Two methods of pruning Benders' cuts and their application to the management of a gas portfolio

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application to the management of a gas portfolio

Laurent Pfeiffer, Romain Apparigliato, Sophie Auchapt

To cite this version:

Laurent Pfeiffer, Romain Apparigliato, Sophie Auchapt. Two methods of pruning Benders’ cuts and

their application to the management of a gas portfolio. [Research Report] RR-8133, INRIA. 2012,

pp.23. �hal-00753578�

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RESEARCH

REPORT

N° 8133

November 2012

Two methods of pruning

Benders’ cuts and their

application to the

management of a gas

portfolio

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RESEARCH CENTRE

SACLAY – ÎLE-DE-FRANCE

Parc Orsay Université

portfolio Laurent Pfeier

∗

, RomainApparigliato

†

, SophieAu hapt

‡

Proje t-TeamCommands

Resear hReport n°8133November201223pages

Abstra t: Inthisarti le,wedes ribeagasportfoliomanagementproblem,whi hissolvedwith theSDDP(Sto hasti DualDynami Programming)algorithm. Wepresentsomeimprovementsof thisalgorithm andfo usonmethodsofpruningBenders' uts,that istosay,methods ofpi king outthemostrelevant utsamongthosewhi hhavebeen omputed. Ourterritoryalgorithm allows aqui ksele tionandagreatredu tionofthenumberof uts. Ourse ondmethodonlydeletes uts whi hdonot ontributetotheapproximationofthevaluefun tion,thankstoatestofusefulness. Numeri alresultsarepresented.

Key-words: SDDPalgorithm,sto hasti programming,energymanagement,pruningmethods, Benders' uts.

Thisstudywassupportedbythe Centre ofExpertiseinE onomi s,Modelling,andStudiesofGDF-Suez. TheauthorsaregratefultoFrédéri BonnansandYvesSmeersforprovidingthemusefuladvi e.

∗

INRIA-Sa layandCMAP,E olePolyte hnique,91128PalaiseauCedex,Fran e (laurent.pfeierpolyte hnique.edu)

†

GDF-Suez,CentreofExpertiseinE onomi s,Modelling,andStudies,1rued'Astorg,75392ParisCedex08, Fran e(romain.apparigliatogdfsuez. om)

‡

GDF-SuezCentreofExpertiseinE onomi s,Modelling,andStudies,1rued'Astorg,75392ParisCedex08, Fran e(sophie.au haptgdfsuez. om)

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Résumé : Dans et arti le, nous dé rivonsun problème de gestiond'un portefeuille gazier, résolu ave l'algorithme SDDP (Sto hasti Dual Dynami Programming). Nous présentons quelquesaméliorationsde ettealgorithmeetnousnous on entronssurdesméthodesd'élagage des oupesdeBenders, 'est-à-dire,desméthodespourséle tionnerles oupeslespluspertinentes parmi elles déjà al ulées. Notrealgorithme des territoires permetune séle tionrapideetune grande rédu tion du nombre de oupes. Notre se onde méthode ne supprime que les oupes quine ontribuentpasàl'approximationdelafon tionvaleur,àl'aided'untest d'utilité. Nous présentonsdesrésultatsnumériques.

Mots- lés : Algorithme SDDP, programmationsto hastique, gestionénergétique, méthodes d'élagage, oupesdeBenders.

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1 Introdu tion

Themanagementofagasportfolioisamajoroptimizationproblemwhi harisesinanun ertain environment, sin e gas onsumption and pri es are random. The goal of the problem is to minimize the osts required to supply gas on dierent zones of onsumption, ea h day of the year. Thegas omes from delivery ontra tsand thedemand anbe regulatedthanks to gas storages. Over one year, the total amount of gas bought on ea h ontra t is limited. This onstraint, as well as the level of ea h storage, introdu e a strong link between the de isions takenatea htimestep. Thislinkwillbetakenintoa ountthankstoahigh-dimensionalstate variable.

Inthisarti le,werstgiveasimpleformulationof thisproblem asamulti-stagesto hasti linearproblemandwegiveades riptionoftheSDDPalgorithm[4,16℄usedtosolvetheproblem. Thiswell-knownalgorithmapproximatesthevaluefun tionwiththesupremumofanefun tions alled Benders' utsor optimality uts. It hasbeenmainly usedin energy managementandis stilla tivelystudied[8℄.

Se ondly,wefo usonmethodsofpi kingoutthemostrelevantoptimality utsamongthose generated by the algorithm. Indeed, the rst uts whi h are omputed may be ome obsolete afterafewiterations. Thoughtheydonot ontributetothe urrentapproximationofthevalue fun tion,theyslowdowntheexe utionofthealgorithm. Thus,thereisaneedto prunetheset ofoptimality uts.

Thisquestionhasnotbeenmu h adressedin thesto hasti programmingliterature. Inthe regularizedde ompositionmethod des ribedin[18℄, onlythea tive utsaresele ted,that isto say,thosehavingapositiveLagrangemultiplierasso iatedatthelastiterationofthealgorithm. Thismethod anbeusedformulti-stageproblemsifwe onsiderthevaluefun tionsasso iated withea hnodeofthes enariotree,asin[19℄. Inourstudy,thankstothestagewiseindepen eof therandomvariable,weonlyneedto des ribeonevaluefun tionforea htimestep. Therefore, the a tive uts ofthe valuefun tion maydierfrom one realizationof thedemand to another and the method annot be applied in this situation. The problem of ut sele tion has been re entlystudied in theframework ofmax-plusbasedapproximationmethods [12℄. Similarlyto SDDP, these methods approximate the value fun tion of a nonlinear optimal ontrol problem by a supremum of basis fun tions. In this ase, the basis fun tions are quadrati fun tions. Pruning methods of these basisfun tions, dierent from ours, havebeenproposed in [15℄ and morere entlyin [14℄.

In a re ent preprint [10℄, de Matos, Philpott, and Finardi des ribe a heuristi to sele t optimality uts alled theLevel 1Dominan e. Beforesolvingalinearprogram,theysele tthe utstobeused. Consideringasetofpoints

(x

i

)

i=1,...,K

,theyonlykeepthe utswhi hdominate the others on at least onepoint

x

i

. Our territory algorithm uses the sameheuristi to delete denitively utspreviouslygenerated.

