Cost-allocation Models in Electricity Systems
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1 8 Cost-allocation Models in Electricity Systems Presented by Athena Wu Supervisor: Andy Philpott Co-supervisor: Golbon Zakeri
2 Cost Recovery Problem Extract payments for shared resource Public utility cost Example: airlines building runway -Sep-8
3 Cost-allocation Objectives Complete cost recovery Incentives for efficiency Symmetry (non-discriminate) Easy to calculate / implement -Sep-8
4 Cost Recovery for Transmission Investment Open-access with competition Share to compete Cost recovery schemes can alter their behaviours Recovery through congestion revenues Discrete investments Economy of scale Security constraints (creates redundancy) Networks appear under-used Only recover ~% -Sep-8
5 Cost Recovery for Transmission Investment (New Zealand) New Zealand North: population South: resource Transmission Charge Large proportion Important in decision making (e.g. wind generation, HVDC) -Sep-8
6 A few definitions Sets of agents N all agents, S subset of N Cost functions c(n), c(s), opportunity cost Value functions v(n), v(s), any valuation function Payments P i is the payment for i th agent -Sep-8
7 Economics Concept the Core The Core An allocation x is in the core of v(n), iff x is feasible, and no coalition improve on x. Runway problem N={A, B, C} S, subset of N c(s) C B A -Sep-8
8 The Core runway example S empty set {A} {B} {C} {A, B} {B, C} {A, C} {A, B, C} c(s) Conditions to satisfy: P A <= c({a}) =, P B <= c({b}) =, P C <= c({c}) =, P + P B C <= c({b,c}) =, P + P A C <= c({a,c}) =, P + P A B <= c({a,b}) =, P + P A B + P C = c({a,b,c}) =, -Sep-8
9 The Core runway example S empty set {A} {B} {C} {A, B} {B, C} {A, C} {A, B, C} c(s) Conditions to satisfy: P A <= c({a}) =, P B <= c({b}) =, P C <= c({c}) =, P + P B C <= c({b,c}) =, P + P A C <= c({a,c}) =, P + P A B <= c({a,b}) =, P + P A B + P C = c({a,b,c}) =, -Sep-8
10 The Core runway example S empty set {A} {B} {C} {A, B} {B, C} {A, C} {A, B, C} c(s) Conditions to satisfy: P A <= c({a}) =, P B <= c({b}) =, P C <= c({c}) =, P + P B C <= c({b,c}) =, P + P A C <= c({a,c}) =, P + P A B <= c({a,b}) =, P + P A B + P C = c({a,b,c}) =, -Sep-8
11 The Core runway example B= A= C= P A <= c({a}) =, P B <= c({b}) =, P C <= c({c}) =, P + P B C <= c({b,c}) =, P + P A C <= c({a,c}) =, P + P A B <= c({a,b}) =, P + P A B + P C = c({a,b,c}) =, -Sep-8
12 The Core runway example A= P A <= c({a}) =, P B <= c({b}) =, P C <= c({c}) =, P + P B C <= c({b,c}) =, P + P A C <= c({a,c}) =, B= B= C= P + P A B <= c({a,b}) =, P + P A B + P C = c({a,b,c}) =, -Sep-8
13 The Core runway example A= P A <= c({a}) =, C= P B <= c({b}) =, P C <= c({c}) =, P + P B C <= c({b,c}) =, P + P A C <= c({a,c}) =, B= B= C= P + P A B <= c({a,b}) =, P + P A B + P C = c({a,b,c}) =, -Sep-8
14 The Core runway example A= P A <= c({a}) =, C= P B <= c({b}) =, P C <= c({c}) =, B+C= P + P B C <= c({b,c}) =, P + P A C <= c({a,c}) =, B= B= C= P + P A B <= c({a,b}) =, P + P A B + P C = c({a,b,c}) =, -Sep-8
15 The Core runway example A= P A <= c({a}) =, C= P B <= c({b}) =, P C <= c({c}) =, B+C= P + P B C <= c({b,c}) =, P + P A C <= c({a,c}) =, B= B= C= P + P A B <= c({a,b}) =, P + P A B + P C = c({a,b,c}) =, -Sep-8
16 The Core runway example A= P A <= c({a}) =, C= P B <= c({b}) =, P C <= c({c}) =, B+C= P + P B C <= c({b,c}) =, P + P A C <= c({a,c}) =, B= B= C= P + P A B <= c({a,b}) =, P + P A B + P C = c({a,b,c}) =, -Sep-8
17 The Core runway example A= P A <= c({a}) =, The Core C= P B <= c({b}) =, P C <= c({c}) =, B+C= P + P B C <= c({b,c}) =, P + P A C <= c({a,c}) =, B= B= C= P + P A B <= c({a,b}) =, P + P A B + P C = c({a,b,c}) =, -Sep-8
