Laboratory battery test systems must identify Uoc, Ro, Rp, and Cp separately for charge and discharge because a battery does not exhibit identical voltage dynamics in both current directions. Charge and discharge can produce different apparent open-circuit voltage behavior, ohmic resistance, polarization resistance, and polarization time constants across the same SOC. An EKF that uses only one parameter set therefore introduces systematic model error, reducing SOC accuracy and degrading voltage prediction.
The central reason is directional hysteresis and nonlinear electrochemical behavior: the battery’s response depends not only on SOC, but also on whether current is entering or leaving the cell. Separate charge and discharge parameter maps allow the EKF to use a model that matches the battery’s present operating direction.
Why One Parameter Set Is Insufficient
Battery voltage depends on operating history
A Thevenin equivalent circuit represents terminal voltage using an open-circuit component and dynamic loss components. A simplified form is:
[ V_t = U_{oc}(SOC) - I R_o - V_p ]
where the polarization voltage evolves according to the resistance-capacitance pair:
[ \tau = R_p C_p ]
The same SOC can produce different terminal voltages depending on the preceding charge or discharge path. This behavior is commonly described as voltage hysteresis.
Charge and discharge activate different dynamics
During discharge, current leaves the cell and internal concentration gradients, reaction overpotentials, and polarization losses develop in one direction. During charge, the gradients and reaction processes reverse, but they do not necessarily retrace the same path.
Consequently, Ro, Rp, Cp, and the resulting time constant can differ by direction, even at the same SOC and temperature.
The issue is especially important across the SOC range
These parameters are strongly nonlinear over the usable SOC window. Characterizing them only at a few points, or only in one direction, can leave the EKF with inaccurate behavior near low, mid-range, or high SOC.
Laboratory systems should therefore identify parameters across the relevant SOC range, such as approximately SOC 0.10 to 0.95, under both charging and discharging conditions.
How Each Parameter Affects EKF Estimation
Uoc determines the voltage reference
(U_{oc}) provides the model’s baseline voltage as a function of SOC. If the charge and discharge voltage curves differ because of hysteresis, a single (U_{oc}(SOC)) relationship creates a persistent voltage residual.
The EKF may interpret that residual as an SOC error. It can then correct SOC in the wrong direction even when the actual error comes from using the wrong directional voltage curve.
Ro controls the immediate voltage step
(R_o) describes the near-instantaneous voltage drop associated with current flow. An inaccurate value causes the predicted terminal voltage to jump too far or not far enough when current changes.
Because the EKF uses the difference between measured and predicted voltage, an incorrect (R_o) produces a biased innovation immediately after load or charging-current transitions.
Rp and Cp control transient polarization
(R_p) and (C_p) determine how the polarization voltage builds and relaxes over time. Their product defines the dominant time constant:
[ \tau = R_p C_p ]
Incorrect directional values cause the model to predict the wrong transient response during current pulses, rest periods, and profile changes.
The EKF depends on local derivatives
An EKF linearizes a nonlinear battery model around its current estimate. Its estimation matrix depends on local relationships such as:
[ \frac{dU_{oc}}{dSOC} \quad\text{and}\quad \frac{dR_o}{dSOC} ]
If the parameter functions do not match the current direction, these partial derivatives are also wrong. The filter then calculates an inappropriate sensitivity between SOC, current, and terminal voltage.
Why Directional Parameter Maps Improve Estimation
They reduce model mismatch
The EKF assumes that the model describes the system closely enough for local linearization to be useful. Separate charge and discharge maps reduce the difference between predicted and measured voltage.
This gives the filter a more credible innovation signal: residual voltage is more likely to represent actual state error rather than an unmodeled directional effect.
They improve SOC tracking
SOC is not measured directly in a conventional battery test. The EKF estimates it by combining current integration with voltage-based correction.
When the voltage model reflects the correct charging or discharging behavior, the correction step is less likely to compensate for parameter errors by incorrectly shifting SOC.
They improve voltage prediction
Accurate (R_o), (R_p), and (C_p) values allow the model to reproduce both immediate voltage changes and slower polarization behavior. This matters for pulse tests, dynamic drive cycles, power capability studies, and continuous experimental profiles.
