Moving Bits Not Watts: Geographically Coordinated Frequency Control

ZhenhuaLiu

JointworkwithJoshuaComden,TanN.Le,YueZhao,BongJunChoiStonyBrookUniversity,SUNYKorea

LBNLApril2018

Short bio

•  AssistantProfessoratStonyBrookUniversitysince2014•  Operationsresearch,computerscience,SmartEnergyTechnologiesCluster•  Onleaveduring2014.6-2015.8(ITRI-RosenfeldPostdoctoralFellowwithMaryAnn)

•  PhDinComputerScience2014,CaliforniaInstituteofTechnology•  MS&BEinComputerScienceandControlfromTsinghuaUniversity•  DoubledegreeBSinEconomicsfromPekingUniversity

•  Research•  Sustainablecomputing,Demandresponse,Onlineanddistributedoptimization,Schedulingandresourceallocation,BigDataSystems

•  4NSFgrantsduringthepast3years•  ~30publicationsincludingSIGMETRICS,NSDI,~1,600citations•  4USpatentsfiled,2ofwhichhavebeenawarded

My Wishlist

• Comments/suggestionsonmyresearch•  Exploringopportunities

•  Research:realdata,systems,domainknowledge,etc•  Funding:NYSERDA,DoE,otherfederalagency

•  Wecanleadorsub

•  Long-termcollaborations

Research examples

• DistributedOptimization•  GeographicalLoadBalancing+DistributedFrequencyControl•  DemandResponseProgramDesign

• OnlineOptimization•  SmoothedOnlineConvexOptimization•  CoincidentPeakPricing,Multi-scaleelectricitymarkets

• BigDataSystems•  Multi-resourceallocation•  BoundedPriorityFairness,InterchangeableResourceAllocation

4

DataCenterabilityforFrequencyControlUSDataCenterElectricityConsumption

UnitedStatesDataCenterEnergyUsageReport,LawrenceBerkeleyNationalLaboratory,2016.

2000 20142005 20100

20

40

60

80

Year

2%TotalUSConsumption

DataCenter

Fast&PreciseEnergyControlSystems

E.g.DynamicFrequencyScalingofCPUsEnergyStoragewithinUPSs

LargepotentialforFrequencyControl

CloudComputingisInterdependent

PowerNetwork

SharedComputationWorkload

DelayedorUnprocessedWorkload

SharedCost

PowerConsumptionDecisionsareDependent

Requiresfrequencystabilizationinseconds

Communication SpeedPrivacyInfeasibleforlarge

networksAgentsdon’twanttoshareprivate

objectives

IssueswithCentralizedControl PrimaryFrequencyControl

DistributedControlforPFCOptimalDecentralizedPrimaryFrequencyControlinPowerNetworks

C.ZhaoandS.Low-CDC2014

AssumesIndependentCostsàSeparableObjectiveFunction(validforsomeapplications)

CloudComputingCostsareInterdependentàInseparableObjectiveFunction

TheneedforDistributioninPrimaryFC

+

8

Goal:DesignPrimaryFCthatusesaNetworkofDataCenters

Approach:IncorporateInterdependentCosts

DistributedControlLaws SimulationResults

CloudComputingModel

WorkloadIncomingRate

𝑊

DataCenterProcessingRates

𝑟↓1 

ExcessProcessingRateoftheNetwork𝑠≔∑𝑗↑▒𝑟↓𝑗  −𝑊

𝑠<0

Workloadunprocessedordelayed

𝑟↓2 

𝑔(𝑠)

Network-wide(Interdependent)Cost

SingleDataCenterPowerConsumption𝑑↓𝑗 ≔ 𝑑 ↓𝑗 + 1/𝑎↓𝑗  𝑟↓𝑗 

Linearpowerusageprofile: 𝑑 ↓𝑗 constantoverhead 𝑎↓𝑗 conversioncoefficient

Costtoconvertelectricalàcomputing

power

𝑐↓𝑗 ( 𝑑↓𝑗 )

