# Control-Oriented Electrochemical Model and Parameter Estimation for an All-Copper Redox Flow Battery

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## Abstract

**:**

## 1. Introduction

## 2. CuRFB Dynamical Model

#### 2.1. Copper Concentration Dynamics

#### 2.2. Cell Voltage

#### 2.3. State of Charge

#### 2.4. Short-Term State of Health

#### 2.5. Long-Term State of Health

## 3. Materials and Methods

## 4. Genetic Algorithm

- $mu{t}_{rate}$: Rate of mutations.
- $mu{t}_{int}$: Mutation intensity.
- $ma{x}_{gen}$: Maximum allowed number of generations.
- $paramete{r}_{max}$: Maximum allowed value for a given parameter.
- $paramete{r}_{min}$: Minimum allowed value for a given parameter.
- ${\mathrm{\Delta}}_{par}$: ${\mathrm{\Delta}}_{par}=paramete{r}_{maximum}-paramete{r}_{minimum}$

## 5. Experimental Verification

#### 5.1. Single Cell

#### 5.2. Diffusion Cell

#### 5.3. Diffusion Cell, Long Trajectory

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Ruetschi, P. Energy storage and the environment: The role of battery technology. J. Power Sources
**1993**, 42, 1–7. [Google Scholar] [CrossRef] - Ren, M.; Mitchell, C.R.; Mo, W. Managing residential solar photovoltaic-battery systems for grid and life cycle economic and environmental co-benefits under time-of-use rate design. Resour. Conserv. Recycl.
**2021**, 169, 105527. [Google Scholar] [CrossRef] - Ahmed, A.; Ge, T.; Peng, J.; Yan, W.C.; Tee, B.T.; You, S. Assessment of the renewable energy generation towards net-zero energy buildings: A review. Energy Build.
**2022**, 256, 111755. [Google Scholar] [CrossRef] - Vazquez, S.; Lukic, S.M.; Galvan, E.; Franquelo, L.G.; Carrasco, J.M. Energy storage systems for transport and grid applications. IEEE Trans. Ind. Electron.
**2010**, 57, 3881–3895. [Google Scholar] [CrossRef] - Zhang, D.; Liu, Q.; Li, Y. Design of flow battery. In Reactor and Process Design in Sustainable Energy Technology; Elsevier: Amsterdam, The Netherlands, 2014; pp. 61–97. [Google Scholar]
- Zhang, H.; Li, X.; Zhang, J. Redox Flow Batteries: Fundamentals and Applications; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- Ye, R.; Henkensmeier, D.; Yoon, S.J.; Huang, Z.; Kim, D.K.; Chang, Z.; Kim, S.; Chen, R. Redox flow batteries for energy storage: A technology review. J. Electrochem. Energy Convers. Storage
**2018**, 15, 010801. [Google Scholar] [CrossRef] - Aluko, A.; Knight, A. A Review on Vanadium Redox Flow Battery Storage Systems for Large-Scale Power Systems Application. IEEE Access
**2023**, 11, 13773–13793. [Google Scholar] [CrossRef] - Sanz, L.; Badenhorst, W.D.; Lacarbonara, G.; Faggiano, L.; Lloyd, D.; Kauranen, P.; Arbizzani, C.; Murtomäki, L. All-copper Flow Batteries. In Flow Batteries; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2023; Chapter 38; pp. 855–873. [Google Scholar] [CrossRef]
- Faggiano, L.; Lacarbonara, G.; Badenhorst, W.; Murtomäki, L.; Sanz, L.; Arbizzani, C. Short thermal treatment of carbon felts for copper-based redox flow batteries. J. Power Sources
**2022**, 520, 230846. [Google Scholar] [CrossRef] - Sanz, L.; Lloyd, D.; Magdalena, E.; Palma, J.; Kontturi, K. Description and performance of a novel aqueous all-copper redox flow battery. J. Power Sources
**2014**, 268, 121–128. [Google Scholar] [CrossRef] - Zanzola, E.; Gentil, S.; Gschwend, G.; Reynard, D.; Smirnov, E.; Dennison, C.; Girault, H.H.; Peljo, P. Solid electrochemical energy storage for aqueous redox flow batteries: The case of copper hexacyanoferrate. Electrochim. Acta
**2019**, 321, 134704. [Google Scholar] [CrossRef] - Badenhorst, W.D.; Sanz, L.; Arbizzani, C.; Murtomäki, L. Performance improvements for the all-copper redox flow battery: Membranes, electrodes, and electrolytes. Energy Rep.
**2022**, 8, 8690–8700. [Google Scholar] [CrossRef] - Lloyd, D.; Magdalena, E.; Sanz, L.; Murtomäki, L.; Kontturi, K. Preparation of a cost-effective, scalable and energy efficient all-copper redox flow battery. J. Power Sources
**2015**, 292, 87–94. [Google Scholar] [CrossRef] - Cruz-Zabalegui, A.; Antaño, R.; Rivera, F. Experimental Evaluation of Copper Redox Couples in Aqueous and Aprotic Electrolytes, for Their Application in A Flow Battery. Electrochim. Acta
**2023**, 448, 142189. [Google Scholar] [CrossRef] - Badrinarayanan, R.; Zhao, J.; Tseng, K.; Skyllas-Kazacos, M. Extended dynamic model for ion diffusion in all-vanadium redox flow battery including the effects of temperature and bulk electrolyte transfer. J. Power Sources
**2014**, 270, 576–586. [Google Scholar] [CrossRef] - Clemente, A.; Montiel, M.; Barreras, F.; Lozano, A.; Costa-Castello, R. Vanadium redox flow battery state of charge estimation using a concentration model and a sliding mode observer. IEEE Access
**2021**, 9, 72368–72376. [Google Scholar] [CrossRef] - Liu, S.; Jiang, J.; Shi, W.; Ma, Z.; Guo, H. Butler–volmer-equation-based electrical model for high-power lithium titanate batteries used in electric vehicles. IEEE Trans. Ind. Electron.
**2015**, 62, 7557–7568. [Google Scholar] [CrossRef] - Bosworth, J.; Foo, N.Y.; Zeigler, B.P. Comparison of Genetic Algorithms with Conjugate Gradient Methods; Technical Report; NASA: Washington, DC, USA, 1972. [Google Scholar]
- Golub, M. An Implementation of Binary and Floating Point Chromosome Representation in Genetic Algorithm. 1996. Available online: http://www.zemris.fer.hr/~golub/clanci/iti96.pdf (accessed on 17 March 2023).
- Janikow, C.Z.; Michalewicz, Z. An Experimental Comparison of Binary and Floating Point Representations in Genetic Algorithms. In Proceedings of the International Conference on Genetic Algorithms, San Diego, CA, USA, 13–16 July 1991. [Google Scholar]
- Kirkpatrick, S.; Gelatt, C.D., Jr.; Vecchi, M.P. Optimization by simulated annealing. Science
**1983**, 220, 671–680. [Google Scholar] [CrossRef] [PubMed] - CuBER Parameter Estimation in Rust. Available online: https://github.com/uffejak/cuber_rust_ga (accessed on 12 March 2023).
- Kuldeep; Manzanares, J.A.; Kauranen, P.; Mousavihashemi, S.; Murtomäki, L. Determination of Ionic Diffusion Coefficients in Ion-Exchange Membranes: Strong Electrolytes and Sulfates with Dissociation Equilibria. ChemElectroChem
**2022**, 9, e202200403. [Google Scholar] - Krzewska, S. Impedance investigation of the mechanism of copper electrodeposition from acidic perchlorate electrolyte. Electrochim. Acta
**1997**, 42, 3531–3540. [Google Scholar] [CrossRef]

