# Implementation of Equilibrium Strategy Aiming at Throughput Maximization of Series Battery Pack

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Parameter Identification for Series Batteries

#### 2.1. M+D Model

#### 2.2. Parameter Identification Based on RLS

#### 2.3. Parameter Identification Results

## 3. Equalization Strategy Design Based on Model Predictive Control

#### 3.1. State Space Model of Series Batteries

_{ij}, indicating the state of the ith battery in the battery pack at the j moment. The matrix is a diagonal matrix formed by the coefficient matrix in the cell model, which can be written as:

#### 3.2. Objective Function and Solution Method of Equilibrium Strategy

_{act}/C

_{init}, where C

_{init}is the initial capacity of the battery, which can be regarded as a constant, define K = dc/dn as the battery aging rate, define n as the number of life cycles, define at the time that c = 0.8, the number of life cycles as the maximum number of battery cycles, Q represents the throughput of the battery life cycle [3]. Where

## 4. Simulation Result

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Feng, F.; Hu, X.; Liu, K.; Che, Y.; Lin, X.; Jin, G.; Liu, B. A practical and comprehensive evaluation method for series-connected battery pack models. IEEE Trans. Transp. Electrif.
**2020**, 6, 391–416. [Google Scholar] [CrossRef] - Liu, X.; Ai, W.; Naylor, M.; Patel, Y.; Wu, B. The effect of cell-to-cell variations and thermal gradients on the performance and degradation of lithium-ion battery packs. Appl. Energy
**2019**, 248, 489–499. [Google Scholar] [CrossRef] - Schmalstieg, J.; Käbitz, S.; Ecker, M.; Sauer, D.U. A holistic aging model for Li(NiMnCo)O2 based 18650 lithium-ion batteries. J. Power Sources
**2014**, 257, 325–334. [Google Scholar] [CrossRef] - Yang, S.; He, R.; Zhang, Z.; Cao, Y.; Gao, X.; Liu, X. CHAIN: Cyber hierarchy and interactional network enabling digital solution for battery full-lifespan management. Matter
**2020**, 3, 27–41. [Google Scholar] [CrossRef] - Weng, C.; Feng, X.; Sun, J.; Peng, H. State-of-health monitoring of lithium-ion battery modules and packs via incremental capacity peak tracking. Appl. Energy
**2016**, 180, 360–368. [Google Scholar] [CrossRef] [Green Version] - Gao, X.; Liu, X.; He, R.; Wang, M.; Xie, W.; Brandon, N.P.; Wu, B.; Ling, H.; Zang, S. Designed high-performance lithium-ion battery electrodes using a novel hybrid model-data driven approach. Energy Storage Mater.
**2021**, 36, 435–458. [Google Scholar] [CrossRef] - Wu, B.; Widanage, W.D.; Yang, S.; Liu, X. Battery digital twins: Perspectives on the fusion of models, data and artificial intelligence for smart battery management systems. Energy AI
**2020**, 1, 100016. [Google Scholar] [CrossRef] - Che, Y.; Deng, Z.; Lin, X.; Hu, L. Learning and online model Correction. IEEE Trans. Veh. Technol.
**2021**, 70, 1269–1277. [Google Scholar] [CrossRef] - Yang, S.; Gao, X.; Li, Y.; Xie, W.; Guo, B.; Zhang, L.; Liu, X. Minimum lithium plating overpotential control based charging strategy for parallel battery module prevents side reactions. J. Power Sources
**2021**, 494, 229772. [Google Scholar] [CrossRef] - Feng, F.; Hu, X.; Liu, J.; Lin, X.; Liu, B. A review of equalization strategies for series battery packs: Variables, objectives, and algorithms. Renew. Sustain. Energy Rev.