LoadBalancer

PowerNetworkModel

Bus𝒋RealPowerInjection

𝑃↓𝑗 ≔ 𝑝↓𝑗 − 𝐷↓𝑗 𝜔↓𝑗 − 𝑑↓𝑗 

Non-controllableloadsControllableloads(DataCenter)

Frequency-sensitive

FrequencyDeviation

𝜔↓𝑗 = 𝑑𝜃↓𝑗 /𝑑𝑡 

Voltagephaseangle(w.r.t.nominal 𝜃↓0 )

RealPowerFlows

𝐹↓𝑗 (𝜽)≔∑𝑘:(𝑗,𝑘)𝜖𝐸↑▒𝑌↓𝑗𝑘 sin�(𝜃↓𝑗 − 𝜃↓𝑘 )  −∑𝑖:(𝑖,𝑗)𝜖𝐸↑▒𝑌↓𝑖𝑗 sin�(𝜃↓𝑖 − 𝜃↓𝑗 )  Maximumpowerflowacrossline(𝑗,𝑘)𝜖𝐸

SystemDynamicsModel

SwingEquation𝑀↓𝑗 𝑑𝜔↓𝑗 /𝑑𝑡 = 𝑝↓𝑗 − 𝐷↓𝑗 𝜔↓𝑗 − 𝑑↓𝑗 − 𝐹↓𝑗 (𝜽)

PhysicalInertia

Equilibrium(𝜽↑∗ , 𝝎↑∗ , 𝑷↑∗ , 𝒅↑∗ ,𝑠)

𝑑𝜔↓𝑗↑∗ /𝑑𝑡 =0

𝑑𝑃↓𝑗↑∗ /𝑑𝑡 =0

𝜔↓𝑗↑∗ =𝜔

∀𝑗: (nochangeinfrequency)

(nochangeinpowerinjection)

(allbusesatsamefrequency)

GeographicFrequencyControlProblem

Minimize CostofPFCtoDataCenters CostofFrequencyDeviations+𝑠,𝒅,𝝎 𝑔(𝑠)+∑𝑗↑▒𝑐↓𝑗 (𝑑↓𝑗 )  ∑𝑗↑▒𝐷↓𝑗 /2 𝜔↓𝑗↑2  

ComputationalProcessingPowerBalances.t.

ElectricalPowerNetworkBalance

𝑠=∑𝑗↑▒𝑎↓𝑗 𝑑↓𝑗  −𝑊−∑𝑗↑▒𝑎↓𝑗 𝑑 ↓𝑗  

0=∑𝑗↑▒(𝑝↓𝑗 − 𝐷↓𝑗 𝜔↓𝑗 − 𝑑↓𝑗 ) 

13

Goal:DesignPrimaryFCthatusesaNetworkofDataCenters

Approach:IncorporateInterdependentCosts

DistributedControlLaws SimulationResults

DistributedControlLaws

𝜇(𝑡)= 𝜇(0)+ 𝛽∫0↑𝑡▒(𝑠(𝜏)− (𝑔′)↑−1 (𝜇(𝜏)))𝑑𝜏 

𝑑↓𝑗 (𝑡)= [(𝑐↓𝑗 ′)↑−1 (𝜔↓𝑗 (𝑡)− 𝑎↓𝑗 𝜇(𝑡))]↓𝑑 ↓𝑗 ↑¯𝑑 ↓𝑗  PowerNetwork

FrequencyDeviation 𝜔↓𝑗 

LoadControlleratDataCenter𝒋

Load 𝑑↓𝑗 

WorkloadBalancer

ProcessingRate𝑟↓𝑗 

Workload𝑊

Supply/DemandDisturbance

ExcessRate𝑠WorkloadBalancerSignalController

ControlSignal𝜇

ControlLawsConvergetoanOptimalPoint

Theorem:Thetrajectoryof(𝝎,𝑷,𝒅,𝑠,𝜇)asymptoticallyconvergestoanequilibriumpoint(𝝎↑∗ , 𝑷↑∗ , 𝒅↑∗ , 𝑠↑∗ , 𝜇↑∗ ).

