The design and performance of lithium-ion battery cells, modules and packs is
important for a wide variety of battery applications. The thermo-electric battery
solution enables detailed analysis of thermal and electric design and performance
for these applications under a wide array of scenarios. The analysis is enabled
through four core modeling components:
Battery cell core modeling
The cell is modeled as a homogenized region, where the battery voltage
response and associated heat generation is determined using the solution
of an ODE system. The solution utilizes two approaches: single potential
and multi-scale multi-dimensional (MSMD).
Charge conservation
In electrically conducting components, for example, connectors, busbars,
and terminals, the charge conservation equation is solved and coupled to
the energy equation as a source term.
Transient thermal conduction
Transient thermal conduction in all cell/module/pack components is
solved using the energy equation.
Cooling
Optionally, the Navier-Stokes equation can be included to design and
enhance cooling systems.
Collectively, the components allow comprehensive analysis and optimization of the
thermal and electrical performance of battery systems.
Two overarching methods are employed in the thermo-electric battery solution: single
potential and multi-scale multi-dimensional (MSMD). The former solution is
applicable for all battery types, for example, cylindrical-cells, prismatic and
pouch-type cells, while the latter is only applicable for pouch cells. Each approach
solves a variant of charge conservation and the energy equation for the
batteries:
Single potential
This approach uses equivalent circuit models (ECM) to characterize the
voltage-current response. From the ECM, a source term can be derived and
applied to the energy equation and boundary conditions based on current
and terminal voltage drop for charge conservation in the connected
components of the batteries, for example, terminals/tabs, busbars and
any other electrically conducting component.
MSMD
This approach employs a multi-scale method, with a cell scale and a
sub-domain scale. On the cell scale, two potential fields represent the
potentials of the negative and positive current collectors. The
sub-domain model, currently only supporting ECM models, represents the
voltage-current response of the battery derived from the solution of an
ODE system. Inter-domain coupling between the sub-domain and cell scale
is achieved via averaging source terms to eliminate any spatial
dependence, in the dual electric potential fields and the energy
equation. Cell scale to sub-domain coupling is achieved using spatially
resolved variables directly in the sub-domain equations.
The ECM models supported are first, second and third order, where the former has the
advantage of simplicity, for example, fewer parameters, and the latter two give a
more accurate battery cell voltage response.
A more detailed summary of the models is given in the following sections.
Equivalent Circuit Modeling
In an equivalent circuit model (ECM), the electric behavior, for example, the
voltage-current response, is modeled using a phenomenological electric circuit
approach. For example, in a 2nd order ECM (see Figure 1), that is, two RC pairs, the governing set of ODEs are
given by
Where is the terminal voltage, soc is
the state of charge, T is the temperature,
is the open circuit voltage,
is the ohmic resistance,
and
are the polarization resistances of
the first and second RC pairs respectively,
and
are the polarization capacitance of
the first and second RC pairs respectively,
is the input current, and t is the
time.
The output terminal voltage is therefore dependent on the input current and the ECM
parameters ( and
) which can depend on both state of
charge and temperature.
In AcuSolve, the sign convention dictates that current is
positive during discharge. AcuSolve supports three types
of ECM models: 1st, 2nd, and 3rd order. The choice of model order depends on the
desired dynamics, such as dynamic current conditions. The 2nd and 3rd order models
offer enhanced accuracy for pulse-discharge experiments and dynamic current
scenarios. Typically, the 2nd order model is considered the optimal choice, striking
a balance between accuracy and ease of parameter determination.
The battery state of charge (soc) is updated using coulomb counting:
where is the current flux. State of
charge (soc) represents the amount of charge in a battery relative to its nominal
capacity ().
