efficiency and marginal cost pricing in dynamic competitive markets with friction

Dynamics of Prices in Electric Power Networks

Sean Meyn

Department of ECEand the Coordinated Science Laboratory University of Illinois

Joint work with M. Chen and I-K. Cho

NSF support: ECS 02-17836 & 05-23620 Control Techniques for Complex Networks

Prices

Normalized demand

Reserve

DOE Support: http://www.sc.doe.gov/grants/FAPN08-13.htmlExtending the Realm of Optimization for Complex Systems: Uncertainty, Competition and DynamicsPIs: Uday V. Shanbhag, Tamer Basar, Sean P. Meyn and Prashant G. Mehta

What is the valueof improved transmission?More responsive ancillary service?

How does a centralized planneroptimize capacity?

Is there an efficient decentralized solution?

OREGON

NEVADA

MEXICO

SANFRANCISCO

LEGEND

COAL

GEOTHERMAL

HYDROELECTRIC

NUCLEAR

OIL/GAS

BIOMASS

MUNICIPAL SOLID WASTE (MSW)

SOLAR

WIND AREAS

California’s 25,000Mile Electron Highway

Forecast DemandForecast of the demand expected today. The procurement of energy resources for the day is based on this forecast

Actual DemandToday's actual system demand

Revised Demand ForecastThe current forecast of the system demand expected throughout the remainder of the day.This forecast is updated hourly.

Available ResourcesThe current forecast of generating and import resources available to serve the demand for energy within the California ISO service area

Meeting Calendar | OASIS | Employment | Site Map | Contact Us

HOME | Search

Hour BeginningDay Ahead Demand Forecast

Available Resources Forecast

Revised Demand Forecast Actual Demand

http://www.caiso.com/outlook/outlook.html September 28, 2008

4,000Megawatts

40,000 MW

30,000 MW

20,000 MW

http://www.caiso.com

Emergency Notices

Generating reserves less than requirements

(Continuously recalculated. Between 6.0% & 7.0%)

Generating reserves less than 5.0%

Generating reserves less than largest contingency

(Continuously recalculated. Between 1.5% & 3.0%)

Generating Reserves

7.0%

6.0%

5.0%

4.0%

3.0%

2.0%

1.0%

0.0%

Stage 1EmergencyStage 1

Emergency


Emergency


Emergency

Meeting Calendar | OASIS | Employment | Site Map | Contact Us

HOME | Search

0

1000

2000

3000

4000

5000

1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21

Monday Tuesday Wednesday Thursday Friday

Purchase Price $/MWh

One Hot Week in Urbana ...

Spinning Reserve Prices PX Prices

0

50

100

150

200

250

10

20

30

40

50

60

70

Weds Thurs Fri Sat Sun Mon Tues Weds

Southern California, July 8-15, 1998 ...

First Impressions:

July 1998: first signs of "serious market dysfunction" in California

FERC ....authorized the ISO to "[reject]...bids in excess of whatever price levels it believes are appropriate ... file additional market-monitoring reports".

Lessons From the California “Apocalypse:” Jurisdiction Over Electric UtilitiesNicholas W. Fels and Frank R. LindhEnergy Law Journal, Vol 22, No. 1, 2001

Mon Tues Weds Thurs Fri Sat Sun

100

150

0

50

200

250

300

350

400

Prices (Eur/MWh) Week 25Week 26

APX Europe, June 2003

vr za zo ma di wo do

4000

3000

2000

1000

0

800

600

400

200

0

Previous weekCurrent week (10/24/05)

Volume (MWh)

Price (Euro)

APX Europe, October 2005

Ancillary service contract clause:Minimum overall ramp rate of 50 MW/min.

Projected power pricesreached $2000/MWh

Ontario, November 2005

Australia January 16 2007

Tasmania

Victoria

0

Australia January 16 2007

Tasmania

Victoria

Volu

me

(MW

h)

Pric

e (A

us $

/MW

h)Pr

ice

(Aus

$/M

Wh)

Volu

me

(MW

h)

- 1,000

- 500

0

+ 500

- 5000

4,000

6,000

8,000

800

1,000

1,200

1,400

+ 10,000

ICentralized Control

3

16

D(t

t

) = demand - forecast

Centered demand:

Reserve options for servicesbased on forecast statistics

On-line capacity

Forecast

Actual demand

Revised forecast

Dynamic model

Stochastic model:

Normalized cost as a function of Q:

G Goods available at time t

D Normalized demand

Excess/shortfall:

c−

c+

Shortfall

Excess production

q

Q(t) = G(t) − D(t)

Dynamic Single-Commodity Model

Sean Meyn

Text Box

Scarf, Arrow, Bellman, ...

