Optmzaton under uncertant Antono J. Conejo The Oho State Unverst 2014
Contents Stochastc programmng (SP) Robust optmzaton (RO) Power sstem applcatons A. J. Conejo The Oho State Unverst 2
Stochastc Programmng (SP) A. J. Conejo The Oho State Unverst 3
SP References Two-stage stochastc programmng Decson framework Decson tree Scenaro formulaton Node formulaton EVPI VSS A. J. Conejo The Oho State Unverst 4
SP References J. R. Brge and F. Louveaux. Introducton to Stochastc Programmng. Sprnger New York 1997. J. L. Hgle. Tutorals n Operatons Research INFORMS 2005. Chapter 2: Stochastc Programmng: Optmzaton When Uncertant Matters. INFORMS Hanover Marland 2005. A. J. Conejo M. Carrón J. M. Morales Decson Makng Under Uncertant n Electrct Markets Sprnger New York. 2010 Chapters 23 and 4. A. J. Conejo The Oho State Unverst 5
Expectaton Two-stage SP CVaR Stochastc vector mnmze subject to x O h g f x x 0 x 0 A. J. Conejo The Oho State Unverst 6
Decson framework Decsons x are made (here & now) Stochastc vector λ realzes n a scenaro λ Gven x decsons (xλ) are made for each realzaton of λ λ (wat & see) A. J. Conejo The Oho State Unverst 7
Decson tree Scenaro 1 Scenaro Scenaro n 1st stage decsons here & now 2nd stage decsons wat & see Recourse A. J. Conejo The Oho State Unverst 8
Scenaro formulaton 1 2 3 2 3 1 hgh average low Realzaton Probablt A. J. Conejo The Oho State Unverst 9
Scenaro formulaton 3 2 1 3 1 123 0 123 0 subject to mnmze S 3 2 1 3 2 1 x x x x g x h x f Z x x x A. J. Conejo The Oho State Unverst 10 Here & Now Non-antcpatvt Expectaton
Non-antcpatvt constrants x x x 1 2 3 Decsons cannot depend on the unknown future! A. J. Conejo The Oho State Unverst 11
Node formulaton 3 12 0 123 0 subject to mnmze 3 1 3 2 1 x g x h x f Z S x Non-antcpatvt constrants are mplct! A. J. Conejo The Oho State Unverst 12 Just one x
EVPI Expected Value of the Perfect Informaton Measure of the value of perfect nformaton A. J. Conejo The Oho State Unverst 13
EVPI 3 12 0 123 0 subject to mnmze 3 1 3 2 1 3 2 1 x g x f x f Z P x x x No non-antcpatvt constrants: we perfectl foresee the future A. J. Conejo The Oho State Unverst 14
EVPI EVPI S Z Z P EVPI s non-negatve A. J. Conejo The Oho State Unverst 15
VSS (onl expectaton) Value of the Stochastc Soluton Measure of the relevance (gan) of usng a stochastc approach A. J. Conejo The Oho State Unverst 16
VSS maxmze x f x avg subject to h g x x avg avg Soluton x D Average! A. J. Conejo The Oho State Unverst 17
VSS A. J. Conejo The Oho State Unverst 18 3 12 0 123 0 subject to mnmze 3 1 3 2 1 x g x h x f Z D D D D Ths problem decomposes b scenaro
VSS Z D 3 1 f x D We evaluate the determnstc soluton n all scenaros A. J. Conejo The Oho State Unverst 19
VSS D VSS Z Z VSS s non-negatve S A. J. Conejo The Oho State Unverst 20
Robust Optmzaton A. J. Conejo The Oho State Unverst 21
Outlne Wh Robust Optmzaton (RO)? RO wthout recourse RO wth recourse Schedulng energ and reserve A. J. Conejo The Oho State Unverst 22
Wh RO? A. J. Conejo The Oho State Unverst 23
References Bertsmas D. Brown D.B. Caramans C. Theor and applcatons of robust optmzaton. SIAM Rev. vol. 53 pp. 464 501 2011. Bertsmas D. Sm M. Robust dscrete optmzaton and network flows Math. Program Ser. B vol. 98 no. 13 pp. 49 71 2003. Bertsmas D. Ltvnov E. Sun X. A. Zhao J. Zheng T. Adaptve robust optmzaton for the securt constraned unt commtment problem. IEEE Transactons on Power Sstems n press 2012. A. J. Conejo The Oho State Unverst 24
