Electricity Markets
Tuesday April 1st
Matt Davison
Departments of Applied Mathematics
and Statistical & Actuarial Sciences
The University of Western Ontario
Collaborators
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Former PhD students (Lindsay Anderson, now dept of
Biological Eng, Cornell, Matt Thompson, now Faculty
of Business, Queen's University).
Former M.Sc. students (Abu Bah, Rizwan Mukadam,
Karen Anderson)
Current research team (Guangzhi Zhao, Sharon
Wang, Natasha Kirby)
Private sector (Peter Vincent, OPG, Peter Stabins,
Dydex, Ligong Kang, Transalta)
UWO Collaborators (Brock Fenton, Dmitri
Karamanev)
Thanks
Financial support provided by
„ MITACS
„ NSERC
„ Canada Research Chair Program
„ Ontario Power Generation
„ Dydex Research and Capital Ltd
Electricity Units
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Energy has units Joules – a Joule is a NewtonMetre, or a Coulomb-Volt, or a Watt-Second.
Lift a small (100g) apple 1 meter. Heat 1 g of
cool dry air 1 degree Celcius.
Power has units Watts. (think of a 40 Watt bulb)
Electricity is measured as power but traded as
energy – unit is kilowatt-Hour (3600 joules) for
retail, or MWh (3.6 million joules) for wholesale.
Electricity is *cheap* -- order 100,000 joules per
dolllar.
Outline
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Deregulated electricity markets
A hybrid model for price spikes
A control model for generating facilities
Future work and lessons for public policy
Deregulated Electricity Markets
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Ideological approach to deregulation
Some Ontario data
Deregulated markets as an engineering
and planning tool.
1. Why Deregulate?
Why should we deregulate?
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The idea of a “natural monopoly”
debunked
Ideological reasons (private sector is
always more efficient than the public
sector)
(Ontario) – power utility was
¾ out of control
¾ nuclear cult
¾ sea of red ink
1. Why Deregulate?
Legal Framework
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In 2002 the former monopolist Ontario Hydro
was divided into three main entities: Ontario
Power Generation (OPG), Hydro One, and
the Ontario Energy Board (OEB).
OPG does generation, Hydro One long
distance (high voltage) transmission, and the
OEB licenses all market participants
Types of Market Participants
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The OEB issues 6 classes of licences
1) Generators (OPG 65%, Bruce Power 25%)
2) Transmission (Hydro One – monopoly)
3) Distributors (Local Distribution Companies
LDC)
4) Wholesalers (GM, Dofasco, Adjacent
Markets)
5) Retailers (eg. Direct Energy)
6) the Independent Electricity Systems Operator
(IESO) which administers the market.
How does the market work?
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Power is traded at each of 24 hours per day.
Generators offer power; users bid for power.
Bids/Offers are prepared by 11PM the previous
night for each hour but can be revised up until 4
hours ahead of the beginning of each hour.
Each participant submits one or more ordered
pairs into the market for that hour – (amount bid,
price bid) or (amount offered, price offered).
Aggregating bids and offers
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Bid stack is constructed in decreasing order of
price. (Prices range from -$2000 to $2000
/MWh)
Offer stack is constructed in increasing order of
price
See whiteboard sketch
Where supply meets demand – is price that
everyone pays for that hour.
Pesky details buried in “market uplift charge”
about which more later
Bid/Ask strategy
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If you “have to have it” you bid a very high
amount. For instance GM – cost of power
is tiny compared to cost of running assembly
line.
If you “have to sell it” you bit a very low, often
a negative, amount. For instance Nuclear
Power Plant.
Price is usually set by the flexible people.
Game Theory
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Obviously lots of game theory going on here
– see work by Scott Rogers (Toronto) and his
co-workers.
Oligopoly pricing theory from economics –
Cournot Equilibria, etc.
Special players
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Solar power generators are guaranteed
$420/MWh for all power they sell; wind and
special green microhydro $140/MWh. All of
this power is bid in at -$2000 to guarantee it
is taken.
Who pays for this? Uplift Charge.
Other special features
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Line losses. It isn’t free, although it is cheap,
to transmit (or ‘wheel’) power long distances.
Intuition – about 1% loss per 500km
travelled.
