# Reduced-Order Electro-Thermal Battery Model Ready for Software-in-the-Loop and Hardware-in-the-Loop BMS Evaluation for an Electric Vehicle

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## Abstract

**:**

## 1. Introduction

## 2. Electrochemical Model

## 3. Reduced-Order Electro-Thermal Model

- Firstly, the equivalent circuit models are intensively studied in the literature. They commonly share a similar structure with electrical or electrochemical elements: (1) a voltage source to represent the open circuit voltage; (2) an ohmic resistance to represent the instantaneous voltage drop when a current is applied to the battery; (3) a diffusion impedance to capture the dynamic behavior related to the diffusion of lithium-ion inside the battery. The diffusion impedance can be represented by a network of RC circuits connected in series [7,8,9,10,11,12,13,14], by an electrochemical impedance such as the constant phase element (CPE) or Warburg impedance [15,16,17], or by other circuit structures including resistances and capacitors [11,18]. The equivalent circuit models are mostly calibrated with temporal test data. Frequency data from the electrochemical impedance spectroscopy (EIS) has also been used to identify the diffusion impedance [16,17]. The equivalent circuit models can be easily coupled with a simplified thermal model to estimate the temperature change during the battery operation [7,9,10,11,14]. The simplified thermal models reported in the literature commonly include one or several equivalent thermal capacitors to represent the specific heat of the battery cell, one or several equivalent thermal resistances to represent the heat conduction inside the cell and the heat convection with the ambient environment. The equivalent circuit models are compatible with aging modeling [19] and thermal runaway modeling [20]. These models have also been implemented in some of the simulation software, such as Simcenter Amesim which is a multi-physical simulation software of Siemens.
- Secondly, the single particle (SP) model, which is a simplified version of the p2D electrochemical model, is also widely studied in the literature [21,22,23]. The SP model shares many parameters with the p2D model. Compared to the equivalent circuit model, the SP model requires considerable effort in experimental tests and parameter identification. The computation cost of the SP model is higher than that of the equivalent circuit model.
- Finally, some black box models based on the neural network have been reported in the literature [24,25]. While they could capture the battery dynamic behavior, specific sets of data are needed to train the neural network model so that the model gives correct estimation in the expected operating range. Since the parameter values of the black box models do not have explicit physical meaning like in the equivalent circuit model and the electrochemical model, the black box models require higher effort to find out the exact reason when there is a significant difference between the model estimation and the battery test data.

#### 3.1. Proposed Model

_{ohm}to represent the ohmic resistance responsible for the instantaneous voltage drop when the cell current changes; and (3) several RC circuits R

_{dff}[i]/C

_{diff}[i] to represent the dynamic behavior of the cell mainly related to the diffusion of lithium-ion inside the battery cell. Different numbers of RC circuits were used in the literature, e.g., one in [7,10], two in [9] and five in [8]. Having more RC circuits helps to improve the model accuracy at the cost of complexity in parameter identification and higher calculation time [8,9]. The OCV is a function of the state of charge (SoC) and the temperature. Other elements in Figure 1 are functions of the SoC, the temperature and the current.

_{init}is the initial SoC; I is the current; Q

_{cell}is the cell capacity; η

_{cell}is the faradic efficiency which is set to 1 in our study since the faradic efficiency of lithium-ion cells is very close to 1. However, for example the faradic efficiency for NiMH cells should not be neglected. The voltage of the cell is calculated by the following equation:

- The OCV is calculated with the following equation to consider the hysteresis behavior of the OCV in battery:

_{c}and OCV

_{d}are the open circuit voltages measured in charge and in discharge at the reference temperature T

_{ref}; $\frac{dU}{dT}$ is the entropic coefficient; F

_{hys}is the hysteresis factor with value varying between 0 and 1 and is calculated by the following equations:

_{hys}is the state of charge variation necessary for full charge/discharge open circuit voltage transition. An example of the open circuit voltage transition is shown in Figure 2: the initial open circuit voltage starts from point (1); during a charge, the open circuit voltage approximates its high boundary defined by OCV

_{c}; during a discharge, the open circuit voltage joins progressively the low boundary defined by OCV

_{d}.

