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Decision Making Under Uncertainty
By: Alireza Soroudi
03/23/15 [email protected]://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/ 1
Topics to be covered in this seminar:
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Introduction
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Introduction
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is the chance, within a specified time frame, of an adverse event with specific (negative) consequences
Risk
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Robustness and Opportuneness
Uncertainty
Undesired
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Favorable
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Uncertain events
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• Weather changes – Solar radiation – Wind speed
• Load values • Market prices • Gas network failures
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Decison Makers
• Policy makers / regulators• Indepandent System Operators • Gencos (self scheduling problem) • DNO (DG units, demand ,… ) • Aggregators (energy procurement) • Prosumers (demand response)
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Christiaan Huygens
•Pierre de Fermat, Blaise Pascal, and Christiaan Huygens gave the earliest known scientific treatment of probability. Blaise Pascal
Pierre de Fermat Jacob Bernoulli
Stochastic techniques
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Game of flipping a coin:
Let’s flip the coin one hundred times and count how many heads or Tails.
What are the results ?
Heads: Tails:
Stochastic techniques
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Stochastic techniques
General representation :
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Min y=f(U,x)
Where
• X is the control vector {decision variable set}• U is the input uncertain parameter vector
• Can we obtain the pdf of y knowing the PDF of U?• Can we optimize this PDF using X?
PD
F
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Monte Carlo Simulation Model Output
Ui : Uncertain inputs
Input
U1
U2
…
U3
…1 2 n
…
U4
Uk
y
( , )y f x U= r
)(yp
Stochastic techniques
Probabilistic dynamic multi-objective model for renewable and non-renewable distributed generation planning, A Soroudi, R Caire, N Hadjsaid, M Ehsan,IET generation, transmission & distribution 5 (11), 1173-1182
•Can we obtain the pdf of y knowing the PDF of x?
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Head
Tail
100$
0$
Number * 10$
The money you earn ?
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Dealing with Uncertainties
Stochastic techniques
03/23/15 Binary PSO-based dynamic multi-objective model for distributed generation planning under uncertainty, alireza soroudi, mozhgan afrasiab, Renewable Power Generation, IET 6 (2), 67-78 13
Scenario based optimization
Min y=f(U,x)
y=f(Us,x)
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Risk Measures
Conejo, Antonio J., Miguel Carrión, and Juan M. Morales. Decision making under uncertainty in electricity markets. Vol. 153. Springer, 2010.
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03/23/15Energy Hub Management with Intermittent Wind PowerA Soroudi, B Mohammadi-Ivatloo, A Rabiee, Large Scale Renewable Power Generation, 413-438 15
Risk measures in stochastic techniques
Dealing with Uncertainties
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Stochastic techniques Multi-stage Scenario based decision making
Suppose a newsboy wants to maximize his profit . He has to decide how many newspapers to buy from a distributor to satisfy demand .
d Demand
S Units sold
left-over newspapers are stored in an inventory at a holding cost of h per unit.
I Units stored
X buy
Profit.. Z =e= v*S - c*X - h*I - p*L;Row1.. d =e= S + L;Row2.. I =e= X - S;
distributor
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D=60 0.3
D=63 0.1
D=68 0.1
D=40 0.1
D=80 0.1
D=10 0.3
X
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c Purchase costs per unit /30/p Penalty shortage cost per unit / 5 /h Holding cost per unit leftover /10/v Revenue per unit sold /60/d Demand /63/;
Stochastic techniques Multi-stage Scenario based decision making
Demand=63 X=63 bought
X=60 bought
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D=60 0.3
D=63 0.1
D=68 0.1
D=40 0.1
D=80 0.1
D=10 0.3
X
, Exp(profit)= 594.500
Expected Value
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Variance
D=60 0.3
D=63 0.1
D=68 0.1
D=40 0.1
D=80 0.1
D=10 0.3
X
X=14 , Exp(profit)= 113
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D=60 0.3
D=63 0.1
D=68 0.1
D=40 0.1
D=80 0.1
D=10 0.3
X
Shortfall Probability
=500
X=26 , Exp(profit)=23
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D=60 0.3
D=63 0.1
D=68 0.1
D=40 0.1
D=80 0.1
D=10 0.3
X
Expected shortage
=500
X=23 , Exp(profit)=282
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D=60 0.3
D=63 0.1
D=68 0.1
D=40 0.1
D=80 0.1
D=10 0.3
X
Value at risk
X=43 , Exp(profit)=509.5
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D=60 0.3
D=63 0.1
D=68 0.1
D=40 0.1
D=80 0.1
D=10 0.3
X
CVaR
X=41 , Exp(profit)=499.5
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Risk – Profit tradeoff
594.5
499.5
509.5
113
28223
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Risk – Profit tradeoff
Stochastic Real-Time Scheduling of Wind-Thermal Generation Units in an Electric Utility
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Soroudi, A.; Rabiee, A.; Keane, A., "Stochastic Real-Time Scheduling of Wind-Thermal Generation Units in an Electric Utility," Systems Journal, IEEE , vol.PP, no.99, pp.1,10
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500M
W
Hydro power
Windpower
Thermal power
T
Thermal power
T
600MW
720MW
720MW
720M
W
720M
W
P
600MW
600MW
Thermal power T
T Thermal power
HPool
power
Soroudi, A.; Rabiee, A., "Optimal multi-area generation schedule considering renewable resources mix: a real-time approach," Generation, Transmission & Distribution, IET , vol.7, no.9, pp.1011,1026, Sept. 2013
Dealing with Uncertainties
Fuzzy techniques
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Lotfi Aliaskerzadeh
• A fuzzy set is a set whose elements have degrees of membership.