Cost of this is absorbed into uplift charge.
Derivative Markets in Ontario
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Currently all OTC
Discussions have been ongoing with the TSX
about getting some contracts online.
Other types of Electricity Markets
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Markets for Ancillary Services such as:
Spinning Reserve
Standby Reserve
Reactive Power
Day Ahead Markets (quasi-spot) to put teeth
into 24 hour advance bids.
Ontario Open Market Price
1. Why Deregulate?
Load Shapes
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Daily loads
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30000
Weekly loads
25000
25000
20000
20000
L o ad
L o ad
15000
15000
10000
10000
Daily Load
Weekly Load
5000
5000
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0
Sunday
Monday
Tuesday
Wednesday
12/22/2005
Friday
Saturday
Date
Hour
08/17/2005
Thursday
Peak Load Day (07/13/2005)
Week of 08/17/2005
Week of 12/22/2005
1. Why Deregulate?
Why should we deregulate?
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Two things going on here:
Desire to break up large “lazy” utility
monolith
But that could happen without hourly prices,
couldn’t it?
Controversial: The whole point of an hourly
market is the price spikes!!
Price spikes – flatten load shape –
encourage market entry
1. Why Deregulate?
What causes price spikes?
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Hybrid model overview
Sub-Models: electrical load and system
capacity
Spot price results
¾ Applications: derivatives pricing and
risk measures
Optimal maintenance schedules
2. Understanding Price Spikes
Why Is Electricity Different?
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Electricity cannot be stored
Demand for electricity is inelastic
Electricity produced must be dispatched
What appears to be a complication can be a
modeling advantage..
2. Understanding Price Spikes
How To Model A Non-Existent Market?
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Time series is:
¾ Short
¾ Volatile
¾ Non-stationary
Benefit from knowledge in “regulated” setting
Underlying drivers are stationary
Markets are highly regional
2. Understanding Price Spikes
Stack-Based Pricing
Price
$100
$30
8
Coal
Hydro
$20
Nuclear
$25
20
Gas Turbine
Peaker
$40
26 28 29
Load(MW)
2. Understanding Price Spikes
Price Model Desiderata
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What do we want to use the model for?
Price spikes
Two distinct price regimes
Prices don’t drift indefinitely
Seasonal pattern of price spikes
*A two-regime switching model can incorporate these
characteristics*
For a discussion of the modelling philosophy and early implementation,
see MD, L. Anderson et al. IEEE Transactions on Power Systems 17(2):
257-264 2002
2. Understanding Price Spikes
A Two-Regime Switching Model
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Switching variable controls the process
What controls the switching variable?
¾ When do spikes typically occur?
¾ Seasonal (summer, winter)
¾ Some spiking in shoulder months as
well
2. Understanding Price Spikes
The α-Ratio and Spike Probability
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The primary driver of the switching variable is
Load(t)
Demand(t)
α(t) =
=
Capacity(t) Supply(t)
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The following should be true:
lim Pr( price spike) = 0
α →0
lim Pr( price spike) = 1
α →1
Pr(High price) vs. α
„ The probability of a spike increases rapidly near
α = 0 .8 5
2. Understanding Price Spikes
A Hybrid Model
Simulated
Price (t)
e (t) = f(a (t))
α (t) =
Electricity
Load M odel
L o a d (t)
C a p a c ity (t)
G enerating Capacity
M odel
2. Understanding Price Spikes
Modelling Generating Capacity
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Generating system has fixed maximum capacity
Available operating capacity is the maximum, less;
¾ Planned (maintenance) outages
¾ Unplanned (forced) outages
Build a probabilistic model of system-wide capacity
¾ Aggregate exponential
¾ Sequential simulation
¾ Aggregate Weibull
2. Understanding Price Spikes
Modelling Unplanned Outages
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Each generating unit has Weibull distributed
Time to failure (TTF) and Time to repair (TTR)
Weibull CDF is given by:
Pr(t > D) = 1 − e
−(
D
ηi
) βi
2. Understanding Price Spikes
Power System Assumptions
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All generators are either operational or failed
(under repair)
Only a single unit can change state in any
instant
All TTFs and TTRs are independent and
Weibull distributed
2. Understanding Price Spikes
A System Model of Forced Outages
System changes state whenever a unit changes state
Pr(TTSC i < D ) = Fi ( D ), and
N
TTSC s = min(TTSC i )
i =1
Therefore
Pr(TTSCi < D) = Pr (NO units change state before time D)
N
= ∏(1− Fi (D))
i =1
Whole State vs. Remaining State issues.
