, , CORE, UCL

25 , 2013

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1

2 Lagrange

3

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:

: (robust) , ( , )

: (duality gap)

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Monticelli et al. (1987): Benders (SCOPF)Pereira et al. (1990): , SCOPF, ,

Falcao (1997):

Kim, Baldick (1997): , (2003) et al. (2005):

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PSR Cloud

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Lagrange

: ( ) ( )

Stochastic model(renewable energy,

demand,contingencies)

Scenario selection

Stochastic UC

Economic dispatch

Outcomes

Representative outcomes

Slow gen UC schedule

Outcomes

Min load, startup, fuel cost

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Lagrange

D / , , Kirchhoff, (Gf ) (Gs)

(UC) : mingG

tT

(Kgugt + Sgvgt + Cgpgt )

s.t .

gGn

pgt = Dnt

Pg ugt pgt P+g ugtelt = Bl(nt mt )(p,e,u,v) D

.

Lagrange

(SUC) : mingG

sS

tT

s(Kgugst + Sgvgst + Cgpgst )

s.t .

gGn

pgst = Dnst ,

Pgsugst pgst P+gsugstelst = Bls(nst mst )(p,e,u,v) Dsugst = wgt , vgst = zgt ,g Gs

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Lagrange

Lagrange

: (Takriti et al., 1996), (Carpentier etal., 1996), (Nowak and Rmisch, 2000), (Shiina and Birge,2004) : -(non-anticipativity) 1 -2

Lagrangian:

L =gG

sS

tT

s(Kgugst + Sgvgst + Cgpgst )

+

gGs

sS

tT

s(gst (ugst wgt ) + gst (vgst zgt ))

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Lagrange

z gt*

v gst*

wgt*

gst gst

gst*u

Dual multiplier update

Second-stage subproblems P2 s

1 2 N s. . . .

P1First-stage subproblem

Second-stage feasibility runs ED s

1 2 N s. . . .

N c1 2 . . . .

Monte Carlo economic dispatch EDc

Lawrence Livermore National Laboratory Hera cluster:13,824 864 , 2.3 Ghz, 32GB/MPI CPLEX Java

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Lagrange

: (Grwe-Kuska et al., 2002),(Dupacova et al., 2003), (Heitsch and Rmisch, 2003),(Morales et al., 2009) (importance sampling)

1 S , M = |S|. CD() S

, C =M

i=1

CD(i )M

.

2 N S, CD()/(MC).

3 s = CD()1 s .

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Lagrange

: (Brown et al, 1984), (Torres et al.,2005), (Morales et al, 2010)

1 :

ySkt =ykt kmtkmt

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2 Gauss:

yGSkt = N1(Fk (ySkt )).

3 kj Yule-Walker.

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Lagrange

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WECC

130 (88 , 42 ), 225 , 375

Arizona

Utah

IdahoWyoming

Montana

WashingtonOregon

CanadaHumboldt

Nevada

North Valley

LADWP

Orange County

SanDiego

Imperial Valley

Sierra

East Bay

North Coast/Geysers

Fresno

SanFrancisco

SouthBay

ZP26(CentralCoast)

So Cal Edison(Other)

Mexico

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(MW) 2 4,499 94 20,595.6 6 285.9 5 252 23 4,599 22 12,691 6 10,842 3 558 2 1,193 () 10 14,143 88 11,006.1 42 19,225.4

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: SAA 1000 :

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bundling

(ARPA-E) Lagrange Benders

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;

: anthony.papavasiliou@uclouvain.be

http://perso.uclouvain.be/anthony.papavasiliou/public_html/

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Eisagwg`hMont`eloDi`aspash LagrangeEpilog`h Senar`iwnMont`elo Aiolik`hs Paragwg`hs

Apotel`esmata