587 P2 = 125.0000 P3 = 175.0000 P4 = 40.0000 P5 = 278.3413
F1 = 604.2 F2 = 343.2 F3 = 1343.5 F4 = 2415.2 F5 = 2293.9
P1 = 29.6943 P2 = 118.9272 P3 = 175.0000 P4 = 40.0000 P5 = 286.3785
F1 = 459.9 F2 = 504.4 F3 = 1605.2 F4 = 1618.0 F5 = 2812.6
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M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901 6
6
x 10
1.09
1.105 GSO PSO EP
1.08
1.095
1.075
1.09
1.07
1.085
1.065
1.08
1.06
1.075
1.055
1.07 0
20
40
60
80
100
120
140
160
180
GSO PSO EP
1.1
Cost ($)
Cost ($)
1.085
x 10
200
0
20
40
60
80
100
120
140
160
180
200
Iteration
Iteration
Fig. 4. Cost convergence obtained from case 3 of test system 1.
Fig. 3. Cost convergence obtained from case 2 of test system 1.
Parameter setting During each iteration, some of group members are selected as scroungers. The scroungers will keep searching for opportunities to join the resources found by the producer. At the kth iteration, the area copying behavior of the ith scrounger can be modeled as a random walk toward the producer using Eq. (14)
X kþ1 ¼ X ki þ r 3 ðX kP X ki Þ i
ð14Þ
where X ki is the position of the ith scrounger at the kth iteration. r3 is a uniform random number in the range of (0, 1). Operator ‘‘’’ is the Hadamard product, which calculates the entry wise product of the two vectors. At each iteration, some of group members are selected as rangers. Rangers are dispersed from their positions and randomly walk at search space. At the kth iteration, a ranger generates a random head angle /i using Eq. (14), and then it chooses a random distance using Eq. (15) and move to the new point using Eq. (16).
li ¼ br1 lmax
ð15Þ
¼ X ki þ li Dki ð/kþ1 Þ X kþ1 i
ð16Þ
where b is a constant and r1 is a normally distributed random number with mean 0 and standard deviation 1.
The initial population of GSO is generated uniformly at random in the search space. The initial head angle /0 of each individual pffiffiffiffiffiffiffiffiffiffiffiis ffi set to be (P/4, . . ., P/4). The constant a is given by roundð n þ 1Þ 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
qX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ðU i Li Þ2 i¼1
ð17Þ
where Li and Ui are the lower and upper bounds for the ith dimension. The most important control parameter that affects the search performance 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. Simulation results The proposed method has been applied to two test systems. In order to show the effectiveness of the proposed GSO approach, these two test systems are solved by using particle swarm optimi-
Table 3 Economic generation scheduling for case 3 of test system 1 with initial fuel storage (tons) V 01 ¼ 2000; V 02 ¼ 5000; V 03 ¼ 5000; V 04 ¼ 8000; V 05 ¼ 500. Interval
GSO
PSO
EP
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
1
P1 = 74.6392 P2 = 124.6857 P3 = 174.9533 P4 = 112.0461 P5 = 213.6754
F1 = 0 F2 = 1000.0 F3 = 0 F4 = 3000.0 F5 = 3000.0
1,070,340
P1 = 60.7327 P2 = 124.8970 P3 = 174.9973 P4 = 102.6182 P5 = 236.7549
F1 = 596.0 F2 = 890.7 F3 = 1684.8 F4 = 836.9 F5 = 2991.6
1,071,476
P1 = 75.0000 P2 = 121.8625 P3 = 175.0000 P4 = 123.5708 P5 = 204.5667
F1 = 264.2 F2 = 631.7 F3 = 1580.8 F4 = 1544.8 F5 = 2978.5
1,071,174
2
P1 = 75.0000 P2 = 124.9925 P3 = 174.8302 P4 = 144.6305 P5 = 280.5466
F1 = 985.4 F2 = 1000.0 F3 = 2000.0 F4 = 16.9 F5 = 2997.5
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 166.6609 P5 = 258.3391
F1 = 14.5 F2 = 506.2 F3 = 1443.7 F4 = 2035.7 F5 = 3000.0
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 140.5695 P5 = 284.4305
F1 = 542.7 F2 = 810.9 F3 = 1724.7 F4 = 926.9 F5 = 2994.8
3
P1 = 55.0912 P2 = 124.2366 P3 = 175.0000 P4 = 65.7868 P5 = 229.8853
F1 = 1000.0 F2 = 1000.0 F3 = 2000.0 F4 = 0 F5 = 3000.0
P1 = 55.9316 P2 = 125.0000 P3 = 175.0000 P4 = 63.8360 P5 = 230.2324
