A numerical method to calculate Nash-Cournot equilibria in electricity markets. Victoria University of Wellington

Transcription

1 A numerical method to calculate Nah-Cournot equilibria in electricit market Javier Contrera UCLM Ciudad Real Spain Jacek B. Krawczk Victoria Univerit of Wellington Wellington New Zealand

2 Summar Introduction Definition and concept The relaation algorithm Duopol eample Cae tud Cae tud Concluion

3 Introduction Power tem retructuring foter competition Cot minimization cheme replaced b bidding algorithm Nah Equilibria in network-contrained electricit market 3

4 Introduction Nikaido-Ioda Relaation Algorithm NIRA Convergence to a Nah-Cournot equilibrium Unique olution under certain condition NIRA olve coupled contraint game Applicable to electricit market 4

5 Definition and concept A game repreent: a et of individual interacting trategic interdependence To decribe a game ou need: plaer rule of the game outcome paoff 5

6 Definition and concept Information et: information about hi own and other plaer action. It contain information on what i obervable Strateg: A rule that tell the plaer wich action he hould take according to hi information et Action: a choice made b a plaer a a reult of hi trateg 6

7 Definition and concept i... n plaer in a game Individual action given b vector i Collective action given b vector... n X i i the action et of plaer i Ø i : X R i the paoff function X i the collective action et 7

8 Definition and concept... n and... n are element of the collective action et X An element i L i i i L of X i a et of action where the ith plaer pla i while the remaining agent pla j A point * *... n* i called the Nah equilibrium point if for each i φ i * ma * i X n φ i i 8

9 Definition and concept Nikaido-Ioda function NI: Ψ n i [ i i φi ] φ Ψ 0 Each ummand of the NI function repreent the improvement in paoff that a plaer will receive if he change hi action from i to i Each ummand can be at mot zero at Nah equilibrium 9

10 Definition and concept * An element X i referred to a Nah normalized equilibrium point if ma Ψ X Given the concavit condition a Nah normalized equilibrium i alo a Nah equilibrium point An iterative algorithm uing the NI function can converge to a Nah equilibrium * 0 0

11 Definition and concept The optimum repone function at the point i Z arg ma Ψ Z X All plaer tr to unilaterall maimize their paoff B plaing action Z rather than the plaer approach the equilibrium X

12 The relaation algorithm It i applied to the optimum repone function Given concavit condition inglevaluedne of Z and an initial etimate 0 then α α Z 0K 0 < α

13 The relaation algorithm An iterative tep i contructed a a weighted average of the improvement point Z and the current point The optimum repone function Z i a reult of maimizing the NI function The averaging enure convergence of the algorithm under certain condition 3

14 The relaation algorithm Taking a ufficient number of iteration the NIRA converge to the Nah equilibrium * The problem calculate the ucceive action taken b the plaer at each tage when optimizing their repone unilaterall 4

15 The relaation algorithm To enure convergence of NIRA the mot important condition in coupled contraint game i Diagonal Strict Concavit DSC DSC mean that each plaer ha more control over hi paoff than the other plaer have over it Coupled contraint game can be due for eample to Kirchhoff law 5

16 6 Duopol eample Two identical firm elling a product Market price: Profit made b firm i: The Nikaido-Ioda function : p ρ α i i i i p ] [ ρ λ α λ φ ] [ ] [ ] [ ] [ ρ λ α ρ λ α ρ λ α ρ λ α Ψ

17 7 Duopol eample The optimum repone function Z: ma arg X Ψ ρ λ α ; where Z ρ λ α ρ λ α

18 Duopol eample Convergence to Nah equilibrium if: Q Ψ Ψ i poitive definite 8

19 9 Duopol eample ρ Q Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ

20 Cae tud Three-bu tem Unlimited capacit generator at bue and Demand at bue and 3 Each pair of bue connected b a tranmiion line No tranmiion line flow limit 0

21 Cae tud Demand function: p i q i *q i for bue i p 3 q *q 3 for bu 3 Generator contant marginal cot are $5/MWh and $0/MWh repectivel Bilateral market: ij mean that generator compan i ell MW to conumer at node j P fgi i the production of generator g of compan f placed at node i

22 Cae tud Each generator compan maimize profit: ale revenue generation cot ma ubject to j [ P jo P jo / Q jo fj ] kj fj k f i g C P fgi P fgi P fgi P ma fgi node i generator g j fj i g P fgi f fj q j fj P fgi 0

23 3 Cae tud Firm Firm ubject to: } 5 ] [3 ] 0.08 [40 ] 0.08 ma {[ P } 0 ] [3 ] 0.08 [40 ] 0.08 ma{[ P

24 4 Cae tud q S S q S S P q S S P q q q P P bae bae bae bae bae bae θ θ θ θ θ θ θ θ θ θ θ θ

25 Reult cae tud 5

26 Reult cae tud q 87.5 MW price $5/MWh q 87.5 MW price $5/MWh q MW price 3 $.3/MWh Flow -: MW Flow -3: MW Flow -3: 56.7 MW Profit $/h 354. Profit $/h

27 Cae tud Tranmiion line flow limit of 5 MW on line - binding contraint New contraint added to the one in cae tud : θ θ 5 θ θ 5 Sbae S bae θ θ 5 S bae 7

28 Reult cae tud 8

29 Reult cae tud 9

30 Reult cae tud q 99. MW price $4./MWh q 75.9 MW price $5.9/MWh q MW price 3 $.3/MWh Flow -: 5 MW Flow -3: 06.5 MW Flow -3: 8.5 MW Profit $/h 985 Profit $/h

31 Cae tud : contraint enforcement Aume an authorit i empowered to charge agent from deviation of the line flow limit The Lagrange multiplier of the active contraint Kirchhoff law and line flow limit calculated b NIRA can be ued to enforce contraint 3

32 Cae tud : contraint enforcement The Lagrange multiplier decouple a game where the plaer action are coupled through contraint The tep of the procedure are a follow: 3

33 Cae tud : contraint enforcement. The regulator olve a coupled contraint game i.e. compute production level which do not violate contraint and the Lagrange multiplier. The regulator modifie the plaer paoff b telling them the value of the Lagrange multiplier 3. The plaer olve the new game with new paoff and miraculoul obtain the ame production level a the regulator 33

34 Cae tud : contraint enforcement The authorit announce that for a unit of contraint violation each plaer will be charged: λ λ λ ma0 P 3 4 ma0 S ma0 S bae bae S bae θ θ θ θ 3 θ 3 θ S bae 5 S bae θ θ θ θ 3 q 3 q 34

35 Cae tud : contraint enforcement The previou etra term i added to the original individual paoff function of firm and lide 3 The ame reult a in cae i obtained without coupling contraint decoupled Nah equilibrium 35

36 Concluion A Nikaido-Ioda Relaation Algorithm to find unique Nah-Cournot equilibria with coupled contraint Either a centralized or a ditributed optimization viewpoint Application to electricit trading Enforcement of contraint decouple the problem and find the ame olution 36

37 End of the preentation Thank ou!