Step 6: Mutation Apply the mutation operator to each gene in the individuals with a mutation probability pm.
Step 7: Termination test If a stopping condition is satisfied, stop the algorithm. Otherwise, go to Step 2.
V. RESULTSANDDISCUSSIONS A. Simulation Environment and Parameter Settings
This power system considers a type of conventional power unit, cogeneration unit and heat-alone unit, respectively. The power generation limits of the conventional power unit are 0 and 150 MW and heat generation limits of heat-alone units are 0 and 2695.2 MWth. The feasible operating regions of the cogeneration unit are given in figure 1. The value of emission coefficients α, β and γ are given as 13.85932, 0.32767 and 0.00419, respectively. The emission factors of heat-alone units are obtained from the average heat generation from residential boilers in urban areas, with an equivalent fuel mix as input [16].
The emission factors μNOx, μCO2 and μCO are given as 0.2 kg/MW, 0.27 kg/MW and 0.04 kg/MW, respectively.
The feasible operating regions of the cogeneration unit from Figure 1 can be expressed as inequality constraints as follows:
0 90 105.744680
-O -4H
1.78191489 ≤
(21)0 247.0 -O 8H
0.17777777 + ≤
(22)0 98.8 O -8H 0.16984732
- + ≤
(23)104.8 180
81 215 247
Heat(MWth)
Power(MW)
Figure 1. Feasible operating regions of cogeneration unit.
Based on the given environment and constraints, three benchmark problems “demand (200, 115)”, “demand (700, 615)” and “demand (2000, 1115)” are designed to validate our approach. The notation “demand (P, H)” represents that the power demand is P and the heat demand is H.
The parameter settings of MOGA are listed as follows:
population size Npop=50, recombination probability pc=0.9, mutation probability pm=0.01, the number of maximum generations Gmax=100. Thirty independent runs are conducted for each problem.
Figures 2-4 shows the distributions of non-dominated solutions in four objectives by means of boxplot. The results indicate that the proposed approach is capable of obtaining a set of wide-spread and non-dominated solutions.
Figures 5-8 depict a typical run of MOGA in solving
“demand (2000, 1115)”. The maximum, mean and minimum objective values of individuals during a typical run are shown in the figures. The results indicate that the proposed approach converge steadily and rapidly.
cost 1
1.5 2 2.5
x 104
objectives
value(US$/hr)
emission 0.05
0.1 0.15 0.2 0.25 0.3
objectives
value(tonne/hr)
heat overhead 0
100 200 300 400
objectives
value(MWth/hr)
heat overhead 0
100 200 300 400
objectives
value(MWth/hr)
Figure 2. Boxplot of non-dominated solutions in solving “demand (200,115)” problem.
cost 3.4
3.6 3.8 4 4.2 4.4 4.6 4.8
x 104
objectives
value(US$/hr)
emission 0.2
0.25 0.3 0.35 0.4 0.45 0.5 0.55
objectives
value(tonne/hr)
power overhead 0
50 100 150 200
objectives
value(MWe/hr)
heat overhead 0
50 100 150 200 250
objectives
value(MWth/hr)
Figure 3. Boxplot of non-dominated solutions in solving “demand (700,615)” problem.
cost 6
7 8 9 10
x 104
objectives
value(US$/hr)
emission 0.4
0.6 0.8 1 1.2
objectives
value(tonne/hr)
power overhead 0
100 200 300 400
objectives
value(MWe/hr)
heat overhead 0
200 400 600 800
objectives
value(MWth/hr)
Figure 4. Boxplot of non-dominated solutions in solving “demand (2000,1115)” problem.
0 20 40 60 80 100
0.5 1 1.5 2 2.5
3x 105
Generation
Fuel Cost (US$/hr)
maximum mean minimum
Figure 5. The maximum, mean, and minimum fuel cost of a typical run in solving “demand (2000,1115)” problem.
0 20 40 60 80 100
0.5 1 1.5 2 2.5 3
Generation
Emission (tonne/hr) maximummean
minimum
Figure 6. The maximum, mean, and minimum emission of a typical run in solving “demand (2000,1115)” problem.
0 20 40 60 80 100
0 500 1000 1500 2000 2500 3000
Generation
Power Overhead (MWe/hr)
maximum mean minimum
Figure 7. The maximum, mean, and minimum power overhead of a typical run in solving “demand (2000,1115)” problem.
