Optimization under uncertaintyegon.cheme.cmu.edu/ewo/docs/EWO_Seminar_10_02_2014.pdfOptimization under uncertainty Antonio J. Conejo The Ohio State University 2014 Contents •Stochastic

Optimization under uncertainty

Antonio J. Conejo

The Ohio State University

2014

Contents

• Stochastic programming (SP)

• Robust optimization (RO)

• Power system applications

A. J. Conejo, The Ohio State University 2October 2, 2014

Stochastic Programming (SP)

October 2, 2014 A. J. Conejo, The Ohio State University 3

SP

• References

• Two-stage stochastic programming

• Decision framework

• Decision tree

• Scenario formulation

• Node formulation

• EVPI

• VSS


SP References

— J. R. Birge and F. Louveaux. “Introduction to StochasticProgramming”. Springer, New York, 1997.

— J. L. Higle. Tutorials in Operations Research, INFORMS 2005.Chapter 2: “Stochastic Programming: Optimization WhenUncertainty Matters”. INFORMS, Hanover, Maryland, 2005.

— A. J. Conejo, M. Carrión, J. M. Morales, “Decision MakingUnder Uncertainty in Electricity Markets”, Springer, New York.2010, Chapters 2,3 and 4.


http://link.springer.com/book/10.1007/978-1-4419-7421-1/page/1

Two-stage SP

0,,

0,, subject to

,,O minimize

yxg

yxh

yxf x,y

Expectation CVaR

A. J. Conejo, The Ohio State University 6

Stochasticvector

October 2, 2014

Decision framework

— Decisions x are made (here & now)

— Stochastic vector λ realizes in a scenario λi

— Given x, decisions yi(x,λ) are made for each realization of λ, λi (wait & see)


Decision tree


Scenario 1

Scenario i

Scenario n

1st stage decisionshere & now

2nd stage decisionswait & see

Recourse

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Scenario formulation

low

average

high

33

22

1

1

Realization

Probability


Scenario formulation

321

3

1

,,,,,

3,2,1 0,,

3,2,1 0,, subject to

,, minimizeS

321321

xxx

iyxg

iyxh

yxfZ

iii

iii

iii

i

iyyyxxx


Here & NowNon-anticipativity

Expectation

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Non-anticipativity constraints

321 xxx


Decisions cannot depend on the unknown future!

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Node formulation

3,2,1 0,,


,, minimize3

1

,,, 321

iyxg

iyxh

yxfZ

ii

ii

ii

i

i

S

yyyx

Non-anticipativity constraints are implicit!


Just one x

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EVPI

Expected

Value of the

Perfect

Information

Measure of the value of “perfect” information


EVPI

3,2,1 0,,


,, minimize3

1

,,,,, 321321

iyxg

iyxf

yxfZ

iii

iii

iii

i

i

P

yyyxxx

No non-anticipativity constraints:

we perfectly foresee the future


EVPI

PS ZZ EVPI

EVPI is non-negative


VSS(only expectation)

Value of the

Stochastic

Solution

Measure of the relevance (gain) of using astochastic approach


VSS

D

avg

avg

avg

x

,y,xg

,y,xh

,y,xf

Solution

subject to

maximize yx,


Average!

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VSS


3,2,1 0,,


,, minimize3

1

,, 321

iyxg

iyxh

yxfZ

ii

D

ii

D

ii

D

i

i

D

yyy

This problem decomposes by scenario

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VSS

3

1

,,i

ii

D

i

D yxfZ


We evaluate the “deterministic” solution

in all scenarios

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VSS

SD ZZ VSS

VSS is non-negative


Robust Optimization

October 2, 2014 A. J. Conejo, The Ohio State University 21

Outline

• Why Robust Optimization (RO)?

• RO without recourse

• RO with recourse

• Scheduling energy and reserve


Why RO?


References

— Bertsimas D., Brown D.B., Caramanis C. Theory andapplications of robust optimization. SIAM Rev., vol. 53, pp.464–501, 2011.

— Bertsimas D., Sym M., Robust discrete optimization andnetwork flows, Math. Program, Ser. B, vol. 98, no. 13, pp. 49–71, 2003.

— Bertsimas D., Litvinov E., Sun X. A., Zhao J., Zheng T. Adaptiverobust optimization for the security constrained unitcommitment problem. IEEE Transactions on Power Systems, inpress, 2012.


RO without recourse


Uncertainty set

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RO without recourse


RO without recourse

Deterministic problem Robust counterpart

LP Larger LP

MILP Larger MILP

NLP Larger NLP


Under certain conditions over the robust set:

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RO with recourse

• Make scheduling decisions (min)

• Uncertainty realizes (max)

• Make operation (recourse) decisions (min)


RO with recourse


RO with recourse: Example


RO with recourse

• Make scheduling decisions “x” with a prognosis of the future

• The uncertainty “w” realizes

• Make operation (recourse) decisions “y”


Power system applications


ISO

• ISO market clearing: large-scale, stochastic?

Maximize Expected Social Welfare

subject to:

Market equilibrium

Producer constraints

Consumer constraints


Producer

• Offering by non-strategic producers: stochastic

Maximize Expected Profit

subject to:



Stochastic producer

• Offering by non-dispatchable producers: stochastic


subject to:



Producer

• Futures market involvement (forward contracts and options)


subject to:


Contracting constraints


Producer

• Insurances

If selling through forward contracts and the production units fail… an insurance is advisable


Consumer

• Consumer energy procurement

Maximize Expected Cost

subject to:

Consumer constraints

Contracting constraints


Producer• Capacity investment by non-dispatchable producers


UPPER LEVELMaximize Profit from Wind Generation

LOWER LEVELMaximize SW

MARKET CLEARING 1

MARKET CLEARING 2

MARKET CLEARING N

…

INVESTMENT DECISIONS

LMPs

Different load and wind conditions!

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TSO

• Transmission capacity investment


TSO• Transmission capacity investment


Social Welfare Maximization(Market Clearing)

Trade Maximization

subject toLines built

Upper-Level

Lower-Level

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ISO

• Transmission maintenance


Social Welfare Maximization(Market Clearing)

Security Maximization

subject to Lines in maintenance

Upper-Level

Lower-Level

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Conclusions

(Electrical) Energy problems are important!


Conclusions

How many coal plants are currently being built in planet Earth?


ConclusionsIf renewables are considered:

– No such thing in the past (just demand uncertainty)

– No such thing in models for industry (production facilities are generally deterministic)

• Major uncertainty: stochastic production facilities



– Spatial correlations (i) among production facilities, (ii) among demands, and (iii) among demands and production facilities.

– Temporal correlations for demands and production facilities

• Complex uncertainty:multiple dependencies



– No two-stage stochastic models

– No adaptive robust optimization

• Multi-stage modeling is a must: future investment cost in stochastic sources is highly uncertain: the technology is not mature



Documents

Optimization under uncertaintyegon.cheme.cmu.edu/ewo/docs/EWO_Seminar_10_02_2014.pdfOptimization under uncertainty Antonio J. Conejo The Ohio State University 2014 Contents •Stochastic