1

Itfastensthesolvingoftheproblemandavoidsex eeding the storagespa e apa ity of the omputerwith auseless a umulationof uts. However,the territory algorithm may delete some useful uts. Our se ond method ombines the territory algorithm with a test of usefulness, whi h examines if a given ut brings information. This methodavoidsadamagingoftheapproximationofthevaluefun tion.

Theaimofthearti leistoshowtheinterestofpruningmethods,thankstosimulationsona real-worldproblem. Theterritoryalgorithmallowsaqui ksele tionofthe utsandaredu tion byafa torof10ofthenumberof uts. Althoughourse ondmethodisslow,itdoesnotdamage theapproximationof thevaluefun tion and showsthat aredu tionbyafa tor5is rea hable. We observe that the time needed to realise forwardpasses is redu ed by afa tor 3when the

1

This algorithm was presented for the rst time in 2007 at the ROADEF ongress by David Game and GuillaumeLeRoy(GDF-Suez).

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territoryalgorithmisapplied.

Theorganisationofthearti leisasfollows. Wedes ribetheprobleminse tion2andgivea mathemati alformulationofitin se tion3. Inse tion4, wedes ribetheSDDPalgorithm and in se tion5,wegiveashort reviewof omputationalimprovementswith pre isereferen es. In se tion6,wedes ribetheterritoryalgorithmandthetestofusefulness. Inse tion7,weprovide somenumeri alresultswithatest asewithahigh-dimensionalstatespa eofdimension19.

2 Gas portfolio management

2.1 General des ription of assets

In this part, we des ribe in general terms the gas assets of our problem. The gas network is omposed of a set of balan ing zones, representinggeographi pla es of onsumption. All the balan ing zonesarelinkedbypipes, and theirgasinowsand outowsmust be equalin order tosatisfythedemand. Tothatpurpose,various assets anbeused.

Figure1: An exampleof GasNetwork

Long-term (LT) delivery ontra ts They are usually signed for periods up to 20 or 30 yearsandin ludetake-or-pay lauses(ToP):thebuyeragreesto buyaminimumamountofgas atoneorseveraldeliverypointsduringpre-spe iedperiodsoftime(annual/quarterly/monthly onstraints).Thisallowsthesellertohedgehisvolume-riskandtopayoandse urehuge invest-mentsin produ tion elds. Minimal amountsof deliveredgasperperiod are hard onstraints: violatingsu ha onstraintwouldleadtonan ialpenalizationandunusedgaswouldbelost. In ex hange, to redu e his volume-risk,the seller guaranteesa ompetitive pri efor thegas sold. These ontra ts an beseenasaswapofvolume-riskagainstpri e-risk.

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Storages Ingeneral,the gasavailable from LT ontra tsisnot su ientto fa e winter on-sumptionpeaks. On the ontrary,when thedemand is lowin summer,take-or-pay lausesare su hthattheminimalgasdeliveriesarehigherthanthe onsumption. Thisla kofexibility an be ounterbalan edbystoragefa ilities whi h dealwith the strongseasonality ofthe demand: storagesarelledinsummer,whenthe onsumptionislow. Inwinter,theyprovideanadditional resour eto satisfyhigh loads,thus preventing alls to themarket whenpri esare expe ted to get veryhigh. Storages are mainly of twotypes: aquifers and salt averns. Aquifers typi ally havehigh apa itiesbut lowerinje tionandextra tionrates. Theyareusedtosmooththeload annual seasonality. By ontrast, salt averns have low apa ity but higher deliveryrates and maybeusedto overpeakloadsortoperformdailyarbitragesonthemarket.

Markets Spotmarketpla esarenamedhubs. Physi alhubsrepresentabiginter onne tion pointbetweenseveralgasassets. Virtualhubsarepartsofthenetworkonwhi hex hangepoints ouldnotbeidentied. We anmentionanEnglish virtualhub,theNationalBalan ingPoint, andtheBelgiumphysi alhubZeebrugge.

2.2 Optimization variables and onstraints

Wedes ribeinthispartthegasportfolioproblem. Fromthepointofviewoftheoperationalgas portfoliomanager, themain un ertainty omes from the onsumptionlevel. Its main obje tive is to balan e supply and demand whatever the demand is. This is why we onsider pri es as deterministi andwhywedonottakeinto a ountthemarketsintheoptimizationproblem.

2.2.1 Data

Inthisse tion,weusethefollowingnotations:

t

timeindex

T

i

of ontra t

a

b

a,i

_{, e}

Thesto hasti pro essasso iatedwiththedemandandthede isionpro esswillbedes ribedin se tion3. Forthemoment,wesimplydes ribethevariablesandthe onstraintsinvolvedforone parti ularrealizationofthedemand.

2.2.2 Variables

State variables Thestatevariableisdenoted by

x

t

. Itis omposed ofthefollowingassets:

x

s

t

levelofstorage

s

attime

t

x

a,i

t

umulatedgasquantityusedfromthebeginningofquota

i

of ontra t

a

untiltime

t

.

at time

t

trans

z,z

z

attime

t

.

(2)

Notethatthesla kvariableswillenableusto satisfythesupply-demand onstraint.

2.2.3 Constraints

Bounded ex hanges All the ex hanges, withdrawals, inje tions, pur hases and sells, are bounded. Thefollowingboundsaredened forea h

t

∈ [0, T − 1]

asfollows:











0 ≤ inj

s

t

≤ inj

s

t

∀s ∈ S,

0 _{≤ wth}

s

t

≤ wth

s

t

∀s ∈ S,

q

a

t

≤ q

a

t

≤ q

a

t

∀a ∈ A,

0 ≤ trans

z,z

t

′

≤ trans

z,z

′

t

∀z, z

′

∈ Z,

0 _{≤ g}

z+

_t

_{∀z ∈ Z,}

0 ≤ g

z−

q

a

t

< 0

).

