RCR Model

The RCR model uses a simple equivalent circuit consisting of a resistance $R_{o}$ in series with a resistor $R_{p}$ and capacitor $C$ in parallel, along with a state-of-charge dependent voltage source $V_{o}$ to represent the behavior of the battery cell.

The RCR models were developed to provide accurate state of charge (SOC) information on electric vehicle batteries. The algorithms are based on simple-circuit representations with least-square regression of the parameters. These algorithms are robust enough to handle potential inaccuracies with specified initial states of charge and maintain reasonably accurate results. However, for best results, you regress the model based on empirical test data.

Two versions of the RCR model are provided in Simcenter STAR-CCM+ Batteries:

A 0D RCR model is available through user-defined battery cells.
A 3D RCR model is available when importing the battery descriptions from a .tbm file.

0D RCR Model

The following diagram displays an example of the 0D RCR model for two RC elements:

Simcenter STAR-CCM+ Batteries computes the working voltage of the cell

V_{L}

by:

Figure 1. EQUATION_DISPLAY

V_{L} = V_{o} - I \cdot R_{o} - (\frac{I}{C}) Δ t + {(V_{L} - V_{o} + I \cdot R_{o})}_{t - Δ t} exp [\frac{- Δ t}{τ}]

(4394)

where

V_{o}

represents zero current or open circuit voltage,

I

is current,

R_{o}

is the series resistance,

C

is the capacitance,

t

is time,

τ

is a time constant, and

R_{p}

is the polarization resistance.

The RCR table model within Simcenter STAR-CCM+ Batteries uses tables to specify the parameters as a function of SOC. Each set of $V_{o}$ , $R_{o}$ , $R_{p}$ , and $τ$ parameters are functions of the SOC for a particular temperature. There are no equations to express the evolution of each parameter, as they are measured data which vary from one cell to another. The evolution of each parameter is specified as a table and Simcenter STAR-CCM+ Batteries calculates values between table points by either linear interpolation or Bezier curve interpolation.

The 0D RCR model accounts for the rate-dependent resistance that is found through the following relationship:

Figure 2. EQUATION_DISPLAY

R_{p} = R_{p, 0} (\frac{1}{\frac{| i |}{| i_{1} |} + \exp (- \frac{| i |}{i_{0}})})

(4395)

in which

R_{p}

is a polarization resistance that can be physically interpreted as a charge-transfer resistance.

i

is the local unit cell current (A),

i_{0}

and

i_{1}

are user-specified constants (A).

For further information, refer to [863] and [864].

3D RCR Model

Due to the presence of the capacitor within the model, the 3D RCR model is more suited (than the NTG model) to battery cells which contain high discharge or charge peaks, allowing for the temporal effects within the battery cell. This model can be created using the regression process within Simcenter Battery Design Studio, depending on pulse power curves presented from testing. The five input parameters to the model can be represented by fifth-order polynomials. The coefficients should then be normalized by the active area of the battery cell [863].

Figure 3. EQUATION_DISPLAY

V_{o} = \sum_{} a_{i} \cdot S O C^{i}

(4396)

Figure 4. EQUATION_DISPLAY

R_{o} = e^{- \frac{E_{o}}{R T}} \sum_{} R_{o, i} \cdot S O C^{i} E_{o} = \sum_{} E_{o, i} \cdot S O C^{i}

(4397)

Figure 5. EQUATION_DISPLAY

R_{p} = e^{- \frac{E_{p}}{R T}} \sum_{} R_{p, i} \cdot S O C^{i} E_{p} = \sum_{} E_{p, i} \cdot S O C^{i}

(4398)

Figure 6. EQUATION_DISPLAY

\begin{matrix} τ = e^{- \frac{E_{τ}}{R T}} \sum_{} τ_{i} \cdot S O C^{i} \\ E_{τ} = \sum_{} E_{τ, i} \cdot S O C^{i} \end{matrix}

(4399)

Figure 7. EQUATION_DISPLAY

V_{L} = V_{o} - I \cdot R_{o} - (\frac{I}{C}) Δ t + {(V_{L} - V_{o} + I \cdot R_{o})}_{t - Δ t} exp [\frac{- Δ t}{τ}]

(4400)

Figure 8. EQUATION_DISPLAY

\frac{ⅆ V_{o}}{d T} = \sum_{} \frac{ⅆ V_{o}}{d T_{i}} \cdot S O C^{i}

(4401)

Figure 9. EQUATION_DISPLAY

Q = I \cdot (V_{o} - V_{L} + T \frac{d V_{o}}{d T})

(4402)

Diffusion Resistance

The polarization resistance modifies the parallel resistance

R_{p, 0}

Figure 10. EQUATION_DISPLAY

R_{d} = A_{d} e^{(\frac{E_{a, w}}{R T})} \sqrt{\frac{t - t_{w}}{e^{B_{d}} - e^{V_{o c}}}}

(4403)

where

A_{d}

and

B_{d}

are user-specified constants,

E_{a, w}

is the Warburg activation energy,

t

is time,

t_{w}

is the Warburg time offset, and

V_{o c}

is the open circuit voltage at time

t

Polarization Resistance

The polarization resistance modifies the parallel resistance $R_{p, 0}$ :

Figure 11. EQUATION_DISPLAY

R_{p, 1} = \frac{R_{p, 0}}{\frac{| i |}{| i_{p, 1} |} + \exp (- \frac{| i |}{| i_{p, 0} |})}

(4404)

where

i

is the local unit cell current (A),

i_{p, 0}

and

i_{p, 1}

are user-specified constants (A).