2. Understanding Price Spikes
The System Wide Failure Model
Pr(ts > D) = e
−(
t
ηc
) βc
ηi
1 D β
[
(
, ( ) )]
Γ
∏
β i ηi
i =1 β1 M i
N
i
i ≠c
Here Γ(a, x) is the incomplete Gamma function
Γ (a, x) =
∫
∞
x
− t a −1
e t
dt
For the details, see LA & MD, IEEE Trans on Power Systems 20 (4)1783- 1790 (2005)
2. Understanding Price Spikes
Simulating Electrical Load
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Load is a well-studied problem
Predictable annual and diurnal load cycles
Strongly linked to weather, daylight, culture
2. Understanding Price Spikes
Simulating Mean Load
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Double sinusoid for base load
Lb (t ) = A0 + A1 sin(ω1t + φ1 ) + A2 sin(ω 2t + φ2 )
2. Understanding Price Spikes
Simulating Load Volatility
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Seasonal volatility given by AR(1) model
R(t ) = α i + β i R(t − 1) + Zσ i , Where Z ∼ N(0, 1)
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The resulting electrical load is then
Lˆ ( t ) = Lb ( t ) + R ( t )
2. Understanding Price Spikes
Simulating Electrical Load
Observed and predicted loads (January 2001 – December 2002)
2. Understanding Price Spikes
Sample Spot Price Results (1)
Observed and Simulated Prices for PJM
2. Understanding Price Spikes
Sample Spot Price Results (2)
Log Histogram of Observed and Simulated Prices for PJM (2000)
2. Understanding Price Spikes
Derivative Pricing Results
Forward Values($/MWh)
Delivery
Market
Simulated
Std Error
Realized
J/F
40
36
0.24
28
4Q
31
31
0.20
40
Summer
91
46
0.28
33
Call Option Value, Strike = $100
Expiry
Market
Simulated
Std Error
Realized
Summer
35 - 50
0.8
0.18
~0
Market and Simulated Forward Prices for September 12, 2000
2. Understanding Price Spikes
Discussion of Options Pricing Results
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Simulated spot prices are a good proxy for
observed
For derivative contracts, simulated prices
are lower
Highly illiquid market for derivatives
Huge risk premia
Contract sellers and purchasers are highly
risk-averse
This makes more sense if we view it as an insurance-like
market
2. Understanding Price Spikes
Flattening the load shape
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Amory Lovins “negawatts”
Sell uses of energy, not energy itself
Show retail hydro bill
Discuss industrial users
Supply, not demand, side solutions?
Pump storage facilities
3. Managing Load Shape
An Ontario Electricity Bill
3. Managing Load Shape
Industrial/Commercial Users
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Industrial users have very flat load shape
They also have significant political clout
Commercial users have peaked load shapes
But for them energy costs are comparatively
minor (mostly cooling)
Supply-side solutions?
3. Managing Load Shape
Pump Storage Facilities
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Conversion of mechanical to electrical
energy is efficient
Can get 80% round trip efficiency from
electricity Æ running water Æ electricity
So pump water when power price is low
Use water to run turbine when power price is
high
What is the best way of doing this?
3. Managing Load Shape
Pump Storage II
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Pump storage plant
3. Managing Load Shape
Stochastic Optimal Control
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Valuation and Optimal Operation of electric
power plants in competitive markets
Continuous time model for power prices
including Poisson jumps
Price dynamics
N
dP = μ1 ( P, t )dt + σ 1 ( P, t )dX 1 + ∑ γ k ( P, t , J k )dqk ,
k =1
where μ , σ and the γ k can be any arbitrary functions of price and/or time.
For detailed discussion, see M. Thompson, MD & H. Rasmussen
(2004), Operations Research 52, 546-562.
