robust optimization based decision making in energy systems
TRANSCRIPT
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Robust Optimization application in Smart Energy Systems
By: Alireza Soroudi
9/6/2016 1
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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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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
Information gap decision theory
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Min y=f(u,x)
G(u,x)<=0
H(u,x) =0
Scenarios
Stochastic
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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
)(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
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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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Uncertainty modelling tools
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Min y=f(u,x)
G(u,x)<=0
H(u,x) =0
Fuzzy Arithmetic Stochastic
Fuzzy arithmetic
Robust optimization
Information gap decision theory
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Uncertainty modelling tools
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Fuzzy Arithmetic Stochastic
Fuzzy arithmetic
Robust optimization
Information gap decision theory
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Uncertainty modelling tools
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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
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Uncertainty modelling tools
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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.
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Uncertainty modelling tools
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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
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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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Uncertainty modelling tools
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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
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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
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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
Δ𝑎𝑖−𝑤𝑖𝑥𝑖 ≥ 𝑏
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
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
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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
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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
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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;
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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)
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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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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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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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