1 systems analysis advisory committee (saac) thursday, october 24, 2002 michael schilmoeller john...
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Systems Analysis Advisory Committee (SAAC)
Thursday, October 24, 2002Michael Schilmoeller
John Fazio
Northwest Power Planning Council
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Original Agenda
• Metrics– Stakeholders– Risk measures– Timing
• Representations in the portfolio model– thermal generation– hydro generation– conservation and renewables– loads– contracts– reliability– ** Plan Issues **
Northwest Power Planning Council
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Plan Issues
• incentives for generation capacity• price responsiveness of demand• sustained investment in efficiency• information for markets• fish operations and power• transmission and reliability• resource diversity• role of BPA• global change
Northwest Power Planning Council
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Revised Agenda
• Approval of the Oct 4 meeting minutes• Price Processes• Representations in the portfolio model
– thermal generation
• Metrics– Stakeholders– Risk measures– Timing
• Representations in the portfolio model– ** Plan Issues ** : price responsive demand
Northwest Power Planning Council
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Revised Agenda
• Approval of the Oct 4 meeting minutes• Price Processes• Representations in the portfolio model
– thermal generation
• Metrics– Stakeholders– Risk measures– Timing
• Representations in the portfolio model– ** Plan Issues ** : price responsive demand
Northwest Power Planning Council
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Price Processes
• Problem: There are mathematical difficulties with describing prices and price processes statistically.
• To show: The natural logarithm of prices (or price ratios) provides a solution to the problem
Price Processes
Northwest Power Planning Council
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Price Processes
• Problem: Naïve attempts to describe prices statistically lead to nonsense. For example, a symmetric distribution, unbounded on the high side, must be unbounded on the low side
Price Processes
Probability density
0
0.05
0.1
0.15
0.2
0.25
-5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Price
den
sity
Oops
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Price Processes
• Problem: More seriously, if we try to describe variation in a price process from historical data, we run into seasonality problems. Suppose we wanted to estimate the daily variation in prices from a price series over 90 days:
Price Processes
Standard deviation of the price curve (black) would be quite
large and would not describe the daily variation (red)
Clearly, we want something that more
closely resembles daily price returns
Northwest Power Planning Council
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Price Processes
• Problem: Returns themselves, however, have bad statistical properties. For example, what is the meaning of the average of a 50% increase in prices and a 50% decrease in prices?
Price Processes
75.5.05.1
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Price Processes
• Solution: What does work is the logarithm of returns (and the inverse transformation, exponentiation)
Price Processes
period over thereturn the/)/ln(
)ln()ln(
)ln()ln()ln()ln()ln()ln(
)/ln()/ln()/ln(
:returns of Sum
11
1
12312
12312
pppp
pp
pppppp
pppppp
nn
n
nn
nn
• If price ends where it started out, the ratio of the prices is one, and the logarithm of returns is zero.
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Price Processes
• Other nice properties of the logarithm of returns
Price Processes
rategrowth period the/))/ln((
)/ln(1
)/ln()/ln()/ln(1
:returns of Average
1
1
1
1
12312
n pppp
ppn
ppppppn
nn
n
n
nn
returnregular the,1~/)~/ln(
,~ As
pppp
pp
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GBM• Geometric Brownian Motion (GBM)
– independent draws of ln(p) are made from a normal distribution
– grows as
-12
-8
-4
0
4
8
12
0 2 4 6 8 10 12 14 16 18 20
19
16
T
Time T-->
• Makes sense, because the standard deviation of the sum of T draws from a
distribution is
),0( 2N
TT
T
2
222
21
Price Processes
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GBM• And the normal distribution of ln(p) gives us a reasonable
distribution of prices
Lognormal Prices
0
0.05
0.1
0.15
0.2
0.25
0 50 100 150 200 250 300 350 400
Prices $/MWh
Pro
b d
ensi
ty
Price Processes
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Conclusion
• The natural logarithm of prices (or price ratios) overcome the mathematical difficulties with describing prices and price processes statistically
• Normal log returns produce lognormal price distributions, which have desirable distribution
• GBM is perhaps the simplest description of a stochastic process, where draws are independent, random, normal. May describe processes like stock prices well, where daily returns should be about normal
Price Processes
Northwest Power Planning Council
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Revised Agenda
• Approval of the Oct 4 meeting minutes• Price Processes• Representations in the portfolio model
– thermal generation
• Metrics– Stakeholders– Risk measures– Timing
• Representations in the portfolio model– ** Plan Issues ** : price responsive demand
Northwest Power Planning Council
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Thermal Generation
• Objectives– We need a way to quickly estimate the dispatch factor for
thermal generation, so that we can calculate variable cost.– Should have certain basic properties
• If average monthly prices ($/MWh) for gas are about the same as average monthly prices for electricity, the dispatch factor should be about 50 percent.
