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A Theory of Probabilistic Power Provisioning for Data Centers with Distributed Energy Storage

Can Emre Koksal, Richard A. Barry, Artun Sel

arXiv:2608.12993v1cs.ITmath.IT

TL;DR

The paper addresses how to size provisioned power and distributed storage to meet a target overdraw probability under stochastic demand. It develops a probabilistic framework for this tradeoff and reports distinct short- and long-timescale operating regimes, with production-trace measurements supporting its models’ predictions.

  • Problem

    The central question is how much supplied or stored power is needed to keep the overdraw rate below a target while accounting for group size.

    Source: D. Overdraw, Capacity, and Margin · p. 6

  • Method

    The paper develops a probabilistic framework relating provisioned power, battery capacity, and target overdraw probability, and models demand variation across multiple timescales.

    Source: §I · p. 1, §X · p. 22

  • Results

    Small batteries address short-lived excursions, whereas large batteries absorb sustained fluctuations; production-data-center measurements support the stochastic models’ predictions of provisioning tradeoffs.

    Source: D. Discussion · p. 9, §X · p. 22

  • Takeaways & Limitations

    The framework provides guidance for battery dimensioning and identifies diminishing returns as storage increases, with provisioning efficiency depending jointly on storage and statistical multiplexing.

    Source: §X · p. 22

Abstract

from arXiv · show

The growing power demands and variability of AI workloads make electrical power delivery a critical constraint in data-center operation. Distributed energy storage can reduce the power capacity required to support stochastic loads, but its benefits depend fundamentally on the statistics and time scales of demand. This paper develops a probabilistic framework that jointly characterizes provisioned power, energy-storage capacity, and the probability of overdraw. We show that storage-assisted provisioning separates into two operating regimes. In the Small Battery Region, overdraw is dominated by short-lived demand excursions and storage provides nearly linear reductions in the required power margin. In the Large Battery Region, overdraw results from sustained demand fluctuations over longer spans of time, and the required margin exhibits diminishing returns with storage. For this regime we introduce effective power, an analogue of effective bandwidth that captures the temporal statistics of the demand and gives an asymptotically tight characterization of the required power. We further quantify how temporal correlation and spatial aggregation affect storage requirements and statistical multiplexing gains, and extend the analysis to loads with multiple demand time scales. Finally, we evaluate the framework using power-demand traces from three production data centers spanning HPC, GPU-training, and cloud-service workloads. Despite their heterogeneous, cyclo-stationary and multi-modal behavior, the measured workloads exhibit the predicted regimes, and a simple four-parameter two-state model captures the dynamics governing their storage-power tradeoffs. The resulting framework provides both a probabilistic foundation and practical dimensioning principles for storage-assisted power provisioning in next-generation AI data centers.

I. INTRODUCTION

Data centers manage rare overdraws under fixed power limits, but prior work lacked a probabilistic account of how distributed storage trades battery capacity against required power margin. The paper develops that framework and describes two operating regimes, while situating the model among demand-side controls, storage systems, and network-theory tools.

I. INTRODUCTION

  • Fixed power capacity can leave resources underused, while rare demand peaks can exceed the provisioned supply; existing approaches largely manage this risk on the demand side.The paper frames storage as a way to use supply-side flexibility to meet statistical overdraw constraints with reduced power margins.Source: §I · p. 1
  • The framework relates provisioned power, distributed storage, and statistical overdraw guarantees, introducing effective power to predict required margin for larger batteries.For small batteries and short overdraw times, the paper instead relies on bounds, Markov analysis, and simulations.Source: §I · pp. 1–2
  • In the Small Battery Region, short-lived demand spikes dominate overdraw and each storage unit saves approximately one unit of provisioned power; in the Large Battery Region, sustained excess demand drains the battery and savings diminish.The relative size and transition of battery sub-regions depend primarily on demand statistics and less on the overdraw guarantee.Source: §I · p. 2
  • Combining spatial and temporal multiplexing can yield additive, neutral, or mildly subtractive gains, depending on how storage scales with group size.Larger groups can reduce margin per server through spatial aggregation but require larger batteries.Source: §I · p. 2