Theorem:Anequilibriumpoint(𝝎↑∗ , 𝑷↑∗ , 𝒅↑∗ , 𝑠↑∗ , 𝜇↑∗ )isoptimaltotheGeographicFrequencyControlProblem.

Stability

Optimality

16

Goal:DesignPrimaryFCthatusesaNetworkofDataCenters

Approach:IncorporateInterdependentCosts

DistributedControlLaws SimulationResults

SimulationSetupNewEnglandIEEE39-bus

=25MWDataCenter

=Disturbanceof50MWDropinPower

10DataCenters=1.8%TotalDemand

PowerSystemToolbox(Matlab)

EachDataCentersharesafractionofa100MWequivalentcomputationalworkload.

Interdependentcost:𝑔(𝑠)=𝛾𝑠↑2 

Eachwithdifferentefficiency

http://icseg.iti.illinois.edu/ieee-39-bus-system/

ProposedControlLawsStabilizetheSystem

60

59.95

59.90 10 20 30 40 50

Time(sec)

Freq

uency(Hz)

5

0

-100 10 20 30 40 50

Time(sec)

Power(M

W)

-5

OLC

Proposed

NoControl

OLC(NoInterdependentCosts)OptimalDecentralizedPrimaryFrequencyControlinPowerNetworks

C.ZhaoandS.Low-CDC2014

Disturbanceof50MWDropinPowerDataCenterDemandChange

Convergestoequilibriumwithin30seconds

MostefficientDataCenter

CostofProposedControlLawsisnearOptimal

50

0

100

OLC Proposed Optimal

Totalcost($/sec)

Costs=Interdependent=Independent

ProposedincorporatesInterdependentcosts,whereasOLCdoesnot.

20

Goal:DesignPrimaryFCthatusesaNetworkofDataCenters

Approach:IncorporateInterdependentCosts

DistributedControlLaws SimulationResults

AsymptoticallyConvergestoward

OptimalCost

ConvergestoanEquilibriumthatisnearOptimalCost

Bonus:CommunicationandimplementationDelay

DistributedControlLaws

𝜇(𝑡)= 𝜇(0)+ 𝛽∫0↑𝑡▒(𝑠(𝜏)− (𝑔′)↑−1 (𝜇(𝜏)))𝑑𝜏 

𝑑↓𝑗 (𝑡)= [(𝑐↓𝑗 ′)↑−1 (𝜔↓𝑗 (𝑡)− 𝑎↓𝑗 𝜇(𝑡))]↓𝑑 ↓𝑗 ↑¯𝑑 ↓𝑗  PowerNetwork

FrequencyDeviation 𝜔↓𝑗 

LoadControlleratDataCenter𝒋

Load 𝑑↓𝑗 

WorkloadBalancer

ProcessingRate𝑟↓𝑗 

Workload𝑊

Supply/DemandDisturbance

ExcessRate𝑠WorkloadBalancerSignalController

ControlSignal𝜇

Outdated𝜇(𝑡-𝚫))

Systemwithdelay

Equilibrium(𝜽↑∗ , 𝝎↑∗ , 𝑷↑∗ , 𝒅↑∗ ,𝑠)

𝑑𝜔↓𝑗↑∗ /𝑑𝑡 =0

𝑑𝑃↓𝑗↑∗ /𝑑𝑡 =0

𝜔↓𝑗↑∗ =𝜔

∀𝑗: (nochangeinfrequency)

(nochangeinpowerinjection)

(allbusesatsamefrequency)

Andholdforatleast𝚫oftimeoftime

Stability:if𝛽issmallenough,thetrajectoryasymptoticallyapproachesanequilibriumpoint.