3. Managing Load Shape
The PIDE
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Merton-style portfolio optimization problem
Plus lots of engineering fluid mechanics
Leads to PIDE with initial and boundary conditions:
1
3600 c
V h − ( r + λup ( P ) + λ down ( P ))V + H ( c , h ) P
σ ( P )V pp + μ ( P , t )V p −
2
20000
∞
1
− ( S − 700) 2
exp(
) dJ 1
+ λ up ( P ) ∫ V ( J 1 , h , t )
2
−∞
2(10)
100 2π
Vt +
1
− ( S − 100) 2
exp(
) dJ 2 = 0,
+ λ down ( P ) ∫ V ( J 2 , h , t )
2
−∞
2(10)
10 2π
Initial condition: V ( P , h , T ) = 0,
Boun dary conditions: V PP → 0 (for P large),
∞
V PP → 0 ( as P → 0).
3. Managing Load Shape
The value surface
Solve the equation numerically using flux limiters to get:
3. Managing Load Shape
The control surface
3. Managing Load Shape
What if Power Prices are Predictable?
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Price depends on Load,
Load depends on Temperature
Temperature can be predicted fairly well up
to a week into the future (NASA/NOAA is
aiming to increase 5 day forecast accuracy to
90%)
Keep in mind prices are usually formed ca.
24 hours before the fact
Optimal Operation with Predictability
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What if a storage facility is small (relative to
inflow).
For instance there is a pump storage facility
at Niagara falls that can store just one day’s
mean water inflow.
For such a facility, to a decent first
approximation, the price might be considered
deterministic.
Interesting Deterministic Optimal Control
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The engineering assumptions used are crucial
If water inflow rates are assumed exogeneous but
nonzero, we obtain the counterintuitive result that
pump release cycles will be used even with a
constant price
Value of hydrological EP I
Low water inflow case; From Zhao and Davison, 2007
Value of hydrological EP II
48 hour time horizon with and without perfect EP: High water inflow case; From Zhao and
Davison, 2007
Additional Sales expected from optimal use of
hydrological EP – as function of inflow variance
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From Zhao and Davison (2007)
Wind Power
Wind power promising but with a lot of problems
„ Non-dispatchable
¾ couple with microhydro or pump storage;
¾ optimal construction of such pairs
„ Uneconomic
¾ use better economic metrics than total energy
produced to optimize wind turbine design and siting
„ Siting
¾ Use investment under uncertainty techniques for
optimal siting
¾ Constraints on siting: aesthetic, bird/bat safety
(with Brock Fenton)
4. Lessons & Future Work
Microhydro
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Power sites formerly thought too small to
develop
New technology and new need is bringing
these into play
Small scale watersheds have long memory
overflows
Mathematical techniques known as
fractional Brownian Motion
Investigate joint siting of Wind/Water hydro
sites
4. Lessons & Future Work
Lessons for Public Policy
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Goals of deregulation must be communicated
in realistic, non-ideological terms
Ubiquitous time of day metering is essential
There is a business niche for someone to
“vacuum up the pennies” in saving
homeowner and commercial users money
4. Lessons & Future Work
Some Basic Renewal Theory
Renewal Theory ⇒ arbitrary inspection time,
which requires Remaining State Distribution
Pr(ti < D | St = St −1 ) =
1
μi
∫
∞
D
(1 − Fi (t ))dt
Where μ i is the mean duration of state i.
2. Understanding Price Spikes
The System Wide Failure Model
Using TTSC as the inspection time, the distribution
of the time to the next state change is given by;
Pr(t s > D ) = e
−(
t
ηc
)βc
N
∏[
i =1
i≠c
1
μi
∫
∞
D
−(
e
t
ηi
) βi
dt ]
Completing the integration yields our distribution for
TTSC of the system
next
2. Understanding Price Spikes
PJM Market Details
The PJM market
„ Serves: 25 million people
„ Peak load: 64,000 MW
„ Maximum capacity: 76,000 MW
Type
Steam Turbine (Coal)
Generators
Capacity
(MW)
Percentage
10
750
146
Steam Turbine (Oil)
124
28, 000
46
Combustion Turbine (Oil)
275
13, 000
21
Nuclear
13
13, 000
21
Hydro
60
3,600
6
Others (various renewable)
42
2,700
4
524
61,050
Total
Capacity Profile 2000/2001
2. Understanding Price Spikes