applying high performance computing to multi-area...

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Applying High Performance Computing toMulti-Area Stochastic Unit Commitment for

Renewable IntegrationINFORMS 2012

Anthony Papavasiliou, Shmuel Oren

Department of Industrial Engineeringand Operations Research

U.C. Berkeley

October 17, 2012

A. Papavasiliou, S. Oren Parallel Stochastic Unit Commitment

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Outline

1 Introduction

2 ModelLagrangian DecompositionScenario SelectionWind Production Model

3 Results


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Motivation and Research Objective

Increased computational burden in power systemsoperations due to:

renewable penetration anddemand response integration

Potential applications of distributed computation:Stochastic optimizationRobust optimizationTopology control, system expansion planningOptimization of storage (deferrable loads/demandresponse, hydro-scheduling)

Want to quantify sensitivity of:unit commitment policyduality gapscost performance

on number of scenarios.


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Parallel Computing Literature in Power Systems

Monticelli et al. (1987): Benders decomposition algorithmfor SCOPFPereira et al. (1990): Applications of parallelization invarious applications including SCOPF, composite(generator, transmission line) reliability, hydrothermalschedulingFalcao (1997): Survey of HPC applications in powersystemsKim, Baldick (1997): Distributed OPFBakirtzis, Biskas (2003) and Biskas et al. (2005):Distributed OPF


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PSR Cloud

Industry practice for hydrothermal scheduling in 8000 nodes


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Lagrangian DecompositionScenario SelectionWind Production Model

Full Model

Application: stochastic unit commitment for large-scalerenewable energy integrationTwo-stage model representing DA market (first stage)followed by RT market (second stage)

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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Unit Commitment Model

Domain D represents min up/down times, ramping rates,Kirchhoff voltage/current laws, thermal limits of linesGenerator set partitioned between fast (Gf ) and slow (Gs)generators

(UC) : min∑g∈G

∑t∈T

(Kgugt + Sgvgt + Cgpgt )

s.t .∑

g∈Gn

pgt = Dnt

P−g ugt ≤ pgt ≤ P+g ugt

elt = Bl(θnt − θmt )

(p,e,u,v) ∈ D


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Stochastic Unit Commitment Model

(SUC) : min∑g∈G

∑s∈S

∑t∈T

πs(Kgugst + Sgvgst + Cgpgst )

s.t .∑

g∈Gn

pgst = Dnst ,

P−gsugst ≤ pgst ≤ P+gsugst

elst = Bls(θnst − θmst )

(p,e,u,v) ∈ Ds

ugst = wgt , vgst = zgt ,g ∈ Gs


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Lagrangian Decomposition Algorithm

Past work: (Takriti et al., 1996), (Carpentier et al., 1996),(Nowak and Römisch, 2000), (Shiina and Birge, 2004)Key idea: relax non-anticipativity constraints on both unitcommitment and startup variables

1 Balance size of subproblems2 Obtain lower and upper bounds at each iteration

Lagrangian:

L =∑g∈G

∑s∈S

∑t∈T

πs(Kgugst + Sgvgst + Cgpgst )

+∑

g∈Gs

∑s∈S

∑t∈T

πs(µgst (ugst − wgt ) + νgst (vgst − zgt ))


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Parallelization

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 ED

c

Lawrence Livermore National Laboratory Hyperion cluster:1152 nodes, 2.4 Ghz, 8 CPUs/node, 10 GB/nodeMPI calling on CPLEX Java callable library


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Scenario Selection

Past work: (Gröwe-Kuska et al., 2002), (Dupacova et al.,2003), (Heitsch and Römisch, 2003), (Morales et al., 2009)Scenario selection algorithm inspired by importancesampling

1 Generate a sample set ΩS ⊂ Ω, where M = |ΩS| isadequately large. Calculate the cost CD(ω) of each sampleω ∈ ΩS against the best deterministic unit commitment

policy and the average cost C =M∑

i=1

CD(ωi )

M.