A. Demand-side power oversubscription and capping

E. Effective bandwidths and large deviations

  • Prior storage solutions and network-theory tools did not provide a closed-form probabilistic tradeoff between storage size and provisioned-power margin under realistic demand.Effective-bandwidth and large-deviations methods were developed for communication systems rather than storage sizing.Source: E. Effective bandwidths and large… · p. 3

A. System Description

  • The model focuses on a single energy group with shared supply and storage, while noting that interactions across layers, non-greedy battery use, and orchestration effects are not addressed.The broader architecture assigns storage to groups at different layers, with longer-term variation suited to higher layers and faster dynamics to lower layers.Source: A. System Description · p. 4

B. Energy Group Model

  • The energy-group model uses time slots, a constant power supply, summed server demands, and a bounded battery whose charge changes with excess or deficient demand.Battery charge is expressed in power units by normalizing energy capacity by the slot length.Source: B. Energy Group Model · pp. 4–5

C. Statistical Assumptions

The model assumes statistically identical, stationary, ergodic, nonnegative, finite individual demand processes, while allowing temporal correlation. It asks how to provision power and storage to keep the long-term overdraw rate below a target ε.

C. Statistical Assumptions

  • Individual demand processes are statistically identical, stationary, ergodic, nonnegative, and finite, with log moment generating functions in a region around zero.The assumptions allow temporal correlations; demand statistics depend on slot length, with longer slots tending to smooth demand.Source: C. Statistical Assumptions · p. 5

D. Overdraw, Capacity, and Margin

  • Stationarity and ergodicity make the probability of overdraw in an arbitrary slot equal to the long-term fraction of slots that overdraw.Overdraw occurs when group demand exceeds the available supplied or stored power.Source: D. Overdraw, Capacity, and Margin · p. 5
  • When overdraws are rare, average consumed power is approximately average demand, while batteries smooth demand variations passed to higher system layers.The framework treats overdraw as a period when demand must be curtailed.Source: D. Overdraw, Capacity, and Margin · pp. 5–6
  • The target is to find the minimum supplied power for battery size b that keeps the overdraw rate below ε, and to characterize its margin above average demand.Required power and margin depend on ε and group-demand statistics; both are nonincreasing with battery size, and the margin approaches zero as battery size grows without bound.Source: D. Overdraw, Capacity, and Margin · p. 6

E. Choke-Point Utilization and Oversubscription Ratio

The section relates oversubscription to deployed power and frames storage as a way to reduce required margin. Without a battery, the required margin is governed by the demand tail, with approximations whose accuracy depends on the distribution.

E. Choke-Point Utilization and Oversubscription Ratio

  • Storage reduces required margin and can therefore allow a data center to operate at a higher oversubscription ratio while staying below its maximum overdraw rate.OSR is nameplate IT power divided by supplied power; higher OSR improves efficiency but is limited by the risk of hardware overload.Source: E. Choke-Point Utilization and… · p. 6

IV. NO-BATTERY REGIME (NBR) BASELINE

  • Without a battery, required power and margin depend on the demand distribution’s quantile, and the margin also represents the maximum tolerable slot burst over the mean.The no-battery case provides a baseline for evaluating spatial and temporal multiplexing.Source: §IV · pp. 6–7

A. Estimate of No-Battery Baseline

  • At overdraw target ε = 10^-3, standard deviations of 32W, 79W, and 158W require no-battery margins of approximately 98W, 241W, and 481W, respectively.At ε = 10^-3, the system requires about 3.05 standard deviations of power demand above average power.Source: A. Estimate of No-Battery Baseline · p. 7
  • For multi-modal demand, the Chernoff approximation can be widely inaccurate, while a separated high-demand mode can determine the tail and required margin.For skewed distributions, heavier tails require higher margins and lighter tails lower margins when the third-order expansion applies.Source: A. Estimate of No-Battery Baseline · p. 7

B. Multi-modal Distributions

For separated multi-modal demand, the high state can determine the tail and required margin. In the small-battery region, storage offsets isolated demand spikes nearly one-for-one until demand statistics determine a transition toward diminishing returns.