Corollary:Anequilibriumpoint(𝝎↑∗ , 𝑷↑∗ , 𝒅↑∗ , 𝑠↑∗ , 𝜇↑∗ )isoptimaltotheGeographicFrequencyControlProblem.

0 10 20 30 40 50

Time (sec)

59.9

59.95

60

Fre

qu

en

cy (

Hz)

OLC

proposed

none

(a) Frequency

0 10 20 30 40 50Time (sec)

-10

-5

0

5

Po

we

r (M

W)

(b) Power demand change, �j

0 10 20 30 40 50Time (sec)

0

50

100

tota

l co

st (

US

D/s

ec)

OLC

proposed

(c) Cost change

Fig. 4. Trajectories of the system state variables: (a) Bus frequencies for each of the ten buses containing a datacenter under the three different controlschemes; (b) changes in load for each of the ten datacenters under the proposed control; (c) Total cost changes under OLC and the proposed control.

OLC proposed Lower bound0

20

40

60

80

100

tota

l co

st (

US

D/s

ec)

independent interdependent

(a) Total cost

1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

tota

l co

st (

US

D/s

ec)

OLC proposed

(b) Independent costs

1 2 3 4 5 6 7 8 9 10

-10

-5

0

5

Po

we

r (M

W)

OLC proposed

(c) Load deviations

Fig. 5. (a) Total equilibrium cost separated by independent and interdependent costs of OLC, the proposed control, and the lower bound. For each datacenter(b) and (c) give the independent costs and equilibrium load deviations respectively under OLC and the proposed control. Note: The datacenters are numberedin decreasing order of efficiency aj .

0.05 0.1 0.15 0.2 0.25 0.3

γ

0

10

20

30

40

cost

savi

ngs

(%)

(a) Impact of � (interdependent cost)

40 60 80 100

flexibility (%)

0

10

20

30

40

cost

savi

ngs

(%)

(b) Impact of demand flexibility

0 50 100

Time (sec)

59.96

59.97

59.98

59.99

60

Fre

quency

(H

z)

No delay

∆=0.5 second

∆=1 second

(c) Impact of communication delay

Fig. 6. Sensitivity analysis in terms of cost savings by using the proposed control instead of OLC for: (a) Interdependent cost coefficient; (b) Demandflexibility. Stability analysis with the presence of (c) Communication delay and PFC updating every 0.1 second.

The results presented in this paper open up four distinctfuture research directions. The first is to explore how otherinterdependent systems (e.g. electric mass transit, thermalgrids) can be used to help increase the reliability of the grid.The second is to investigate how a network of datacentersthat are located in multiple disjoint power grids can utilizetheir interconnectedness to enhance the reliability in thosegrids. The third is to separate the computational workloadinto different resource demands, each with a different inter-dependent cost and computational efficiency. The fourth is toapply the distributed control laws to a system with a higher-order transient stability model. For our future work, we plan toextend the proposed control laws to take into account furthernetwork effects such as power flow constraints across linesand network losses.

X. ACKNOWLEDGMENTSWe would like to thank Dr. Changhong Zhao for his valuable

and insightful comments on this paper and the reviewers fortheir suggestions. This research is supported by NSF grantsCNS-1464388, CNS-1617698, CNS-1730128, CNS-1717558,and was partially funded by MSIP, Korea, grants IITP-2017-R0346-16-1007 and NRF-2015R1C1A1A01053788.

REFERENCES

[1] H. Shayeghi, H. Shayanfar, and A. Jalili, “Load frequency control strate-gies: A state-of-the-art survey for the researcher,” Energy Conversionand management, vol. 50, no. 2, pp. 344–353, 2009.

[2] A. Molina-Garcia, F. Bouffard, and D. S. Kirschen, “Decentralizeddemand-side contribution to primary frequency control,” IEEE Trans-actions on Power Systems, vol. 26, no. 1, pp. 411–419, 2011.

