Mechanism design and allocation algorithms for energy-network markets with piece-wise linear costs and quadratic externalities

Transcription

1 1 / 45 Mechanism design and allocation algorithms for energy-network markets with piece-wise linear costs and quadratic externalities Alejandro Jofré 1 Center for Mathematical Modeling & DIM Universidad de Chile Optimization and Equilibrium in Energy Economics, IPAM UCLA,January In collaboration with N. Figueroa and B. Heymann

2 2 / 45 Outline Introduction and motivation Modeling market and Equilibrium. Discontinuous Games Nash and beyond Intrinsic market Power Efficient regulations and Extended Mechanism Design Conclusions

3 3 / 45 1 Introduction and motivation 2 Modeling Market Equilibrium: Nash 3 Intrinsic Market Power 4 Efficient regulations and mechanism design The benchmark game Comparing Benchmark with Optimal Mechanism

4 4 / 45 Motivations Most of ISOs have few generation companies: oligopoly Transmission networks highly congested in some areas Intrinsic market power produced by externalities and information asymmetries

5 5 / 45 Transmission Europe

6 6 / 45 Transmission US

7 7 / 45 Transmission Chile

8 8 / 45 1 Introduction and motivation 2 Modeling Market Equilibrium: Nash 3 Intrinsic Market Power 4 Efficient regulations and mechanism design The benchmark game Comparing Benchmark with Optimal Mechanism

9 9 / 45 A generation short term market: day-ahead mandatory pool Today: generators taking into account an estimation of the demand bid increasing piece-wise linear cost functions or equivalently piece-wise constant price. Even general convex cost functions. Tomorrow: the (ISO) using this information and knowing a realization of the demand, minimizes the sum of the costs to satisfy demands at each node considering all the transmission constraints: dispatch problem. Tomorrow: the (ISO) sends back to generators the optimal quantities and prices (multipliers associated to supply = demand balance equation at each node)

10 10 / 45 ISO problem or dispatch DP (c, d) The (ISO) knows a realization of the demand d IR V, receives the costs functions bid (c i ) i G and compute: (q i ) i G, (λ i ) i G c i (q i ). (1) min (h,q) i G r e 2 h2 e + d i q i + e K i e K i h e sgn(e, i), i G (2) q i [0, q i ], i G, (3) 0 h e h e (4)

11 11 / 45 We denote Q(c, d) IR G the generation component of the optimal solution set associated to each cost vector submitted c = (c i ) and demand d. We denote Λ(c, d) IR G the set of multipliers associated to the supply=demand in the ISO problem.

12 12 / 45 Modeling Generators 1 At each node i G we have a generator with payoff u i (λ, q) = λq c i (q) c i is the real cost. 2 The strategic set for each player i denoted S i : {c i : IR IR + convex, nondecreasing, bounded subgradients or c i [0, p ], p is a price cap.

13 13 / 45 Equilibrium: Nash Equilibrium An equilibrium is (q, λ, m) such that q is a selection of Q(, ) and λ is a selection of Λ(, ) and m = (m i ) i G is a mixed-strategy equilibrium of the generator game in which each generator submits costs c i S i with a payoff Eu i (λ i (c, ), q i (c, )) = [λ i (c, d)q i (c, d) c i (q i (c, d))]dp(d), Neutral or risk averse D

14 14 / 45 Equilibrium: Nash Literature In some cases, for example, using a supply function equilibria approach there are previous works by Anderson, Philpott, or using variational inequality approach by Pang, Ralph, Ferris or also using game theory by Hogan, Smeers, Wilson, Joskow, Tirole, Hobbs, Oren, Borestein, Wolak...

15 15 / 45 Equilibrium: Nash Nash equilibrium Consider a game G = (X i, u i ) N that consists of N players where each player i = 1,..., N has a strategy set X i and a payoff function u i : X IR, where X = Π i N X i. Nash equilibrium (x i ) x i argmax {u i (x i, x i) x i X i }

16 16 / 45 Equilibrium: Nash Nash equilibrium For the sake of simplicity, we assume that each X i is contained in a metric vectorial space: If for all i the strategy set X i is a compact set, and u i is a bounded function, we say that G is a compact game. If for all i the set X i is convex and for each x i X i, u i (, x i ) is a (concave) quasiconcave function, then we say that G is a convex game (quasiconvex game)

17 17 / 45 Equilibrium: Nash Nash equilibrium existence A convex compact game G = (X i, u i ) N satisfying: u i (, ) is upper semicontinuous u i (x i, ) is lower semicontinuous for all x i has a Nash equilibrium point. Extensions: generalized games, convergence-stabilty lopsided convergence

18 18 / 45 Equilibrium: Nash Discontinuos games: tie-breaking rules Consider the following two-player game: Let the payoff for the i player be given by l i (x i ) if x i < x i, u i (x i, x i ) = ϕ(x i ) if x i = x i, m i (x i ) if x i > x i, where x i [0, 1]. Assume that for all i and x [0, 1] (a) l i and m i are continuous functions, l i is nondecreasing ϕ(x) is a convex combination of l i (x) and m i (x); sign [l i (x) ϕ(x)] = sign [ϕ i (x) m i (x)]. (5)

19 19 / 45 Equilibrium: Nash Existence discontinuos games Reny (1999) Econometrica Theorem A compact quasiconcave game possesses a Nash equilibrium if it is also a better reply secure game. Bagh and Jofre (2006) Econometrica Theorem If (X i, u i ) N is weakly reciprocally upper semicontinuous and payoff secure, then it is better reply secure.

