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Integrated Equivalent Circuit and Thermal Model for Simulation of Temperature-Dependent LiFePO_{4} Battery in Actual Embedded Application

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## Abstract

**:**

_{4}battery working in a real environment. A cell balancing strategy applied to the proposed temperature-dependent battery model balanced the SOC of each cell to increase the lifespan of the battery. The simulation outputs are validated by a set of independent experimental data at a different temperature to ensure the model validity and reliability. The results show a root mean square (RMS) error of 1.5609 × 10

^{−5}for the terminal voltage and the comparison between the simulation and experiment at various temperatures (from 5 °C to 45 °C) shows a maximum RMS error of 7.2078 × 10

^{−5}.

## 1. Introduction

_{4}is popular due to its high capacity, low self-discharge current, wide temperature operation range, and long service life that make them better candidates for many applications. However, lithium-ion batteries are sensitive to overcharging or discharging that could deteriorate the performance resulting in a shorter lifetime [4]. The accurate SOC estimation is, therefore, necessary for a properly functioning LiFePO

_{4}battery power system.

_{4}cells in a series are used for the modeling and parameter identification process. The parameters are estimated online by a series of lookup tables to provide a good compromise between the high fidelity and computational effort for integrated BMS implementation, where the lookup table uses an array of data to map input values to output values, approximating a mathematical function. If the lookup table encounters an input that does not match any of the table’s pre-defined input values, the block interpolates or extrapolates the output values based on nearby table values. Since table lookups and simple estimations can be faster than mathematical function evaluations, using the lookup table method can result in a faster computational time. The SOC estimation algorithm of the battery pack and cell balancing strategy are implemented and validated using the experimental data collected in a laboratory. The battery model under different temperatures is included to improve the battery model. The experiment results show the feasibility of the proposed model for simulation prototyping before the actual implementation.

## 2. Battery Cell Model Description

#### 2.1. Equivalent Circuit Model

_{4}battery cell test. It is obvious that a higher number of RC increases the computational resources without significantly improving the model accuracy. Therefore, 1 RC battery cell model is proposed for the embedded applications in this paper, the model structure is shown in Figure 2.

_{4}battery used in this experiment, $\mathsf{\eta}>0.994$ under room temperature [1]. In this paper, $\mathsf{\eta}=1$, and ${S}_{d}=0$ are assumed.

#### 2.2. Lumped-Capacitance Thermal Model of the Battery Cell

_{4}26650 cylindrical cells are selected to be the research objects, which are constructed in a multilayer structure in which the radial thermal conductivity is lower than the axial one. Nevertheless, the thermal resistance by the radial conduction is still much less than the convective thermal resistance, as air is used as the coolant (i.e., the Biot number, $\mathrm{Bi}={L}_{c}{h}_{f}/{k}_{s}<0.1$). Therefore, a lumped-capacitance thermal model for battery cells assuming a uniform temperature in each cell is sufficient without compromising accuracy of the numerical analysis. The thermal energy balance of the battery cell is modeled by using the first law of thermodynamics:

- Convective heat transferThe convective heat transfer ${Q}_{conv}$ from the cell to the surrounding is determined by$${Q}_{conv}={h}_{conv}{S}_{area}\left({T}_{cell}-{T}_{air}\right)$$
- Conductive heat transferThe convective heat transfer ${Q}_{cond}$ represents the thermal diffusion through cell to cell electric connector. It can be modeled by$${Q}_{cond}=\frac{{T}_{cel{l}_{2}}-{T}_{cel{l}_{1}}}{{R}_{cond}}$$

#### 2.3. Coupled Equivalent Circuit Model (ECM) and Thermal Battery Model

_{4}battery is proposed. In this model, the inputs are the current battery I and the ambient temperature, ${T}_{air}$. In the coupled model, both thermal and electrical are considered since the temperature affects the four main parameters (OCV, ${R}_{0}$, ${R}_{1}\text{}\mathrm{and}\text{}{C}_{1}$). As shown in Figure 3, the parameters at different temperatures provide two-dimensional lookup tables for the ECM to compute the terminal voltage and SOC of each cell while the thermal model determines the temperature within the cells due to convection.