Supply-demand onstraint Onea hzone

z

∈ Z

,atea htime

t

∈ [0, T − 1]

,theequilibrium onstraintisgivenby

X

s∈SZ(z)

(wth

s

t

− inj

t

s

) +

X

a∈AZ(z)

q

a

t

+

X

z

′

_∈Z

z

′

_6=z

trans

z

t

′

,z

= d

z

t

+ g

t

z+

− g

t

z−

+

X

z

′

_∈Z

z

′

_6=z

trans

z,z

t

′

.

(4)

Dynami of the state variables Theevolutionofthestoragesisdes ribedby

x

s

t+1

= x

s

t

+ inj

t

s

− wth

s

t

,

∀s ∈ S, ∀t ∈ [0, T − 1],

(5) andtheinitiallevel

x

s

0

isgiven,forall

s

in

S

. Theevolutionofthegas ontra tsisdes ribedby

x

a,i

_t+1

= x

a,i

_t

+ q

a

t

,

∀a ∈ A, ∀i ∈ I(a), ∀t ∈ [b

a,i

, e

a,i

− 1],

(6)

x

a,i

_b

a,i

= 0,

∀a ∈ A, ∀i ∈ I(a).

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State onstraints Allthestoragelevelsandallthe umulatedlevelsonthelongterm ontra ts areupperand lowerboundedasfollows:

(

x

s

t

≤ x

s

t

≤ x

s

t

∀s ∈ S, ∀t ∈ [1, T ],

x

a,i

t

≤ x

a,i

t

≤ x

a,i

t

∀a ∈ A, ∀i ∈ I(a), ∀t ∈ [b

a,i

, e

a,i

].

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Originally, thesebounds aregiven forsomeparti ulartime steps. Forthe longterm ontra ts, weonlyhavealowerandanupperbound attheendofea hquota,attime

e

a,i

,andtheinitial stateat time

b

T

−1

,thelevelmustbeatleastequalto

x

s

T

−inj

s

T−1

, otherwiseit is impossibleto inje tenoughgas to satisfythenal onstraint. We an ompute minimalstate onstraintssothatforea hfeasiblestateattime

t

,forea hdemand

d

t

,thereexists ade isionwhi h leadsto afeasiblestateattime

t + 1

. Wedonotdes ribethe omputationof thesestate onstraints. Notethatthesla kvariablesallowtosatisfytheequilibrium onstraint whateverthedemandis.

2.2.4 Cost

Thedaily ostdepends linearlyonallthede isionvariables. Thetotal ostasso iatedwiththe s enarioofdemand

(d

0 , .., d

T−1

)

is

T

_X

−1

t=0

X

s∈S

(pwth

s

t

.wth

s

t

+ pinj

t

s

.inj

t

s

) +

X

a∈A

pq

a

t

.q

a

t

(9)

+

X

z,z

′

∈Z

ptrans

z,z

t

′

.trans

z,z

′

t

+

X

z∈Z

(pg

z+

t

.g

z+

t

+ pg

t

z−

.g

t

z−

)

,

where theunitarypri es(

pinj

s

t

,

pwth

s

t

,

pq

a

t

,

ptrans

z

′

_,z

t

,

pg

z+

t

, and

pg

z−

t

)are knownand deter-ministi . Thedaily ostdoesnotdependonthestatevariables: this approximationis madeto keepthe onvexityof theproblem. Notethat inpra ti e,

pg

z+

t

and

pg

z−

t

are hosenveryhigh, sothat thesla kvariablesareusedultimately.

3 Problem formulation

In this se tion, we give a mathemati al formulation of the gas supply problem introdu ed in se tion 2. The problem is des ribed by amultistage sto hasti linear program with relatively ompletere ourse. Forsurveysonsto hasti programming,wereferto[6,21℄.

Variables and onstraints Let us formalize the variables and the onstraints involved at stage

t

oftheglobalproblem. Wedenoteby:

x

t

trans

z,z

′

t

,

g

z+

t

and

g

z−

t

,des ribedby(2)

d

t

,thedemand ofgasforthedierentzones. Weintrodu ethefollowingnotationsforthe onstraints:

Bu

t

≤ b

t

,the onstraintsonthe ontrolvariable

u

t

,des ribedby(3)

Eu

t

= d

t

,thebalan ebetweensupplyand demandgivenby(4)

(11)

x

t+1

= x

t

+ A

t

u

t

,thedynami ofthestate

x

t

givenby (5)and (6)

x

t+1

∈ X

t+1

,thestate onstraintattime

t + 1

,whi hmustbeanti ipatedattime

t

.

X

t+1

isapolyhedrondes ribedby(8) .

Thepartofthe ostfun tionasso iatedtostage

t

isdenotedby

c

t

u

t

. Theparameter

c

t

isdened by(9).

t

and

d

t

′

are independent. Inthe sequel, we use thesame notationforarandomvariableand aparti ularrealizationofthisvariable. Thedistin tionwill be learfromthe ontext. We onsideradis retepro ess: forall

t

,

d

t

takesitsvaluein

D

t

=

{d

1 t

, ..., d

i

t

, ..., d

K

t

},

where

K

isthe ardinalof

D

t

. Theasso iatedprobabilitiesaredenoted by

p

1 t

, ..., p

i

t

, ..., p

K

t

. We denoteby

d

[t]

= (d

0 , ..., d

t

)

thehistoryofthepro essuntiltime

t

d

0 −→

de isionof

u

1 −→ x

1 −→

observationof

d

1 −→ ...

−→ x

T

−1

−→

observationof

d

T−1

−→

de isionof

u

T−1

−→ x

min

E

P

T−1

t=0

c

t

u

t

(d

[t]

)

.

x

t+1

(d

[t]

), u

t

(d

[t]

),

∀t ∈ {0, ..., T − 1},

s.t.

x

t+1

(d

[t]

) = x

t

(d

[t−1]

) + A

t

u

t

(d

[t]

),

Bu

t

(d

[t]

)

≤ b

t

,

Eu

t

(d

[t]

) = d

t

,

x

t+1

(d

[t]

)

∈ X

t+1

,

∀t ∈ {0, ..., T },

(

P

)

The sto hasti pro ess

d

t

being dis rete, problem

P

is nite-dimensional. However, sin e the numberof variablesin reases exponentially withthe numberof stages,it annot be solved di-re tly.

