Model Voltage Hysteresis in Battery

Open Model

Open the cellHysteresis model.

modelName = 'cellHysteresis';
open_system(modelName);

Simulate Cell with no Active Hysteresis Model

If the hysteresis model is not active, the Battery Equivalent Circuit block uses the value of the Open-circuit voltage, OCV (SOC) parameter during the simulation. The open-circuit voltage is typically the average between the charge and discharge open-circuit voltages. To improve the simulation accuracy, you can define the open-circuit voltage depending on the load case. To measure the charge open-circuit voltage, you normally perform a charge-only potentiometric experiment, or galvanostatic intermittent titration technique (GITT). During a GITT, you charge the battery cell to a given state of charge and then you allow the battery to rest at zero current over a period of time. Therefore, the charge OCV is measured only after a previous charge step with its associated voltage relaxation time. Conversely, you measure the discharge open-circuit voltage exclusively after a discharge step in a discharge-only experiment.

If you do not select an active hysteresis model, the OCV always stays the same whether you charge or discharge the battery cell. Run a simulation without the hysteresis and plot the behavior of the open-circuit voltage. To setup the simulation, ensure that the hysteresis model is not active.

set_param([modelName,'/Cell'],'HysteresisModel','simscape.battery.enum.cells.HysteresisModel.none');

Additionally, the initial condition for hysteresis must not be active either.

set_param([modelName,'/Cell'],'hysteresisState_specify','off');

Simulate a full charge and discharge cycle with 1C.

ssc_hysteresis = sim(modelName,'StopTime','3600*2');

Plot the results.

cellNoHysteresisPlot;

The light blue surface represents the area where hysteresis occurs. The upper and lower limits of the surface represent the charge OCV and the discharge OCV. The upper limit (red dotted line) is the maximum hysteresis voltage when charging. The lower limit (black dotted line) is the maximum hysteresis voltage when discharging. The black line shows the OCV of the battery model which is the average between charge and discharge OCV.

Although the model performs a full charge followed by a full discharge, the OCV of the cell always remains on the same line since the hysteresis model is not active. Depending on the required model accuracy, using the average OCV might not yield satisfactory results.

Simulate Cell with Active Hysteresis Model

Now run a simulation by activating the simplest hysteresis model. Set the Hysteresis model parameter of the Battery Equivalent Circuit block to oneStateModel.

set_param([modelName,'/Cell'],'HysteresisModel', 'simscape.battery.enum.cells.HysteresisModel.oneStateModel');

This hysteresis model implements a first-order model for the OCV of the cell. The block calculates the new OCV of the cell by using this equation,

$$\mathrm{ocvCell}\left(\mathrm{SOC}\right)=\mathrm{ocvVec}\left(\mathrm{SOC}\right)+{\mathit{U}}_{\mathrm{hyst}}$$,

where${\mathit{U}}_{\mathrm{hyst}}$accounts for the cell hysteresis:

${\mathit{U}}_{\mathrm{hyst}}=\mathrm{hysteresisState}*\mathrm{maximumHysteresisVoltage}\left(\mathrm{SOC}\right)$.

Since no thermal model is currently active for the cell, the maximum hysteresis voltage depends only on the SOC. The maximumHysteresisVoltage(SOC) term in the equation is equal to half the difference between the charge and discharge OCV curves:

$$\mathrm{maximumHysteresisVoltage}\left(\mathrm{SOC}\right)=\frac{\mathrm{ChargeOCV}\left(\mathrm{SOC}\right)-\mathrm{DischargeOCV}\left(\mathrm{SOC}\right)}{2}$$.

The hysteresisState term describes the memory effect of the cell hysteresis.

hysteresisState is equal to this first-order characteristic equation,

$$\stackrel{\cdot }{\mathrm{hysteresisState}}=\frac{\mathrm{rateHysteresis}}{{\mathit{C}}_{\mathrm{cell}}}*\left({\mathit{I}}_{\mathrm{cell}}-\mathrm{abs}\left({\mathit{I}}_{\mathrm{cell}}\right)*\mathrm{hysteresisState}\right)$$,

where ${\mathit{C}}_{\mathrm{cell}}$ is the current cell capacity, in ampere-hour, and ${\mathit{I}}_{\mathrm{cell}}$ is the cell current and rateHysteresis is a constant that determines how fast the value of hysteresisState converges. The bigger rateHysteresis, the quicker the convergence.

