Pricing Transmission

Transcription

1 1 / 47 Pricing Transmission Quantitative Energy Economics Anthony Papavasiliou

2 2 / 47 Pricing Transmission 1 Locational Marginal Pricing 2 Congestion Rent and Congestion Cost 3 Competitive Market Model for Transmission Capacity 4 Zonal Pricing Zonal Pricing with Re-Dispatch Gaming Zonal Pricing

3 3 / 47 Table of Contents 1 Locational Marginal Pricing 2 Congestion Rent and Congestion Cost 3 Competitive Market Model for Transmission Capacity 4 Zonal Pricing Zonal Pricing with Re-Dispatch Gaming Zonal Pricing

4 Recall DCOPF 4 / 47 (DCOPF) : max l L dl 0 MB l (x)dx g G pg 0 MC g (x)dx (λ + k ) : f k T k (λ k ) : f k T k (ψ k ) : f k F kn r n = 0 n N (ρ n ) : r n p g + d l = 0 g G n l L n (φ) : r n = 0 n N p g, d l 0

5 5 / 47

6 Complicating Constraints of OPF 6 / 47 Energy balance constraint: d l p g = 0. g G l L Transmission network limits: T k F kn p g F kn d l T k n N g G n n N l L n

7 7 / 47 Locational Marginal Pricing Locational marginal pricing/nodal pricing: uniform price auction conducted as follows: Sellers and buyers submit price-quantity pairs Market operator solves (DCOPF) and announces ρ n as market clearing price for bus n

8 8 / 47 Efficiency of LMP Auction If agents bid truthfully, LMP auction reproduces optimal solution of OPF Proof: Follows from KKT conditions of OPF Locational marginal prices (LMPs): Prices ρ n produced for bus n from OPF

9 Example 9 / 47 All lines have identical electrical characteristics (reactance)

10 10 / 47 Price Splitting in Neighboring Nodes Suppose T 1 2 = T 2 3 = T 1 3 = 50 MW Lines 1-3, 2-3 should be used fully (why?) Optimal dispatch: p 1 = 50 MW, p 2 = 150 MW, p 3 = 100 MW Optimal flows: f 1 2 = 0 MW, f 2 3 = f 1 3 = 50 MW ρ 1 = 40 $/MWh, ρ 2 = 80 $/MWh, ρ 3 = 140 $/MWh (why?) Observe: f 1 2 < T 1 2, but ρ 2 > ρ 1

11 Settlement of the LMP auction: Bid Cleared Payment ($/h) G1 + MW at 40 $/MWh 50 MW at 40 $/MWh 2000 G2 + MW at 80 $/MWh 150 MW at 80 $/MWh G3 + MW at 140 $/MWh 100 MW at 140 $/MWh L2 100 MW at + $/MWh 100 MW at 80 $/MWh L3 200 MW at + $/MWh 200 MW at 140 $/MWh How much surplus is left over to the auctioneer? 11 / 47

12 12 / 47 LMP Can Be Different From Fuel Cost Suppose T 1 2 = 50 MW, T 2 3 = 100 MW, T 1 3 = 120 MW Optimal dispatch: p 1 = 160 MW, p 2 = 140 MW, p 3 = 0 MW Optimal flows: f 1 2 = 40 MW, f 2 3 = 80 MW, f 1 3 = 120 MW ρ 3 = 120 $/MWh (use sensitivity) Observe: ρ 3 is different from marginal cost of all generators

13 13 / 47 Non-Uniqueness of LMPs Suppose T 1 2 = 50 MW, T 2 3 = 100 MW, T 1 3 = 100 MW Optimal dispatch: p 1 = 100 MW, p 2 = 200 MW, p 3 = 0 MW Optimal flows: f 1 2 = 0 MW, f 2 3 = f 1 3 = 100 MW ρ 3 = 140 $/MWh is a valid LMP (use sensitivity) ρ 3 = 120 $/MWh is a valid LMP (use sensitivity) Observe: 120 $/MWh ρ $/MWh are all valid LMPs

14 14 / 47 Efficiency of LMP Pricing If agents bid truthfully, 1 locational marginal pricing maximizes welfare, and 2 the resulting allocation maximizes the profit of agents given the market clearing price Proof of item 1: LMP auction is solving welfare maximization problem

15 Proof of item 2: Decomposition of KKT conditions of DCOPF 15 / 47

16 Nodal Pricing in PJM (February 15, 2014) Figure: 05:40 (upper left), 08:40 (upper right), 09:20 (lower left), 09:55 (lower right). 16 / 47

17 17 / 47 Table of Contents 1 Locational Marginal Pricing 2 Congestion Rent and Congestion Cost 3 Competitive Market Model for Transmission Capacity 4 Zonal Pricing Zonal Pricing with Re-Dispatch Gaming Zonal Pricing

