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Parameter Estimation of Power Electronic Converters with Differentiable Physics Simulation

Pingjunjin Tan, Chunlin Lv, Jinjun Liu, Yang Li

arXiv:2608.30915v1eess.SY

TL;DR

Estimating converter parameters from measured signals is challenging because explicit mappings are nontrivial under nonlinear operation, while direct measurement schemes may require auxiliary sensors. This paper embeds buck-converter simulation in a differentiable computational graph for parameter estimation and reports accurate tracking of critical-component variation trends in hardware.

  • Problem

    Explicit mappings from component parameters to observed signals are nontrivial for nonlinear switched-mode power supplies, while direct measurement schemes may use auxiliary sensors.

    Source: §I · p. 1

  • Method

    The method embeds converter time-stepping simulation in a differentiable computational graph, enabling gradient-based parameter optimization through converter dynamics.

    Source: §I · p. 3, §V · p. 15

  • Results

    Experiments show that the method accurately and robustly tracks relative variation trends in critical buck-converter components.

    Source: B. Experiment Results Analysis · p. 14, C. Discussion · p. 15

  • Takeaways & Limitations

    Under practical hardware conditions, estimated critical parameters provide meaningful condition indicators for the dc–dc buck converter.

    Source: C. Discussion · p. 15

    Some parameters have limited identifiability because of weak sensitivity to measured states and strong mutual coupling.

    Source: C. Discussion · p. 15

Abstract

from arXiv · show

This article proposes a differentiable physics simulation (DP simulation)-based parameter estimation method for the condition monitoring of power electronic converters. In the proposed method, the time-domain simulation of converter dynamics is embedded into a differentiable computational graph, directly linking device parameters to observed voltage and current trajectories. By formulating differentiable time-stepping operators, the nonlinear dynamics of the converter across different circuit topologies are simulated in a unified, differentiable manner. A dc-dc buck converter is used as a representative case study. Using sparse transient samples from existing sensing channels, the method enables noninvasive parameter estimation without additional sensing hardware. Comprehensive simulation studies are conducted to evaluate the impacts of time-stepping schemes, regularization constraints, and various uncertainty sources on estimation accuracy and robustness. Subsequently, 30 distinct hardware configurations are experimentally tested for validation. The results show that the proposed method can effectively track the relative variations of health-related parameters across the critical components. This DP simulation framework provides a novel perspective for physics-informed machine learning in power electronic applications.

I. INTRODUCTION

The introduction argues for converter condition monitoring through electrical health indicators and proposes differentiable physics simulation to estimate circuit parameters directly through converter dynamics, including across topology changes.

Motivation

  • Electrical changes can reveal otherwise hard-to-observe component aging, making intrinsic parameters such as MOSFET on-state resistance and capacitor capacitance useful health indicators.The introduction frames monitoring as a way to track degradation and support timely intervention before failures, especially in safety-critical applications.Source: §I · p. 1

Prior methods

  • Prior estimation methods span model-based, hybrid, and data-driven categories, but direct measurement schemes add sensing hardware while simplified models can assume other parameters are known and constant.Those assumptions are problematic when multiple components degrade together, motivating whole-converter rather than isolated-component monitoring.Source: §I · pp. 1–2
  • Data-driven estimators can be fast after training, but power electronics is data-scarce and distribution shifts can produce catastrophic, physically inconsistent predictions.The introduction notes that these approaches rely on datasets and extracted time- or frequency-domain features.Source: §I · pp. 2–3
  • Time-domain hybrid methods use switching dynamics, but simulator-based implementations face deployment and recompilation costs, while adaptive observers often ignore parasitic elements.Numerical propagation applies Runge–Kutta discretization across arbitrary topologies, yet derivative-free optimization can require extensive evaluations and be sensitive to hyperparameters.Source: §I · p. 3

Prior methods and contribution

  • Existing PINN constraints generally apply only within topology-invariant intervals, forcing samples to switching instants, while joint neural-weight and parameter optimization can destabilize convergence.The proposed DP approach instead backpropagates through discrete physical evolution without neural-network surrogates, preserving physical consistency and computational efficiency.Source: §I · p. 3

Contributions

  • The proposed DP framework differentiates numerical integration for end-to-end parameter optimization, enabling sampling away from noisy switching instants and simultaneous estimation of critical parameters across topologies.The authors present a dc-dc buck converter as an example and describe the design as extensible to other architectures, including dc-ac inverters.Source: §I · p. 3

II. METHODOLOGY · A. Dynamic Modeling of the Buck Converter in CCM

The buck converter’s CCM dynamics are modeled as two linear stages selected by semiconductor switching, with a discrete switch function unifying the topology-dependent behavior. Its state-space formulation uses inductor current and output voltage as states and includes relevant parasitic and load parameters.

