· 5 Oren - October 29, 2008 9 0 200 400 600 800 1000 1200 0 10000 20000 30000 40000 50000 60000...

1

Oren - October 29, 2008 1

PSERC

Shmuel [email protected]

IEOR Dept. , University of California at Berkeley

and Power Systems Engineering Research Center (PSerc)(Based on joint work with Yumi Oum and Shijie Deng)

Centre for Energy and Environmental Markets

University of New South WalesOctober 29, 2008


Power

Generation

Micro-Grids

Retail

Providers

Competitive

State

Regulated

FERC

Regulated

Customers

Demand

Management

Transmission Distribution

SC - Scheduling Coordinator

PX - Power Exchange

ISO - Indep. System Operator

TO - Transmission Owner

LSE - Load Serving Entity

Generation

SCSCPXPX

LSELSE TOTO

ISOISO

2


CAISO (1998)

Population: 30 Million

Peak Load: 45,500MW

Annual Total Energy: 236,450GWh

Generation Capacity: 52,000MW

Average Net Imports: 6,500MW

(Peak 8,300MW)

Wholesale Average price: $56.7/MWh

Annual Wholesale Market: $14 Billion

PJM (1999)

Population: 51 million

Peak load: 144,644 MW


Generating capacity: 164,905 MW

Transmission lines - 56,250 miles

Members/customers - 450+


ERCOT (2001) (not under FERC)

Population: 18 Million (85% of Texas)

Peak Load: 63,000MW


Generation Capacity: 80,000MW

Average Net Imports: Non

Wholesale Average price: ~$70/MWh



$0

$10

$20

$30

$40

$50

$60

$70

$80

$90

$100

0 20000 40000 60000 80000 100000

Demand (MW)

Marginal Cost

($/MWh)Marginal Cost

Resource stack includes:

Hydro units;

Nuclear plants;

Coal units;

Natural gas units;

Misc.

3


ERCOT Energy Price – On peak

Balancing Market vs. Seller’s ChoiceJanuary 2002 thru July 2007

$0

$50

$100

$150

$200

$250

$300

$350

$400Jan

-02

Mar-

02

May-0

2Ju

l-02

Sep

-02

No

v-0

2Jan

-03

Mar-

03

May-0

3Ju

l-03

Sep

-03

No

v-0

3Jan

-04

Mar-

04

May-0

4Ju

l-04

Sep

-04

No

v-0

4Jan

-05

Mar-

05

May-0

5Ju

l-05

Sep

-05

No

v-0

5Jan

-06

Mar-

06

May-0

6Ju

l-06

Sep

-06

No

v-0

6Jan

-07

Mar-

07

May-0

7Ju

l-07

En

erg

y P

rice (

$/M

Wh

)

Balancing Energy (UBES) Price

Spot Price Energy- ERCOT SC

UBES 2/25/03

$523.45

Shmuel Oren, UC Berkeley 06-19-2008


4



Wholesale electricity market

(spot market)

LSE(load serving

entity

Customers(end users servedat fixed regulated

rate)

Generators

Similar exposure is faced by a trader with a fixed price load following obligation(such contracts were auctioned off in New Jersey and Montana to cover defaultservice needs)

5


0

200

400

600

800

1000

1200

0 10000 20000 30000 40000 50000 60000

Actual load (MW)

Real T

ime P

rice (

$/M

Wh

) Period: 4/1998 – 12/2001


Electricity Demand and Price in California

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

9, July ~ 16, July 1998

Dem

an

d (

MW

)

0

20

40

60

80

100

120

140

160

180

Ele

ctric

ity P

rice ($

/MW

h)

Load

Price

Correlation coefficients:

0.539 for hourly price and load from 4/1998 to 3/2000 at Cal PX

0.7, 0.58, 0.53 for normalized average weekday price and load in Spain, Britain, andScandinavia, respectively

6


Properties of electricity demand (load)Uncertain and unpredictable

Weather-driven volatile

Sources of exposureHighly volatile wholesale spot price

Flat (regulated or contracted) retail rates & limited demandresponse

Electricity is non-storable (no inventory)

Electricity demand has to be served (no “busy signal”)

