PURDUE ENERGY MODELING
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A Load Factor Based MeanVariance Analysis for Fuel
Diversification
Third Annual Electricity Conference at Carnegie Mellon
March 13, 2007
Dr. Doug Gotham
Prof. Kumar Muthuraman
Prof. Ron Rardin
Suriya Ruangpattana
School of Industrial Engineering, Purdue University
Funded in part by the Indiana Utility Regulatory Commission
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Outline
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Introduction
Model formulation
Parameter estimates
Results and discussion
Future work
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Introduction
• Fuel diversification: the selection of a mix of
electric generation technologies in a way
that strikes a good balance between reduced
cost and reduced risk.
• Short-term perspective: scheduling problem
• Long-term perspective: resource planning
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Benefits of Fuel Diversity
See Costello (2005)
• Lower long-term prices and price risk
• Less dependency on foreign sources of energy
• Higher power reliability and a cleaner environment
Impact: may cause different strategic decisions to be made
• When reviewing a generation resource plan for a utility in
isolation or when compared to what the policy decision
may be when considering the statewide or a regional
perspective
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Fuel Diversity Advocates
• National Association of Regulatory Utility
Commissioners (NARUC)
• National Regulatory Research Institute
(NRRI)
• Commissioners identified fuel diversity in
electric generation as one of the top issues
(see Costello (2005))
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Mean-variance for Fuel
Diversification
PEMRG
• Risk: high fuel price and variability; variance as
the measure of risk
• Probable cause: reliance on inadequate mix of
fuel sources or generation technologies
• Impact of exposure: future high fuel and
externalities costs resulting in increase and
variability of unit cost of electricity
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Approach for Fuel
Diversification
PEMRG
• Estimate of the means, variances and covariances
of per unit generation costs using various
technologies/fuels from historical data
• Deterministic fixed and operating costs versus
non-deterministic fuel and operating costs
• Optimization traces solutions (mixes) as frontier
in mean-variance plot; exact points correspond to
risk preferences
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Issue with Direct Meanvariance Analysis
PEMRG
• Results obtained often dismissed by
practitioners
• Important aspect of the underlying problem:
investment in generation assets pays off by
utilizations which depend on variation of the
load
• Natural gas generation infrastructure costs
are low; fuel costs considerably high…
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Related Literature
• Costello (2005)
• Markowitz (1959)
• Humphreys and McClain (1998)
• Awerbuch and Berger (2003)
• DeLaquil et al. (2005)
• Krey and Zweifel (2006)
• Bar-Lev and Katz (1976)
• Yu (2003)
still counting…
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Model Formulation
• Load duration curve illustrates the
distribution of demand for electricity, power
requirement plotted against time
• Capacity factor (CF) measures a plant
utilization
• Load factor (LF) measures the entire system
utilization
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Load Factor Based Model
energy produced
Load Factor =
peak load*time period
• Low LF means large variations occur in the load, whereas
high LF for small
• LF unity implies constant load curve through the period
• Reflects how generation plants are dispatched to cover load
duration
• Generating system can have one or more units
• Used as factor to scale fixed costs for the system; thus,
more realistic consideration of investment payoffs
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Power Production Mix
• A portfolio is a mix of power generation
plants, e.g., a mix of production by fuel
sources or plant technologies to provide the
best means of hedging future risks,
assuming risk reduction is achieved by
diversification of production mix.
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Optimization Model
Minimize [
∑ ∑
i=1,...,I l=1,...,L
(
Fn i
yn il + (OM i + Vi )(yeil + yn il )] + β *[ ∑
LFil
i=1,...,I
(Fixed costs terms)
(Variable costs terms)
∑ ∑ (σ
ij
* (yeil + yn il )(ye jl + yn jl ))]
j=1,...,I l=1,...,J
(Covariance terms)
Assumptions:
- Existing generation capacity has sunk costs
- Future demand by load types are known
- No upper bounds for building new generation capacity
β= cost reduction versus variance tradeoff
– How much decrease in $ costs the investor would be
willing to trade for one unit increase of risk
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Optimization Model
Constraints
∑ ye
il
+
i=1,...,I
∑
yeil
l=1,...,L LFil
yeil
yn il
∑ yn
il
= Dl
for l = 1,...,L
,
i=1,...,I
≤ Ui ,
≥ 0 ,
≥ 0 ,
for i = 1,...,I
for i = 1,...,I;l = 1,...,L
for i = 1,...,I;l = 1,...,L
• Demand by load types
must be met by existing
and/or new generation
capacity
• Existing capacity by fuel
types/technologies) each
has upper bound
• Non negativity
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Data
• Cost attributes: expected investment costs, fuel costs,
variable O&M costs, --expressed in $/Watt*hour--
• Historical hourly load and future projection factor
• Load by types, predetermined by specific cutoffs
• Covariance between fuel costs is source of
variability, estimated from historical data
• Existing capacity in Indiana by fuel sources
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Variance as Measure of Risk
Fuel costs’ variance rankings, from least to most
volatile: nuclear, coal, natural gas, oil.
