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Impedance-Aware Optimized Pulse Patterns for Reconfigurable Battery Systems

Julian Estaller, Johannes Buberger, Andreas Wiedenmann, Wolfgang Grupp, Tobias Hoegerl, Thomas Weyh

arXiv:2609.16828v1eess.SYcs.SY

TL;DR

Classical OPP assumes an ideal or configuration-independent source, while RBS path resistance changes with voltage level. IA-OPP incorporates that impedance and an event budget into pattern optimization; at identical budget n = 48, the classical OPP baseline shows elevated harmonics at h = 7 to 11.

  • Problem

    Classical OPP presumes an ideal or configuration-independent source, although each RBS voltage level corresponds to a distinct path resistance.

    Source: §I · p. 1

  • Method

    IA-OPP embeds level-dependent source impedance in the objective and uses an exact event budget with free step sizes and a dual switching-cost metric.

    Source: §I · p. 2

  • Results

    At identical budget n = 48, the classical OPP baseline shows elevated harmonics at h = 7 to 11.

    Source: B. Spectral Distribution · p. 6

  • Takeaways & Limitations

    RBS switching cost is described by both level-change bus events and submodule switching actions, rather than one pulse count alone.

    Source: D. Evaluation Metrics · p. 4

    The study is simulation-based, assumes purely ohmic cell impedance and ideal switching transitions, and does not examine transitions between lookup-table patterns.

    Source: §I · p. 2, B. Limitations · p. 10

Abstract

from arXiv · show

Reconfigurable battery systems (RBS) synthesize the converter output voltage by inserting and bypassing battery cell groups, so every switching state changes not only the source voltage but also the source impedance. This paper shows that this configuration-dependent impedance breaks a tacit assumption of optimized pulse patterns (OPP): patterns designed with the customary linear plant model overestimate their achievable current quality by up to an order of magnitude and converge to a distortion floor set by the resistance modulation. We propose impedance-aware optimized pulse patterns (IA-OPP), which embed the level-dependent source impedance in the optimization objective, combined with an event-budget formulation that treats the reconfiguration rate of the communication bus as a managed resource. An exact closed-form evaluation of the piecewise-exponential load current renders the objective smooth in the switching instants. In a simulated cascaded-bridge reference system, IA-OPP attains 0.52% current total harmonic distortion (THD) at a reconfiguration rate of 2.4 kHz, outperforming nearest-level modulation, phase-shifted carrier pulse-width modulation (PWM) at up to five times the switching rate, and a classical OPP baseline at every event budget; the advantage persists across the modulation-index range and under +/-30% parameter mismatch. For three-wire star-connected systems, masking triplen orders in the objective restores the full advantage, with the impedance-aware patterns beating their linear-model counterparts by about 20% at equal budget.

I. INTRODUCTION

RBS offer freely selectable voltage levels, but their configuration-dependent impedance and event-based switching costs violate assumptions behind classical OPP. IA-OPP incorporates both properties, while prior modulation approaches either omit impedance from the pattern objective or use different operating structures.

I. INTRODUCTION

  • RBS voltage levels and step instants are both selectable, creating a richer modulation actuator than classical multilevel converters with fixed level sequences.For motor-type loads, OPP can compute full-period patterns offline for slowly varying operating points and replay them from lookup tables.Source: §I · p. 1
  • Classical OPP assumes an ideal or configuration-independent source, whereas each RBS voltage level selects a distinct path resistance that varies with switching.This makes the mismatch between the optimized model and real plant systematic, motivating impedance-aware optimization.Source: §I · p. 1
  • RBS switching costs require an event budget for level changes plus a separate measure of submodule switching actions, rather than a single fixed-height pulse count.A level change counts once as a string reconfiguration and bus telegram even when multiple submodules switch.Source: §I · pp. 1–2
  • IA-OPP models level-dependent impedance in its objective and uses an exact event budget with free step sizes, assessed by bus reconfigurations and submodule actions.Its closed-form switched-RL evaluation supplies exact Fourier coefficients and loss figures for comparisons on a common footing.Source: §I · p. 2

II. RELATED WORK

  • Prior multilevel OPP work assumes constant sources, while battery-integrated modulation has mainly targeted battery management; IA-OPP couples variable impedance to event-based cost.The freedom to choose voltage levels is shared with recent OPP formulations [9], but its coupling to RBS event cost is identified as new.Source: §II · p. 2

III. SYSTEM MODEL AND EVALUATION METRICS

The reference setup is a single-phase cascaded-bridge RBS feeding a stationary RL load, with five modules and twelve cell submodules per module.