18 The Core runway example The Core is defined by a set of linear constraints But how to find a solution that: satisfies objectives is unique The Core A= C= B+C= B= C= -Sep-8 B=
19 The Shapley Value Lloyd Shapley (9) Fair allocation of gains / costs By cooperation among several agents The Intuitive Idea Average incremental cost for each player Computed over all possible orders in total set - N -Sep-8
20 The Shapley Value Properties Efficiency complete cost allocation Symmetry treat identical players the same Dummy no use no pay Additivity additive function -> additive rules Cost non-decreasing returns to scale The Core of the function is nonempty and contains the Shapley Value (Shapley, 97) -Sep-8
21 The Shapley Value the runway problem Permutations C C C PA PB PC (A, B, C) A -Sep-8
22 The Shapley Value the runway problem Permutations C C C PA PB PC (A, B, C) B A -Sep-8
23 The Shapley Value the runway problem Permutations C C C PA PB PC (A, B, C) C B A -Sep-8
24 The Shapley Value the runway problem Permutations C C C PA PB PC (A, B, C) C B A -Sep-8
25 The Shapley Value the runway problem Permutations C C C PA PB PC (A, B, C) (A, C, B) -Sep-8
26 The Shapley Value the runway problem Permutations C C C PA PB PC (A, B, C) (A, C, B) (B, A, C) B -Sep-8
27 The Shapley Value the runway problem Permutations C C C PA PB PC (A, B, C) (A, C, B) (B, A, C) B A -Sep-8
28 The Shapley Value the runway problem Permutations C C C PA PB PC (A, B, C) (A, C, B) (B, A, C) C B A -Sep-8
29 The Shapley Value the runway problem Permutations C C C PA PB PC (A, B, C) (A, C, B) (B, A, C) C B A -Sep-8
30 The Shapley Value the runway problem Permutations C C C PA PB PC (A, B, C) (A, C, B) (B, A, C) (B, C, A) -Sep-8
31 The Shapley Value the runway problem Permutations C C C PA PB PC (A, B, C) (A, C, B) (B, A, C) (B, C, A) (C, A, B) -Sep-8
32 The Shapley Value the runway problem Permutations C C C PA PB PC (A, B, C) (A, C, B) (B, A, C) (B, C, A) (C, A, B) (C, B, A) Average: 9/6 7/6 / -Sep-8
33 The Shapley Value runway example The Shapley Value solution is in the Core, and satisfies all the properties as promised. A= C= / A= 9/6 B+C= B= C= A = B = C = Total Cost Allocation /6 /6 / B= 7/6 -Sep-8
34 The Shapley Value General formulation Properties Efficiency Symmetry Dummy Additivity -Sep-8
35 The Shapley Value in electricity market (Chile []) Assumptions Inelastic demand Power flow on link as measure of cost Minimise generation cost subject to Generation = Demand for all subset S of grand coalition N [] J. M. Zolezzi and H. Rudnick, Transmission cost allocation by cooperative games and coalition formation, IEEE Trans. Power Syst., vol. 7, no.4, pp. 8-, Nov.. -Sep-8
36 The Shapley Value in electricity market (NZ) The two-node NZ model Generations North: G =, South: G =, G = Demands North: D =4, South: D = G D G D G -Sep-8
37 The Shapley Value in electricity market (NZ) Permutations T T T P P P (,, ) G = D =4 D = -Sep-8
38 The Shapley Value in electricity market (NZ) Permutations T T T P P P (,, ) G = D =4 G = D = -Sep-8
39 The Shapley Value in electricity market (NZ) Permutations T T T P P P (,, ) G = D =4 G = G = D = -Sep-8
40 The Shapley Value in electricity market (NZ) Permutations T T T P P P (,, ) G = D =4 G = G = D = -Sep-8
41 The Shapley Value in electricity market (NZ) Permutations T T T P P P (,, ) (,, ) -Sep-8
42 The Shapley Value in electricity market (NZ) Permutations T T T P P P (,, ) (,, ) (,, ) -Sep-8
43 The Shapley Value in electricity market (NZ) Permutations T T T P P P (,, ) (,, ) (,, ) (,, ) -Sep-8
44 The Shapley Value in electricity market (NZ) Permutations T T T P P P (,, ) (,, ) (,, ) (,, ) (,, ) -Sep-8
45 The Shapley Value in electricity market (NZ) Permutations T T T P P P (,, ) (,, ) (,, ) (,, ) (,, ) (,, ) Average: -Sep-8
