Tutorial for Energy Systems Week - Cambridge 2010

Issues and ApproachesReliability by Design

Dynamic Markets and their EquilibriaWho Commands the Wind?

Research FrontiersReferences

Dynamic Models and Dynamic Markets

for Electric Power Networks

Sean P. Meyn

Joint work with: In-Koo Cho, Matias Negrete-Pincetic, Gui Wang,

Anupama Kowli, and Ehsan Shafieeporfaard

Coordinated Science Laboratory

and the Department of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign, USA

May 25, 2010

1 / 66

Outline

1 Issues and Approaches

2 Reliability by Design

3 Dynamic Markets and their Equilibria

4 Who Commands the Wind?

5 Research Frontiers

6 References

2 / 66

Demand

Prices

19,000

10,000

- 1,000

- 1,500

- 1,000

10,000

Tasmania

Victoria

Australia - January 16, 2007

Victoria

Tasmania

Issues and Approaches

3 / 66

Issues: Political Mandates come, and go...

Renewable Portfolio Standards

State renewable portfolio standard

State renewable portfolio goal

www.dsireusa.org / February 2010

Solar water heating eligible * †

Extra credit for solar or customer-sited renewables

Includes non-renewable alternative resources

WA: 15% x 2020*

CA: 33% x 2020

NV: 25% x 2025*

AZ: 15% x 2025

NM: 20% x 2020 (IOUs)

10% x 2020 (co-ops)

HI: 40% x 2030

Minimum solar or customer-sited requirement

TX: 5,880 MW x 2015

UT: 20% by 2025*

CO: 20% by 2020 (IOUs) 10% by 2020 (co-ops & large munis)*

MT: 15% x 2015

ND: 10% x 2015

SD: 10% x 2015

IA: 105 MW

MN: 25% x 2025 (Xcel: 30% x 2020)

MO: 15% x 2021

WI: Varies by utility; laog 5102 x %01

MI: 10% + 1,100 MW x 2015*

OH: 25% x 2025†

ME: 30% x 2000 New RE: 10% x 2017

NH: 23.8% x 2025

MA: 15% x 2020 + 1% annual increase

(Class I RE)

RI: 16% x 2020

CT: 23% x 2020

NY: 29% x 2015

NJ: 22.5% x 2021

PA: 18% x 2020†

MD: 20% x 2022

DE: 20% x 2019*

DC: 20% x 2020

VA: 15% x 2025*

NC: 12.5% x 2021 (IOUs)

10% x 2018 (co-ops & munis)

VT: (1) RE meets any increase in retail sales x 2012;

(2) 20% RE & CHP x 2017

KS: 20% x 2020

OR: 25% x 2025 (large utilities)*

5% - 10% x 2025 (smaller utilities)

IL: 25% x 2025 WV: 25% x 2025*†

29 states +

DC have an RPS (6 states have goals)

4 / 66

Issues: Dynamics Coupled generators and consumers

Time in Seconds

0 20 40 60 80

Mass-spring model

Instability in the North-West U.S.

5 / 66

Issues: Complexity

Interconnected

network of

ENTSO-E

European Network of TransmissionSystem Operators for Electricity

6 / 66

Issues: Market Power Risk to consumers

Spinning reserve prices PX prices $/MWh

Mon Tues WedsWeds Thurs Fri Sat Sun

Enron traders openly discussed manipulating

California’s power market during profanity-laced

telephone conversations in which they merrily

gloated about ripping o! “those poor

grandmothers” during the state’s energy crunch

in 2000-01 ...

California, July 2000

[AP by Kristen Hays, 06/03/04]

NRON As Je!rey Skilling, the "rst but sadly not the last guy

to package up a big sample of nothing and sell it to a greedy

public, Samuel West oozes self-belief; his boss, Ken Lay (Tim

Piggott-Smith) and stooge, Andy Fastow (Tom Goodman-Hill)

end up smeared with it. But Lucy Prebble's play is rather more

than a simple tale ... -Time Out, London 2010

Enron traders Enron traders Enron traders

California’s power market during California’s power market during California’s power market during California’s power market during California’s power market during California’s power market during California’s power market during California’s power market during California’s power market during California’s power market during California’s power market during California’s power market during

telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations in which they merrily telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations

gloated about ripping o! “those poor gloated about ripping o! “those poor gloated about ripping o! “those poor gloated about ripping o! “those poor gloated about ripping o! “those poor gloated about ripping o! “those poor gloated about ripping o! “those poor gloated about ripping o! “those poor

grandmothers” during the state’s energy crunch grandmothers” during the state’s energy crunch grandmothers” during the state’s energy crunch grandmothers” during the state’s energy crunch grandmothers” during the state’s energy crunch

