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Sensorless Battery Internal Temperature Estimation using a Kalman Filter with Impedance Measurement

Robert R. Richardson, David A. Howey

arXiv:1501.06160v1eess.SY

TL;DR

The paper addresses the need to estimate battery internal temperatures without relying on surface sensors or impedance-only methods that provide only average temperature. It couples impedance measurements with a reduced-order thermal model and Kalman filtering, achieving accurate core and surface estimates and comparable performance to surface-temperature-based estimation.

  • Problem

    Conventional thermal estimation can drift without sensor feedback, while impedance-only temperature detection does not uniquely identify the cell temperature distribution and surface sensing increases instrumentation cost.

  • Method

    The study couples current, voltage, and periodic impedance measurements with a polynomial thermal model in an EKF, and uses a DEKF to estimate the convection coefficient when unknown.

  • Results

    0.47 ◦C (2.4% of the core temperature increase) agreement was obtained, with impedance-based estimation comparable to equivalent surface-temperature-based estimation.

  • Takeaways & Limitations

    Impedance temperature detection can support core and surface temperature estimation without temperature sensors and can identify the convection coefficient during operation.

Abstract

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This study presents a method of estimating battery cell core and surface temperature using a thermal model coupled with electrical impedance measurement, rather than using direct surface temperature measurements. This is advantageous over previous methods of estimating temperature from impedance, which only estimate the average internal temperature. The performance of the method is demonstrated experimentally on a 2.3 Ah lithium-ion iron phosphate cell fitted with surface and core thermocouples for validation. An extended Kalman filter, consisting of a reduced order thermal model coupled with current, voltage and impedance measurements, is shown to accurately predict core and surface temperatures for a current excitation profile based on a vehicle drive cycle. A dual extended Kalman filter (DEKF) based on the same thermal model and impedance measurement input is capable of estimating the convection coefficient at the cell surface when the latter is unknown. The performance of the DEKF using impedance as the measurement input is comparable to an equivalent dual Kalman filter using a conventional surface temperature sensor as measurement input.

I. INTRODUCTION

Accurate battery temperature estimation is important for safe, effective electric-vehicle operation, but conventional model-based approaches require thermal knowledge and costly surface sensing. The study combines impedance-based measurement with a thermal model and Kalman filtering to estimate internal temperature distributions without a surface sensor on every cell.

  • Motivation: Battery management systems require accurate on-board temperature estimates because lithium-ion cells can experience temperature differences during vehicle operation.These differences matter for safe and optimal battery performance.
  • Existing approaches: Conventional thermal models depend on cell properties, heat generation, and boundary conditions, while models without sensor feedback may drift from true temperatures.Online surface and coolant-temperature measurements can improve accuracy but add instrumentation requirements.
  • Existing approaches: Large battery packs may contain several thousand cells, making surface-temperature sensors on every cell substantially costly.This motivates measurement strategies that reduce dependence on per-cell surface sensors.
  • Impedance-based estimation: Impedance-Temperature Detection infers a volume-averaged cell temperature from electrochemical impedance, but impedance alone does not uniquely determine the internal temperature distribution.A thermal model can use this scalar measurement to estimate core temperature and the broader distribution.
  • Proposed approach: The proposed approach uses current, voltage, and periodic real-part impedance measurements with an extended Kalman filter and a polynomial thermal model, while a dual filter can identify the convection coefficient.The study validates predicted core and surface temperatures against thermocouple measurements.

III. THERMAL-IMPEDANCE MODEL

The thermal-impedance model combines a one-dimensional cylindrical heat equation with a reduced polynomial approximation and impedance-related temperature information. Its two-state formulation represents the cell’s thermal behavior using volume-average temperature and temperature gradient.

  • Thermal model: The thermal model represents unsteady radial heat conduction in a cylindrical cell using density, specific heat capacity, thermal conductivity, cell volume, heat generation, and convection boundary conditions.The heat source includes electrical and entropic contributions, although the study neglects entropic heat under its operating conditions.
  • Heat generation: The heat-generation model uses online current and voltage measurements, approximates open-circuit voltage as constant, and neglects entropic heat for the studied SOC range.The operating SOC range is 47–63%, and the temperature derivative of open-circuit voltage is reported as small.
  • Polynomial approximation: The polynomial approximation assumes a parameterized temperature distribution and reduces the heat equation to two ordinary differential equations.The reduced model uses the volume-average temperature and temperature gradient as its two states.
  • Polynomial approximation: The temperature distribution and surface temperature are expressed from the reduced states together with the cell surface and environmental temperatures.The surface temperature is computed from the reduced thermal states and thermal parameters.
  • State-space formulation: The reduced model is represented in state-space form with defined states, inputs, outputs, and system matrices.This formulation provides the model structure used to connect thermal dynamics with measurements.