A. Dynamic Modeling of the Buck Converter in CCM

  • In CCM, MOSFET switching divides the buck converter’s nonlinear dynamics into two distinct linear circuit stages.The stages are depicted in Fig. 2(a) and Fig. 2(b).Source: A. Dynamic Modeling of the Buck… · p. 4
  • A discrete switching function S ∈ {0, 1} represents the active switch state and unifies buck-converter dynamics across topologies.PWM governs the semiconductor switching behavior.Source: A. Dynamic Modeling of the Buck… · p. 4
  • The state-space formulation selects inductor current iL and output voltage vo to align with sparsely sampled observations.It incorporates input voltage vin, diode forward drop vF, MOSFET on-state resistance Rdson, inductance L, capacitance C, their series resistances RL and RC, and load resistance R.Source: A. Dynamic Modeling of the Buck… · p. 4

B. Generalized Runge–Kutta Time-stepping Scheme

The section motivates numerical Runge–Kutta time stepping as a practical alternative to rigid closed-form solutions for converter ODEs, then distinguishes explicit and implicit stage-solving methods.

B. Generalized Runge–Kutta Time-stepping Scheme

  • Rather than derive an intractable closed-form solution, numerical stepping discretizes the continuous ODE and iteratively advances the state using step size h and increment function Φ.The state at step n + 1 is deduced from the known state at step n.Source: B. Generalized Runge–Kutta… · p. 4
  • Runge–Kutta time stepping replaces integration with weighted evaluations at multiple intermediate stages, with its coefficients organized in a Butcher tableau.The tableau contains the stage matrix, weights, and nodes.Source: B. Generalized Runge–Kutta… · pp. 4–5
  • ERK's strictly lower-triangular coefficient matrix enables sequential stages, requiring exactly ν function evaluations and O(ν) time per step.Its stages depend only on earlier stages.Source: B. Generalized Runge–Kutta… · p. 5
  • For IRK, dense stage coupling can be solved by Gaussian elimination because the buck converter is LTI within each switching interval, yielding O(ν^3) time complexity.The figure depicts the buck converter’s MOSFET off-state and on-state circuits.Source: Figure 2 · p. 4, B. Generalized Runge–Kutta… · p. 5

C. Differentiable Physics Simulation for Buck Converter

The framework turns converter time-stepping into a differentiable composition of operators, enabling gradient-based parameter estimation from sparse buck-converter observations. It handles switching by changing operators at known topology transitions while preserving gradient propagation with respect to the physical parameters.

C. Differentiable Physics Simulation for Buck Converter

  • Each time step is modeled as a differentiable operator, and q successive steps form a state-evolution composition whose operators provide exact gradients with respect to parameters and state inputs.The target parameter vector is θ = [Rdson, vF, L, RL, C, RC, R]T.Source: C. Differentiable Physics Simulation… · p. 5
  • The buck-converter model uses sparsely sampled state vectors u_n = [iL(n), vo(n)]T as training data, linking observations to simulated dynamics.The framework minimizes mean squared error between measured data and model-derived states to identify physical parameters with gradient-based optimization.Source: C. Differentiable Physics Simulation… · pp. 5–6
  • At switching nodes, the framework swaps on-state and off-state operators, preserving parameter-gradient propagation through the buck converter’s piecewise-smooth dynamics.Sampling instants and switching nodes are accessible from the digital controller; each interval is advanced using its corresponding discrete steps.Source: C. Differentiable Physics Simulation… · pp. 5–6, Figure 3 · p. 6
  • The framework’s physical interpretability ensures inferred target parameters conform to the converter’s underlying dynamic constraints.The authors characterize this design as bridging classical numerical integration with machine learning.Source: C. Differentiable Physics Simulation… · p. 6

D. Automatic Differentiation and Backpropagation

The section explains how automatic differentiation computes parameter gradients through the converter simulation and how backpropagation through time accumulates those gradients across shared parameters. These gradients support gradient-based and higher-order optimization.