**2019**, 116, 109464. [Google Scholar] [CrossRef] - Li, W.; Sengupta, N.; Dechent, P.; Howey, D.; Annaswamy, A.; Sauer, D.U. Online capacity estimation of lithium-ion batteries with deep long short-term memory networks. J. Power Sources
**2021**, 482, 228863. [Google Scholar] [CrossRef] - Nguyen, N.; Oruganti, S.K.; Na, K.; Bien, F. An adaptive backward control battery equalization system for serially connected lithium-ion battery packs. IEEE Trans. Veh. Technol.
**2014**, 63, 3651–3660. [Google Scholar] [CrossRef] - Ugle, R.; Li, Y.; Dhingra, A. Equalization integrated online monitoring of health map and worthiness of replacement for battery pack of electric vehicles. J. Power Sources
**2013**, 223, 293–305. [Google Scholar] [CrossRef] - Zheng, Y.; Ouyang, M.; Lu, L.; Li, J.; Han, X.; Xu, L. On-line equalization for lithium-ion battery packs based on charging cell voltages: Part 1. Equalization based on remaining charging capacity estimation. J. Power Sources
**2014**, 247, 676–686. [Google Scholar] [CrossRef] - Zou, C.; Manzie, C.; Ne, D. Multi-time-scale observer design for state-of-charge and state-of- health of a lithium-ion battery. J. Power Sources
**2016**, 335, 121–130. [Google Scholar] [CrossRef] - Dai, H.; Wei, X.; Sun, Z.; Wang, J.; Gu, W. Online cell SOC estimation of Li-ion battery packs using a dual time-scale Kalman filtering for EV applications. Appl. Energy
**2012**, 95, 227–237. [Google Scholar] [CrossRef] - Wu, T.; Qi, Y.; Liao, L.; Ji, F.; Chen, H. Research on equalization strategy of lithium-ion batteries based on fuzzy logic control. J. Energy Storage
**2021**, 40, 102722. [Google Scholar] [CrossRef] - Crespo, M.; Georgious, R.; Garcia, P.; Villa, G. Active equalization of series/parallel Li-ion battery modules including no-load conditions. In Proceedings of the IEEE Energy Conversion Congress and Exposition, Detroit, MI, USA, 11–15 October 2020; pp. 4431–4437. [Google Scholar]
- Li, K.; Zong, X.; Liu, Q.; Sun, Y.; Xue, F. Design of an active battery equalization circuit with DC-DC converter. In Proceedings of the 3rd Asia Energy and Electrical Engineering Symposium, Chengdu, China, 26–29 March 2021; pp. 863–866. [Google Scholar]
- Xiong, H.; Song, D.; Shi, F.; Wei, Y.; Jinzhen, L. Novel voltage equalisation circuit of the lithium battery pack based on bidirectional flyback converter. IET Power Electron.
**2020**, 13, 2194–2200. [Google Scholar] [CrossRef] - Zhang, S.; Qiang, J.; Yang, L.; Zhao, X. Prior-knowledge-independent equalization to improve battery uniformity with energy efficiency and time efficiency for lithium-ion battery. Energy
**2016**, 94, 1–12. [Google Scholar] [CrossRef] - Diao, W.; Xue, N.; Bhattacharjee, V.; Jiang, J.; Karabasoglu, O. Active battery cell equalization based on residual available energy maximization. Appl. Energy
**2018**, 210, 690–698. [Google Scholar] [CrossRef] - Wang, B.; Qin, F.; Zhao, X.; Ni, X.; Xuan, D. Equalization of series connected lithium-ion batteries based on back propagation neural network and fuzzy logic control. Int. J. Energy Res.