Adversely correlated wholesale price and load

Covering expected load with forward contracts will result in acontract deficit when prices are high and contract excess whenprices are low resulting in a net revenue exposure due to loadfluctuations


Electricity derivatives

Forward or futures

Plain-Vanilla options (puts and calls)

Swing options (options with flexible exercise rate)

Temperature-based weather derivatives

Heating Degree Days (HDD), Cooling Degree Days (CDD)

Power-weather Cross Commodity derivatives

Payouts when two conditions are met (e.g. both hightemperature & high spot price)

Demand response Programs

Interruptible Service Contracts

Real Time Pricing

7


One-period model

At time 0: construct a portfolio with payoff x(p)

At time 1: hedged profit Y(p,q,x(p)) = (r-p)q+x(p)

Objective

Find a zero cost portfolio with exotic payoff which maximizesexpected utility of hedged profit under no credit restrictions.

Spot market

LSE Load (q)

p r

Portfolio

x(p)

for a delivery at time 1


Objective function

Constraint: zero-cost constraint

[ ]

+

+

dqdpqpfpxqprU

pxqprUE

px

px

),()]()[(max

)]()[(max

)(

)(

Utility function over profit

Joint distribution of p and q

0)]([1

=pxEB

Q

risk-neutral probability measure

price of a bond paying $1 at time 1

! A contract is priced as an expecteddiscounted payoff under risk-neutral measure

8


( )[ ])(

)(*|)(*)('

pf

pgppxqprUE

p

=+

)(2

1][)]([ YaVarYEYUE =

The Lagrange multiplier is determined so that the constraint is satisfied

Mean-variance utility function:

]|),([

)(

)(

)(

)(

]]|),([[

)(

)(

)(

)(

11

)(* pqpyE

pf

pgE

pf

pg

pqpyEE

pf

pgE

pf

pg

apx

p

Q

pQ

p

Q

p+=


-5 0 5

x 104

0

1

2

3

4

5

6

7

8x 10

-5

profit: y = (r-p)q + x(p)

Probability = 0

= 0.3 = 0.5 = 0.7 = 0.8

rate) retail(flat /120$

60)(,300][

/56$)(,/70$][

Q&Punder ),2.0,69.5,7.0,4(~)log,(log 22

MWhr

MWhqMWhqE

MWhpMWhpE

Nqp

=

==

==

Bivariate lognormal distribution:Dist’n of profit

Optimal exotic payoff

Mean-Var

E[p]

Note: For the mean-varianceutility, the optimal payoff is linear inp when correlation is 0,

0 50 100 150 200 250

-2

0

2

4

6

8x 10

4

Spot price p

x*(p)

=0 =0.3 =0.5 =0.7 =0.8

9


-1 -0.5 0 0.5 1 1.261.5 2 2.5 3

x 104

0

0.5

1

1.5

2

2.5

3x 10

-4

Profit

Probability

After price & quantity hedge

After price hedge

Before hedge

Bivariate lognormal for (p,q)

Comparison of profit distribution for mean-variance utility ( =0.8)

Price hedge: optimal forward hedge

Price and quantity hedge: optimal exotic hedge


0 50 100 150 200 250-2

-1

0

1

2

3

4

5

6

7x 10

4

Spot price p

x*(p)

m2 = m1-0.1m2 = m1-0.05m2 = m1m2 = m1+0.05m2 = m1+0.1

With Mean-variance utility (a =0.0001)

=

=

=

=

=

=

1.0,1.77

05.0,3.73

0,8.69

05.0,4.66

1.0,1.63

][

12

12

12

12

12

mmif

mmif

mmif

mmif

mmif

pEQ][log2 pEmQ

=

10


0 50 100 150 200 250-5

0

5

10

15

20x 10

4

Spot price p

x*(p)

a =5e-006a =1e-005a =5e-005a =0.0001a =0.001

with mean-variance utility (m2 = m1+0.1)

)(2

1][)]([ YaVarYEYUE =

(Bigger ‘a’ = more risk-averse)

Note: if m1 = m2 (i.e., P=Q), ‘a’ doesn’t matterfor the mean-variance utility.