They support real-time computation
First-principles electrochemical models can represent transport and reaction mechanisms in detail, but they are often too computationally demanding for continuous real-time estimation. A direction-dependent lumped model captures the dominant electrical behavior with much lower computational cost.
Polynomial or similarly compact functions can represent the measured parameter surfaces over SOC. The EKF can then select the appropriate charge or discharge function during operation.
What Laboratory Characterization Must Capture
Test the full operating window
Parameter identification should cover the SOC region relevant to the application rather than relying on a single nominal SOC. Low- and high-SOC regions often exhibit stronger nonlinearity and more pronounced voltage sensitivity.
A sparse map can cause interpolation errors that appear as estimation drift or transient prediction error.
Use both current directions
The test plan should include controlled charging and discharging sequences at representative currents. Rest periods and current steps are useful for separating immediate ohmic behavior from slower polarization behavior.
The resulting data should be fitted into distinct parameter relationships for charge and discharge.
Preserve the operating conditions
Battery parameters also depend on temperature, current magnitude, aging state, and sometimes rest history. Directional separation does not eliminate these dependencies; it addresses one major source of variation.
For high-accuracy systems, parameter maps may need to be indexed by additional variables such as temperature or SOH.
Validate against dynamic profiles
A good fit to isolated laboratory pulses does not automatically guarantee accurate performance in an EKF. The identified model should be validated against representative mixed charge-discharge profiles.
Validation should compare both SOC estimation and terminal-voltage prediction, particularly around current reversals.
Understanding the Trade-offs
Separate maps increase model complexity
Maintaining charge and discharge functions requires more data, more calibration effort, and additional logic to select the active direction. The parameter set may also become larger if temperature and aging are included.
This complexity is justified when estimation accuracy and voltage prediction are important, but it may be unnecessary for rough monitoring applications.
Direction alone may not explain every discrepancy
A charge/discharge split can capture directional hysteresis, but it cannot fully represent every history-dependent effect. Two tests with the same current direction and SOC may still differ because of rest time, prior current magnitude, temperature, or aging.
The model should therefore be treated as a calibrated approximation, not as a complete electrochemical description.
Polynomial fitting requires care
Polynomial functions are computationally convenient, but poorly chosen orders or extrapolation beyond the tested SOC range can produce nonphysical parameter values. Fits should remain within the characterized domain and be checked for smoothness and plausibility.
The fitted functions should also preserve physically meaningful constraints, such as positive resistance and capacitance values.
More parameters can amplify poor data quality
Directional identification cannot compensate for noisy voltage measurements, insufficient rest periods, uncontrolled temperature, or inadequate excitation. If the test profile does not separate fast and slow dynamics, (R_p) and (C_p) may be poorly identifiable even when the mathematical fit appears good.
Experimental design is therefore as important as the choice of EKF equations.
Making the Right Choice for Your Goal
The appropriate level of directional characterization depends on the required accuracy and operating conditions.
- If your primary focus is accurate SOC estimation: Identify separate (U_{oc}(SOC)), (R_o(SOC)), (R_p(SOC)), and (C_p(SOC)) relationships for charge and discharge across the full usable SOC range.
- If your primary focus is transient voltage prediction: Give particular attention to directional (R_o), (R_p), (C_p), and the time constant (\tau = R_pC_p) under representative current pulses.
- If your primary focus is computational efficiency: Use compact, validated directional parameter functions in the lumped model rather than replacing the EKF with a computationally intensive first-principles model.
- If your primary focus is robustness across test conditions: Characterize direction together with temperature, current level, rest history, and aging state where those variables materially affect the results.
Separate charge and discharge parameter characterization gives the EKF the directional information it needs to distinguish true state error from predictable battery hysteresis.
Summary Table:
| Parameter | Charge vs. Discharge | Impact on EKF |
|---|---|---|
| Uoc (Open-Circuit Voltage) | Different due to hysteresis | Baseline voltage reference; mismatch causes persistent residual |
| Ro (Ohmic Resistance) | Can vary | Immediate voltage step; bias after current transitions |
| Rp (Polarization Resistance) | Can vary | Transient voltage response; affects dynamic accuracy |
| Cp (Polarization Capacitance) | Can vary | Time constant tau = Rp*Cp; transient dynamics |
| Combined | Must be characterized separately | Reduces model mismatch, improves SOC and voltage prediction |
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