[AP by Kristen Hays, 06/03/04]in 2000-01 ... in 2000-01 ... in 2000-01 ... [AP by Kristen Hays, 06/03/04]in 2000-01 ... [AP by Kristen Hays, 06/03/04][AP by Kristen Hays, 06/03/04][AP by Kristen Hays, 06/03/04][AP by Kristen Hays, 06/03/04]

NRON NRON NRON As Je!rey Skilling, the "rst but sadly not the last guy As Je!rey Skilling, the "rst but sadly not the last guy NRON As Je!rey Skilling, the "rst but sadly not the last guy As Je!rey Skilling, the "rst but sadly not the last guy As Je!rey Skilling, the "rst but sadly not the last guy As Je!rey Skilling, the "rst but sadly not the last guy As Je!rey Skilling, the "rst but sadly not the last guy

to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy

to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy

public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim

Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Andy Fastow (Tom Goodman-Hill) Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Andy Fastow (Tom Goodman-Hill)

up smeared with it. But Lucy Prebble's play is rather more

Piggott-Smith) and stooge, Piggott-Smith) and stooge, Andy Fastow (Tom Goodman-Hill) Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge,

end up smeared with it. But Lucy Prebble's play is rather more

than a simpl

Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge, Andy Fastow (Tom Goodman-Hill) Piggott-Smith) and stooge, Piggott-Smith) and stooge,

end up smeared with it. But Lucy Prebble's play is rather more end up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more end up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more

e tale ...

e tale ... than a simplthan a simple tale ... than a simple tale ... e tale ...

telephone conversations in which they merrily telephone conversations telephone conversations

California’s power market during

openly discussed manipulating

openly discussed manipulating Enron traders openly discussed manipulating

telephone conversations

in 2000-01 ... [AP by Kristen Hays, 06/03/04]

in 2000-01 ...

telephone conversations telephone conversations

gloated about ripping o! “those poor gloated about ripping o! “those poor

NRON to package up a big sample of nothing and sell it to a greedy

in 2000-01 ...

Enron traders

Enron traders Enron traders

Enron traders

openly discussed manipulating

California’s power market during California’s power market during California’s power market during

in which they merrily telephone conversations

telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations telephone conversations in which they merrily telephone conversations telephone conversations telephone conversations

telephone conversations telephone conversations telephone conversations

in 2000-01 ... in 2000-01 ...

[AP by Kristen Hays, 06/03/04][AP by Kristen Hays, 06/03/04][AP by Kristen Hays, 06/03/04][AP by Kristen Hays, 06/03/04]

in 2000-01 ...

in 2000-01 ... in 2000-01 ...

grandmothers” during the state’s energy crunch grandmothers” during the state’s energy crunch

in 2000-01 ... in 2000-01 ... in 2000-01 ... [AP by Kristen Hays, 06/03/04]in 2000-01 ... in 2000-01 ... in 2000-01 ... in 2000-01 ... [AP by Kristen Hays, 06/03/04]in 2000-01 ... in 2000-01 ... [AP by Kristen Hays, 06/03/04][AP by Kristen Hays, 06/03/04]

As Je!rey Skilling, the "rst but sadly not the last guy

to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy

NRON to package up a big sample of nothing and sell it to a greedy

Piggott-Smith) and stooge,

than a simple tale ...

Piggott-Smith) and stooge, Piggott-Smith) and stooge, Piggott-Smith) and stooge,

NRON As Je!rey Skilling, the "rst but sadly not the last guy As Je!rey Skilling, the "rst but sadly not the last guy NRON As Je!rey Skilling, the "rst but sadly not the last guy As Je!rey Skilling, the "rst but sadly not the last guy As Je!rey Skilling, the "rst but sadly not the last guy

NRON to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy

public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim

to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy to package up a big sample of nothing and sell it to a greedy

Andy Fastow (Tom Goodman-Hill)

-Time Out, London 2010

public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim

public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim

Andy Fastow (Tom Goodman-Hill) Piggott-Smith) and stooge, Piggott-Smith) and stooge, Andy Fastow (Tom Goodman-Hill)

up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more

up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more

in 2000-01 ...

public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim public, Samuel West oozes self-belief; his boss, Ken Lay (Tim

than a simpl

Enron traders

in 2000-01 ...

than a simple tale tale ...

up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more up smeared with it. But Lucy Prebble's play is rather more

“Ripping off those poor grandmothers”

7 / 66

Issues: Risks to suppliers

Mon Tues Weds Thurs Fri SatSun0

Wind generation (MW)

Balancing Authority Load (MW) January 16-22, 2010

Who will buy my power if the wind is blowing?