[3] N. Jaleeli, L. S. VanSlyck, D. N. Ewart, L. H. Fink, and A. G. Hoffmann,“Understanding automatic generation control,” IEEE transactions onpower systems, vol. 7, no. 3, pp. 1106–1122, 1992.

Futurework

Includeothergeographicinterdependentsystems,e.g.thermalgrids,electricmasstransit,naturalgas.

Research examples

• DistributedOptimization•  GeographicalLoadBalancing+DistributedFrequencyControl•  DemandResponseProgramDesign

• OnlineOptimization•  SmoothedOnlineConvexOptimization•  CoincidentPeakPricing,Multi-scaleelectricitymarkets

• BigDataSystems•  Multi-resourceallocation•  BoundedPriorityFairness,InterchangeableResourceAllocation

25

Solar+WindenergyoutpacesDRadoption

0

2

4

6

8

10

12

14

2003 2005 2007 2009 2011 2013

Percen

tofG

enerationCapacity

California

Solar+Wind CAISODR

CaliforniaStateLegislaturesolution(2013):1,325MWofgridstorageby2020

http://energyalmanac.ca.gov/electricity/electric_generation_capacity.htmlFERCAssessmentofDemandResponseandAdvancedMeteringStaffReports:2010-2015.CAISODemandResponseBarriersStudy2009.

26

Expensive

CustomerLoadReductions

Mandatory Voluntary

BecauseDRProgramshavevariousLevelsofCommitment

BaseInterruptibleProgram(BIP)EmergencyResponseService(ERS)

SpecialCaseResources(SCR)

DemandBiddingProgram(DBP)VoluntaryLoadReduction(VLR)EmergencyDRProgram(EDRP)

CAISOERCOTNYISO

27

DRprogramsallocatecustomeruncertainties

Customerisresponsible

LSEisresponsible

Customeruncertainties

Baseline

MandatoryDRtobaselinemaybecostly

UMassTraceRepository:Smart*DataSet 28

VoluntaryDRisnotdispatchableforLSE.

vs.

29

Challengesinhandlingcustomeruncertainties

LSEdoesnotknoweachofthecustomer’suncertainties.

DRprogramsthathavecustomerstakesomeresponsibilityaremandatory.

30

Goal:IncreasereliableDRadoption

Approach:IncorporateCustomerUncertainties

DistributedAlgorithm LeverageRandomness

Linearcontract

𝑥↓𝑖 (𝐷, 𝛿↓𝑖 )

Within10%OfflineOptimalsolution

31

LoadReduction

= ++Aggregatemismatch

Individualmismatch

Constantreduction

𝛼↓𝑖 𝐷 𝛽↓𝑖 𝛿↓𝑖  𝛾↓𝑖 = + +

Real-timeOptimal:Quadraticcostfunctions

Simpleandeasytoimplement

32

Customercostfunctionsraisechallenges

Accuracy:LSEdoesnotknowtheircostfunctions

Privacy:Customersdonotwanttogiveuptheirinformation

àSeparatetheDRdecisionproblem

DistributedAlgorithm

33

Customer1LSE

Customer2

Customer𝑖( 𝛼↓𝑖 , 𝛽↓𝑖 , 𝛾↓𝑖 )

( 𝛼↓2 , 𝛽↓2 , 𝛾↓2 )

( 𝛼↓1 , 𝛽↓1 , 𝛾↓1 )

(𝜋↓𝑖 , 𝜆↓𝑖 , 𝜇↓𝑖 )

(𝜋↓2 , 𝜆↓2 , 𝜇↓2 )

(𝜋↓1 , 𝜆↓1 , 𝜇↓1 )(𝝅,𝝀,𝝁)

SubgradientMethod

DistributedAlgorithmconvergestoCentralizedTheorem:Thedistributedalgorithm’strajectoryofpricesconvergestotheoptimalcentralizedprices.