2

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Dynami programming The SDDP algorithm uses dynami programming in order to de- ompose thegeneralprobleminto

T

sub-problems. Letusintrodu etwovaluefun tions

V

t

(x

t

)

and

V

t

(x

t

, d

t

)

, denedre ursivelyby,forall

t

in

{0, ..., T − 1}

,forall

x

t

in

X

t

,and forall

d

t

in

R

,

V

T

(x

T

) = 0,

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V

t

(x

t

, d

t

) =

min

u

t

, x

t+1

,

s.t.

x

t+1

=x

t

+A

t

u

t

,

Bu

t

≤b

t

,

Eu

t

=d

t

,

x

t+1

∈X

t+1

,

c

t

u

t

+

V

t+1

(x

t+1

),

(11)

V

t

(x

t

) =

K

X

i=1

p

i

t

V

t

(x

t

, d

i

t

).

(12)

Inthesequel,wewillalso usethenotation

V

i

t

(x

t

) = V

t

(x

t

, d

i

t

)

, for

d

i

t

in

D

t

. By onvention,we set

V

t

(x

t

, d

t

) =

V

t

(x

t

) = +

∞

if

x

t

∈ X

/

t

. Thevaluefun tion

V

t

(x

t

)

standsfortheexpe tedvalue of theproblem from

t

to

T

before the observation of

d

t

(d

[t]

)

areoptimal solutions of

P

if andonly if, for all possible realizations of

d(t)

, for all

t

,

0 ≤ t < T

,

(x

t+1

(d

[t]

), u

t

(d

[t]

))

is an optimal solution toproblem (11) .

t

. However,the number of dis retization points in reasesexponentially with the dimension of the state. This fa t is the well-known urse of dimensionality and preventsus from solvingour problem with su hanapproa h.

4 SDDP algorithm

GeneralPrin iple TheSDDPalgorithmstandsonBenders'de omposition,proposedinitially in [2℄. VanSlykeandWetsusedthiste hniquetodeveloptheL-Shaped methodfortwo-stage sto hasti linearproblemsin[22℄andBirgeextendedtheirmethodtomultistageproblemsin[4℄. PereiraandPintoappliedthese te hniquesin[16℄ tosolveaproblemofoptimals hedulingofa hydrothermalgeneratingsystemanddevelopedSDDP,thealgorithmweused.

SDDP isan approximate dynami programmingalgorithm, based onthe followingtwokey ideas.

Thevaluefun tions

V

t

k

iterations,thealgorithmhas omputed

k

utsforthevaluefun tion

V

t

,whi haredenoted by

H

j

t

x

t

+ h

j

t

, and indexed by

j

,

1 ≤ j ≤ k

. Thus, the orresponding outer approximation, denotedby

V

t

is

V

t

(x

t

) = max

1≤j≤k

{ H

j

t

x

t

+ h

j

t

} ≤ V

t

(x

t

).

Thethreestepsofthe

k

-thiterationofthealgorithmarethefollowing. Simulate oneparti ularrealization

d

∗

t

of

d

t

.

Forwardpass: fortimes

t

from

to

0

, omputeanoptimality ut

H

k

t

x

t

+ h

k

t

,relevant intheneighborhoodof

x

∗

t

.

x

∗

t+1

aresolutionstothefollowingproblem:

min

u

t

, x

t+1

,

s.t.

x

t+1

= x

t

+ A

t

u

t

,

Bu

t

≤ b

t

,

Eu

t

= d

t

,

x

t+1

∈ X

t+1

,

c

t

u

t

+

V

t+1

(x

t+1

),

(

P

t

(x

t

, d

t

)

) for

x

t

= x

∗

t

and

d

t

= d

∗

t

. Thisproblemisequivalentto thefollowinglinearprogram:

min

u

t

, x

t+1

, v,

s.t.

x

t+1

= x

t

+ A

t

u

t

,

Bu

t

≤ b

t

,

Eu

t

= d

t

,

x

t+1

∈ X

t+1

,

1v

≥ H

t+1

x

t+1

+ h

t+1

,

c

t

u

t

+ v,

where

v

is an additionalvariable whi h standsfor theexpe ted future ost and

1

is a olumn ve tor omposed of asmany ones asoptimality uts. Thesequen es of statesobtained at the dierent iterations an be onsidered as realisti in so far as the approximations are getting loser to the exa t value fun tions

V

t

. Moreover, the uts provide a good approximation of thevaluefun tion in theneighborhoodof thepointsforwhi h theyhavebeen omputed. Ina way, the algorithm on entrates onthe most probableareasof the statespa e to improvethe approximationofthevaluefun tions,whereneeded.

(14)

Ba kward pass First,letusjustify the onvexityofthevaluefun tions.

Lemma2. Forall

t

,

0 ≤ t < T

,the value fun tions

V

t

(x

t

, d

t

)

and

V

t

(x

t

)

are onvex fun tions of

(x

t

, d

t

)

and

x

t

respe tively.

Proof. Weprovethislemmabyba kwardindu tion. Thefun tion

V

T

(x

T

)

is onvex,beingequal to

V

t−1

(x

t−1

, d

t−1

)

is onvex,sin eitis thevalueofa onvexoptimizationproblem(11)where

(x

t−1

, d

t−1

)

appearsattheright-hand-side ofthe onstraints. Then,

V

t−1

(x

t−1

)

is onvex,asasumof onvexfun tions.

Now, let us x

t

,

0 ≤ t < T

, and let us fo us on the linear program

P

t

(x

t

, d

t

)

. It is feasible,by onstru tionof

X

t

. For onvenien e,thelinear onstraint

x

t+1

∈ X

t+1

isdenotedby

G

t+1

x

t+1

≤ g

t+1

. We onsiderthedual problemof

P

t

(x

t

, d

t

max

π, β

_{≤ 0, ε, γ ≤ 0, η ≤ 0,}

πA

t

+ βB + εE = c

t

,

η1 =

−1,

π + γG

t+1

+ ηH

t+1

= 0,

πx

t

+ βb

t

+ εd

t

+ γg

t+1

− ηh

t+1

.