## 3. Experiment Tests for Battery Characterizations

_{4}battery cells (ANR26650M1-B from A123 System with Nanophosphate

^{®}lithium-ion chemistry) were used in the experiments. The key specification of the battery cell is tabulated in Table 1. Battery cell or battery pack was placed in the temperature chamber as seen in Figure 5 to perform a series of tests under different controlled temperatures. The ambient temperatures 5 °C, 15 °C, 25 °C, 35 °C and 45 °C were used to determine the model parameters of the 12-cell battery. The load current is created using a programmable DC electronic load, and a programmable DC power supply for charging the battery cells. The power supply is utilized as a controlled voltage or current source with the output voltage from 0 to 36 V and current from 0 to 20 A. A current sensor LEM 50-P is used to measure the charge and discharge current. The NTC temperature sensors are utilized to measure the temperatures of the battery cells and the ambient temperature. The National Instruments DAQ device controlled all input and output data. The host PC communicates with the DAQ device to monitor the power supply and charge and discharge status of the battery in real-time. As the data acquisition rate is limited in the embedded system, it is one sample per second. The host PC performs the model simulation and algorithm development using the battery’s data received. A custom-designed pulse relaxation that includes the transient part and non-transient part (rather than simple constant current cycles often adopted in the literature) is employed in the SOC estimation as seen in Figure 7.

#### 3.1. Static Capacity Test

- Charge the battery at 0.8 C rate (2 A) to the fully charged state in CCCV mode under the specified temperature. The battery is fully charged to 3.6 V when the current reaches 1 mA.
- Apply a 15-hour relaxation period before discharging the battery cell.
- Discharge at a constant current 0.8 C rate until the voltage reaches the battery minimum limit of 2.5 V.
- Record the data and calculate the static capacity as follows.$${Q}_{d}=\frac{1}{3600}{{\displaystyle \int}}_{0}^{{t}_{d}}{I}_{d}\left(\tau \right)\mathrm{d}\tau \text{}\left(\mathrm{Ah}\right)$$

#### 3.2. Pulse Discharge Test

- Charge the battery to a fully charged state, follow step 1 in Section 3.1.
- Apply a 15-hour relaxation period before discharging the battery cell.
- Discharge the battery cell at a pulse current 0.8 C rate with 450 s discharging time and 45 min relaxation period, until the terminal voltage reaches the cut-off voltage 2.5 V.
- Record the data and proceed to model validation and simulation.

#### 3.3. Cycling Aging Test

- Charge the battery to a fully charged state, follow step 1 in Section 3.1.
- Allow the battery to rest for 15 min until its temperature stabilized.
- Discharge at a constant current 0.8 C rate until the voltage reaches the battery minimum limit of 2.5 V.
- Record the data and proceed to another cycle after the battery rests for 15 min.

## 4. Battery Model Identification and Results

#### 4.1. Temperature-Dependent Battery Cell Parameters Identification

_{4}battery.

#### 4.2. Temperature-Dependent Battery Cell Parameters Validation

^{−5}. It shows the battery cell indeed operating quite poorly at a lower temperature (a common characteristic of a battery cell). From the figures, it is evident that the terminal voltage errors due to the suddenly changed current can be converged to around 0 quickly (e.g., within 1.2 × 10

^{−5}s); this means the model has a certain degree of robustness, which is relevant to the further study of the advanced algorithms. To test the robustness of the model under different ambient temperatures, a set of experimental data under 35 °C (not used for the parameter identification) was compared with the simulation model. The results in Figure 21 show that the RMS error of the terminal voltage between the simulation and experiment is approximately 1.5609 × 10

^{−5}. It indicates that the temperature-dependent battery model output can estimate the terminal voltage at a different ambient temperature with an acceptable error for the embedded applications.