2 Choose N scenarios from ΩS, where the probability ofpicking a scenario ω is CD(ω)/(MC).

3 Set πs = CD(ω)−1 for all ωs ∈ Ω.


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Wind Production Model

Relevant literature: (Brown et al, 1984), (Torres et al.,2005), (Morales et al, 2010)Calibration steps

1 Remove systematic effects:

ySkt =

ykt − µkmt

σkmt.

2 Transform data to obtain a Gaussian distribution:

yGSkt = N−1(Fk (yS

kt )).

3 Estimate the autoregressive parameters φkj and covariancematrix Σ using Yule-Walker equations.


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Data Fit


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WECC Model

124 units (82 fast, 42 slow), 225 buses, 375 transmissionlines

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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Unit Characteristics

Type No. of units Capacity (MW)Nuclear 2 4,499Gas 88 18,745.6Coal 6 285.9Oil 5 252Dual fuel 23 4,599Import 22 12,691Hydro 6 10,842Biomass 3 558Geothermal 2 1,193Wind (deep) 10 14,143Fast thermal 82 9,156.1Slow thermal 42 19,225.4


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Sensitivity of Optimal Policy on Number of Scenarios

Table: Day-ahead reserve capacity (MW)

S10 S50 S100 S1000WinterWD 8846 8575 8885 8600SpringWD 9173 8639 9077 8572

SummerWD 12185 12327 12261 12497FallWD 10039 10182 9771 9989

WinterWE 7700 8074 6978 7170SpringWE 7588 7001 7105 7032

SummerWE 11041 10545 10795 10810FallWE 9476 8669 8665 8637Average 9744 9542 9538 9485


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Unit Commitment: Weekdays

S10 tends to commit the most capacity, S1000 tends to beencapsulated


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Unit Commitment: Weekends

S10 tends to commit the most capacity, S1000 tends to beencapsulated


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Sensitivity of Bounds on Number of Scenarios

Table: Lower and Upper Bound ($ 1000)

S10 S50 S100 S1000WinterWD (254, 325) (100, 169) (-165, -93) (-180, -105)SpringWD (1073, 1123) (28, 135) (97, 154) (115, 164)

SummerWD (-367, -234) (48, 87) (187, 304) (-76, 62)FallWD (-146, -45) (-292, 397) (-191, -77) (-108, 7)

WinterWE (185, 295) (-323, 413) (-504, -411) (-84, 17)SpringWE (668, 783) (-121, 202) (-228, -153) (52, 128)

SummerWE (-57, 99) (-283, 438) (-150, 93) (-108, 50)FallWE (810, 913) (-530, 624) (-304, -207) (-92, 7)


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Improving the Gap Versus Adding More Scenarios

Sxx derived by terminating LR when S1000 bound attainedSxx? derived by running LR for 100 iterations

Policy Gap ($) Cost ($M)S10 97827 7.303S10? 70559 7.300S50 92413 7.308S50? 62190 7.286S100 93711 7.299S100? 67069 7.289S1000 98485 7.301

More benefits accrued from reducing gapS10 does as well as S1000, thereby validating scenarioselection policy


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Running Time


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Conclusions and Perspectives

ConclusionsValidation of scenario selection algorithm: Theimportance sampling scenario selection algorithm with 10scenarios performs as well as naive scenario selection with1000 scenariosDecreasing the duality gap versus increasing thenumber of scenarios: Reducing the duality gap seems toyield superior benefits relative to adding more scenarios

PerspectivesStochastic unit commitment versus security constrainedunit commitmentParallelization of Benders decomposition for dealing withN-1 reliability and zero-weight scenariosSelf-scheduling and self-commitment of slow units inday-ahead electricity markets for dealing with priceuncertainty


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Thank you

Questions?

Contact: [email protected]

http://www3.decf.berkeley.edu/∼tonypap/publications.html


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