Unit Commitment under Market Equilibrium Constraints

Transcription

1 Uni Commimen under Marke Equilibrium Consrains Luce Brocorne 1 Fabio D Andreagiovanni 2 Jérôme De Boeck 3,1 Bernard Forz 3,1 1 INOCS, INRIA Lille Nord-Europe, France 2 Universié de Technologie de Compiègne, France 3 Déparemen d informaique, Universié libre de Bruxelles, Belgium PGMO Days, EDF Lab Paris Saclay, November 14, 2017

2 Inroducion Inegraed model MIP reformulaion MIP numerical experimens Rolling Horizon Heurisic Concluding remarks

3 Inroducion Inroducion

4 Inroducion Uni Commimen Uni Commimen problem Esablish he energy oupu of a se of generaion unis over a muli-period ime horizon, in order o saisfy a demand for energy, while minimizing he cos of generaion and respecing echnological resricions of he unis

5 Inroducion Uni Commimen Uni Commimen problem Esablish he energy oupu of a se of generaion unis over a muli-period ime horizon, in order o saisfy a demand for energy, while minimizing he cos of generaion and respecing echnological resricions of he unis Large scale mixed ineger program Deerminisic and robus versions sudied in he lieraure Uncerainy models focus on renewable power oupu See Tahanan e al. (4OR 2015) for a survey

6 Inroducion Day-ahead Elecriciy Markes Two-sided aucions Paricipans submi orders o buy (reailers) or sell (producers) elecric power during some hours of he following day Marke clearing: compued prices should ideally suppor a marke equilibrium Difficulies wih non-convex bids (e.g. block bids) See Madani and Van Vyve (EJOR 2015) for a survey

7 Inroducion Objecive Assuming he producer sells (par of) he energy produced on he day-ahead marke, how can we simulaneously decide a price bidding sraegy and an opimal UC sraegy aking he marke reacion ino accoun?

8 Inroducion Objecive Assuming he producer sells (par of) he energy produced on he day-ahead marke, how can we simulaneously decide a price bidding sraegy and an opimal UC sraegy aking he marke reacion ino accoun? Problems wih decoupled decisions Opimal producion from deerminisic UC model no sold a he desired price on he marke possible loss Bid for higher quaniies on he marke o increase profi infeasibiliies in he UC

9 Inegraed model Inegraed model

10 Inegraed model (Simplifying) Hypoheses Coninuous bids welfare maximizaion problem is an LP Opimisic assumpions: perfec knowledge of he oher players bids price maker: always able o sell a clearing price Deerminisic daa

11 Inegraed model Daa T : se of ime periods c (p ): cos of generaing p unis in period P: se of feasible soluions o he uni commimen problem (Carrión e al, IEEE Trans Power Sysems 2006) S: se of compeiors (sellers on he marke) B: se of buyers on he marke Qs : quaniy offered by seller s in period Qb : quaniy offered by buyer b in period πs: price offered by seller s in period πb : price offered by buyer b in period

12 Inegraed model Variables p 0: Energy offered in period λ 0: Clearing price in period xs : proporion of quaniy Qs cleared in period xb : proporion of quaniy Q b cleared in period ys : welfare of seller s in period yb : welfare of buyer b in period

13 Inegraed model Marke equilibrium Source: A. Ehsani, A.M. Ranjbar, M. Fouhi-Firuzabad (2009), A proposed model for co-opimizaion of energy and reserve in compeiive elecriciy markes, Applied Mahemaical Modelling 33(1),

14 Inegraed model Welfare maximizaion problem Assuming he quaniy p offered on he marke in each period is known, he marke clearing LP is: max s.. πbq bx b πsq s xs b B s S Qbx b Qs xs = p (λ ) b B s S 0 xb 1 b B (yb) 0 xs 1 s S (ys )

15 Inegraed model Clearing price compuaion The dual problem allows o compue he marke clearing price and oher players welfare: min λ p + yb + ys b B s S s.. Qbλ + yb πbq b b B Qs λ + ys πsq s s S yb 0 b B ys 0 s S

16 Inegraed model Combined model Objecive of he leader: maximize profi Profi: difference beween revenue (selling producion a marke clearing price) and producion cos Consrains: echnical consrains from UC problem

17 Inegraed model Bilevel formulaion max p s.. ( λ p c (p ) ) T (p ) T P min λ p + y λ,yb,y s b + ys T b B s S s.. Qbλ + yb πbq b b B Q s λ + y s π sq s y b 0 y s 0 s S b B s S

18 MIP reformulaion MIP reformulaion

19 MIP reformulaion Model properies Bilevel bilinear/linear model As he second level is linear: use dualiy o ransform he problem ino a single level one Replace he second level objecive by dual consrains (i.e. he welfare maximizaion problem) complemenariy consrains