Single Potential Model
In the single potential model, the equivalent circuit model (ECM) computes the
voltage drop and current within the homogenized section of the battery cell,
automatically applying associated boundary/interface conditions. Briefly, the ECM
model is stepped in time and the terminal voltage at time k is updated. For a
1st order ECM model this would be:
where is the current flux on the negative terminal of the
battery (from the charge conservation PDE solution), is the open circuit voltage, is the ohmic resistance, is the polarization resistance of the 1st
RC pair, is the polarization capacitance of the
1st RC pair, is diffusion resistor current at time step k-1, and is the time step size.
The terminal voltage is applied to the positive terminal of the battery for the
solution of charge conservation in the electrically conducting components of the
battery pack. Proper distribution of current is determined from the pack load and
how this distributes to individual cells in the modules that make up the pack. This
distribution is based on the properties of a cell, for example, SOC level or
temperature. The current distribution is applied as a flux condition at the negative
terminals of the battery cells.
The governing equation for the thermal (energy equation) and electrical solution
(charge conservation) are given by:
where is the heat generation (see later
for a description of this heat source) from the battery and
is the joule heat generated in
electrically conducting components, is
the conductivity and is the electric
potential.
This source term, derived from Joule’s first law, denotes the heat generated per unit
volume, calculated as the product of current density and electric field ():
MSMD Model
For pouch batteries the thermal-electric response can be modeled using a MSMD
approach which bridges the scale between the cell domain and the electrode scale. In
this approach electric potential fields are modeled in a cell composite, or more
specifically the current collectors, together with the thermal field. This is then
coupled using a multi- scale approach down to the sub-scale, where the
electrochemical phenomena, that is, the inner workings of the battery, are
modeled.
Cell scale model
At the cell scale, the temperature and a pair of electric potential fields are
solved. More specifically, the 3D battery geometry is treated as a homogenized
medium with orthotropic electrical and thermal conductivities. Positive and negative
electric potential is computed in the battery from charge conservation at the
current collectors of the cell composite. For the negative current collector, the
governing equation is:
Equivalently at the positive current collector the governing equation
is:
where is the transfer current density and is modeled
through an electrochemical sub-model, for example, the ECM model. Additionally, is subject to the constraint:
where is the current through the cell. The above system of
equations comes with associated boundary conditions such as current, for example,
constant current discharge, or voltage.
The energy equation in AcuSolve is modified by an
additional source term that couples both sources of ohmic (or joule) heating from
the current collectors and electrochemical reaction heat. The total heat source term
is given by:
where the first two terms are ohmic heating and the last term, , is the heat generated by the ECM model.
Note: The
ECM model can be closer tied to an electrochemical model by fitting parameters
to virtual experimental data provided by a DFN (Doyle Fuller Newman)
model.
MSMD multi-cell
formulation
In a battery module, batteries are connected in a nSmP pattern, for example, 8s6p,
where m batteries are connected in parallel and then n of these parallel connected
units are connected in serial to form a pack. In the MSMD method, two potential
equations are solved for the and electrical field. Depending on battery connectivity, and could be the potential from the positive current
collector or the negative one. Therefore, in a battery module simulation, the two
potential equations are solved with the transfer current density changing sign
dependent on the location in the module. The cells are alternating as odd and even
stages, see Figure 2. In tabs and busbars, you only solve for one of the
fields. In Figure 2, the and on the left indicates which field was solved in the
tabs/busbars in the line. The governing equations for the cell scale
become:
For in an odd stage:
For in an even stage:
For in an odd stage:
For in an even stage:
Bernardi’s Heat Generation Equation
Bernardi’s heat generation equation represents the irreversible and reversible heat
generation from the battery and is included in the energy equation as a source term
derived from the solution of the ECM model, given by
The first term of the RHS represents the cell over potential and includes
irreversibilities, such as ohmic losses and mass transport limitations, while the
second term represents entropic heat generation. The last two terms represent ohmic
losses and are enabled only for the MSMD approach. Bernardi’s heat generation
equation is automatically included in the energy equation as a source term for all
battery models. The entropic heat can be both temperature and state_of_charge
dependent.