III. SYSTEM MODEL AND EVALUATION METRICS

  • The reference string has five cascaded modules, each with twelve half-bridge submodules and one 3.7 V battery cell per submodule.The submodule layer provides up to 60 voltage levels over a 222 V range, and the H-bridges set polarity.Source: A. Reference System and Operating Point · p. 3
  • The operating point uses a 50 Hz, 80 V-amplitude reference, corresponding to modulation index ma ≈ 0.36 against the 222 V maximum string voltage.The load surrogate has R = 2 Ω and L = 10 mH; no back-EMF is modeled.Source: A. Reference System and Operating Point · p. 3
  • The RL surrogate represents a motor-type load at partial load and captures only its stationary current response.The reference describes periodic operation without modeling back-EMF.Source: A. Reference System and Operating Point · p. 3

B. Level-Synchronous Resistance Modulation

RBS path resistance varies synchronously with voltage level, so switching changes both the applied voltage and the RL plant. The paper evaluates this switched plant exactly, without time-stepping.

B. Level-Synchronous Resistance Modulation

  • At the nominal operating point, path resistance ranges from 2.020 Ω at zero level to 2.264 Ω, a 12% modulation relative to zero level.The submodule resistance contribution jumps module-wise, while the inserted-cell resistance grows linearly with level magnitude.Source: B. Level-Synchronous Resistance… · p. 3
  • A 99.5 mΩ second-harmonic resistance component predicts a 0.107 A third current harmonic for a voltage-faithful staircase; exact evaluation gives 0.114 A.Even resistance modulation mixes with the odd fundamental to produce odd current harmonics, with the second-order resistance component contributing at the third order.Source: B. Level-Synchronous Resistance… · p. 3

C. Closed-Form Evaluation of the Switched RL Plant

  • Exact segment-wise integration of the piecewise-exponential current yields Fourier coefficients, RMS values, power, and losses without time-stepping.The metrics are smooth in the switching instants, and applying the same evaluation to every method makes comparisons attributable to the patterns.Source: C. Closed-Form Evaluation of the… · p. 4

D. Evaluation Metrics

Current quality is measured by load-current THD, while switching cost is represented by separate bus-event and submodule-action counts.

D. Evaluation Metrics

  • Current THD uses exactly computed harmonics up to order 50, with the staircase source-voltage fundamental required to match the 80 V reference within 1%.In this reference system, the load inductance strongly attenuates higher-frequency components, so in-band and wide-band THD practically coincide.Source: D. Evaluation Metrics · p. 4
  • The number of level changes per period, n, measures string reconfigurations and bus load; at 50 Hz, n = 48 corresponds to a 2.4 kHz bus rate.A level change spanning multiple levels remains one bus event but can switch multiple submodules.Source: D. Evaluation Metrics · p. 4
  • Submodule switching actions per period capture loss effort, so both that count and level changes are needed to describe RBS switching cost.The two quantities decouple systematically in an RBS.Source: D. Evaluation Metrics · p. 4

IV. IMPEDANCE-AWARE OPTIMIZED PULSE PATTERNS

IA-OPP uses a constrained staircase pattern and a two-stage search to account for level-dependent impedance while managing switching events. The dynamic program initializes each budgeted candidate, and continuous refinement evaluates current distortion on the exact plant model.

A. Pattern Structure and Search Space

B. Two-Stage Optimization

  • A dynamic program sets the level sequence and initial edges; coordinate descent then refines edge times on the exact plant using current THD and a weak fundamental-error penalty.Source: B. Two-Stage Optimization · pp. 4–5
  • The quarter-period dynamic program uses 90 adaptive intervals, with denser sampling near zero crossings and a time-weighted squared-error cost against an impedance-compensated reference.Source: B. Two-Stage Optimization · p. 4
  • The additive dynamic-programming surrogate targets ideal voltage plus the level-dependent resistance drop at ideal current; exact current THD is assessed during refinement and selection.Source: B. Two-Stage Optimization · pp. 4–5
  • Edge searches stay between neighboring edges, with minimum spacing T/7200 ≈ 2.8 µs and 100 ns resolution, and stop when a full pass improves by less than 10^-3 percentage points.Source: B. Two-Stage Optimization · p. 4
  • Ten jittered starts at n = 48 finish at 0.513 to 0.522% THD; noticeable single-start scatter occurs only at n ≤20, below the n ≥24 operating range.Source: B. Two-Stage Optimization · p. 5

C. Selection Rule and Operating Modes

The selection rule chooses the lowest-THD feasible candidate at a prescribed budget, with a tie-break favoring the smaller budget. Patterns are precomputed offline for operating points and replayed online from a lookup table.