46 The Shapley Value in electricity market (NZ) P = G = Cost = D =4 G = P = G = P = D = -Sep-8
47 From Shapley Value to Aumann- Shapley prices Aumann-Shapley prices Marginal cost averaged over all vectors [] Consider the progression of cost function rather than just two states [] H. P. Young, Cost allocation, in Handbook of Game Theory, R. J. Aumann and S. Hart, Eds. New York: Elsevier, 994, vol., ch. 4. -Sep-8
48 Aumann-Shapley continued the Brazilian idea Model from PSR, Brazil [] Divide agents into very small agents Entrance order does not matter Number of calculation step = number of small agents Calculate using same model as in Shapley Value Price to pay is the sum of all small agents [] M. Junqueira, L. C. Costa, Jr., L. A. Barroso, G. C. Oliveira, L. M. Thomé and M. V. Pereira An Aumann Shapley approach to allocate transmission service cost among network users in electricity markets, IEEE Trans. Power Syst., vol., pp. -46, Nov. 7. -Sep-8
49 Aumann-Shapley continued New Zealand example Again, the two-node NZ model Divide each agents into small agents and enters the market sequentially G = D =4 G = D = G = -Sep-8
50 Aumann-Shapley continued New Zealand example Step G G G P P P Demand Supplied D D 4 -Sep-8
51 Aumann-Shapley continued New Zealand example Step G G G P P P Demand Supplied D D 4 8 -Sep-8
52 -Sep-8 Aumann-Shapley continued New Zealand example D D Demand Supplied P P P G G G Step
53 -Sep-8 Aumann-Shapley continued New Zealand example D D Demand Supplied P P P G G G Step
54 -Sep-8 Aumann-Shapley continued New Zealand example D D Demand Supplied P P P G G G Step
55 -Sep-8 Aumann-Shapley continued New Zealand example D D Demand Supplied P P P G G G Step
56 -Sep-8 Aumann-Shapley continued New Zealand example 7 Total: D D Demand Supplied P P P G G G Step
57 Aumann-Shapley continued New Zealand example (P = ) P' = G = Cost = D =4.7/unit G = P' = 7 (P = ) G = P' = (P = ) D = -Sep-8
58 Where in the Core? Only if define by cost function Chile & Brazil markets are central planned New Zealand Competitive Oligopoly G = P' = 7 G = D = G = D =4 Some passon cost -Sep-8
59 In the Core? Depend on demand / supply functions D & G can form coalition Not in the core Perverse incentives G = D = G = D =4 G = -Sep-8
60 The Benefit Game Cost Games vs Benefit Games [4] Taking gain as well as cost into consideration Defining the characteristic function B i is the max value consumer i is willing to pay r i is the cost component [4] W.W.Sharkey, "Suggestions for a game theoretic approach for public utility pricing and cost allocation," The Bell Journal of Economics, pp. 7 68, 98 -Sep-8
61 The Benefit Game The net surplus for each agent The allocation (outcome) is in the core if [4] W.W.Sharkey, "Suggestions for a game theoretic approach for public utility pricing and cost allocation," The Bell Journal of Economics, pp. 7 68, 98 -Sep-8
62 Benefit Games and beyond Cannot re-allocate welfare gains in competitive market Allocate cost in proportion to benefit game outcomes doesn t necessarily lie in the Core gives a good echo of pricing signal -Sep-8
63 Benefit Games and beyond All these charging schemes assumed no change to agents behaviour, however HVDC link charge Charge on historical maximum injection In fact a capacity constraint in the long term Cournot Equilibrium Models Non-cooperative game theory Explored together with the cost-allocation scheme (cooperative) -Sep-8
64 Thank You! -Sep-8
65 What s Happening Elsewhere? From European Transmission System Operators (ETSO) Report Benchmarking on transmission pricing in Europe: Synthesis -Sep-8
66 What s Happening Elsewhere? From European Transmission System Operators (ETSO) Report Benchmarking on transmission pricing in Europe: Synthesis -Sep-8
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