2 4 6 8 100

1000

2000

3000

4000

5000

iteration,k

DistCent

34

2 4 6 8 100

0.2

0.4

0.6

0.8

iteration,k

CUST

Cent

1

∑𝑖↑▒𝛼↓𝑖  

LSE

iteration,k

2 4 6 8 100

0.002

0.004𝛼↓1 

𝛼↓2 

𝛼↓3 2 4 6 8 10

0

2 4 6 8 100

0.002

0.004

0.002

0.004

Customer1

Customer2

Customer3

𝛽↓𝑖 , 𝛾↓𝑖 (remainat0forcentralizedanddistributedsolutions)

35

Customercostuncertainty0.1 0.2 0.3 0.4

0.2

0.4

0.6

0.8

1

1.2

0.1 0.2 0.3 0.4

0.5

1

1.5

2

Linearcontractdrawback:Itismandatory.

RequiredcomplianceduringhighcustomercostperiodsCustomercostuncertainty

Customercost

time

Ex.Importanttasksthatneedtobecompleted

𝐶↓𝑖 (𝑥↓𝑖 )

LIN+()

36

Aimtoavoidhighcustomercostperiods

0 1

LevelofCommitment

LIN

𝜌

Customercommitsto𝜌fractionoftimeslotstofollowLIN

Commitmentdecisionbasedonlyonrealizedcostfunction 𝐶↓𝑖 (𝑥↓𝑖 )= 𝑎↓𝑖 𝑥↓𝑖↑2 

Ex.Quadratic

Customeronlyknows 𝑎↓𝑖 beforedecidingtocommittoDR

LIN+(𝝆)reducescostfurtherthanLINonly

𝜌 LIN10.80.60.40.20

2300

2400

2500

2600

2700

2800

2900

RSD(a)=0.05RSD(a)=0.56

𝐶↓𝑖 (𝑥↓𝑖 (𝑡);𝑡)= 𝑎↓𝑖 (𝑡) (𝑥↓𝑖 (𝑡))↑2 

LargercustomercostuncertaintyàlargersavingsfromLIN+(𝝆)

37

Decreasefrom𝜌=1=1àavoidshighcustomercosts

Furtherdecreaseof𝜌àlargermismatchforLSE

LevelofCommitment

38

Goal:IncreasereliableDRadoption

Approach:IncorporateCustomerUncertainties

Convergestocentralizedsolution

LowerSocialCostclosertoOffline

Optimal

DistributedAlgorithm LeverageRandomness

39

Futurework:Incorporatepowernetworkconstraints

CongestedlinesàNecessarytocontrollocalmismatches 𝛿↓𝑖 locally(with 𝛽↓𝑖 )

Ex.Lineconstraints

Research examples

• DistributedOptimization•  GeographicalLoadBalancing+DistributedFrequencyControl•  DemandResponseProgramDesign

• OnlineOptimization•  SmoothedOnlineConvexOptimization•  CoincidentPeakPricing,Multi-scaleelectricitymarkets

• BigDataSystems•  Multi-resourceallocation•  BoundedPriorityFairness,InterchangeableResourceAllocation

40

Artificial Intelligence

Robotics

AutonomousVehicles

SmartGrid

1. Real-timedecisions2. Changingenvironment3. SwitchingCosts4. Utilizepastinformation

ContinualLearningStoica,etal.(2017)

41

Example: Dynamic Capacity Provisioning

Demand𝑦↓𝑡 

DataCenter

𝑥↓𝑡  =#serverson

OperatingCostℎ(𝑥↓𝑡 , 𝑦↓𝑡 )

SwitchingCost𝛽‖𝑥↓𝑡 − 𝑥↓𝑡−1 ‖

+

Futuredemandpredictedat𝑡𝑦↓𝑡+1|𝑡 , 𝑦↓𝑡+2|𝑡 ,…, 𝑦↓𝑡+𝑤−1|𝑡 

Howmanyserversshouldbeturnedon/offrightnow?

