Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. SAND NO. 2011-XXXXPSandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. SAND NO.

SolvingtheMixedIntegerNon-linearProgrammingProblemofUnitCommitmentonACPowerSystems

AnyaCastilloSandiaNationalLaboratoriesAlbuquerque,NMUSA

CarlLaird– SandiaNationalLaboratoriesJianfeng Liu– PurdueUniversityJean-PaulWatson– SandiaNationalLaboratories

FERCSoftwareConference June27,2017

§ Today’sPractice§ Contributions

§ Overview§ LocalSolutionMethod§ GlobalSolutionMethod§ UC+ACOPFResults

§ OngoingWork

Overview

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TODAY’SPRACTICE

§ OperationalChallenges§ CommittingLeastCost+MaintainingReliability§ Out-of-MeritReliabilityCommitments§ Improvingconvergencebetweenday-aheadandreal-timeprices

§ AlgorithmicChallenges§ Accountingforreliabilityneedsindispatchandpricingoptimization§ Betterphysicalrepresentationofthegeneratingunitsandunderlying

network

IssuesinDay-AheadMarkets

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UnitCommitmentintheDay-AheadMarket

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Current Practices Proposed ApproachUC/Security-Constrained UC• Copper-plate (no network/single node)• Ignores congestion; requires cutsets to

proxy capacity limits on network• Most tractable

SCUC DCOPF• Real power flows only (proportional to

current)• BΘ (full) or PTDF (compact) approach

Extensions:• Accounts for losses • Nomograms/cutsets to proxy reliability

requirements

SCUC ACOPF• Co-optimizes real and reactive power

dispatch• Accounts for commitments needed for

blackstart service, reactive support, voltage support, and interface control

• Nonlinear, nonconvex on meshed networks

Thelinkbetweenphysicsandprices§ Locationalmarginalpricing(LMP)isthespotpriceofelectricity§ Dualvariable/Lagrangemultiplier(λn)torealpowerbalancingatallbuses

TheLMPincorporatesthemarginalcostofsupplyingthenextMWofloadforagivenlocationintime;includes1.marginalunitcost,2.costofnetworkcongestion(duetothermallinelimits),and3.costofrealpowerlossesonthenetwork

⇡ |vn|X

m2N|vm| (Bnm✓nm)

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CONTRIBUTIONS

CONTRIBUTIONSOVERVIEW

UC+ACOPF:MINLP

AC Network LimitsReal power balancingReactive power balancingVoltage magnitude boundsThermal line limitsSpinning reserves

Apparent Power Production Limits §

Max/min real/reactive power generation Ramp up/down rates on real powerMinimum up/down time

§ Extends Morales-España, Latorre, and Ramos, “Tight and compact MILP formulation for the thermal unit commitment problem,” IEEE Trans. on Power Syst., vol. 28, no. 4, pp. 4897–4908, 2013.

System DataNodal voltage limits

Reserve requirementsReal/reactive power load

Transformer tap ratio and phase-shiftersThermal line limits and line R/X/B

Shunts

Generator DataSynchronous condensersT0 state and startup lags

Minimum up/down timeRamp up/down limits

Startup/shutdown ramp limitsMin/max real/reactive power limits

Min Production Costs + Startup Costs + No-Load Costs

subject to

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§ PolarPower-VoltagePowerFlowFormulation(PSV)

§ RectangularPower-VoltagePowerFlowFormulation(RSV)

§ RectangularCurrentInjectionFormulation(RIV)

NodalPowerBalancingisNonconvex

|vn,t|X

m2N|vm,t| (Gnm cos ✓nm,t +Bnm sin ✓nm,t)� p+n,t + p�n,t = 0, 8n 2 N

|vn,t|X

m2N|vm,t| (Gnm sin ✓nm,t �Bnm cos ✓nm,t)� q+n,t + q�n,t = 0, 8n 2 N

vrn,tX

m2N

⇣Gnmvrm,t �Bnmvjm,t

⌘+ vjn,t

X

m2N

⇣Gnmvjm,t +Bnmvrm,t

⌘� p+n,t + p�n,t = 0, 8n 2 N

vjn,tX

m2N

⇣Gnmvrm,t �Bnmvjm,t

⌘� vrn,t

X

m2N

⇣Gnmvjm,t +Bnmvrm,t

⌘� q+n,t + q�n,t = 0, 8n 2 N

irn,t �⇣ X

k(n,·)2F

irk(n,m),t +Gsnv

rn,t �Bs

nvjn,t

⌘= 0,

⇣vrn,ti

rn,t + vjn,ti

jn,t

⌘� p+n,t + p�n,t = 0, 8n 2 N

ijn,t �⇣ X

k(n,·)2F

ijk(n,m),t +Gsnv

jn,t +Bs

nvrn,t

⌘= 0,

⇣vjn,ti

rn,t � vrn,ti

jn,t

⌘� q+n,t + q�n,t = 0, 8n 2 N

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MINLPsolvedbyOuterApproximation§(OA)