Stochastic model:

Generation is rate-constrained:

G Goods available at time t

D Normalized demand

Excess/shortfall: Q(t) = G(t) − D(t)

Q

q

(t

t

)

High cost

ζ +

- ζ -


Average cost: density when

Optimal hedging-point: solves

q

c−

c+

pQ(q )

c−P{ Q ≤ 0} + c+P{ Q ≥ 0} = 0

∗q

∗q

− ∗q

∗q

∫c( q−q ) p

Q(dq)=

=

E[c(Q)]

pQ 0

¯

−


Average cost: density when

Optimal hedging-point: solves

q

c−

c+

ccc−

ccc+c−

c+

pQ(q )

c−P{ Q ≤ 0} + c+P{ Q ≥ 0} = 0

∗q

∗q

− ∗q

∗q

∫c( q−q ) p

Q(dq)=

=

E[c(Q)]

pQ RBM model:

exponentialpQ

0

¯

∗ = 12

σ2

ζ+ logq−


Sean Meyn

Text Box

Wein '92

The two goods are substitutable, but

1. primary service is available at a lower price

2. ancillary service can be ramped up more rapidly

G(t)

G (t)a

Ancillary service

Excess capacity:

Power flow subject to peak and rate constraints:

K

G(t )G (t )a

Q(t) = G(t) + Ga (t) − D (t), t≥ 0 .

−ζa− ≤ d

dtGa (t) ≤ ζa + −ζ− ≤ d

dtG(t) ≤ ζ +

Ancillary service

Policy: hedging policy with multiple thresholds

K

G(t )G (t )a

Q(t) = G(t) + Ga (t) − D (t)

q

t

G (t ) = 0

Downward trend:

Blackout

a

G (t ) = - ζ

q

-ddt

ζ +

ζ +

ζ a ++

- ζ -

Ancillary service

Relaxations: instantaneous ramp-down rates:

Cost structure:

Control: design hedging points to minimize average-cost,

−∞ ≤ d

dtG (t) ≤ ζ +, −∞ ≤ d

dtGa (t) ≤ ζa +.

c(X(t)) = c1G (t) + c2Ga (t) + c3 |Q(t)| 1 {Q(t) < 0}

minEπ [c(Q(t))] .

( )X(t) =

( Q(t)

Ga (t)

)Diffusion model & control

( )X(t) =

( Q(t)

G a (t)

)Markov model:Markov model: Hedging-point policy:Hedging-point policy:

q2 q1

Ancillary serviceis ramped-up whenexcess capacity fallsbelow

Ancillary serviceis ramped-up whenexcess capacity fallsbelow q2

Diffusion model & control

( )X(t) =

( Q(t)

Ga (t)

)Markov model:Markov model: Hedging-point policy:Hedging-point policy:

q2 q1

γ0 = 2ζ++ζa+

σ2D

, γ1 = 2 ζ+

σ2D

.

Optimal parameters:Optimal parameters:

q∗2 =1

γ0log

c3c2

q∗1 − q∗2 =1

γ1log

c2c1

Markov model & control

Sean Meyn

Text Box

(proof)

Discrete Markov model: Optimal hedging-pointsfor RBM:

q1 − q2 = 14.978

q2 = 2.996E(k) i.i.d. Bernoulli.

Q(k + 1) − Q(k)

= ζ(k) + ζa(k) + E(k + 1)

ζ(k), ζa(k) allocation increments.

Simulation

Discrete Markov model:

q1 − q2

q1 − q2q2 q2

Optimal hedging-pointsfor RBM:

q1 − q2 = 14.978

q2 = 2.996

Q(k + 1) − Q(k)

= ζ(k) + ζa(k) + E(k + 1)

18

20

22

24

7

18

20

22

24

3

16

Average cost: CRW Average cost: Diffusion model

Simulation

IIRelaxations

(skip to market)

Line 1 Line 2

Line 3

E1D1

E2

D2E3

D3

Gp1

Ga2

Ga3

Resource pooling from San Antonio to Houston?

Texas model

Line 1 Line 2

Line 3

single producer/consumer model

Assume Brownian demand, rate constraints as before

Provided there are no transmission constraints,

QA(t) =∑

Ei(t) − Di(t)

GaA(t) =

=

=∑

Gai (t)

extraction - demand

aggregate ancillary

XA = (QA GaA) ≡,

Aggregate model

Line 1 Line 2

Line 3Given demand and aggregate statefind the cheapest consistentnetwork configuration subject to transmission constraints

min

s.t.