20 20 / 45 Equilibrium: Nash Assumptions S1. For all d D, there exists δ d > 0 such that Ω(d), ˆd d δ d. S2. D is compact S3. (1) Either P is non atomic; or (2) given two convex sets M, N IR G, u(m N) is convex. S4 u i : IR 2 IR is continuous.

21 21 / 45 Equilibrium: Nash Equilibrium existence Theorem If each S v is a nonempty closed set for the point-wise convergence, then there exists an equilibrium (q, λ, m) for the bid-based generator pool game. Example: In real system... increasing piece-wise constant cost functions

22 22 / 45 1 Introduction and motivation 2 Modeling Market Equilibrium: Nash 3 Intrinsic Market Power 4 Efficient regulations and mechanism design The benchmark game Comparing Benchmark with Optimal Mechanism

23 Outline Introduction and motivation Modeling Market Intrinsic Market Power Efficient regulations and mechanism design Two-node case Two nodes case Symmetric Nash equilibrium Profit = mul@plier quan@ty cost quan@ty d d r d < 1/2r Nash = c/(1 2rd) $ q 23 / 45

24 24 / 45 the ISO Problem: two-node case Given that each generator reveals a cost c i, the (ISO) solves: min q,h 2 c i q i i=1 s.t. q i h i + h i r 2 [h2 1 + h2 2 ] + d for i = 1, 2 q i, h i 0 for i = 1, 2

25 25 / 45 Result Escobar and J. (ET (2010)) equilibrium exists but producers charge a price above marginal cost: Nash = c/(1 2rd)

26 26 / 45 Sensitivity formula Proposition Let c i G S i and c i ĉ i a Lipschitz function with constant κ. Then, Q i (c, d) Q i (ĉ i, c i, d) κη, (1+r i h i ) 2 where η = 2 ]0, + [ and min i G r i c + i (0) c + i (0) = lim y 0+ c i(y) c i (0) y. Why? losses => the second-order growth

27 27 / 45 Market Power formula Proposition The equilibrium prices p i satisfy where η i = 2 K i 2 ( E p i γ E[Q i(p i, p i, d)] η i ) 2 1+max{r eh e : e K i } p min e K r e γ(p i, d) is a measurable selection of c i (Q i (p i, p i, d)).

28 28 / 45 Market Power formula Proposition Linear case: c i (q) = c i q, then p i c i E[Q i(p i, p i, d)]. η

29 29 / 45 1 Introduction and motivation 2 Modeling Market Equilibrium: Nash 3 Intrinsic Market Power 4 Efficient regulations and mechanism design The benchmark game Comparing Benchmark with Optimal Mechanism

30 30 / 45 The Questions In an electric network with transmission costs and private information: Does the usual (price equal Lagrange multiplier) regulation mechanism minimize costs for the society? If not, what is the mechanism that achieves this objective? How does the performance of both systems compare? Methodology: Bayesian Game Theory Mechanism Design

31 31 / 45 Framework A network with demand d at each node. One producer at each node, with piece-wise linear cost of production c i F i [c i, c i ]. Common knowledge! Transmission costs rh 2, with h the amount sent from one node to another.

32 32 / 45 ISO for piece-wise linear cost functions Problem minimize (q,h) n i=1 j=1 N q i,j c i,j N q i,j + j=1 i V (i) (i, i ) E : h i,i 0 (γ i,i ) i I, j J : q i,j 0 (µ i,j ) i I, j J : q i,j q (ν i,j ). h i,i h i,i h2 i,i + h 2 i,i r i,i d i (λ i ) 2 (6)

33 33 / nodes network

34 34 / 45 The ISO Problem: two-node case Given that each generator reveals a cost c i, the ISO solves: min q,h 2 c i q i i=1 s.t. q i h i + h i r 2 [h2 1 + h2 2 ] + d for i = 1, 2 q i, h i 0 for i = 1, 2

35 35 / 45 The Solution for ISO problem If we define and H(x, y) = d + 1 2r ( ) x y 2 1 x + y r [ ] 1 1 2dr q = 2 r ( ) x y x + y then the solution to this problem can be written as H(c i, c i ) if H(c i, c i ) 0 and H(c i, c i ) 0 q i (c i, c i ) = q if H(c i, c i ) < 0 0 if H(c i, c i ) < 0 λ i (c i, c i ) p i (c i, c i ) = c i if H(c i, c i ) 0

36 36 / 45 The benchmark game The Bayesian Game: benchmark The game: 2 players. Strategies c i C i = [c i, c i ], i=1,2. Payoff u i (c i, c i ) = (λ i (c i, c i ) c i )q i (c i, c i ), where c i is the real cost. The Equilibrium: A strategy b i : [c i, c i ] IR + (convex at equilibrium!) In a Nash equilibrium b(c) arg max [λ i (x, b(c i )) c]q i (x, b(c i ))f i (c i )dc i x C i (7)

