# State-of-Charge Prediction Model for Ni-Cd Batteries Considering Temperature and Noise

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Optimal Feature Selection

## 3. APSO-GRNN Algorithm

#### 3.1. The GRNN Model

- Input layerThe number of neurons in the input layer is equal to the vector dimension of the sample dataset, and each neuron in the input layer is equivalent to an identical function, which directly passes the input variable to the pattern layer.
- Pattern layerThe pattern layer is fully connected and the number of neurons is set to the total number of training samples n. Each neuron corresponds to a different training sample. The value of the Gassu function for any training sample $tr{x}_{i}$ and test sample $te{x}_{j}$ is shown in Equation (2).$$Gauss(te{x}_{i}-tr{x}_{j})={e}^{-\frac{\u2225te{x}_{i}-tr{x}_{j}\u2225}{2{\sigma}^{2}}}$$
- Summation layerThe number of neurons in the summation layer is one more than the sample dimension. The summation layer has two outputs: the first neuron output is the summation of the pattern layer outputs. For any summation layer neuron with inputs $\{{g}_{1},{g}_{2},{g}_{3},\dots ,{g}_{i}\}$, the transfer function for the first summation layer neuron is as follows:$${S}_{D}=\sum _{i=1}^{m}{g}_{i}$$The remaining neuron outputs are the weighted sum of the pattern layer outputs, with a transfer function as follows:$${S}_{Nj}=\sum _{i=1}^{m}{k}_{j}{g}_{i}$$
- Output layerThe number of neurons in the output layer is equal to the output vector dimension of the training samples, and the output is a vector consisting of the quotient of each ${S}_{N}j$ and ${S}_{D}$. The transfer function is shown in Equation (5), and the structure of the GRNN is shown in Figure 2.$${y}_{i}=\frac{{S}_{Nj}}{{S}_{D}}$$

#### 3.2. Optimization Process of APSO Algorithm

#### 3.3. Prediction Process of Battery SOC Based on APSO-GRNN Model

- Initialize the particle swarm algorithm parameters. The initial particle population size is m × n, where m is the number of particle vectors, n is the number of neurons in the summation layer, and the number of iterations is T. The smoothing factor of each neuron in the summation layer is taken as the position attribute of the particle ${x}_{i}$, and the particle position parameters and matrix parameters are randomly initialized.$$x=\left[\begin{array}{ccc}{\sigma}_{11}& ...& {\sigma}_{1n}\\ \vdots & \vdots & \vdots \\ {\sigma}_{m1}& \cdots & {\sigma}_{mn}\end{array}\right]\phantom{\rule{0.277778em}{0ex}}v=\left[\begin{array}{ccc}{v}_{11}& ....& {v}_{1n}\\ \vdots & \vdots & \vdots \\ {v}_{m1}& ...& {v}_{mn}\end{array}\right]$$
- Assuming that the GRNN model is a fitness calculation model, the error matrix of the particle swarm is calculated using Equations (9) and (13) as e, and the error vector of a single particle is ${E}_{i}$.$$e=\left[\begin{array}{ccc}{E}_{11}& \cdots & {E}_{1n}\\ \vdots & \cdots & \vdots \\ {E}_{m1}& \cdots & {E}_{mn}\end{array}\right]{E}_{i}=\left[\begin{array}{c}{E}_{i1}\\ \vdots \\ {E}_{in}\end{array}\right]\phantom{\rule{4pt}{0ex}}$$
- The local optimal value ${p}_{i}$ and each particle’s global optimal value ${g}_{M}$ are updated according to the error matrix. The particle swarm velocity weight matrix w is modified using Equation (13), and the particle swarm position vector and velocity vector are updated using Equations (7) and (8).
- Repeat Steps 2 and 3 until the error of the global optimal particle is less than the target error ${E}_{target}$ or the maximum number of iterations ${T}_{max}$ and then end the iteration. The algorithm’s structure is shown in Figure 3.

## 4. Analysis of SOC Prediction Results of Ni-Cd Battery Using APSO-GRNN

#### 4.1. Test Platform and Experimental Method

#### 4.2. SOC Prediction Algorithm Results and Discussion

#### 4.2.1. APSO-GRNN Model Training Results

#### 4.2.2. Model Robustness Test Results for Noise

#### 4.2.3. Model Robustness Test Results for Temperature

## 5. Conclusions

- The advantages and disadvantages of SOC prediction models proposed in the literature were analyzed, and a GRNN model with adaptive adjustment was proposed based on the characteristics of nickel-cadmium batteries. This contribution enriches the research methods for the SOC prediction of nickel-cadmium batteries and provides a new theoretical reference for developing energy management strategies for train battery packs.
- A comparison of the prediction results of the APSO-GRNN and GRNN models showed that APSO can enhance the diversity of mode-layer smoothing factors and improve the accuracy of SOC prediction. It was demonstrated that APSO can filter the time series and adjust the size of each neuron smoothing factor based on the correlation of the time series. Thus, the model prioritizes historical data that has a more significant impact on the current moment.
- Two experimental scenarios were designed to verify the robustness of the APSO-GRNN model against noise interference and temperature changes. The prediction results were compared with those of the GRNN, SVR, and XGBoost models. The experimental results demonstrated that the APSO-GRNN model can maintain a high prediction accuracy, even under changing experimental conditions, temperatures, and noise interference.
- The APSO-GRNN model is suitable for the online prediction of the SOC of nickel-cadmium batteries due to its reduced number of parameters, shorter training time, and stronger real-time performance. It can be deployed in an onboard battery management system (BMS) to provide a theoretical basis for battery energy management strategies. In future research, we plan to broaden the scope of this study by incorporating real-world applications of train Ni-Cd batteries.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