• If average monthly prices ($/MWh) for gas are well above the average monthly prices for electricity, but there is a good deal of uncertainty in the prices, the plant should dispatch, albeit a small amount
• If average monthly prices ($/MWh) for gas are well below the average monthly prices for electricity, but there is a good deal of uncertainty in the prices, the plant should run close to, but not quite 100% capacity factor (disregarding maintenance and forced outage)
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Thermal Generation
• Objectives– To show: Thermal dispatch can be reasonably well using a
spread call option on electricity and gas– To show: The monthly capacity factor of the thermal unit is
provided by the “delta” of the option, that is, the change in the option’s price with respect to the underlying spread
– To show: The standard Black-Scholes model for option pricing gives a good estimate of the capacity factor, with these adjustments:
• Discount rate r = 0• Volatility incorporates terms for the uncertainty in and the
expected variation of the spread over the specified time frame
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Typical Dispatchable
• Example of 1 MW Single Cycle Combustion Turbine (no dispatch constraints)
• Natural Gas price, $3.33/MBTU
• Heat rate, 9000 BTU/kWh
• Price of electricity generated from gas, $30/MWh
Representations - thermal
0
5
10
15
20
25
30
35
40
1 25 49 73 97 121
145
169
193
217
241
265
289
313
337
361
385
409
433
457
481
505
529
553
577
601
625
649
Hour
Mar
ket
Pri
ce $
/MW
h
Value:note: we assuming the
entire capacity is switched on when the turbine runs
case)our in MW (1 turbine theofcapacity theis
($/MWh) rateheat fixed a assuming
hour, in this gas of price theis )(
($/MWh)hour in thisy electricit of price theis )(
case) in this (672 hours ofset theis
where
)))()((,0max(
C
hp
hp
H
hphpCV
g
e
Hhge
Northwest Power Planning Council
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Price Duration Curve
• If we assume each hour’s dispatch is independent, we can ignore the chronological structure. Sorting by price yields the market price duration curve (MCD)
0
5
10
15
20
25
30
35
40
1 25 49 73 97 121
145
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Hour
Mar
ket
Pri
ce $
/MW
h
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
Count of Hours
$/M
Wh
Value V is this area
Representations - thermal
operatorn expectatio theis (672) period in the hours ofnumber theis
where])()(,0[max
or
)()(,0max
)()(,0max
EN
hphpECNV
N
hphpCN
hphpCV
H
geH
H
Hhge
H
Hhge
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Variability viewed as CDF
• Turning the MCD curve on its side, we get something that looks like a cumulative probability density function (CDF)
Value V is this area
factorcapacity or the CDF theof value theis )(where
)(
Calculus) of Thm (Fund
)(
e
eHe
e
P
eH
pf
pfNdp
dV
dppfNVg
Cumulative Frequency
0
100
200
300
400
500
600
37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20
$/MWh
Co
un
t o
f h
ou
rs
0%10%20%30%40%50%60%70%80%90%100%
Cap
acit
y F
acto
r
Representations - thermal
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Cumulative FrequencySingle, fixed hour
0
0.2
0.4
0.6
0.8
1
40.00 38.00 36.00 34.00 32.00 30.00 28.00 26.00 24.00 22.00 20.00
$/MWh
Pro
b o
f p
ric
e
ex
ce
ed
ing
p
Uncertainty
• To this point, we have assumed we know what the hourly electricity price will be in each hour. However, we could similarly calculate the expected capacity factor for fixed hour using a CDF that described our uncertainty about prices within that hour. The preceding results still apply.
