Competitive Equilibria for Stochastic Dynamic Markets

7/30/2019 Competitive Equilibria for Stochastic Dynamic Markets

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Dynamics of Power*LCCC Lund University, Sweden

Based on tutorial & panel lectures at Energy Systems Week, Cambridge UK

Sean P. Meyn

Joint work with: In-Koo Cho, Anupama Kowli, Matias Negrete-Pincetic,Ehsan Shafieeporfaard, Uday Shanbhag, and Gui Wang

Coordinated Science Laboratoryand the Department of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign, USA

Thanks to NSF, AFOSR, and DOE / TCIPG

*Dynamik i Makten (i grossistledet elmarknaderna)

May 19, 2011
http://www.newton.ac.uk/programmes/SCS/scsw07p.htmlhttp://www.newton.ac.uk/programmes/SCS/scsw07p.htmlhttp://find/http://goback/http://www.newton.ac.uk/programmes/SCS/scsw07p.html


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Outline

1 Can You Spot the Competitive Equilibrium?

2 Competitive Equilibria in Dynamic Markets

3 Coping with Uncertainty and Constraints

4 Conclusions

5 References

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Can You Spot the Competitive Equilibrium?

Spot the Competitive Equilibrium

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Competitive EquilibriumStandard economic setting

Perfect competition

Long-run setting with uncertainty:

KD(GD) = E

etWD(t) dtKS(GS) = E

etWS(t), dt

Consumers and suppliers each wish to maximize their individual welfare.Welfare functions defined with a price-process {P(t) : t 0}

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C Y S h C i i E ilib i ?
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Competitive EquilibriumStandard economic setting

Perfect competition

Long-run setting with uncertainty:

KD(GD) = E

etWD(t) dtKS(GS) = E

etWS(t), dt

Consumers and suppliers each wish to maximize their individual welfare.Welfare functions defined with a price-process {P(t) : t 0}

Price-taking assumption

Key assumption of equilibrium theory: The price of power P(t) does notdepend on the decisions of the market agents.

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Competitive EquilibriumEfficiency

Efficient Equilibrium

Social Planners Problem:

max K(G) = E etWS(t) + WD(t) dtSuppose that there exists a price process {P(t)} that forms anequlibrium: The consumers and suppliers agree, GS = GD = G.

Suppose the agreed upon decisions G solve the SPP.

Then, the market is called efficient.

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Competitive EquilibriumEfficiency


Social Planners Problem:

max K(G) = E etWS(t) + WD(t) dtSuppose that there exists a price process {P(t)} that forms anequlibrium: The consumers and suppliers agree, GS = GD = G.

Suppose the agreed upon decisions G solve the SPP.

Then, the market is called efficient.

Lets look for examples of efficient equilibria!

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p p q

Efficient Equilibrium?Australia, January 16, 2007

Price(Aus$/M

Wh)

Volume(MW)

Demand

Demand

Prices

Prices

24:00

23:00

22:00

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7,000

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5,000

4,000

3,000

2,000

1,000

0

Tasmania

Victoria

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Efficient Equilibrium?Australia, January 16, 2007

Price(Aus$/M

Wh)

Volume(MW)

Demand

Demand

Prices

Prices

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23:00

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6,000

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3,000

2,000

1,000

0

Tasmania

Victoria

Fires cause chaos Is this efficient?

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Efficient Equilibrium?Midwest ISO today: Friday afternoon, March 4, 2011 3:30 p.m.

-2000.00

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-762.55

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Efficient Equilibrium?Texas today: Winter of 2011

5am 10am 3pm 8pm500

0

1000

2000

$/MWh

February 2, 2011$/MWh

10

0

10

20

40

60

80

5am 10am 3pm 8pm

Power Prices in TexasJanuary 31, 2011

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5am 10am 3pm 8pm500

0

1000

2000

$/MWh


10

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There will be multiple autopsies of the causes for the latest power breakdowns ... Who profited

off this near-meltdown and what can be done to incentivize power producers to maintainadequate reserve capacity for emergencies rather than waiting for emergency windfalls? HOUSTON CHRONICLE, Feb 12, 2011

New report hits ERCOT, electricity deregulation: A report released Monday concludes thatelectric deregulation has cost Texas residential consumers more than $11 billion in higherrates... Dallas Morning News, Feb 14, 2011

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What do theexperts say?

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000$/MWh


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...long-term contracts could reduce market power... ...thelarger the proportion of total demand auctioned in advance, the lower areboth the contract and the average spot price of energy...

M. Soledad Arellano and Pablo Serra, 2010

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5am 10am 3pm 8pm500

0

1000

2000

000$/MWh


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0

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...long-term contracts could reduce market power... ...thelarger the proportion of total demand auctioned in advance, the lower areboth the contract and the average spot price of energy...