F1 = 665.0 F2 = 549.1 F3 = 1669.7 F4 = 1116.2 F5 = 3000.0
P1 = 57.5784 P2 = 123.2328 P3 = 175.0000 P4 = 61.6197 P5 = 232.5691
F1 = 827.3 F2 = 692.0 F3 = 832.6 F4 = 1656.5 F5 = 2991.7
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M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901
Table 4 Economic generation scheduling for case 1 of test system 2 without considering fuel constraints. Interval
GSO
PSO
EP
Generation (MW)
Cost ($)
Generation (MW)
Cost ($)
Generation (MW)
Cost ($)
1
P1 = 50.4239 P2 = 125.0000 P3 = 174.9994 P4 = 48.0101 P5 = 300.0000 P6 = 50.8705 P7 = 124.9943 P8 = 174.9596 P9 = 50.7613 P10 = 299.9810
2,115,313
P1 = 50.2034 P2 = 125.0000 P3 = 175.0000 P4 = 49.9442 P5 = 300.0000 P6 = 49.6952 P7 = 125.0000 P8 = 175.0000 P9 = 50.1572 P10 = 300.0000
2,115,350
P1 = 51.6145 P2 = 124.8992 P3 = 174.5670 P4 = 53.2483 P5 = 300.0000 P6 = 50.6754 P7 = 125.0000 P8 = 175.0000 P9 = 46.1097 P10 = 298.8860
2,115,573
2
P1 = 74.9165 P2 = 124.9975 P3 = 174.9939 P4 = 125.3525 P5 = 300.0000 P6 = 74.9245 P7 = 124.9785 P8 = 175.0000 P9 = 124.9377 P10 = 299.8988
P1 = 74.9996 P2 = 125.0000 P3 = 175.0000 P4 = 128.9239 P5 = 300.0000 P6 = 74.9439 P7 = 124.9775 P8 = 175.0000 P9 = 121.1551 P10 = 300.0000
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 129.2501 P5 = 300.0000 P6 = 72.4004 P7 = 125.0000 P8 = 175.0000 P9 = 123.4143 P10 = 299.9352
3
P1 = 32.5830 P2 = 124.3235 P3 = 174.5465 P4 = 40.0123 P5 = 288.7465 P6 = 30.3859 P7 = 118.7845 P8 = 175.0000 P9 = 40.0000 P10 = 275.6178
P1 = 30.9489 P2 = 125.0000 P3 = 175.0000 P4 = 40.0000 P5 = 300.0000 P6 = 30.9489 P7 = 125.0000 P8 = 175.0000 P9 = 40.0000 P10 = 258.1022
P1 = 25.5707 P2 = 118.5388 P3 = 171.9169 P4 = 40.0144 P5 = 300.0000 P6 = 28.5973 P7 = 118.9537 P8 = 173.2292 P9 = 40.0000 P10 = 283.1789
zation (PSO) and evolutionary programming (EP). All the algorithms i.e. GSO, PSO and EP used in this paper for solving economic fuel scheduling problem are implemented by using MATLAB 7.0 on a PC (Pentium-IV, 80 GB, 3.0 GHz).
6
2.19
GSO PSO EP
2.18 2.17
Test system 1
2.16
Cost ($)
This test system considers five coal-burning generating units which remain on line for a 3-week period. All the generator data containing coefficients of cost and coal consumption, fuel delivery limits, fuel storage limits, load demand and fuel delivered during the scheduling period are given in Tables A.1 and A.2 in Appendix A. Here, three cases are considered. In case 1, economic generation scheduling is done without considering fuel constraints. In case 2, economic generation scheduling is done considering fuel constraints when all the units have sufficient coal. Case 3 is purposely structured to show the interaction of the fuel deliveries and economic dispatch of the generating units when there is fuel shortage at unit 5. The problem is solved by using GSO. Here, the population size (NP) and the maximum iteration number (Nmax) have been selected as 100, and 200 respectively for the test system under consideration. The same problem is solved by using PSO and EP. In case of PSO, parameters are selected as NP = 100, Nmax = 200, wmax = 0.9, wmin = 0.4, c1 = 0.5 and c2 = 0.5. The population size (NP), scaling factor (F) and the maximum iteration number (Nmax) have been selected as 100, 0.25 and 200 respectively in case of EP. Table 1 shows the economic generation scheduling of case 1 without considering fuel constraints. Fig. 2 depicts cost convergence obtained from case 1. Table 2 summarizes the economic generation scheduling of case 2 considering fuel constraints when all the units have sufficient coal. Cost convergence obtained from case
x 10
2.15 2.14 2.13 2.12 2.11
0
20
40
60
80
100
120
140
160
180
200
Iteration Fig. 5. Cost convergence obtained from case 1 of test system 2.