B. Multi-modal Distributions

  • A separated high-demand state can determine the tail of a multi-modal distribution, so provisioning must meet an overdraw target scaled by that state’s time fraction.For a two-state ON/OFF process, the high state’s target overdraw rate is ε/a(1), where a(1) is its average time fraction.Source: B. Multi-modal Distributions · p. 7

V. SMALL BATTERY REGIME (SBR)

  • The SBR covers isolated single-slot excess demands, but not excess demand sustained over multiple slots.The transition point for processes other than i.i.d. Gaussian remains future work in this analysis.Source: §V · p. 8

A. Ideal Bound

  • Required power cannot fall faster than linearly with battery size; in the SBR, each added unit of battery charge reduces required power by one unit.For battery energy be = Ts b, each joule can save at most 1/Ts of provisioned power.Source: A. Ideal Bound · p. 8

B. Characteristics of the SBR

  • In the SBR, the battery is usually full, making the ideal bound tight; the one-to-one power saving ends as the system leaves the SBR and returns diminish.The transition battery size depends on demand-tail statistics and temporal correlations; btx cannot exceed the no-battery margin m(0).Source: B. Characteristics of the SBR · p. 8

C. Gaussian i.i.d. Group Power

For the Gaussian i.i.d. examples, the SBR transition occurs near the midpoint of the no-battery margin, while positive temporal correlation shortens the SBR. In the LBR, effective power characterizes the required supply through an asymptotically tight approximation.

C. Gaussian i.i.d. Group Power

  • For ε = 10^-3, the three Gaussian examples transition near βtx ≈ 0.5, after storage reduces the margin requirement by about 50%.Their normalized curves coincide exactly and are invariant to mean shifts and scaling; the remaining margin reduction occurs in the LBR.Source: Figure 2 · p. 9, C. Gaussian i.i.d. Group Power · p. 9

D. Discussion

  • Positive temporal correlation reduces the probability of full battery recharge immediately after an overdraw and shrinks the SBR.The SBR accounts for about half the margin reduction for i.i.d. Gaussian demand; short-term correlations shorten it, while heavier-tailed distributions have wider SBRs but require larger margins.Source: C. Gaussian i.i.d. Group Power · p. 9, D. Discussion · p. 9

A. Theory of Effective Power for Dimensioning the Demand

  • Effective power translates the overdraw constraint into a linear condition on demand statistics and accurately predicts required margin for large batteries.Its LBR approximation is asymptotically tight as battery size grows; the analysis removes finite battery boundaries because they affect lower-order, sub-exponential terms.Source: §VI · p. 9, A. Theory of Effective Power for… · p. 10

VI. LARGE BATTERY REGIME (LBR)

  • In the LBR, batteries absorb sustained multi-slot demand variation, and as battery size grows, typical overdraw duration converges to a characteristic duration.The typical duration is determined by load statistics and system parameters.Source: §VI · p. 9, D. Discussion · p. 9

B. Gaussian Demands and the Second-Order Approximation

The second-order effective-power approximation captures the large-battery margin decay, while clarifying how demand variability and temporal correlation shape required margin. Its accuracy depends on the demand distribution and, for general processes, on battery size relative to correlation timescales.

