Theory for the Lumped Battery Interface
The Lumped Battery Interface makes use of a small set of lumped parameters for adding contributions for the sum of all voltage losses in the battery, stemming from ohmic resistances and (optionally) change transfer or diffusion processes. These lumped parameters are defined for the cell in the case of the Lumped cell model, and are defined individually for the positive and negative electrodes in the case of the Two electrodes model.
The Lumped cell model defines a single cell state of charge (SOC) dependent variable to define the battery charge level, whereas the Two electrodes model defines individual degrees of conversion (DOC, denoted as dimensionless variable χ) dependent variables for each electrode.
For the Lumped cell model, the battery cell voltage Ecell (SI unit: V) is defined as
where ηIR is the ohmic overpotential, ηact is the activation overpotential, and ηconc is the concentration overpotential. EOCV is the cell open circuit voltage which is dependent on the state of charge SOC and temperature T. EOCV can be specified directly as a function of SOC and T, or can be calculated according to,
where EOCV,ref is the open circuit voltage at a reference temperature Tref.
The time evolution of SOC is defined as
where Icell (SI unit: A) is the applied current and Qcell (SI unit: C) is the battery cell capacity that is set equal to the initial battery cell capacity Qcell,0 (SI unit: C). The initial cell state of charge is specified by SOCcell,0. If the concentration overpotential is calculated based on particle diffusion, SOC is replaced by the average state of charge, SOCaverage, in the above equations.
For the Two electrodes model, the battery cell voltage Ecell (SI unit: V) is defined as
where ηIR is the ohmic overpotential, ηact,pos and ηact,neg are the activation overpotentials for the positive and negative electrode, respectively, and ηconc,pos and ηconc,neg are the concentration overpotentials for the positive and negative electrode, respectively. Eeq,pos and Eeq,neg are the respective half cell equilibrium potential curves of the positive and negative electrode materials and are dependent on the respective degree of conversion χ and temperature T.
The degree of conversion χ represents some physical quantity like degree of lithiation (lithium-ion batteries), degree of hydration (nickel-metal hydride batteries) or electrode volume fraction (lead-acid batteries), and it should be defined so that it varies linearly with the electrode charge level. Typically χ = 0 corresponds to the electrode being in its fully oxidized state, and χ = 1 corresponds to the electrode being in its fully reduced state.
By convention, Icell is defined to be positive during charge and negative during discharge. With the additional convention that anodic currents are positive and cathodic currents are negative, this results in the following definition of the individual electrode currents Ineg and Ipos
In addition, by convention, when the cell state of charge increases, the degree of conversion of the negative electrode increases, whereas the degree of conversion of the positive electrode decreases.
The time evolution of χneg and χpos is defined as
where Qhost,neg and Qhost,pos (SI unit: C) are the respective electrode host capacities, and are set equal to the respective initial electrode host capacities Qhost,neg,0 and Qhost,pos,0 (SI unit: C). If the concentration overpotential is calculated based on particle diffusion, χ is replaced by the average degree of conversion, χaverage, in the above equations. The respective initial degrees of conversion are calculated from the respective equilibrium degree of conversion specified as a function of the initial electrode potential. The initial negative and positive electrode potentials are defined internally based on the settings at the interface level (Initial Cell Charge Distribution section and State-of-Charge Definition section). Also refer to the discussion in the Cell Capacity and State of Charge section.
Voltage Losses
The lumped voltage loss associated with ohmic process in the electrolyte and electrodes is calculated in the same way for both the Lumped cell and Two electrodes model, and is given as
where ηIR,1C is the ohmic overpotential at 1C, and
For the Lumped cell model, the lumped voltage loss associated with activation overpotential on both the positive and negative electrode surfaces is defined as the following
where R denotes the molar gas constant, T the temperature, F Faraday’s constant, and J0 the dimensionless charge exchange current.