2 is shown in Fig. 3. The dispatch solution for case 3 considering fuel constraints when there is fuel shortage at unit 5 is shown in Table 3. Fig. 4 depicts cost convergence obtained from case 3. It is seen from Tables 1–3 that minimum fuel cost obtained from GSO is less than that obtained from PSO and EP. It is seen from Tables 1 and 2 that minimum fuel cost obtained from economic generation scheduling without considering fuel constraints and economic generation scheduling considering fuel constraints when all the units have sufficient coal is same in case of EP and quite close to each other in case of GSO and PSO. It is seen from Table 3 that minimum fuel cost obtained from case 3 is more compared to case 1 and case 2.
899
M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901
Table 5 Economic generation scheduling for case 2 of test system 2 with initial fuel storage (tons) V 01 ¼ 2000; V 02 ¼ 5000; V 03 ¼ 5000; V 04 ¼ 8000; V 05 ¼ 8000; V 06 ¼ 2000; V 07 ¼ 5000; V 08 ¼ 5000; V 09 ¼ 8000; V 010 ¼ 8000. Interval
GSO
PSO
EP
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
1
P1 = 55.4326 P2 = 124.9783 P3 = 175.0000 P4 = 47.7339 P5 = 299.9991 P6 = 47.9198 P7 = 124.9998 P8 = 175.0000 P9 = 48.9865 P10 = 299.9501
F1 = 432.7 F2 = 817.8 F3 = 1412.8 F4 = 30,000 F5 = 2999.0 F6 = 493.2 F7 = 873.5 F8 = 1962.0 F9 = 0 F10 = 2009.0
2,115,341
P1 = 52.6067 P2 = 125.0000 P3 = 175.0000 P4 = 54.8588 P5 = 300.0000 P6 = 52.5346 P7 = 125.0000 P8 = 175.0000 P9 = 40.0000 P10 = 299.9999
F1 = 1000.0 F2 = 999.3 F3 = 896.9 F4 = 2887.2 F5 = 3000.0 F6 = 622.2 F7 = 1000.0 F8 = 983.8 F9 = 1749.2 F10 = 861.4
2,115,402
P1 = 53.0949 P2 = 125.0000 P3 = 175.0000 P4 = 52.8567 P5 = 300.0000 P6 = 52.9295 P7 = 125.0000 P8 = 175.0000 P9 = 41.8547 P10 = 299.2641
F1 = 997.3 F2 = 580.9 F3 = 1884.7 F4 = 2853.8 F5 = 2066.3 F6 = 848.2 F7 = 368.2 F8 = 1780.6 F9 = 993.9 F10 = 1626.2
2,115,672
2
P1 = 74.8972 P2 = 124.9988 P3 = 174.9942 P4 = 125.2755 P5 = 300.0000 P6 = 75.0000 P7 = 125.0000 P8 = 175.0000 P9 = 124.9284 P10 = 299.9059
F1 = 907.9 F2 = 5.7 F3 = 1874.1 F4 = 2999.4 F5 = 3000.0 F6 = 0 F7 = 260.2 F8 = 1981.1 F9 = 911.5 F10 = 2059.9
P1 = 75.0000 P2 = 125.0000 P3 = 174.9999 P4 = 124.8507 P5 = 300.0000 P6 = 75.0000 P7 = 125.0000 P8 = 174.9998 P9 = 125.1496 P10 = 300.0000
F1 = 685.1 F2 = 513.6 F3 = 1807.4 F4 = 2490.6 F5 = 1199.4 F6 = 944.6 F7 = 76.4 F8 = 1553.9 F9 = 2069.2 F10 = 2659.8
P1 = 75.0000 P2 = 125.0000 P3 = 175.0000 P4 = 126.8406 P5 = 300.0000 P6 = 75.0000 P7 = 125.0000 P8 = 174.5579 P9 = 124.1440 P10 = 299.4576
F1 = 159.1 F2 = 741.9 F3 = 1212.5 F4 = 2279.2 F5 = 2804.7 F6 = 42.3 F7 = 699.4 F8 = 970.3 F9 = 2817.8 F10 = 2272.8
3
P1 = 32.2989 P2 = 119.1383 P3 = 175.0000 P4 = 40.0000 P5 = 281.8965 P6 = 32.5553 P7 = 124.2220 P8 = 174.9636 P9 = 40.0000 P10 = 279.9255
F1 = 606.1 F2 = 1000.0 F3 = 1548.2 F4 = 271.8 F5 = 2294.2 F6 = 949.2 F7 = 406.0 F8 = 1911.9 F9 = 3000.0 F10 = 2012.6
