Effect of Aluminium Layer on Induction Heating Control Case Study ...

Journal <strong>of</strong> electrical systems Special Issue N° 01 : November 2009For comparison purpose, we use 03 cases studies:1- In the first case market, agents compete in thebilateral forward market only.2- In the second case, agents compete in the spot marketonly.3- In the third case, agents compete in both markets.Table 2: Consumers demand function parametersConsumer0 eD i0 eiD i$/MWh Consumer i$/MWhMWhMWh1 62.5 1.00 11 50.0 0.902 40.0 0.80 12 40.0 0.803 50.0 0.80 13 50.0 1.004 75.0 1.00 14 50.0 1.005 60.0 0.90 15 37.5 0.706 50.0 1.00 16 25.0 0.607 60.0 1.00 17 25.0 0.908 40.0 0.70 18 40.0 0.809 25.0 0.60 19 37.5 0.8010 40.0 0.80 20 50.0 1.001) Bilateral future electricity market equilibriumIn this case, we consider a bilateral future contractsmarket where exchanges are made directly between suppliersand consumers. In this paper, we assume that each supplierhas a portfolio <strong>of</strong> clients, for instance we assume that eachsupplier has 06 clients, and each consumers has one or moresuppliers. Suppliers’ portfolio is depicted on Table 3, eachsupplier has 6 consumers randomly chosen.Table 3 locations <strong>of</strong> Suppliers’ clients.Location <strong>of</strong> consumers (bus)1 2 3 4 5 6supplier 1 9 18 5 14 11 6supplier 2 10 8 12 1 4 7supplier 3 8 3 13 11 15 16supplier 4 17 20 14 19 3 15supplier 5 6 9 19 4 1 13supplier 6 7 2 16 5 18 12To compute the market equilibrium, we use the proposedcompetitive coevolutionary algorithm the results are reportedon Tables 4 and 5.Table 4: Suppliers’ equilibrium pointquantities contracted (MWh)1 2 3 4 5 6Supp 1 9.27 7.33 13.24 10.44 10.39 9.94Supp 2 9.95 6.83 5.49 11.83 15.89 10.12Supp 3 8.80 11.12 9.42 9.75 7.78 6.93Supp 4 9.26 13.08 7.73 5.77 9.85 6.45Supp 5 8.67 8.45 6.69 18.18 13.86 8.75Supp 6 14.55 11.64 7.86 14.04 7.99 8.93Table 5: Consumers’ load and price at equilibrium pointD PD PConsumer MWh $/MWh Consumer [MWh] [$/MWh]1 25.69 36.81 11 20.13 33.182 11.64 35.45 12 14.42 31.983 20.97 36.29 13 18.17 31.834 34.08 40.92 14 18.18 31.825 27.28 36.36 15 14.23 33.246 18.61 31.39 16 14.79 33.687 24.66 35.34 17 9.26 34.168 15.63 34.81 18 15.32 30.859 17.72 37.13 19 12.47 31.2910 9.95 37.57 20 13.08 36.922) Spot market EquilibriumIn the second case, we consider that agents make bids inthe spot market only. We assume that suppliers submit linearaffine supply function to the ISO who performs an OPF tocompute the quantity and price to be dispatched to eachsupplier. By using the proposed algorithm, the spot marketequilibrium is reported on table 6.Table 6 Suppliers’ equilibrium point in spot marketx [$/ MW 2 i h]Pg [MWh]Price[$/MWh]supplier 1 0.11 177.15 51.47supplier 2 0.16 129.67 51.50supplier 3 0.13 154.07 51.51supplier 4 0.16 126.75 51.52supplier 5 0.13 159.85 51.51supplier 6 0.08 200.00 51.523) Day-ahead electricity market equilibriumIn this case study, agents take part on both bilateralforward market and spot market. Using the developedcompetitive coevolutionary algorithm, we compute theEquilibrium point <strong>of</strong> the mixed bilateral/pool market. Theobtained results are shown on Tables 7, 8 and 9.Table 7 Suppliers’ optimal strategies in the bilateral marketquantities contracted (MWh)1 2 3 4 5 6Supp 1 2.76 0.00 4.67 0.00 1.16 0.00Supp 2 0.00 1.56 0.00 3.64 7.93 2.81Supp 3 1.35 2.63 0.00 1.53 0.36 1.50Supp 4 0.00 0.00 0.00 0.00 3.11 0.56Supp 5 0.00 3.11 0.01 7.99 4.22 0.00Supp 6 3.04 0.00 1.46 4.39 0.00 0.00Table 8: Consumers’ load and price at equilibrium pointConsumerD PMWh $/MWh Consumer D PMWh $/MW1 7.86 54.64 11 2.69 52.562 0.00 50.00 12 0.00 50.003 5.74 55.33 13 0.00 50.004 15.91 59.09 14 0.00 50.005 9.07 56.59 15 0.92 52.266 0.00 50.00 16 2.97 53.397 5.85 54.15 17 0.00 44.448 2.91 52.98 18 0.00 50.009 5.86 56.9 19 0.01 46.8610 0.00 50.00 20 0.00 50.00-----------------------------------------------------------------------Paper presented at the Third International Conference on Electrical Engineering-ICEE'0919-21 May 2009, USTHB, Algiers, Algeria and selected for publication in "JES" by the ICEE 2009 Editorial Boardpp:67-72

Journal <strong>of</strong> electrical systems Special Issue N° 01 : November 2009VII. BIBLIOGRAPHY[1]. Pinto, L. and Szcrupak, J., An Evolutionary Game Approach toEnergy Markets, IEEE Proceedings <strong>of</strong> International Symposium onCircuits and Systems, ISCAS, Vol. III., pp 324-27, 2003.[2]. Samuelson, L, Evolution and Game Theory, Journal <strong>of</strong> EconomicPerspectives, Volume 16, Number 2, Spring, 47–66., 2002.[3]. Crawford, V.P, Theory and Experiment in the Analysis <strong>of</strong> StrategicInteraction, Invited Symposium Lecture at the Econometric SocietySeventh World Congress, Tokyo, 1995.[4]. Maynard Smith, J., Evolution and the Theory <strong>of</strong> Games, CambridgeUniversity Press, ISBN 0-521-28884-3, 1982.