B. Gaussian Demands and the Second-Order Approximation

  • The approximation uses effective variance, combining demand variance with temporal-correlation effects; for i.i.d. demands, effective variance equals demand variance.For Gaussian demands, including potentially correlated ones, the second-order approximation is exact; for general statistics, it becomes accurate when the battery averages out correlation-driven fluctuations.Source: B. Gaussian Demands and the… · pp. 10–11
  • In the large-battery region, the second-order effective-power approximation captures the exact margin’s 1/˜b decay and conservatively slightly overestimates required margin.For the three Gaussian i.i.d. examples, exact curves reach 5% remaining margin near ˜b ≈5.3.Source: Figure 3 · p. 11, B. Gaussian Demands and the… · p. 11
  • For positive overdraw constraints, required margin rises with effective variance and falls as 1/b, rather than declining linearly as in the small-battery region.Margin is not very sensitive to the overdraw rate because δ = −log(ε).Source: B. Gaussian Demands and the… · p. 11

C. Typical Duration for an Overdraw Event

In the large-battery region, overdraws have a characteristic duration that grows with battery capacity, and their probability decays according to a duration-scaled rate. The section then relates spatial and temporal multiplexing, showing that their combined efficiency gains are generally less than cumulative.

C. Typical Duration for an Overdraw Event

  • In the large-battery region, overdraws occur over a typical finite duration rather than as instantaneous bursts or arbitrarily long, slight excesses.At the exact required power, typical overdraw duration scales as O(b^2); for fixed provisioned power, it is proportional to battery capacity.Source: C. Typical Duration for an Overdraw… · p. 11

D. Decay Rate of Overdraw Probability

  • Without a battery, overdraw probability decays at rate I_P, whereas with a battery it scales as exp(−Ttyp(b)I_B); battery overdraws require a positive excess sustained over a duration that scales with b.Both rate functions increase with the margin.Source: D. Decay Rate of Overdraw Probability · pp. 11–12

VII. STATISTICAL MULTIPLEXING OF ENERGY DEMAND

  • Spatial pooling and battery smoothing both reduce margin per server, but their combined efficiency gains are generally less than cumulative because each form has diminishing returns.Spatial gains are greatest for independent server demands, while temporal gains are greatest for independent time slots.Source: §VII · p. 12
  • The relative growth of battery capacity determines whether spatial multiplexing dominates, temporal gains persist as a multiplier, or both forms contribute asymptotically.The three cases are battery growth slower than, at, or faster than the natural rate relative to the SBR/LBR transition.Source: §VII · p. 12

A. The Combined Effects of Spatial and Temporal Multiplexing

The margin per server depends on demand statistics and how battery capacity scales with group size; storage gains are shaped by temporal correlations and the SBR/LBR transition. For independent Gaussian demands, fixed total storage eventually loses its temporal-multiplexing benefit as the group grows.

A. The Combined Effects of Spatial and Temporal Multiplexing

  • For general demand groups, margin per server depends on demand tails and inter- and intra-correlations, while battery scaling can move the system between SBR and LBR.Spatial margin gains depend on single-slot joint demand statistics; the natural battery growth rate for constant temporal-multiplexing gains varies from O(1/√n) for independent demands to O(1) for perfectly correlated demands.Source: A. The Combined Effects of Spatial and… · pp. 12–13

B. Independent Gaussian i.i.d. Demands

  • For independent, temporally uncorrelated Gaussian demands, the natural total battery growth rate is O(√n).The constant δ/(2α^2) ranges from .45 to .3 for target overflow rates between 10−2 and 10−6.Source: B. Independent Gaussian i.i.d. Demands · p. 13

1) Scenario 1: Fixed b, increasing n:

  • With fixed total storage, battery per server falls as 1/n, so temporal-multiplexing gains eventually vanish and margin per server approaches the no-battery behavior.For finite groups, adding battery can substantially reduce margin per server, but the curves converge as the battery effect is diluted.Source: 1) Scenario 1: Fixed b, increasing n · p. 13

2) Scenario 2:

The balance between spatial and temporal multiplexing depends on battery growth with group size, while the storage needed to preserve temporal gains also depends on demand correlation. The paper models multi-scale demand with Markov-modulated states and notes that a second-order Chernoff approximation can be inaccurate for highly skewed, multi-modal no-battery demand.