For the Two electrodes model, the activation overpotential voltage losses, ηact,neg and ηact,pos, are specified individually for the negative and positive electrodes. They are defined in the same way as for the Lumped cell model, except that the respective negative and positive lumped parameters are used instead of the lumped cell parameters. Additionally, the individual electrode currents Ineg and Ipos are used instead of cell current Icell, and I1C,neg and I1C,pos are used instead of I1C,cell. The individual electrode currents Ineg and Ipos are defined based on the corresponding initial electrode host capacities, and are defined as follows
For fast transients or for frequency domain simulations, it may also be of interest to include the double-layer capacitance between the electrode and electrolyte phases in the battery. Note that the Two electrodes model does not support including a double-layer capacitance.
For the Lumped cell model, when including a double-layer capacitance, one cannot derive an analytical expression for the activation overpotential. Instead, the activation overpotential is solved for as a dependent variable fulfilling
where Ict is the charge transfer current and Idl is the double-layer current. For the Lumped cell model, they are defined as
and
where the normalized 1C double-layer capacitance, Cdl,1C, relates the double-layer current to the time derivative of activation potential. Note that the above equation assumes that the time derivative of the equilibrium potential is orders of magnitude lower than the time derivative of the activation potential.
Concentration overpotential effects can be modeled either based on diffusion in an idealized particle or by using an RC pair (a linear resistor coupled in parallel with a capacitor), for both the Lumped cell and Two electrodes models.
In the particle diffusion case for the Lumped cell model, Fickian diffusion of a dimensionless SOC variable is solved for in 1D on an interval of length 1 with X as the dimensionless spatial variable according to
where τ is the diffusion time constant. The diffusion equation is solved either globally or locally (depending on the selection of either global or local formulation, respectively) in a 1D pseudo extra dimension corresponding to the particle dimension. The gradient is calculated in Cartesian, cylindrical, or spherical coordinates, depending on if the particles are assumed to be best described as flakes, rods, or spheres, respectively.
The boundary conditions at the center and surface of the particle are as follows:
where Nshape is 1 for Cartesian, 2 for cylindrical, and 3 for spherical coordinates. The initial cell state of charge is specified by SOCcell,0. The surface state of charge, SOCsurface, is defined at the surface of the particle. The average state of charge, SOCaverage, is defined by integrating over the volume of the particle, appropriately considering Cartesian, cylindrical, or spherical coordinates. Note that, as mentioned above, SOCaverage is used in the definition of Ecell.
The lumped voltage loss associated with concentration overpotential is defined as
In the particle diffusion case for the Two electrodes model, the Fickian diffusion expressions are specified individually for the negative and positive electrode degrees of freedom, χneg and χpos, similar to the Lumped cell model. The respective negative and positive lumped parameters are used instead of the lumped cell parameters. Additionally, Ineg and Ipos are used instead of Icell, and Qhost,neg and Qhost,pos are used instead of Qcell. The concentration overpotential voltage losses, ηconc,neg and ηconc,pos, are specified individually for the negative and positive electrodes as follows:
In the RC pair case for the Lumped cell model, the concentration overpotential ηconc is defined as
where the RC time constant, τRC = RC, and the RC potential at 1C, ERC,1C = RI1C.
In the RC pair case for the Two electrodes model, concentration overpotential voltage losses, ηconc,neg and ηconc,pos, are specified individually for the negative and positive electrodes, similar to the Lumped cell model. The respective negative and positive lumped parameters are used instead of the lumped cell parameters. Additionally, Ineg and Ipos are used instead of Icell. The RC time constant and the RC potential at 1C are defined individually for the negative and positive electrodes, similar to the Lumped cell model, with the respective negative and positive lumped parameters.
Battery Cell Volume
In 0D, the battery cell volume Vcell is specified by the user. In higher dimensions, the cell volume Vcell, is defined according to
where Ω is the selected domain where the Lumped Battery interface is active, and dvol is the cell cross-sectional area in 1D, the out-of-plane-thickness in 2D and 1D with axial symmetry, and is equal to 1 in 3D and 2D with axial symmetry space dimensions. In 1D with axial symmetry and 2D with axial symmetry, the expressions computing the volume integrals are also multiplied by 2πr. Note that for some cases, Vcell is explicitly only needed to calculate the heat source variables.
