Demand Response with Linear Bidding: Efficiency vs. Risk. Munther A. Dahleh MIT Institute for Data, Systems, and Society

Transcription

1 Demand Response wth Lnear Bddng: Effcency vs. Rsk Munther A. Dahleh MIT Insttute for Data, Systems, and Socety

2 Collaoraton Na L : Harvard Unversty Ljun Chen: Unversty of Colorado at Boulder Qngqng Huang: MIT Mardavj Roozehan: MIT

3 Demand Response: Demand Electrcty demand: hghly tmevaryng Provson for peak load q Low load factor Natonal load factor s aout 55% q Underutlzed 10% of generaton and 25% of dstruton facltes are used less than 5% of the tme A way out: Shape the demand q Reduce the peak q Smooth the varaton Source: DoE, Smart Grd Intro, 2008

4 Demand Response: Generaton Supply ecomes hghly tme-varyng q steady rse of renewale energy resources Intermttent generaton q Large storage s not avalale A way out: Match the supply Ths work

5 Demand Response Use ncentve mechansms such as real-tme prcng to nduce customers to shft usage or reduce (even ncrease) consumpton Match Clmate - the Weathe Supply r Specal actvte s Prce (Market) Shape the Else Demand

6 generaton Overall structure customer wholesale market retal market utlty company

7 Man ssues The role of as an ntermedary q Play n mul@ple wholesale markets to provson aggregate power to meet demands day- ahead, alancng, ancllary servces q Resell, wth approprate prcng, to the end users q Provde two mportant values Aggregate demand at the wholesale level so that overall system s more effcent Asor large uncertanty/complexty n wholesale markets and translate them nto a smoother envronment (oth n prces and supply) for the end users. How to quantfy these values and prce them n the form of approprate contracts/prcng schemes?

8 Man ssues users q Desgn Welfare- maxmzng, proft- maxmzng q Prce- takng (Compe@@ve) vs Prce- an@cpa@ng (Game) q Prce of Anarchy q Rsk assessment (possle value of Anarchy)

9 The ascs of supply and demand Supply demanded at gven prces Demand suppled at gven prces Market equlrum: q= S( p) q = D( p) * * ( q, p ) such that q No surplus, no shortage, prce clears the market * * * q = S( p ) = D( p ) * p supply demand * q

10 Prolem settng Supply defct (or surplus) on electrcty: d weather change, unexpected events, Supply s nelas@c Prolem: How to allocate the defct among demand- responsve customers?

11 Supply functon ddng Customer load to shed: q Customer supply func@on (SF): q (, p) p = q the amount of load that the customer s commzed to shed gven prce p Market- clearng prcng: q (, p) = d p= d / p q1 customer 1: q = p 1 1 u@lty company: defct d p q n customer n: q = p n n

12 Parameterzed supply functon Adapts ezer to changng market than does a smple commtment to a fxed prce of quan@ty (Klemper & Meyer 89) q wdely used n the analyss of the wholesale electrcty markets q Green & Newery 92, Rudkevch et al 98, Baldck et al 02, 04, Parameterzed SF q easy to mplement q control nforma@on revela@on

13 Compettve market: Optmal demand response Customer cost (or C q ) ( q con@nuous, ncreasng, and strctly convex u@lty company: defct d Compe@@ve market and prce- takng customers p 1 p n Op@mal demand response max pq (, p) C ( q (, p)) max customer : pq (, p) C ( q (, p))

14 Compettve equlrum Theorem: There exst a unque CE. Moreover, the CE s effcent,.e., maxmzes socal welfare: max q C (q ) s.t. q = d Corollary (Indvdual Ran7onalty):

15 Proof Proof Idea: Compare the equlrum wth the (KKT) of the prolem. Equlrum max pq (, p) C ( q (, p)) q (, q) = d Socal Welfare op@mza@on max C ( q ) s.t. q = q d

16 Compettve equlrum Theorem (A water- fllng structure): q Corollary (Indvdual Ran7onalty):

17 Iteratve supply functon ddng Upon recevng the prce each customer updates ts supply Upon gatherng ds from the customers, the company updates prce p( k Requres 1 ( Cʹ ) ( p( k)) ( k) = [ ] p( k) + 1) = [ p( k) γ ( ( k) p( k) d) ] two- way communca@on q certan computa@onal capalty of the customers + + p( k + 1) = [ p( k) γ ( ( k) p( k) d) ] u@lty company: defct d pk ( ) pk+ ( 1) q ( k) 1 q ( k+ 1) 1 customer 1: 1 ( Cʹ 1) ( p( k)) 1 ( k) = [ ] pk ( ) + +