[5]. Ireland, N.J., Evolutionarily Stable Strategies: who does the mutanthave to beat?, Department <strong>of</strong> Economics, University <strong>of</strong> Warwick, UK,2006.[6]. Cressman, R., Continuously Stable Strategies, NeighborhoodSuperiority and Two-Player Games with Continuous Strategy Space,Department <strong>of</strong> Mathematics. Wilfrid Laurier University Canada,2006.[7]. Cressman, R., Evolutionarily stable strategies, prepared for the NewPalgrave Dictionary <strong>of</strong> economics, 2nd Edition, 2006.VI. CONCLUSION[8]. G. Stamtsis ‘Power transmission cost calculation in deregulatedelectricity market’. PhD thesis, University <strong>of</strong> Duisburg-Essen,Germany, 2003.[9]. R. Baldick, Electricity market equilibrium models:The effect <strong>of</strong>parametrization, Department <strong>of</strong> Electrical and Computer EngineeringThe University <strong>of</strong> Texas at Austin,http:// users.ece.utexas.edu/~baldick/papers/parametrization.pdf.[10]. A.A Ladjici, M. Boudour, Electricity market equilibrium usingcompetitive coevolutionary algorithms with transmission constraints,IEEE SSD 2008, 2008, pp 1—6[11]. Wiegand, R.P., An Analysis <strong>of</strong> Cooperative CoevolutionaryAlgorithms, PhD Thesis, George Mason University, 2003.[12]. Stanley, K.O. and R. Miikkulainen, Competitive Coevolution throughEvolutionary Complexification, Journal <strong>of</strong> Artificial IntelligenceResearch, 63-100, 2004.[13]. Johnson et al., Coevolutionary Optimization <strong>of</strong> Fuzzy LogicIntelligence for Strategic Decision Support, IEEE Transactions onEvolutionary Computation, Vol. 9, No. 6. 2005.[14]. Service T et al., Infrastructure Hardening: A CompetitiveCoevolutionary Methodology Inspired by Neo-Darwinian ArmsRaces, 31st Annual International Computer S<strong>of</strong>tware andApplications Conference, COMPSAC, Vol.1., 101-104. 2007[15]. Rosin, C.D. and. Belew R.K, New methods for CompetitiveCoevolution, Evolutionary Computation, Vol. 5, 1-29, 1997[16]. Bäck T. Evolutionary Algorithms and Their Standard Instances,Handbook <strong>of</strong> Evolutionary Computation, Oxford University Press, ChB1.1, 1997.[17]. Mahfoud, S.W, Niching methods, Handbook <strong>of</strong> EvolutionaryComputation, Oxford University Press, ch. C6.1, 1997.[18]. Angeline, P.J. and J.B. Pollack, Competitive Environments EvolveBetter Solutions for Complex Tasks, Proceedings <strong>of</strong> the FifthInternational Conference on Genetic Algorithms, 1993.Table 9 Suppliers’ strategies and outcome <strong>of</strong> spot marketx [$/ MW 2 i h]D [MWh]Price[$/MWh]supplier 1 0.11 178.23 51.08supplier 2 0.15 133.85 51.08supplier 3 0.13 149.94 51.08supplier 4 0.16 127.97 51.09supplier 5 0.13 157.52 51.08supplier 6 0.08 200.00 51.09It is clearly observed from Tables 7 and 8 that in order toincrease their pr<strong>of</strong>it, suppliers should withdraw some <strong>of</strong> theirbids in the bilateral market and choose spot market whereprices are higher.This paper presents a new approach to find out themarket equilibrium in deregulated electricity markets using acompetitive coevolutionary algorithm based on the concept<strong>of</strong> Evolution Stable Strategies. In the latter, each agent isrepresented by a population <strong>of</strong> strategies and uses nestedevolution strategies as learning method to find the bestbehaviors to overcome opponents’ strategies.In our approach, we consider the day-ahead market as amixed bilateral/pool market, each agent acts strategically inorder to maximize his pr<strong>of</strong>it: it submits bids in terms <strong>of</strong>quantity to be sold in the bilateral forward market, and asupply function in the spot market in order to maximize itspr<strong>of</strong>it.To validate the proposed algorithm, an IEEE 30 bus testsystem is used as a case study. The proposed algorithm isapplied to analyze strategic behavior <strong>of</strong> market’s agents inboth bilateral forward market, spot market and day-aheadmarket. In the different cases considered, the proposedalgorithm has succeeded in finding the ESS set and hencethe market equilibrium point. Furthermore, the obtainedresults show that suppliers’ strategic behavior increases themarket clearing price by withdrawing quantities to be sold inthe bilateral market to increase the market price and there fortheir pr<strong>of</strong>its.From the studied cases, one can notice that the mainadvantage <strong>of</strong> the proposed approach is the possibility tohandle different market structures and pricing schemes,which makes it an appropriate simulation and analysis toolfor deregulated electricity markets.pp:67-72-----------------------------------------------------------------------Paper presented at the Third International Conference on Electrical Engineering-ICEE'0919-21 May 2009, USTHB, Algiers, Algeria and selected for publication in "JES" by the ICEE 2009 Editorial Board