C. Selection Rule and Operating Modes

V. SIMULATION RESULTS

  • The nominal-point comparison evaluates all methods with the same closed-form plant model; Fig. 2 depicts the selected n = 48 pattern over one fundamental period against ideal references.Source: §V · p. 5, Figure 2 · p. 5

A. Selected Solution

At the nominal operating point, IA-OPP achieves low current THD with a modest bus load, while alternatives trade distortion against switching effort. Its spectrum suppresses low-order components, although a third harmonic remains prominent across methods.

A. Selected Solution

B. Spectral Distribution

  • About 0.5% third harmonic is common to all candidates and accounts for the bulk of their THD; IA-OPP suppresses low orders and shifts residual energy toward medium orders.Source: Figure 3 · p. 6, B. Spectral Distribution · p. 6
  • Submodule-level PS-PWM leaves the considered band almost empty but uses 2.7 times IA-OPP's submodule switching rate and 4.9 times its bus load.Source: B. Spectral Distribution · p. 6

C. Value of Plant Modeling

The linear-model OPP baseline substantially underpredicts real-plant current THD, while resistance modulation creates a distortion floor for voltage-oriented methods. IA-OPP improves current quality by matching the pattern to the plant.

3) Implicit Predistortion:

  • With optimizer, pattern structure, budgets, refinement, and selection held constant, IA-OPP and classical OPP differ only in optimization plant model, so outcome differences are causally attributable to that model.Source: C. Value of Plant Modeling · p. 6

1) Prediction Gap of the Linear Model:

2) Resistance Modulation as a THD Floor:

  • Voltage-oriented methods converge to about 0.565 % THD as rising switching effort reproduces the voltage more faithfully, but level-synchronous resistance modulation mixes harmonics into current distortion.The dominant second harmonic of path resistance mixes with the fundamental onto the third current harmonic, creating a floor that further switching cannot break.Source: 2) Resistance Modulation as a THD Floor · p. 7
  • On a constant-resistance plant, the floor vanishes: submodule PS-PWM reaches 0.0002 % and 1S-NLM reaches 0.073 % THD.IA-OPP instead reaches 0.245 % there because its predistortion corrects for resistance modulation that is absent.Source: 2) Resistance Modulation as a THD Floor · p. 7

3) Implicit Predistortion:

At equal budget, IA-OPP places edges and step sizes to reduce current error even when its voltage staircase resembles classical OPP. Across the nominal THD-versus-bus-load comparison, it forms the Pareto front.

3) Implicit Predistortion:

  • At identical budget n = 48, IA-OPP and classical OPP have nearly congruent voltage staircases but clearly different current errors.Classical OPP errors oscillate at about ±1.1 A, while IA-OPP stays below ±0.85 A; edge placement and step sizes drive the difference.Source: 3) Implicit Predistortion · p. 7, Figure 4 · p. 7
  • At n = 48, IA-OPP attains 0.521 % versus 0.607 % for classical OPP, a relative advantage of about 14 %, and undercuts the voltage-oriented-method floor by close to 8 %.At n = 12, the methods attain 1.07 % and 2.00 % respectively; at n = 8, they attain 1.99 % and 5.00 %.Source: 3) Implicit Predistortion · p. 7

1) The THD Plane: Current Quality Versus Bus Load:

  • IA-OPP forms the Pareto front, attaining the lowest THD at every bus load and the lowest bus load for every demanded THD.It reaches 0.521 % at 2.4 kHz, compared with 0.561 % for 1S-NLM at 4.4 kHz; the THD plane plots both logarithmic axes against reconfiguration frequency.Source: 1) The THD Plane: Current Quality… · p. 7, Figure 5 · p. 8

2) Operating-Point Sweep:

IA-OPP remains the best method across the modulation-index sweep and retains its advantage under parameter deviations of ±30 %, with a few small reversals in the most benign combined-perturbation cases.