F | Multidisciplinary Digital Publishing Institute. |

${e}_{i}$ | The out-of-bag error after random rearrangement of the feature ${f}_{i}$. |

${e}_{T}$ | The error of the sample in the decision tree T. |

$tr{x}_{i}$ | The i-th training sample. |

$te{x}_{j}$ | The j-th testing sample. |

$\sigma $ | The model smoothing factor. |

${g}_{i}$ | The output of the i-th pattern layer neuron. |

${k}_{j}$ | The corresponding eigenvalue of any training sample. |

${x}_{i}$ | The position vector of the particle. |

${v}_{i}$ | The velocity vector of the particle. |

${p}_{i}$ | The historical optimal value of the particle. |

${g}_{M}$ | The group optimal value in the current particle population. |

${y}_{k}$ | The true value of the training set. |

${E}_{k}$ | The prediction error matrix of the k-th iteration of the model. |

h | The error score matrix. |

${E}_{target}$ | The error threshold that the particle swarm should reach to stop the iteration. |

${E}_{gbest}$ | The prediction error of the global optimal particle. |

${E}_{mbest}$ | The average optimal fitness of all particles in the current epoch. |

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**Figure 5.**Voltage and current under different conditions: (

**a**) DST voltage; (

**b**) DST current; (

**c**) FUDS voltage; (

**d**) FUDS current; (

**e**) cyclic pulse voltage; (

**f**) cyclic pulse current.

**Figure 6.**Prediction results based on test set: FUDS prediction results (

**a**), pulse condition prediction results (

**b**), FUDS prediction error (

**c**), pulse condition prediction error (

**d**).

**Figure 7.**Convergence curve of polymerization degree (

**a**), smoothing factor distribution diagram (

**b**).

Item | Parameter |
---|---|

Nominal capacity | 190 Ah |

Initial weight | 6.1 kg |

Initial voltage | 1.35 V |

Resistance | $1.35\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

Initial density | 1.23 g/cm${}^{3}$ |

Test Type | Experimental Process |
---|---|

DST | Step 1: The battery is discharged to the cutoff voltage (1 V) at a constant current rate of 0.5 C and a set temperature of 20 °C. Step 2: Static battery for one hour. Step 3: The battery is charged to the cutoff voltage (1.52 V) at a rate of 1 C and then charged to the cutoff current (20 A) at a constant voltage. Step 4: The charge-discharge cycle is set to 6 min and the charge-discharge experiment is carried out according to the DST power meter. |

FUDS | Step 1: The battery is discharged to the cutoff voltage (1 V) at a constant current rate of 0.5 C and a set temperature of 20 °C. Step 2: Static battery for one hour. Step 3: The battery is charged to the cutoff voltage (1.52 V) at a rate of 1 C and then charged to the cutoff current (20 A) at a constant voltage. Step 4: Discharge under FUDS conditions. |

Cyclic Pulse | Step 1: The battery is discharged to the cutoff voltage (1 V) at a constant current rate of 0.5 C and set temperature. Step 2: Static battery for one hour. Step 3: The battery is charged to the cutoff voltage (1.52 V) at a rate of 1 C and then charged to the cutoff current (8.5 A) at a constant voltage. Step 4: The discharge cycle is set to 15 min and the pulse-discharge experiment is carried out. |

Model | Parameter |
---|---|

SVR | Kernel function: radial basis; Penalty factor: 2.5 |

XGBoost | Number of decision trees: 200; Learning rate: 0.05; Depth of a single tree: 6 |

GRNN | Smoothing factor: 0.2; Number of neurons: 100 |

APSO-GRNN | Initial particle position: random numbers ranging from 0.01 to 0.5; Initial velocity weight: 1; Learning factor: 2 |

$\mathit{\delta}$ | ${\mathit{\delta}}_{\mathbf{RMSE}}$ | ${\mathit{\delta}}_{\mathit{p}}$ | |
---|---|---|---|

SVR | 4.58 | 0.0237 | 0.9597 |

XGBoost | 3.59 | 0.0184 | 0.9656 |

GRNN | 3.86 | 0.0189 | 0.9534 |

APSO-GRNN | 1.70 | 0.0107 | 0.9789 |

$\mathit{\delta}$ | ${\mathit{\delta}}_{\mathbf{RMSE}}$ | ${\mathit{\delta}}_{\mathit{p}}$ | |
---|---|---|---|

SVR | 6.86 | 0.0378 | 0.9421 |

XGBoost | 5.09 | 0.0326 | 0.9561 |

GRNN | 7.11 | 0.0418 | 0.9452 |

APSO-GRNN | 2.56 | 0.0176 | 0.9746 |

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**MDPI and ACS Style**

Xu, H.; Yu, T.; Chen, C.; Wu, X. State-of-Charge Prediction Model for Ni-Cd Batteries Considering Temperature and Noise. *Appl. Sci.* **2023**, *13*, 6494.
https://doi.org/10.3390/app13116494

**AMA Style**

Xu H, Yu T, Chen C, Wu X. State-of-Charge Prediction Model for Ni-Cd Batteries Considering Temperature and Noise. *Applied Sciences*. 2023; 13(11):6494.
https://doi.org/10.3390/app13116494

**Chicago/Turabian Style**

Xu, Haiming, Tianjian Yu, Chunyang Chen, and Xun Wu. 2023. "State-of-Charge Prediction Model for Ni-Cd Batteries Considering Temperature and Noise" *Applied Sciences* 13, no. 11: 6494.
https://doi.org/10.3390/app13116494