How can battery test systems leverage the least-squares fitting method to analyze post-discharge polarization voltage relaxation, and what level of fitting accuracy can be achieved? Achieve ±1 mV Accuracy

Battery test systems can use least-squares fitting to rapidly model polarization-voltage relaxation after dynamic discharge, typically achieving a measured-to-fitted voltage error within ±1 mV. The system records the voltage recovery during a defined rest period, fits the relaxation curve to a battery polarization model, and uses the extracted parameters to improve SOC and SOH estimation. In representative tests following FUDS discharge at 60%, 70%, and 80% depth of discharge, fitting can be completed in approximately 235–378 seconds.

The practical value is not merely a better curve fit. Least-squares analysis converts post-discharge voltage relaxation into reliable polarization-model parameters, allowing battery test systems to reduce fitting time while preserving approximately millivolt-level accuracy for battery-model and state-estimation work.

Why Post-Discharge Relaxation Matters

Polarization Distorts the Measured Voltage

During dynamic discharge, the measured terminal voltage contains more than the battery’s equilibrium voltage. It also includes ohmic voltage drop and transient polarization effects caused by electrochemical and concentration processes.

When the load is removed, the polarization component gradually decays. Until that decay is sufficiently complete, the measured terminal voltage does not accurately represent the battery’s true open-circuit voltage, or OCV.

Relaxation Reveals Battery Dynamics

A controlled rest period allows the polarization voltage to move toward zero and the terminal voltage to approach OCV. In some cells, relaxation can remove voltage offsets on the order of 20–30 mV.

This makes the relaxation curve useful for identifying the resistance, capacitance, and time-constant characteristics of an equivalent-circuit battery model.

Dynamic Tests Create the Relevant Data

A battery test system can apply a dynamic profile, such as a FUDS simulated driving cycle, and then initiate a controlled rest period. The same procedure can be repeated at selected depths of discharge, including 60%, 70%, and 80%, to evaluate how relaxation behavior changes with battery state.

How Least-Squares Fitting Is Applied

Capture the Recovery Curve

The test system samples terminal voltage throughout the rest period after discharge. The resulting data set contains time samples and corresponding voltage values:

[ {(t_i,V_i)} ]

The measured recovery curve is then related to the polarization voltage, either directly or through the difference between the measured voltage and an estimated equilibrium-voltage baseline.

Select a Polarization Model

For a Thevenin-type equivalent-circuit model, polarization relaxation is commonly represented through an RC response. A simplified single-time-constant form can be written as:

[ U_P(t)=U_{P,0}e^{-t/\tau} ]

where (U_{P,0}) is the initial polarization voltage and (\tau=R_PC_P) is the polarization time constant.

The exact model may include a voltage offset, multiple exponential terms, or additional battery states. The fitting principle remains the same: choose parameters that make the calculated relaxation curve agree as closely as possible with the measured data.

Minimize the Voltage Residuals

For each measurement point, the system calculates a residual:

[ e_i=V_i-V_{\text{fit}}(t_i,\theta) ]

where (\theta) represents the unknown model parameters. Least squares determines the parameter set that minimizes the sum of squared residuals:

[ \min_{\theta}\sum_{i=1}^{N}e_i^2 ]

This gives the test system an objective method for estimating polarization parameters rather than relying on a manually selected relaxation endpoint or a single voltage sample.

Automate the Procedure in the Tester

A battery test system can automate the full sequence:

1. Apply the dynamic discharge profile.
2. Stop the current at the selected discharge depth.
3. Record the voltage during the rest period.
4. Run the least-squares calculation.
5. Store the fitted parameters and residual error.
6. Repeat the procedure at other states of discharge or aging conditions.

Automation makes the method suitable for repeated characterization across cells, operating conditions, and aging stages.

Improving Computational Efficiency

Use Recursive Polarization Tracking

For dynamic operating conditions, the polarization state can be tracked without retaining the complete current history. A discrete recurrence relation can be used:

[ M_k(k)=I_{\text{Dch}}(k)+\alpha M_k(k-1) ]

with:

[ \alpha=1-\frac{T}{R_PC_P}<1 ]

Here, (T) is the sampling interval, while (R_P) and (C_P) describe the polarization branch.

The polarization voltage can then be approximated as:

[ U_P(k)\approx\frac{T}{C_P}M_k(k) ]

Reduce Memory and Processing Requirements

This recurrence requires the system to retain only the previous value of (M_k), rather than the full historical load-current sequence. That reduces computational overhead in battery monitoring software and supports real-time implementation in test systems or battery-management algorithms.

Benefit From Error Convergence

Because (\alpha<1), an initial error in the recurrence state decreases approximately as:

[ \alpha^N\rightarrow 0 ]

as the number of samples (N) increases. Consequently, moderate uncertainty in the initial polarization state does not necessarily prevent the algorithm from converging toward the correct dynamic behavior.