(

D

t

(x

t

, d

t

)

Theimportantpointin theexpli itformulationofthis dual problemis thatthe parameters

x

t

and

d

t

donotappearinthe onstraints. Indeed,aswealreadymentioned,the ostfun tion (in theprimal problem)doesnotdependonthestatevariables. Wedenoteby

(π

i

_{, β}

i

_{, ε}

i

_{, γ}

i

_{, η}

i

₎

an optimal solution to the problem for

x

t

= x

∗

t

and

d

t

= d

i

t

and wedenote by

W

i

t

(x

t

)

the nite valueofthedual problemfor

d

t

= d

i

t

andforagivenvalueof

x

t

. Proposition3. Forall

x

t

in

X

t

,the followinginequalities hold:

W

t

i

(x

t

)

≥ π

i

x

t

− π

i

x

∗

t

+ W

t

i

(x

∗

t

),

(13)

V

i

t

(x

t

)

≥ W

t

i

(x

t

),

(14)

V

t

(x

t

)

≥

_X

K

i=1

p

i

t

π

i

x

t

+

_X

K

i=1

p

i

t

− π

i

x

∗

t

+ W

t

i

(x

∗

t

)

.

(15)

The rstinequalityistight for

x

t

= x

∗

t

.

i

_{, η}

i

₎

is afeasible pointofthedualproblemwith

d

t

= d

i

t

whateverthevalueof

x

t

. Therefore,

W

t

i

(x

t

)

≥ π

i

x

t

+ β

i

b

t

+ ε

i

d

i

t

+ γ

i

g

t+1

− η

i

h

t+1

= π

i

_x

t

− π

i

x

∗

t

+ W

t

i

(x

∗

t

),

when e inequality (13). Let us proveinequality (14).

V

i

t

(x

t

)

is the valueof problem (11)and weknowfromthelinearprogrammingtheorythat

P

t

(x

t

, d

t

)

andits dualhavethesamevalue,

W

i

t

(x

t

)

. Problems(11)and

P

t

(x

t

, d

t

)

havethesame onstraintsbuttherstonehasagreater ostfun tion,when einequality(14) . Wenallyobtain:

V(x

t

) =

K

X

i=1

p

i

t

V

t

i

(x

t

)

≥

K

X

i=1

p

i

t

W

t

i

(x

t

)

≥

_X

K

i=1

p

i

t

π

i

|

{z

}

:=H

k

t

x

t

+

_X

K

i=1

p

i

t

− π

i

x

∗

t

+ W

t

i

(x

∗

t

)

|

{z

}

:=h

k

t

,

when einequality(15) .

(15)

Proposition 3 provides an expli it method to ompute anew optimality ut

H

k

t

(x

t

) + h

k

t

. Noti ethatthe omputationrequirestheresolutionoftheproblem(

P

t

(x

t

, d

t

)

)forthe

K

possible valuesof

d

t

. Moreover,forthepoint

x

∗

t

,theequality

H

k

t

x

∗

t

+ h

k

t

=

P

K

i=1

p

i

t

W

i

(x

∗

t

)

holds,whi h meansthatthenew uthasthehighestpossiblevaluein

x

∗

t

,giventhe urrentapproximationof

V

t+1

.

Figure 2 illustrates how the

T

sub-problemsex hange information during the forward and ba kwardpasses. Theprimalproblemasso iatedwithstage

t

passesonthenextstate

x

∗

t+1

to stage

t + 1

whereasthedual problempassesonanoptimality uttostage

t

− 1

.

Figure2: Aforwardpassandaba kwardpass

Stopping riterion Weuse the riterion des ribedin [16℄. It onsists in omputing alower boundandanupperoneofthevalue

V

0 (x

0 )

ofproblem

P

. Alowerbound

v

isdire tlyobtained with

V

0 (x

0 )

. Arelevantupperbound of

V

0 (x

0 )

ouldbethemean ost ofthestrategydened bytheapproximatevalue fun tions

V

t

. To omputethisbound, weshould solvethe problems

P

t

(x

t

, d

t

)

for all the possible s enarios of demand

d

t

, whi h is impossible. Instead, we an omputea onden eintervalofthisupperboundwithaMonte-Carlomethod. Wexanumber

N

su iently large, we simulate

N

realizations of

d

t

. We solve the asso iated approximate solutionsand ostswithproblems

P

t

(x

t

, d

t

−

1, 96σ

√

N

, v +

1, 96σ

√

N

i

.

Then,the riterionisthefollowing: stopthealgorithmwhen

v

belongsthe onden ealgorithm. Thistestmustberegularlyperformedduringthealgorithm.

Convergen e It is proved in [6, hapter 7, theorem 1℄ and [17℄ that the SDDP algorithm onvergesinanitetimeto anoptimalsolution. Theproofrelies onthefa t that thereisonly anitenumberofoptimality uts. Seealso[1℄foraproofof onvergen einamoregeneral ase and[20℄forastatisti alanalysisofthe onvergen eofthealgorithmanda riti ismoftheabove stopping riterion(remark1).

5 Review of omputational improvements

Sto hasti programming te hniques have been widely studied in literature and numerous en-han ementshavebeendesignedforvarioussto hasti problems. Inthisse tion,wegiveashort review of someof the enhan ements that were proposed. We donot aim at being exhaustive but wetry to onfront them to our parti ular problem. We refer to [6℄ for a survey onthese te hniques.

(16)

Similaritiesofsub-problems Duringtheba kwardpass,atstage

t

,onehastosolve

K

very similar problems, sin ethe onstraintof equilibrium betweensupply anddemand,

Eu

t

= d

t

is theonlyoneto hange. Te hniqueshavebeenproposedtoredu e thenumberofsimplexpivots required to solvelinearprograms dieringonly onthe right-hand-side of the onstraints. We referagainto [6,se tion5.6℄ formoredetails.

For our appli ation, we used a ommer ial solver, CPLEX. The software in ludes e ient te hniquesofhotstarts,redu ing onsiderablythenumberofsimplexpivots. Roughlyspeaking, in theba kwardpass,whenthevalueof

d

t

hanges,thesolverinitializesthesimplexalgorithm withthepreviousoptimalsimplexbasis.

