Online charging polarization voltage can be tracked by subtracting the estimated OCV and ohmic voltage drop from the measured terminal voltage. During the test, calculate SOC in real time, obtain the corresponding OCV from a piecewise-linear OCV–SOC curve, estimate dynamic resistance from current and voltage changes, and then apply (U_P = U_O - OCV(SOC) - I R_d) at every sampling point.
The tracking calculation method provides a low-computation estimate of polarization voltage without least-squares fitting or long rest periods. Its accuracy depends mainly on reliable SOC, a representative OCV–SOC curve, consistent current-sign conventions, and a valid estimate of dynamic resistance.
How the Tracking Calculation Method Works
Separate the measured battery voltage
The measured terminal voltage can be represented conceptually as the sum of several components:
[ U_O = OCV(SOC) + I R_d + U_P ]
where:
- (U_O) is the measured terminal voltage.
- (OCV(SOC)) is the open-circuit voltage corresponding to the current SOC.
- (I R_d) is the instantaneous or dynamic ohmic voltage component.
- (U_P) is the polarization voltage.
Rearranging this relationship gives the online calculation:
[ \boxed{U_P = U_O - OCV(SOC) - I R_d} ]
For a charging test, use the current-sign convention consistently. If charging current is defined as negative rather than positive, the corresponding signs in the voltage model must be adjusted consistently.
Use SOC to obtain the OCV estimate
The battery’s SOC is updated online from the test current and the initial SOC. The calculated SOC is then used as the input to a piecewise-linear OCV–SOC lookup curve.
If the current SOC lies between two reference points, interpolate the OCV between those points:
[ OCV(SOC)=OCV_i+ \frac{OCV_{i+1}-OCV_i}{SOC_{i+1}-SOC_i} (SOC-SOC_i) ]
This avoids requiring a continuous analytical OCV function while still providing an OCV estimate at each sampling instant.
Estimate dynamic resistance from voltage and current changes
At identifiable current step points, determine the dynamic DC resistance from the corresponding terminal-voltage and current changes:
[ \boxed{R_d=\frac{\Delta U}{\Delta I}} ]
The voltage change should be taken consistently with the current change and the selected sign convention. This resistance represents the rapid voltage response associated with the battery’s dynamic internal resistance.
The method assumes that current changes are sufficiently continuous and gradual for the extracted resistance to remain meaningful during the tracking process.
Online Calculation Procedure
Step 1: Measure the operating variables
At each sampling interval, record:
- Terminal voltage (U_O).
- Battery current (I).
- Sampling time.
- SOC or the data needed to update SOC.
Accurate synchronization between voltage, current, and SOC measurements is important because the three quantities are combined in the same polarization-voltage calculation.
Step 2: Update SOC
Use the measured current to update SOC according to the test’s battery model or coulomb-counting procedure. The resulting SOC is the operating point used to identify the corresponding OCV.
SOC error directly affects the OCV estimate and therefore appears as error in the calculated (U_P).
Step 3: Interpolate the OCV
Use the updated SOC in the piecewise-linear OCV–SOC curve. The result is the estimated equilibrium voltage that the battery would approach without current-induced ohmic and polarization effects.
Step 4: Determine or update (R_d)
Use voltage and current step information to calculate:
[ R_d=\frac{\Delta U}{\Delta I} ]
The resistance may be updated when a new usable step is detected. The calculation should exclude intervals where the voltage change is dominated by unrelated measurement disturbances or where the current change is too small to provide a reliable ratio.
Step 5: Calculate polarization voltage
Apply the tracking equation:
[ U_P(k)=U_O(k)-OCV(SOC(k))-I(k)R_d(k) ]
This produces an online estimate at sample (k), allowing the charging system or test software to display and monitor polarization voltage during the test.
Model-Based Recurrence for Dynamic Operation
Track polarization using an RC equivalent model
For dynamic current profiles, polarization can also be tracked using a Thevenin-equivalent RC model. Define a cumulative current coefficient recursively as:
[ M_k(k)=I_{\mathrm{Dch}}(k)+\alpha M_k(k-1) ]
with:
[ \alpha=1-\frac{T}{R_P C_P}<1 ]
where:
- (T) is the sampling interval.
- (R_P) is the polarization resistance.
- (C_P) is the polarization capacitance.
- (R_P C_P) is the polarization time constant.
- (I_{\mathrm{Dch}}) is the defined discharge-current quantity under the selected sign convention.
