Optimization under uncertainty. Antonio J. Conejo The Ohio State University 2014

Transcription

1 Optmzaton under uncertant Antono J. Conejo The Oho State Unverst 2014

2 Contents Stochastc programmng (SP) Robust optmzaton (RO) Power sstem applcatons A. J. Conejo The Oho State Unverst 2

3 Stochastc Programmng (SP) A. J. Conejo The Oho State Unverst 3

4 SP References Two-stage stochastc programmng Decson framework Decson tree Scenaro formulaton Node formulaton EVPI VSS A. J. Conejo The Oho State Unverst 4

5 SP References J. R. Brge and F. Louveaux. Introducton to Stochastc Programmng. Sprnger New York J. L. Hgle. Tutorals n Operatons Research INFORMS Chapter 2: Stochastc Programmng: Optmzaton When Uncertant Matters. INFORMS Hanover Marland A. J. Conejo M. Carrón J. M. Morales Decson Makng Under Uncertant n Electrct Markets Sprnger New York Chapters 23 and 4. A. J. Conejo The Oho State Unverst 5

6 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

7 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

8 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

9 Scenaro formulaton hgh average low Realzaton Probablt A. J. Conejo The Oho State Unverst 9

10 Scenaro formulaton subject to mnmze S 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

11 Non-antcpatvt constrants x x x Decsons cannot depend on the unknown future! A. J. Conejo The Oho State Unverst 11

12 Node formulaton subject to mnmze 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

13 EVPI Expected Value of the Perfect Informaton Measure of the value of perfect nformaton A. J. Conejo The Oho State Unverst 13

14 EVPI subject to mnmze 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

15 EVPI EVPI S Z Z P EVPI s non-negatve A. J. Conejo The Oho State Unverst 15

16 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

17 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

18 VSS A. J. Conejo The Oho State Unverst subject to mnmze x g x h x f Z D D D D Ths problem decomposes b scenaro

19 VSS Z D 3 1 f x D We evaluate the determnstc soluton n all scenaros A. J. Conejo The Oho State Unverst 19

20 VSS D VSS Z Z VSS s non-negatve S A. J. Conejo The Oho State Unverst 20

21 Robust Optmzaton A. J. Conejo The Oho State Unverst 21

22 Outlne Wh Robust Optmzaton (RO)? RO wthout recourse RO wth recourse Schedulng energ and reserve A. J. Conejo The Oho State Unverst 22

23 Wh RO? A. J. Conejo The Oho State Unverst 23

24 References Bertsmas D. Brown D.B. Caramans C. Theor and applcatons of robust optmzaton. SIAM Rev. vol. 53 pp Bertsmas D. Sm M. Robust dscrete optmzaton and network flows Math. Program Ser. B vol. 98 no. 13 pp 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 A. J. Conejo The Oho State Unverst 24

25 RO wthout recourse Uncertant set A. J. Conejo The Oho State Unverst 25

26 RO wthout recourse A. J. Conejo The Oho State Unverst 26

27 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

28 RO wth recourse Make schedulng decsons (mn) Uncertant realzes (max) Make operaton (recourse) decsons (mn) A. J. Conejo The Oho State Unverst 28

29 RO wth recourse A. J. Conejo The Oho State Unverst 29

30 RO wth recourse: Example A. J. Conejo The Oho State Unverst 30

31 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

32 Power sstem applcatons A. J. Conejo The Oho State Unverst 32

33 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

34 Producer Offerng b non-strategc producers: stochastc Maxmze Expected Proft subject to: Producer constrants A. J. Conejo The Oho State Unverst 34

35 Stochastc producer Offerng b non-dspatchable producers: stochastc Maxmze Expected Proft subject to: Producer constrants A. J. Conejo The Oho State Unverst 35

36 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

37 Producer Insurances If sellng through forward contracts and the producton unts fal an nsurance s advsable A. J. Conejo The Oho State Unverst 37

38 Consumer Consumer energ procurement Maxmze Expected Cost subject to: Consumer constrants Contractng constrants A. J. Conejo The Oho State Unverst 38

39 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

40 TSO Transmsson capact nvestment A. J. Conejo The Oho State Unverst 40

41 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

42 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

43 Conclusons (Electrcal) Energ problems are mportant! A. J. Conejo The Oho State Unverst 43

44 Conclusons How man coal plants are currentl beng bult n planet Earth? A. J. Conejo The Oho State Unverst 44

45 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

46 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

47 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

48 A. J. Conejo The Oho State Unverst 48