How the RPM Meets the Requirements for a Risk Model

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


6

Estimating Energy Generation

Price duration curve (PDC)


7



8

Estimating Energy Value


Price of fuel pg(h)

Set of hours H={h}

Price of electricity pe(h)

9

Gross Value of Resources


Then for a turbine with capacity C MW, the value is

10

Gross Value of Resources


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


12


*

*

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.


13

Variable Fuel Price

• Assume lognormal distribution• Include information about price volatility

and correlation with electricity price

gegege pppppp

ge dNpdNpV

,222

21

2

)()(


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


15

Application of PDC to Energy-Limited Resources


16

Energy-Limited Dispatch


17


2/)((exp*

)(

1

12

ee

eg fN

pp

fNd


18


• If pg* > pg then use energy and value associated with pg*

• Otherwise, use energy and value associated with pg


19

Application of Revenue Curve Equilibrium Prices


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



curve


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



curve


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



curve


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



curve


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


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


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


47

End

Documents

How the RPM Meets the Requirements for a Risk Model