If the cell is charging, the value of the hysteresisState converges towards 1 and the charge OCV curve. If the cell is discharging, the value of hysteresisState converges towards -1 and the discharge OCV curve. If the current across the cell goes to zero, hysteresisState assumes the last value when the cell was either charging or discharging.

Assign the initial conditions for the hysteresis model by specifying the initial value of hysteresisState. First, activate the initial condition by setting the property hysteresisState_specify to "on".

set_param([modelName,'/Cell'],'hysteresisState_specify','on');

In this example, the initial value of hysteresisState is set to 0. Therefore, the initial OCV for the cell is equal to the mean OCV.

set_param([modelName,'/Cell'],'hysteresisState','0');

To ensure that the initial condition is fulfilled, set the hysteresisState_priority property to "High".

set_param([modelName,'/Cell'],'hysteresisState_priority','High');

Set the value of the Hysteresis Rate, HysteresisRate(SOC) parameter to 3.

rateHysteresis = 3;

Disable the instantaneous hysteresis voltage by setting the Instantaneous hysteresis voltage, InstantaneousHysteresisVoltage(SOC) parameter to 0.

instantaneousHysteresisVoltage = zeros(21,1);

Simulate the model with a full charge and discharge cycle with 1C.

ssc_hysteresis = sim(modelName,'StopTime','3600*2');

Plot the results.

cellHysteresisPlot;

The OCV starts at a depth of discharge (DOD) of 0 on the blue dotted line, which is the mean OCV. In the subsequent discharging phase, the OCV converges towards the hysteresis voltage discharge curve. When the approaches the value of 1, a charging procedure starts. You can observe the hysteresis in the OCV because the OCV does not reach the charge hysteresis curve until the cell is fully recharged.

Simulate Cell with Active Hysteresis Model and Higher Hysteresis Rate

To highlight the impact of the value of the hysteresis rate on the OCV, double the value of the Hysteresis rate, HysteresisRate(SOC) parameter.

rateHysteresis = 6;

Disable the instantaneous hysteresis voltage by setting the Instantaneous hysteresis voltage, InstantaneousHysteresisVoltage(SOC) parameter to 0.

instantaneousHysteresisVoltage = zeros(21,1);

Simulate the model.

ssc_hysteresis = sim(modelName,'StopTime','3600*2');

Plot the results.

cellHysteresisPlot;

The convergence to the discharge and charge hysteresis curves occurs much faster because you doubled the hysteresis rate.

Simulate Cell with Active Hysteresis Model and Instantaneous Hysteresis

You can extend the hysteresis model by adding an instantaneous hysteresis component. To specify the instantaneous hysteresis, set the Instantaneous hysteresis voltage, InstantaneousHysteresisVoltage(SOC) parameter accordingly. If you enabled a thermal model for the battery cell, set the Instantaneous hysteresis voltage, InstantaneousHysteresisVoltageThermal(SOC,T) parameter instead. In this example, you set the instantaneous voltage drop to 10 mV regardless of the SOC value. For very low or very high SOC values, the instantaneous voltage drop is zero:

instantaneousHysteresisVoltage = [0;0;ones(17,1)*0.01;0;0];

To calculate the hysteresis voltage, the Battery Equivalent Circuit block uses this equation:

$${\mathit{U}}_{\mathrm{hyst}}=\mathrm{stateHysteresis}*\mathrm{maximumHysteresisVoltage}\left(\mathrm{SOC}\right)+\mathrm{sign}\left({\mathit{I}}_{\mathrm{cell}}\right)*\mathrm{instantaneousHysteresisVoltage}\left(\mathrm{SOC}\right)$$.

The instantaneousHysteresisVoltage(SOC) term introduces an instantaneous component to the hysteresis response. Set the rateHysteresis value back to the original value of 3.

rateHysteresis = 3;

Simulate the model to observe the effect of the instantaneous hysteresis component.

ssc_hysteresis = sim(modelName,'StopTime','3600*2');

Plot the results.

cellHysteresisPlot;

The instantaneous hysteresis voltage occurs much faster than the relatively slower dynamic hysteresis of the previous simulations. Note that in this example, the summation of the individual dynamic and instantaneous hysteresis factors do not add up to the maximum hysteresis voltage.

References