Track polarization voltage with a one-state recurrence, not a stored current history. For a discrete-time RC Thevenin battery model, update a cumulative current state as
[ M_k(k)=I_{\mathrm{Dch}}(k)+\alpha M_k(k-1), \qquad \alpha=1-\frac{T}{R_P C_P}, ]
then estimate polarization voltage from
[ U_P(k)\approx \frac{T}{C_P}M_k(k). ]
This requires storing only the previous value of (M_k), rather than the full dynamic current record. With a stable coefficient, errors caused by the initial state decay exponentially, although uncertainty in the actual model parameters can create a persistent estimation bias.
Core takeaway: The recurrence converts the RC polarization calculation into a low-memory, constant-computation update suitable for dynamic-current testing and BMS execution. Its robustness is strong against initial-state uncertainty, but it depends on a stable sampling configuration and reasonably accurate values of (R_P), (C_P), and the associated model parameters.
How the Low-Memory Tracking Method Works
Replace the current history with a recursive state
Polarization voltage in a first-order RC model depends on the battery’s recent current history. Direct evaluation would require retaining or repeatedly processing that history.
The recurrence compresses the relevant history into one state variable:
[ M_k(k)=I_{\mathrm{Dch}}(k)+\alpha M_k(k-1). ]
At each sample, the system needs only the new current measurement and the previous value of (M_k).
Convert the recursive state into polarization voltage
Once (M_k(k)) is updated, polarization voltage is calculated as
[ U_P(k)\approx \frac{T}{C_P}M_k(k). ]
The computation therefore consists of one multiplication, one addition, and a voltage scaling operation. Memory usage is constant with respect to test duration, which is important for embedded battery monitors and long dynamic drive-cycle tests.
Interpret the coefficient physically
The coefficient
[ \alpha=1-\frac{T}{R_P C_P} ]
represents the sampled decay of the polarization state. The product (R_P C_P) is the RC polarization time constant, so it determines how quickly the battery’s polarization response relaxes.
A larger time constant produces slower decay, while a smaller time constant produces faster decay between current updates.
Why This Is Suitable for Dynamic-Current Testing
It follows changing current without resting periods
Because the state is updated at every sample, the method can track polarization during variable-current charging or discharging, including drive-cycle-like profiles.
It does not require the battery to rest long enough for polarization to disappear before estimating the voltage component.
It avoids repeated least-squares fitting
The recurrence is a direct online update rather than a batch identification procedure. This avoids storing long datasets and reduces the computational burden associated with repeatedly fitting a model over a moving data window.
That makes it practical for real-time test systems, embedded controllers, and BMS algorithms with limited processing and memory resources.
It can be combined with terminal-voltage measurements
A broader online estimation scheme can use the measured terminal voltage, current, SOC, and an OCV–SOC relationship. For example,
[ U_P=U_o-\mathrm{OCV}(\mathrm{SOC})-I R_d, ]
where (R_d) may be estimated from voltage and current changes at current step points:
[ R_d=\frac{\Delta U}{\Delta I}. ]
This voltage-based calculation can provide an observation or validation signal for the recursive polarization estimate.
How Robust Is the Calculation to Initial Uncertainty?
Initial-state errors decay exponentially
Suppose the estimated recursive state starts with an error (e_0). Under the recurrence, the error after (N) updates is approximately
[ e_N=\alpha^N e_0. ]
Therefore, if
[ |\alpha|<1, ]
then
[ \alpha^N\rightarrow 0 ]
as the number of samples increases. The influence of an incorrect initial polarization state consequently becomes negligible over time.
This is the central robustness property of the method.
The result is stable only under the correct sampling condition
The statement (\alpha<1) alone is not sufficient for discrete-time stability. The relevant condition is
[ |\alpha|<1. ]
For the stated first-order discretization, this requires the sampling interval and time constant to be selected so that the recursive coefficient remains inside the unit circle. If the sample interval is too large relative to (R_P C_P), (\alpha) can become less than (-1), causing numerical instability rather than convergence.
In practice, the test-system sampling interval should therefore be chosen with the polarization time constant in mind.
Initial parameter guesses are not the same as parameter errors
The recurrence is robust to an incorrect initial state, such as an incorrect (M_k(0)). However, errors in the actual values of (R_P), (C_P), or (T) affect (\alpha) and the voltage scaling factor (T/C_P).
Those model-parameter errors do not generally disappear merely because the recursion is stable. They can produce a persistent gain or dynamic-response error, even after the initial-condition error has decayed.