Representations - thermal
??
Northwest Power Planning Council
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Transformation of variables
• Everything we have done to this point still holds if we used transformed prices
Representations - thermal
price base fixedbut arbitrary, some is ~where
)~/ln(
)~/ln(
p
ppzp
ppzp
ggg
eee
g
ge
eHH
e
z
f
zz
V
dp
dVfNfN
dz
dV
for
CDF theof value theis '
ofleft the tofor
CDF under the area theis '
where
''
z is dual to p, for positive p
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5
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Simplification #1
• The expected capacity factor over the time period will be a function of expected variation over the period and the uncertainty associated with each hour. It will be determined by the CDF of
Representations - thermal
0
5
10
15
20
25
30
35
40
1 25 49 73 97 121
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433
457
481
505
529
553
577
601
625
649
Hour
Mar
ket
Pri
ce $
/MW
h
)(hpe
)()()( hhph ee
)(h
)(he
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-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
1 25 49 73 97 121
145
169
193
217
241
265
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313
337
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457
481
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553
577
601
625
649
Hour
ln(pe/
pg
)
Simplification #1• Assume distributions of uncertainty in , say ,
are identical across all hours of the time period (e.g., month). Note is still a vector and has covariance structure
Representations - thermal
)(hze
)()()( hhzh eee
)(he
)(he)(hze
)(he
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Simplification #1
•
• Note that constant uncertainty for z implies greater price uncertainty during times of high prices than during times of low prices (cool)
Representations - thermal
with themnscalculatio perform easier tomuch be it will
t,independen now are (h) and (h) Because eez
0
5
10
15
20
25
30
35
40
1 25
Hour
$/M
Wh
)(hze
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Digression to Options
• European call option: Confers the right, but not the obligation to purchase a specified quantity of an instrument or commodity at a fixed price at a specified time in the future
Representations - thermal
• Example of a call option on a stock with a strike price of $30
• Below $30, the option is worthless
• For each dollar over $30 that the price of the stock reaches, the value of the option increases a dollar
European Call Option
0
2
4
6
8
10
12
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Price of underlying
Val
ue
of
Op
tio
n (
$) Intrinsic value
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Digression to Options
• Because the option will expire at some specified time in the future and because we are uncertain what the value of the stock will be when the option expires, the value of the option is greater than the intrinsic value
Representations - thermal
European Call Option
0.00
2.00
4.00
6.00
8.00
10.00
12.00
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Price of underlying
Val
ue
of
Op
tio
n (
$)
max(0,p-X)
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Digression to Options
• The value of an option is the expected value of the discounted payoff (See Hull, 3rd ed., p. 295)
Representations - thermal
price strike theis stock theof price theis
(years) expiration to time theis ratediscount annual theis
operatorn expectatio theis where
)],0max([
XpTrE
XpeEV rT
Northwest Power Planning Council
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Digression to Options
• The challenge in solving for the value explicitly (in closed form) is determining the discount rate r. The problem was essentially solved otherwise much earlier by A.J. Boness in his Ph.D. thesis.
