Robust optimization based decision making in Energy systems
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This presentation describes the Robust Optimization method and its application in Energy systems. The GAMS codes are provided.
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Robust optimization based decision making in Energy systems
- 1. 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
- 3. 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
- 4. 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
- 5. 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
- 8. 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
- 9. 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
- 10. 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
- 11. 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
- 12. 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
- 13. 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 U 𝑼 𝟏 𝑼 𝟐
- 14. 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.
- 15. 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
- 16. Alireza.soroudi@ucd.ie http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/ 0 𝛼 𝛼 𝜶 𝒎𝒂𝒙 Maximum possible uncertainty IGDT Maximum tolerable uncertainty based on 𝛽 Risky region Safe region 0 ≤ 𝛼 ≤ 𝛼 𝑚𝑎𝑥 Prediction techniques ≤ 𝛼
- 17. 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
- 18. 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-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
- 19. Alireza.soroudi@ucd.ie http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 19 Robust optimization min x 𝑥1 + 2𝑥2 + 0.3𝑥3 𝑥1 + 2𝑥2 + 𝑥3 ≥ 4 set 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; min x 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑖 𝑥𝑖 ≥ 𝑏 𝑎 = 1 2 1 , 𝑏 = 4, c = 1 2 0.3
- 20. Alireza.soroudi@ucd.ie http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 20 Robust optimization min x 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑖 𝑥𝑖 ≥ 𝑏 min x 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑖 𝑥𝑖 ≥ 𝑏 𝑎 = 1 2 1 𝑏 = 4 c = 1 2 0.3 a 𝐚 𝐦𝐢𝐧 𝐚 𝐦𝐚𝐱𝒂 𝑎𝑖= 𝑎𝑖 + (Δ𝑎𝑖 + −Δ𝑎𝑖 − )𝑤𝑖 Δ𝑎𝑖 +Δ𝑎𝑖 − 0 ≤ 𝑤𝑖 ≤ 1 Δ𝑎𝑖 + ∗ Δ𝑎𝑖 − = 0 min x 𝑖 𝑐𝑖 𝑥𝑖 𝑖 [ 𝑎𝑖+(Δ𝑎𝑖 + −Δ𝑎𝑖 − )𝑤𝑖]𝑥𝑖 ≥ 𝑏
- 21. Alireza.soroudi@ucd.ie http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 21 Robust optimization min x 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑖 𝑥𝑖 − Δ𝑎𝑖 − 𝑤𝑖 𝑥𝑖 ≥ 𝑏 LP or NLP ? min x 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑖 𝑥𝑖 − 𝑖 Δ𝑎𝑖 − 𝑤𝑖 𝑥𝑖 ≥ 𝑏 0 ≤ 𝑤𝑖 ≤ 1 min x 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑖 𝑥𝑖 − max wi 𝑖 Δ𝑎𝑖 − 𝑤𝑖 𝑥𝑖 ≥ 𝑏 0 ≤ 𝑤𝑖 ≤ 1 Difficulties ? NLP Bi-level optimization Can we solve it in a single level ?