Table 1. Fuel Price Covariance Estimates for the Electric Sector, Indiana (Prices
in million $ per GWh)
Coal
Oil
Gas
Nuclear (MI)
0.00001877226
0.00006815061
0.00004736821
0.00000653022
Oil
0.00006815061
0.00037299100
0.00021565344
0.00002063233
Gas
0.00004736821
0.00021565344
0.00018436456
0.00001697626
Nuclear
0.00000653022
0.00002063233
0.00001697626
0.00000363517
Coal
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Base Scenario
10.0
beta =inf
9.9 cents, s.d.=0.35
PEMRG
beta = 0.25
5.4 cents, s.d.=0.35
9.0
8.0
7.0
6.0
5.0
beta = 1.0
6.6 cents, s.d.=0.35
4.0
beta = 0
4.1 cents, s.d.=0.4
3.0
beta = 0.001
4.1 cents, s.d.=0.39
2.0
1.0
0.0
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
Production Mix Risk (standard deviation), in cents per kWh
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Base Scenario
• What we saw:
– From β=0 (risk neutral) to the β=inf (most risk
averse), there is an increase in production from
oil and gas-fired units so as to reduce variance.
There is a drop of production from coal units (83
to 80%). Increase for natural gas (7 to 8%).
Increase for oil (1 to 4%). Thus, an increase of
total production costs from 4.1 to 9.9 cents/kWh
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Scenario 1: extreme coal price
variance (x30)
PEMRG
18.0
16.0
beta = 0.25
13.1 cents, s.d.=0.75
beta =inf
16.6 cents, s.d.=0.75
beta = 0.001
6.2 cents, s.d.=1.17
14.0
12.0
10.0
8.0
beta = 1.0
14.3 cents, s.d.=0.75
beta = 0
4.1 cents, s.d.=1.94
6.0
4.0
2.0
0.0
0.00
0.50
1.00
1.50
2.00
Production Mix Risk (standard deviation), in cents per kWh
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Scenario 1: extreme coal price
variance
• What we saw:
– As expected for coal having 30 times higher variance,
the production from coal sharply decreases as we
become more risk averse. At β=inf only 13% of energy
is provided by coal. The model simply backs away
production from coal with high variance fuel costs,
instead uses more higher cost alternatives.
– As a result, substantial increase in uses of oil (19%),
natural gas (40%), and nuclear (28%).
– At β=0 the mix of generation and the cost are the same
as in the base case, but the variance is higher.
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Scenario 2: extreme fuel cost (coal
price increases 10 times)
PEMRG
25.0
beta = 0.25
16.1 cents, s.d.=0.35
beta =inf
21.2 cents, s.d.=0.35
20.0
beta = 0.001
11.8 cents, s.d.=0.78
15.0
beta = 1.0
17.0 cents, s.d.=0.35
10.0
beta = 0
11.3 cents, s.d.=1.09
5.0
0.0
0.00
0.20
0.40
0.60
0.80
1.00
1.20
Production Mix Risk (standard deviation), in cents per kWh
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Scenario 2: extreme fuel cost (coal
price increases 10 times)
• What we saw:
– For β=inf the mix of generation and the variance
are exactly the same as in the base scenario.
The only difference is the price of coal, which
has no effect on the optimal mix.
– For β=0, much less of the high cost coal is used
(10%) with natural gas picking up much of the
slack (79%). This results in a much higher cost
and variance when compared to the base case.
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Scenario 3: extreme natural gas price
variance (x30)
PEMRG
18.0
16.0
beta =inf
11.4 cents, s.d.=0.37
beta = 0.25
7.8 cents, s.d.=0.37
14.0
beta = 0.001
4.1 cents, s.d.=0.54
12.0
10.0
8.0
beta = 0
4.1 cents, s.d.=0.55
6.0
4.0 beta = 1.0
10.2 cents, s.d.=0.37
2.0
0.0
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
Production Mix Risk (standard deviation), in cents per kW
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Scenario 3: extreme natural gas price
variance
• What we saw:
– Similar results to the high coal price variance
scenario. For β=0, the generation mix and the
cost are the same as in the base scenario, while
the variance increases.
– For β=inf, production from natural gas drops
from 7% to less than 1%. The drop in
production is made up in oil (1% to 4%) and coal
(83% to 87%).
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Conclusion
• Load Factor-based optimization model is a tool to
generate costs-risk efficient frontiers, which could
be used in analyzing strategic planning for leastcoast production.
• Model ran with other varying input assumptions like
coal and gas variance, and found the model to
behave intuitively.
• Extreme scenarios perform as expected and tend
to validate the method.
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Current and Future Work
• Will include renewable
resources or emissions
controls
–
–
–
–
Wind
Solar
Biomass
Possibly emissions limits
and/or availability factors
• Zero-start scenarios
• Want to choose and run
more realistic scenarios
• Technologies
– Simple cycle combustion gas
turbine
– Combined cycle combustion
gas turbine
– Internal combustion engine
– Pulverized coal, fluidizedbed combustion, and
integrated gas combinedcycle
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Questions or Comments?
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Thank you
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