Goal:DesignanOnlineAlgorithmwithPerformanceGuarantees.

42

1-Dimensional Multi-Dimensional

sup┬{𝑦↓𝑡 , 𝑦 ↓𝑡|𝑡 ,…, 𝑦 ↓𝑇|𝑡 }↓𝑡=1↑𝑇  � cost(ALG)/cost(OPT)  CompetitiveRatio:

Lin,etal.(2011):3-competitiveLazyCapacityProvisioning

Bansal,etal.(2015):3-competitiveMemoryless

Bansal,etal.(2015):2-competitive

IGCC12:1+𝑶(𝟏⁄𝒘 )-competitiveRecedingHorizonControl

1-StepPrediction

IGCC12:1+𝛀(𝟏)-competitiveRecedingHorizonControl

IGCC12:1+𝑶(𝟏⁄𝒘 )-competitiveAveragingFixedHorizonControl

43

RHC vs AFHC

44

FHC with limited commitment 𝑣

1.  Every𝒗≤𝑤rounds,receivethepredictions𝑦 ↓𝑡|𝑡 ,…, 𝑦 ↓𝑡|𝑡+𝑤−1 2.  Solvemin┬𝑥↓𝑡 ,…, 𝑥↓𝑡+𝑤−1  �∑𝜏=𝑡↑𝑡+𝑤−1▒[𝑐(𝑥↓𝜏 , 𝑦↓𝜏|𝑡 )

+‖𝑥↓𝜏 − 𝑥↓𝜏−1 ‖]  3.  Implement𝑥↓𝑡 ,…, 𝑥↓𝑡+𝑣−1 forthenext𝒗rounds

Use𝑣ofthecalculatedactionsbeforeusingnewpredictions.

LevelofCommitment,𝑣

45

Committed Horizon Control

1.  Run𝑣FHCalgorithmswithlimitedcommitment𝑣,eachstartingatadifferentround.2.  UseFHC(k)todetermine𝑥↓𝑡↑(𝑘) ,…, 𝑥↓𝑡+𝑣−1↑(𝑘) 3.  Implement𝑥↓𝑡 = 1/𝑣 ∑𝑘=0↑𝑣−1▒𝑥↓𝑡↑(𝑘)  

Averagethedecisionsbetweenasetofdifferent𝑣FHCalgorithms.

46

CHC generalizes RHC and AFHC (Sigmetrics16)

AFHC(𝑣=𝑤)RHC(𝑣=1)

47

Average-case Analysis Theorem:Let‖𝑓↓𝑘 ‖beameasureofthepredictionerrorfor𝑘stepsintothefuture,then𝐸[cost(𝐶𝐻𝐶)−cost(𝑂𝑃𝑇)]≤ 2𝑇/𝑣 (𝐷+𝐺∑𝑘=0↑𝑣−1▒‖𝑓↓𝑘 ‖↑𝛼  )Optimal𝒗dependsonhow‖𝒇↓𝒌 ‖growswith𝒌dependsonhow‖𝒇↓𝒌 ‖growswith𝒌

48

Different properties give different optimal 𝑣

𝒗↑∗ =𝟏00(AFHC)00(AFHC)𝒗↑∗ =𝟏(RHC)

𝒗↑∗ ≈𝟕

49

𝛼-Höldercontinuity-Höldercontinuity|𝑐(𝑥, 𝑦↓1 )−𝑐(𝑥, 𝑦↓2 )|≤𝐺‖𝑦↓1 − 𝑦↓2 ‖↓2↑𝛼 

||𝑓(𝑠)||↓𝐹 =𝑐,𝐿≥𝑠>0

Rangelimitingcorrelationerror

||𝑓(𝑠)||↓𝐹 =0,𝑠>𝐿

Whitenoisevariancetrace(Cov(𝑒))= 𝜎↑2 

IllustrationofTheorem