Expert Systems with Applications 38 (2011) 13527–13531
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Bee colony optimization for combined heat and power economic dispatch M. Basu Department of Power Engineering, Jadavpur University, Kolkata 700098, India
a r t i c l e
i n f o
Keywords: Cogeneration Combined heat and power economic dispatch Bee colony optimization
a b s t r a c t This paper presents a novel optimization approach to the combined heat and power economic dispatch problem by using bee colony optimization algorithm. The algorithm is a swarm-based algorithm inspired by the food foraging behavior of honey bees. The performance of the proposed algorithm is validated by illustration with a test system. The results of the proposed approach are compared with those of particle swarm optimization, real-coded genetic algorithm and evolutionary programing techniques. From numerical results, it is seen that bee colony optimization based approach is able to provide a better solution at a lesser computational effort. Ó 2011 Published by Elsevier Ltd.
1. Introduction The conversion of fossil fuel into electricity is an inefficient process. Even the most modern combined cycle plants are between 50% and 60% efficient. Most of the energy wasted in the conversion process is heat. The principle of combined heat and power, known as cogeneration, is to recover and make beneficial use of this heat and as a result the overall efficiency of the conversion process is increased. Combined heat and power generation has higher energy efficiency and less green house gas emission as compared with the other forms of energy supply. Recently, cogeneration units have been extensively used in utility industry. The heat production capacity of most cogeneration units depends on the power generation and vice versa. The mutual dependencies of heat and power generation introduce a complication in the integration of cogeneration units into the power economic dispatch. The objective of the combined heat and power economic dispatch (CHPED) is to find the optimal point of power and heat generation with minimum fuel cost such that both heat and power demands and other constraints are met while the combined heat and power units are operated in a bounded heat versus power plane. Non-linear optimization methods, such as dual and quadratic programming (Rooijers & Van Amerongen, 1994), and gradient descent approaches, such as Lagrangian relaxation (Guo, Henwood, & van Ooijen, 1996), have been applied for solving CHPED. However, these methods cannot handle nonconvex fuel cost function of the generating units. The advent of stochastic search algorithms has provided alternative approaches for solving CHPED problem. Improved ant colony search algorithm (Song, Chou, & Stonham, 1999), evolutionary programming (Wong & Algie, 2002) Genetic algo-
E-mail address:
[email protected] 0957-4174/$ - see front matter Ó 2011 Published by Elsevier Ltd. doi:10.1016/j.eswa.2011.03.067
rithm (Su & Chiang, 2004), harmonic search algorithm (Vasebi, Fesanghary, & Bathaee, 2007) and multiobjective particle swarm optimization (Wang & Singh, 2008) have been applied to solve CHPED problem. But these methods did not consider transmission loss. Swarm intelligence (Bonabeau, Dorigo, & Theraulaz, 1999; Camazine et al., 2003; Eberhart, Shi, & Kennedy, 2001), a branch of natural inspired algorithms, focuses on the behavior of insect in order to develop some meta-heuristics algorithms. Bee colony optimization (BCO) algorithm (Karaboga, 2005) is a new member of swarm intelligence and it mimics the food foraging behavior of honey bees. This algorithm is simple, robust and capable to solve difficult combinatorial optimization problems. This paper proposes BCO algorithm for solving the CHPED problem. Here, transmission loss is considered. In order to show the validity of the proposed approach, the developed algorithm is illustrated on a test system (Guo et al., 1996). Results obtained from the proposed approach are compared with those obtained from particle swarm optimization (PSO), real-coded genetic algorithm (RCGA) and evolutionary programming (EP). The comparison shows that the proposed BCO based approach achieves lower production cost and CPU time.
2. Formulation of CHPED problem The system under consideration has conventional thermal generators, cogeneration units, and heat-only units. Fig. 1 shows the heat–power feasible operation region of a combined cycle cogeneration unit. The feasible operation is enclosed by the boundary curve ABCDEF. Along the boundary curve BC, the heat capacity increases as the power generation decreases, the heat capacity declines along the curve CD.