The polarization voltage can then be approximated by:
[ U_P(k)\approx \frac{T}{C_P}M_k(k) ]
Why the recurrence is computationally efficient
The recurrence requires the current value and the previous value of (M_k). It does not require the software to retain the entire historical current waveform.
This makes it suitable for battery test systems and BMS algorithms operating under variable-current driving schedules or other dynamic charging profiles.
Understand the convergence behavior
Because (\alpha<1), an initial error in (M_k(0)) decreases approximately as:
[ \alpha^N\rightarrow 0 ]
over successive samples. Consequently, the influence of an uncertain initial polarization state diminishes with time, although the rate of convergence depends on the sampling interval and polarization time constant.
Why This Method Is Useful During Charging Tests
Avoid prolonged rest periods
Off-line approaches often require the battery to rest so that its terminal voltage can approach OCV. The tracking method estimates OCV from SOC and the OCV–SOC curve instead, allowing polarization voltage to be estimated while the battery remains under test.
Avoid repeated least-squares fitting
The direct calculation based on measured voltage, SOC-derived OCV, and dynamic resistance avoids the computational burden of repeatedly fitting a complete model using least-squares methods.
The RC recurrence provides a similarly efficient alternative when a polarization model and its parameters are available.
Support fast-charging evaluation
Polarization voltage reflects internal electrochemical limitations, including reaction-rate and ion-concentration effects. Tracking it during charging helps identify operating conditions in which the applied current produces excessive internal voltage response.
This information can be used to compare charging protocols, evaluate dynamic-rate capability, and investigate conditions associated with reduced efficiency or accelerated degradation.
Understanding the Trade-offs
The method depends on OCV–SOC accuracy
The OCV–SOC curve must represent the tested cell, chemistry, temperature, and relevant operating condition. Hysteresis, temperature variation, aging, and inaccurate SOC estimation can cause the calculated OCV to differ from the battery’s actual equilibrium voltage.
Because (U_P) is obtained by subtraction, an OCV error transfers directly into the polarization-voltage estimate.
Dynamic resistance is not universally constant
The relationship
[ R_d=\frac{\Delta U}{\Delta I} ]
is an effective dynamic estimate, not necessarily a constant physical resistance. It can vary with SOC, temperature, aging, current level, and the time window used to measure the step response.
Using an outdated or poorly measured (R_d) can cause ohmic voltage to be incorrectly attributed to polarization voltage.
Charging and discharging polarization are not identical
Polarization may behave differently during discharge and charging. In particular, polarization built up during discharge can decay during the early stage of charging before charging polarization becomes established.
Therefore, the transition between discharge and charge should be interpreted carefully rather than assuming that the polarization voltage immediately follows a single steady-state charging response.
The RC recurrence requires valid parameters
The recurrence method is efficient only if (R_P), (C_P), the sampling interval, and the current definition are appropriate for the battery and operating condition.
Incorrect parameters may still produce a numerically stable result, but the result may not represent the actual polarization voltage accurately.
How to Apply This to Your Test System
The practical implementation is a repeated calculation executed at every sampling interval.
- If your primary focus is a simple online estimator: Update SOC, interpolate (OCV(SOC)), estimate (R_d) from voltage/current step changes, and calculate (U_P=U_O-OCV(SOC)-IR_d).
- If your primary focus is variable-current dynamic testing: Use the RC recurrence (M_k(k)=I_{\mathrm{Dch}}(k)+\alpha M_k(k-1)) and (U_P(k)\approx(T/C_P)M_k(k)), provided the polarization parameters are identified.
- If your primary focus is fast-charging safety or degradation analysis: Track (U_P) together with SOC, current, temperature, and charging history so that excessive polarization can be related to the operating condition that caused it.
- If your primary focus is measurement accuracy: Validate the OCV–SOC curve, current sign convention, resistance extraction interval, sampling rate, and RC parameters before interpreting the calculated polarization voltage.
With reliable SOC, OCV, resistance, and model parameters, polarization voltage can be monitored continuously during charging without interrupting the test for long rest periods.
Summary Table:
| Component | Description |
|---|---|
| OCV(SOC) | Open-circuit voltage from SOC via piecewise-linear curve |
| Rd | Dynamic resistance from voltage/current changes |
| Up | Polarization voltage = Uo - OCV - I*Rd |
| SOC | State of charge updated online |
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