8 / 66

Issues: Friction

Demand

Prices

19,000

10,000

- 1,000

- 1,500

- 1,000

10,000

Tasmania

Victoria

Power flows according toKirchoff’s Laws, and subjectto constraints ...

9 / 66

Issues: Friction

Demand

Prices

19,000

10,000

- 1,000

- 1,500

- 1,000

10,000

Tasmania

Victoria

... Mechanical andthermal constraints, rampconstraints, securityconstraints, ...

9 / 66

Issues: Friction

Demand

Prices

19,000

10,000

- 1,000

- 1,500

- 1,000

10,000

Tasmania

Victoria

... Mechanical andthermal constraints, rampconstraints, securityconstraints, ...

Market and physical system

are tightly coupled

9 / 66

Issues: Uncertainty

00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Hour

Actual Demand (GW)Day Ahead Demand Forecast

Available Resources Forecast (GW)

California ISO www.caiso.com

CAISO Mission:

• Operate the grid reliably and efficiently• Provide fair and open transmission access• Promote environmental stewardship• Facilitate effective markets ...

10 / 66

Control solution: Model basedSimplest model that leads to useful policy

Listed in order of complexity:

1. Generation is modeled via overall ramp-rate:

G(t1)−G(t0)t1−t0

≤ ζ+, 0 ≤ t0 < t1 < ∞

We may also impose lower bounds.

11 / 66

G(t1)−G(t0)t1−t0

≤ ζ+, 0 ≤ t0 < t1 < ∞

2. Distinguish primary and ancillary service:

Gp(t1)−Gp(t0)t1−t0

≤ ζp+, Ga(t1)−Ga(t0)t1−t0

≤ ζa+

11 / 66

G(t1)−G(t0)t1−t0

≤ ζ+, 0 ≤ t0 < t1 < ∞

≤ ζa+

3. Introduce transmission constraints:Power flow on transmission lines = linear function of nodalpower values

F (t) = ∆Z(t), ∆ = distribution factor matrix

11 / 66

G(t1)−G(t0)t1−t0

≤ ζ+, 0 ≤ t0 < t1 < ∞

≤ ζa+

3. Introduce transmission constraints:Power flow on transmission lines = linear function of nodalpower values

F (t) = ∆Z(t), ∆ = distribution factor matrix

F (t) ∈ F := {x ∈ Rℓt : −f+ ≤ x ≤ f+}

11 / 66

Market Analysis: Perfect competition

A beautiful world...

In this lecture,Dynamic market equilibriaunder the most ideal circumstances:

12 / 66

Price manipulation is excluded

12 / 66

Price manipulation is excludedNo “ENRON games” – no “market power”

12 / 66

Externalities are disregardedNo government mandates

12 / 66

Externalities are disregardedNo government mandates

Consumers own available wind generation resources

12 / 66

00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Hour

Reliability by Design

13 / 66

Reliability: How to achieve the CAISO Mission?

00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Hour

Begin with,• Operate the grid reliably and efficiently

14 / 66

Reliability: How to achieve the CAISO Mission?

00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Hour

Begin with,• Operate the grid reliably and efficiently

These can wait:

• Provide fair and open transmission access• Promote environmental stewardship• Facilitate effective markets ...

14 / 66

A diffusion modelDynamic model for reserves

R(t) = Available power − Demand = G(t) − D(t)

15 / 66

D(t) = Actual demand − Forecast demand

G(t): Deviation in on-line capacity from day-ahead market

15 / 66

Volatility

Deviation in demand D is modeled as a driftless Brownian motionwith instantaneous variance σ2 (imposed for computation)

15 / 66

Volatility

Deviation in demand D is modeled as a driftless Brownian motionwith instantaneous variance σ2 (imposed for computation)

Friction

Generation cannot increase instantaneously:

G(t′) − G(t)

t′ − t≤ ζ , for all t ≥ 0, and t′ > t.

