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arxiv: 2511.17099 · v2 · submitted 2025-11-21 · 💻 cs.CE

Multivariate Sensitivity Analysis of Electric Machine Efficiency Maps and Profiles Under Design Uncertainty

Aylar Partovizadeh , Sebastian Sch\"ops , Dimitrios Loukrezis This is my paper

Pith reviewed 2026-05-17 20:42 UTC · model grok-4.3

classification 💻 cs.CE
keywords multivariate sensitivity analysiselectric machinesefficiency mapsdesign uncertaintymodel simplificationpolynomial chaos expansionSobol indices
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The pith

Multivariate sensitivity analysis assigns one importance index per design parameter for the full efficiency map or profile of an electric machine.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that applying global sensitivity analysis to an efficiency map or profile as a single object, rather than computing it separately at every operating point, produces one sensitivity index per uncertain design parameter. That single index ranks how much each parameter contributes to variation across the whole map. The approach is tested on permanent magnet synchronous machine models of different fidelity levels. Computations use both Monte Carlo sampling and polynomial chaos expansions to compare costs. The resulting indices then guide model simplification by fixing non-influential parameters to nominal values, with validation through direct comparison of uncertainty estimates from the original and reduced models.

Core claim

Multivariate global sensitivity analysis provides a single sensitivity index per parameter, allowing a holistic estimation of parameter importance over the full efficiency map or profile, in contrast to applying variance-based sensitivity analysis elementwise.

What carries the argument

Multivariate global sensitivity analysis, which computes a single sensitivity index per uncertain parameter over the entire efficiency map or profile instead of separately at each operating point.

If this is right

  • Simplified models that vary only the influential parameters produce uncertainty estimates comparable to those from the full models.
  • Polynomial chaos expansions require less computation than Monte Carlo sampling while delivering the same sensitivity indices.
  • The method applies equally to electric machine models of low and high fidelity.
  • Model simplification guided by the single indices per parameter maintains the overall uncertainty behavior of the maps and profiles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same single-index approach could rank parameters for other machine outputs such as torque or loss maps.
  • Design optimization loops could restrict random sampling to the influential parameters identified here, lowering overall cost.
  • Checking the indices against measured data from prototype machines would test whether nominal values adequately capture real manufacturing scatter.

Load-bearing premise

That fixing non-influential parameters to their nominal values after the analysis preserves the overall uncertainty behavior of the efficiency maps, as confirmed only by comparing uncertainty estimates between full and reduced models.

What would settle it

A case in which the reduced model, after fixing parameters according to the multivariate indices, produces clearly different uncertainty distributions or bounds on the efficiency map compared with the full model under identical input variations.

Figures

Figures reproduced from arXiv: 2511.17099 by Aylar Partovizadeh, Dimitrios Loukrezis, Sebastian Sch\"ops.

Figure 1
Figure 1. Figure 1: PMSM ECM. TABLE I PARAMETERS OF THE PMSM ECM. Parameter Symbol Nominal value Units Phase resistance Rs 8.9462 Ω Magnet flux linkage λ 0.1144 Wb d-axis inductance Ld 0.2055 H q-axis inductance Lq 0.332 H 0 2000 4000 6000 8000 Speed (rpm) −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 Torque (Nm) 0.20 0.40 0.60 0.70 0.80 0.85 0.90 0.92 (a) Mean. 0 2000 4000 6000 8000 Speed (rpm) −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 … view at source ↗
Figure 3
Figure 3. Figure 3: illustrates the results of Sobol’ GSA applied elementwise, which yields one sensitivity map per uncertain parameter. Only first-order Sobol’ indices are displayed; total￾order indices are almost identical, suggesting negligible pa￾rameter interactions. The Sobol’ GSA results reveal that Rs and λ have a significant influence on efficiency. In contrast, the contribution of Ld is negligible across the entire … view at source ↗
Figure 2
Figure 2. Figure 2: Mean and standard deviation of the ECM’s efficiency map, along with pointwise absolute errors between MCS- and PCE-based estimates. which corresponds to a sample size NPCE s = 30. Figures 2c and 2d show the pointwise errors for mean and standard deviation, respectively. Similar errors are obtained for the GSA results, therefore, only one set of results is presented in the following. For the aforementioned … view at source ↗
Figure 6
Figure 6. Figure 6: presents the elementwise Sobol’ GSA results for the efficiency profile [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mean and standard deviation of the ECM’s efficiency profile, along with pointwise absolute errors between MCS￾and PCE-based estimates. To validate the GSA results, mean absolute errors (MAEs) in the mean and standard deviation estimates obtained with the full ECM, i.e., with random variations in all four parameters, and with a reduced ECM with fixed Lq and Ld, are computed and presented in Table II. The er… view at source ↗
Figure 9
Figure 9. Figure 9: Mean and standard deviation of theIGA model’s efficiency map, with MAEs for increasing sample sizes. The MAEs are computed with respect to reference values obtained with a PCE trained on the dataset of maximum size Ns,max = 3300. TABLE V MAES IN EFFICIENCY MAP MEAN AND STANDARD DEVIATION ESTIMATES BETWEEN THE FULL AND REDUCED IGA MODEL. Fixed parameters MAE, mean MAE, st.d. SF, HSO, WSO, HM, RRI 5.77 · 10−… view at source ↗
Figure 8
Figure 8. Figure 8: PMSM half-geometry in two dimensions. to vary uniformly within ±5% of its nominal value. The parameters and their nominal values are listed in Table IV. The geometry of the PMSM is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: Multivariate GSA of the IGA model’s efficiency map. the shortcomings of the Sobol’ GSA method applied elemen￾twise, as well as the benefits of multivariate GSA. The latter overcomes the limitations of Sobol GSA with the use of gener￾alized sensitivity indices tailored to multidimensional QoIs. In addition, we compare GSA based on MCS and PCE, and find that the latter is be much more computationally effici… view at source ↗
Figure 10
Figure 10. Figure 10: Elementwise Sobol’ GSA of the IGA model’s ef￾ficiency map. Only first-order indices are shown, due to negligible higher-order interactions. parameters on efficiency maps and profiles. Using two models of a PMSM, namely, an ECM and an IGA model, we showcase 0.0 0.1 0.2 0.3 0.4 Sensitivity index SRO HY HSO WT WSO HM RRI MR SF Rs Br First-order Total-order [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
read the original abstract