D. Automatic Differentiation and Backpropagation

  • AD applies the chain rule across simulation arithmetic operations instead of symbolically differentiating its deep composition to obtain target-parameter gradients.Reverse mode is especially efficient for a scalar objective with many parameters.Source: D. Automatic Differentiation and… · p. 6
  • The forward phase evaluates functions and records dependencies, while the backward phase propagates adjoints from outputs to inputs to compute derivatives.An adjoint represents an output’s sensitivity to an intermediate variable.Source: D. Automatic Differentiation and… · pp. 6–7
  • Because outputs depend on past states and simulation operators share θ, BPTT unrolls the dynamics over time and sums stepwise gradient contributions for shared parameters.The unrolled graph is illustrated in Fig. 5.Source: D. Automatic Differentiation and… · p. 7, Figure 5 · p. 7
  • The resulting gradient flow enables gradient-based optimization and efficient higher-order derivatives, including the Hessian, for higher-order optimization strategies.Source: D. Automatic Differentiation and… · p. 7

E. Sampling Strategy and Data Configuration

The section prioritizes sparse, low-frequency sampling during transients and explains how the differentiable simulation handles switching events between observations. It also describes reusing sampled points to extend the simulation horizon and reduce sampling frequency.

E. Sampling Strategy and Data Configuration

  • Carrier-synchronized samples at carrier valleys and peaks capture transient dynamics away from switching events, avoiding ringing that compromises peak-to-peak measurements.Transient responses are more sensitive to parameter variations than steady-state operation, and carrier-synchronized sampling is compatible with standard digital control systems.Source: E. Sampling Strategy and Data… · p. 7
  • Deterministic controller timing lets the framework simulate across topological switching events between adjacent sparse observations, improving sampled-data utilization.Source: E. Sampling Strategy and Data… · p. 7
  • A sampled point can serve as both the preceding interval’s ending state and the next interval’s initial state, or predict Nh −1 subsequent points over a horizon of length Nh.This reuse can reduce sampling from 2fsw to fsw or lower, including one sample over multiple switching cycles.Source: E. Sampling Strategy and Data… · pp. 7–8
  • The subsequent study first considers one-step prediction, then examines extended-horizon simulation and other constraints.Source: E. Sampling Strategy and Data… · p. 8

III. SIMULATION VERIFICATION AND ROBUSTNESS ANALYSIS · A. Accuracy and Computation Complexity

Simulation studies assess converter-parameter identification across configurations and practical uncertainties. They find that C, L, and Rload are accurately identifiable, while RL and Rdson are better monitored through their equivalent resistance; a single-step classical fourth-order Runge–Kutta scheme balances accuracy and computational cost.

III. SIMULATION VERIFICATION AND ROBUSTNESS ANALYSIS

  • The simulation study evaluates generalization across converter configurations and operating conditions, and examines accuracy–complexity tradeoffs and aleatoric and epistemic uncertainties.The uncertainties include measurement noise, ADC quantization, synchronization offset, and model–plant mismatch.Source: §III · p. 8

A. Accuracy and Computation Complexity

  • Training uses transient samples of inductor current and output voltage, which are commonly available during startup, shutdown, load switching, and control-mode transitions.These observations are also feedback signals used in the control loop, making them readily accessible without additional sensing hardware.Source: A. Accuracy and Computation Complexity · p. 8, Figure 8 · p. 8
  • Shorter transients as switching frequency and control algorithms improve make data collection harder, but steady-state samples can partly alleviate data scarcity.Slow component degradation, typically over months or years, provides time for data collection given the method’s low data demand.Source: A. Accuracy and Computation Complexity · pp. 8–9
  • The simulation framework combines samples from three load transitions, each collected over 30 switching cycles, and trains differentiable converter dynamics using Adam followed by full-batch L-BFGS.The framework is implemented in Python with PyTorch, enabling gradient-based optimization through its differentiable computational graph.Source: A. Accuracy and Computation Complexity · p. 9
  • C, L, and Rload are accurately identified, but RL and Rdson are difficult to estimate individually, so their sum RD is used to monitor degradation.RL and Rdson have nearly identical effects on the observed output voltage and inductor current.Source: A. Accuracy and Computation Complexity · p. 9
  • A single-step classical fourth-order Runge–Kutta method is selected because higher orders or smaller steps yield marginal identification gains, while substeps more directly increase computational cost.Numerical integration error is already much smaller than errors from observation noise and model–plant mismatch in practical monitoring.Source: A. Accuracy and Computation Complexity · p. 9

B. Regularization Constraint

The section addresses ill-posed parameter estimation with converter-dynamics-based regularization rather than hard-to-standardize prior bounds. It introduces bidirectional and long-horizon consistency constraints, which generally improve identifiability while trading off accuracy against data and computational cost.