15 / 66

Social planner’s objective

Discounted welfare: For given initial generation and demand,

K(g, d) = E[

e−γt(

WS(t) + WD(t) + WE(t))

Welfare functions:

Supplier, WS(t): payments - cost

16 / 66

K(g, d) = E[

e−γt(

Welfare functions:

Consumer, WD(t): utility - payments - cost of b/o

16 / 66

K(g, d) = E[

e−γt(

Welfare functions:

External, WE(t): environmental, political

16 / 66

K(g, d) = E[

e−γt(

Welfare functions:

External, WE(t): environmental, political

Goal: Maximize the discounted social welfare

16 / 66

Social planner’s objectivePiecewise linear welfare

Linear costs and values

WS(t) := P (t)G(t) − cG(t)

WD(t) := v min(D(t), G(t))

− cbo max(0,−R(t)) − P (t)G(t)

17 / 66

Linear costs and values

WS(t) := P (t)G(t) − cG(t)

WD(t) := v min(D(t), G(t))

− cbo max(0,−R(t)) − P (t)G(t)

=⇒ discounted welfare is a function of reserves

K(g, d) = (v − c)d + E[

e−γtw(R(t)) dt]

w(R(t)) = v min(0, R(t)) + cbo min(R(t), 0) − c(R(t))

17 / 66

Discounted welfare is a function of reserves

K(g, d) = (v − c)d + E[

e−γtw(R(t)) dt]

w(R(t)) = v min(0, R(t)) + cbo min(R(t), 0) − c(R(t))

Threshold policy

Optimal threshold: r̄∗ =1

θ+log

cbo + v

where θ+ > 0 solves, 12σ2θ2 − ζθ − γ = 0

18 / 66

Threshold policy

Optimal threshold: r̄∗ =1

θ+log

cbo + v

Average cost case:1

θ+→ 1

ζ, as γ ↓ 0.

19 / 66

Social planner’s objectiveReserve Dynamics

Optimal reserves = reflected Brownian motion:

Reserves

Normalized demand

Optimal threshold r̄∗ = 1θ+

cbo+vc

20 / 66

Primary and Ancillary ServiceMultiple thresholds

R(t) = Available power − Demand = Gp(t) + Ga(t) − D(t)

21 / 66

Social Planner’s Problem solved using an affine policy:

21 / 66

The cheaper resource Gp(t) ramps up when R(t) < r̄p

21 / 66

Ancillary service Ga(t) ramps up when R(t) < r̄a

21 / 66

(R constrained

by affine policy

(t), Ga(t))

Trajectory of the two-dimensional model under an affine policy

21 / 66

(R constrained

by affine policy

(t), Ga(t))Ga(t))

D = 6 σ2

D = 18 σ2

D = 24

qp∗qp

qaqa∗

CRW model CBM model

10- 10 0 20- 20 30

Solidarity between optimal policies for non-Brownian demand

22 / 66

Reliability by Design in NetworksThree-Node Example

Line 1 Line 2

Line 3

D Dallas1

D Houston3

D Austin2

F (t) = ∆Z(t)

23 / 66

Line 1 Line 2

Line 3

D Dallas1

D Houston3

D Austin2

F (t) = ∆Z(t)

Power flow on transmission lines F = (F1, F2, F3)T

Nodal power Z = (Gp1 − E1, G

a2 − E2, G

a3 − E3)

23 / 66

a2 − E2, G

a3 − E3)

Distribution factor matrix

∆ = 13

1 −1 02 1 0

−1 −2 0

24 / 66

a2 − E2, G

a3 − E3)

Distribution factor matrix

∆ = 13

1 −1 02 1 0

−1 −2 0

Constraints: F (t) ∈ F := {x ∈ Rℓt : −f+ ≤ x ≤ f+}

24 / 66

Reliability by Design in NetworksExtension to general topology

Primary and ancillary capacity processes at each node aresubject to ramp constraints

Deviation in demand is a multi-dimensional stochastic process– Brownian motion for computation.

Cost in social planner’s problem,

pi + ca

i gai + cbo

i (ei − di)−)

25 / 66

Reliability by Design in NetworksExtension to general topology

Primary and ancillary capacity processes at each node aresubject to ramp constraints

Deviation in demand is a multi-dimensional stochastic process– Brownian motion for computation.

Cost in social planner’s problem,

pi + ca

i gai + cbo

i (ei − di)−)

Optimization problem is now intractable!

25 / 66

Reliability by Design in NetworksRelaxations

Consider aggregate reserves and ancillary generalization:

RA(t) =∑

Ri(t) , GaA(t) =

Gai (t).

26 / 66

RA(t) =∑

Ri(t) , GaA(t) =

Gai (t).

For given

Aggregate state xA = (rA, gaA)T ∈ XA := R × R+

Demand vector d ∈ Rℓn

Effective cost: Cost of cheapest state vector consistent with xA:

26 / 66

RA(t) =∑

Ri(t) , GaA(t) =

Gai (t).