C. Impedance Measurement

The impedance measurement is incorporated as a function of the reduced thermal state and environmental temperature. With known thermal parameters, the model maps the state variables to the cell’s electrical admittance.

  • Impedance measurement: For a sufficiently small inner radius, the annular cell model is treated as a solid cylinder.This simplifies the cylindrical admittance and thermal expressions used by the model.
  • Impedance measurement: The real admittance is expressed as a function of surface temperature, volume-average temperature, and temperature gradient.The surface temperature is itself determined by the reduced thermal states and environmental temperature.
  • Impedance measurement: With known cell and convection parameters, impedance becomes a function of the reduced cell state and environmental temperature.The relevant parameters include ro, kt, cp, ρ, and h.

IV. FREQUENCY DOMAIN ANALYSIS

The frequency-domain analysis compares the polynomial approximation and a quadratic-assumption model with the analytical cylindrical heat-transfer solution. The polynomial approximation agrees with the analytical solution over a broader frequency range, especially for the studied slow cooling-fluid variation.

  • Frequency-domain comparison: The analysis evaluates approximation error by comparing the polynomial and quadratic models with the analytical solution of the cylindrical heat equation.The comparison uses frequency responses of the thermal models and their errors relative to the analytical solution.
  • Frequency-domain comparison: Figure 3 compares frequency responses for the analytical solution, the study’s polynomial approximation, and the quadratic assumption from [17].The responses cover effects of heat generation and cooling-fluid temperature on core and surface temperatures.
  • Frequency-domain comparison: The polynomial and quadratic models agree well with the analytical solution for heat-generation responses, while the quadratic model has zero surface-response error because it uses measured surface temperature.The zero-error result applies to the quadratic model’s H21 and H22 responses.
  • Frequency-domain comparison: Above approximately 10^-3 Hz, the quadratic model’s H12 response shows a rapid increase in error relative to the analytical solution.H12 represents the core-temperature response to changes in cooling-fluid temperature.
  • Frequency-domain comparison: The polynomial approximation remains satisfactory to higher frequencies and has a broader agreement range than the quadratic model, although its H21 and H22 errors become unsatisfactory above approximately 10^-2 Hz.The reported range is considered satisfactory given the slow rate of cooling-fluid variation.

V. EXPERIMENTAL

The experiments identified and validated a reduced thermal model for a 2.3 Ah LiFePO4 cell using measured electrical and temperature data. The parameterised model reproduced core and surface temperatures closely across separate current excitation profiles.

  • Experimental setup: Experiments used a 2.3 Ah cylindrical LiFePO4 cell instrumented with surface and core thermocouples for validation.The core thermocouple was inserted through a drilled hole in the positive-electrode end.
  • Experimental protocol: Two 3500 s HEV drive-cycle current profiles were used for model parameterisation and validation, with impedance measured every 24 s.Applied currents ranged from −23 A to +30 A, and temperatures were monitored throughout.
  • Parameter identification: Thermal parameters kt, cp, and h were estimated offline by minimising the Euclidean distance between measured and estimated core and surface temperatures.The first excitation profile included current, voltage, surface, core, and chamber-temperature data.
  • Parameterisation results: 0.19 °C and 0.18 °C were the surface and core RMSEs, respectively, for the parameterised model.Measured and predicted temperatures were compared in Fig. 5.
  • Validation results: 0.21 °C and 0.16 °C were the core and surface RMSEs, respectively, on the second excitation profile used for validation.These errors were only marginally greater than those in the parameterisation test.

VII. STATE ESTIMATION

The state-estimation framework uses impedance as the model output within a dual extended Kalman filter, allowing temperatures and the convection coefficient to be estimated jointly. A standard EKF is obtained by fixing the convection coefficient and omitting parameter updates.

  • Dual estimation: The DEKF estimates core temperature, surface temperature, and the convection coefficient when the latter is unknown.The convection coefficient is particularly influential and depends strongly on thermal-management settings.
  • Measurement model: The discrete-time model treats impedance Z′ as the measured output while computing core and surface temperatures from the identified states and parameter.Those temperatures are computed for validation against thermocouple measurements.
  • State update: The nonlinear impedance measurement relationship is linearised about the predicted observation at each measurement before the state update.The state update uses the Jacobian of the measurement function with respect to the state.
  • Parameter update: The parameter filter separately performs time and measurement updates for the convection coefficient and its error covariance.The parameter-update equations use the Jacobian of the measurement function with respect to h.
  • Baseline comparison: The DEKF reduces to a standard EKF when parameter updates are omitted and the convection coefficient is fixed.The study compares the baseline EKF and full DEKF algorithms.