𝑓(𝑥),𝑔 𝑥 ≤ 0,𝑥 ∈ 𝑋,𝑥𝑖 ∈ ℤ, ∀𝑖 ∈ 𝐼

§ Outer Approximation Algorithm (Duran and Grossman, 1986); Graphics (Belotti et al., 2013)

/𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒4𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑡𝑜

𝑓:ℝ? → ℝ, 𝑔:ℝ? → ℝAare twice continuously differentiable functions,𝑋 ⊂ ℝ?is a bounded polyhedral set, and𝐼 ⊆ {1, … , 𝑛} is the index set of integer variables

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CONTRIBUTIONSLOCALSOLUTIONMETHOD

MIN Piecewise linear cost function with penalty factors

Line Current Flows

Network Current Balancing

s.t.

ik (n,m)r = Re Y1,1

k vn +Y1,2k vm( ), ik (m,n)

r = Re Y2,1k vn +Y2,2

k vm( ) ∀k ∈K

ik (n,m)j = Im Y1,1

k vn +Y1,2k vm( ), ik (m,n)

j = Im Y2,1k vn +Y2,2

k vm( ) ∀k ∈K

inr − ik (n,m)

r +Gnshvn

r − Bnshvn

jk (n,⋅)∑( ) = 0 ∀n∈N

inj − ik (n,m)

j +Gnshvn

j + Bnshvn

rk (n,⋅)∑( ) = 0 ∀n∈N

Nodal Voltage Magnitude LimitsOuter approximation,

First-order Taylor series,Step-size bounds,

Tangential cutting planes, &Inequality constraints with

slack variables

Nodal Power InjectionsFirst-order Taylor series

Generator LimitsInequality constraints with

slack variables

Thermal Line (Flowgate) LimitsSet reduction, Outer approximation,

First-order Taylor series,Tangential cutting planes, & Inequality constraints with

slack variables

SuccessiveLinearProgramming(SLP)[R1]

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(2)

ACOPF Feasible

(1)

ACOPF Optimal

(3) SLP Feasible

(4) SLP Infeasible

SLPConvergenceProperties§

(1)AKKTpointtotheACOPFisfound(2)TheSLPoptimalsolutionisACOPFfeasiblebutnotoptimal§ Stillausefulsolution;maybebetterthana

DCOPFwithACfeasibilityordecoupledOPFsolution

(3) TheSLPoptimalsolutionisACOPFinfeasible§ Activepenaltiespresent§ Solutionmaybeusefuldependingupon

whethertheviolatedlimitsare“soft”or“hard”

(4) TheSLPisinfeasible§ TheACOPFmayhavenosolution§ TheSLPrequiresabetterinitialization

§ Extends Theorem 10.3.1 of Bazaraa et al. (2006)

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TimeComplexityPerformance

§ Runningtimeincreaseslinearlywiththenetworksize(p=1correspondstoalinearalgorithmicscaling)fortheSLPalgorithm

§ Potentiallyapplicableinthestricttimeframesofthereal-timemarkets

Best-Case Simulations All Converged SimulationsBaseline p R2 RMSE (s) p R2 RMSE (s)

NLP/KNITRO 1.42 0.83 1.46 1.47 0.82 1.40NLP/Ipopt 1.13 0.95 0.60 1.34 0.97 0.50SLP/CPLEX 0.97 0.99 0.20 1.01 0.98 0.33SLP/Gurobi 1.01 0.99 0.21 1.03 0.98 0.33

Thermally ConstrainedNLP/KNITRO 1.39 0.88 1.13 1.39 0.89 1.08NLP/Ipopt 1.11 0.98 0.36 1.22 0.97 0.50SLP/CPLEX 0.99 0.99 0.17 1.00 0.98 0.31SLP/Gurobi 1.06 0.99 0.23 1.05 0.97 0.36

⇥ (|N |p)

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