∑(cp

i gpi + ca

i gai + cbo

i q−i )

qA =∑

(ei − di)gaA =

∑gai

0 =∑

(gpi + ga

i − ei)

q = e d−f = ∆p

f ∈ F

consistency

vector reserves

extraction = generation

power flow equations

transmission constraints

c(xA, d)Effective cost

Line 1 Line 2

Line 3

c(xA, d)

- 50- 50 - 40- 40 - 30- 30 - 20- 20 - 10- 10 00 1010 2020 3030 4040 505000

1010

2020

3030

4040

5050

1

2

3 4

ggaaAA

qqAA

XX++

xxAA

xxAA

xxAA xxAA

RRWhat do theseaggregate states sayabout the network?

Effective cost

qq11 == −−4646, q, q22 = 3= 3..05640564, q, q33 = 1= 122..94369436

gg11 == −−7711, g, gaa22 = 3= 333..05640564, g, gaa

33 = 6= 6..94369436

ff11 == 1313, f, f22 == −−55,, ff33 == −−88

City 1 in blackout:City 1 in blackout:

Insufficient primary generation:Insufficient primary generation:

Transmission constraints binding:Transmission constraints binding:

Line 1 Line 2

Line 3

c(xA, d)

- 50- 50 - 40- 40 - 30- 30 - 20- 20 - 10- 10 00 1010 2020 3030 4040 505000

1010

2020

3030

4040

5050

ggaaAA

qqAA

1xxAA

Effective cost

Controlled work-release, controlled routing,uncertain demand.

demand 1

demand 2

Inventory model

−(1 − ρ)−(1 − ρ)

WW++Resource 1is idle

Resource 2is idle

w2

w1

∗= 12

σ2

ζ+q

Asymptotes:

c−

c+log

Inventory model:Workload relaxation

IIIDecentralized Control

Supply & Demand

Cost of generation depends on source

Nuclear ($6)

Coal ($10 -$15)

Gas Turbine ($20-$30)

Supply Curve

Price

Quantity

Supply & Demand

Demand for power is not flexible

High Priority Customers $5,000/MW?

Customers with interruptible services

Demand Curve

Price

Quantity

Supply & Demand

Equilibrium Price & Quantity: Intersection of supply & demand curves

High Priority Customers

Consumer Surplus

Supplier Surplus

Equilibrium Price$20 - $30

Equilibrium Quantity 45,000MW


Demand Curve

Price

Quantity

Nuclear ($6)

Coal ($10 -$15)


Supply Curve

Supply & Demand

Equilibrium Price & Quantity: Intersection of supply & demand curves

High Priority Customers

Consumer Surplus

Supplier Surplus

Equilibrium Price$20 - $30

Equilibrium Quantity 45,000MW


Demand Curve

Price

Quantity

Nuclear ($6)

Coal ($10 -$15)


Supply Curve

1000

2000

3000

4000

5000

1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21

Where do ramp constraints appear?Variability?Where is hedging?

0

50

100

150

200

250

300

350

400

500 MWh

1000 MWh

1500 MWh

2000 MWh

2500 MWh

Mon Tue Wed Thu Fri Sat Sun

Prices (Eur/MWh)

Volumes (MWh)

Week 25

Week 26

Welcome to APX!

APX is the first electronic energy tradingplatform in continental Europe. The daily spotmarket has been operational since May 1999.The spot market enables distributors,producers, traders, brokers and industrialend-users to buy and sell electricity on aday-ahead basis.

The APX-index will be published daily around12h00 (GMT +01:00) to provide transparencyin the market. Prices can be used as abenchmark.

www.apx.nl

Main Page Market Results

1000

2000

3000

4000

5000

1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21

Second Welfare Theorem

Each player independently optimizes ...

Supplier: profits from two sources of generation

Consumer: value of consumption minus prices paid minus disaster

WD(t) := v min D(t),Gp(t) + Ga(t) − ppGp(t) + paGa(t) + cboQ−(t)

WS(t) := pp − cp Gp(t) + pa − ca)Ga(t)

1000

2000

3000

4000

5000

1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21

Is there an equilibrium price functional?

Is the equilibrium efficient??


1000

2000

3000

4000

5000

1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21 1 5 9 13 17 21

Is there an equilibrium price functional?

Is the equilibrium efficient??

Yes to all !

Q(t) = q

D(t) = d

c cost of insufficient service

v value of consumption

reserve

demand

bo


pe(re, d) = (v + cbo) }{I re < 0

0

2000

4000

6000

8000

10000

Prices

Normalized demand

Reserve

Efficient Equilibrium

Spinning Reserve Prices PX Prices

0

50

100

150

200

250

10

20

30

40

50

60

70


Southern California, July 8-15, 1998 ...