RO wthout recourse Uncertant set A. J. Conejo The Oho State Unverst 25
RO wthout recourse A. J. Conejo The Oho State Unverst 26
RO wthout recourse Under certan condtons over the robust set: Determnstc problem LP MILP NLP Robust counterpart Larger LP Larger MILP Larger NLP A. J. Conejo The Oho State Unverst 27
RO wth recourse Make schedulng decsons (mn) Uncertant realzes (max) Make operaton (recourse) decsons (mn) A. J. Conejo The Oho State Unverst 28
RO wth recourse A. J. Conejo The Oho State Unverst 29
RO wth recourse: Example A. J. Conejo The Oho State Unverst 30
RO wth recourse Make schedulng decsons x wth a prognoss of the future The uncertant w realzes Make operaton (recourse) decsons A. J. Conejo The Oho State Unverst 31
Power sstem applcatons A. J. Conejo The Oho State Unverst 32
ISO ISO market clearng: large-scale stochastc? Maxmze Expected Socal Welfare subject to: Market equlbrum Producer constrants Consumer constrants A. J. Conejo The Oho State Unverst 33
Producer Offerng b non-strategc producers: stochastc Maxmze Expected Proft subject to: Producer constrants A. J. Conejo The Oho State Unverst 34
Stochastc producer Offerng b non-dspatchable producers: stochastc Maxmze Expected Proft subject to: Producer constrants A. J. Conejo The Oho State Unverst 35
Producer Futures market nvolvement (forward contracts and optons) Maxmze Expected Proft subject to: Producer constrants Contractng constrants A. J. Conejo The Oho State Unverst 36
Producer Insurances If sellng through forward contracts and the producton unts fal an nsurance s advsable A. J. Conejo The Oho State Unverst 37
Consumer Consumer energ procurement Maxmze Expected Cost subject to: Consumer constrants Contractng constrants A. J. Conejo The Oho State Unverst 38
Producer Capact nvestment b non-dspatchable producers UPPER LEVEL Maxmze Proft from Wnd Generaton INVESTMENT DECISIONS LMPs LOWER LEVEL Maxmze SW MARKET CLEARING 1 MARKET CLEARING 2 MARKET CLEARING N Dfferent load and wnd condtons! A. J. Conejo The Oho State Unverst 39
TSO Transmsson capact nvestment A. J. Conejo The Oho State Unverst 40
TSO Transmsson capact nvestment Upper-Level Trade Maxmzaton subject to Lnes bult Lower-Level Socal Welfare Maxmzaton (Market Clearng) A. J. Conejo The Oho State Unverst 41
ISO Transmsson mantenance Upper-Level Securt Maxmzaton subject to Lnes n mantenance Lower-Level Socal Welfare Maxmzaton (Market Clearng) A. J. Conejo The Oho State Unverst 42
Conclusons (Electrcal) Energ problems are mportant! A. J. Conejo The Oho State Unverst 43
Conclusons How man coal plants are currentl beng bult n planet Earth? A. J. Conejo The Oho State Unverst 44
Conclusons If renewables are consdered: Major uncertant: stochastc producton facltes No such thng n the past (just demand uncertant) No such thng n models for ndustr (producton facltes are generall determnstc) A. J. Conejo The Oho State Unverst 45
Conclusons If renewables are consdered: Complex uncertant: multple dependences Spatal correlatons () among producton facltes () among demands and () among demands and producton facltes. Temporal correlatons for demands and producton facltes A. J. Conejo The Oho State Unverst 46
Conclusons If renewables are consdered: Mult-stage modelng s a must: future nvestment cost n stochastc sources s hghl uncertan: the technolog s not mature No two-stage stochastc models No adaptve robust optmzaton A. J. Conejo The Oho State Unverst 47
A. J. Conejo The Oho State Unverst 48