Regularization rationale

  • Small waveform mismatch can still produce large errors in weakly identifiable parameters, even when training-loss MSE is on the order of 10−5, indicating an ill-posed inverse problem.The section argues that narrow prior bounds are difficult to justify because converter parameters may vary widely over a device’s lifetime.Source: B. Regularization Constraint · p. 9

Bidirectional consistency

  • Bidirectional consistency requires a parameter set to predict the next state from the current state and reconstruct the current state from the next state over a short sampling interval.It imposes local consistency in both time directions on dynamics that physically evolve forward in time.Source: B. Regularization Constraint · p. 9, Figure 9 · p. 10

Long-horizon consistency

  • Long-horizon consistency reduces compensation among coupled parameters by matching the measured trajectory across an extended window, where small modeling errors can accumulate and become more evident.One-step prediction can yield low loss despite inaccurate estimates of parameters such as RL, Rdson, and vF because their effects on observable states are coupled.Source: B. Regularization Constraint · p. 10, Figure 9 · p. 10

Regularization results

  • The proposed constraints generally improve parameter identifiability; on clean simulation data, longer prediction windows improve accuracy and help distinguish strongly coupled RL and Rdson.Accurate estimation of RL and Rdson remains challenging in Cases V and VI, which have small parasitic resistances and relatively small load variations.Source: B. Regularization Constraint · pp. 10–11, Table VI · p. 11
  • At Nh = 16, training takes under one minute on an NVIDIA GeForce RTX 5060 Ti, while ˆθ1, RC, VF, and RD have errors below 1% in most cases.The paper selects this window length for subsequent robustness analysis to balance estimation accuracy and computational cost.Source: B. Regularization Constraint · p. 11
  • Longer prediction windows require more transient samples and increase differentiable-simulation computation approximately linearly with window length.The section cautions that excessively long windows may impose unnecessary training cost.Source: B. Regularization Constraint · p. 11

C. Robustness Analysis

The robustness analysis finds that estimation remains accurate for strongly identifiable parameters under tested uncertainties, while some parameters are more sensitive or difficult to estimate separately because of coupling.

Robustness Analysis

  • Tests introduced aleatoric measurement noise, ADC quantization, inter-channel synchronization offset, and epistemic model–plant mismatch into clean simulation data.Noise settings represented ±0.5% and ±1% full-scale accuracy; the 12-bit ADC model quantized to nearest code, delays were sampled up to 0.02Tsw (1 µs at 20 kHz), and mismatch used added parasitic components.Source: C. Robustness Analysis · pp. 11–12, Figure 10 · p. 11
  • ADC quantization and model–plant mismatch caused limited degradation, while measurement noise and synchronization offset had more influence; C, L, and Rload remained accurate, unlike sensitive RC/vF and unreliable separate RL/Rdson estimates.These comparisons summarize the Case I uncertainty tests.Source: Table VII · p. 12, C. Robustness Analysis · p. 12
  • Across 1,000 trials under the most severe uncertainty configuration, 95% of vF estimates fell between −11% and 17%, while most RL/Rdson estimates stayed within the ±20% band for RD = RL + Rdson.L, C, and Rload errors were tightly concentrated; RL and Rdson estimates were strongly correlated along an approximately constant equivalent-resistance direction.Source: C. Robustness Analysis · pp. 12–13, Figure 11 · p. 12
  • With only 30 switching cycles per training set and sampling at twice the switching frequency, the framework remained robust for strongly identifiable parameters and provided reliable health indicators for coupled degradation-related parameters.Source: C. Robustness Analysis · p. 13

IV. EXPERIMENT VERIFICATION

Experimental validation uses a buck-converter platform with replaceable components, onboard sensing, and calibrated acquisition channels. Capacitor degradation is emulated by combining capacitors and series resistors with different values.

IV. EXPERIMENT VERIFICATION

  • A buck-converter test platform supports hardware variation through replaceable capacitors, inductors, MOSFETs, and diodes.Capacitor aging is emulated by combining capacitors and series resistors with different values to vary capacitance and ESR.Source: §IV · p. 13
  • Onboard sensors acquire signals using the Section III-A sampling strategy, and sensing channels are calibrated against oscilloscope measurements before data collection.Calibration is performed to reduce acquisition errors.Source: §IV · p. 13

A. Experiment Configuration · B. Experiment Results Analysis

Experiments use transient data from 30 hardware configurations to assess parameter estimation against relative changes from an estimated baseline. The method tracks key component trends, with accuracy varying by parameter.