The hedging point (affine) policyis average cost optimal

Amazing solidarity between CRW and CBM models

Deregulation is a disaster!

Future work?

ConclusionsSpinning Reserve Prices PX Prices

0

50

100

150

200

250

10

20

30

40

50

60

70


0

2000

4000

6000

8000

10000

Complex models:Workload or aggregate relaxations

Price caps: No!

Responsive demand: Yes!

Is ENRON off the hook: ?

Extensions and future work

0

2000

4000

6000

8000

10000

Complex models:Workload or aggregate relaxations

Price caps: No!

Responsive demand: Yes!

Is ENRON off the hook: ?

Extensions and future work

What kind of society isn't structured on greed? The problem of social organization is how to set up an arrangement under which greed will do the least harm; capitalism is that kind of system. -M. Friedman

0

2000

4000

6000

8000

10000Epilogue

Fundamentally, there are only two ways of coordinating the economic activities of millions. One is central direction involving the use of coercion - the technique of the army and of the modern totalitarian state. The other is voluntary cooperation of individuals - the technique of the marketplace.

-Milton Friedman

0

2000

4000

6000

8000

10000Epilogue

Justification: 1. Economic systems are complex 2. Regulators cannot be trusted

0

2000

4000

6000

8000

10000Epilogue


Airplanes, highway networks, cell phones... all complex

0

2000

4000

6000

8000

10000Epilogue


Airplanes, highway networks, cell phones... all complex

Complexity is only inherent in the uncontrolledsystem: In each of these examples, thebehavior of the closed loop system is very simple, provided appropriate rules of use, and appropriate feeback mechanisms are adopted.

3

16q1 − q2

q2

References

• M. Chen, I.-K. Cho, and S. Meyn. Reliability by design in a distributed power transmission network. Automatica 2006 (invited)

• I.-K. Cho and S. P. Meyn. The dynamics of the ancillary service prices in power networks. 42nd IEEE Conference on Decision and Control. De-cember 2003

• I.-K. Cho and S. P. Meyn. Efficiency and marginal cost pricing in dy-namic competitive markets. Under revision for J. Theo. Economics. 46th IEEE Conference on Decision and Control 2006

• P. Ruiz. Reserve Valuation in Electric Power Systems. PhD disserta-tion, ECE UIUC 2008

q ∗1 − q ∗

2 =1

γ1log

c2c1

q ∗2 =

1

γ0log

c3c2

First reflection times,

τp:=inf{t ≥ 0 : Q(t) = qp}, τa:=inf{t ≥ 0 : Q(t) ≥ qa}

h(x) = Ex

[∫ τp

0

(c(X(s)) − φ

)ds

]

Poisson’s Equation

Sean Meyn

Text Box

return

First reflection times,

Solves martingale problem,

τp:=inf{t ≥ 0 : Q(t) = qp}, τa:=inf{t ≥ 0 : Q(t) ≥ qa}

h(x) = Ex

[∫ τp

0

(c(X(s)) − φ

)ds

]

M(t) = h(X(t)) +

∫ t

0

(c(X(s)) − φ

)ds


X(t) =( Q(t)

Ga(t)

)

qa qp

h(x) = Ex

[h(X(τa))+

h(X(τa))

∫ τa

0

(c(X(s))−φ

)ds

]


qa qp

h(x) = Ex

[h(X(τa))+

X(τa)

∫ τa

0

(c(X(s))−φ

)ds

]

〈∇h(x),(11

)〉 = =0, x

Derivative Representations

qa qp

h(x) = Ex

[h(X(τa))+

X(τa)

∫ τa

0

(c(X(s))−φ

)ds

]

〈∇h(x),(11

)〉 = =0, x

λa(x) = 〈∇c(x),(11

)〉= ca − I{q ≤ 0}cbo


〉 = Ex

[∫ τa

0

λa(X(t)) dt]

= caE[τa] − cboEx

[∫ τa

0

I{Q(t) ≤ 0} dt]

Computable basedon one-dimensional height (ladder) process,

Ha(t) = qa − Q(t)

〈∇h(x),(11

)


qp∗qa∗

〈∇3.

If

Then h solves the dynamic programming equations,

qp = qp∗ and a = qq a∗

h(x),(10

)〉 < 0, x ∈ Rp

〈∇h(x),(11

)〉 < 0, x ∈ Ra

1. Poisson's equation

2.

Dynamic Programming Equations

Sean Meyn

Text Box

return

efficiency and marginal cost pricing in dynamic competitive markets with friction

Education