- 22. Alireza.soroudi@ucd.ie http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 22 Robust optimization min x 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑖 𝑥𝑖 − max wi 𝑖 Δ𝑎𝑖 − 𝑤𝑖 𝑥𝑖 ≥ 𝑏 0 ≤ 𝑤𝑖 ≤ 1 𝑖 𝑤𝑖 ≤ Γ Degree of conservativeness max wi 𝑖 Δ𝑎𝑖 − 𝑥𝑖 𝑤𝑖 0 ≤ 𝑤𝑖 ≤ 1 𝑖 𝑤𝑖 ≤ Γ max 𝑤 𝑑 𝑇 𝑊 𝐴𝑊 ≤ 𝑄 min 𝑦 𝑄 𝑇 𝑌 𝐴 𝑇 𝑌 ≤ 𝑑
- 23. Alireza.soroudi@ucd.ie http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 23 Robust optimization max wi 𝑖 Δ𝑎𝑖 − 𝑥𝑖 𝑤𝑖 0 ≤ 𝑤𝑖 ≤ 1 𝑖 𝑤𝑖 ≤ Γ max 𝑤 𝑑 𝑇 𝑊 𝐴𝑊 ≤ 𝑄 min 𝑦 𝑄 𝑇 𝑌 𝐴 𝑇 𝑌 ≤ 𝑑 max wi [Δ𝑎1 − 𝑥1 Δ𝑎2 − 𝑥2 Δ𝑎3 − 𝑥3] 𝑤1 𝑤2 𝑤3 1 0 0 0 1 0 0 0 1 1 1 1 𝑤1 𝑤2 𝑤3 ≤ 1 1 1 Γ min 𝑦i, 𝛽 [1 1 1 Γ] 𝑦1 𝑦2 𝑦3 𝛽 1 0 0 0 1 0 0 0 1 1 1 1 𝑦1 𝑦2 𝑦3 𝛽 ≤ Δ𝑎1 − 𝑥1 Δ𝑎2 − 𝑥2 Δ𝑎3 − 𝑥3 𝐦𝒊𝒏 𝒚 𝒊,𝜷 𝒊 𝒚𝒊 + 𝚪 ∗ 𝜷 𝒚𝒊 + 𝜷 ≤ 𝜟𝒂𝒊 − 𝒙𝒊
- 24. Alireza.soroudi@ucd.ie http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 24 Robust optimization min x 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑖 𝑥𝑖 − max wi 𝑖 Δ𝑎𝑖 − 𝑤𝑖 𝑥𝑖 ≥ 𝑏 0 ≤ 𝑤𝑖 ≤ 1 𝑖 𝑤𝑖 ≤ Γ 𝐦𝒊𝒏 𝒚 𝒊,𝜷 𝒊 𝒚𝒊 + 𝚪 ∗ 𝜷 𝒚𝒊 + 𝜷 ≤ 𝜟𝒂𝒊 − 𝒙𝒊 min x 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑖 𝑥𝑖 − 𝐦𝒊𝒏 𝒚 𝒊,𝜷 𝒊 𝒚𝒊 + 𝚪 ∗ 𝜷 ≥ 𝑏 𝒚𝒊 + 𝜷 ≤ 𝜟𝒂𝒊 − 𝒙𝒊 min x,yi,𝛽 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑖 𝑥𝑖 − ( 𝒊 𝒚𝒊 + 𝚪 ∗ 𝜷) ≥ 𝑏 𝒚𝒊 + 𝜷 ≤ 𝜟𝒂𝒊 − 𝒙𝒊
- 25. Alireza.soroudi@ucd.ie http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 25 Robust optimization min x,yi,𝛽 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑖 𝑥𝑖 − ( 𝒊 𝒚𝒊 + 𝚪 ∗ 𝜷) ≥ 𝑏 𝒚𝒊 + 𝜷 ≤ 𝜟𝒂𝒊 − 𝒙𝒊 min x 𝑥1 + 2𝑥2 + 0.3𝑥3 𝑥1 + 2𝑥2 + 𝑥3 ≥ 4 𝑎 = 1 2 1 , 𝑏 = 4, c = 1 2 0.3 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;
- 26. Alireza.soroudi@ucd.ie http://www.ucd.ie/research/people/electricalelectroniccommseng/dralirezasoroudi/9/6/2016 26 Robust optimization min x,yi,𝛽 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑗𝑖 𝑥𝑖 − 𝒊 𝒚𝒋𝒊 + 𝚪𝐣 ∗ 𝜷𝒋 ≥ 𝑏𝑗 ∀𝑗 𝒚𝒋𝒊 + 𝜷𝒋 ≤ 𝜟𝒂𝒋𝒊 − 𝒙𝒊 ∀𝑖,𝑗 min x 𝑖 𝑐𝑖 𝑥𝑖 𝑖 𝑎𝑖𝑗 𝑥𝑖 ≥ 𝑏𝑗 ∀𝑗 Robust counterpart
- 28. 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)
- 30. 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.
- 31. 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.
- 32. 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.