• Fischer Black and Myron Scholes in 1973 showed that if certain assumptions held, the discount rate r should be the risk-free discount rate:– the stock pays no dividends, markets are efficient, interest rates are
known– returns are normally distributed (Geometric Brownian Motion), so
prices are lognormally distributed– prices change continuously so the option can be hedged
Representations - thermal
Northwest Power Planning Council
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Digression to Options
• The Black Scholes pricing formula
Representations - thermal
TddrTXpd
NN
dNXedpNV rT
12
21
21
)2/()/ln()p~ln(p/ ofdeviation standard is variablerandom )1,0( afor CDF theis
where
)()(
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Simplification #2
• We will discount the payoff externally, so let r = 0
Representations - thermal
price strike theis stock theof price theis operatorn expectatio theis
where
)],0[max(
XpE
XpEV
• This now closely resembles our calculation for the value of a thermal plant, assuming the CDF is the CDF of
)()()( hhzh eee
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Simplification #3
• We assume that the distributions of
Representations - thermal
can be adequately approximated by normal distributions.
• Seems reasonable that could be described by a multivariate normal, because our uncertainty is largely symmetric, continuous, and unbounded above and below
)( and )( hhz ee
)(he
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Frequency Distribution
0
1
2
3
4
-0.1
2306
54
-0.1
0653
61
-0.0
9027
55
-0.0
7427
52
-0.0
5852
68
-0.0
4302
26
-0.0
2775
52
-0.0
1271
73
0.00
2097
79
0.01
6696
59
0.03
1085
32
0.04
5269
96
0.05
9256
2
0.07
3049
52
0.08
6655
18
0.10
0078
2
0.11
3323
42
0.12
6395
5
Price
Co
un
t
0
0.05
0.1
0.15
0.2
0.25
Idea
l n
orm
al
Simplification #3
• Case of
Representations - thermal
)(hze
Prices
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Hours on peak
pri
ce l
n($
/MW
h/p
)
30
31
32
33
34
35
36
ln(p/p') prices
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Almost there...
• Then the B-S formula for the value the plant is
Representations - thermal
TddTXpd
NN
dXNdpNV
12
21
21
2/)/ln()p~ln(p/ ofdeviation standard is variablerandom )1,0( afor CDF theis
where
)()(
with the variance of playing the role of)()()( hhzh eee
2T
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Option “Delta”
• The change in value of the option with respect to the price of the underlying is the option “Delta.” It is just the slope of the price curve at a specified price
Representations - thermal
European Call Option
0.00
2.00
4.00
6.00
8.00
10.00
12.00
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Price of underlying
Val
ue
of
Op
tio
n (
$)
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The payoff
• The B.S. formula for the capacity factor the plant is
Representations - thermal
2/)()/ln(2/)/ln(
where
)(
22
21
1
ezge epp
TXpd
dNp
Vf
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• Two issues remain– Gas prices are not constant (X is not fixed)
– Most of what we may think we know about future price uncertainty might be expressed in terms of average monthly prices
• Solution– Use a European “spread” option instead of a standard
European call option
– Try to estimate the volatility of the hourly spread from the monthly volatilities and correlations
Ah, Darn It
Representations - thermal
Northwest Power Planning Council
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• Use a European spread call option instead of a standard European call option
European spread option
Representations - thermal
European Spread Option
0.00
2.00
4.00
6.00
8.00
10.00
12.00
-10 -8 -6 -4 -2 0 2 4 6 8 10
Spread in Price (pe-pg)
Val
ue
of
Op
tio
n (
$)0
5
10
15
20
25
30
35
40
12
54
97
39
71
21
14
51
69
19
32
17
24
12
65
28
93
13
33
73
61
38
54
09
43
34
57
48
15
05
52
95
53
57
76
01
62
56
49
Hour
$/M
Wh
-30
-25
-20
-15
-10
-5
0
5
10
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European spread option
• The Margrabe pricing formula for the value of a spread option, assuming no yields
Representations - thermal
212,12
22
12
12
212
1
2112
2
2/)/ln(where
)()(
TddT
Tppd
dNpdNpV
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European spread option
• The delta for the Margrabe spread option, assuming no yields
Representations - thermal
212,12
22
12
12
212
1
1
2
2/)/ln(where
)(
TddT
Tppd
dNf
Northwest Power Planning Council
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Hourly Volatilities from Monthly
• We are dealing with expected variation of electricity and gas price over the specific time period and with uncertainties in these, as well. Using our assumption that the hourly uncertainties are constant and independent of the temporal variations in the respective commodities,
Representations - thermal
)2(
)2(
implies assumption ceindependenour by which
)()()()()(
,22
)()(,2
)(2
)(2
gegege
gegege hzhzzzhzhz
ggee hhzhhzh
Northwest Power Planning Council
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Hourly Volatilities from Monthly
• The first term is determined by the expected temporal covariance in commodity prices over the period, due to normal “seasonality”
Representations - thermal
)2( )()(,2
)(2
)( hzhzzzhzhz gegege
-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
1 25 49 73 97 121
145
169
193
217
241
265
289
313
337
361
385
409
433
457
481
505
529
553
577
601
625
649
Hour
ln(pe/
pg
)
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Hourly Volatilities from Monthly
• Question: Is unlikely new information will become available that would influence our view of temporal structure?