This work introduces the use of multivariate global sensitivity analysis for assessing the impact of uncertain electric machine design parameters on efficiency maps and profiles. Contrary to the common approach of applying variance-based (Sobol') sensitivity analysis elementwise, multivariate sensitivity analysis provides a single sensitivity index per parameter, thus allowing for a holistic estimation of parameter importance over the full efficiency map or profile. Its benefits are demonstrated on permanent magnet synchronous machine models of different fidelity. Computations based on Monte Carlo sampling and polynomial chaos expansions are compared in terms of computational cost. The sensitivity analysis results are subsequently used to simplify the models, by fixing non-influential parameters to their nominal values and allowing random variations only for influential parameters. Uncertainty estimates obtained with the full and reduced models confirm the validity of model simplification guided by multivariate sensitivity analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces multivariate global sensitivity analysis (GSA) for assessing the impact of uncertain electric machine design parameters on efficiency maps and profiles. It contrasts this with the common elementwise application of variance-based Sobol' analysis by providing a single sensitivity index per parameter, enabling a holistic evaluation across the full map or profile. The approach is demonstrated on permanent magnet synchronous machine (PMSM) models of varying fidelity, with comparisons between Monte Carlo sampling and polynomial chaos expansions regarding computational cost. Sensitivity results are used to simplify models by fixing non-influential parameters to nominal values, with validation via comparison of uncertainty estimates between full and reduced models.

Significance. If the multivariate GSA successfully yields a single per-parameter index that captures holistic importance and the subsequent model reduction preserves uncertainty behavior, the work could facilitate more efficient uncertainty quantification and design exploration in electric machines by reducing the number of random parameters without sacrificing map-level fidelity. The demonstrations across model fidelities and the explicit MC-versus-PCE cost comparison are practical strengths that support broader adoption in computational engineering workflows.

major comments (1)
  1. [§5] §5 (model reduction and validation): The claim that uncertainty estimates from the full and reduced models confirm the validity of simplification guided by multivariate GSA is load-bearing for the central result. However, it is not clear whether the compared quantities include pointwise variance fields, quantile maps, or distributional distances across the torque-speed plane, or are limited to scalar summaries such as total output variance or mean efficiency. If the latter, this does not fully substantiate preservation of the spatial structure and operating-point dependence of the efficiency map uncertainty.
minor comments (2)
  1. [Abstract and results section] The abstract states that computations based on Monte Carlo and polynomial chaos are compared, but specific quantitative metrics (e.g., number of samples, convergence rates, or wall-clock times) for the two methods on each fidelity model are not tabulated or plotted in the results section.
  2. [Methodology section] Notation for the multivariate sensitivity indices (e.g., how the total-effect index is aggregated over the map) should be defined explicitly with reference to the underlying Sobol' decomposition before the numerical demonstrations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback. We address the major comment point by point below and have made revisions to the manuscript to clarify and strengthen the validation section.

read point-by-point responses

  1. Referee: [§5] §5 (model reduction and validation): The claim that uncertainty estimates from the full and reduced models confirm the validity of simplification guided by multivariate GSA is load-bearing for the central result. However, it is not clear whether the compared quantities include pointwise variance fields, quantile maps, or distributional distances across the torque-speed plane, or are limited to scalar summaries such as total output variance or mean efficiency. If the latter, this does not fully substantiate preservation of the spatial structure and operating-point dependence of the efficiency map uncertainty.

    Authors: We appreciate the referee's comment highlighting the importance of specifying the uncertainty quantities used for validation. In the original submission, the comparisons between full and reduced models were based on scalar summaries, including total output variance and mean efficiency. We agree that this alone does not fully address the spatial structure. In the revised manuscript, we have expanded Section 5 to include pointwise variance fields and quantile maps across the torque-speed plane. New figures have been added to show these comparisons, demonstrating that the reduced model preserves the operating-point dependence of the uncertainty estimates. We believe this addresses the concern and strengthens the central result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper applies standard multivariate global sensitivity analysis (via Monte Carlo or polynomial chaos) to obtain per-parameter indices over the full efficiency map, ranks parameters, fixes non-influential ones to nominal values, and then independently validates the reduced model by comparing its uncertainty estimates against the full model. No step reduces a claimed prediction or result to its own inputs by construction, no load-bearing uniqueness theorem or ansatz is imported via self-citation, and the validation comparison is presented as an external check rather than a definitional tautology. The central claim therefore rests on the independent numerical evidence of the sensitivity indices and the subsequent uncertainty comparison.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work builds on standard variance-based sensitivity analysis extended to multivariate outputs; no new free parameters are introduced, and the central claim relies on domain assumptions about model fidelity and the validity of parameter fixing.

axioms (2)
  • domain assumption Variance-based (Sobol') sensitivity analysis can be meaningfully extended from scalar to multivariate outputs such as efficiency maps
    Paper explicitly contrasts this with the common elementwise approach.
  • domain assumption Fixing non-influential parameters to nominal values does not materially alter the uncertainty quantification of the efficiency maps
    Used to justify model simplification and confirmed only by comparing uncertainty estimates.

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