Optimization under uncertainty
Antonio J. Conejo
The Ohio State University
2014
Contents
• Stochastic programming (SP)
• Robust optimization (RO)
• Power system applications
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Stochastic Programming (SP)
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SP
• References
• Two-stage stochastic programming
• Decision framework
• Decision tree
• Scenario formulation
• Node formulation
• EVPI
• VSS
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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.
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Two-stage SP
0,,
0,, subject to
,,O minimize
yxg
yxh
yxf x,y
Expectation CVaR
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Stochasticvector
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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)
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Decision tree
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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
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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
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Here & NowNon-anticipativity
Expectation
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Non-anticipativity constraints
321 xxx
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Decisions cannot depend on the unknown future!
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Node formulation
3,2,1 0,,
3,2,1 0,, subject to
,, minimize3
1
,,, 321
iyxg
iyxh
yxfZ
ii
ii
ii
i
i
S
yyyx
Non-anticipativity constraints are implicit!
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Just one x
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EVPI
Expected
Value of the
Perfect
Information
Measure of the value of “perfect” information
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EVPI
3,2,1 0,,
3,2,1 0,, subject to
,, minimize3
1
,,,,, 321321
iyxg
iyxf
yxfZ
iii
iii
iii
i
i
P
yyyxxx
No non-anticipativity constraints:
we perfectly foresee the future
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EVPI
PS ZZ EVPI
EVPI is non-negative
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VSS(only expectation)
Value of the
Stochastic
Solution
Measure of the relevance (gain) of using astochastic approach
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VSS
D
avg
avg
avg
x
,y,xg
,y,xh
,y,xf
Solution
subject to
maximize yx,
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Average!
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VSS
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3,2,1 0,,
3,2,1 0,, subject to
,, 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
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We evaluate the “deterministic” solution
in all scenarios
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VSS
SD ZZ VSS
VSS is non-negative
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Robust Optimization
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Outline
• Why Robust Optimization (RO)?
• RO without recourse
• RO with recourse
• Scheduling energy and reserve
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Why RO?
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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.
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RO without recourse
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Uncertainty set
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RO without recourse
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RO without recourse
Deterministic problem Robust counterpart
LP Larger LP
MILP Larger MILP
NLP Larger NLP
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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)
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RO with recourse
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RO with recourse: Example
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RO with recourse
• Make scheduling decisions “x” with a prognosis of the future
• The uncertainty “w” realizes
• Make operation (recourse) decisions “y”
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Power system applications
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ISO
• ISO market clearing: large-scale, stochastic?
Maximize Expected Social Welfare
subject to:
Market equilibrium
Producer constraints
Consumer constraints
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Producer
• Offering by non-strategic producers: stochastic
Maximize Expected Profit
subject to:
Producer constraints
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Stochastic producer
• Offering by non-dispatchable producers: stochastic
Maximize Expected Profit
subject to:
Producer constraints
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Producer
• Futures market involvement (forward contracts and options)
Maximize Expected Profit
subject to:
Producer constraints
Contracting constraints
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Producer
• Insurances
If selling through forward contracts and the production units fail… an insurance is advisable
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Consumer
• Consumer energy procurement
Maximize Expected Cost
subject to:
Consumer constraints
Contracting constraints
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Producer• Capacity investment by non-dispatchable producers
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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
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TSO• Transmission capacity investment
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Social Welfare Maximization(Market Clearing)
Trade Maximization
subject toLines built
Upper-Level
Lower-Level
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ISO
• Transmission maintenance
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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!
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Conclusions
How many coal plants are currently being built in planet Earth?
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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
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ConclusionsIf renewables are considered:
– 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
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ConclusionsIf renewables are considered:
– 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
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