#### 4.3. Temperature-Dependent 12-Cell Battery Model with Convective Heat Transfer Simulation

## 5. Conclusions

_{4}battery pack for a more realistic simulation instead of a single battery cell. As a trade-off between the high fidelity and computation effort, the conductive thermal transfer is neglected in this paper. Instead of using the temperature as an external disturbance acting on the battery power system, the thermal influence due to convective heat transfer of each cell was included as parameters to couple both the equivalent circuit model (ECM) and the thermal model. Also, the temperature-dependent battery model was included to estimate the SOC that was balanced by an automatic cell balancing scheme. As compared with the experimental results, there exists a minimal root mean square error of the terminal voltage at a different ambient temperature (from 5 °C to 45 °C). The proposed simulation model allows SOC and temperature estimation of the battery cells for the embedded implementation. It can be used to develop and validate any advanced algorithms using the proposed battery cell/pack model.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 8.**Comparison of C/50 low-rate discharge profile and the 10% state-of-charge (SOC) step incremental open-circuit voltage (OCV) curve (25 °C).

**Figure 10.**(

**a**) OCV-SOC relationship curves under different temperatures; (

**b**) detailed view from SOC 0.1 to 0.4.

**Figure 23.**Current battery pack’s physical connection (

**Left**); and proposed final compact enclosure design (

**Right**).

**Figure 28.**Battery pack simulation result without cell balancing: (

**a**) current pulse (active when charging and negative when discharging); (

**b**) voltage response of each cell; (

**c**) SOC of each cell; (

**d**) temperature of each cell.

**Figure 29.**Simulation result with cell balancing: (

**a**) comparison of cell #1 to #4; (

**b**) error between cell voltage and average voltage of cell #1 to #4 after balancing.

Cell Dimensions (mm) | Ø 26 × 65 |

Cell Weight (g) | 76 |

Cell Capacity (nominal/minimum) (0.5 C Rate) | 2.5/2.4 |

Voltage (nominal, V) | 3.3 |

Recommended Standard Charge Method | 2.5 A to 3.6 V CCCV for 60 min |

Cycle Life at 20 A Discharge, 100% DOD | >1000 cycles |

Maximum Continuous Discharge | 50 A |

Operating Temperature | −30 °C to 55 °C |

Storage Temperature | −40 °C to 60 °C |

Specific Heat Capacity of the Cell ${C}_{p}\text{}\left(\mathrm{J}/\mathrm{kg}/\mathrm{K}\right)$ | 810.53 |

Convective Heat Transfer Coefficient ${h}_{conv}$ (W/m^{2}/K) | 5 |

Surface Area of Heat Exchange ${S}_{area}$ (m^{2}) | 0.0149 |

Ambient Temperature ${T}_{air}$ (°C) | 25 |

Temperature (°C) | 5 | 15 | 25 | 35 | 45 |
---|---|---|---|---|---|

Static capacity (Ah) | 2.2369 | 2.4474 | 2.5642 | 2.5693 | 2.5706 |

**Table 3.**Comparison results between adaptive least square (ALS), extended Kalman filter (EKF) and lookup table methods.

R_{0} | ALS Method | EKF Method | Lookup Table |
---|---|---|---|

RMS | 0.0055 | 0.0042 | 0.0058 |

Computation time | 1.35 s | 1.25 s | 0.021 s |

**Table 4.**Root mean square (RMS) error of terminal voltage between simulation and experimental results.

Temperature (°C) | 5 | 15 | 25 | 45 |
---|---|---|---|---|

RMS error | 6.8736 × 10^{−5} | 7.2078 × 10^{−5} | 2.2671 × 10^{−5} | 9.5907 × 10^{−6} |

© 2017 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Gao, Z.; Chin, C.S.; Woo, W.L.; Jia, J. Integrated Equivalent Circuit and Thermal Model for Simulation of Temperature-Dependent LiFePO_{4} Battery in Actual Embedded Application. *Energies* **2017**, *10*, 85.
https://doi.org/10.3390/en10010085

**AMA Style**

Gao Z, Chin CS, Woo WL, Jia J. Integrated Equivalent Circuit and Thermal Model for Simulation of Temperature-Dependent LiFePO_{4} Battery in Actual Embedded Application. *Energies*. 2017; 10(1):85.
https://doi.org/10.3390/en10010085

**Chicago/Turabian Style**

Gao, Zuchang, Cheng Siong Chin, Wai Lok Woo, and Junbo Jia. 2017. "Integrated Equivalent Circuit and Thermal Model for Simulation of Temperature-Dependent LiFePO_{4} Battery in Actual Embedded Application" *Energies* 10, no. 1: 85.
https://doi.org/10.3390/en10010085