**2020**, 44, 4812–4826. [Google Scholar] [CrossRef] - Song, L.; Liang, T.; Lu, L.; Ouyang, M. Lithium-ion battery pack equalization based on charging voltage curves. Int. J. Electr. Power Energy Syst.
**2020**, 115, 105516. [Google Scholar] [CrossRef] - Lin, Y.; Xu, X.; Wang, F.; Xu, Q. Active equalization control strategy of Li-ion battery based on state of charge estimation of an electrochemical-thermal coupling model. Int. J. Energy Res.
**2020**, 44, 3778–3789. [Google Scholar] [CrossRef] - Zhang, S.; Yang, L.; Zhao, X.; Qiang, J. Electrical power and energy systems a GA optimization for lithium-ion battery equalization based on SOC estimation by NN and FLC. Int. J. Electr. Power Energy Syst.
**2015**, 73, 318–328. [Google Scholar] [CrossRef] - Einhorn, M.; Guertlschmid, W.; Blochberger, T.; Kumpusch, R.; Permann, R.; Conte, F.V.; Kral, C.; Fleig, J. A current equalization method for serially connected battery cells using a single power converter for each cell. IEEE Trans. Veh. Technol.
**2011**, 60, 4227–4237. [Google Scholar] [CrossRef] - Zheng, Y.; Ouyang, M.; Lu, L.; Li, J.; Han, X.; Xu, L.; Ma, H.; Dollmeyer, T.A.; Freyermuth, V. Cell state-of-charge inconsistency estimation for LiFePO4 battery pack in hybrid electric vehicles using mean-difference model. Appl. Energy
**2013**, 111, 571–580. [Google Scholar] [CrossRef] - Hu, L.; Hu, X.; Che, Y.; Feng, F.; Lin, X.; Zhang, Z. Reliable state of charge estimation of battery packs using fuzzy adaptive federated filtering. Appl. Energy
**2020**, 262, 114569. [Google Scholar] [CrossRef] - Feng, T.; Yang, L.; Zhao, X.; Zhang, H.; Qiang, J. Online identification of lithium-ion battery parameters based on an improved equivalent-circuit model and its implementation on battery state-of-power prediction. J. Power Sources
**2015**, 281, 192–203. [Google Scholar] [CrossRef] - Altaf, F.; Egardt, B. Gain-scheduled control of modular battery for thermal and SOC balancing. Engineering
**2016**, 49, 62–69. [Google Scholar] - Liu, J.; Chen, Y.; Fathy, H.K.; Chen, Y. Nonlinear model-predictive optimal control of an active cell-to-cell lithium-ion battery pack balancing circuit. IFAC
**2017**, 50, 14483–14488. [Google Scholar] [CrossRef] - Liu, J.; Duan, Q.; Chen, H.; Sun, J.; Wang, Q. An optimal multistage charge strategy for commercial lithium ion batteries. Sustain. Energy Fuels
**2018**, 2, 1726–1736. [Google Scholar] - Han, W.; Zhang, L. Battery cell reconfiguration to expedite charge equalization in series-connected battery systems. IEEE Robot. Autom. Lett.
**2018**, 3, 22–28. [Google Scholar] [CrossRef] - Burgos-Mellado, C.; Orchard, M.E.; Kazerani, M.; Cárdenas, R.; Sáez, D. Particle-filtering-based estimation of maximum available power state in Lithium-Ion batteries. Appl. Energy
**2016**, 161, 349–363. [Google Scholar] [CrossRef] - Birkl, C.R.; Roberts, M.R.; McTurk, E.; Bruce, P.G.; Howey, D.A. Degradation diagnostics for lithium ion cells. J. Power Sources
**2017**, 341, 373–386. [Google Scholar] [CrossRef]

