The Anode state of charge measures the momentary Li-ion content of the anode relative to the maximal Li-ion content of the anode. The Li-ion content is the volume integral of the Li-ion concentration.

I.e. for the fully resolved model, for each active material voxel, the SOC is the voxel’s Li-ion concentration divided by the voxel’s maximum Li-ion concentration. The anode SOC is the mean value of the voxel SOCs of all active-material voxels within the anode.

For the homogenized model, the anode SOC is the integral of the active material’s SOC over all anode voxels.

The Cathode state of charge measures the Li-ion content of the cathode relative to the maximal Li-ion content of the cathode. The Li-ion content is the volume integral of the Li-ion concentration.

I.e. for the fully resolved model, for each active material voxel, the SOC is the voxel’s Li-ion concentration divided by the voxel’s maximum Li-ion concentration. The cathode SOC is the mean value of the voxel SOCs of all active-material voxels in the cathode.

For the homogenized model, the cathode SOC is the integral of the active material’s SOC over all cathode voxels.

The Cell state of charge describes the overall state of charge of the battery cell. In BatteryDict, it is 0% if one of the following conditions is met: the anode is empty, or the cathode is full. It is 100% if one of the following conditions is met: the anode is full or the cathode is empty.

For a half cell simulation the lithium reservoir has always the same capacity as the other electrode. To get a numerically stable solution, the state of charge can be set in GeoDict between 5% and 95%, see also Fully Resolved.

In BatteryDict’s fully resolved model on voxel scale, each voxel has a unique material ID. In simulations with the homogenized model using the BESTmeso solver, a voxel can represent a porous electrode with electrolyte and solid volume fractions. Lithium ions intercalate or deintercalate from active materials during lithiation or delithiation, therefore the lithium composition x of these materials changes during the (dis)charge process. Each material has a maximum lithium composition and a minimum lithium composition.

For anode materials, the minimum lithium composition usually is the status without intercalated lithium ions (e.g. for graphite). Analogously, the maximal lithium composition is the maximum amount of lithium ions that can reversibly be intercalated into the material.

For cathode materials, the maximum lithium composition is usually the stoichiometric lithium composition (e.g. x=1 for LixNi0.33Mn0.33Co0.33O2). The minimum lithium composition is the lithium composition where the maximal amount of lithium ions is removed reversibly from the structure. Often the minimum lithium composition does not mean that all lithium was taken out of the material because the potential rises to values where the material would get intrinsically unstable while some lithium is still present in the material.

For both anode and cathode materials, BatteryDict works with lithium concentrations instead of lithium compositions on a voxel scale basis. BatteryDict estimates for each active material voxel (in the fully resolved simulation) or each porous electrolyte voxel (in the homogenized model) a certain “material state of charge” by comparing the current lithium concentration cs in the material to the maximum lithium concentration cmax of the material. This material state of charge is defined as

where is the (removable) lithium concentration in the active material and is the maximum (removable) lithium concentration of the material. Due to this definition, the material SOC in an active material’s voxel will always have a value between 0% (minimum lithium composition) and 100% (maximum lithium composition) and the removable lithium concentration assigned to the voxel is in the range between 0 and . The active material voxels in fully-resolved model or the porous electrode voxels in homogenized model obtain always material SOC between 0% (i.e., ) and 100% (i.e., ).

When time steps are set too large and a voxel’s lithium concentration is close to the maximum lithium concentration, the lithium concentration might exceed the maximum lithium concentration in this time step and prevent convergence; analogously, voxels that are close to a lithium concentration of 0 might result in negative lithium concentrations with large time steps and prevent convergence.

In such circumstances, the solver usually tries smaller time steps to converge to a meaningful result.

Note that the material SOC will usually not coincide with the lithium composition x because the minimum (removable) lithium concentration for cathode materials might be x>0 and the maximum (removable) lithium composition for anode materials might be x<1.