Stochastic models
and optimization for
the Energy Industry
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
Commodities quant, Scotiabank; Guangzhi Zhao, now
at Alberta Treasury Branches).
Current research team (Natasha Kirby)
Practitioners (Peter Vincent, OPG/RBC, Peter
Stabins, Dydex/Cdn Energy Wholesalers, Ligong
Kang, Transalta, Ozgur Gurtuna, Turquoise
Technologies, Brian Mills & Claude Masse,
Environment Canada)
U Windsor collaborators (Rupp Carriveau, David Ting
& James Konrad, Mech Eng; Frank Simpson
Geology)
Funding
Financial support provided by
 MITACS
 NSERC
 Canada Research Chair Program
 Ontario Power Generation
 Dydex Research and Capital Ltd
 Ontario Energy Centre of Excellence
Outline
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Deregulated electricity markets
A hybrid model for price spikes
A control model for generating facilities
with applications to valuing hydrological
forecasts
Current electricity storage work preview
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
 out of control
 nuclear cult
 sea of red ink
1. Why Deregulate?
Ontario Open Market Price
1. Why Deregulate?
Load Shapes

Daily loads

30000
Weekly loads
25000
25000
20000
20000
Load
Load
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
 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:
 Short
 Volatile
 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*
MD, L. Anderson et al. IEEE Transactions on Power Systems 2002, LA,
MD IEEE Trans PES 2008, LA, MD HERA 2009
2. Understanding Price Spikes
A Two-Regime Switching Model
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Switching variable controls the process
What controls the switching variable?
 When do spikes typically occur?
 Seasonal (summer, winter)
 Some spiking in shoulder months as
well
2. Understanding Price Spikes
The -Ratio and Spike Probability

The primary driver of the switching variable is
Load(t )
Demand(t )
 (t ) 

Capacity(t ) Supply(t )

The following should be true:
lim Pr( price spike)  0
 0
lim Pr( price spike)  1
 1
Pr(High price) vs. 
 The probability of a spike increases rapidly near
  0.85
2. Understanding Price Spikes
A Hybrid Model
Simulated
Price (t)
e(t) = f(a(t))
α(t) =
Electricity
Load Model
Load(t)
Capacity(t)
Generating Capacity
Model
2. Understanding Price Spikes
Modelling Generating Capacity
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Generating system has fixed maximum capacity
Available operating capacity is the maximum, less;
 Planned (maintenance) outages
 Unplanned (forced) outages
Build a probabilistic model of system-wide capacity
 Aggregate exponential
 Sequential simulation
 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
i
) i
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(TTSCi  D)  Fi ( D), and
N
TTSCs  min(TTSCi )
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
c
) c
i
1 D 
[

(
, ( ) )]

 i i
i 1 1M i
N
i
i c
Here a, x) is the incomplete Gamma function

(a, x)   e t
x
 t a 1
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(1t  1 )  A2 sin(2t  2 )
2. Understanding Price Spikes
Simulating Load Volatility
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Seasonal volatility given by AR(1) model
R(t )  i  i R(t  1)  Z i , Where Z  N(0, 1)

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  running water  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  1 ( P, t )dt   1 ( P, t )dX 1    k ( P, t , J k )dqk ,
k 1
where  ,  and the  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
3600c
Vt   ( P )V pp   ( P, t )V p 
Vh  (r  up ( P )  down ( P ))V  H (c, h) P
2
20000

1
( S  700) 2
 up ( P )  V ( J1 , h, t )
exp(
) dJ1
2

2(10)
100 2
1
( S  100) 2
 down ( P )  V ( J 2 , h, t )
exp(
) dJ 2  0,
2

2(10)
10 2
Initial condition: V ( P, h, T )  0,
Boundary conditions: VPP  0 (for P large),

VPP  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 quite predictable a week into
the future (NASA/NOAA sees 90% 5 day
forecast accuracy within reach)
Prices are usually formed ca. 24 hours
before the fact
Optimal Operation with Predictability
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Some storage facilities small rel. to inflow.
The pump storage facility at Niagara Falls
can store just one day’s mean water inflow.
For such a facility the price might be
considered deterministic.
Two steps forward, one step back
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G. Zhao & MD, “Optimal Control of
Hydroelectric Facilities incorporating Pump
Storage” Renewable Energy 2009
-- deterministic market prices (step back)
-- inflows (deterministic variable, then
random) (step forward)
-- more realistic engineering (friction losses,
realistic turbine efficiency curves)
Zhao & Davison (2009a)
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Used dynamic programming to optimize
discounted cash flows at each time step
Solution gave value (not interesting) and
optimal control (very interesting) as a
function of various price and inflow models
and utilizing realistic turbine efficiency
parameterizations.
Typical Result
Interesting/Counterintuitive results
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Even with no randomness in price or inflow
exercise is very interesting
Depending on the reservoir geometry, the
physics of the system are very important
For instance for an upper reservoir with a
surface area low compared to the inflow rate,
the steady state is periodic even with fixed
price:
Cycling even with constant prices
Zhao/Davison 09a conclusions
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The optimal control of real hydro facilities,
don`t care much about (deterministic) price
variations
They are driven by the physics of head
maintenance and by the need to operate
hydromachinery near optimal design points.
Next slide shows efficiency ellipse
Turbine efficiency plot
Valuing Hydrological forecasts
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Organizations such as Environment Can ada
like to know where they are adding value.
Environment Canada Study on the value of
Environmental Predictions to the Canadian
Energy Sector (see Davison, Gurtuna,
Masse, Mills, paper submitted to IJER, 2009)
Use above model to value hydrological
forecasts
What’s a forecast worth?
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A forecast adds value when
-- it is actionable
-- it allows for better decisions than use of a
naïve information set
Value of Perfect forecast upper bound for
value of imperfect forecast
Valuing short term inflow forecasts
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Solve stochastic optimal control problem for
hydro plant with random water inflows.
Solve deterministic optimal control problem
for hydro plant with deterministic water
inflows (drawn from same random model)
Value of perfect forecast = Value(perfect
exercise) – Value (usual optimal exercise)
Zhao, Davison, J Hydrology 2009
Some more details:
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Random inflow model was 48 inflows which
were equally likely to be “Low” or “High”
Deterministic inflow was a realization from
this random process
Low inflow base case -- results
EP value versus inflow variance
Economic Analysis for CAES
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“Wind needs a dance partner”
Dance partner is electricity storage
Which storage should be chosen?
Battery storage (work in progress with Kirby
& Anderson)
Compressed Air Energy Storage (OCE
funded work in progress with Carriveau,
Simpson, Ting, and Konrad)
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
Thank You !