Robust optimization based decision making in Energy systems

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

Robust Optimization application in Smart Energy Systems

By: Alireza Soroudi

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9/6/2016 1

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http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 2

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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Uncertain events

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• Weather changes – Solar radiation – Wind speed

• Load values • Market prices • Gas network failures

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Introduction

Stochastic

Fuzzy arithmetic

Robust optimization

Information gap decision theory

Power system applications

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Introduction

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Introduction

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Introduction

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Uncertainty modelling tools

Stochastic

Fuzzy arithmetic

Robust optimization


9/6/2016 8

Min y=f(u,x)

G(u,x)<=0

H(u,x) =0

Scenarios

Stochastic

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http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/ 9

Monte Carlo Simulation Model Output

Ui : Uncertain inputs

Input

U1

U2

…

U3

…1 2 n

…

U4

Uk

y

( , )y f x U

)(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

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

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Wind uncertainty modelling

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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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9/6/2016 11

Min y=f(u,x)

G(u,x)<=0

H(u,x) =0

Fuzzy Arithmetic Stochastic

Fuzzy arithmetic

Robust optimization


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Fuzzy Arithmetic Stochastic

Fuzzy arithmetic

Robust optimization


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9/6/2016 13

Min y=f(u,x)

G(u,x)<=0

H(u,x) =0

Robust Optimization Stochastic

Fuzzy arithmetic

Robust optimization


U

Uncertainty set

U𝑼𝟏

𝑼𝟐

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9/6/2016 14

Min y=f(u,x)

G(u,x)<=0

H(u,x) =0

Robust Optimization Stochastic

Fuzzy arithmetic

Robust optimization


A. J. Conejo, J. M. Morales and L. Baringo, "Real-Time Demand Response

Model," in IEEE Transactions on Smart Grid, vol. 1, no. 3, pp. 236-242,

Dec. 2010.

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9/6/2016 15

Min y=f(u,x)

G(u,x)<=0

H(u,x) =0

IGDT Stochastic

Fuzzy arithmetic

Robust optimization


U

Uncertainty set

𝜶

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0

𝛼

𝛼𝜶𝒎𝒂𝒙

Maximum possible

uncertainty

IGDT

Maximum tolerable

uncertainty based on 𝛽

Risky

regionSafe

region

0 ≤ 𝛼 ≤ 𝛼𝑚𝑎𝑥

Prediction

techniques

≤ 𝛼

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9/6/2016 17

Min y=f(u,x)

G(u,x)<=0

H(u,x) =0

IGDT Stochastic

Fuzzy arithmetic

Robust optimization


• K. Zare, M. P. Moghaddam and M. K. Sheikh-El-Eslami, "Risk-Based Electricity Procurement for Large

Consumers," in IEEE Transactions on Power Systems, vol. 26, no. 4, pp. 1826-1835, Nov. 2011.

• A. Soroudi and M. Ehsan, "IGDT Based Robust Decision Making Tool for DNOs in Load Procurement

Under Severe Uncertainty," in IEEE Transactions on Smart Grid, vol. 4, no. 2, pp. 886-895, June 2013.

• A. Rabiee, A. Soroudi and A. Keane, "Information Gap Decision Theory Based OPF With HVDC

Connected Wind Farms," in IEEE Transactions on Power Systems, vol. 30, no. 6, pp. 3396-3406, Nov.

2015.

• S. Shafiee; H. Zareipour; A. M. Knight; N. Amjady; B. Mohammadi-Ivatloo, "Risk-Constrained Bidding

and Offering Strategy for a Merchant Compressed Air Energy Storage Plant," in IEEE Transactions on

Power Systems , vol.PP, no.99, pp.1-1

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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-TalArkadi Nemirovski

Dimitris Bertsimas

The Price of RobustnessDimitris Bertsimas and Melvyn Sim, Operations Research, Vol. 52,