- The ohmic voltage drop ΔU
_{ohm}is calculated with equation:

- The voltage drop ΔU
_{diff}_{_i}for each of the RC circuits (i = 1, 2, …, N_{RC}) is calculated with the following equation:

_{gen}) during its operation includes the heat related to the entropic coefficient (${\varnothing}_{\frac{dU}{dT}}$), the hysteresis loss (${\varnothing}_{hys}$), the ohmic loss (${\varnothing}_{ohm}$) and the diffusion loss (${\varnothing}_{diff}$). The heat generated is calculated with equations as follows:

_{amb}. The cell temperature T

_{cell}is calculated with equation:

_{cell}is the cell mass and C

_{p}is the cell specific heat. If the temperature gradient inside the battery cell needs to be considered, additional RC elements could also be added to the model in Figure 3 at the cost of increasing simulation time and calibration effort. For example, one additional set of RC elements could be added to the model in Figure 3 to take into account the heat capacity of the cell casing and the heat conduction between the cell core and the casing.

#### 3.2. Model Calibration

- Test 1: Pulses test. As shown in Figure 4a, this profile discharges the cell from 100% to 0% SoC. It includes several groups of short-duration (<2 s) charge and discharge pulses at different current levels. Between two groups of pulses, a long-duration discharge with a constant current is used to decrease 5% SoC of the cell. Two levels of current are used alternatively for the long-duration discharge (1 C and 0.5 C). Each long-duration discharge is followed by a long rest period (1800 s) to stabilize the cell voltage and temperature.
- Test 2: Charge test. As shown in Figure 4b, this profile consists of long-duration charges with two levels of current used alternatively (1 C and 0.5 C) to charge the cell from 0% to 95% SoC. Each long-duration charge increases the SoC of the cell by 5% and is followed by a long rest period (1800 s).

#### 3.2.1. OCV_{d} and OCV_{c}

_{d}and OCV

_{c}are identified with the relaxation phase after each long-duration discharge or charge. The value at the end of each relaxation after the long-duration discharge (Figure 4a is considered as the OCV value in discharge. Figure 5 shows an example of the OCV

_{d}identified at the three temperatures. The same process allows us to get the OCV

_{c}at the three temperatures by using the charge profile in Figure 4b. A temperature of 25 °C is chosen as the reference temperature in our study to calculate the open circuit voltage with Equation (3).

#### 3.2.2. ΔSoC_{hys}

_{d}and OCV

_{c}at 25 °C. The experimental test results of a Li-ion cell in [27] show that the state of charge variation (ΔSoC

_{hys}) necessary for full charge/discharge OCV transition is from 15% to 25% in most of the case. In the absence of tests to identify ΔSoC

_{hys}, its value is set arbitrarily to 15% in our study. For the future studies, it would be interesting to investigate several things in the hysteresis behavior for different battery chemistries, such as the impact of the temperature on the value of ΔSoC

_{hys}, as well as a fast and simple method to identify the value of ΔSoC

_{hys}from limited test data.

#### 3.2.3. dU/dT

_{1}and T

_{2}are two different temperatures; OCV

_{x}can be OCV

_{d}or OCV

_{c}. In our case, the OCV is identified at three temperatures in discharge and in charge. There are therefore six possible combinations to calculate six curves of entropic coefficient as illustrated in Table 2. The average curve of the six entropic coefficient curves is the one to be used in the reduced-order circuit model. The average curve is shown in Figure 7.

