Robust optimization based decision making in Energy systems
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Transcript of Robust optimization based decision making in Energy systems
http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/
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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β’ Weather changes β Solar radiation β Wind speed
β’ Load values β’ Market prices β’ 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
β¦
U3
β¦1 2 n
β¦
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
UπΌπ
πΌπ
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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
πΆ
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0
πΌ
πΌπΆπππ
Maximum possible
uncertainty
IGDT
Maximum tolerable
uncertainty based on π½
Risky
regionSafe
region
0 β€ πΌ β€ πΌπππ₯
Prediction
techniques
β€ πΌ
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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
β’ 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.
β’ 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.
β’ 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.
β’ 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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βThe decision-maker constructs a solution that is optimal for any realization of
the uncertainty in a given setβ
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
π₯1 + 2π₯2 + 0.3π₯3
π₯1 + 2π₯2 + π₯3 β₯ 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
π
πππ₯π
π
πππ₯π β₯ π
π =121
, π = 4, c =120.3
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Robust optimization
minx
π
πππ₯π
π
πππ₯π β₯ π
minx
π
πππ₯π
π
πππ₯π β₯ π π =121
π = 4 c =120.3
a
ππ¦π’π§ ππ¦ππ± π ππ= ππ + (Ξππ
+βΞππβ)π€π
Ξππ+Ξππ
β0 β€ π€π β€ 1
Ξππ+ β Ξππ
β = 0
minx
π
πππ₯π
π
[ ππ+(Ξππ+βΞππ
β)π€π]π₯π β₯ π
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Robust optimization
minx
π
πππ₯π
π
πππ₯π β Ξππβπ€ππ₯π β₯ π LP or NLP ?
minx
π
πππ₯π
π
πππ₯π β
π
Ξππβπ€ππ₯π β₯ π
0 β€ π€π β€ 1
minx
π
πππ₯π
π
πππ₯π β maxwi
π
Ξππβπ€ππ₯π β₯ π
0 β€ π€π β€ 1
Difficulties ?
NLP
Bi-level
optimization
Can we solve it in a single level ?
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Robust optimization
minx
π
πππ₯π
π
πππ₯π β maxwi
π
Ξππβπ€ππ₯π β₯ π
0 β€ π€π β€ 1
π π€π β€ Ξ Degree of conservativeness
maxwi
π
Ξππβπ₯ππ€π
0 β€ π€π β€ 1
π π€π β€ Ξ
maxπ€
πππ
π΄π β€ π
minπ¦
πππ
π΄ππ β€ π
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Robust optimization
maxwi
π
Ξππβπ₯ππ€π
0 β€ π€π β€ 1
π π€π β€ Ξ
maxπ€
πππ
π΄π β€ π
minπ¦
πππ
π΄ππ β€ π
maxwi
[Ξπ1βπ₯1 Ξπ2
βπ₯2 Ξπ3βπ₯3]
π€1
π€2
π€3
1 0 00 1 00 0 11 1 1
π€1
π€2
π€3
β€
111Ξ
minπ¦i, π½
[1 1 1 Ξ]
π¦1
π¦2
π¦3
π½
1 0 00 1 00 0 1
111
π¦1
π¦2
π¦3
π½
β€
Ξπ1βπ₯1
Ξπ2βπ₯2
Ξπ3βπ₯3
π¦ππππ,π·
π
ππ + πͺ β π·
ππ + π· β€ πππβππ
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Robust optimization
minx
π
πππ₯π
π
πππ₯π β maxwi
π
Ξππβπ€ππ₯π β₯ π
0 β€ π€π β€ 1
π π€π β€ Ξ
π¦ππππ,π·
π
ππ + πͺ β π·
ππ + π· β€ πππβππ
minx
π
πππ₯π
π
πππ₯π β π¦ππππ,π·
π
ππ + πͺ β π· β₯ π
ππ + π· β€ πππβππ
minx,yi,π½
π
πππ₯π
π
πππ₯π β (
π
ππ + πͺ β π·) β₯ π
ππ + π· β€ πππβππ
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Robust optimization
minx,yi,π½
π
πππ₯π
π
πππ₯π β (
π
ππ + πͺ β π·) β₯ π
ππ + π· β€ πππβππ
minx
π₯1 + 2π₯2 + 0.3π₯3
π₯1 + 2π₯2 + π₯3 β₯ 4
π =121
, π = 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,π½
π
πππ₯π
π
ππππ₯π β
π
πππ + πͺπ£ β π·π β₯ ππ βπ
πππ + π·π β€ ππππβππ βπ,π
minx
π
πππ₯π
π
ππππ₯π β₯ ππ βπ
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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