The recursive method supports real-time polarization tracking, while least-squares fitting provides the parameter-identification mechanism. Used together, they improve both computational efficiency and parameter reliability.

What Accuracy Can Be Achieved?

Fitting Error

The referenced test results indicate that the difference between measured voltage and fitted voltage remains within ±1 mV. This is a high level of agreement for empirical polarization-voltage characterization.

The reported fitting time is approximately 235–378 seconds, depending on the test condition, including the selected discharge depth and relaxation behavior.

Interpretation of the Accuracy

An error band of ±1 mV means that the fitted curve closely reproduces the measured relaxation voltage over the analyzed period. It does not mean that every battery parameter, such as resistance or capacity, has an independent ±1 mV-equivalent uncertainty.

Parameter accuracy still depends on sensor quality, sampling interval, model structure, temperature control, and the quality of the OCV reference.

Use in SOC and SOH Estimation

Accurate polarization characterization helps separate transient voltage effects from equilibrium voltage. This reduces the risk that an SOC estimator interprets temporary polarization as a genuine change in battery charge.

Repeated fitting across aging conditions also provides empirical information for SOH models, particularly when tracking changes in resistance, relaxation time constants, or usable capacity.

Connecting Relaxation to OCV–SOC Characterization

Allow Sufficient Rest

A structured rest schedule after controlled discharge or Coulomb-counted charge allows polarization to decay before OCV is recorded. The resulting voltage is more representative of the battery’s equilibrium state.

This is important because OCV–SOC curves are used as a foundation for many model-based state estimators.

Account for Cell-to-Cell Capacity Differences

Cells with different capacities may reach the same nominal discharge step at different actual SOC values. Test systems should therefore account for individual cell capacity when aligning OCV–SOC data.

Otherwise, apparent differences in polarization behavior may actually reflect mismatched SOC conditions.

Improve Pack Consistency Analysis

Aligned OCV–SOC curves help researchers distinguish genuine cell differences from measurement artifacts caused by residual polarization. This supports more reliable evaluation of pack consistency and cell balancing requirements.

Understanding the Trade-offs

Accuracy Depends on the Model

A simple single-RC model is computationally efficient, but it may not represent every relaxation process over a long rest period. A more complex model can capture additional dynamics, but it increases fitting complexity and may require more data.

The model should therefore be no more complicated than necessary for the intended SOC, SOH, or control application.

Rest Time and Test Time Must Be Balanced

Longer rest periods generally provide more information about slow relaxation processes and improve separation between polarization and OCV. However, they increase test duration and reduce laboratory throughput.

The reported 235–378-second fitting interval demonstrates rapid processing, but it should not be confused with the total time required for the discharge, rest, stabilization, and measurement sequence.

Polarization Must Not Be Confused With Ohmic Resistance

A voltage-drop resistance test must use a tightly controlled short pulse, typically 5–50 milliseconds, when the objective is to isolate ohmic internal resistance. Longer pulses allow polarization to develop and can make the measured resistance appear artificially high.

In simplified form, total observed resistance may include both ohmic and polarization contributions:

[ R_T=R_{ir}+R_P ]

Least-squares relaxation fitting is useful for characterizing the transient component, but it does not eliminate the need to design resistance tests with appropriate pulse durations.

Measurement Quality Still Matters

A ±1 mV fitting result is meaningful only when the voltage measurement system, timing, current control, and temperature conditions are sufficiently stable. Noise, drift, poor synchronization, or an inaccurate OCV baseline can limit the validity of the fitted parameters.

How to Apply This to Your Project

Least-squares fitting is most effective when the test protocol and the model are designed together.

• If your primary focus is rapid battery-model parameterization: Apply a controlled dynamic discharge and rest sequence, fit the relaxation curve automatically, and use recursive polarization tracking to reduce computation and memory requirements.
• If your primary focus is SOC accuracy: Use the fitted polarization response to distinguish transient voltage from equilibrium voltage, then combine it with properly rested and capacity-aligned OCV–SOC data.
• If your primary focus is SOH evaluation: Repeat the same relaxation and fitting procedure across aging conditions and compare changes in fitted polarization parameters and relaxation time constants.
• If your primary focus is internal-resistance measurement: Keep resistance pulses within the appropriate short duration and avoid using longer pulses that mix ohmic resistance with polarization effects.
• If your primary focus is test throughput: Use automated fitting and evaluate whether the approximately 235–378-second processing window meets the required characterization schedule without sacrificing the ±1 mV agreement target.

With a controlled relaxation protocol, an appropriate polarization model, and automated least-squares analysis, battery test systems can obtain fast, high-precision data for more dependable battery state estimation.

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