M. Soledad Arellano and Pablo Serra, 2010

...forward markets do not mitigate market power... ...forwardmarkets systematically enhance market power in some symmetriccapacity-constrained markets...

Frederic Murphy and Yves Smeers, 2010

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Efficient Equilibrium?New Zealand today: March 25, 2011

A typical day in the New Zealand power market on the N. Island

Stratford

Otahuhu

http://www.electricityinfo.co.nz/

0

50

100

Nodal Power Prices in NZ: $/MWh

4am 9am 2pm 7pm

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$25 million dollars extracted by the generators in just six hours

Stratford

Otahuhu

http://www.electricityinfo.co.nz/4am 9am 2pm 7pm0

10,000

20,000


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Stratford

Otahuhu

http://www.electricityinfo.co.nz/4am 9am 2pm 7pm0

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Efficient? Energy consultant Bryan Leyland said the high wholesale prices showed howdysfunctional the electricity market is.

Jacking the prices up sends no worthwhile signal to anyone it is nothing to do with ashortage of generating capacity, he said. It just exposes the nonsense of it all.

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Stratford

Otahuhu

http://www.electricityinfo.co.nz/4am 9am 2pm 7pm

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Efficient! Preliminary view of NZ Electrical Authority: Genesis was not guiltyof manipulative, ... or deceptive conduct.

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Stratford

Otahuhu

http://www.electricityinfo.co.nz/4am 9am 2pm 7pm

0

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Efficient! Preliminary view of NZ Electrical Authority: Genesis was not guiltyof manipulative, ... or deceptive conduct. However, high prices threatened to

undermine confidence in, and ... damage the integrity and reputation of the

wholesale electricity market 3:59 PM Friday May 6, 2011 www.nzherald.co.nz

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Efficient Equilibrium?Its not just about power!

Retail prices o onions in IndiaIndian Rupee per kg

Source: Ministry o Consumer Afairs

Jan Mar May July Sep Nov Jan Mar May July Sep Nov Jan

40

50

30

10

-10

20

0

Chennai

Delhi

Mumbai

Patna

2009 2010 2011

Delhi

Mumbai

Rs. 60

Rs. 55

Rs. 50Rs. 44

Rs. 37Rs. 43

Rs. 25

Rs. 13

Rs. 14

Rs. 9

Rs. 15.5

Rs. 29

Dec 20 Dec 27 Jan 10 Jan 24 Feb 7 Feb 11

Sharp Rise & Sharp Fall

Indias ban on onion exports to Nepal ... caused

wholesale price to jump 50% in a week. NepaliEconomy.com, Jan 3, 2011

Soaring onion prices pushed food inflation

again to a double-digit mark... Indian Express, Feb 12, 2011

In India, onion prices are as politically sensitive as mortgage rates are inAustralia. ... rising cost of the staple has helped change governments inIndia. Fidelity International, Jan 2011

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Competitive Equilibria in Dynamic Markets
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El i i M k T d
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Electricity Markets TodayTwo coupled markets

Day-ahead market (DAM):

Cleared one day prior to the production and delivery of energy: The ISOgenerates a schedule of generators to supply specific levels of power foreach hour over the next 24 hour period.

Facilitates the scheduling of generating units, and allows for hedging against

uncertainty

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El i i M k T d
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Electricity Markets TodayTwo coupled markets

Day-ahead market (DAM):

Cleared one day prior to the production and delivery of energy: The ISOgenerates a schedule of generators to supply specific levels of power foreach hour over the next 24 hour period.

Facilitates the scheduling of generating units, and allows for hedging against

uncertainty

Real-time market (RTM):

As supply and demand are not perfectly predictable, the RTM plays therole of fine-tuning this resource allocation process

RTM is the focus here

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RTM M d l
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RTM ModelDynamic model for reserves

Cho & Meyn model: Math model explains volatile prices in power markets. SIAM News, Robinson. 2005.

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

D(t) = Actual demand ForecastFor computation: Deviation in demand D is modeled as Brownian motion

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

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RTM M d l
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Economic Friction

Generation cannot increase instantaneously:

For all t 0 and t > t, G(t

)G(t)t t

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RTM M d l
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Economic Friction

Generation cannot increase instantaneously:

For all t 0 and t > t, G(t

)G(t)t t

In recent work we also impose lower bounds on generation, as well asnetwork constraints.

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Market Analysis: A beautiful world


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Market Analysis: A beautiful world...Perfect competition

Dynamic market equilibria under the most ideal circumstances:

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Price manipulation is excluded

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Price manipulation is excludedNo ENRON games no market power

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Externalities are disregardedNo government mandates

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When wind generation is included, the consumers own and operatethese resources

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When wind generation is included, the consumers own and operatethese resources

What does an efficient equilibrium look like?