2) Operating-Point Sweep:

  • Across modulation indices ma = 0.2 to 0.9, fixed-budget IA-OPP is best at every operating point, including 0.279 % versus 0.319 % for 1S-NLM at ma = 0.2.At ma = 0.9, IA-OPP reaches 1.25 % versus 1.41 % for 1S-NLM, which uses 216 events there, 4.5 times the bus load.Source: 2) Operating-Point Sweep · p. 8, Figure 6 · p. 8

E. Robustness

  • Across single-parameter deviations of ±30 %, IA-OPP remains best at every tested point, with a worst THD of 0.661 % versus at least 0.703 % for the best baseline at the same point.Cell-voltage deviations leave THD invariant but affect fundamental amplitude.Source: E. Robustness · p. 8
  • Across 16 simultaneous parameter-perturbation corners, IA-OPP is best in 12; its worst case is 0.945 % versus 0.993 % for the best baseline.The four exceptions are benign corners with increased inductance and decreased cell resistance, where 1S-NLM leads by at most 0.017 percentage points.Source: E. Robustness · pp. 8–9
  • Within ±30 % parameter uncertainty, lookup-table patterns require neither adaptation nor re-identification, and occasional offline recomputation can absorb systematic drift.The advantage is retained in nearly all tested constellations; where it is not, the deficit is much smaller than the nominal-point gain.Source: E. Robustness · p. 9

VI. THREE-PHASE OPERATION

In three-wire star systems, triplen harmonics are common-mode and do not drive line current, so masking them in the objective restores IA-OPP’s advantage. The impedance advantage persists against a linear-model counterpart.

VI. THREE-PHASE OPERATION

  • Triplen orders are common-mode in three-wire star and drive no line current; the analysis removes them from single-string spectra while neglecting phase-coupling feedback.Triplen voltages persist as common mode and shift the star point.Source: §VI · p. 9
  • For single-phase-optimized patterns, only about 0.09 % remains of the 0.565 % floor in three-wire star, and IA-OPP reaches 0.109 % versus 0.177 % non-triplen THD for classical OPP at n = 48.The floor is 88 to 98 % triplen distortion energy; submodule-level PS-PWM and double-budget classical OPP narrowly beat these unmasked patterns.Source: §VI · p. 9
  • With triplen-masked THD in the objective, IA-OPP reaches 0.076 % at n = 48, beating every method, including submodule-level PS-PWM at 0.082 %, at one fifth of its bus load.With free budget selection, it reaches 0.051 % at n = 92 versus 0.063 % at n = 96 for the best feasible linear-model counterpart.Source: §VI · p. 9, Table IV · p. 9

A. Summary of Findings

  • With the grid topology included alongside impedance in the objective, the impedance advantage grows from 14 % to about 21 % at n = 48 and remains 15 to 30 % for n ≥24.The linear model still promises 0.031 % but delivers 0.098 % at n = 48, a factor of 3.Source: §VI · p. 9, A. Summary of Findings · p. 9

B. Limitations

The results are bounded by a simulated stationary RL setting and unresolved practical extensions. Transferability depends on mapping voltage levels to path resistance and accounting for topology.

B. Limitations

  • The evidence is simulation-only, and the stationary RL surrogate excludes back-EMF and torque formation.Hardware-demonstrator measurements remain future work.Source: B. Limitations · p. 9
  • At budgets n ≤20, timing refinement can scatter from local optima; the stated n ≥24 operating range is unaffected.A multi-start probe at n = 48 scatters by 0.004 percentage points.Source: B. Limitations · p. 9
  • The plant model assumes purely ohmic cell resistance and ideal switching, omitting frequency- and state-of-charge-dependent impedance, dead times, and commutation transients.The reported robustness analysis covers resistance mismatch, not the full impedance spectrum.Source: B. Limitations · p. 10
  • IA-OPP optimizes current THD rather than losses; cell assignment for balancing and exact coupled three-phase evaluation remain unexamined.The three-phase analysis is zeroth order.Source: B. Limitations · p. 10

C. Transferability

  • The formulation transfers through a mapping from voltage level to path resistance, including nonmonotone mappings and systems without a polarity stage.The optimization core and quarter-wave structure remain unchanged when polarity mirroring is dropped.Source: C. Transferability · p. 10
  • IA-OPP is presented as a stationary RBS strategy incorporating impedance, event-based cost, and grid topology, with current quality beyond voltage-oriented methods.The authors make this claim for both single-string and three-phase arrangements.Source: C. Transferability · p. 10
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