18 18 / 47 Congestion Rent Congestion rent: Surplus from locational price differences ρ n ( d l l L n n N g G n p g ) Congestion cost: excess cost due to finite capacity of transmission lines Congestion rent Congestion cost

19 19 / 47 Example: Congestion Rent Congestion cost Suppose D 2 = 50 MW, T 1 2 = 50 MW Competitive market clearing prices: ρ 1 = 20 $/MWh, 20 $/MWh ρ 2 40 $/MWh Congestion rent: 0 $/h $/h Congestion cost: 0 $/h

20 20 / 47 Example: Congestion Rent > Congestion cost Suppose D 2 = 60 MW, T 1 2 = 50 MW Competitive market clearing prices: ρ 1 = 20 $/MWh, ρ 2 = 40 $/MWh Congestion rent: 1000 $/h Congestion cost: 200 $/h

21 Congestion Rent Is Non-Negative 21 / 47 Congestion rent is non-negative, and given by the following expression: ρ n ( d l p g ) = (λ + k + λ k )T k n N l L n g G n k K Proof: If identity is true, then since λ + k, λ k is non-negative 0, congestion rent

22 22 / 47 ρ n ( d l definition of r n l L n n N g G n p g ) = ρ n r n = from ρ n = φ + F kn (λ k λ+ k ) n N k K and r n = 0 n N k K(λ + k λ k ) F kn r n = definition of f k n N (λ + k λ k )f k = from 0 λ + k T k f k 0 k K and 0 λ k T k + f k 0 (λ + k + λ k )T k k K

23 Congestion Rent and FTR Payments 23 / 47 Financial transmission rights (coming later) pay to their holders n N ρ n r n where r n is a feasible (not necessarily optimal) dispatch Congestion rent is adequate to cover FTR payments: n N ρ n r n n N ρ n r n

24 Proof: From previous proof, n N ρ n (r n r n ) = k K(λ + k λ k )(f k f k ) where λ + k, λ k are dual optimal multipliers, f k are flows corresponding to r n fk are flows corresponding to r n Consider three cases: f k = T k (which implies λ k = 0) f k = T k (which implies λ + k = 0) T k < f k < T k (which implies λ + k = λ k = 0) 24 / 47

25 25 / 47 Table of Contents 1 Locational Marginal Pricing 2 Congestion Rent and Congestion Cost 3 Competitive Market Model for Transmission Capacity 4 Zonal Pricing Zonal Pricing with Re-Dispatch Gaming Zonal Pricing

26 Separable Optimization 26 / 47 (Sep) : max x n f i (x i ) i=1 (ρ i ) : g i (x i ) 0 n (λ) : h i (x i ) 0 i=1 n agents, action variables x i R n i Coupling/complicating constraint i h i(x i ) 0 with h i : R n i R m convex g i : R n i R a i is a convex private constraint function f i : R n i R is a concave private benefit function

27 Competitive Equimibrium Competitive market equilibrium: pair of prices and quantities (λ, xi, qi ) such that: (x i, q i ) maximize profit given λ : (Profit-i) : max x i,q i (f i (x i ) (λ ) T q i ) g i (x i ) 0, h i (x i ) = q i, market clearing (supply demand): n i=1 q i 0 27 / 47

28 28 / 47 Competitive Market Model with Transmission 1 Agents: power producers, power consumers 2 Scarce resources (commodities): power, transmission 3 Profit maximization (quantity adjustment) of agents 4 Market clearing (price adjustment) of commodities Complication: usage of line capacity depends on location of producer and consumer Insight: producers responsible for shipping power to hub consumers responsible for shipping power from hub

29 29 / 47 Denote φ: price of power λ + k, (λ k ): price of transmission rights in (opposite to) reference direction Producer profit maximization: max φ p g λ + k F knp g + pg λ k F knp g MC g (x)dx k K k K 0 p g P g p g 0

30 30 / 47 Consumer profit maximization: max dl 0 d l D l d l 0 MB l (x)dx φ d l + λ + k F knd l λ k F knd l k K k K Market clearing for power: g G p g = l L d l, Market clearing for transmission capacity: 0 λ + k T k f k 0 0 λ k T k + f k 0.