20 MIP reformulaion Single level reformulaion max s.. ( λ p c (p ) ) T (p ) T P b B Q bx b s S Q s x s = p T Qbλ + yb πbq b Qs λ + ys πsq s xb ( Q b λ + yb πbq b ) ( = 0 x s Q s λ + ys + πsq s ) = 0 ( ) ( ) 1 x b = 0 y s 1 x s = 0 y b yb 0 ys 0 0 xb 1 0 xs 1 b B, T s S, T

21 MIP reformulaion Eliminaion of y b and y s Bilinear erms x b y b can be eliminaed by using and herefore can be rewrien as y b x b ( 1 x b ) = 0 x b y b = y b ( Q b λ + yb πbq b ) = 0 y b = π bq bx b Q bλ x b so yb can be eliminaed. A similar ransformaion allows o eliminae ys.

22 MIP reformulaion Discree choice model Lemma There exiss an opimal soluion such ha λ {π s} s S {π b } b B, for all T

23 MIP reformulaion Discree choice model Lemma There exiss an opimal soluion such ha λ {π s} s S {π b } b B, for all T Le: I := {1,..., {π s} s S {π b } b B } {λ i } i I := {π s} s S {π b } b B.

24 MIP reformulaion Discree choice model Lemma There exiss an opimal soluion such ha λ {π s} s S {π b } b B, for all T Le: I := {1,..., {π s} s S {π b } b B } {λ i } i I := {π s} s S {π b } b B. New variables: z i = { 1 if λ = λ i 0 oherwise

25 MIP reformulaion Linearizaion Subsiue λ by i I λ i z i wih i I z i = 1

26 MIP reformulaion Linearizaion Subsiue λ by i I λ i z i wih i I z i = 1 The producs of wo coninuous variables λ p, λ x b and λ x s are replaced by producs of a binary and a coninuous variable: z i p, z i x b and z i x s

27 MIP reformulaion Linearizaion Subsiue λ by i I λ i z i wih i I z i = 1 The producs of wo coninuous variables λ p, λ x b and λ x s are replaced by producs of a binary and a coninuous variable: z i p, z i x b and z i x s Classical linearizaion echnique: P i = z i p X ib = z i x b Xis = z i x s

28 MIP reformulaion Srenghening balance consrains b B Q bx b s S Q s x s = p T can be replaced by he sronger se of equaions b B QbX ib Qs Xis = Pi s S i I, T i I P i = p T

29 MIP reformulaion (Srenghened) linearized model max s.. ( T i I λ i P i ) c (p ) (p ) T P QbX ib Qs Xis = Pi b B s S i I, T i I P i = p T i I z i = 1 T 0 P i Q z i i I, T P i p Q (1 z i ) i I, T z i {0, 1} i I, T

30 MIP reformulaion λ i (zi Xib) πb(1 xb) 0 i I i I λ i X ib + π bx b 0 Xib = xb i I i I λ i (z i X is) + π s(1 x s ) 0 λ i Xis πsx s 0 i I Xis = xs i I 0 Xib zi i I 0 Xis zi Xib xb + zi 1 i I Xis xs + zi 1 0 xb 1 0 xs 1 b B, T i I i I s S, T

31 MIP numerical experimens MIP numerical experimens

32 MIP numerical experimens Insances J: number of generaors S: number of bids p: peneraion of he GC in he marke 24 ime periods Demand following a classical duck curve UC daa from Carrión 5 insances for each ( J, p, S ) combinaion

33 MIP numerical experimens Small insances Insance No srenghening Srenghened model J p S LP Gap Solved Time LP Gap Solved Time 10 5% % % %

34 MIP numerical experimens Big insances J p S LP gap Roo gap(%) Solved Final gap Time(s) Nodes % % % % % % % % %

35 Rolling Horizon Heurisic Rolling Horizon Heurisic

36 Rolling Horizon Heurisic Rolling Horizon Heurisic Classical approach for muli-period problems Saic roll s, dynamic roll d Ieraion : Binary variables for periods 0,..., s fixed from he soluion of ieraion 1 Solve he problem wih variables for s + 1,..., s + s + d considered as binary, variables for periods > s + s + d coninuous Ierae unil s T

37 Rolling Horizon Heurisic Numerical resuls Insance MIP RH s = 6 d = 1 J p S Time(s) Time(s) Roo gap Min gap Bes gap % % % % % %

38 Concluding remarks Concluding remarks

39 Concluding remarks Summary Bilevel bilinear/linear model for inegraion of UC and marke clearing sraegies Reformulaion and srenghening as a racable MIP Efficien rolling horizon heurisic for large insances

40 Concluding remarks Fuure Research Srenghen he model (valid inequaliies) Robus version of he model: Uncerain demand curves in he day-ahead marke model Uncerainy in renewable energy in he UC model More realisic marke mechanisms (non-convex bids)

41 12-14 Sepember 2018 MCE Conference & Business Cenre, Brussels hp://