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Market Analysis
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Market AnalysisSecond Welfare Theorem


max K(G) = E

etWS(t) + WD(t)

dt

.

s.t. GS(t) = GD(t) for all t

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Market Analysis
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max K(G) = E

etWS(t) + WD(t)

dt

.


Special case: Welfare functions are piecewise linear,

WS(t) := P(t)GS(t) cGS(t)

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

cbo max(0,RD(t)) P(t)GD(t)

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Market Analysis
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max K(G) = E

etWS(t) + WD(t)

dt


Key component of equilibrium theory: Perfect competition

Price-taking assumption:The price of power P(t) in the RTM is assumed to be exogenous (it doesnot depend on the decisions of the market agents).

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Market Analysis
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Second Welfare Theorem Lagrangian Decomposition

max K(G) = maxGS

E

etWS(t) + (t)GS(t)

dt

+ maxGD E

etWD(t) (t)GD(t)

dt

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Market Analysis
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ySecond Welfare Theorem


max K(G) = maxGS

E

etWS(t) + (t)GS(t)

dt

+ maxGD E

etWD(t) (t)GD(t)

dt

Assume: Social planners problem has a solution,and there is no duality gap.

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Market Analysis
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max K(G) = maxGS

E

etWS(t) + (t)GS(t)

dt

+ maxGD E

etWD(t) (t)GD(t)

dt


Then P(t) = P(t) + (t) provides an efficient equilibrium.

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Market Analysis
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max K(G) = maxGS

E

etWS(t) + (t)GS(t)

dt

+ maxGD E

etWD(t) (t)GD(t)

dt


Then P(t) = P(t) + (t) provides an efficient equilibrium.


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qAnswer: Marginal value

Equilibrium price

The equilibrium price process is a function of equilibrium reserves:

P(t) = p(Re(t))

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Answer: Marginal value

Equilibrium price


P(t) = p(Re(t))

P(t) = p(Re(t)) is the marginal value of power to the consumer

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Answer: Marginal value

Equilibrium price


P(t) = p(Re(t))

P(t) = p(Re(t)) is the marginal value of power to the consumer

For linear cost/utility, marginal value is piecewise constant,

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

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Market Equilibrium
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Price dynamics

v + cbo

0

r

Prices

Reserves

Normalized demand

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

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Market Equilibrium
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Price dynamics

v + cbo

0

r

Prices

Reserves

Normalized demand

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

Smoother prices obtained when cost/utility are strictly convex

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Familiar, right?
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Real-world price dynamics

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Sustainable business?
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Marginal value of electricitymay be

v + cbo

= $100,000/MWh!

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v + cbo

= $100,000/MWh!

However,

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v + cbo

= $100,000/MWh!

However,

Theorem 2: In this equilibrium, the average price isprecisely the marginal cost c.Proof: Lagrangian relaxation of initial condition.

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v + cbo

= $100,000/MWh!

However,

Theorem 2: In this equilibrium, the average price isprecisely the marginal cost c.Proof: Lagrangian relaxation of initial condition.

Is this a sustainable business?

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Coping with Uncertainty and Constraints
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Network ConstraintsEntropic prices


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Entropic prices

What is marginal value?

It is not always obvious. With the introduction of network constraints,

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Entropic prices



Prices can go well beyond marginal value

Prices can go well below zero See [Wang et. al., 2011]

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Entropic prices



Prices can go well beyond marginal value

Prices can go well below zero See [Wang et. al., 2011]

Without price-caps, Australia might look like an efficient equilibrium:

Price(Aus$/

MWh)

Price(Aus$/M

Wh)

Volume(M

W)

Volume(MW)

Demand

Demand

Prices

Prices

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3,000

2,000

1,000

0

TasmaniaVictoria

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The Value of Volatile ResourcesWhat is the impact of wind solar or tidal generation?


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What is the impact of wind, solar, or tidal generation?

Requires consideration of coupled RTM and DAM. In summary:

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The Value of Volatile ResourcesWhat is the impact of wind solar or tidal generation?
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cv0 0.1 0.2 0.3 0.4

k = 0

k = 5

k = 10

k = 15

k = 20Optimal Reserve Level

0

20

40

60

80

100

120

140x 10

3

Optimal reserves rise:

Optimal reserves increase with in-creasing penetration k, or coefficientof variation cv.

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The Value of Volatile ResourcesWhat is the impact of wind, solar, or tidal generation?


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cv0 0.1 0.2 0.3 0.4

k = 0

k = 5

k = 10

k = 15

k = 20Optimal Reserve Level

0

20

40

60

80

100

120

140x 10

3

Optimal reserves rise:

Optimal reserves increase with in-creasing penetration k, or coefficientof variation cv.