#### 3.2.4. R_{ohm_d} and R_{ohm_c}

#### 3.2.5. RC Circuits

_{dff}[i]/C

_{diff}[i] are identified with the Battery Identification Assistant tool of Simcenter Amesim. Instead of identifying directly several RC circuits, the tool identifies a Warburg impedance represented by two parameters (a diffusion resistance R

_{ss}and a time coefficient T

_{c}). The identification process of the Warburg impedance is similar to the ones in [15,16]. Once the Warburg impedance is identified, it can be easily approximated by a different number of RC circuits in the Simcenter Amesim battery model with the help of the RC transformation tool [26]. The parameters R

_{ss}and T

_{c}of the Warburg impedance are identified at three different temperatures (5, 25 and 45 °C). Their values are in function of the current, the SoC and the temperature. Figure 9 shows the R

_{ss}and T

_{c}identified at 25 °C.

_{ss}and T

_{c}in order to capture the eventual dependency of the battery dynamic behavior on the current amplitude and direction. However, the current dependency increases the complexity of the lookup table for the R

_{ss}and T

_{c}, which results in a longer simulation time. Additional effort is also required to implement the high dimensional lookup table in the battery model if one needs to develop in-house battery model instead of using a dedicated simulation software such as Simcenter Amesim. Several further improvements in the Warburg impedance identification, which are not covered by this paper, could be carried out in the future studies to:

- Evaluate the possibility of removing the current dependency, so that faster simulation time and simpler lookup table implementation could be achieved;
- Include the relaxation phase after the current pulse into the parameter identification, so that the dynamic behavior during the relaxation phase could also be properly represented in case the battery dynamic behavior in the relaxation phase is significantly different from the one during the current pulse;
- Search for an optimal function, such as a polynomial used in [9], to represent the parameter value in function of the SoC or the current. It could help to guarantee smooth parameter value change along the SoC and current axis. The smooth parameter value change during the simulation has several benefits. It will allow to have faster model simulation. It will also minimize the risk of simulation failure due to the discontinuity issue in which there is a sharp change of the parameter value during the simulation [29].

#### 3.2.6. Thermal Parameters

#### 3.3. Model Validation

_{RC}= 1 to 5) have therefore been done. Figure 11b–d show the comparison of the two models on the estimation of battery voltage and temperature at respectively 5, 25 and 45 °C, with N

_{RC}= 1 in the reduced-order circuit model. These figures show that the reduced-order circuit model (ROC model) can reproduce correctly the voltage and temperature behavior of the electrochemical model (EC model).

_{ECM,k}is the voltage or temperature of the electrochemical model; V

_{ROM,k}is the voltage or temperature of the reduced-order circuit model; N

_{s}is the number of the voltage or temperature data sample. Table 4 and Table 5 present the RMS errors of the voltage and temperature estimation according to the number of RC circuits used. The results show that the reduced-order circuit model with one RC circuit is sufficient to correctly simulate the electrochemical model. Adding additional RC circuits does not significantly improve the estimation precision. The RMS errors of the reduced-order circuit model are similar to the ones reported in other works in the literature [7,9].

## 4. Battery Pack Model

#### 4.1. Battery Pack Model in a Virtual Test Bench

#### 4.2. Scenarios for Battery Pack Cooling

_{conv}equal to 20 W/m

^{2}/K) to forced convection (with h

_{conv}equal to 200 W/m

^{2}/K). The active air cooling is turned off when all the 48 temperatures are less than 30 °C. For both scenarios, the battery pack is charged during the first 2 h. Then the vehicle follows two WLTC (Worldwide harmonized Light-duty vehicles Test Cycles) driving cycles. The comparison of the simulation results for the two scenarios shows that the active cooling helps to increase the effective charge time and significantly decrease the battery cell temperatures.