Electrochemical Heat Sources
In order to model the energy balance of an electrochemical cell, a Heat Transfer interface is typically used to model the heat transport and the Lumped Battery interface is used to generate the heat sources and sinks due to the electrochemical currents.
For the Lumped cell model, if the temperature derivative of open circuit voltage is specified, the heat source expressions are as follows:
The battery heat source (SI unit: W) is defined as follows, if concentration overpotential is not included:
If concentration overpotential is included and modeled based on diffusion in an idealized particle, the battery heat source (SI unit: W) is defined as
where the heat of mixing, Qmix, is defined as
with the thermoneutral voltage, Etherm, appropriately evaluated on the particle extra dimension as follows,
If concentration overpotential is included by using an RC pair, the battery heat source (SI unit: W) is defined as
where Qmix is defined as
For the Lumped cell model, if the thermoneutral voltage is specified, the heat source expressions are as follows:
The battery heat source (SI unit: W) is defined as follows, if concentration overpotential is not included:
If concentration overpotential is included and modeled based on diffusion in an idealized particle, the battery heat source (SI unit: W) is defined as
where the heat of mixing, Qmix, is defined as
with the thermoneutral voltage, Etherm, directly evaluated on the particle extra dimension.
If concentration overpotential is included by using an RC pair, the battery heat source (SI unit: W) is defined as
where Qmix is defined as
For the Two electrodes model, the heat source expressions are as follows:
The battery heat source (SI unit: W) is defined as follows, if concentration overpotential is not included:
If concentration overpotential is included and modeled based on diffusion in an idealized particle, the heat of mixing contribution is not included, and the battery heat source (SI unit: W) is defined as above with χ being replaced by the average degree of conversion, χaverage.
If concentration overpotential is included by using an RC pair, the battery heat source (SI unit: W) is defined as
where QRC,pos and QRC,neg are defined, respectively, as
Capacity Loss
Capacity loss in batteries is a complex process, encompassing a multiple of different phenomena. Depending on the battery chemistry, the aging mechanisms will be different.
For the Lumped cell model, the accumulated charge corresponding to the parasitic reactions Qloss can be written as
where Iloss (SI unit: A) is the loss current. The initial battery loss capacity is set to 0. The remaining battery cell capacity Qcell (SI unit: C) is defined as
For the Two electrodes model, capacity may be lost due to either loss of negative or positive cyclable charge inventory, or due to loss of negative or positive host capacity.
For the case of loss of cyclable charge inventory, the negative or positive loss current affects the negative or positive degree of conversion, respectively, as
If the concentration overpotential is calculated based on particle diffusion, χ is replaced by the average degree of conversion, χaverage, in the above equations.
For the case of loss of host capacity, the negative or positive loss current affects the negative or positive host capacity, respectively, as
The initial negative and positive loss host capacity is set to 0. The remaining negative and positive host capacities, Qhost,neg and Qhost,pos (SI unit: C), are defined respectively as
Using a lumped modeling approach, assuming limited knowledge about the internal processes occurring inside the battery, any aging model will have to be empirical, not being able to discriminate between different phenomena. The framework used to define the loss current is similar for both the Lumped cell model and the Two electrodes model, except that for the Two electrodes model the equations are defined individually for the negative and positive electrodes, similar to the Lumped cell model, with the respective negative and positive lumped parameters.
Typically, capacity loss and aging may be affected by the battery voltage, the capacity throughput, aging history and temperature. To define a lumped aging rate, sum all parasitic currents in the battery to obtain the capacity loss
where, τloss is a calendar aging time constant defining the rate of the parasitic reactions. The factors fE, fI, faged, and fT are dimensionless aging factors, depending on the battery voltage, battery current, aging history and temperature, respectively. Setting all aging factors to 1 would result in a constant capacity loss from t = 0, reaching 0 remaining capacity when t = τloss, independent of battery SOC, capacity throughput, aging history and temperature. For the Two electrodes model, the above equation applies except that it is defined individually for the negative and positive electrodes with the respective negative and positive lumped parameters. Also, the respective initial electrode host capacities Qhost,neg,0 and Qhost,pos,0 are used instead of the initial battery cell capacity Qcell,0.
