Work supported in part byNSF Grants ECS 02 17836, DMI 00-85165

and SES 00-04315

Dynamics of Reserve Service PricesIn Power Networks

Sean MeynDepartment of ECEand the Coordinated Science Laboratory

Joint work with

Prof. I-K Cho, Dept. EconomicsM. Chen, Coordinated Science Lab, Urbana

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

Variability in demand and operating conditionsare critical in determining allocations ...

Link at the California ISO website:

How do I tell whenI am going to be blacked out?

Dynamics and Congestion

Mission includes creation of efficient wholesale electric markets

"The California ISO determines a level of load reduction that is needed to maintain grid reliability ...

Please be advised that system conditions are dynamic and this information is subject to change without further notice!"

http://www.caiso.com

PG&E: "THERE ARE NO GRID CONSTRAINTS AT THIS TIME!"

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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 with historical trend

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

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Glossary (paraphrased)Glossary (paraphrased)

Surplus/Shortfall This number represents the amount of capacity available tomeet the customer demand. If the number is negative, there is not enough capacity to meet the forecasted demand.

Ancillary Services The services other than scheduled energy which are required to maintain system reliability

Spinning/Non-Spinning Reserves The portion of generating capacity, controlled by the ISO, which is capable of being loaded in 10 minutes, and which is capable of running for at least two hours

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Glossary (paraphrased)Glossary (paraphrased)

Surplus/Shortfall This number represents the amount of capacity available tomeet the customer demand. If the number is negative, there is not enough capacity to meet the forecasted demand.

Ancillary Services The services other than scheduled energy which are required to maintain system reliability

Spinning/Non-Spinning Reserves The portion of generating capacity, controlled by the ISO, which is capable of being loaded in 10 minutes, and which is capable of running for at least two hours

Minimum Operating Reliability Criteria The sum of a) regulating reserve plus b) contingency reserve plus c) addtional reserve for interruptible imports plus d) additional reserve for on-demand obligations

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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

Stage 2EmergencyStage 2

Emergency

Stage 3EmergencyStage 3

Emergency

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1000 MWh

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Mon Tue Wed Thu Fri Sat Sun

Prices (Eur/MWh)

Volumes (MWh)

Week 25

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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

D(t

t

) = demand - forecast

Centered demand:

Reserve options for servicesbased on forecast statistics

On-line capacity

Forecast

Actual demand

Revised forecast

Capacity planning

Excess capacity:

Power flow subject to peak and rate constraints:

K

Z(t)Z (t)a

Q(t) = Z(t) + Za(t) − D(t), t ≥ 0 .

−ζa− ≤ ddt

Za(t) ≤ ζa+ −ζ− ≤ ddt

Z(t) ≤ ζ+

One producer, one consumer model

Excess capacity: hedging policy

K

Z(t)Z (t)a

Q(t) = Z(t) + Za(t) − D(t)

q

t

Z (t) = 0

Downward trend:

Blackout

a

Z (t) = - ζ

q

-ddt

ζ +

ζ +

ζ a ++

- ζ -

One producer, one consumer model

Relaxations: instantaneous ramp-down rates:

Cost structure:

Control: design hedging points to minimize average-cost,

−∞ ≤ ddt

Z (t) ≤ ζ +, −∞ ≤ ddt

Za (t) ≤ ζa +.

c(X(t)) = c1Z(t) + c2Za(t) + c3|Q(t)|1{Q(t) < 0}

minEπ [c(Q(t))] .

X(t) =( Q(t)

Za(t)

)X(t) =

( Q(t)Za(t)

)Diffusion model & control

X(t) =( Q(t)

Za(t)

)X(t) =

( Q(t)Za(t)

)

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

q̄2 q̄1

Ancillary serviceis ramped-up whenexcess capacity fallsbelow

Ancillary serviceis ramped-up whenexcess capacity fallsbelow q̄2

Diffusion model & control

X(t) =( Q(t)

Za(t)

)X(t) =

( Q(t)Za(t)

)

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

q̄2 q̄1

γ0 = 2ζ++ζa+

σ2D, γ1 = 2

ζ+

σ2D.

Optimal parameters:Optimal parameters:

q̄∗2 =1

γ0log

c3c2

q̄∗1 − q̄∗2 =1

γ1log

c2c1

Markov model & control

Discrete Markov model:

q̄1 − q̄2q̄2

Average cost

12

34

56

12

13

14

15

1617

18

18

20

22

24

Optimal hedging-pointsfor RBM:

q̄1 − q̄2 = 14.978

q̄2 = 2.996

E(k) i.i.d. Bernoulli.

Q(k + 1) − Q(k)= ζ(k) + ζa(k) + E(k + 1)

ζ(k), ζa(k) allocation increments.

Simulation

Line 1 Line 2

Line 3

E1D1

E2D2

E3D3

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)

ZaA(t) =

=

=∑

Zai (t)

extraction - demand

aggregate ancillary

XA = (QA, ZaA) ≡

Aggregate model

Line 1 Line 2

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

min

s.t.

∑(cpi z

pi + c

ai z

ai + c

boi q

−i )

qA =∑

(ei − di)zaA =

∑zai

0 =∑

(zpi + zai − ei)

q = e d−f = ∆p

f ∈ F

consistency

vector reserves

extraction = generation

power flow equations

transmission constraints

c̄(xA, d)Effective cost

Line 1 Line 2

Line 3

c̄(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

zzaaAA

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, gaa33 = 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

c̄(xA, d)

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

1010

2020

3030

4040

5050

zzaaAA

qqAA

1xxAA

Effective cost

xxAA

Line 1 Line 2

Line 3

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

1010

2020

3030

4040

5050

zzaaAA

qqAA

Will a competitivemarket supportsufficient reserves?

Dynamic Prices & Reserves

Line 1 Line 2

Line 3

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

1010

2020

3030

4040

5050

zzaaAA

qqAA

Prices

Reserves

Market prices risewith demand!

xxAA

q̄∗ = 1γ0

logc3p2

Dynamic Prices & Reserves

Sensitivity to transmission constraints?

Game among suppliers?

Design of financial contracts?

Policy structure as a function of ramp-rates

Workload formulation for model reduction

Solidarity between fluid, RBM,and other optimal MDP solutions

Future work:

Contributions:

3

16q̄1 − q̄2

q̄2

q̄ ∗2 =1

γ0log

c3c2

q̄ ∗1 − q̄ ∗2 =1

γ1log

c2c1

Conclusions & Extensions

3

16q̄1 − q̄2

q̄2

q̄ ∗2 =1

γ0log

c3c2

q̄ ∗1 − q̄ ∗2 =1

γ1log

c2c1

• M. Chen, C. Pandit, and S. Meyn. In search of sensitivity innetwork optimization. Queueing Systems, 2003

• I.-K. Cho and S. Meyn. Dynamics of ancillary service prices inpower distribution networks. CDC, 2003

• M. Chen, R. Dubrawski, and S. Meyn. Management of demanddriven production systems. IEEE TAC, 2004

References