how the rpm meets the requirements for a risk model michael schilmoeller tuesday, february 2, 2011...
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How the RPM Meets the Requirements for a Risk Model
Michael SchilmoellerTuesday, February 2, 2011
SAAC
2
Overview• Statistical distributions
– Estimating hourly cost and generation– Application to limited-energy resources– The price duration curve and the revenue
curve
• Valuation costing• An open-system models• Unit aggregation• Performance and precision
3
Computation Cost Distribution AssociatedWith a Plan
Hourly demand
Coal
Buy in Market
Buy in Market
Sell in Market
Gas Fired
Price-driven generation
Hydro
Contracts
HydroTotal
Resources
Year 1Summer Winter
Year 2Summer Winter
Background
Coal
4
Statistical Distributions• Alternative strategies for speeding up
calculation– More computer processing power
• Previous presentation raises concerns about the limitations of this approach
– Using selected hours of each week• A type of statistical sampling
– Statistical distributions• Origins in older production cost models that used
load duration curves
Statistical distributions
5
Dispatchable Resources
Statistical distributions
6
Estimating Energy Generation
Price duration curve (PDC)
Statistical distributions
7
Estimating Energy Generation
Statistical distributions
8
Estimating Energy Value
Statistical distributions
Price of fuel pg(h)
Set of hours H={h}
Price of electricity pe(h)
9
Gross Value of Resources
Statistical distributions
Then for a turbine with capacity C MW, the value is
10
Gross Value of Resources
Statistical distributions
11
Gross Value of Resources Using Statistical Parameters of
Distributions
e
ee
ge
ee
g
e
ge
dd
ppd
(h))(p
p
p
NN
dNpdNpc
12
1
21
2/)/ln(
ln ofdeviation standard is
price gas theis
pricey electricit average theis
variablerandom )1,0( afor CDF theis
where
(4) )()( Assumes:
1) prices are lognormally distributed
2) 1MW capacity
3) No outages
V
Statistical distributions
12
Estimating Energy Generation
*
*
1)(CDFcf
)(CDF
Calculus) of Thm (Fund
)(CDF
*
*
gg
gg
g
ppgHg
gH
ppg
e
P
eH
p
V
NCp
pNCp
V
dppNCV
Applied to equation (4), this gives us a closed-form evaluation of the capacity factor and energy.
Statistical distributions
13
Variable Fuel Price
• Assume lognormal distribution• Include information about price volatility
and correlation with electricity price
gegege pppppp
ge dNpdNpV
,222
21
2
)()(
Statistical distributions
14
Implementation in the RPM
• Distributions represent hourly prices for electricity and fuel over hydro year quarters, on- and off-peak– Sept-Nov, Dec-Feb, Mar-May, June-Aug– Conventional 6x16 definition– Use of “standard months”
• Easily verified with chronological model• Execution time <30µsecs• 56 plants x 80 periods x 2 subperiods
Statistical distributions
15
Application of PDC to Energy-Limited Resources
Statistical distributions
16
Energy-Limited Dispatch
Statistical distributions
17
Energy-Limited Dispatch
2/)((exp*
)(
1
12
ee
eg fN
pp
fNd
Statistical distributions
18
Energy-Limited Dispatch
• If pg* > pg then use energy and value associated with pg*
• Otherwise, use energy and value associated with pg
Statistical distributions
19
Application of Revenue Curve Equilibrium Prices
Statistical distributions
Cu
mu
lati
ve M
ark
et
Pri
ce
(mil
ls/k
W)
Time (hours) 8760
Diesel ECC
SCCT ECC
CCCT ECC
Net revenue for the diesel (negative)
h* for diesel
Source: page 5, Figure 3, Q:\MS\Markets and Prices\Market Price Theory MJS\Price Relationships in Equilibrium2.doc
20
Overview• Statistical distributions
– Estimating hourly cost and generation– Application to limited-energy resources– The price duration curve and the revenue
curve
• Valuation costing• An open-system models• Unit aggregation• Performance and precision
21
Challenges Using DistributionsComplications arise when we use extended time periods
price
Loads (solid) & resources (grayed)
Valuation Costing
22
Average loads and resources are the same, but in the first case, our system has net cost and in the second it has net benefit.
