Pricing Transmission

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1 1 / 37 Pricing Transmission Quantitative Energy Economics Anthony Papavasiliou

2 2 / 37 Pricing Transmission 1 Equilibrium Model of Power Flow Locational Marginal Pricing Congestion Rent and Congestion Cost 2 Alternative Market Designs for Congestion Zonal Pricing with Re-Dispatch Gaming Zonal Pricing

3 Separable Optimization 3 / 37 (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

4 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 (λ, x i, h i (x i )) is a competitive market equilibrium exist ρ i such that (λ, ρ i, x i ) is a solution to KKT conditions of (Sep) 4 / 37

5 5 / 37 Table of Contents 1 Equilibrium Model of Power Flow Locational Marginal Pricing Congestion Rent and Congestion Cost 2 Alternative Market Designs for Congestion Zonal Pricing with Re-Dispatch Gaming Zonal Pricing

6 6 / 37

7 Recall OPF 7 / 37 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

8 Complicating Constraints of OPF 8 / 37 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

9 9 / 37 Locational Marginal Pricing Locational marginal pricing/nodal pricing: uniform price auction conducted as follows: Sellers and buyers submit price-quantity pairs Market operator solves (OPF 1) and announces ρ n as market clearing price for bus n

10 10 / 37 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 (LMP): Prices ρ n produced for bus n from OPF

11 Example 11 / 37 All lines have identical electrical characteristics (reactance)

12 12 / 37 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

13 13 / 37 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

14 14 / 37 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

15 15 / 37 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

16 16 / 37 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

17 17 / 37 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

18 Congestion Rent Is Non-Negative 18 / 37 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

19 19 / 37 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

20 Congestion Rent and FTR Payments 20 / 37 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

21 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) 21 / 37

22 22 / 37 Table of Contents 1 Equilibrium Model of Power Flow Locational Marginal Pricing Congestion Rent and Congestion Cost 2 Alternative Market Designs for Congestion Zonal Pricing with Re-Dispatch Gaming Zonal Pricing

23 23 / 37 Criticisms of Nodal Pricing Criticisms of nodal pricing Too complicated (implementation, too many prices) Local markets are too small reduced liquidity Local markets are too small opportunities for manipulation of prices Zonal pricing with re-dispatch: an alternative approach intended to simplify pricing of transmission 1 aggregate nodes into a single zone with a common price (zonal pricing) 2 system operator re-dispatches generators to prevent violation of transmission constraints (re-dispatch)

24 Formulation of Zonal Pricing Model 24 / 37 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 g G z l L z a=(,z) ATC a f a ATC a p g, d l 0 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

25 25 / 37 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

26 6-Node Example 26 / 37

27 27 / 37 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

28 28 / 37 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 prices?

29 29 / 37 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

30 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 30 / 37

31 31 / 37 Flow-based zonal pricing: uniform price auction that maximizes welfare subject to Zonal prices Flow-based constraints (FBP) : dl 0 max MB l (x)dx l L g G 0 p g MC g (p g ) ρ z(g) 0 pg 0 d l MB l (d l ) + ρ z(l) 0 p g d l = 0 g G l L T k F kn ( p g d l ) T k n N g G n l L n 0 MC g (x)dx

32 32 / 37 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?

33 33 / 37 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 suppliers are willing to pay to market operator for decreasing supply relative to zonal pricing auction Inc bids cleared to minimize payment to bidders Dec bids cleared to maximize payment to market operator

34 34 / 37 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

35 35 / 37 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

36 36 / 37 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

37 37 / 37 Gaming Zonal Pricing Zonal pricing with re-dispatch can be gamed by price-taking agents Consider generators in node 1 whose marginal cost exceeds 56 $/MWh Pure strategy Nash equilibrium: bid 56 + ɛ $ MWh Bids will be in the money in the zonal auction Dec bids will be in the money in the re-dispatch auction These generators earn profit for doing nothing!

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