In many battery systems, it has been seen that high SOC values accelerate capacity loss. Since a high SOC typically also results in a high battery voltage, one way of defining fE for such systems, for the Lumped cell model, is hence
fE defined as above would correspond either to a parasitic electrochemical reduction reaction occurring on the negative electrode, or an oxidation reaction occurring on the positive electrode. The transfer coefficient α and offset potential Eoffset parameters relate how the rate of the parasitic reactions changes when the battery voltage deviates from the average open circuit voltage,, defined as
For the Two electrodes model, the voltage dependency is a function either the positive or negative electrode potential (instead of the cell voltage) and fE,neg and fE,pos are accordingly defined as
and the average individual electrode equilibrium potentials are defined respectively as
For many battery systems it is also often observed that the lifetime is closely related to the amount of cycled equivalent full cycles (capacity throughput). A current dependence term, resulting in a linear relation between the capacity fade and the number of full cycles can be defined, for the Lumped cell model, as
where H defines the additional (dimensionless) capacity loss induced by cycling. Note that 2Qcell,0 corresponds to the capacity throughput of one full charge-discharge cycle. For the Two electrodes model, a similar expression holds, defined individually for the negative and positive electrodes with the respective negative and positive lumped parameters. Additionally, Ineg and Ipos are used instead of Icell, and Qhost,neg,0 and Qhost,pos,0 are used instead of Qcell,0.
The rate of the capacity fade may be slowed down as a result of products formed by the parasitic reactions, for example by the formation of a mass transport limiting film on the electrode particles. One way of defining a decelerating aging rate is to use the following definition of faged, for the Lumped cell model:
where G defines how many times the capacity fade rate has been reduced when all capacity has been lost. Again, for the Two electrodes model, a similar expression holds, defined individually for the negative and positive electrodes with the respective negative and positive lumped parameters. Additionally, Qhost,neg,0 and Qhost,pos,0 are used instead of Qcell,0.
Finally, temperature is also known to be a crucial factor for aging. The temperature factor fT is defined using an Arrhenius expression according to
where Ea is the activation energy and Tref is a reference temperature. Again, for the Two electrodes model, a similar expression holds, defined individually for the negative and positive electrodes with the respective negative and positive lumped parameters.
As the capacity fade reactions give rise either to change in cell capacity (for the case of Lumped cell model) or to changes in cyclable charge inventory and electrode host capacities (for the case of Two electrodes model), the state of health of the cell will change. The state of health, SOHcell, is defined as
For the Lumped cell model, Qcell,0 is the specified initial cell capacity. For the Two electrodes model, Qcell,0 refers to the capacity of the cell as computed for the provided initial values of the cell inventory and host capacities (refer to Cell Capacity and State of Charge).
Finally, it is reasonable to assume that the activation overpotential is not affected by the parasitic reactions, and there is no need to explicitly include the parasitic reactions when dealing with the main reaction potentials. The loss currents are assumed so small that the polarization effects are neglected when computing the battery cell voltage. This means that it is possible to treat the activation overpotential independently of the parasitic reactions. The following then holds good:
Short Circuit
In order to model a short circuit scenario inside a lumped battery, with known short circuit resistance (inverse of conductance), one can use this node to define the short circuit conductance Gshort. The short circuit current associated with this according to the Ohm’s law is
The corresponding Ishort is seen as a discharge current for the overall cell operation and is subtracted from the applied current:
Cell Capacity and State of Charge
For the Two electrodes model, the cell capacity and cell state of charge at any point in time, is defined implicitly by specifying cell voltages corresponding to 0% SOC and 100% SOC.
For both 0% SOC and 100% SOC, the corresponding degree of conversion levels for each electrode are computed by solving the following:
and
respectively.
The battery cell capacity Qcell (SI unit: C) can be defined as
Subsequently, the cell state of charge, SOCcell (1), can be defined as
For a more detailed explanation, refer to sections Cell Capacity and State of Charge and Defining Initial Cell Charge Distributions Based on Cell Voltage or Cell State of Charge in Theory for the Lithium-Ion Battery Interface.