Initialization ofthe algorithm Forourproblem,thesto hasti dataappearlinearlyatthe right-hand side of the onstraints only. As wementioned in lemma 2, the intermediate value fun tions

V

t

(x

t

, d

p

i

t

d

i

t

. Theproblem obtainedis deterministi and anbe a urately solved withforwardand ba kwardpasses. Denoting by

V

e

t

(x

t

)

the asso iated value fun tions,thefollowinginequality:

e

V

t

(x

t

)

≤ V

t

(x

t

),

∀t, 0 ≤ t ≤ T, ∀x

t

,

holds and an be shown by ba kwardre ursion, by using Jensen's inequality. Therefore, the optimality uts omputedforthedeterministi problemremainvalidforthesto hasti problem. These utsenableustoredu ethenumberofiterationsneededtorea h onvergen e. A omplete proofofthisproperty an befoundin [3℄. Thete hniquewasalsousedsu essfullyin[11℄.

Forourappli ation,thisinitializationprovede ientandallowsanimportantde reasingof theforward ost.

Sequen ing proto ols Thealgorithm wedes ribedisknownasthefast-forward,fast-ba k pro edure. Roughlyspeaking,oneiterationofouralgorithmgoesfromstage

0

tostage

T

− 1

in theforwardpass, thenitgoesfromstage

T

− 1

We followed the on lusions of the numeri al results of Gassmann [13℄, showing that the fast-forwardfast-ba kpro edure wasmoree ient.

Sampling TheSDDPalgorithmbelongstothebroad lassofsamplingalgorithmsinsofaras itworkswithonlyasmallpartofthesetofallthepossiblerealizationsofthesto hasti pro ess

d

t

. The s enarios involved are those obtained after the forward pass. In [11℄, Donohue and BirgedevelopedtheAbridgedNestedDe ompositionalgorithm,whi husesmorerepresentative realizationsofthesto hasti pro essintheforwardpass. LetusalsomentiontheCutting-Plane andPartial-Samplingalgorithmof Chenand Powell[9℄andtheDynami OuterApproximation SamplingAlgorithmsof Philpott andGuan [17℄, whi h usesampling to avoidthe resolutionof allthe

K

subproblemsatea hstageintheba kwardpass.

Wehavenottestedthese te hniques.

Multi uts Thiste hniquewasproposed byBirgeand Louveauxin [7℄. Inourdes riptionof thealgorithm,wesolveintheba kwardpassalltheproblemsasso iatedwiththedierentvalues

(17)

of

d

t

,to obtainanoptimality utofthevaluefun tion

V

i

t

(x

t

)

for all

i

,followingproposition3. Then, weonly keep themeanvalue ofthese uts, whi h is a utfor thevalue fun tion

V

t

(x

t

)

. Onthe ontrary, theidea ofthe multi ut algorithm isto keepin memory all theintermediate uts. Wedenotethemby

H

j,i

t

x

t

+ h

j,i

t

,thenewproblemstosolvearethefollowing:

min

u

t

, x

t+1

, v

x

t+1

=x

t

+A

t

u

t

,

Bu

t

≤b

t

,

Eu

t

=d

t

,

x

t+1

∈X

t+1

v

i

_{≥ H}

i

t+1

x

t+1

+ h

i

t+1

,

∀i

c

t

u

t

+

K

X

i=1

p

i

t

v

i

.

(16)

Withthismethod,weobtainabetterapproximationofthevaluefun tions. Ontheotherhand, thenumberofoptimality utsismultipliedby

K

.

This te hnique ouldnotbetestedfor ourappli ationproblem, giventhat itrequires atoo widespa ememory. Onthe ontrary,wetriedtolimitthenumberofoptimality uts.

Parallelization SDDP an be e iently parallelized. A rst approa h onsists in realizing severalforwardandba kwardpassesondierentpro essors. Anotherapproa h onsistsin par-titioningthe time period. Ea h pro essorsolvesthe problems asso iatedto the stages of one subsetofthepartition. Itisthenpossibletolaun hanewiterationofthealgorithmevenifthe previousoneisnotnishedyet. Wereferto[5,19℄fordetails.

Wehavenottestedsu h te hniques.

6 Pruning optimality uts

Whilethe algorithmisrunning,thenumberofoptimality utsin reases. Optimality uts om-puted during the rst iterations of the algorithm an qui kly be ome obsolete, espe ially the onesasso iatedwiththerststages,be ausetheyareobtainedwithaverybadapproximation ofthevaluefun tions. Thea umulationof uts anbepenalizing fortworeasons. First,ifthe numberof iterations to rea h the onvergen e is high, one risksex eeding the apa ity of the storagespa e. Moreover,itslowsdownthe solvingof linearprograms. Inthisse tion, wedeal withthequestionofpi kingouttheoptimality utswhi h anbe onsideredasrelevant.

Givenanapproximation

V

t

(x) = max

k

H

k

t

x + h

k

t

,wesaythatone ut

j

isuseless ifforall

x

in

X

t

,

H

t

j

x + h

j

t

≤ max

k,k6=j

[H

k

t

x + h

k

t

],

(17) or,equivalently,if

V

t

(x) = max

k,k6=j

[H

k

t

x + h

k

t

].

(18)

Ifagiven ut

j

isuseless,we anremoveitfromthesetoptimality utswithoutwithoutdamaging theapproximation ofthe valuefun tion

V

t

. On gure3, ut

1

is useless, while theothers are useful.

(18)

Figure3: Cut

1

isuseless.

Figure4: Testofusefulnessfor ut

0

.

Testofusefulness Arstpropositiontodetermineifa ut

j

isuseless onsistsinsolvingthe followinglinearprogram,thatwe alltest ofusefulness:

max

x∈X

t

H

t

j

x + h

j

t

− max

k,k6=j

[H

k

t

x + h

k

t

]

(19)

= max

x∈X

t

min

k,k6=j

[H

j

t

x + h

j

t

− H

t

k

x

− h

k

t

]

(20)

=

max

x

_{∈ X}

t

, y

∈ R

y

≤ (H

t

j

− H

t

k

)x + (h

j

t

− h

k

t

),

∀k, k 6= j

y.