13528
M. Basu / Expert Systems with Applications 38 (2011) 13527–13531
3. The bee colony optimization
Power(MW)
B
A
Max. fuel
C
Max. Heat Extraction
F
E Min. fuel
D
Fig. 1. Heat-power feasible operation region for a cogeneration unit.
The power output of the power units and the heat output of heat units are restricted by their own upper and lower limits. The power is generated by conventional thermal generators and cogeneration units while the heat is generated by cogeneration units and heat-only units. The CHPED problem is to determine the unit power and heat production so that the system’s production cost is minimized while the power and heat demands and other constraints are met. It can be mathematically stated as:
" a X
Minimize
b X
F ti ðPi Þ þ
F ci ðPi ; Hi Þ þ
i¼aþ1
i¼1
n X
# F hi ðhi Þ :
ð1Þ
i¼bþ1
Subject to the equilibrium constraints of electricity and heat production, and the capacity limits of each unit: a X
Pi þ
Pi ¼ PD þ PL ;
ð2Þ
i¼aþ1
i¼1 b X
b X
Hi þ
i¼aþ1
n X
Hi ¼ H D ;
ð3Þ
i¼bþ1
6 Pi 6 Pmax Pmin i i
i 2 1; 2; . . . ; a;
ð4Þ
ðHi Þ 6 Pi 6 Pmax ðHi Þ i 2 a þ 1; a þ 2; . . . ; b; Pmin i i
ð5Þ
ðPi Þ 6 Hi 6 Hmax ðPi Þ i 2 a þ 1; a þ 2; . . . ; b; Hmin i i
ð6Þ
6 Hi 6 Hmax Hmin i i
ð7Þ
i 2 b þ 1; b þ 2; . . . ; n:
The active power transmission loss PL can be calculated by the network loss formula:
PL ¼
b X b X i¼1
Pi Bij Pj ;
ð8Þ
j¼1
where Fti, Fci, Fhi are the respective fuel characteristics of the conventional thermal generators, cogeneration units and heat-only units. P is the unit power generation. H is the unit heat production. i 2 ½1; 2; . . . ; a denotes conventional thermal generators. i 2 [a + 1, a + 2, . . . , b] denotes cogeneration units. i 2 [b + 1, b + 2, . . . , n] denotes heat-only units. The operation ranges of conventional thermal generators and heat- only units are expressed in Eqs. (4) and (7) and those for cogeneration units are in Eqs. (5) and (6). The heat and power outputs of the cogeneration units are non-separable and one output will affect the other. HD and PD are the system heat and power demands respectively. Bij the loss coefficient for a network branch connected between generators i and j. Pmin and P max are the unit power capacity limits. Hmin and Hmax are the unit heat capacity limits. Pmin(H), Pmax(H), Hmin(P) and Hmax(P) are the linear inequalities that define the feasible operating region of the cogeneration units.
Social insects have lived on earth for million of years, building nests, organizing production and procuring food. The colonies of social insects are very flexible and can adapt well to the changing environment. This flexibility allows the colony of social insects to be robust and maintain its life in spite of considerable disturbances. The dynamics of the social insect population is a result of the different actions and interactions of individual insects with each other as well as with their environment. The interactions are executed via multitude of various chemical and/or physical signals. The final product of different actions and interactions represents social insect colony behavior. The examples of interaction between individual insects in the colony of social insects are bee dancing during the food procurement, ants’ pheromone secretion and performance of specific acts which signal the other insects to start performing the same action. These communication systems between individual insects contribute to the formation of swarm intelligence.