#### 4.3. Real-Time Capability

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 11.**(

**a**) Validation current and SoC profiles. (

**b**) Validation test at 5 °C. (

**c**) Validation test at 25 °C. (

**d**) Validation test at 45 °C.

**Figure 12.**Integration of the battery pack model in a virtual test bench of an electric vehicle in Simcenter Amesim.

Parameter | Description | Unit |
---|---|---|

Q_{cell} | Cell capacity | Ah |

OCV_{c} | OCV in charge at the reference temperature | V |

OCV_{d} | OCV in discharge at the reference temperature | V |

ΔSoC_{hys} | SoC variation for full charge and discharge open circuit voltage transition | % |

dU/dT | Entropic coefficient | V/K |

R_{ohm} | Ohmic resistance | Ohm |

R_{diff}[i] | Diffusion resistance | Ohm |

C_{diff}[i] | Diffusion capacitance | F |

C_{p} | Specific heat of the cell | J/kg/K |

h_{conv} | Convective heat exchange coefficient | W/m^{2}/K |

S_{conv} | Convective heat exchange area | m^{2} |

m_{cell} | mass of the cell | kg |

OCV_{x} | OCV_{d} | OCV_{d} | OCV_{d} | OCV_{c} | OCV_{c} | OCV_{c} |
---|---|---|---|---|---|---|

T_{1} (°C) | 5 | 25 | 45 | 5 | 25 | 45 |

T_{2} (°C) | 25 | 45 | 5 | 25 | 45 | 5 |

${\frac{dU}{dT}|}_{x}^{{T}_{1},{T}_{2}}$ | ${\frac{dU}{dT}|}_{d}^{5,25}$ | ${\frac{dU}{dT}|}_{d}^{25,45}$ | ${\frac{dU}{dT}|}_{d}^{45,5}$ | ${\frac{dU}{dT}|}_{c}^{5,25}$ | ${\frac{dU}{dT}|}_{c}^{25,45}$ | ${\frac{dU}{dT}|}_{c}^{45,5}$ |

Parameter | Value | Unit |
---|---|---|

C_{p} | 791.86 | J/kg/K |

h_{cov} | 27.5087 | W/m^{2}/K |

S_{conv} | 0.00421525 | m^{2} |

m_{cell} | 0.04622 | kg |

RMS Error of Voltage | 1 RC (mV) | 2 RC (mV) | 3 RC (mV) | 4 RC (mV) | 5 RC (mV) |
---|---|---|---|---|---|

5 °C | 36 | 36 | 35 | 35 | 36 |

25 °C | 29 | 29 | 29 | 29 | 29 |

45 °C | 29 | 28 | 28 | 28 | 28 |

RMS Error of Temperature | 1 RC (°C) | 2 RC (°C) | 3 RC (°C) | 4 RC (°C) | 5 RC (°C) |
---|---|---|---|---|---|

5 °C | 0.55 | 0.55 | 0.54 | 0.54 | 0.54 |

25 °C | 0.32 | 0.32 | 0.32 | 0.32 | 0.31 |

45 °C | 0.34 | 0.34 | 0.34 | 0.34 | 0.34 |

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**MDPI and ACS Style**

Li, A.; Ponchant, M.; Sturm, J.; Jossen, A.
Reduced-Order Electro-Thermal Battery Model Ready for Software-in-the-Loop and Hardware-in-the-Loop BMS Evaluation for an Electric Vehicle. *World Electr. Veh. J.* **2020**, *11*, 75.
https://doi.org/10.3390/wevj11040075

**AMA Style**

Li A, Ponchant M, Sturm J, Jossen A.
Reduced-Order Electro-Thermal Battery Model Ready for Software-in-the-Loop and Hardware-in-the-Loop BMS Evaluation for an Electric Vehicle. *World Electric Vehicle Journal*. 2020; 11(4):75.
https://doi.org/10.3390/wevj11040075

**Chicago/Turabian Style**

Li, An, Matthieu Ponchant, Johannes Sturm, and Andreas Jossen.
2020. "Reduced-Order Electro-Thermal Battery Model Ready for Software-in-the-Loop and Hardware-in-the-Loop BMS Evaluation for an Electric Vehicle" *World Electric Vehicle Journal* 11, no. 4: 75.
https://doi.org/10.3390/wevj11040075