Equilibrium in Wholesale
Electricity Markets
Nathan Larson
UCL and NERA
David Salant
NERA
Agenda
Introduction
Description of California energy markets
Existence of SFEs
Lower bounds on expected prices
Price adjustment process and stability with discrete units
!
!
Even when equilibrium is unique at Bertrand/competitive levels, prices
may be persistently above those levels
Outcomes of games with learning may not converge to equilibrium of
the underlying game
2
California Energy Markets
Three utilities – PGE, SDG&E/Sempra and Southern California Edison
Peak demands of approximately 45K MW in summer months
!
18% of supplies from imports
!
Five main generators controlled nearly half of fossil fuel generation within CA
Summer 2000 crisis began and continued past the close of the CALPX in
Feb. 2001 with
!
Stage 2 emergencies declared on scattered days in May – September 2000
!
Stage 3 emergencies in Dec. 2000 (1 day), January 2001 (18 days), February
2001 (16 days), March 2001 (2 days) and May 2001 (2 days).
!
Average loads of < 33K MW through the latter part of the crisis
AB 1890 mandate IOUs purchase energy through CALPX/CAISO
3
Some Explanations of What Happened?
Strategic withholding of supplies (JK & BBW) and less than
perfectly competitive behavior (Puller)
Lack of forward contracts
!
!
!
CALPX had block forward market,
IOUs were permitted to purchase much more block forward than they
did and they could hedge with derivatives
Ex post regulatory review of forward/derivative purchases placed all the
downside risk with IOU, so this was not a factor
Ability of other WECC buyers to be active in forward markets
and inability of CA IOUs placed burden on latter (Allaz and
Vila/Salant&Loxley – NJ experience)
Lack of capacity credits/markets or regulation
Nature of equilibrium in CALPX/CAISO SFE type auctions
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NJ vs. California
CA
NJ
!
3 large utilities
!
3 large utilities
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Moderate HHI on generation
!
High HHI > 3200 in generation
!
!
!
!
!
Capacity and other reserves needed to
respond to demand peaks no longer
mandated
CAISO/WECC – provides links
between zones
!
!
!
High price spikes
CALPX/CPUC did not give utilities
much of an out
!
Theoretically unsound Supply Function
Auction format
!
Significant transmission constraints to
the remain of PJM
Capacity credit market mandates
reserve margins
Outcome competitive average 5.1¢ for
entire year
Combination of primary long term
contracts and options for spot and
short term contracts gave more
flexibility for buyers to get a more
competitive price
Clock auction designed to attenuate
market power of the sellers
5
The CALPX/CAISO
The CALPX/CAISO consisted of a set of markets
!
!
!
In the main day-ahead CALPX market
– Participants were required to submit energy supply/demand schedules for
each hour of the next day
– Bids were required to be piecewise linear supply or demand schedules
– The CALPX, as a scheduling coordinator with the CAISO, would submit
the aggregate schedules.
CALPX also conducted
– A post-close quantity match for parts of bids that were close to clearing
price in the day-ahead market.
– A block forward market for future hours, up to nine months in advance,
and a limited fraction of the load.
– Hour ahead market (essentially CALPX passed along bids to CAISO)
CAISO managed an hour ahead/real time market for Ancillary Services
(replacement, non-spin, spin, regulation)
– Bids would include capacity price for availability and energy price for
dispatch
– Separate processing of capacity bids and energy bids
6
Equilibrium in the CALPX/CAISO Markets
CALPX – supply function game
!
!
!
Supply functions were restricted to be piecewise linear functions with a
limited number of segments
Equilibrium need not exist in pure strategies
Payoffs can have wrong type of jump discontinuities near q = 0, but
assuming no initial jump discontinuities or flats in supply, payoffs are
continuous and Glicksberg result applies
CAISO
!
!
!
!
Bids are step functions
Well known no pure strategy equilibrium exists (e.g.,
Kreps/Scheinkman)
However, discontinuities are quite poorly behaved.
Mixed strategy equilibrium exists (with some assumptions, DasguptaMaskin (1986)) – but profits may not be very competitive, and though
limit of discrete games, may not be stable.
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Bounds on Average Prices
! Equilibrium prices in individual hours can vary substantially, due to
variations in load, availability of supply and random behaviour. This makes
predicting prices hour-by-hour is difficult.