3.1. Bees in nature A colony of honey bees can extend itself over long distances and in multiple directions simultaneously to exploit a large number of food sources. A colony prospers by deploying its foragers to good fields. In principle, flower patches with plentiful amounts of nectar can be collected with less effort and should be visited by more bees whereas patches with less nectar should receive fewer bees. The foraging process begins in a colony by scout bees that are sent to search for promising flower patches. Scout bees move randomly from one patch to another. When they return to the hive, those scout bees that found a patch which is rated above a certain quality threshold deposit their nectar and go to the dance floor to perform a dance known as waggle dance. This dance is essential for colony communication and contains three pieces of information regarding a flower patch: the direction in which it will be found, its distance from the hive and its quality rating. This information helps the colony to send it bees to flower patches precisely. After waggle dancing on the dance floor, the dancer goes back to the flower patch with follower bees that are waiting inside the hive. More follower bees are sent to more promising patches. This allows the colony to gather food quickly and efficiently. While harvesting from a patch, the bees monitor its food level. If the patch is still good enough as a food source, then it will be advertised in the waggle dance and more bees will be recruited to that source.
3.2. Bee colony optimization algorithm Bee colony optimization (BCO) algorithm is proposed by Karaboga for numerical optimization in 2005. This algorithm mimics the food foraging behavior of honey bees. In BCO algorithm, the colony of bees consists of two groups, scout and employed bees. The scout bees seek a new food source and the employed bees look for a food source within the neighborhood of the food source in their memories. Both scout and employed bees share their information with other bees within the hive. Fig. 2 shows the flowchart of Bee Colony Optimization algorithm. The algorithm starts with ns scout bees randomly distributed in the search space. The nectar amounts of sites visited by ns scout bees are calculated. Sites (m) that have the highest nectar amounts are chosen for neighborhood search. Recruit nb bees for each selected site to explore neighborhood search. The nectar amounts of all (nb m) sites are calculated. Select m sites which have the highest nectar amounts from (nb m) sites to form the next bee population.
13529
M. Basu / Expert Systems with Applications 38 (2011) 13527–13531
The elements of pk should satisfy the constraints given by Eqs. (2)– (7). The BCO algorithm implemented to solve CHPED problem is stated in the following subsections.
Start
Generate randomly n s scout bees
4.1. Initialization The initial scout bee vector pk, k = 1, 2, . . . , n, is determined by setting P U(Pmin, Pmax) and H U(Hmin, Hmax). U(a, b) denotes a uniform random variable ranging over [a, b].
Calculate the nectar amount of each site visited by the scout bee Sort the visited sites in descending order according to their nectar amounts
4.2. Evaluation of fitness Evaluate the fitness of the initial population using the following equation:
Select the best m visited sites out of n s visited sites
fitness ¼ "
Generate nb neighborhood search for each selected site
1 a P
F ti ðP i Þ þ
i¼1
Calculate the nectar amount of each neighborhood search site
b P i¼aþ1
#:
n P
F ci ðP i ; Hi Þ þ
ð11Þ
F ðH i Þ
i¼bþ1
4.3. Selection of initial population
Sort the ( nb × m ) sites in descending order according to their nectar amounts
Select m best solutions on the basis of highest fitness for neighborhood search and determine the size of neighborhood search of each best solution.
Select the best m visited sites out of (nb × m ) visited sites
4.4. Generation of neighborhood solution Generate nb solutions around each selected solution within neighborhood search using the following equation:
Is maximum generation reached?
P0kj ¼ Pij þ F Nð0; 1Þ Pmax Pmin ; j j
No
i ¼ 1; . . . ; m; k
¼ 1; . . . ; nb ; j ¼ 1; . . . ; b; H0kj ¼ Hij þ F Nð0; 1Þ Hmax Hmin ; j j
Yes
¼ 1; . . . ; nb ; j ¼ a þ 1; . . . ; n:
Stop
ð12Þ i ¼ 1; . . . ; m; k ð13Þ
where F is the scaling factor and N(0, 1) represents a Gaussian random variable with mean 0 and standard deviation 1.
Fig. 2. Flowchart of bee colony optimization algorithm.
4.5. Selection 4. Bee colony optimization based combined heat and power economic dispatch
Evaluate the fitness of m nb solutions using Eq. (11) and select the best m solutions on the basis of highest fitness.