37 37 / 45 The benchmark game Numerical Approximation For simplicity C i = [1, 2]. Let k {0,..., n 1}, and b(c) = b k for c [ k n, k+1 n ]. The weight of each interval is given by w k = F ( k+1 n ) F ( k n ). The approximate equilibrium is characterized by: n 1 b k arg max [λ i (x, b l ) r k ]q i (x, b l )w l for all k {0,..., n 1} x l=0 (8)

38 The benchmark game Optimal Mechanism. Principal Agent Model (Myerson) A direct revelation mechanism M = (q, h, x) consists of an assignment rule (q 1, q 2, h 1, h 2 ) : C R 4 and a payment rule x : C R 2. The ex-ante expected profit of a generator of type c i when participates and declares c i is U i (c i, c i; (q, h, x)) = E c i [x i (c i, c i ) c i q i (c i, c i )] A mechanism (q, h, x) is feasible iff: U i (c i, c i ; (q, h, x)) U i (c i, c i; (q, h, x)) for all c i, c i C i U i (c i, c i ; (q, h, x)) 0 for all c i C i q i (c) h i (c) + h i (c) r 2 [h2 1(c) + h 2 2(c)] + d for all c C q i (c), h i (c) 0 for all c C 38 / 45

39 39 / 45 The benchmark game The Regulator s Problem Using the revelation principle, the regulator s problem can be written as: min C 2 x i (c)f(c)dc (9) i=1 subject to (q, h, x) being feasible Existence: Knuster-Tarski fixed point theorem (monotone relations)

40 40 / 45 The benchmark game The Regulator s Problem (II) It can be rewritten as min s.t 2 q i (c)[c i + F i(c i ) f i (c i ) ]f(c)dc C i=1 q i (c i, c i )f i (c i )dc i is non-increasing in c i C i q i (c) h i (c) + h i (c) r 2 [h2 1 (c) + h2 2 (c)] + d for all c C q i (c), h i (c) 0 for all c C We denote by J i (c i ) = c i + F i(c i ) f i (c i ) the virtual cost of agent i. We assume it is increasing (Monotone likelihood ratio property: true for any log concave distribution)

41 41 / 45 The benchmark game Solution An optimal mechanism is given by H(J i (c i ), J i (c i )) if H(J i (c i ), J i (c i )) 0 an ˆq i (c i, c i ) = q if H(J i (c i ), J i (c i )) < 0 0 if H(J i (c i ), J i (c i )) < 0 c i ˆx i (c i, c i ) = c iˆq i (c i, c i ) + c i ˆq i (s, c i )ds Such mechanism is dominant strategy incentive compatible.

42 42 / 45 Comparing Benchmark with Optimal Mechanism Comparing Benchmark with Optimal Mechanism We consider the family of distributions with densities { a(x 1) + (1 a f a (x) = 4 ) if x 1.5 a(x 1) + (1 + 3a 4 ) if x 1.5

43 Comparing Benchmark with Optimal Mechanism Asymmetric information 1.8 Density Functions ga 1 Cumulatives Functions Ga a =3 a= a =3 a =2 a =1 1.2 a =1 0.6 a = a = Marginal Cost Marginal Cost 43 / 45

44 Comparing Benchmark with Optimal Mechanism Social costs for different mechanisms 7 6 Optimal 0 Standard 0 Optimal 2 Standard 2 Optimal 4 Standard 4 5 Social Cost r 44 / 45

45 Comparing Benchmark with Optimal Mechanism Robustness and Practical Implementation The optimal mechanism is detail free. If the designer is wrong about common beliefs, then the mechanism is still not bad: X f X f x 1 f f c q f f The assignment rule is computationally simple to implement. It requires solving once the dispatcher problem, with modified costs. However, the payments are computationally difficult c i c iˆq i (c i, c i ) + c i ˆq i (s, c i )ds 45 / 45

46 Comparing Benchmark with Optimal Mechanism Bagh, Adib; Jofre, Alejandro. Weak reciprocally upper-semicontinuity and better reply secure games: A comment. Econometrica, Vol. 74 Issue 6 (2006), Figueroa, N. Jofre, A. and Heymann B. Cost-Minimizing regulations for a wholesale electricity market. Submitted (2015) Heymann B. and Jofre, A. Mechanism design and allocation algorithms for network markets with piece-wise linear costs and quadratic externalities. (2016) Escobar, Juan and Jofre, Alejandro. Monopolistic Competition in Electricity Markets. Economics Theory, 44, Number 1, (2010) Escobar, Juan and Jofre, Alejandro. Equilibrium analysis of electricity auctions. Submitted. 45 / 45

47 45 / 45 Comparing Benchmark with Optimal Mechanism Escobar, Juan F.; Jofre, Alejandro. Equilibrium analysis for a network market model. Robust optimization-directed design, 63 72, Nonconvex Optim. Appl., 81, Springer, New York, (2006).