# An Adaptive Peak Power Prediction Method for Power Lithium-Ion Batteries Considering Temperature and Aging Effects

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## Abstract

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## 1. Introduction

_{4}, LiCoO

_{2}, ternary lithium material, and so on. The negative electrode of a battery is generally made of graphite material [9]. SOP describes the maximum power that lithium-ion batteries can release or absorb over a period of time, which can be used to determine whether the power battery can meet the power requirements of electric vehicles during acceleration and climbing, or whether the battery can recover energy to the maximum extent during braking, thereby avoiding overcharging or discharging of the battery pack during operation and extending its service life [10]. I It should be noted that SOP cannot be directly measured through sensors on the battery, as it will vary depending on factors such as battery operating temperature, battery SOC and SOH [11]. For power lithium-ion batteries, the power that can be released or absorbed is limited by the internal resistance. In addition, the working temperature environment and aging of the battery can also affect the power supply capacity. Therefore, accurately identifying the model parameters of batteries under different operating conditions is of great significance for power prediction.

## 2. Battery Modeling and Parameter Identification

#### 2.1. Battery Equivalent Circuit Model

_{oc}(SOC) is the open circuit voltage of the battery, which can be expressed as a function of SOC; R

_{0}is the ohmic internal resistance of the battery; R

_{p}is the polarization internal resistance of the battery; C

_{p}is the polarization capacitance of the battery; U

_{p}is the polarization voltage; I

_{L}is the current flowing through the battery. Here, it is specified that the charging current is positive, and U

_{t}is the terminal voltage of the battery.

_{0}is the initial SOC of the battery, t is the duration of the charging and discharging process.

_{oc}(SOC) between OCV and SOC can be fitted using the function as follows [26]:

_{0}~K

_{6}are coefficients obtained by fitting the OCV-SOC curve through software. The method for obtaining the OCV-SOC curve is introduced in Section 4.2.

#### 2.2. Online Identification of Model Parameters

_{L}(s) as shown in Equation (9). The discrete domain transfer function obtained by z-transformation of Equation (9) is shown in Equation (10). In Formulas (10) and (11), τ

_{1}is the time constant, a

_{1}~a

_{3}is the coefficient of the intermediate variable, T

_{s}is the sampling time. Further, the battery parameters can be obtained as shown in Formula (12).

**h**(k) is the observation vector,

**θ**(k) is a parameter vector:

- (1)
- Parameter initialization. ${\widehat{\mathit{\theta}}}_{\mathit{L}\mathit{S}}$ represents the parameters identified by VFF-RLS,
**I**is the identity matrix,_{0}**P**is the covariance matrix, δ is a constant:$$\left\{\begin{array}{c}{\widehat{\mathit{\theta}}}_{\mathrm{L}\mathrm{S}}(0)=\frac{1}{\delta}{[1,1,1,1,1]}^{T}\\ \mathit{P}(0)=\delta {\mathit{I}}_{0}\end{array}\right.$$ - (2)
- Calculate estimation error e(k):$$e(k)=y(k)-{\mathit{h}}^{\mathit{T}}(k){\widehat{\mathit{\theta}}}_{\mathrm{L}\mathrm{S}}(k-1)$$
- (3)
- Calculate gain matrix $\mathit{K}\left(k\right)$, among λ It’s a forgetting factor ($0<\lambda \le 1$):$$\mathit{K}(k)=\frac{\mathit{P}(k-1)\mathit{h}(k)}{\lambda +{\mathit{h}}^{T}(k)\mathit{P}(k-1)\mathit{h}(k)}$$
- (4)
- Update Covariance matrix $\mathit{P}\left(k\right)$:$$\mathit{P}(k)=\frac{1}{\lambda}[{I}_{0}-\mathit{K}(k){\mathit{h}}^{\mathrm{T}}(k)]\mathit{P}(k-1)$$
- (5)
- Parameter estimation:$${\widehat{\mathit{\theta}}}_{\mathrm{L}\mathrm{S}}(k)={\widehat{\mathit{\theta}}}_{\mathrm{L}\mathrm{S}}(k-1)+e(k)\mathit{K}(k)$$
- (6)
- Update forgetting factor:$${\mathrm{\lambda}}_{k}=1-\frac{e(k)e(k)}{1+\mathit{K}{(k)}^{\mathrm{T}}\mathit{P}(k)\mathit{K}(k)}$$

_{p}will decrease with the increase of current when the SOC of the battery is constant, and there is a dependence on current. During the online parameter identification process, the battery is in a low current condition, while in power prediction, the current flowing through the battery is relatively large. If the dependence of polarization resistance on the battery current is not considered, it will lead to inaccurate SOP estimation.

_{b}and k are the fitting coefficients of polarization internal resistance R

_{p}with respect to current. Discharge tests were conducted on the battery under 50% SOC state using pulse currents of different magnification, and R

_{p}of the battery was identified using the least squares method. Table 1 shows the polarization internal resistance of the battery obtained through pulse testing at different current rates, where I

_{R}= ln(|I

_{L}|+1)/|I

_{L}|.

_{p}and I

_{R}at 50% SOC is shown in Figure 2.

_{b}=1.79 and k = 0.0139 in Equation (20).

## 3. Battery State Estimation and Power Prediction

#### 3.1. SOC Estimation Method

- (1)
- For state variable ${\widehat{\mathit{x}}}_{0}^{+}$ assign initial value, Assign initial value to error covariance matrix ${\mathit{P}}_{0}^{+}$, ${\mathit{Q}}_{0}$ and ${\mathit{R}}_{0}$. ${\mathit{Q}}_{k-1}$ is the process excitation noise covariance matrix of the state vector; ${\mathit{R}}_{k-1}$ is the observation noise Covariance matrix of the state vector.