No. 1 (Jan. - Feb., 2004), pp. 35-53

http://epubs.siam.org/doi/abs/10.1137/080734510

http://scholar.google.com/citations?user=prKmkzMAAAAJ&hl=en&oi=sra

http://scholar.google.com/citations?user=LUFYHREAAAAJ&hl=en&oi=sra

http://scholar.google.com/citations?user=47YTUrEAAAAJ&hl=en&oi=sra

https://www.google.ie/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCoQFjAA&url=http://web.mit.edu/dbertsim/www/&ei=1l5eU5tViZ7sBoGBgcgM&usg=AFQjCNFlz-vbma0jaNSYIKrHGWbQ1xD60g

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Robust optimization

minx

𝑥1 + 2𝑥2 + 0.3𝑥3

𝑥1 + 2𝑥2 + 𝑥3 ≥ 4set i /1*3/;

positive variables x(i);

parameter c(i)

/ 1 1

2 2

3 1/;

variable of1;

equations

eq1,eq2;

eq1 .. of1=e=x('1')+2*x('2')+0.3*x('3');

eq2 .. sum(i,c(i)*x(i))=g=4;

model primal /eq1,eq2/;

solve primal us lp min of1;

minx

𝑖

𝑐𝑖𝑥𝑖

𝑖

𝑎𝑖𝑥𝑖 ≥ 𝑏

𝑎 =121

, 𝑏 = 4, c =120.3

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Robust optimization

minx

𝑖

𝑐𝑖𝑥𝑖

𝑖

𝑎𝑖𝑥𝑖 ≥ 𝑏

minx

𝑖

𝑐𝑖𝑥𝑖

𝑖

𝑎𝑖𝑥𝑖 ≥ 𝑏 𝑎 =121

𝑏 = 4 c =120.3

a

𝐚𝐦𝐢𝐧 𝐚𝐦𝐚𝐱 𝒂 𝑎𝑖= 𝑎𝑖 + (Δ𝑎𝑖

+−Δ𝑎𝑖−)𝑤𝑖

Δ𝑎𝑖+Δ𝑎𝑖

−0 ≤ 𝑤𝑖 ≤ 1

Δ𝑎𝑖+ ∗ Δ𝑎𝑖

− = 0

minx

𝑖

𝑐𝑖𝑥𝑖

𝑖

[ 𝑎𝑖+(Δ𝑎𝑖+−Δ𝑎𝑖

−)𝑤𝑖]𝑥𝑖 ≥ 𝑏

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Robust optimization

minx

𝑖

𝑐𝑖𝑥𝑖

𝑖

𝑎𝑖𝑥𝑖 − Δ𝑎𝑖−𝑤𝑖𝑥𝑖 ≥ 𝑏 LP or NLP ?

minx

𝑖

𝑐𝑖𝑥𝑖

𝑖

𝑎𝑖𝑥𝑖 −

𝑖

Δ𝑎𝑖−𝑤𝑖𝑥𝑖 ≥ 𝑏

0 ≤ 𝑤𝑖 ≤ 1

minx

𝑖

𝑐𝑖𝑥𝑖

𝑖

𝑎𝑖𝑥𝑖 − maxwi

𝑖


0 ≤ 𝑤𝑖 ≤ 1

Difficulties ?

NLP

Bi-level

optimization

Can we solve it in a single level ?

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Robust optimization

minx

𝑖

𝑐𝑖𝑥𝑖

𝑖


𝑖


0 ≤ 𝑤𝑖 ≤ 1

𝑖 𝑤𝑖 ≤ Γ Degree of conservativeness

maxwi

𝑖

Δ𝑎𝑖−𝑥𝑖𝑤𝑖

0 ≤ 𝑤𝑖 ≤ 1

𝑖 𝑤𝑖 ≤ Γ

max𝑤

𝑑𝑇𝑊

𝐴𝑊 ≤ 𝑄

min𝑦

𝑄𝑇𝑌

𝐴𝑇𝑌 ≤ 𝑑

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Robust optimization

maxwi

𝑖

Δ𝑎𝑖−𝑥𝑖𝑤𝑖

0 ≤ 𝑤𝑖 ≤ 1


max𝑤

𝑑𝑇𝑊

𝐴𝑊 ≤ 𝑄

min𝑦

𝑄𝑇𝑌

𝐴𝑇𝑌 ≤ 𝑑

maxwi

[Δ𝑎1−𝑥1 Δ𝑎2

−𝑥2 Δ𝑎3−𝑥3]