In this section, an algorithm based on BCO for solving CHPED problem is described below. Let pk = [P1, P2, . . . , Pa, Pa+1, Pa+2, . . . , Pb, Ha+1, Ha+2, . . . , Hb, Hb+1, Hb+1, . . . , Hn]T be the initial vector designating the kth scout bee of a population to be evolved. The elements of pk are the real power outputs of conventional thermal generators and cogeneration units and heat outputs of cogeneration units and heat-only units. In order to meet exactly the power demand and heat demand dependent power generating unit and heat generating unit are selected. Let Pd and Hd be the power output and heat output of the dependent units:
Pd ¼ PD þ PL
b X
Pi ;
ð9Þ
i¼1 i–d
Hd ¼ HD
n X i¼aþ1 i–d
Hi :
ð10Þ
4.6. Termination A maximum number of generations Nmax is given. The search process is stopped as the count of generations reaches Nmax otherwise select best m solutions to generate neighborhood solutions. 5. Simulation results In this paper the performance of the proposed BCO-based CHPED problem is implemented using MATLAB 7 on a P-IV, 80 GB, 3.0 GHz personal computer. The proposed method has been applied to a test system which consists of four conventional thermal generators, two cogeneration units and a heat-only unit. Unit data has been modified from Guo et al. (1996). System data containing coefficients of fuel cost equations, B loss coefficients and heat-power feasible regions are given in the Appendix A. The power and heat demands of the test system are 600 MW and 150 MWth respectively.
13530
M. Basu / Expert Systems with Applications 38 (2011) 13527–13531
The problem is solved by using BCO algorithm. Here, parameters are selected as n5 = 50, m = 20, nb = 10, F = 0.01 and Nmax = 100 for the test system under consideration. To validate the proposed BCO based approach, the same test system is solved using evolutionary programming (EP), particle swarm optimization (PSO), and real-coded genetic algorithm (RCGA). The population sizes (Np) and scaling factor (F) have been
Power(MW) 125.8
Table 1 Results obtained from BCO, EP, PSO and RCGA.
1.18
x 10
EP
PSO
RCGA
43.9457 98.5888 112.9320 209.7719 98.8000 44.0000 12.0974 78.0236 59.8790 8.0384 10317 5.1563
61.3610 95.1205 99.9427 208.7319 98.8000 44.0000 18.0713 77.5548 54.3739 7.9561 10390 5.2750
18.4626 124.2602 112.7794 209.8158 98.8140 44.0107 57.9236 32.7603 59.3161 8.1427 10613 5.3844
74.6834 97.9578 167.2308 124.9079 98.8008 44.0001 58.0965 32.4116 59.4919 7.5808 10667 6.4723
4
BCO EP PSO RCGA
1.14 1.12 Cost($)
C
F
E
40
BCO
1.16
D
15.9 32.4
75
135.6 Heat(MWth)
Fig. 5. Heat-power feasible operation region for the cogeneration unit 2.
selected as 100, and 0.25, respectively in case of EP. In case of PSO parameters are taken as Np = 100, wmax = 0.2, wmin = 0.05, c1 = 0.35 and c2 = 0.35. In case of RCGA, the population size, crossover and mutation probabilities have been selected as 100, 0.07 and 0.5 respectively. Maximum number of generations has been selected 100 for EP, PSO and RCGA. Table 1 compares the four computational results of this test system obtained from BCO, EP, PSO and RCGA. It is found that the proposed approach provides lower production cost and CPU time. Fig. 3 shows the cost convergence obtained from BCO, EP, PSO and RCGA. Fig. 3 and Table 1 show the best convergence rate as well as the best solution time among the four is achieved by BCO, followed by EP. RCGA is the worst performer, followed by PSO. 6. Conclusion
1.1
This paper has presented a novel approach based on bee colony optimization for solving combined heat and power economic dispatch problem. It is evident from the comparison that the proposed bee colony optimization based approach provides better performance in terms of optimal solution as well as computation effort.
1.08 1.06 1.04 1.02
B
110.2
44
P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) H5 (MWth) H6 (MWth) H7 (MWth) PL (MW) Cost ($) CPU time (s)
A
Appendix A 0
10
20
30
40
50 60 Generation
70
80
90
100
A.1. Cost function of each unit
Fig. 3. Cost convergence.