- (2)
- Calculate Kalman gain ${\mathit{K}}_{k}$:$${\mathit{K}}_{k}={\mathit{P}}_{k}{\mathit{C}}_{k}^{T}{({\mathit{C}}_{k}{\mathit{P}}_{k}^{-}{\mathit{C}}_{k}^{T}+{\mathit{R}}_{k-1})}^{-1}$$
- (3)
- Posteriori estimation of state variables,${\widehat{\mathit{x}}}_{k}^{+}$ is posteriori estimation of state variable:$${\widehat{\mathit{x}}}_{k}^{+}={\widehat{\mathit{x}}}_{\mathit{k}}^{-}+{\mathit{K}}_{k}{e}_{k}$$
- (4)
- Posteriori estimate of the error Covariance matrix, ${\mathit{P}}_{k}^{+}$ is prior estimation of error covariance matrix:$${\mathit{P}}_{k}^{+}={\mathit{P}}_{k}^{-}-{\mathit{K}}_{k}{\mathit{C}}_{k}{\mathit{P}}_{k}^{-}$$

**A**

_{k},

**B**

_{k}and

**C**

_{k}in the EKF algorithm are:

#### 3.2. Battery Power Prediction Method

#### 3.2.1. Voltage Constraint

_{k}is input of the system:

_{p, k+L}on the polarization internal resistance during the k + L sampling period can be calculated:

_{p}and battery current in Formula (20) and removing the exponential term in Formula (28), the terminal voltage U

_{t, k+L}at the k + L sampling cycle can be obtained:

_{t, k+L}obtained from Formula (28) after L cycles, assuming that the voltage is discharged to the lower voltage limit or charged to the upper voltage limit after L cycles, the maximum current that the battery can release within the predicted time can be calculated. In this paper, the Newton iterative method is used to optimize the maximum current under voltage constraints, set the optimization objective function as shown in Formula (29), and the algorithm flow chart is shown in Figure 4.

_{oc, k+L}are the open circuit voltage of the battery at the end of the predicted time, which can be obtained by combining the ampere hour integration method with U

_{oc}’s fitting function on SOC. In order to reduce the computational complexity of the optimization calculation process and ensure the accuracy of the calculation, the calculation of U

_{oc, k+L}is achieved by expanding the Taylor formula on U

_{oc}(SOC).

#### 3.2.2. SOC Constraint

_{min}, during the charging process, it should not exceed the specified SOC

_{max}, to avoid overcharging and discharging the battery.

#### 3.2.3. SOP under Multiple Constraints

## 4. Experimental Verification

#### 4.1. Experimental Subjects and Platforms

#### 4.2. Battery OCV-SOC Curve

#### 4.3. Online Parameter Identification and SOC Estimation Results

_{0}and polarization capacitance C

_{p}parameters identified using the VFF-RLS algorithm are shown in Figure 8:

_{0}of the battery increases as the battery discharges, which will lead to a decrease of the battery power release ability. In addition, the polarization capacitance increases with battery discharge.

#### 4.4. SOP Prediction Verification

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 8.**Parameter identification results. (

**a**) Polarization capacitance Cp. (

**b**) Ohmic internal resistance R

_{0.}

Current | 4 | 20 | 40 | 60 | 80 | 100 | 120 |
---|---|---|---|---|---|---|---|

I_{R} (lnA/A) | 0.402 | 0.152 | 0.0928 | 0.0685 | 0.0549 | 0.0461 | 0.0399 |

R_{p} (mΩ) | 7.16 | 4.67 | 3.11 | 2.58 | 2.24 | 2.32 | 2.35 |

Items | Parameter |
---|---|

Working voltage | 3.2 V |

Nominal capacity | 8 Ah |

Charging cutoff voltage | 3.65 V |

Discharge cutoff Voltage | 2.5 V |

Maximum charging current | 10 C |

Maximum discharge current | 30 C |

Operating temperature range | Discharge: 0~30 °C Charge: −20~60 °C |

SOC (%) | Discharge Current (A) | Discharge Time (s) |
---|---|---|

70 | 129 | 115 |

127 | 117 | |

124 | 119 | |

120 | 124 | |

117 | 126 | |

115 | 129 | |

50 | 120 | 105 |

110 | 107 | |

105 | 114 | |

103 | 116 | |

100 | 125 | |

95 | 138 | |

20 | 45 | 85 |

42 | 100 | |

40 | 113 | |

38 | 122 | |

36 | 132 | |

34 | 144 |

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**MDPI and ACS Style**

Ye, J.; Wu, C.; Ma, C.; Yuan, Z.; Guo, Y.; Wang, R.; Wu, Y.; Sun, J.; Liu, L.
An Adaptive Peak Power Prediction Method for Power Lithium-Ion Batteries Considering Temperature and Aging Effects. *Processes* **2023**, *11*, 2449.
https://doi.org/10.3390/pr11082449

**AMA Style**

Ye J, Wu C, Ma C, Yuan Z, Guo Y, Wang R, Wu Y, Sun J, Liu L.
An Adaptive Peak Power Prediction Method for Power Lithium-Ion Batteries Considering Temperature and Aging Effects. *Processes*. 2023; 11(8):2449.
https://doi.org/10.3390/pr11082449

**Chicago/Turabian Style**

Ye, Jilei, Chao Wu, Changlong Ma, Zijie Yuan, Yilong Guo, Ruoyu Wang, Yuping Wu, Jinlei Sun, and Lili Liu.
2023. "An Adaptive Peak Power Prediction Method for Power Lithium-Ion Batteries Considering Temperature and Aging Effects" *Processes* 11, no. 8: 2449.
https://doi.org/10.3390/pr11082449