! In contrast, average prices are easily observable, and less susceptible to
unobserved idiosyncratic shocks. And theoretical predictions using average
prices are tighter.
Supply
!
!
Two symmetric generators
Marginal costs = 0 and each has capacity constraint of K
Demand
!
!
Price cap pc
Demand at price cap = Qc (possibly uncertain)
Expected profit bound
Let , πc = pc E(min{Qc – K, K} | Qc > K) Pr(Qc > K)
Then, a lower bound on expected value of equilibrium prices is
E(p)
E(p) > πc / K
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Smoothness vs. Discrete Supply Functions
In practice, supply schedules are discrete, but this may not matter much. Multi-step
supply schedules, aggregated across generators can be nearly smooth.
Smooth supply functions
! Supply Function Equilibrium
(Klemperer and Meyer) approach is
appropriate.
! The Idea: A generator chooses an
infinite number of price-quantity pairs (a
continuous schedule) to solve a
pointwise optimization. (Each p-q pair is
optimal for some level of demand.)
Equilibrium is characterized by a system
of differential equations.
!Prediction:
!
!
Range of equilibria. Markups (above
MC) range from low (Bertrand-like) to
high (Cournot-like).
Some evidence that affine equilibrium
(with intermediate markups) is stable.
Discrete supply functions
! Multi-unit Auctions (von der Fehr and
Harbord) use this approach.
!The idea:
A generator submits a finite list
of prices – one for each discrete unit.
!Prediction:
Undercutting implies Bertrandlike pricing in simple cases and no
equilibrium in pure strategies for more
complex cases.
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An Open Question:
Do the two approaches converge?
In other words, when is the “approximate smoothness”
assumption valid?
Baldick and Hogan show that when bids are piecewise linear
rather than continuous, a subset of SFE outcomes are stable
(outcomes near the affine equilibrium), but
!
!
!
Limited notion of stability.
Requires quadratic costs.
No results for step function bids.
Two Models
!
!
Model 1 (MUA friendly) - Simple, two discrete unit per bidder
setting with a unique, Bertrand-like equilibrium
Model 2 (SFE friendly) - “Quasi-continuous” ten unit per bidder
setting.
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Model 1
Generators
Two identical generators (A,B).
Each has two discrete units of capacity.
Constant marginal cost across each unit: c1 = 0 , c2 = 1.
The Market
A bid is a single price for power from each unit: (p1, p2).
Demand = 2 units with certainty up to a price cap of 3.
Market-clearing:
!
!
p1A , p2A , p1B , p2A are ranked in a merit order: p(1) ≤ p(2) ≤ p(3) ≤ p(4).
The units corresponding to p(1) and p(2) are dispatched and paid p(2)
(uniform price).
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Nash Equilibrium Prediction
(Essentially) unique pure strategy Nash equilibrium:
!
Both generators bid (1,1)
!
Only most efficient units are dispatched
!
Price = marginal cost of most competitive idle unit
Doubts about the NE prediction?
!
(1,1) is not a strict NE – since unit 2 is never dispatched,
a generator has no incentive to price it competitively.
!
Could a model of learning out of equilibrium help to
motivate p2 = 1 ?
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The Model of Learning
f t = (1 − α ) f
i
+ αg
i
t
i
Generator i’s strategy in period t
(distribution over bids)
i
t
Update term based on market
outcomes in period t-1
ft
g
i
t −1
α
“Inertia” parameter
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Updating: Noisy Best Reply
g
−i
π ( p, ft −1 )
κ
i
t −1
∝e
κπ ( p , f t −−1i )
Expected payoff to bidding p, given an
−i
opponent using f t −1
Controls the level of noise
κ = ∞ : pure best reply
κ = 0 : random updating
κ ∈ (0, ∞) : favors better replies,
but allows for mistakes, experimentation, ...
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Distribution of bidding “errors”
Relative frequency of bidding strategy
1
0.9
0.8
0.7
0.6
0.5
κ = 30
0.4
0.3
κ = 100
0.2
0.1
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
∆π : Deviation from profit - maximizing response
0.09
0.1
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Price Adjustment Steady State
If a steady state exists, it satisfies:
f ( p) = g ( p)
κπ ( p, f )
=e
This is a logit equilibrium (McKelvey and Palfrey).
As κ " ∞ (perfect profit maximization), logit equilibrium approaches a
Nash equilibrium.
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The Noiseless Limit
One might conjecture that as κ " ∞, behavior should approach
the pure strategy equilibrium at (1,1).
