Traction battery modeling

Sitio:	Moodle ZČU
Curso:	KEV/EMB Elektromobilita - podklady k výuce
Libro:	Traction battery modeling

Imprimido por:	Nepřihlášený host
Día:	lunes, 9 de noviembre de 2026, 03:16

Tabla de contenidos

1. Electrical properties and modeling

1. Electrical properties and modeling

Several types of battery models can be distinguished:

Electric model (SoC)
Temperature model
Aging model (SoH)
…

From an electrical point of view, the battery is a source with significatn internal resistance, ie. the voltage drops under load.
As the voltage drops, the power consumption decreases.

_{http://batteries2020.eu}

1.1. Electric model

The electric model helps to determine the SoC or SoH.
Determination of SoC from simple voltage measurement is practically impossible!
The 1^st order model provides sufficient accuracy if the parameters are f (T, SoC).
The internal resistance R₀ is critical.
RC cells model the dynamic processes under load and unloading of the cell.

First order Thévenin model

Second order Thévenin model

1.2. Open circuit voltage

The open circuit voltage (OCV) curve is the dependence of the steady voltage (without the effect of the drop on the internal resistance, etc.) on the state of charge of the cell.
OCV measurement is time consuming - stabilization of voltage after loading in the order of tens of minutes.
Alternatively, it can be determined from the discharge with a very small constant current, but R₀ and T are biased.

OCV (open circuit voltage) curve of Li-ion cell

OCV change depending on temperature and SoC

1.3. Parameter identification

Determination of parameters using discharge characteristics during pulse discharge.
Dependences of parameters on state of charge and temperature: R, C = f (SOC, T).

Discharge characteristics of the cell during pulse discharge with a current of 23 A at a temperature of 25° C

Identification of internal resistance R₀

Internal resistance is a key parameter of the battery.
It causes a voltage drop.
It affects the efficiency of charging and discharging, i.e. losses and range.
Heavily dependent on temperature - huge differences between summer and winter operation of electric cars!

Current pulse load of the cell

Identification of internal resistance R₀ depending on temperature and SoC

1.4. Parameter identification II

Determination of RC cell parameters

Thévenin models of first and second order

First vs. second order model

The voltage stabilization transient does not correspond to a single exponential function → better modeled by a combination of two or more exponential functions.
On the other hand, $y = C_1 e^{α_1t}$ is an unambiguous problem => the dependence of R₁ and C₁ on T and SOC can be determined.
Instead, $y = C_1e^{α_1t} + C_2e^{α_2t}$ there is no unambiguous problem, ie. more combinations R₁, C₁, R₂, C₂ lead to the same curve => very problematic determination of dependence on T and SOC.
However, the first-order model provides sufficient accuracy.

Derivation

We derive the model for the RC cell by solving a first-order differential equation. Assume an RC cell to which we connect a constant current source I at time t = 0. We compile a differential equation where the total current I is divided into a current by a resistance Uc divided by R and a capacitor current, ie C duc according to dt. We convert the first-order differential equation into a standard form. We assume a general solution in the form

$u_c = Ke^{\lambda t}$ .

We solve the homogeneous equation

$RC\lambda + 1 = 0$ from where lambda equals -1 / RC and add it to the general solution.

For a particular solution, we use a steady state, where the voltage across the capacitor equals the resistance drop, i.e.

$u_c = RI$ . So we get the solution

$u_c = Ke ……$

We determine the constant K using the state at time zero, when the voltage on the capacitor is equal to zero. Substituting into the equation we get the constant