2) Scenario 2:

  • When battery capacity grows as √n, temporal multiplexing supplies a persistent multiplicative efficiency reduction while margin per server still scales as 1/√n.The system remains in the SBR or LBR depending on the battery-growth constant.Source: 2) Scenario 2 · pp. 13–14

3) Scenario 3:

  • When battery capacity grows linearly with n, spatial and temporal multiplexing each contribute a 1/√n efficiency factor asymptotically..Source: 3) Scenario 3 · p. 14

C. Discussion

  • To preserve temporal-multiplexing gains, battery growth must match the no-battery margin scale; uncorrelated demands need only O(1/√n) battery per server, while highly correlated demands require total storage growing about linearly with group size..Source: C. Discussion · p. 14

VIII. MULTI-SCALE TEMPORALLY CORRELATED POWER DEMANDS

  • The multi-state Markov-modulated model represents slow transitions among operating regimes and fast, within-state fluctuations, thereby capturing persistent high-demand periods.State dwell times affect battery dynamics even when stationary distributions match; long dwell times model longer-term cyclo-stationarity, while short dwell times between closer states model intra-period correlations.Source: §VIII · p. 14, A. Multi-state Model · p. 14

B. No-battery Margin Requirements

  • For multi-state demand, a second-order Chernoff approximation may misestimate no-battery margin because the power distribution can be highly skewed and multi-modal.When the high state is well separated and other states occur often enough, the analysis approximates margin using the high-state distribution; the margin depends on time spent in the high state, not its dwell time.Source: B. No-battery Margin Requirements · p. 14
  • The no-battery margin has components for the high-state mean above the long-term average and for high-state fluctuations above their mean.The high-state time fraction has only a small effect on the first component and none on the second.Source: B. No-battery Margin Requirements · pp. 14–15

C. SBR Margin Requirements

Storage-assisted provisioning has a small-battery regime with nearly linear power-margin reductions and a large-battery regime governed by sustained fluctuations. In the ON/OFF example, effective power closely tracks required power in the large-battery region except near the transition.

C. SBR Margin Requirements

  • For small batteries, required power falls approximately one-for-one with storage, and the transition out of this regime depends on demand statistics.Overdraws are associated with short-lived excess demand; the battery is typically full and rapidly recharges.Source: C. SBR Margin Requirements · p. 15

D. LBR Margin Requirements

  • In the large-battery region, required power is approximately effective power, but multiple subregions can occur and general calculation may require numerical computation.The cited approximations characterize the start and deep end of the region but do not describe the full large-battery region.Source: D. LBR Margin Requirements · p. 15

E. ON/OFF Example

  • Both ON/OFF examples show small- and large-battery regions, with effective power closely matching required power except near their transition.The examples use high-variance, long-dwell and low-variance, short-dwell demand; the two-phase analysis distinguishes suppression of ON-state variation from averaging across ON/OFF cycles.Source: E. ON/OFF Example · pp. 15–16

1) Region 1a (SBR):

The ON/OFF example progresses from linear savings in the small-battery region to ON-state variation suppression and then mean reduction. The region boundaries and achievable savings depend on dwell times and the overdraw target.

1) Region 1a (SBR):

  • For ε = 10−3, α1 = 2.84 versus α = 3.05 for an equivalent always-ON system; without a battery, the required margin is reduced by approximately 7%.In Region 1a, the green dashed line indicates the approximation's accuracy, and margin requirements are primarily determined by ON-state statistics.Source: 1) Region 1a (SBR) · p. 16

2) Region 1b (ON state LBR):

  • Region 1b reduces ON-state variations toward the ON mean, with larger regions for longer ON dwell times and smaller ON-state variations.For extremely short ON dwell times, such as L = 2, Region 1b can be small or essentially absent.Source: 2) Region 1b (ON state LBR) · p. 16

3) Region 2a - Mean Reduction, part a:

  • Exact calculations show Region 2a ends 13.2% (L = 1000) and 14.2% (L = 10) below the mean; after btx,3, additional storage reduces the mean toward its long-term average with diminishing returns.For both examples, the Region 2a approximation lines up very well with exact Markov calculations and numerical effective-power computation.Source: 3) Region 2a - Mean Reduction, part a · p. 17, 4) Region 2b - Mean Reduction, part b · p. 17
  • The Region 2a approximation matches the exact calculations, and its endpoint reduction is smaller for the stricter overdraw target ε = 10−6 than for ε = 10−3.The reported endpoint reduction is about 6.7% below the mean at ε = 10−6, compared with 13.2% and 14.2% for the two examples at ε = 10−3.Source: 3) Region 2a - Mean Reduction, part a · p. 17

4) Region 2b - Mean Reduction, part b:

In the final large-battery subregion, temporal correlations shape effective power and the storage needed to approach long-term mean demand. The measured workloads span distinct scales and distributions, while a simple two-state model captures the dynamics governing their storage-power tradeoff.

4) Region 2b - Mean Reduction, part b:

  • Longer dwell times or higher ON means raise effective power and require larger batteries to bring required power toward the long-term mean.The σeff approximation is accurate only for fractional margins of about 20–30% or less; otherwise the full effective-power calculation is needed.Source: 4) Region 2b - Mean Reduction, part b · p. 17

F. Discussion

  • The small-battery-region/large-battery-region formulation applies to a wider range of demand statistics, and the large-battery region can contain regimes with changing energy-to-power efficiencies.Because margins are invariant to shifts in the mean, the ON/OFF analysis also applies to well-separated HIGH/LOW demands.Source: F. Discussion · p. 17

IX. DATA INTEGRATION AND ANALYSIS

  • Across three production workloads, short-time-scale fluctuations within bursts and longer-term transitions between bursts form a common multi-timescale structure that a two-state HIGH/LOW model captures.The model is intended to capture the dynamics governing storage-power tradeoffs, not to reproduce the complete statistical structure of real demand.Source: §IX · p. 18

A. Data Sets

  • The three datasets span mean demands of 288–2,841 kW, have multimodal single-slot distributions, and show strong temporal correlation with multiple time scales.Long-term correlation times vary by about a factor of 4.4; burst persistence and maximum burst length also differ across the traces.Source: A. Data Sets · pp. 18–19

B. 2-State High/Low Model

A four-parameter HIGH/LOW model is used to predict storage-power tradeoffs, then tested against three data-center traces. The empirical results show the predicted two-phase, four-subregion behavior, while revealing transition-point and correlation-related discrepancies.

B. 2-State High/Low Model

  • The four-parameter model uses Gaussian i.i.d. demands in HIGH and LOW states, with equal dwell times and variances, to predict storage-power savings.The simplifications reflect the finding that margin requirements depend more on the HIGH-state demand and its separation from the LOW state than on dwell times.Source: B. 2-State High/Low Model · p. 19

C. Data Driven Simulations

  • The OSC trace supports testing only the first 46% of potential margin savings because it contains too few demand cycles with long busy periods.Empty- and full-battery simulations diverged significantly sooner for OSC than for Philly and Alibaba.Source: C. Data Driven Simulations · p. 21
  • Phase 1 storage suppresses HIGH-state variation, producing margin savings of about 30% for OSC, 18% for Philly, and 25% for Alibaba.In Phase 2, further storage reduces required power below the HIGH-state mean toward the long-term average.Source: C. Data Driven Simulations · p. 21
  • Observed SBR slopes were 53 kW/kWh for OSC and 28 kW/kWh for Philly and Alibaba, roughly half the ideal slopes; time-permuted traces came within 2.2% of the ideal bound.This supports temporal correlations as the primary cause of the reduced SBR slopes.Source: C. Data Driven Simulations · p. 21
  • The empirical traces exhibit two phases, each with two subregions, and the model predicts the provisioning curves well, especially for Philly and Alibaba.Transition points were determined empirically; remaining inaccuracies are attributed mainly to unmodeled HIGH-state temporal correlations.Source: C. Data Driven Simulations · p. 21