(21)

The utis useless ifand only if thesolution

y

is non-positive. It should then bedeleted. On the ontrary, if the value of

y

is high, it means that the examined ut widely ontributes to theapproximationof thevaluefun tion. Noti e that inour ase,

X

t

is bounded and thus, the problemhasanite value. Figure4showsthesolution

(x, y)

to thelinearprogramfor ut

0

.

Thetestofusefulnessprovidesuswithanexa tmethodtodetermineifa utisuseless,with onelinearprogram. However,it isimpossibletoperformthistest foralltheoptimality utsat ea hiterationofthealgorithm,this wouldtaketoomu htime.

Territoryalgorithm In[15℄,thevalueofthetestofusefulnessisdes ribedastheimportan e metri (see theirproblem(10),whi h isequivalentto ourproblem (21)). Letus mentionthat in [14,se tion6℄,thenumberof utstosele tisxed andtheproblemofpruningisformulated asa ombinatorialoptimizationproblem. Dierentmethods aretested.

Ourapproa h hereis dierent, in sofaraswedonot x thenumberof sele ted uts. Our territory algorithm enablesus to sele t the uts whi h are potentially useless. In the test of usefulness,wetestforallthepointsof

X

t

ifagiven utis aboveorunder theotherones. The idea here is to perform the same kind of test for a dis rete set of points of

X

t

. The method onsists inasso iatingwithea h ut

j

asetof pointsof

X

j

t

≥ H

t

k

x + h

k

t

. This setof pointsiswhat we allaterritory: anarea of

X

t

where ut

j

ishigherthan theotherones. Whena ut

j

is omputed forapoint

x

t

,the algorithmisthefollowing:

Set

P

j

t

=

{x

t

}

.

Sear hforthe ut

k

whi h maximizes

H

k

t

x

t

+ h

k

t

. If

H

k

t

x

t

+ h

k

t

> H

j

t

x

t

+ h

j

t

,thendo:

P

j

t

← ∅

(19)

P

k

t

← P

t

k

∪ {x

t

}

Forall ut

k

,forall

x

in

P

k

t

,if

H

j

t

x + h

j

t

> H

t

k

x + h

k

t

, thendo:

P

j

t

← P

t

j

∪ {x}

P

k

t

← P

t

k

\{x}

Ifforsome ut

k

,

P

k

t

=

∅

,thenremove ut

k

fromtheset ofoptimality uts.

, H

j

t

x + h

j

t

< H

t

k

x + h

k

t

,

thereexists

x

in

X

t

su has

H

j

t

x + h

j

t

≥ max

_k,k6=j

[H

t

k

x + h

k

t

].

However, in su h a situation, it is possible that the algorithm omputes again a ut whi h hasbeeneliminatedpreviously,with anotherpointasso iated. Asa onsequen e,theterritory algorithmdoesnotprevent SDDP from onverging. Noti e that thetotal numberof pointsin theterritories

P

j

t

in reasesthroughoutthealgorithm,therefore,thetestbe omesmoreandmore pre ise.

Theadvantagesofthismethod arethefollowing:

thesele tionofthe utstoberemovedisverysimpleanddoesnottakemu htime

themethodfo useson utswhi h areuseful inarealisti areaof

X

t

,sin ethepointsof theterritoriesareobtainedwithforwardpasses.

Figure 5 showsoptimality uts with their territories. In gures 6and 7, a point

x

of the territoryof ut

1

,delete the ut,itwasa tuallyuseless.

If

y > 0

,keepthe utandset:

P

j

(20)

Figure5: Cutsandterritories

Figure6: Transferofapoint

x

from

P

1 t

to

P

3 t

Figure7: Eliminationofanon-useless ut, ut

1

7 Numeri al results

7.1 Case study

Themainfeaturesofour asestudyarepresentedintable1. Theprogramusedis

C♯

,thesolver Dimensionofthestate

x

t

19

Numberofzones 1 Numberofstorages 1 Numberof ontra ts 18 Numberoftimesteps 365 Numberofrealizationsof

d

at time

t

10

Table1: Casestudy

used isCplexv

12.2

and thetestshavebeenperformedonapi-pro essor omputerwith a2.13 GHzquad- oreCPUand16GBytemainmemory. Inthenumeri alteststhatwehaverealized, we ompare three methods. In the rst, there is no sele tion of the uts. In the se ond,the territoryalgorithmisperformedandinthethird,the ombinationoftheterritoryalgorithmand thetest ofusefulnessisperformed. Seepages20 and21thedierentgraphsand tablesforthe numeri alresults.

(21)

Notethatwedistinguishanoptimizationphaseandasimulationphase. Thealgorithmisrun duringtheoptimizationphase. No onvergen etestisrealizedduringthisphase. The optimiza-tionphasetakesintoa ountthetime neededto sele tthe uts. Duringthesimulationphase, we omputethesolutionsasso iated with1000dierents enariosof demand. Like onvergen e tests,thisphaseisonly omposedofforwardpasses.

7.2 Comparison of methods of sele ting uts, without initialization of the algorithm

Optimality uts Figure 10 shows the evolutionof the number of ane minorants. As was expe ted, thenumberof utsis thesmallestwiththeterritoryalgorithmand thelargestwhen thereis nosele tionof the uts. Figure 12showsthemean numberof utspertimestepafter the500iterationsofthealgorithm. Morepre isely,gure11representsthenumberof utsfor ea htimestepoftheproblem. Weobservethat whenthereisasele tion,thenumberof utsis dire tlylinkedto thesizeoftheareaofthestatespa ewhi h hasbeenexplored. Theterritory algorithm enables to divide by 10 the numberof uts, yet a small loss of information has to beobservedsin etheforward ost isslightlyhigherthantheforward ostwhen nosele tionis realized. Figure 13 showsthe forward ostwith respe t to the number of hyperplanes for the threemethods. Itappears thatthe territory algorithm provides themost e ientrepresentation ofthe value fun tion,with thesmallestnumberof uts.

Forward ost Figure 14 shows the evolution of the forward ost when no initialization is performed. The number of iterations of the algorithm is xed to 500. The forward ost is de reasing atsimilarratesfor the threemethods.

