# Fuzzy Logic Control of External Heating System for Electric Vehicle Batteries at Low Temperature

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. System Modeling

#### 2.1. System Configuration

#### 2.2. Equivalent Circuit Model

_{o}is the ohmic resistance which represents the static character of the battery cell that satisfies the Ohm’s law. The two pairs of RC circuits represent the dynamic character of the battery cell which is caused by the polarization phenomenon, in which R

_{1}is the activation polarization resistance, R

_{2}is the concentration polarization resistance, and C

_{1}and C

_{2}are the polarization capacitances, respectively. U

_{OCV}is the open-circuit voltage of the cell, and U is the output voltage of the cell. U

_{o}is the voltage drop on R

_{o}, U

_{1}and U

_{2}are the polarization voltages on R

_{1}and R

_{2}, respectively, and I is the current. The dynamics of the model can be represented by the following equations.

_{OCV}was given a fixed number since the corresponding SOE was known. R

_{o}, R

_{1}, R

_{2}, C

_{1}and C

_{2}were the parameters to be identified. The least squares approach was used for parameter estimation. The results show the two major properties of the ohmic resistance. One property is that the ohmic resistance is highly affected by the cell temperature, and has a significant increase as the cell temperature goes below 0 °C, as shown in Figure 4. The other property is that the SOE does not affect the ohmic resistance too much, which almost remains as a constant at a certain temperature.

_{1}and R

_{2}increase gradually, and the latter significantly increase when the SOE approaches 0%.

_{t}) is determined by the calibration shown in Figure 6 with known cell temperature. Second, the instantaneous discharging power of the cell is the summation of load power (P

_{load}) the Joule heat generated by the internal resistances of the cell, and the heating power consumed by the external heating system (P

_{heat}). Third, the consumed energy (E

_{c}) is the integral of the instantaneous discharging power over time, as shown in Equation (2).

_{0}is the initial state of energy. Note that in the final controller implementation, SOE refers to the state of energy of the entire battery pack, P

_{load}is the load power of the vehicle, and P

_{heat}is the total heating power of the heater; thus, the Joule heat terms in Equation (2) should be multiplied by the cell numbers in a pack. Since the total available cell capacity (E

_{t}) is a function of cell temperature, it will be updated in Equation (3) for when the cell temperature changes. This implies the possibility that the SOE increases even if the battery is discharging, when the rising cell temperature brings in the recovery of the cell capacity.

#### 2.3. Thermal Model

_{j}. The other part is the entropic heat denoted by Q

_{e}, which is relatively small and negligible in the simulation. The heat generated externally comes from the heat transfer between the cell and the electrothermal film, and can be modeled as a heat convection process, as shown in Equation (5):

_{tr}is the total transferred heat, and h

_{1}and h

_{2}are the heat transfer coefficients of the cell to the electrothermal film and of the cell to the air, respectively. A

_{1}and A

_{2}are the area of the film and the area of the cell exposed to air, respectively. The temperature of the cell, the film, and the environmental air are denoted by T, T

_{h}and T

_{∞}, respectively. Combining internal heat and external heat together, the final thermal model which represents the dynamic behavior of the cell and film temperature can be expressed by Equations (6) and (7):

_{h}are the mass of the cell and the electrothermal film, respectively. C and C

_{h}are the thermal capacity of the cell and the film, respectively. η is the thermal efficiency of the film. Similarly, Equations (6) and (7) should be modified to describe an entire battery pack in the final controller implementation.

## 3. Fuzzy Logic External Heating Control

#### 3.1. Control Structure

_{load}), SOE and cell temperature (T). The output of the control system is the heating power (P

_{heat}). The P

_{load}changes from positive (accelerating) to negative (regenerative braking) all the time during a driving cycle, and has the fastest change rate among the three inputs. SOE has a medium change rate, and has an overall trend of depletion with some small fluctuations. The cell temperature is the slowest changing parameter and has a monotonously increasing trend during a driving cycle. Therefore, only the load power and SOE are taken as the inputs of the FLC to reduce the computational load. Fuzzy inference performs within different temperature regions, and the final decision is made through weighted switching between defuzzifying results.

#### 3.2. Fuzzy Logic Control Design

#### 3.2.1. Membership Functions

_{load}) and the SOE of the battery pack. The range of the input variable P

_{load}is between −50 kW and 100 kW, which can cover most of the normal driving conditions of the vehicle. The P

_{load}consists of six membership functions: NB, NS, ZE, PS, PM, and PB, which represent strong regenerative braking, mild regenerative braking, zero power, mild acceleration, and strong acceleration cases, respectively. The range of the input variable SOE is from 0% to 100%, and it consists of five membership functions: L, ML, M, MH, and H, which represent low, medium low, medium, medium high, and high SOEs, respectively.

_{heat}consists of five membership functions: Z, L, ML, MH, and H, which represent non-heating, low power heating, medium low power heating, medium high power heating, and high power heating cases, respectively.

#### 3.2.2. Fuzzy Control Rules

#### 3.2.3. Defuzzification

_{h}

_{1}, P

_{h}

_{2}and P

_{h}

_{3}, which refer to the heating power at low, medium and high temperature regions, respectively. These three output variables will be merged into the final heating power in the weighted switching block.