• If not, we do not expect uncertainties in monthly averages to be affected too much by assumptions about expected temporal variations in price.
Representations - thermal
Northwest Power Planning Council
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Hourly Volatilities from Monthly
• The second term is determined by the hourly uncertainties surrounding the hourly values for gas and electricity and their expected covariance.
• Clearly, uncertainty factors can swamp temporal variations
Representations - thermal
)2( ,22
gegege
05
101520253035404550556065707580859095
100105110115
1 25 49 73 97 121
145
169
193
217
241
265
289
313
337
361
385
409
433
457
481
505
529
553
577
601
625
649
Hour
$/M
Wh )( ee )( gg
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Hourly Volatilities from Monthly
• Assumptions about hourly covariance in uncertainties is important here. They will affect the uncertainty in the monthly average price.
• If the hourly uncertainties for electricity (gas) are perfectly correlated, the relationship between monthly variance and hourly variance is
• If the hourly uncertainties for electricity (gas) are uncorrelated, the relationship between monthly variance and hourly variance is
Representations - thermal
)( in the covariance theis where
/112
h
N
e
He
22
ee
HNee
/22
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Hourly Volatilities from Monthly
• Assignment for the next class:
– If we know the correlations between the monthly average returns for electricity and gas, what can we conclude about the correlation between the hourly returns?
Representations - thermal
Northwest Power Planning Council
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Example
• If we use the example we started out with, and the BS equation for delta, we use a normal distribution on ln(prices) that looks like
• But exact CF is 50% and B-S calculated CF is 49.7% (not bad)
• 100% uncertainty in the log returns (prices could be higher or lower by a factor of 2.7) gives us a 69% CF
Representations - thermal
Distribution of ln(p)
0
10
20
30
40
50
60
70
80
90
0.3
0.2
8
0.2
6
0.2
4
0.2
2
0.2
0.1
8
0.1
6
0.1
4
0.1
2
0.1
0.0
8
0.0
6
0.0
4
0.0
2 0 -0 -0
-0.1
-0.1
-0.1
-0.1
-0.1
-0.2
-0.2
-0.2
-0.2
-0.2
-0.3
-0.3
-0.3
-0.3
-0.3
Fre
qu
en
cy
0
0.01
0.02
0.03
0.04
0.05
0.06
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Conclusion
– Thermal dispatch can be reasonably well using a spread call option on electricity and gas
– The monthly capacity factor of the thermal unit is provided by the “delta” of the option, that is, the change in the option’s price with respect to the underlying spread
– The standard Black-Scholes model for option pricing gives a good estimate of the capacity factor, with these adjustments:
• Discount rate r = 0• Volatility incorporates terms for the uncertainty in and the
expected variation of the spread over the specified time frame
– We need to better understand not only the expected correlation of uncertainties in electricity and gas prices, but how hourly prices are self-correlated over the time period of interest.