dKKpKxdKpKKxFpFxFxpxF

F++

+++= ))((''))((''))(('1)()(0

Forwardpayoff

Put optionpayoff

Call optionpayoff

Bondpayoff

Strike < F Strike > F

Exact replication can be obtained froma long cash position of size x(F)

a long forward position of size x’(F)

long positions of size x’’(K) in puts struck at K, for a continuum of Kwhich is less than F (i.e., out-of-money puts)

long positions of size x’’(K) in calls struck at K, for a continuum of Kwhich is larger than F (i.e., out-of-money calls)

Payoff

p

Forward price

F

Payoff

K

Payoff

Strike price

K

Strike price

11


0 50 100 150 200 250-2

-1

0

1

2

3

4

5

6

7x 10

4

Spot price p

payoff

x*(p)Replicated when K = $1Replicated when K = $5Replicated when K = $10

0 50 100 150 200 250

-2

0

2

4

6

8x 10

4

Spot price p

x*(p)

=0 =0.3 =0.5 =0.7 =0.8

0 50 100 150 200 250-500

0

500

Spot price p

x*'(p)

=0 =0.3 =0.5 =0.7 =0.8

0 50 100 150 200 2500

10

20

30

40

50

60

Spot price p

x*''(p)

=0 =0.3 =0.5 =0.7 =0.8

)( px )( px )( px

puts calls

F

Payoffs from discretized portfolio


Objective functionUtility function over profit

(A contract is priced as an expecteddiscounted payoff under risk-neutral measure)

Q: risk-neutral probability measure

zero-cost constraint

(Same as Nasakkala and Keppo)

12


Price and quantity dynamics


13


Optimal


14


*In reality hedging portfolio will be determined at hedging time based on realized quantities and prices at that time.

Oren - October 29, 2008 28Oum and Oren, July 10, 2008 28

ã1í}P{Xthatsuch í(X)VaRwhere

V(Y(x))VaR

0[x(p)]Es.t.

E[Y(x)]max

ã

0ã

Q

x(p)

==

=

• Let’s call the Optimal Solution: x*(p)

• Q: a pricing measure

• Y(x(p)) = (r-p)q+x(p) includes multiplicative term of two risk factors

• x(p) is unknown nonlinear function of a risk factor

• A closed form of VaR(Y(x)) cannot be obtained

15


0)]([E s.t.

kV(Y(x))2

1-E[Y(x)]maxarg(p)x

Q

x(p)

k

=

=

px

• For a risk aversion parameter k

• We will show how the solution to the mean-varianceproblem can be used to approximate the solution tothe VaR-constrained problem


Suppose..

There exists a continuous function h such that

with h increasing in standard deviation (std(Y(x))) and non-increasing in mean (mean(Y(x)))

Then

x*(p) is on the efficient frontier of (Mean-VaR) plane and (Mean-Variance) plane

Oum and Oren, July 10, 2008 30

ã)std(Y(x)), , ))h(mean(Y(x(Y(x))VaR ã =

16


The property holds for distributions such as normal, student-t, andWeibull

For example, for normally distributed X

The property is always met by Chebyshev’s upper bound on theVaR:


( ) ( ) ( )VaR x z std x E x=

12

2

2

1

{ ( ) } ( ) /

{ ( ) ( )} 1/

( ) ( ) ( )t

P x E x k var x k x

P x E x t std x t

VaR x t std x E x


17


Under P

log p ~ N(4, 0.72)

q~N(3000,6002)

corr(log p, q) = 0.8

Under Q:

log p ~ N(4.1, 0.72)

We assume a bivariate normal dist’n for log p and q

Distribution of

Unhedged Profit (120-p)q


xk(p) =(1- ApB)/2k-(r-p)(m + E log p – D) + CpB


18


Once we have optimal xk(p), we can calculate associated VaR bysimulating p and q and

Find smallest k such that VaR(k) V0 (Smallest k => Largest Mean)


19



Risk management is an essential element of competitiveelectricity markets.

The study and development of financial instruments canfacilitate structuring and pricing of contracts.

Better tools for pricing financial instruments and developmentof hedging strategy will increase the liquidity and efficiency ofrisk markets and enable replication of contracts throughstandardized and easily tradable instruments

Financial instruments can facilitate market design objectivessuch as mitigating risk exposure created by functionalunbundling, containing market power, promoting demandresponse and ensuring generation adequacy.

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