(a) Power-only units:
Power(MW)
F t1 ðP 1 Þ ¼ 25 þ 2P1 þ 0:008P21 n o þ 100sin 0:042 Pmin P1 $ 10 6 P1 6 75 MW; 1
A
247
F t2 ðP 2 Þ ¼ 60 þ 1:8P2 þ 0:003P22 n o þ 140 sin 0:04 Pmin P 2 $ 20 2
B
215
6 P2 6 125 MW; 98.8 D 81
F t3 ðP 3 Þ ¼ 100 þ 2:1P3 þ 0:0012P23 n o þ 160 sin 0:038 P min P3 $ 30 3
C
6 P3 6 175 MW; 104.8
180
Heat(MWth)
Fig. 4. Heat-power feasible operation region for the cogeneration unit 1.
M. Basu / Expert Systems with Applications 38 (2011) 13527–13531
F t4 ðP4 Þ ¼ 120 þ 2P4 þ 0:001P24 n o þ 180 sin 0:037 Pmin P4 $ 40 4 6 P4 6 250 MW: (b) Cogeneration units:
F c5 ðP5 ; H5 Þ ¼ 2650 þ 14:5P5 þ 0:0345P 25 þ 4:2H5 þ 0:03H25 þ 0:031P5 H5 $;
F c6 ðP6 ; H6 Þ ¼ 1250 þ 36P6 þ 0:0435P26 þ 0:6H6 þ 0:027H26 þ 0:11P6 H6 $: (c) Heat-only unit:
F h7 ðH7 Þ ¼ 950 þ 2:0109H7 þ 0:038H27 $ 0 6 H7 6 2695:2 MWth: A.2. Operation limits The heat-power feasible regions of the cogeneration units are illustrated in Fig. 4 and 5. A.3. Network loss coefficients These are given below:
2
49 14 15 15 20 25
6 14 6 6 6 15 B¼6 6 15 6 6 4 20
3
45 16 20 18 19 7 7 7 16 39 10 12 15 7 7 107 : 20 10 40 14 11 7 7 7 18 12 14 35 17 5
25 19 15 11 17 39
13531
References Bonabeau, E., Dorigo, M., & Theraulaz, G. (1999). Swarm intelligence: From natural to artificial systems. New York: Oxford University Press. Camazine, S., Deneubourg, J., Franks, N. R., Sneyd, J., Theraula, G., & Bonabeau, E. (2003). Self-organization in biological systems. Princeton: Princeton University Press. Eberhart, R., Shi, Y., & Kennedy, J. (2001). Swarm intelligence. San Francisco: Morgan Kaufmann. Guo, T., Henwood, M. I., & van Ooijen, M. (1996). An algorithm for heat and power dispatch. IEEE Transactions on Power Systems, 11(4), 1778–1784. Karaboga, D. (2005). An idea based on honey bee swarm for numerical optimization, Technical Report-Tr06t. Turkey: Erciyes University, Engineering Faculty, Computer Engineering Department. Rooijers, F. J., & Van Amerongen, R. A. M. (1994). Static economic dispatch for cogeneration systems. IEEE Transactions on Power Systems, 9(3), 1392–1398. Song, Y. H., Chou, C. S., & Stonham, T. J. (1999). Combined heat and power dispatch by improved ant colony search algorithm. Electric Power Systems Research, 52, 115–121. Su, C. T., & Chiang, C. L. (2004). An incorporated algorithm for combined heat and power economic dispatch. Electric Power Systems Research, 69, 187–195. Vasebi, A., Fesanghary, M., & Bathaee, S. M. T. (2007). Combined heat and power economic dispatch by harmony search algorithm. International Journal of Electrical Power Energy Systems, 29, 713–719. Wang, L., & Singh, C. (2008). Stochastic combined heat and power dispatch based on multi-objective particle swarm optimization. International Journal of Electrical Power Energy Systems, 30, 226–234. Wong, K. P., & Algie, C. (2002). Evolutionary programming approach for combined heat and power dispatch. Electric Power System Research, 61, 227–232.