For given

Aggregate state xA = (rA, gaA)T ∈ XA := R × R+

Demand vector d ∈ Rℓn

Effective cost: Cost of cheapest state vector consistent with xA:

min∑

(cpi g

pi + ca

i gai + cbo

i ri−)

subject torA =

i(ei − di) gaA =

0 =∑

i(gpi + ga

i − ei) 0 = e − d − q

f = ∆p ∈ F, e, q, gp ∈ Rℓn , ga ∈ R

26 / 66

Reliability by Design in NetworksRelaxations and Effective Cost

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 500

c(xA, d)

Level sets of

effective cost

27 / 66

Reliability by Design in NetworksAffine Policy for Relaxation

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 500

c(xA, d)

Level sets of

effective cost

Monotone region

Constraint region

under affine policy

Affine policy is an affine enlargement of “monotone region” inwhich Ga

A ramps at maximum rate.

28 / 66

Reliability by Design in NetworksAffine Policy for Relaxation

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 500

c(xA, d)

Level sets of

effective cost

Monotone region

Constraint region

under affine policy

How does an affine policy respond in these four different scenarios?

29 / 66

Reliability by Design in NetworksEffective States in Relaxation

Effective state X ∗(xA):Minimizer in thecomputation of c̄(xA, d).

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 500

X ∗(x1A): Dallas is experiencing black-out!

30 / 66

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 500

Yet ancillary service is ramping down at maximum rate!

30 / 66

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 500

Yet ancillary service is ramping down at maximum rate!Each of the three transmission lines have reached theircapacity limits: Nodes 2 and 3 cannot transfer reserves tonode 1, and node 1 cannot extract more reserve to reduce theseverity of black-out.

30 / 66

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 500

X ∗(x2A) and X ∗(x3

A): Black-out!

31 / 66

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 500

A): Black-out!However, there is sufficient transmission capacity to provideincreased reserves to any node.

31 / 66

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 500

A): Black-out!However, there is sufficient transmission capacity to provideincreased reserves to any node.Policy dictates that each generator ramp up at maximum rate.

31 / 66

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 500

A): Black-out!However, there is sufficient transmission capacity to provideincreased reserves to any node.Policy dictates that each generator ramp up at maximum rate.

X ∗(x4A): High power reserves at each node, so that

generation ramps down.

31 / 66

Purchase Price $/MWh

Previous week

APX Power NL, April 23, 2007

Cho & Meyn, 2006Illinois, July 1998

Ontario, November 2005

Mon Tues Weds Thurs Fri

Tues Weds ThursTime3 6 9 12 15 18 213 6 9 12 15 18 213 6 9 12 15 18 21

Demand in MW Last Updated 11:00 AM Predispatch 1975.11 Dispatch 19683.5

Hourly Ontario Energy Price $/MWh Last Updated 11:00 AM Predispatch 72.79 Dispatch 90.822000

Fri Sat Sun MonTues Wed Thus

Prev. Week Volume

Current Week Volume

Prev. Week Price

Current Week Price

Prices

Reserves

Normalized demand

Dynamic Markets and their Equilibria

32 / 66

Market StructureDay-ahead market

The day-ahead market (DAM) is cleared one day prior to theactual production and delivery of energy.

Economic world...

Forward markets for electricity which improve market efficiency andserve as a hedging mechanism

33 / 66

Market StructureDay-ahead market

The day-ahead market (DAM) is cleared one day prior to theactual production and delivery of energy.

Economic world...

Forward markets for electricity which improve market efficiency andserve as a hedging mechanism

...Physical world

Facilitate the scheduling of generating units

33 / 66

Market StructureReal-time market

At close of the DAM: The ISO generates a schedule of generatorsto supply specific levels of power for each hour over the next 24hour period.

RTM = Balancing Market

As supply and demand are not perfectly predictable, the RTMplays the role of fine-tuning this resource allocation process

34 / 66

Market StructureReal-time market

At close of the DAM: The ISO generates a schedule of generatorsto supply specific levels of power for each hour over the next 24hour period.

RTM = Balancing Market

As supply and demand are not perfectly predictable, the RTMplays the role of fine-tuning this resource allocation process

RTM is the focus here

34 / 66

Market AnalysisFirst Welfare Theorem

Efficient Equilibrium

max K(g, d) = E[

e−γt(

WS(t) + WD(t))

subject to GS(t) = GD(t) for all t

35 / 66

max K(g, d) = E[

e−γt(

WS(t) + WD(t))

Welfare functions defined with a nominal price function P (t)

35 / 66

max K(g, d) = E[

e−γt(

WS(t) + WD(t))

Key component of equilibrium theory:

Perfect competition

The price of power P (t) in the RTM is assumed to be exogenous(it does not depend on the decisions of the market agents).