VIII. RESULTS

The study compares EKF and DEKF temperature estimation with impedance-based and conventional surface-temperature measurements. It also evaluates DEKF performance when the convection coefficient begins with an incorrect estimate.

  • Results comparisons: The experiments compare EKF estimation with fixed convection coefficient against DEKF estimation when its initial value is incorrect.The DEKF is also compared with a DKF using surface temperature rather than impedance as the measurement input.

A. Convection Coefficient Known

With the convection coefficient known, an EKF using impedance measurements accurately estimates core and surface temperatures. The simpler direct-impedance formulation has important scope limitations.

  • A. Convection Coefficient Known: The EKF using Z′ measurement input quickly converges to correct core and surface temperatures throughout the excitation profile.Core and surface RMSEs are 1.35 °C and 1.34 °C, compared with 6.66 °C and 4.42 °C for the open-loop model.
  • A. Convection Coefficient Known: A simpler EKF assuming impedance is directly related to T also achieves similar performance.That assumption may be unsatisfactory for larger-radius cells or larger internal temperature gradients.
  • A. Convection Coefficient Known: The experiment evaluates temperature results for an EKF using Z′ as measurement input.
  • A. Convection Coefficient Known: The direct-impedance assumption is unsuitable for DEKF application because impedance is treated as a function of the state only, not the convection coefficient h.

B. Convection Coefficient Unknown

When the convection coefficient is unknown, the DEKF corrects it and improves temperature prediction using impedance measurements. Its performance is comparable to surface-temperature-based estimation, while impedance enables sensorless core and surface estimation.

  • B. Convection Coefficient Unknown: The DEKF corrects an initially incorrect convection coefficient and improves subsequent core and surface temperature predictions.The EKF overestimates core temperature and underestimates surface temperature when the convection coefficient is too high.
  • B. Convection Coefficient Unknown: The DKF using Tsurf identifies the convection coefficient correctly, while its thermocouple input converges faster because it is less noisy than impedance.
  • B. Convection Coefficient Unknown: Using Z′ as measurement input accurately estimates core and surface temperatures and the convection coefficient.
  • B. Convection Coefficient Unknown: Impedance temperature detection enables core and surface temperature estimation without temperature sensors.The study demonstrates using ITD as measurement input to a cell thermal model.
  • B. Convection Coefficient Unknown: The PA thermal model is robust to cooling-fluid fluctuations near 10^-2 Hz, whereas the earlier QA solution may become inaccurate near 10^-3 Hz or higher.
  • B. Convection Coefficient Unknown: The DEKF using impedance has performance comparable to an equivalent DKF using surface temperature measurements, although the DKF is slightly superior.
  • B. Convection Coefficient Unknown: Future work will examine ITD for multiple cells and combinations with conventional sensors for monitoring, fault detection, and self-calibration.

A. Frequency Domain Analysis of Quadratic Assumption

The frequency-domain analysis derives transfer-function descriptions for the quadratic-assumption model and compares them with the analytical cylindrical heat-equation solution. The QA model requires a quadratic radial temperature constraint to obtain a unique distribution from limited measurements.

  • A. Frequency Domain Analysis of Quadratic Assumption: The derivation substitutes temperature and surface-temperature expressions, integrates them, and forms the QA system model.
  • A. Frequency Domain Analysis of Quadratic Assumption: The continuous-time system uses heat generation and ambient temperature as inputs and core and surface temperatures as outputs.
  • A. Frequency Domain Analysis of Quadratic Assumption: The transfer-function system provides frequency responses for the analytical solution and the QA model.
  • A. Frequency Domain Analysis of Quadratic Assumption: The QA model uses volume-averaged temperature from impedance and surface temperature measurements, plus a constraint on the temperature profile.
  • A. Frequency Domain Analysis of Quadratic Assumption: The imposed QA profile corresponds to the 1D steady-state cylindrical heat-equation solution with uniform heat generation.
  • A. Frequency Domain Analysis of Quadratic Assumption: Setting the QA volume average equal to the true volume average yields an estimate of core temperature from surface and volume-averaged measurements.
  • A. Frequency Domain Analysis of Quadratic Assumption: The analysis treats T EIS = T only to identify frequencies where QA errors relative to the analytical solution become significant.
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