#### 3.3. Weighted Switching

_{h}

_{1}, P

_{h}

_{2}or P

_{h}

_{3}based on which range the actual cell temperature is located in. However, this hard switch could not provide the optimal control result. Since the cell temperature increased gradually, the final heating power was obtained by summing P

_{h}

_{1}, P

_{h}

_{2}and P

_{h}

_{3}with the respective weightings to achieve smooth switching. As shown in Figure 9, w

_{1}, w

_{2}and w

_{3}are the weightings for P

_{h}

_{1}, P

_{h}

_{2}and P

_{h}

_{3}, respectively. The final heating power was calculated by Equation (8).

_{1}= 0.3, w

_{2}= 0.7, w

_{3}= 0, as seen from Figure 8; thus, P

_{heat}= 0.3P

_{h}

_{1}+ 0.7P

_{h}

_{2}. Since the shape of the weighting functions looks similar to the membership function, we can also name this function “fuzzy switching”. Note that the cell available capacity almost reaches the nominal capacity when the cell temperature is above 15 °C, and no necessary external heating is required; thus, all weightings drop to zero in this range.

## 4. Results and Discussions

#### 4.1. Test Setup

#### 4.2. Experimental Results

#### 4.3. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**External heater. (

**a**) Front view of the electrothermal film attached to the surface of a pouch cell; (

**b**) side view sketch of a battery module with the cells and external heater in parallel connection; (

**c**) battery module equipped with an external heater for experimental validation.

**Figure 5.**Polarization resistance at −20 °C. (

**a**) Activation polarization resistance; (

**b**) concentration polarization resistance.

**Figure 8.**Membership functions of input and output variables. (

**a**) Input variable (P

_{load}); (

**b**) input variable (SOE); (

**c**) output variable (P

_{heat}).

**Figure 12.**External heating tests at constant discharging current and constant heating power. (

**a**) Initial cell temperature of −30 °C, initial SOE of 30%, cell discharging current of 5 A and a heating power on each cell of 1 W; (

**b**) initial cell temperature of −10 °C, initial SOE of 100%, cell discharging current of 10 A and a heating power on each cell of 2 W.

**Figure 13.**External heating tests at constant load power. (

**a**) Initial cell temperature of −30 °C, initial SOE of 30%, and cell discharging power of 50 W; (

**b**) initial cell temperature of −10 °C, initial SOE of 50%, and cell discharging power of 30 W.

**Figure 15.**EPA driving cycle tests with an initial cell temperature of −10 °C and initial SOE of 100%.

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

Mass (kg) | 1.2 |

Length (mm) | 227 |

Width (mm) | 161 |

Thickness (mm) | 15 |

Nominal voltage (V) | 3.29 |

Charge cut-off voltage (V) | 3.65 |

Discharge cut-off voltage (V) | 2.00 |

Capacity (Ah) | 60 |

Energy capacity (Wh) | 193.63 |

Operating temperature range (°C) | −30–55 |

Specific heat capacity (J/Kg·K) | 1130 |

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

Mass (g) | 37.5 |

Length (mm) | 150 |

Width (mm) | 120 |

Thickness (mm) | 0.15 |

Rated voltage (V) | 12 |

Rated power (W) | 20 |

Specific heat capacity (J/Kg·K) | 1220 |

Operating temperature range (°C) | −190~205 |

P_{h}_{1} | P_{load} | ||||||
---|---|---|---|---|---|---|---|

NB | NS | ZE | PS | PM | PB | ||

SOE | H | H | MH | Z | L | L | ML |

MH | H | MH | Z | L | L | MH | |

M | H | MH | Z | L | ML | MH | |

ML | H | MH | L | ML | H | H | |

L | H | MH | L | M | H | H |

P_{h}_{2} | P_{load} | ||||||
---|---|---|---|---|---|---|---|

NB | NS | ZE | PS | PM | PB | ||

SOE | H | H | ML | Z | Z | Z | L |

MH | MH | ML | Z | Z | L | ML | |

M | MH | ML | Z | Z | L | ML | |

ML | MH | ML | Z | L | ML | MH | |

L | MH | ML | L | ML | MH | H |

P_{h}_{3} | P_{load} | ||||||
---|---|---|---|---|---|---|---|

NB | NS | ZE | PS | PM | PB | ||

SOE | H | ML | Z | Z | Z | Z | Z |

MH | L | Z | Z | Z | Z | Z | |

M | L | Z | Z | Z | Z | L | |

ML | L | Z | Z | Z | L | ML | |

L | L | Z | Z | Z | ML | MH |

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**MDPI and ACS Style**

Zhang, S.; Li, T.; Chen, L.
Fuzzy Logic Control of External Heating System for Electric Vehicle Batteries at Low Temperature. *World Electr. Veh. J.* **2023**, *14*, 99.
https://doi.org/10.3390/wevj14040099

**AMA Style**

Zhang S, Li T, Chen L.
Fuzzy Logic Control of External Heating System for Electric Vehicle Batteries at Low Temperature. *World Electric Vehicle Journal*. 2023; 14(4):99.
https://doi.org/10.3390/wevj14040099

**Chicago/Turabian Style**

Zhang, Shupeng, Tao Li, and Liqun Chen.
2023. "Fuzzy Logic Control of External Heating System for Electric Vehicle Batteries at Low Temperature" *World Electric Vehicle Journal* 14, no. 4: 99.
https://doi.org/10.3390/wevj14040099