35 / 66

max K(g, d) = E[

e−γt(

WS(t) + WD(t))

Key component of equilibrium theory:

Perfect competition

The price of power P (t) in the RTM is assumed to be exogenous(it does not depend on the decisions of the market agents).

“price-taking assumption”35 / 66

max K(g, d) = E[

e−γt(

WS(t) + WD(t))

Price function P (t) is irrelevant when GS(t) = GD(t):

WS(t) := P (t)GS(t) − cGS(t)

WD(t) := v min(D(t), GD(t))

− cbo max(0,−R(t)) − P (t)GD(t)

36 / 66

First Welfare Theorem ⇐⇒ Lagrangian Decomposition

max K(g, d) = E[

e−γt{

WS(t) + WD(t))

+ λ(t)(

GS(t) − GD(t))

37 / 66

max K(g, d) = maxGS

e−γt(

WS(t) + λ(t)GS(t))

+ maxGD

e−γt(

WD(t) − λ(t)GD(t))

38 / 66

max K(g, d) = maxGS

e−γt(

WS(t) + λ(t)GS(t))

+ maxGD

e−γt(

Assume: Social planner’s problem has a solution, and there is noduality gap.

38 / 66

max K(g, d) = maxGS

e−γt(

WS(t) + λ(t)GS(t))

+ maxGD

e−γt(

Assume: Social planner’s problem has a solution, and there is noduality gap. Then P ∗(t) = P (t) + λ∗(t) provides an efficientequilibrium.

38 / 66

max K(g, d) = maxGS

e−γt(

WS(t) + λ(t)GS(t))

+ maxGD

e−γt(

Assume: Social planner’s problem has a solution, and there is noduality gap. Then P ∗(t) = P (t) + λ∗(t) provides an efficientequilibrium.

What does an efficient equilibrium look like?

38 / 66

What does an efficient equilibrium look like?Market analysis assumptions

Disutility from power loss

The consumer suffers utility loss if demand is not met:Disutility to the consumer = cbo|R(t)| whenever R(t) < 0.

39 / 66

The production cost is a linear function of G(t),of the form cG(t) for some constant c > 0.

39 / 66

The production cost is a linear function of G(t),of the form cG(t) for some constant c > 0.

Value of power

Consumer obtains v units of utility per unit of power consumed:Utility to the consumer = v min(D(t), G(t)).

39 / 66

What does an efficient equilibrium look like?Answer: Marginal value

Equilibrium price

The equilibrium price functional is a piecewise constant function ofthe equilibrium reserve process,

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

40 / 66

What does an efficient equilibrium look like?Answer: Marginal value

Equilibrium price

The equilibrium price functional is a piecewise constant function ofthe equilibrium reserve process,

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

P ∗(t) = pe(Re(t)) is the marginal value of power to the consumerat time t.

40 / 66

Market EquilibriumPrice Dynamics

v + cbo

Prices

Reserves

Normalized demand

P ∗(t) = pe(Re(t)): The marginal value of power to the consumer

41 / 66

Market EquilibriumPrice Dynamics

v + cbo

Prices

Reserves

Normalized demand

P ∗(t) = pe(Re(t)): The marginal value of power to the consumer

Smoother prices obtained when cost/utility are strictly convex

41 / 66

Familiar, right?

Previous week

California, July 2000Illinois, July 1998

Mon Tues Weds Thurs Fri Mon Tues WedsWeds Thurs Fri Sat Sun

Prev. Week Volume

Current Week Volume

Prev. Week Price

Current Week Price

42 / 66

Sustainable business?

Marginal value ofelectricity may bev + cbo =$200,000/MWh!

Previous week

Prev. Week Volume

Current Week Volume

Prev. Week Price

Current Week Price

43 / 66

Previous week

Prev. Week Volume

Current Week Volume

Prev. Week Price

Current Week Price

Yet, in this equilibrium, expected price is preciselythe ‘marginal cost’ c.

43 / 66

Previous week

Prev. Week Volume

Current Week Volume

Prev. Week Price

Current Week Price

Yet, in this equilibrium, expected price is preciselythe ‘marginal cost’ c.

Is this a sustainable business?

43 / 66

Who Commands the Wind?

44 / 66

Extending the dynamic model

Extend the DA/RT market model to differentiate power generatedby wind and by conventional generation units

45 / 66

Assumption

All the wind power available is injected into the system

45 / 66

Assumption

Key issue

Conventional generators serve residual demand.