P1 = 31.0925 P2 = 117.2705 P3 = 174.9753 P4 = 40.0000 P5 = 300.0000 P6 = 31.3586 P7 = 125.0000 P8 = 174.6601 P9 = 40.0000 P10 = 265.6430
F1 = 665.4 F2 = 22.8 F3 = 1906.8 F4 = 1726.3 F5 = 2917.0 F6 = 886.4 F7 = 69.8 F8 = 2000.0 F9 = 2918.9 F10 = 886.5
P1 = 21.1081 P2 = 122.6257 P3 = 174.4971 P4 = 40.0000 P5 = 288.0505 P6 = 33.5099 P7 = 118.4827 P8 = 175.0000 P9 = 40.0000 P10 = 286.7259
F1 = 874.0 F2 = 951.4 F3 = 1943.7 F4 = 2296.3 F5 = 2280.7 F6 = 645.0 F7 = 225.6 F8 = 2000.0 F9 = 658.7 F10 = 2124.5
Test system 2
6
2.19
GSO PSO EP
2.18 2.17 2.16
Cost ($)
This test system considers 10 coal-burning generating units which remain on line for a 3-week period. The data of test system 1 is doubled to obtain a 10 unit system. Load demand and fuel delivered during the scheduling period are also doubled. Here, three cases are considered. In case 1, economic generation scheduling is done without considering fuel constraints. In case 2, economic generation scheduling is done considering fuel constraints when all the units have sufficient coal. Case 3 is purposely structured to show the interaction of the fuel deliveries and economic dispatch of the generating units when there is fuel shortage at unit 2. The problem is solved by using GSO. Here, the population size (NP) and the maximum iteration number (Nmax) have been selected as 100, and 200 respectively for the test system under consideration. The same problem is solved by using PSO and EP. In case of PSO, parameters are selected as NP = 100, Nmax = 200, wmax = 0.9, wmin = 0.4, c1 = 0.5 and c2 = 0.5. The population size (NP), scaling factor (F) and the maximum iteration number (Nmax) have been selected as 100, 0.25 and 200 respectively in case of EP. Table 4 summarizes the economic generation scheduling of case 1 without considering fuel constraints. Fig. 5 shows cost convergence obtained from case 1. Table 5 shows the economic generation scheduling of case 2 considering fuel constraints when all the units have sufficient coal. Cost convergence obtained from case 2 is depicted in Fig. 6. The dispatch solution for case 3 considering fuel constraints when there is fuel shortage at unit 2 is shown in Table 6. Fig. 7 shows cost convergence obtained from case 3.
x 10
2.15 2.14 2.13 2.12 2.11
0
20
40
60
80
100
120
140
160
180
200
Iteration Fig. 6. Cost convergence obtained from case 2 of test system 2.
It is seen from Tables 4–6 that minimum fuel cost obtained from GSO is less than that obtained from PSO and EP. It is seen from Tables 4 and 5 that minimum fuel cost obtained from economic generation scheduling without considering fuel constraints and economic generation scheduling considering fuel constraints when all the units have sufficient coal is quite close to each other. It is seen from Table 6 that minimum fuel cost obtained from case 3 is more compared to case 1 and case 2.