Social welfare falls with wind.

[Value of Volatile Resources, CDC, 2010]

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The Value of Volatile ResourcesDistribution of welfare (consumers command wind resources)
https://netfiles.uiuc.edu/meyn/www/spm_files/MarketsIFAC2011/MarketsIFAC2011.htmlhttps://netfiles.uiuc.edu/meyn/www/spm_files/MarketsIFAC2011/MarketsIFAC2011.htmlhttps://netfiles.uiuc.edu/meyn/www/spm_files/MarketsIFAC2011/MarketsIFAC2011.htmlhttp://find/


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( )

With increased volatility: Consumer welfare falls, supplier welfare rises:

0 0.1 0.2 0.3 0.40 0.1 0.2 0.3 0.4

x 106

Supplier Welfare

k = 0

k = 5

k = 10

k = 15

k = 20

x 106

Consumer Welfare

10

0

5

15

k = 0

k = 5

k = 10

k = 15k = 20

3

4

5

cv

Penetration k, coefficient of variation cv .

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Conclusions
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Conclusions

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Conclusions

Economics and KontrollDistributed control and prices
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The current RTM paradigm is doomed to fail:

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Conclusions

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In the competitive equilibrium nirvana:Volatile prices combined with low average-prices will halt investment.

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Conclusions

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In the real world?Strategic behavior can lead to a new crisis each year!

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Conclusions

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However,

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Conclusions

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However,

No one of us has all the answers!

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Conclusions

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However,


Collaboration is required Between researchers in economics, systemssciences, statistical sciences, and power engineers.

How could two smart people come to such different conclusions?I had to get to the bottom of this. MacKay 2009

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Conclusions

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However,


Collaboration is required Between researchers in economics, systemssciences, statistical sciences, and power engineers.

How could two smart people come to such different conclusions?I had to get to the bottom of this. MacKay 2009

A gentle debate. Please, no shouting!

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Conclusions

Economics and KontrollEconomics for an Entropic Grid
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How to create policies to protect participants on both sides of the market,

while creating incentives for R&D on renewable energy?

Our community must consider long-term planning and policy, along with

traditional systems operations

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Conclusions

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Planning and Policy, includes Markets & Competition

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Conclusions

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Evolution?

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Conclusions

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Evolution? Too slow!

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Conclusions

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Evolution? Too slow!

What we need is Intelligent Design

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Conclusions

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Real-world issues: Incorporate dynamics and uncertainty in a strategic

setting

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Conclusions

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setting

Beyond efficiency: The economic notion of efficiency disregards issuessuch as reliability, emissions, public acceptance, or public safety.

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Conclusions

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setting


Long-term incentives: How to create policies to protect participants on

both sides of the market, while creating incentives for R&D on renewableenergy?

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Conclusions

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setting




Approach: Follow the example of highway engineering

Analysis of strategic behavior will be possible,

if agent behavior is suitably constrained

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setting




Approach: Follow the example of highway engineering

Analysis of strategic behavior will be possible,

if agent behavior is suitably constrained

The economic security of the region is at stake:We need well-designed lanes and speed limits in the energy highway!

36/40 Conclusions

Concluding Remarks


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An Entropic Grid may emerge as a result of many of the proposed

Smart Grid initiativesDecision & control, simulation & learning provide tools that will helpto design an intelligent grid

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Concluding Remarks
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We must move beyond the traditional static competitive equilibrium

analysis

We must move beyond the traditional myopic goal of economicefficiency, taking into account externalities such as sustainabilityand reliability

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Concluding Remarks
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analysis


The best architecture for tomorrows energy highway?This is the central open question.

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Concluding Remarks


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analysis


The best architecture for tomorrows energy highway?This is the central open question.

Its solution opens many exciting research challenges!

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Thanks!
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Celebrating with Dutch Babies after finishing part of this work

38/40 References

Control Techniques

Pre-publication version for on-line viewing . Monograph available for purchase at your favorite retailerMore information available at http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521884419
http://find/https://netfiles.uiuc.edu/meyn/www/spm_files/book.htmlhttps://netfiles.uiuc.edu/meyn/www/spm_files/CTCN/CTCN.html


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FOR

Complex Networks

Sean Meyn

References

39/40 References

References
https://netfiles.uiuc.edu/meyn/www/spm_files/book.htmlhttps://netfiles.uiuc.edu/meyn/www/spm_files/book.htmlhttps://netfiles.uiuc.edu/meyn/www/spm_files/CTCN/CTCN.htmlhttp://find/


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

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Competitive Equilibria for Stochastic Dynamic Markets