What Determines Practical Accuracy?
Quality of the polarization time constant
The time constant (R_P C_P) determines the rate at which the recursive state forgets past current. If it is poorly identified, the calculated polarization may decay too quickly or too slowly compared with the real cell.
Accurate parameter fitting remains important even though the online calculation itself is inexpensive.
Sampling interval
The sampling interval affects both numerical stability and the ability to resolve fast polarization dynamics. A slow sampling rate can miss rapid changes and can also make the discrete approximation inaccurate.
The recurrence should be evaluated at a rate appropriate to the fastest dynamics represented by the selected RC model.
Current and voltage measurement quality
Noise, offsets, and sign-convention errors in current measurement directly affect (M_k). Voltage-based validation using terminal voltage, OCV, SOC, and dynamic resistance is therefore valuable for detecting inconsistent estimates.
The current direction convention must also remain consistent throughout the recurrence and voltage equations.
SOC and OCV accuracy
When the supplementary voltage relationship is used,
[ U_P=U_o-\mathrm{OCV}(\mathrm{SOC})-I R_d, ]
errors in SOC or in the piecewise-linear OCV–SOC curve appear as polarization-voltage errors. This method can avoid long rest periods, but it does not eliminate the need for reliable SOC and OCV information.
Understanding the Trade-offs
Low memory does not mean parameter-free estimation
The recursive calculation reduces storage and computation, but it still depends on a suitable RC model and parameter set. It is a tracking mechanism, not a substitute for model identification.
Parameters may need to be fitted across relevant SOC, temperature, aging, and current operating ranges.
The first-order model may be insufficient
A single (R_P C_P) branch captures one dominant polarization time scale. Real cells can exhibit multiple dynamic processes, hysteresis, temperature effects, and nonlinear behavior.
If those effects are significant, a higher-order model may be needed, which requires additional recursive states and parameters.
Stability does not guarantee physical accuracy
A stable recurrence can converge smoothly to the wrong result if the model parameters, OCV curve, current sign, or voltage decomposition are incorrect.
Convergence should therefore be checked against measured terminal voltage and, where possible, independent test conditions.
Dynamic resistance can be sensitive to measurement conditions
The estimate
[ R_d=\frac{\Delta U}{\Delta I} ]
is most meaningful when the voltage and current changes are correctly aligned and the step is sufficiently distinguishable from measurement noise. Gradual current changes may not provide a clean step point for this calculation.
The recursive RC estimate and the dynamic-resistance-based voltage estimate are best treated as complementary methods rather than interchangeable formulas.
How to Apply This to a Battery Testing System
The implementation can be kept compact:
- Calibrate or identify (R_P) and (C_P) for the relevant operating condition.
- Select a sampling interval (T) that keeps the recursion stable and resolves the intended dynamics.
- Compute (\alpha=1-T/(R_P C_P)).
- Initialize (M_k(0)), even if the initial polarization state is uncertain.
- At each sample, update (M_k(k)) from the new current measurement.
- Calculate (U_P(k)\approx (T/C_P)M_k(k)).
- Optionally compare the result with the terminal-voltage, OCV, SOC, and (R_d)-based estimate.
- If your primary focus is low-memory real-time execution: Implement the one-state recurrence and retain only the previous (M_k) value, current sample, and calibrated parameters.
- If your primary focus is robustness to unknown initial polarization: Use the recurrence with (|\alpha|<1); initial-state errors will decay geometrically over successive samples.
- If your primary focus is parameter-identification accuracy: Validate (R_P), (C_P), SOC, OCV, and dynamic resistance across the intended operating range, because stable recursion cannot remove persistent model-parameter errors.
- If your primary focus is model validation: Compare recursive (U_P) with the terminal-voltage decomposition (U_o-\mathrm{OCV}(\mathrm{SOC})-I R_d) during representative dynamic tests.
With a stable sampling design and properly identified parameters, this approach provides an efficient and dependable way to track polarization voltage throughout dynamic battery operation.
Summary Table:
| Method | Memory Usage | Computation | Robustness to Initial Error | Key Requirement |
|---|---|---|---|---|
| Recursive State Update | Constant (one state variable) | One addition, multiplication, scaling per sample | Errors decay exponentially if |α|<1 | Stable sampling interval, accurate R_P, C_P |
| Direct Full History | Grows with test duration | High (reprocess all data) | Exact if parameters known | Stores entire current record |
| Batch Least-Squares | High (store dataset) | Heavy (repeated fitting) | Depends on dataset | Offline processing |
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