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M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901
Table 6 Economic generation scheduling for case 3 of test system 2 V 01 ¼ 2000; V 02 ¼ 500; V 03 ¼ 5000; V 04 ¼ 8000; V 05 ¼ 8000; V 06 ¼ 2000; V 07 ¼ 5000; V 08 ¼ 5000; V 09 ¼ 8000; V 010 ¼ 8000. Interval
GSO
with
PSO
initial
fuel
storage
(tons)
EP
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
Generation (MW)
Fuel delivered (tons)
Cost ($)
1
P1 = 64.3155 P2 = 80.8564 P3 = 175.0000 P4 = 72.0319 P5 = 299.8533 P6 = 52.7862 P7 = 125.0000 P8 = 175.0000 P9 = 60.0350 P10 = 295.1217
F1 = 0 F2 = 996.7 F3 = 2000.0 F4 = 2943.5 F5 = 2997.6 F6 = 0 F7 = 0 F8 = 1999.7 F9 = 2451.8 F10 = 610.6
2,129,740
P1 = 66.1757 P2 = 69.0337 P3 = 171.8788 P4 = 42.8439 P5 = 300.0000 P6 = 58.6313 P7 = 121.2165 P8 = 175.0000 P9 = 95.4076 P10 = 299.8126
F1 = 391.2 F2 = 922.0 F3 = 2000.0 F4 = 2118.4 F5 = 2010.4 F6 = 998.3 F7 = 30.2 F8 = 1632.3 F9 = 3000.0 F10 = 897.2
2,130,525
P1 = 67.2790 P2 = 48.4807 P3 = 175.0000 P4 = 85.4257 P5 = 300.0000 P6 = 58.4337 P7 = 121.7879 P8 = 175.0000 P9 = 73.6433 P10 = 294.9496
F1 = 414.4 F2 = 921.0 F3 = 1504.3 F4 = 2219.5 F5 = 1479.6 F6 = 638.2 F7 = 866.1 F8 = 1894.6 F9 = 1336.5 F10 = 2725.7
2,130,572
2
P1 = 75.0000 P2 = 76.1953 P3 = 175.0000 P4 = 123.3936 P5 = 299.9087 P6 = 75.0000 P7 = 125.0000 P8 = 175.0000 P9 = 175.8587 P10 = 299.6437
F1 = 1000.0 F2 = 1000.0 F3 = 2000.0 F4 = 3000.0 F5 = 2167.2 F6 = 1000.0 F7 = 0 F8 = 0 F9 = 3000.0 F10 = 832.8
P1 = 74.9798 P2 = 84.0597 P3 = 175.0000 P4 = 141.9937 P5 = 300.0000 P6 = 75.0000 P7 = 125.0000 P8 = 175.0000 P9 = 148.9846 P10 = 299.9823
F1 = 0 F2 = 1000.0 F3 = 1991.9 F4 = 2714.7 F5 = 2199.2 F6 = 969.1 F7 = 992.2 F8 = 1335.5 F9 = 1987.4 F10 = 810.0
P1 = 75.0000 P2 = 91.7479 P3 = 175.0000 P4 = 132.8599 P5 = 300.0000 P6 = 73.8834 P7 = 124.9989 P8 = 174.8171 P9 = 152.5742 P10 = 299.1186
F1 = 618.9 F2 = 855.6 F3 = 1360.8 F4 = 1878.7 F5 = 2712.4 F6 = 514.9 F7 = 670.3 F8 = 645.8 F9 = 2398.2 F10 = 2344.5
3
P1 = 45.9335 P2 = 60.1562 P3 = 173.3245 P4 = 43.7764 P5 = 299.9995 P6 = 44.6254 P7 = 121.8623 P8 = 174.8939 P9 = 40.0043 P10 = 295.4239
F1 = 1000.0 F2 = 1000.0 F3 = 1999.8 F4 = 417.5 F5 = 2999.9 F6 = 999.9 F7 = 942.6 F8 = 2000.0 F9 = 3.2 F10 = 2637.0
P1 = 40.8748 P2 = 58.5879 P3 = 175.0000 P4 = 40.0000 P5 = 300.0000 P6 = 63.1483 P7 = 124.1625 P8 = 175.0000 P9 = 40.0000 P10 = 283.2265
F1 = 673.2 F2 = 989.8 F3 = 2000.0 F4 = 87.3 F5 = 2906.3 F6 = 49.2 F7 = 1000.0 F8 = 1043.2 F9 = 2449.8 F10 = 2801.1
P1 = 47.4505 P2 = 57.8164 P3 = 175.0000 P4 = 40.0000 P5 = 300.0000 P6 = 45.5320 P7 = 125.0000 P8 = 172.6829 P9 = 40.0000 P10 = 296.5182
F1 = 831.2 F2 = 982.3 F3 = 1800.5 F4 = 1883.0 F5 = 218.5 F6 = 799.2 F7 = 840.3 F8 = 1223.9 F9 = 2581.5 F10 = 2839.6
6
2.22
x 10
Conclusion
2.21
GSO PSO EP
2.2
This paper examines the usefulness of the group search optimization for solving economic fuel dispatch. The results show that fuel consumption can be adequately controlled to satisfy constraints imposed by suppliers using the proposed method. Sum additional amount of fuel is usually required in this method to serve the same power demand, but the cost of this fuel is generally much less than the penalty that could be imposed for violating the fuel system constraints. The test results are compared with those obtained from particle swarm optimization and evolutionary programming. It has been observed from the comparison that the proposed group search optimization has the ability to converge to a better quality solution than particle swarm optimization and evolutionary programming.