45 / 66

Assumption

Key issue

Conventional generators serve residual demand.

Volatility of the residual demand will result in higher reserves inthe dynamic market equilibrium

45 / 66

Emerging issues

Not only mean energy

The details depend on both volatility of wind and its proportion ofthe overall generation

What about now?

If the penetration of wind resources is low, then the increase inload volatility will be negligible, and hence there will be negligibleimpact on the market outcomes

46 / 66

Emerging issues

Not only mean energy

The details depend on both volatility of wind and its proportion ofthe overall generation

What about now?

If the penetration of wind resources is low, then the increase inload volatility will be negligible, and hence there will be negligibleimpact on the market outcomes

What about in 2020?

Potential negative market outcomes are possible with acombination of high wind generation penetration and highvolatility of wind

46 / 66

Who Commands the Wind?Main findings

Volatility can have tremendous impact on the market outcome

47 / 66

Who Commands the Wind?Main findings

Volatility can have tremendous impact on the market outcome

Asymmetric and surprising result

The supplier can achieve significant gains,even when the consumer commands the wind

47 / 66

Consumers command the wind

Wind generating units are commanded by the demand side:

Consumers’ and suppliers’ welfare expressions

D,W(t) = v min(Dttl(t), Gttl(t) + Gttl

− cbo max(0,−R(t))

− P (t)G(t) − p(t)gda(t)

S,W(t) =(

P (t) − crt)

G(t) +(

p(t) − cda)

gda(t)

48 / 66

Consumers command the wind

Welfare expressions: DAM/RTM decomposition

D,W(t) = Wrt

D,W(t)

vdda(t) − p(t)(dda(t) − gda

(P (t) − p(t))rda

0 + vGW(t)}

S,W(t) = Wrt

S,W(t) +(

p(t) − cda)

gda(t)

D,W(t), Wrt

S,W(t): Welfare obtained in the RTMwith residual demand Dnet(t).

49 / 66

Suppliers command the wind

The other alternative is to consider wind as part of the supply side,

Consumers’ welfare

D,W(t) = v min(Dttl(t), Gttl(t) + Gttl

− cbo max(0,−R(t))

− P (t)(G(t) + GW(t)) − p(t)(gda(t) + gda

D,W(t)

(v − p(t))dda + (P (t) − p(t))rda

+ (v − P (t))GW(t)

50 / 66

Suppliers command the wind

Suppliers’ welfare

S,W(t) =(

P (t) − crt)

G(t) + P (t)GW(t)

p(t) − cda)

gda(t) + p(t)gda

S,W(t)

p(t) − cda)

gda(t) + P (t)GW(t) + p(t)gda

The welfare gain p(t)gda

W(t) is now taken by the suppliers

51 / 66

Numerical Results: Setting the stage I

Parameters used in all experiments:

Parameters [$/MWh]

Cost of blackout and value of consumption:

cbo = 200, 000 v = 50

Cost and prices of generation: crt = 30, and

cda = 0.75 ∗ crt, pda = 0.85 ∗ crt

Discount factor: γ = 1/12 (12 hour time horizon).

52 / 66

Numerical Results: Setting the stage II

Parameters (physical)

Ramp limit: ζ = 200 MW/min

Mean demand: D̄ = 50, 000 MW

Standard deviation: σ = 500 MW.

53 / 66

Numerical Results: Setting the stage III

Penetration and volatility

Coefficient of variation to capture the relative volatility of wind:

cv =σw

E[GttlW

Percentage of wind penetration:

k = 100 ∗ E[Gttl

W(t)]/D̄

54 / 66

Numerical Results: Volatility ImpactsOptimal reserves rise with volatility

k = 10

k = 15

k = 20

0.10 0.2 0.30

80Optimal Reserve Level

55 / 66

Numerical Results: Volatility ImpactsSocial planner’s welfare falls with volatility

0 1000200 400 600 80010

ζ = 200Total Welfare

ζ =0.1

ζ = 500

56 / 66

Numerical Results: Consumers command the windConsumer welfare falls and supplier welfare rises with volatility

0 0.1 0.2 0.3 0.40 0.1 0.2 0.3 0.4

Supplier Welfare

k = 10

k = 15

k = 20

Consumer Welfare

k = 10

k = 15k = 20

57 / 66

Research Frontiers

58 / 66

Research FrontiersStrategic behavior

Building bridges

59 / 66

Building bridges between this research and what was described byHobbs & Ralph — How to incorporate dynamics and uncertainty

in a strategic setting? How could two smart people come to such

different conclusions? I had to get to the

bottom of this. –MacKay 2009

59 / 66

Approach: Follow the example of highway engineering:

Analysis of strategic behavior will be possible if agent

behavior is suitably constrained

59 / 66

Stalinist?