Challenges Using Distributions
Valuation Costing
23
Traditional Costing
trequiremen totalis
energy alefor wholes ($/MWh) price theis
resource of ($/MWh) price theis
resourceby provided (MWh)quantity is
($)cost totalis
(2) )(
Q
p
ip
iq
c
qQppqc
m
i
i
iim
iii
Hourly variable cost calculation:
Valuation Costing
24
Traditional Costing
(1) )()()( pqqpqEpEpqE
N*(N+1)/2 correlations (upper triangular matrix)
Valuation Costing
25
Traditional Costing
trequiremen totalis
energy alefor wholes ($/MWh) price theis
resource of ($/MWh) price theis
resourceby provided (MWh)quantity is
($)cost totalis
(2) )(
Q
p
ip
iq
c
qQppqc
m
i
i
iim
iii
Valuation Costing
26
“Valuation” Costing
)(
*
)(
imi
im
iii
iimm
iim
iii
ppqQp
pqqpQp
qQppqc
Only correlations are now those with the market
Valuation Costing
27
Valuation Costing
• Solves the correlation problem by decoupling fuel price variation
• We get the value term for dispatchable resources from the earlier calculation (V)
• For wind and most renewables, the resource is non-patchable and correlation is fixed (we typically assume zero), which makes an easy calculation
• For the pmQ term, hourly correlation of prices and load is important
Valuation Costing
)( imi ppq
28
Overview• Statistical distributions
– Estimating hourly cost and generation– Application to limited-energy resources– The price duration curve and the revenue
curve
• Valuation costing• An open-system models• Unit aggregation• Performance and precision
29
Closed-System Models
Open-System Models
30
Open-System Models
?
Open-System Models
31
Modeling Evolution
• Problems with open-system production cost models– valuing imports and exports– desire to understand the implications of events
outside the “bubble”
• As computers became more powerful and less expensive, closed-system hourly models became more popular– better representation of operational costs and
constraints (start-up, ramps, etc.)– more intuitive
Open-System Models
32
Open Systems Models• The treatment of the Region as an island seems
like a throw-back– We give up insight into how events and
circumstances outside the region affect us– We give up some dynamic feedback
• Open systems models, however, assist us to isolate the costs and risks of participant we call the “regional ratepayer”
• Any risk model must be an open-system model
Open-System Models
33
Relationship of electricity price to fuel price
fuel price
dispatchprice
energygeneration
energyrequire-ments
market price for electricity
Only one electricity price balances requirements and generation
• In a closed model, there are no imports or exports• (Hourly) electricity price is entirely determined by the
value of other variables, such as fuel price
Open-System Models
34
Closed-system models
• A closed system has by definition certain “constant” relationships, a preserved quantity such as energy
• Introducing uncertainty means introducing additional variables εi for error or uncertain variation
• Doing so creates an “over-specified” system which generally has no solution
Open-System Models
35
Closed-system models• Consequently, when we introduce uncertainty
into systems that are closed with respect to electrical energy, we are actually creating an open-system model with respect to total energy, and
• There is a equal and opposite response among the variables we elect to make dependent, and
• There is a “perfect correlation” among our “sources of uncertainty,” with unknown consequences. (CCCTs are always marginal.)
Open-System Models
36
The New Open-System Model
fuel price+εi
dispatchprice
energygeneration
energyrequire-ments
market price +εi for electricity
Only one electricity price balances requirements and generation
• If fuel price is the only “independent” variable, the assumed source of uncertainty, electricity price will move in perfect correlation
• That is, outside influences drive the results• We are back to an open system
Open-System Models
37
The RPM Convention
• Respect the first law of thermodynamics: energy generated and used must balance
• The link to the outside world is import and export to areas outside the region
• Import (export) is the “free variable” that permits the system to balance generation and accommodate all sources of uncertainty
• We assure balance by controlling generation through electricity price. The model finds a suitable price by iteration.