Cost ($)
2.19 2.18 2.17 2.16 2.15 2.14 2.13 2.12
0
20
40
60
80
100
120
140
160
180
200
Iteration
Appendix A See Tables A.1 and A.2.
Fig. 7. Cost convergence obtained from case 3 of test system 2.
Table A.1 Generator characteristics. Unit
1 2 3 4 5
P min i MW
P max i
ai
bi
ci
di
ei
ai
bi
ci
$/h
$/MW h
$/(MW)2 h
$/h
rad/MW
ton/h
ton/MW h
ton/(MW)2 h
F min i ton
F max i
MW
ton
V min i ton
ton
V max i
20 20 30 40 50
75 125 175 250 300
25 60 100 120 40
2.0 1.8 2.1 2.2 1.8
0.0080 0.0030 0.0012 0.0040 0.0015
10 20 30 40 50
0.012 0.010 0.009 0.008 0.007
0.83612 2.00669 3.34448 4.01338 1.33779
0.066889 0.060200 0.070230 0.073578 0.060200
0.00026756 0.00010033 0.00004013 0.00013378 0.00005017
0 0 0 0 0
1000 1000 2000 3000 3000
0 0 0 0 0
10,000 10,000 20,000 30,000 30,000
M. Basu / Electrical Power and Energy Systems 64 (2015) 894–901 Table A.2 Load demand and fuel delivered during scheduling period. Interval
Duration (h)
Load demand PD (MW)
Fuel delivered FD (ton)
1 2 3
168 168 168
700 800 650
7000 7000 7000
References [1] Trefny FJ, Lee KY. Economic fuel dispatch. IEEE Trans Power Apparatus Syst PAS-100 1981;7:3468–77. [2] Vemuri S, Kumar ABR, Hackett DF, Eisenhauer JT, Lugtu R. Fuel resource scheduling. Part-I – overview of an energy management problem. IEEE Trans Power Apparatus Syst PAS-103 1984;7:1542–8. [3] Kumar ABR, Vemuri S. Fuel resource scheduling, Part-II – constrained economic dispatch. IEEE Trans Power Apparatus Syst PAS-103 1984;7:1549–55. [4] Kumar ABR, Vemuri S, Gibbs LA, Hackett DF, Eisenhauer JT. Fuel resource scheduling, Part-III – the short-term problem. IEEE Trans Power Apparatus Syst PAS-103 1984;7:1556–61.
901
[5] Asgarpoor Sohrab. Comparison of linear, nonlinear, and network flow programming techniques in fuel scheduling. Elect Power Syst Res 1994;30(3):169–74. [6] Wood AJ, Wollenberg BF. Power generation, operation and control. New York: Wiley; 1996. [7] He S, Wu QH, Saunders JR. A novel group search optimizer inspired by animal behavioral ecology. In: IEEE congress on evolutionary computation, CAS; 2006. [8] He S, Wu QH, Saunders JR. Group search optimizer: an optimization algorithm inspired by animal searching behavior. IEEE Trans Evol Comput 2009;13(5):973–90. [9] Shen H, Zhu Y, Zou W, Zhu Z. Group search optimizer algorithm for constrained optimization. Comput Sci Environ Eng EcoInf 2011. [10] Zhong G, Liu F. Optimal design of plate structures with discrete variables by group search optimizer. Adv Sci Lett 2011;4. [11] Yuan B, Ren FM, Zhong GQ, Zhou J. Optimal design of spatial grid structure using group search optimization. Adv Mater Res 2011;243. [12] Mustard D. Numerical integration over the n-dimensional spherical shell. Math Comput 1964;18(88):578–89. [13] Giraldeau LA, Lefebvre L. Exchangeable producer and scrounger roles in a captive flock of feral pigeons – a case for the skill pool effect. Animal Behav 1986;34(3):797–803.