A. Experiment Configuration

  • Each estimation task combines three load transitions, with 60 transient samples per record taken over 30 switching cycles at synchronized PWM-carrier valley and peak instants.The tests use a 20 kHz switching frequency, single-step ERK4, and the specified regularization constraints with Nh = 16 for the long-horizon constraint.Source: A. Experiment Configuration · p. 13
  • Thirty hardware configurations are formed by interchanging passive and active components with different values to simulate degradation.Inductor and capacitor baselines are measured with an LCR meter at switching frequency; MOSFET and diode values come from static measurements at a specified current.Source: A. Experiment Configuration · p. 13, Table VIII · p. 13

B. Experiment Results Analysis

  • Evaluation uses relative percentage changes rather than absolute deviations from offline measurements, which may differ from effective operational parameters.Estimates for L1M1C1 define the 0% reference baseline and are the average of three repeated experiments under the same operating conditions; other configurations are normalized to it and compared with offline-derived changes.Source: B. Experiment Results Analysis · pp. 13–14, Table IX · p. 14, Figure 13 · p. 14
  • Capacitor ESR variation is tracked despite an 18.78% maximum relative-change deviation under L1M3C5, compared with a 161.62% ESR increase.The deviation is relatively small given that an ESR value 2–3 times its initial value is commonly used as a capacitor failure criterion.Source: Table IX · p. 14, B. Experiment Results Analysis · pp. 14–15
  • Inductance, capacitance, and load-resistance variations are tracked with high accuracy and narrow distributions across hardware configurations.The inductance estimates are slightly more dispersed than capacitance estimates, plausibly because current sensors are more susceptible to noise and inaccuracies than voltage sensors.Source: B. Experiment Results Analysis · p. 14, Figure 13 · p. 14
  • The diode forward voltage vF is not emphasized because its weak contribution to measured waveforms makes accurate identification difficult under measurement noise and model mismatch.Source: B. Experiment Results Analysis · p. 15
  • Equivalent resistance RD estimates are less accurate and more dispersed; under L2M2, trial relative-variation rates differ by approximately 8%, with a considerable mean-to-reference discrepancy.RL and Rdson vary with temperature and current although the estimation model treats their sum as constant; offline static measurements and sampling distortion also contribute to error.Source: Figure 13 · p. 14, B. Experiment Results Analysis · p. 15
  • Despite RD estimation errors, its estimated variation generally follows the reference trend, supporting its use as a condition indicator, particularly for detecting severe faults.The paper gives a multifold increase in MOSFET on-state resistance caused by gate-drive anomalies as an example.Source: B. Experiment Results Analysis · p. 15

C. Discussion · V. CONCLUSION

The differentiable-physics framework estimates converter parameters through physical time-stepping and tracks health-related variations in a buck converter. The discussion highlights extensions beyond monitoring and unresolved challenges involving operating conditions, identifiability, and uncertainty.

C. Discussion

  • The parameter-to-waveform mapping may support converter design optimization and model predictive control, while the differentiable solver could integrate with neural-network-based deep learning.The conclusion also notes potential integration with established numerical integration schemes.Source: C. Discussion · p. 15, §V · p. 15
  • Because component characteristics vary with temperature and load current, fixed estimated values compared with static offline references may mismatch operating conditions; health-state representations should account for this behavior.The discussion identifies this mismatch as a challenge for widespread industrial implementation.Source: C. Discussion · p. 15
  • Weak sensitivity to measured states and strong parameter coupling limit identifiability, motivating a rigorous framework to quantify it rather than relying only on empirical experiments.Source: C. Discussion · p. 15
  • Measurement noise and synchronization error can undermine estimation credibility, so uncertainty should be explicitly modeled and propagated through estimation rather than treated only as robustness-test perturbations.Source: C. Discussion · p. 15

V. CONCLUSION

  • L, C, and Rload had maximum experimental variation deviation within 5%, while capacitor ESR RC and equivalent resistance RD had maximum deviations within 20%; the method reliably tracks critical buck-converter parameters under practical hardware conditions.The latter parameters can still serve as useful health indicators for maintenance guidance.Source: §V · p. 15, C. Discussion · p. 15
  • Reformulating numerical time-stepping as a differentiable computational graph enables end-to-end gradient-based parameter optimization through converter physical evolution.The framework establishes a differentiable mapping between critical component parameters and externally measured waveforms.Source: §V · p. 15, C. Discussion · p. 15
  • The framework maintains gradient propagation across converter topology transitions and supports reliable training with sparsely sampled data.Its time-stepping formulation is topology-general and could extend to dc–ac inverters and resonant converters.Source: §V · p. 15, C. Discussion · p. 15

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