0 20 40 60 80 100
0 1000 2000 3000 4000 5000
Generation
Heat Overhead (MWth/hr)
maximum mean minimum
Figure 8. The maximum, mean, and minimum heat overhead of a typical run in solving “demand (2000,1115)” problem.
VI. CONCLUSION
In this paper, a multi-objective evolutionary approach is proposed to solve the combined heat and power environmental/economic dispatch problem. The problem is formulated as multi-objective optimization problem with competing economic and environmental objectives.
Experimental results demonstrated the proposed method is capable of optimizing fuel cost, emission, power overhead and heat overhead simultaneously. Moreover, the proposed approach can provide decision makers a set of non-dominated solutions to choose a suitable dispatch plan.
ACKNOWLEDGMENT
This work was supported by the National Science Council of Taiwan, R.O.C. under Contract NSC-96-2221-E-216-037-MY2, and Chung-Hua University under Contract CHU-96-2221-E-216-037-MY2.
REFERENCES
[1] L. Xuebin, "Study of multi-objective optimization and multi-attribute decision-making for economic and environmental power dispatch,"
Electric Power Systems Research, vol. 79, pp. 789-795, 2009.
[2] P. Subbaraj, R. Rengaraj, and S. Salivahanan, "Enhancement of combined heat and power economic dispatch using self adaptive real-coded genetic algorithm," Applied Energy, vol. 86, pp. 915-921, 2009.
[3] K. P. Wong and C. Algie, "Evolutionary programming approach for combined heat and power dispatch," Electric Power Systems Research, vol. 61, pp. 227-232, 2002.
[4] R. Ramanathan, "Emission constrained economic dispatch," Power Systems, IEEE Transactions on, vol. 9, pp. 1994-2000, 1994.
[5] M. A. Abido, "Multiobjective particle swarm optimization for environmental/economic dispatch problem," Electric Power Systems Research, vol. 79, pp. 1105-1113, 2009.
[6] G. P. Granelli, M. Montagna, G. L. Pasini, and P. Marannino, "Emission constrained dynamic dispatch," Electric Power Systems Research, vol.
24, pp. 55-64, 1992.
[7] A. Farag, S. Al-Baiyat, and T. C. Cheng, "Economic load dispatch multiobjective optimization procedures using linear programming techniques," Power Systems, IEEE Transactions on, vol. 10, pp. 731-738, 1995.
[8] J. S. Dhillon, S. C. Parti, and D. P. Kothari, "Stochastic economic emission load dispatch," Electric Power Systems Research, vol. 26, pp.
179-186, 1993.
[9] C. S. Chang, K. P. Wong, and B. Fan, "Security-constrained multiobjective generation dispatch using bicriterion global optimisation," Generation, Transmission and Distribution, IEE Proceedings-, vol. 142, pp. 406-414, 1995.
[10] R. Yokoyama, S. H. Bae, T. Morita, and H. Sasaki, "Multiobjective optimal generation dispatch based on probability security criteria,"
Power Systems, IEEE Transactions on, vol. 3, pp. 317-324, 1988.
[11] D. Srinivasan, C. S. Chang, and A. C. Liew, "Multiobjective generation scheduling using fuzzy optimal search technique," Generation, Transmission and Distribution, IEE Proceedings, vol. 141, pp. 233-242, 1994.
[12] H. Chao-Ming, Y. Hong-Tzer, and H. Ching-Lien, "Bi-objective power dispatch using fuzzy satisfaction-maximizing decision approach," Power Systems, IEEE Transactions on, vol. 12, pp. 1715-1721, 1997.
[13] D. B. Das and C. Patvardhan, "New multi-objective stochastic search technique for economic load dispatch," Generation, Transmission and Distribution, IEE Proceedings-, vol. 145, pp. 747-752, 1998.
[14] M. A. Abido, "A novel multiobjective evolutionary algorithm for environmental/economic power dispatch," Electric Power Systems Research, vol. 65, pp. 71-81, 2003.
[15] M. A. Abido, "Environmental/economic power dispatch using multiobjective evolutionary algorithms," Power Systems, IEEE Transactions on, vol. 18, pp. 1529-1537, 2003.
[16] G. Chicco, P. Mancarella, and R. Napoli, "Emission Assessment of Distributed Generation in Urban Areas," in Power Tech, 2007 IEEE Lausanne, 2007, pp. 532-537.
A Force-Driven Evolutionary Approach for Multi-objective 3D Differentiated Sensor Network Deployment
Liang-Che Wei, Chih-Wei Kang and Jian-Hung Chen*