59 / 66

Stalinist? The economic security of the region is at stake! Expectationsto market participants must be made clear, and must be enforced!!

59 / 66

Learning

60 / 66

Learning Once a market framework is in place, agents will learnover time to optimize their reward

60 / 66

Learning Once a market framework is in place, agents will learnover time to optimize their reward Q-learning and TD-learning are

potential approaches

60 / 66

Research FrontiersControlling supply and demand

The impact of demand management and storage on the market

61 / 66

The impact of supply management and storage on the market.e.g., modern wind turbines allow pitch angle control.

61 / 66

The impact of supply management and storage on the market.e.g., modern wind turbines allow pitch angle control.

Beyond arithmetic: What are the dynamical properties of thepower grid in LA county with dynamic pricing, hybrid cars, andautomated water pumping?

61 / 66

Research FrontiersFacing volatility

Not just “missing money” —

62 / 66

Not just “missing money” — Mechanisms to reduce consumersand suppliers exposure to volatility.

62 / 66

Not just “missing money” — Mechanisms to reduce consumersand suppliers exposure to volatility.

Especially supply-side volatility with introduction of renewableenergy, such as wind

62 / 66

Research FrontiersLong-term incentives

How to create policies to protect participants on both sides of themarket, while creating incentives for R&D on renewable energy?

63 / 66

The economic notion of efficiency disregards issues such asemissions, or public safety.

63 / 66

The economic notion of efficiency disregards issues such asemissions, or public safety.

We hope that the ideas described here will form a building blockfor constructing and analyzing far more intricate models, takinginto account a broader range of issues.

63 / 66

Thanks!

Celebrating with Dutch Babies after finishing The value of volatile

resources...64 / 66

Complex Networks FOR

Sean Meyn

Control Techniques

References

65 / 66

References

I.-K. Cho and S. P. Meyn. Efficiency and marginal cost pricing in dynamic

competitive markets. To appear in J. Theo. Economics, 2010.

M. Chen, I.-K. Cho, and S. Meyn. Reliability by design in a distributed power

transmission network. Automatica, 42:1267–1281, August 2006.

I.-K. Cho and S. P. Meyn. A dynamic newsboy model for optimal reserve

management in electricity markets. Submitted for publication, SIAM J. Control

and Optimization., 2009.

S. Meyn, M. Negrete-Pincetic, G. Wang, A. Kowli, and E. Shafieepoorfard.

The value of volatile resources in electricity markets. Submitted Proc. of the

49th Conf. on Dec. and Control, Atlanta, GA, 2010.

D. J. C. MacKay. Sustainable energy : without the hot air. UIT, Cambridge,

England, 2009.

L. M. Wein. Dynamic scheduling of a multiclass make-to-stock queue.

Operations Res., 40:724–735, 1992.

66 / 66

Tutorial for Energy Systems Week - Cambridge 2010

Education

MGX3100 Tutorial week 1

Laporan Tutorial Week 1 3.2

Tutorial Edu Week 1

CSCA48 Winter 2018 Week 1: ADT - University of … Winter 2018 Week 1: ADT ... • Attend on your tutorial only • No tutorial this week. ... Assignment, Lecture, Tutorial, etc.)

VBA: Tutorial Week 1

Week 2 Tutorial Slides

CAMBRIDGE HEADLINES - Fultonschools.org · CAMBRIDGE HEADLINES Cambridge High School 11.7.19 Around Cambridge Bell Schedule Week of November 4, 2019 For the latest Cambridge News—See

Week 9 Tutorial - Ali

Tutorial Week 5.pptx

Cambridge lms students tutorial

Week 4 tutorial

Tutorial Week 5

Tutorial/HW Week #7

Tutorial 4.2 Week 4.1

CAMBRIDGE HEADLINES - fultonschools.org...Dec 17, 2019 · CAMBRIDGE HEADLINES Cambridge High School 12.17.19 Around Cambridge Bell Schedule Week of December 16—Final Exam Week

KCB207: WEEK 8 Tutorial

Cambridge NanoTech ALD Tutorial

VBA: Tutorial Week 5

CAMBRIDGE HEADLINES - Fultonschools.org€¦ · CAMBRIDGE HEADLINES Cambridge High School 11.6.19 Around Cambridge Bell Schedule Week of November 4, 2019 For the latest Cambridge

Ansys tutorial week 9