Representations - thermal
Northwest Power Planning Council
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Revised Agenda
• Approval of the Oct 4 meeting minutes• Price Processes• Representations in the portfolio model
– thermal generation
• Metrics– Stakeholders– Risk measures– Timing
• Representations in the portfolio model– ** Plan Issues ** : price responsive demand
Northwest Power Planning Council
50
Objective
• Objectives of this section:– Develop a risk metric for the region
– To show: Risk metric should be minimum total power cost, subject to an annual CVaR constraint
Metrics
Northwest Power Planning Council
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Stakeholders
• Candidate Stakeholders– Load Serving Entity (Investor-Owned Utility
or Public Utility District)– Customer– Regulatory Agency (Public Utility
Commission)– BPA
Metrics
Northwest Power Planning Council
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Stakeholders
• Proposed Stakeholder Perspective– Total societal costs, to include– Capital costs, including those of
transmission– Variable costs,– Internalized emission costs
Metrics
Northwest Power Planning Council
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Metric Candidates
• Candidates– Value at Risk (VaR)
– Standard deviation
– Expected shortfall
– Conditional VaR (CVaR)
– Van Neumann utility functions
– Block maxima
Metrics
Northwest Power Planning Council
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V@R
Metrics
95% one-day V@R
Frequency Distribution
0
0.05
0.1
0.15
0.2
0.25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Price
Co
un
t
V@R=4
costcost
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Example of Power Plants
Metrics
Consider an ensemble of 1MW power plants, each with a forced outage rate of 0.10, equal to that of
the 10MW plant.
A Paradox, because we know a system of smaller
plants are better
V@R is not “subadditive”0.0000
0.1000
0.2000
0.3000
0.4000
0.5000
0 1 2 3 4 5 6 7 8 9 10
Var85=2
0.0000
0.2000
0.4000
0.6000
0.8000
1.0000
0 1
Var85=0
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Desirable Properties
Metrics
• Subadditivity – For all random losses X and Y,
(X+Y) (X)+(Y)
• Monotonicity – If X Y for each scenario, then
(X) (Y)
• Positive Homogeneity – For all 0 and random loss X
(X) = (Y)
• Translation Invariance – For all random losses X and constants
(X+) = (X) +
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A List of Loss Scenarios
Scenario X1 X2 X1+X2 X3 = 2*X1 X4 = X1+11 1.00 0.00 1.00 2.00 2.002 2.00 0.00 2.00 4.00 3.003 3.00 0.00 3.00 6.00 4.004 4.00 1.00 5.00 8.00 5.005 3.00 2.00 5.00 6.00 4.006 2.00 3.00 5.00 4.00 3.007 1.00 4.00 5.00 2.00 2.008 0.00 3.00 3.00 0.00 1.009 0.00 2.00 2.00 0.00 1.00
10 0.00 1.00 1.00 0.00 1.00Maximum Loss 4.00 4.00 5.00 8.00 5.00
Define a measure of risk (X) = Maximum{Xi}Metrics
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Subadditivity
Scenario X1 X2 X1+X2 X3 = 2*X1 X4 = X1+11 1.00 0.00 1.00 2.00 2.002 2.00 0.00 2.00 4.00 3.003 3.00 0.00 3.00 6.00 4.004 4.00 1.00 5.00 8.00 5.005 3.00 2.00 5.00 6.00 4.006 2.00 3.00 5.00 4.00 3.007 1.00 4.00 5.00 2.00 2.008 0.00 3.00 3.00 0.00 1.009 0.00 2.00 2.00 0.00 1.00
10 0.00 1.00 1.00 0.00 1.00Maximum Loss 4.00 4.00 5.00 8.00 5.00
(X+Y) (X)+(Y)Metrics
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Monotonicity