• Full membership : 100%
• Partial membership : 0% - 100%
Boolean Sets Fuzzy Sets
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Fuzzy techniques
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Crisp (Traditional) Variables• Crisp variables represent precise quantities:
– x = 9.989999– Binary numbers ∈{0,1}
• A proposition is either True or False– A ⇒ B– A ∧ B ⇒ D
• A natural number is either even or odd– 2 ∈{even}– 3 ∈{odd}
03/23/15 Fuzzy Logic [email protected]://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Membership Functions
• Temp: {Freezing, Cool, Warm, Hot}• Degree of Truth or "Membership"
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• How cool is 36 F° ?
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Membership Functions
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Membership Functions
• How cool is 36 F° ?• It is 30% Cool and 70% Freezing
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0.7
0.3
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Fuzzy Control
A B
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Fuzzy Control
A B
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Fuzzy Control
http://www.mathworks.com/help/pdf_doc/fuzzy/fuzzy.pdf
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Fuzzy Control
Dealing with Uncertainties
Fuzzy techniques
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Possibilistic evaluation of distributed generations impacts on distribution networks, A Soroudi, M Ehsan, R Caire, N HadjsaidPower Systems, IEEE Transactions on 26 (4), 2293-2301
Robust optimization
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“The decision-maker constructs a solution that is optimal for any realization of the uncertainty in a given set”
Theory and applications of robust optimizationD Bertsimas, DB Brown, C Caramanis - SIAM review, 2011 - SIAM
Aharon Ben-TalArkadi Nemirovski
Dimitris Bertsimas
The Price of RobustnessDimitris Bertsimas and Melvyn Sim, Operations Research, Vol. 52, No. 1 (Jan. - Feb., 2004), pp. 35-53
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Min y=f(u,x)G(u,x)<=0H(u,x) =0
Robust optimization
u
Umin< Ui< Umax
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A Soroudi , Robust optimization based self scheduling of hydro-thermal Genco in smart grids, Energy 61, 262-271
Robust optimization (Example)
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Dealing with Uncertainties
Information Gap Decision Theory (IGDT)
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• Soroudi, A.; Ehsan, M., "IGDT Based Robust Decision Making Tool for DNOs in Load Procurement Under Severe Uncertainty," Smart Grid, IEEE Transactions on , vol.4, no.2, pp.886,895, June 2013
• Rabiee, A.; Soroudi, A.; Keane, A., "Information Gap Decision Theory Based OPF With HVDC Connected Wind Farms," Power Systems, IEEE Transactions on , vol.PP, no.99, pp.1,11 , doi: 10.1109/TPWRS.2014.2377201
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Yakov Ben-Haim
Yakov Ben-Haim, 2006, Info-Gap Decision Theory: Decisions Under Severe Uncertainty, 2nd edition, Academic Press, London, ISBN 0-12-373552-1.
An info-gap is the difference between what is known and what needs to be known in order to make a reliable and responsible decision.
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Distance
Risk Averse strategyRisk Averse strategy
V V
Vα
−≤
(1 )Vα− (1 )Vα[email protected]
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Distance
Risk Seeker strategyRisk Seeker strategy
V V
Vα
−≤
(1 )Vα− (1 )Vα[email protected]
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Dealing with Uncertainties
IGDT Example:
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Risk averse Strategy (Example 1)
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Opportuneness function
Risk seeker Strategy (Example 1)
The profit have a chance to reach
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The probability that the project will be completed within the critical time is
The customer demands that the task complete within duration tc with probability no less than Pc.
(Example 2)
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Risk averse Strategy (Example 2)
The question is : How to find the best decision q that P is always more than Pc ?
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Risk seeker Strategy (Example 2)
The question is : How to find the best decision q that P has the chance to be more than Po ?
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(Example 3)
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(Example 3)
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Transmission network
Electric Demand
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RARA RSRS
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Robustness / Opportuneness
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Robustness/opportuneness costs
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Generation strategy
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Wind curtailment
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Sensitivity analysis
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Verification Analysis
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C∆
O∆
Robustness / Opportunity Regions
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