Risk and Uncertainty modeling with application in energy systems

Decision Making Under Uncertainty
By: Alireza Soroudi
Alireza.soroudi@ucd.ie
03/23/15
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/ 1

Topics to be covered in this seminar:
03/23/15 2
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/

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Introduction

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)
03/23/15 7Alireza.soroudi@ucd.ie

8
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

9
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?
PDF

11
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?

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Head
Tail
100$
0$
Number * 10$
The money you earn ?

Dealing with Uncertainties
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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.

03/23/15
Energy Hub Management with Intermittent Wind Power
A Soroudi, B Mohammadi-Ivatloo, A Rabiee, Large Scale Renewable Power Generation, 413-438
15
Risk measures in stochastic techniques

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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/;
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
282
23

03/23/15 25
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

500MW
Hydro power
Wind
power
Thermal
power
T
Thermal
power
T
600M
W
720MW
720MW
720M
W
720M
W
P
600MW
600MW
Thermal
power T
T Thermal
power
H
Pool
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
03/23/15 28
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 30

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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• How cool is 36 F° ?
• It is 30% Cool and 70% Freezing
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Fuzzy Logic
33
0.7
0.3

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http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/ 34/41
Fuzzy Control
A B

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Fuzzy Control
A B

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36/41
Fuzzy Control
http://www.mathworks.com/help/pdf_doc/fuzzy/fuzzy.pdf

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Fuzzy Control

Fuzzy techniques
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Possibilistic evaluation of distributed generations impacts on distribution netw
, A Soroudi, M Ehsan, R Caire, N Hadjsaid
Power 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 optimization
D Bertsimas, DB Brown, C Caramanis - SIAM review, 2011 - SIAM
Aharon Ben-Tal
Arkadi Nemirovski
Dimitris Bertsimas
The Price of Robustness
Dimitris Bertsimas and Melvyn Sim, Operations Research, Vol. 52,
No. 1 (Jan. - Feb., 2004), pp. 35-53

03/23/15 40
Min y=f(u,x)
G(u,x)<=0
H(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)
03/23/15
• 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
47
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
V
α
−
≤
(1 )Vα− (1 )Vα+V

03/23/15 49
Distance
Risk Seeker strategyRisk Seeker strategy
V V
V
α
−
≤
(1 )Vα− (1 )Vα+V

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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Risk and Uncertainty modeling with application in energy systems

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