**Table 1.**State equations of M+D model. Adapted from ref. [28].

M+D Model | State Equations |
---|---|

CMM | ${\dot{U}}_{P,m}=-\frac{1}{{R}_{P,m}{C}_{P,m}}{U}_{P,m}+\frac{1}{{C}_{P,m}}I$ ${U}_{T,m}={U}_{OCV,m}-{U}_{P,m}-{R}_{O,m}I$ where ${U}_{T,m}$ represents the mean terminal voltage of the battery pack, ${R}_{P,m}$ and ${C}_{P,m}$ are the polarization internal resistance and polarization capacitance. ${U}_{P,m}$ is the polarization voltage, ${R}_{0,m}$ represents the ohm internal resistance, and $I$ is the instantaneous current in the circuit. ${U}_{OCV,m}$ represents the OCV. |

CDM#2 | $\Delta {U}_{T,i}=\Delta {U}_{OCV,i}\left(\Delta SO{C}_{i}\right)-\Delta {R}_{O,i}I$ where $\Delta {R}_{0,i}$ represents difference between ohmic resistance and mean ohmic resistance of each cell and $I$ is the instantaneous current in the circuit |

CMM: (i) Initialization: ${\varphi}_{m}$, ${\theta}_{m}$, ${K}_{m}$, ${P}_{m}$, $\lambda $ (ii) Calculate and measure the mean voltage: ${U}_{T,m,i}={\displaystyle {\displaystyle \sum}_{k=1}^{n}}{U}_{T,k,i}/n$ (iii) Calculate of mean cell gain matrix: ${K}_{m,i}=\left({P}_{m,i-1}{\varphi}_{m,i}^{T}\right)/\left(\lambda +{\varphi}_{m,i}^{T}{P}_{i-1}{\varphi}_{m,i}\right)$ (iv) Calculate the mean cell error covariance matrix: ${P}_{m,i}=\left({P}_{m,i-1}-{K}_{m,i}{\varphi}_{m,i}^{T}{P}_{m,i-1}\right)/\lambda $ (v) Update mean cell parameter matrix: ${\theta}_{m,i}^{}={\theta}_{m,i-1}^{}+{K}_{m,i}\left({E}_{T,m,i}-{\varphi}_{m,i-1}^{T}{\theta}_{m,i-1}^{}\right)$ (vi) Update estimated voltage: CDM#2: ${U}_{T,m,i}^{}={\varphi}_{m,i}^{T}{\theta}_{m,i}^{}+{U}_{OCV,m,i}^{}$ ${U}_{T,k,i}^{}={U}_{T,m,i}^{}+\Delta {U}_{OCV,k,i}^{}+\Delta {R}_{0,k}{I}_{i}$ ${U}_{T,i}^{}={\displaystyle \sum _{k-1}^{n}{U}_{T,k,i}}$ where ${U}_{T,k,i}$ represents the terminal voltage value of the cell k at sampling point i, ${U}_{T,i}$ represents the terminal voltage value of the battery pack at time i. In addition ${E}_{T,i}^{}={U}_{T,i}^{}+{U}_{OCV,N,i}^{}$ ${E}_{T,k,i}^{}={U}_{T,k,i}^{}+{U}_{OCV,k,i}^{}$ ${E}_{T,\mathrm{max},i}^{}={U}_{T,\mathrm{max},i}^{}+{U}_{OCV,\mathrm{max},i}^{}$ ${E}_{T,\mathrm{min},i}^{}={U}_{T,\mathrm{min},i}^{}+{U}_{OCV,\mathrm{min},i}^{}$ ${E}_{T,m,i}^{}={U}_{T,m,i}^{}+{U}_{OCV,m,i}^{}$ The forgetting factor $\lambda $ = 0.95 |

Parameter | Cell #1 | Cell #2 | Cell #3 | Cell #4 | Cell #5 | CMM |
---|---|---|---|---|---|---|

${R}_{O}$ (mΩ) | 1.3626 | 1.3652 | 1.3679 | 1.3706 | 1.3701 | 1.3676 |

${R}_{P}$ (mΩ) | 0.6017 | |||||

${C}_{P}$ (10^{4} F) | 1.1969 |

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**MDPI and ACS Style**

Cao, R.; Liu, X.; Zhang, Z.; Wang, M.; Cheng, H.; Gao, X.; Yan, X.; Yang, S.
Implementation of Equilibrium Strategy Aiming at Throughput Maximization of Series Battery Pack. *World Electr. Veh. J.* **2021**, *12*, 208.
https://doi.org/10.3390/wevj12040208

**AMA Style**

Cao R, Liu X, Zhang Z, Wang M, Cheng H, Gao X, Yan X, Yang S.
Implementation of Equilibrium Strategy Aiming at Throughput Maximization of Series Battery Pack. *World Electric Vehicle Journal*. 2021; 12(4):208.
https://doi.org/10.3390/wevj12040208

**Chicago/Turabian Style**

Cao, Rui, Xinhua Liu, Zhengjie Zhang, Mingyue Wang, Hanchao Cheng, Xinlei Gao, Xiaoyu Yan, and Shichun Yang.
2021. "Implementation of Equilibrium Strategy Aiming at Throughput Maximization of Series Battery Pack" *World Electric Vehicle Journal* 12, no. 4: 208.
https://doi.org/10.3390/wevj12040208