X. CONCLUSION AND FUTURE WORK

  • The framework relates provisioned power, battery capacity, and overdraw probability, and production-trace validation supports simple probabilistic abstractions for provisioning tradeoffs.The paper also identifies battery-dimensioning guidance and quantifies how storage and statistical multiplexing jointly affect provisioning efficiency.Source: §X · p. 22

APPENDIX A MARKOV CHAIN ANALYSIS

The appendix computes stationary overdraw probabilities for battery models and develops effective-power properties used to analyze large-battery behavior. It also gives a geometric derivation of the typical duration governing that analysis.

APPENDIX A MARKOV CHAIN ANALYSIS

  • For i.i.d. aggregate demand, battery charge is a time-homogeneous Markov chain, and its overdraw rate is computed as an exact stationary average.The construction covers general demand distributions and is specialized to integer-valued Poisson and continuous Gaussian cases.Source: Appendix A · p. 23

A. Discretizing the charge level

  • A grid with k + 1 charge states uses relative resolution; increasing k reduces discretization error until the computed overdraw rate converges.For integer-valued demand, choosing h = 1 makes the states exact charges and introduces no approximation.Source: A. Discretizing the charge level · p. 23

B. Transition probabilities

  • Transition probabilities assign demand intervals to next-charge states, with boundary states collecting probability mass clamped to an empty or full battery.The interior transition entries depend only on the difference between state indices, giving the matrix a banded Toeplitz structure.Source: B. Transition probabilities · p. 24

C. Stationary overdraw probability

  • The stationary overdraw probability averages the per-state event probability, evaluated using the exact demand tail; required power is found by solving for the target overdraw probability.The charge discretization affects battery dynamics, not evaluation of the overdraw event.Source: C. Stationary overdraw probability · p. 24

APPENDIX B EFFECTIVE POWER AND ITS PROPERTIES

  • Effective power depends on the time scale and transform variable; it is additive for independent multiplexed loads and constant in time scale for uncorrelated samples.For ergodic processes, the paper defines effective power using the asymptotic log moment generating function.Source: Appendix B · p. 24

APPENDIX C DERIVATION OF Ttyp(b)

  • In the large-battery derivation, the typical duration grows linearly with battery size, and geometric tangent-line construction identifies it through an optimized x-intercept.The tangent line’s y-intercept gives the rate function at the supply power.Source: Appendix C · p. 25, Figure 7 · p. 25

APPENDIX D MODEL OF REGION 2A

The appendix approximates Region 2a using the battery’s charging during OFF periods and its use during subsequent ON periods. It also describes how empirical power curves set the HIGH/LOW model parameters.

APPENDIX D MODEL OF REGION 2A

  • In Region 2a, the battery is assumed to recharge to capacity during an OFF period, then reduce required power toward the ON-state mean during the next ON period.Residual ON-state variation is modeled as zero after storage has reduced power to near the ON mean.Source: Appendix D · p. 25
  • For large dwell time L, the Region 2a capacity-reduction slope is approximately −1/(LδH); longer dwell times and smaller target overdraws reduce its absolute value.Linear mean reduction ends when the battery is too large to fully recharge during an OFF period.Source: Appendix D · p. 26

APPENDIX E SETTING MODEL PARAMETERS FROM EMPIRICAL POWER

  • The HIGH-state mean is set to required power at the end of Phase 1, while the LOW-state mean is chosen to match the empirical overall mean.The model assumes equal HIGH/LOW dwell times and equal state variances.Source: Appendix E · p. 26
  • The HIGH/LOW model parameters are set from five empirical measurements: mean demand, no-battery required power, and the start, end, and slope of Region 2a.The authors state that the fitting procedure is not intended as a generic algorithm for identifying Region 2a in any data set.Source: Appendix E · p. 26
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