Risk and Uncertainty modeling with application in energy systems

Decision Making Under Uncertainty

By: Alireza Soroudi

[email protected]

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


Robustness and Opportuneness

Uncertainty

Undesired

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Favorable


Uncertain events

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• Weather changes – Solar radiation – Wind speed

• Load values • Market prices • Gas network failures


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


http://en.wikipedia.org/wiki/Image:Christiaan_Huygens-painting.jpeg

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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:




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


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?

http://scholar.google.com/citations?view_op=view_citation&hl=en&user=w74f9yAAAAAJ&citation_for_view=w74f9yAAAAAJ:roLk4NBRz8UC

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Head

Tail

100$

0$

Number * 10$

The money you earn ?


Dealing with Uncertainties


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)

http://scholar.google.com/citations?view_op=view_citation&hl=en&user=w74f9yAAAAAJ&citation_for_view=w74f9yAAAAAJ:8k81kl-MbHgC

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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.


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


http://scholar.google.com/citations?view_op=view_citation&hl=en&user=w74f9yAAAAAJ&sortby=pubdate&citation_for_view=w74f9yAAAAAJ:orDZ08hpP44C

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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


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


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


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


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"


• How cool is 36 F° ?

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• How cool is 36 F° ?• It is 30% Cool and 70% Freezing

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0.7

0.3


03/23/[email protected]

http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/ 34/41

Fuzzy Control

A B



Fuzzy Control

A B


http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/36/41

Fuzzy Control

http://www.mathworks.com/help/pdf_doc/fuzzy/fuzzy.pdf



Fuzzy Control


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

http://scholar.google.com/citations?view_op=view_citation&hl=en&user=w74f9yAAAAAJ&citation_for_view=w74f9yAAAAAJ:_FxGoFyzp5QC

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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Robust optimization

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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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Supply

Demand

Upstream network

losses

Energy



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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]



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




Engineering

Risk and Uncertainty modeling with application in energy systems