18 Strategc demand response Prce- customer wth Defn7on: A supply func@on profle s a Nash equlrum (NE) f, for all customers and, p q n p 1 q ))) (, ( ( )) (, ( ) ( ), ( p q C p q p u = ), ( max u * ), ( ), ( * * * u u 0 u@lty company: defct d customer : ), ( max u

19 Nash Equlrum Prce- customer max p (, ) q(, p (, )) C ( q(, p (, ))) Nash equlrum exts and s unque when the numer of customers s larger than 2 Each customer wll shed a load of less than d/2 at the equlrum Solvng another gloal op@mza@on prolem max D ( q ) s.t. q d q = 0 d / 2 q q d D (q ) = (1 + ) C ( q ) C (x )dx d q (d-2x ) p q1 Supply defct d customer : max p (, ) q(, p (, )) p q n C ( q (, p(, )))

20 Theorem Nash equlrum

21 Proof Proof Idea: Compare the equlrum wth the (KKT) of the prolem. NE Equlrum max pq ( p( ), p) C ( q ( p( ), p) 2 2 = d / ( Σjj) C( d / Σjj) q (, q) = d 0 Op@mza@on max D( q ) s.t. q = d q D(q) = (1 + q / d 2 q) C( q) q d / (d - 2x ) C (x )dx 2

22 Nash equlrum Theorem (A water- fllng structure): Corollary (Indvdual Ra7onalty):

23 Iteratve supply functon ddng Each customer supply updates ts 1 ( Dʹ ) ( p( k)) ( k) = [ ] p( k) The u@lty company updates prce p( k + 1) = [ p( k) γ ( ( k) p( k) d) ] + + pk ( ) pk+ ( 1) q ( k) 1 q ( k+ 1) 1 customer 1: p( k + 1) = [ p( k) γ ( ( k) p( k) d) ] u@lty company: defct d 1 ( Dʹ ) ( p( k)) ( k) = [ ] p( k) + +

24 Numercal example Optmal supply functon ddng (upper panels) v.s. strategc ddng (lower panels)

25 Effcency Loss of NE (Prce of Anarchy) Theorem:

26 Corollary: Homogeneous Customers

27 Numercal example Prce Total dsu7lty

28 A Specal Case wth Quadratc Dsutlty Functon Theorem: Message: Both the equlrum and game equlrum are ndependent of the supply defct d!

29 Value of Anarchy Prce of Anarchy: Loss n effcency due to strategc nteractons n contrast to a coordnaton Smple model: one agent wth shftale demand and another wth nstantaneous demand Contrast optmal effcent soluton to a Stackelerg game of strategc ehavor A new tradeoff: Cooperaton can ncrease endogenous rsk 30

30 Setup t Inflexle load Flexle load t+1 t+2 t+3 31

31 Model System state: Aggregate unshftale loads Consumer arrval wth shftale load Load shftng decson: Only 1 decson maker at Splt load nto two perods : the new arrval wth shftale load ased on

32 Prolem Formulaton Deadlne constrants on demands: Endogenous prces couple ndvdual decsons: Non-cooperatve decson makng: Mnmze ndvdual cost Cooperatve decson makng: Mnmze aggregate cost 33

33 Soluton: Strategc Symmetrc Markov Perfect equlrum n dynamc stochastc game t t+1 t +2 Overlappng type 2 consumers Flavor of Stackelerg competton

34 Soluton: Strategc Symmetrc Markov Perfect equlrum n dynamc stochastc game Equlrum strategy

35 Soluton: Cooperatve Bellman equaton for nfnte horzon average cost MDP Optmal statonary polcy

36 Welfare mpacts Under lnear statonary polcy _ Effcency/Welfare Tal proalty Rsk Varance Strategc Cooperatve

37 Prce of Anarchy: what aout rsk? Aggregate demand sample path spkes small tme scale large tme scale Cooperatve Non-cooperatve 38

38 Example I: L = 2 Aggregate demand statonary dstruton Low varance spkes Y axs Lnear scale Y axs Log scale Cooperatve Non-cooperatve

39 Concludng remarks Studed one astract models for demand response q Characterzed compe@@ve as well as strategc equlra q Proposed dstruted demand response algorthms ased on op@mza@on prolem characterza@ons q Characterzed the effcency loss and prce of the game- theore@c equlrum Rsk Analyss: Performance- roustness Tradeoffs Market Mechansm Thank you!