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Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Robust Optimization application in Smart Energy Systems
By: Alireza Soroudi
Alireza.soroudi@ucd.ie
9/6/2016 1
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 2
Introduction
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Introduction
9/6/2016 3
is the chance, within a specified time frame, of an adverse
event with specific (negative) consequences
Risk
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Uncertain events
9/6/2016 4
• Weather changes – Solar radiation – Wind speed
• Load values • Market prices • Gas network failures
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Introduction
Stochastic
Fuzzy arithmetic
Robust optimization
Information gap decision theory
Power system applications
9/6/2016 5
Introduction
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 6
Introduction
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 7
Introduction
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Uncertainty modelling tools
Stochastic
Fuzzy arithmetic
Robust optimization
Information gap decision theory
9/6/2016 8
Min y=f(u,x)
G(u,x)<=0
H(u,x) =0
Scenarios
Stochastic
Alireza.soroudi@ucd.ie
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
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Wind uncertainty modelling
9/6/2016 10
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
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Uncertainty modelling tools
9/6/2016 11
Min y=f(u,x)
G(u,x)<=0
H(u,x) =0
Fuzzy Arithmetic Stochastic
Fuzzy arithmetic
Robust optimization
Information gap decision theory
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Uncertainty modelling tools
9/6/2016 12
Fuzzy Arithmetic Stochastic
Fuzzy arithmetic
Robust optimization
Information gap decision theory
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Uncertainty modelling tools
9/6/2016 13
Min y=f(u,x)
G(u,x)<=0
H(u,x) =0
Robust Optimization Stochastic
Fuzzy arithmetic
Robust optimization
Information gap decision theory
U
Uncertainty set
U𝑼𝟏
𝑼𝟐
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Uncertainty modelling tools
9/6/2016 14
Min y=f(u,x)
G(u,x)<=0
H(u,x) =0
Robust Optimization Stochastic
Fuzzy arithmetic
Robust optimization
Information gap decision theory
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.
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Uncertainty modelling tools
9/6/2016 15
Min y=f(u,x)
G(u,x)<=0
H(u,x) =0
IGDT Stochastic
Fuzzy arithmetic
Robust optimization
Information gap decision theory
U
Uncertainty set
𝜶
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
0
𝛼
𝛼𝜶𝒎𝒂𝒙
Maximum possible
uncertainty
IGDT
Maximum tolerable
uncertainty based on 𝛽
Risky
regionSafe
region
0 ≤ 𝛼 ≤ 𝛼𝑚𝑎𝑥
Prediction
techniques
≤ 𝛼
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Uncertainty modelling tools
9/6/2016 17
Min y=f(u,x)
G(u,x)<=0
H(u,x) =0
IGDT Stochastic
Fuzzy arithmetic
Robust optimization
Information gap decision theory
• 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
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
Robust optimization
9/6/2016 18
“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
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 19
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
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 20
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
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 21
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
Δ𝑎𝑖−𝑤𝑖𝑥𝑖 ≥ 𝑏
0 ≤ 𝑤𝑖 ≤ 1
Difficulties ?
NLP
Bi-level
optimization
Can we solve it in a single level ?
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 22
Robust optimization
minx
𝑖
𝑐𝑖𝑥𝑖
𝑖
𝑎𝑖𝑥𝑖 − maxwi
𝑖
Δ𝑎𝑖−𝑤𝑖𝑥𝑖 ≥ 𝑏
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
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 23
Robust optimization
maxwi
𝑖
Δ𝑎𝑖−𝑥𝑖𝑤𝑖
0 ≤ 𝑤𝑖 ≤ 1
𝑖 𝑤𝑖 ≤ Γ
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
Alireza.soroudi@ucd.ie
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Robust optimization
minx
𝑖
𝑐𝑖𝑥𝑖
𝑖
𝑎𝑖𝑥𝑖 − maxwi
𝑖
Δ𝑎𝑖−𝑤𝑖𝑥𝑖 ≥ 𝑏
0 ≤ 𝑤𝑖 ≤ 1
𝑖 𝑤𝑖 ≤ Γ
𝐦𝒊𝒏𝒚𝒊,𝜷
𝒊
𝒚𝒊 + 𝚪 ∗ 𝜷
𝒚𝒊 + 𝜷 ≤ 𝜟𝒂𝒊−𝒙𝒊
minx
𝑖
𝑐𝑖𝑥𝑖
𝑖
𝑎𝑖𝑥𝑖 − 𝐦𝒊𝒏𝒚𝒊,𝜷
𝒊
𝒚𝒊 + 𝚪 ∗ 𝜷 ≥ 𝑏
𝒚𝒊 + 𝜷 ≤ 𝜟𝒂𝒊−𝒙𝒊
minx,yi,𝛽
𝑖
𝑐𝑖𝑥𝑖
𝑖
𝑎𝑖𝑥𝑖 − (
𝒊
𝒚𝒊 + 𝚪 ∗ 𝜷) ≥ 𝑏
𝒚𝒊 + 𝜷 ≤ 𝜟𝒂𝒊−𝒙𝒊
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 25
Robust optimization
minx,yi,𝛽
𝑖
𝑐𝑖𝑥𝑖
𝑖
𝑎𝑖𝑥𝑖 − (
𝒊
𝒚𝒊 + 𝚪 ∗ 𝜷) ≥ 𝑏
𝒚𝒊 + 𝜷 ≤ 𝜟𝒂𝒊−𝒙𝒊
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;
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 26
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
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 27
Robust optimization
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 28
A Soroudi , Robust optimization based self scheduling of hydro-thermal Genco in smart grids, Energy 61, 262-271
Robust optimization (Example)
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 29
Supply
Demand
Upstream
network
losses
Energy
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 30
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.
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 31
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.
Alireza.soroudi@ucd.ie
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 32
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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