Open-System Models
38
Equilibrium search
Open-System Models
39
Overview• Statistical distributions
– Estimating hourly cost and generation– Application to limited-energy resources– The price duration curve and the revenue
curve
• Valuation costing• An open-system models• Unit aggregation• Performance and precision
40
Unit Aggregation
0.00
2.00
4.00
6.00
8.00
10.00
12.00
4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000 16000 17000
VO
M ($
/MW
h)
Heat Rate (BTU/kWh)
West 1 West 2 West 3
West 4 Beaver East 4
East 5 East 7 East 8
Hermiston Ignore East 1
• Forty-three dispatchable regional gas-fired generation units are aggregated by heat rate and variable operation cost
• The following illustration assumes $4.00/MMBTU gas price for scaling
Source: C:\Backups\Plan 6\Studies\Data Development\Resources\Existing Non-Hydro\100526 Update\Cluster_Chart_100528_183006.xls
Unit Aggregation
41
Cluster Analysis
11
30
12
19
13
05
12
90
11
31
12
46
12
47 1
24
81
02
11
04
10
20
14
67
14
68
16
50
16
51
11
98
11
99
12
01
12
02
10
23
11
36
10
28
14
75
14
43
13
68
12
00
12
28
10
89 15
71
14
11
10
00
12
04
12
03
10
01
05
41
79
71
29
11
29
21
40
21
40
3
01
23
45
Dendrogram of agnes(x = Both_Units, diss = FALSE, metric = "manhattan", stand = TRUE)
Agglomerative Coefficient = 0.98Both_Units
He
igh
t
Source: C:\Backups\Plan 6\Studies\Data Development\Resources\Existing Non-Hydro\100526 Update\R Agnes cluster analysis\Cluster Analysis on units.doc
Unit Aggregation
42
Overview• Statistical distributions
– Estimating hourly cost and generation– Application to limited-energy resources– The price duration curve and the revenue
curve
• Valuation costing• An open-system models• Unit aggregation• Performance and precision
43
Performance
• The RPM performs a 20-year simulation of one plan under one future in 0.4 seconds
• A server and nine worker computers provide “trivially parallel” processing on bundles of futures. A master unit summarizes and hosts the optimizer.
• The distributed computation system completes simulations for one plan under the 750 futures in 30 seconds
• Results for 3500 plans require about 29 hours
Performance and Precision
44
Repeatability Over FuturesTotal Study Costs ($M 2006 NPV)
Single machine
multiple machines Difference
81532 81532 0.0000000000000000000000000000000000000000000000119806 119806 0.0000000000000000000000000000000000000000000000121229 121229 0.0000000000000000000000000000000000000000000000113527 113527 0.0000000000000000000000000000000000000000000000195754 195754 0.0000000000000000000000000000000000000000000000214574 214574 0.0000000000000000000000000000000000000000000000170051 170051 0.0000000000000000000000000000000000000000000000164821 164821 0.0000000000000000000000000000000000000000000000104927 104927 0.0000000000000000000000000000000000000000000000146788 146788 0.000000000000000000000000000000000000000000000096562 96562 0.0000000000000000000000000000000000000000000000
129164 129164 0.0000000000000000000000000000000000000000000000191754 191754 0.0000000000000000000000000000000000000000000000170067 170067 0.000000000000000000000000000000000000000000000064095 64095 0.000000000000000000000000000000000000000000000084783 84783 0.0000000000000000000000000000000000000000000000
140423 140423 0.0000000000000000000000000000000000000000000000139862 139862 0.0000000000000000000000000000000000000000000000117622 117622 0.0000000000000000000000000000000000000000000000
Source: C:\Backups\Olivia\SAAC 2010\101202 SAAC First Meeting\Presentation materials\Reproducibility restored for illustration 101130.xls
Performance and Precision
45
Precision
Source: email from Schilmoeller, Michael, Monday, December 14, 2009 12:01 PM, to Power Planning Division, based on Q:\SixthPlan\AdminRecord\t6 Regional Portfolio Model\L812\Analysis of Optimization Run_L812.xls
Performance and Precision
46
Model Resolution: At Least $10 million NPV
• Typically, plans have over 70 of the 75 high-cost futures in common
• The model results then come to resemble sensitivity analyses, rather than statistical sampling
• Of course, we could not have anticipated this beforehand
• The most interesting results occur when the high-cost futures differ
Performance and Precision
47
End