31 31 / 47 Efficiency of LMP Pricing Nodal pricing produces an allocation of power and market clearing prices that correspond to a competitive market equilibrium. The converse is also true. Proof: Compare KKT conditions of (DCOPF) to competitive market model

32 32 / 47 Table of Contents 1 Locational Marginal Pricing 2 Congestion Rent and Congestion Cost 3 Competitive Market Model for Transmission Capacity 4 Zonal Pricing Zonal Pricing with Re-Dispatch Gaming Zonal Pricing

33 33 / 47 Criticisms of Nodal Pricing Criticisms of nodal pricing Too complicated, i.e. too many prices Local markets are too small reduced liquidity Local markets are too small opportunities for manipulation of prices Alternative proposal: zonal pricing

34 Formulation of Zonal Pricing Model 34 / 47 dl pg (ZP) : max MB l (x)dx MC g (x)dx l L 0 g G 0 (ρ z ) : p g f a + d l + f a = 0, z Z g G z l L z a=(,z) ATC a f a ATC a, a A p g 0, g G, d l 0, l L a=(z, ) Z : set of zones A: set of links between zones G z : generators located in zone z L z : loads located in zone z ATC: capacity of links connecting zones

35 35 / 47 Zonal Pricing Zonal pricing: uniform price auction conducted as follows Sellers and buyers submit price-quantity pairs Market operator solves (ZP) and announces ρ z as market clearing price for zone z Features: Kirchhoff s laws are ignored Congestion within zones is ignored Flows within a zone assumed not to influence flows on interconnections among zones

36 6-Node Example 36 / 47

37 37 / 47 6-Node Example with Flow-Based Pricing Node 1 Node 2 Node 3 Node 4 Node 5 Link Link Suppose T 1 6 = 200 MW, T 2 5 = 250 MW LMP pricing Welfare: $/h Different price at each node: ρ 1 = 25 $/MWh, ρ 2 = 30 $/MWh, ρ 3 = 27.5 $/MWh, ρ 4 = 47.5 $/MWh, ρ 5 = 45 $/MWh, ρ 6 = 50 $/MWh Lines flows: f 1 6 = f 2 5 = 200 MW

38 38 / 47 Zonal model definition: Z = {N, S} A = {N-S} North zone includes nodes 1, 2, 3 South zone includes nodes 4, 5, 6 Zonal pricing with ATC N-S = 200 MW Welfare: $/h ρ N = $/MWh, ρ S = $/MWh Flows: f 1 6 = MW, f 2 5 = MW Zonal pricing with ATC N-S = 450 MW Welfare: $/h ρ N = $/MWh, ρ S = $/MWh Flows: f 1 6 = MW, f 2 5 = MW How would you verify the correctness of these prices?

39 39 / 47 Zonal model is either: too conservative (ATC = 200 MW) Flow constraints are respected... but zonal pricing welfare < nodal pricing welfare too aggressive (ATC = 450 MW) Zonal pricing welfare > nodal pricing welfare... but flow constraints are violated

40 Flow-Based Zonal Pricing Consider a link a A of the zonal model, denote K a as set of lines that correspond to link a K A = a A K a Use PTDFs to account for Kirchhoff laws on K A : T k F kn ( p g d l ) T k, k K A n N g G n l L n 40 / 47

41 41 / 47 Flow-based zonal pricing: uniform price auction that maximizes welfare subject to Zonal prices Flow-based constraints (FBP) : dl 0 pg max MB l (x)dx MC g (x)dx l L g G 0 0 p g MC g (p g ) ρ z(g) 0, g G 0 d l MB l (d l ) + ρ z(l) 0, l L p g d l = 0 g G l L T k F kn ( p g d l ) T k, k K A n N g G n l L n

42 42 / 47 6-Node Example with Flow-Based Pricing Recall T 1 6 = 200 MW, T 2 5 = 250 MW Welfare: $/h ρ N = $/MWh, ρ S = $/MWh Flows: f 1 6 = 200 MW, f 2 5 = MW How do these results compare to LMP pricing? zonal pricing without flow-based constraints?

43 43 / 47 Re-Dispatch Re-dispatch: Pay-as-bid auction conducted after zonal pricing Sellers submit increment (inc) and decrement (dec) bids Inc bids: price producers are asking to provide additional power relative to zonal pricing auction Dec bids: price producers are willing to pay to market operator for decreasing production relative to zonal pricing auction Inc bids cleared to minimize payment to bidders Dec bids cleared to maximize payment to market operator

44 44 / 47 Example Under truthful bidding, zonal pricing followed by re-dispatch achieves the same result as LMP pricing with fewer prices lower charges to consumers above generator costs

45 45 / 47 LMP solution: p 1 = 800 MW, p 2 = 400 MW ρ 1 = 56 $/MWh, ρ 2 = 68 $/MWh 9600 $/h left to market operator Zonal pricing (single zone): p 1 = 1100 MW, p 2 = 100 MW (violates line limit) ρ = 62 $/MWh Zero surplus for market operator

46 46 / 47 Re-dispatching under truthful bidding: 300 MW of inc cleared from node MW of dec cleared from node 1 Payment to market operator from dec bids: $/h Payment from operator to cleared inc bids: $/h Difference: 1800 $/h

47 Gaming Zonal Pricing 47 / 47 Zonal pricing with re-dispatch can be gamed Dec Game, known by Enron traders as Death Star