This is false: in the limit,
!
Bids remain mixed.
!
Average bids, and prices, are bounded away from 1.
!
Inefficient units are dispatched with positive probability.
So what do market outcomes look like? We turn to
simulations…
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Rapid Adjustment: α = 0.3 , κ = 100
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Slow Adjustment: α = 0.01 , κ = 100
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Noisier optimization: α = 0.30 , κ = 30
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Distribution of bids in (noisy) steady state
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Lower price cap: pcap = 2, α = 0.30 , κ = 100
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Bidding with demand uncertainty
(Demand distributed uniformly on [1,3])
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Interpreting the Results
! Volatile prices – reminiscent of Edgeworth cycles –
even though no generator is ever capacityconstrained.
! Average prices well above the unique pure strategy
equilibrium – and well above the cost of the next
available unit.
! Price caps can matter even if they never appear to
be binding.
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Model 2:
Does “Many Units ≈ Smooth”?
Still two identical generators.
Now each has ten units of generation capacity.
MC = 0.5q (quadratic costs)
D = a – 0.1p (linear demand)
a is a random variable
Bids are step functions: (p1, p2, … , p10)
(This is a bit closer to reality – for example, bids in the CAISO
Imbalance Energy market were step functions with up to ten
steps.)
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How to bid if supply and demand
were smooth:
m
Dresidual
D
Rival’s supply
p
q
a
For a particular level of demand, the optimal (p,q) pair satisfies
[p – MC(q)] – q⋅m = 0
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Supply Function Equilibrium: The optimal supply curve
traces out the optimal responses to every level of demand:
a3
a2
a1 a4
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Multiple Equilibria
Price
Range of
SFEs
MC
Quantity
Steeper rival supply " less elastic residual
demand " steeper optimal response " Multiple
Equilibria
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Stability
Price
Affine SFE
MC
Quantity
With “linear deviations” only the affine SFE is stable (Baldick/Hogan). Requires
-- piecewise linear bids (not step functions)
-- quadratic costs
-- no capacity constraints or price caps
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Linear Deviations
C
Price
D
B
Affine SFE
A
Quantity
If deviation looks like ABD …
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Linear Deviations
C
Price
D
E
B
Affine SFE
A
Quantity
… then “second order” response looks like ABE.
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Nonlinear Deviations?
D*C
Price
B
Affine SFE
A
Quantity
Bids may deviate randomly for a variety of reasons (unplanned outages, fuel
price changes, generator mistakes, and so on), but there is no particular
reason to expect these deviations to be linear. For example, a generator
could deviate to ABD*.
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Nonlinear Deviations?
Price
D*C
E*
B
Affine SFE
A
Quantity
In this case, the optimal response by other bidders might be to ABE* - i.e., to
even higher prices. But Baldick and Hogan rule out deviations and responses
like this by assumption.
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Doubts About SFE Stability Result
! Doesn’t apply to step function bids – when
destabilizing “undercutting” is most tempting.
! Applies to a narrow class of perturbations. Bids
may be noisy for many reasons (plant outages,
volatility in fuel prices, mistakes & experimentation).
Why should this noise be linear?
! Restrictive assumptions on costs, capacity, …
Simulations provide a robustness check.
34
Simulations with Step Function Bids
As before:
i
f t : distribution over i' s bids in period t
f t = (1 − α ) f
i
i
t −1
+α g
i
t
But now, calculating best responses is not feasible – too many
dimensions.
The alternative:
Sample locally around current strategy distribution.
Include in update term the sampled bids that perform best
against the current distribution of bids.
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Initial Distribution of Bids
Price
Affine SFE
MC
Unit
36
Period 40
37
Period 70
38
Period 120
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Period 180
40
Average Market-Clearing Price
Price
Average price
In affine SFE
Period
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Conclusions
! The effects of the discrete, step-function bidding format do
not “smooth out” as the number of steps grows larger. Even
with ten steps (as in the CAISO real-time market), bidders do
not converge to a stable equilibrium.
! Price cycles are endemic.
! Baldick and Hogan’s stability result is deceptive – while
prices may be near the affine SFE level on average, price
volatility is very high.
! Moreover, our simulated prices are already averaged over
the load curve. In other words, this volatility has nothing to do
with the variation between high and low demand hours. If we
look at prices on an hour by hour basis, they will be even more
volatile.
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