𝑤1

𝑤2

𝑤3

1 0 00 1 00 0 11 1 1

𝑤1

𝑤2

𝑤3

≤

111Γ

min𝑦i, 𝛽

[1 1 1 Γ]

𝑦1

𝑦2

𝑦3

𝛽

1 0 00 1 00 0 1

111

𝑦1

𝑦2

𝑦3

𝛽

≤

Δ𝑎1−𝑥1

Δ𝑎2−𝑥2

Δ𝑎3−𝑥3

𝐦𝒊𝒏𝒚𝒊,𝜷

𝒊

𝒚𝒊 + 𝚪 ∗ 𝜷

𝒚𝒊 + 𝜷 ≤ 𝜟𝒂𝒊−𝒙𝒊

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Robust optimization

minx

𝑖

𝑐𝑖𝑥𝑖

𝑖


𝑖


0 ≤ 𝑤𝑖 ≤ 1


𝐦𝒊𝒏𝒚𝒊,𝜷

𝒊

𝒚𝒊 + 𝚪 ∗ 𝜷


minx

𝑖

𝑐𝑖𝑥𝑖

𝑖

𝑎𝑖𝑥𝑖 − 𝐦𝒊𝒏𝒚𝒊,𝜷

𝒊

𝒚𝒊 + 𝚪 ∗ 𝜷 ≥ 𝑏


minx,yi,𝛽

𝑖

𝑐𝑖𝑥𝑖

𝑖

𝑎𝑖𝑥𝑖 − (

𝒊

𝒚𝒊 + 𝚪 ∗ 𝜷) ≥ 𝑏


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Robust optimization

minx,yi,𝛽

𝑖

𝑐𝑖𝑥𝑖

𝑖

𝑎𝑖𝑥𝑖 − (

𝒊

𝒚𝒊 + 𝚪 ∗ 𝜷) ≥ 𝑏


minx

𝑥1 + 2𝑥2 + 0.3𝑥3

𝑥1 + 2𝑥2 + 𝑥3 ≥ 4

𝑎 =121

, 𝑏 = 4, c =120.3

set i /1*3/;

scalar gamma /2/;

positive variables x(i),y(i),beta;

parameter c(i)

/ 1 1

2 2

3 1/;

variable of1;

equations

eq1,eq3,eq4;

eq1 .. of1=e=x('1')+2*x('2')+0.3*x('3');

eq3 .. sum(i,c(i)*x(i))- (sum(i,y(i))+gamma*beta)=g=4;

eq4(i) .. y(i)+beta =g=0.1*c(i)* x(i);

model RC /eq1,eq3,eq4/;

solve RC us lp min of1;

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Robust optimization

minx,yi,𝛽

𝑖

𝑐𝑖𝑥𝑖

𝑖

𝑎𝑗𝑖𝑥𝑖 −

𝒊

𝒚𝒋𝒊 + 𝚪𝐣 ∗ 𝜷𝒋 ≥ 𝑏𝑗 ∀𝑗

𝒚𝒋𝒊 + 𝜷𝒋 ≤ 𝜟𝒂𝒋𝒊−𝒙𝒊 ∀𝑖,𝑗

minx

𝑖

𝑐𝑖𝑥𝑖

𝑖

𝑎𝑖𝑗𝑥𝑖 ≥ 𝑏𝑗 ∀𝑗

Robust counterpart

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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)

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

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Supply

Demand

Upstream

network

losses

Energy

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A.Soroudi, P. Siano and A. Keane, "Optimal DR and ESS Scheduling for Distribution Losses Payments Minimization Under Electricity Price Uncertainty," in IEEE Transactions on Smart Grid, vol. 7, no. 1, pp. 261-272, Jan. 2016.

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Engineering

Robust optimization based decision making in Energy systems