Computation time Table15showsthetimeneededto run ea h ofthethree methods. The length of the optimization phase is of the same order for the territory algorithm and when there isno sele tion. Onthe ontrary,the ombinationof theterritoryalgorithm andthe test of usefulness takes mu h time, sin e the sele tion of uts requires the solving of many linear problems. The ombination of the territory algorithm and the test of usefulness is ine ient duringthe optimizationphase.

Thelengthofthesimulationphaseisdire tlylinkedtothenumberofhyperplanes,sin ethere areonlyforwardpasses. Consequently, the simulation phase ismu h shorterwith the territory algorithm thanwiththe other methods.

7.3 Results with an initialization of the algorithm

Wehaverealizedthesametestswithaninitializationofthealgorithm.Theinitializationphase in ludes100iterationswith ademand equaltoitsmeanvalue. Afterthis phase,werealize400 iterationsofthealgorithmwiththe lassi almethod.

Optimality uts Figure16showstheevolutionofthenumberofoptimality uts,whi his lose to thenumber of optimality utswhen there is no initialization. Figure 17 shows thenumber of uts at ea h time step. Figure 18showsthemean numberof utsper timestep. Asshown bygure19, the territory algorithm, used with aninitialization stillprovides the most e ient representation of valuefun tion.

(22)

Forward ost Figure 20 shows the evolution of the forward ost. After 500 iterations, the dieren e between theforwardand the ba kward ostis aboutve timessmaller with an ini-tializationaswithout,asshownbygure9(theba kward ostbeingequaltoapproximately12 350). Notethatitisnormalthatthe ostdoesnotde reaseafterea hiteration,sin etheforward ostisonlyanapproximationofthereal ostofthestrategyasso iatedwiththe omputedvalue fun tions. The initialization is very e ient and permits to start with smaller values, for the three methods.

Computation time Thetimesof omputationasso iatedwiththeoptimizationandthe sim-ulation phases are shown in table 21. The omputation times are similar to the ase without initialization, whereasmu hmorea uratesolutions havebeenobtained.

Note that the initialization phase is not su ient to solve the sto hasti problem, sin e a largeareaofthespa estateis overedbythedierentpossibletraje tories. Seegure8,whi h showsthetraje toriesfollowedbythesto kofgasfordierents enarios,afterthesolvingofthe problem.

0

0.2

0.4

0.6

0.8

1 nov.

jan.

mar. may

july

sep.

nov.

Level of the stock

Time

Figure8: Levelofsto kfordierent s enar-ios

12400

12600

12800

13000

13200

100

200

300

400

500 Forward cost

Number of iterations

Without initialisation

With initialisation

Figure 9: Forward ost with and without initialization

8 Con lusion

Wehavepresentedagasportfolio managementproblem, solvedwiththeSDDP algorithm. We have fo used on methods of sele ting the most relevant uts. We have proposed a territory algorithm whi h eliminatesuseless utsand anexa ttestof usefulness,whi h istra table (but ine ient)whenasso iatedwiththeterritoryalgorithm.

Thenumeri alresultsshowrstthegreatadvantageofusingtheinitializationofthealgorithm des ribed se tion 5. With this te hnique, the forward ost de reases very qui kly at the rst iterations,andwe anobtainmorea uratesolutionsto theproblem.

Se ondly,the onvergen eoftheforward ostissimilarforthreemethods(withoutsele tion, the territoryalgorithm andthe ombinationwith the testof usefulness), whether ornotthere is an initialization. It appears that thesele tion of utshas onlya small impa ton the value of forward osts and on the time of the optimization phase. On the ontrary, the number of hyperplanes anbedividedby10withane ientsele tion,liketheterritoryalgorithm,whi his ofgreatinterestifalongsimulationphaseisperformedorifmanyforwardpassedareperformed during onvergen etests. Letusre allalsothatsele ting utsallowstoavoidmemoryproblems when omputinga uratesolutions.

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Resultswithout initialization of the algorithm

0

100

200

300

400

500

600

700

0

100

200

300

400

500 Number of cuts

Number of iterations

Without selection

Territory algorithm

Combination

Figure10: Evolutionofthemeannumberof optimality uts

0

100

200

300

400

500

600

700 nov.

jan.

mar. may

july

sep.

Number of cuts

Months

Without selection

Territory algorithm

Combination

Figure 11: Number of optimality uts at ea htimestep

Withoutsele tion 490 Territoryalgorithm 220 Combinationofterritoryalgorithm 55

andtestofusefulness

Figure12: Meannumberof utspertimestepafter500iterations

12500

13000

13500

14000

0

100

200

300

400

500 Forward cost

Mean number of cuts

Without selection

Territory algorithm

Combination

Figure13: Forward ostwithrespe ttothe meannumberof uts

12500

13000

13500

14000

0

100

200

300

400

500 Forward cost

Number of iterations

Without selection

Territory algorithm

Combination

Figure 14: Evolution of the forward ost withrespe ttothenumberofiterations

Optimization phase Simulationphase Withoutsele tion 543s 798s Territoryalgorithm 690s 168s Combinationofterritoryalgorithm 2181s 382s

andtestofusefulness

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Resultswith aninitialization ofthe algorithm

0

100

200

300

400

500

600

700

100

200

300

400

500 Number of cuts

Number of iterations

Without selection

Territory algorithm

Combination

Figure16: Evolutionofthemeannumberof optimality uts

0

100

200

300

400

500

600

700 nov.

jan.

mar. may

july

sep.

Number of cuts

Months

Without selection

Territory algorithm

Combination

Figure 17: Number of optimality uts at ea htimestep

Withoutsele tion 490 Territoryalgorithm 275 Combinationofterritoryalgorithm 84

andtestofusefulness

Figure18: Meannumberof utspertimestepafter500iterations

12300

12400

12500

0

100

200

300

400

500 Forward cost

Mean number of cuts

Without selection

Territory algorithm

Combination

Figure19: Forward ostwithrespe ttothe meannumberof uts

12300

12400

12500

100

200

300

400

500 Forward cost

Number of iterations

Without selection

Territory algorithm

Combination

Figure 20: Evolution of the forward ost withrespe tto thenumberofiterations

Optimizationphase Simulationphase Withoutsele tion 772s 649s Territoryalgorithm 792s 205s Combinationofterritoryalgorithm 2090s 408s

andtestofusefulness

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