Scenario X1 X2 X1+X2 X3 = 2*X1 X4 = X1+11 1.00 0.00 1.00 2.00 2.002 2.00 0.00 2.00 4.00 3.003 3.00 0.00 3.00 6.00 4.004 4.00 1.00 5.00 8.00 5.005 3.00 2.00 5.00 6.00 4.006 2.00 3.00 5.00 4.00 3.007 1.00 4.00 5.00 2.00 2.008 0.00 3.00 3.00 0.00 1.009 0.00 2.00 2.00 0.00 1.0010 0.00 1.00 1.00 0.00 1.00
Maximum Loss 4.00 4.00 5.00 8.00 5.00
If X Y for each scenario, then (X) (Y)Metrics
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Positive Homogeneity
Scenario X1 X2 X1+X2 X3 = 2*X1 X4 = X1+11 1.00 0.00 1.00 2.00 2.002 2.00 0.00 2.00 4.00 3.003 3.00 0.00 3.00 6.00 4.004 4.00 1.00 5.00 8.00 5.005 3.00 2.00 5.00 6.00 4.006 2.00 3.00 5.00 4.00 3.007 1.00 4.00 5.00 2.00 2.008 0.00 3.00 3.00 0.00 1.009 0.00 2.00 2.00 0.00 1.0010 0.00 1.00 1.00 0.00 1.00
Maximum Loss 4.00 4.00 5.00 8.00 5.00
For all 0 and random loss X, (X) = (Y)Metrics
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Translation Invariance
Scenario X1 X2 X1+X2 X3 = 2*X1 X4 = X1+11 1.00 0.00 1.00 2.00 2.002 2.00 0.00 2.00 4.00 3.003 3.00 0.00 3.00 6.00 4.004 4.00 1.00 5.00 8.00 5.005 3.00 2.00 5.00 6.00 4.006 2.00 3.00 5.00 4.00 3.007 1.00 4.00 5.00 2.00 2.008 0.00 3.00 3.00 0.00 1.009 0.00 2.00 2.00 0.00 1.00
10 0.00 1.00 1.00 0.00 1.00Maximum Loss 4.00 4.00 5.00 8.00 5.00
For all random losses X and constants (X+) = (X) + Metrics
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Axioms for Coherent Measures
• Subadditivity – For all random losses X and Y,(X+Y) (X)+(Y)
• Monotonicity – If X Y for each scenario, then(X) (Y)
• Positive Homogeneity – For all 0 and random loss X
(X) = (Y)• Translation Invariance – For all random losses X and
constants (X+) = (X) +
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Value at Risk/Probability of Ruinviolates subadditivity
Scenario X1 X2 X1+X2
1 0.00 0.00 0.002 0.00 0.00 0.003 0.00 0.00 0.004 0.00 0.00 0.005 0.00 0.00 0.006 0.00 0.00 0.007 0.00 0.00 0.008 0.00 0.00 0.009 0.00 1.00 1.0010 1.00 0.00 1.00
VaR@85% 0.00 0.00 1.00
1 2 1 20 X X X X 1 Metrics
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Standard Deviationviolates monotonicity
Scenario X1 X2
1 1.00 5.002 2.00 5.003 3.00 5.004 4.00 5.005 5.00 5.006 5.00 5.007 4.00 5.008 3.00 5.009 2.00 5.00
10 1.00 5.00E[Loss] 3.00 5.00
StDev[Loss] 1.41 0.00E[Loss]+2*StDev[Loss] 5.83 5.00
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Conditional Value at Risk - (CVaR)
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Subject Loss
Cu
mu
lati
ve P
rob
abili
ty
Value At Risk
CVaR is the average of all losses above the Value at Risk
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What aboutthe upside potential?
CVaR is coherent
But ! variation from year to year can be large! What about minimizing variation from year to year?
We expect that upside will be sold to minimize cost, and variation will be automatically reduced
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Timing
Study? No
Annual? Yes: Rates are often recalculated annually
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Conclusion
Risk metric is
CVAR<X
Objective function is
Min costs
More on coherent measures:See Artzerner, Delbaen, Eber, Heath, “Coherent Measures of Risk,” July 22, 1998, preprint
http://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf
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