**Figure 1.**Experimental setups used in this study for validation of the proposed electrochemical model and parameter estimation. (

**a**) Diffusion cell (1 cm${}^{2}$). (

**b**) Single cell (25 cm${}^{2}$).

Parameter | Value (Single Cell) |
---|---|

${c}_{1a}$ | 131 mol/m${}^{3}$ |

${c}_{1c}$ | 125 mol/m${}^{3}$ |

${R}_{stack}$ | $1.63$$\mathrm{\Omega}$ |

${k}_{+}$ | $8.3\xb7{10}^{-1}$${\mathrm{ms}}^{-1}$ |

${k}_{-}$ | $7.6\xb7{10}^{-5}$${\mathrm{ms}}^{-1}$ |

D | $6.3\xb7{10}^{-12}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ |

${V}_{off}$ (charge) | $-0.788$ V |

${V}_{off}$ (discharge) | $0.620\mathrm{V}$ |

Parameter | Bounds |
---|---|

${c}_{1a}$ | $[0,1200]$ mol/m${}^{3}$ |

${c}_{1c}$ | $[0,1200]$ mol/m${}^{3}$ |

${R}_{stack}$ | $[0,5.0]\phantom{\rule{4pt}{0ex}}\mathrm{\Omega}$ |

${k}_{+}$ | $[1\xb7{10}^{-4},1.0]\phantom{\rule{4pt}{0ex}}{\mathrm{ms}}^{-1}$ |

${k}_{-}$ | $[1\xb7{10}^{-4},1\xb7{10}^{-7}]\phantom{\rule{4pt}{0ex}}{\mathrm{ms}}^{-1}$ |

D | $[1\xb7{10}^{-11},1\xb7{10}^{-13}]\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ |

${V}_{off}$ (charge) | $[0,1.0]\phantom{\rule{4pt}{0ex}}\mathrm{V}$ |

${V}_{off}$ (discharge) | $[0,1.0]\phantom{\rule{4pt}{0ex}}\mathrm{V}$ |

Parameter | Value (Diffusion Cell) |
---|---|

${c}_{1a}$ | 870 mol/m${}^{3}$ |

${c}_{1c}$ | 883 mol/m${}^{3}$ |

${R}_{stack}$ | $1.4\phantom{\rule{4pt}{0ex}}\mathrm{\Omega}$ |

${k}_{+}$ | $6.7\xb7{10}^{-1}\phantom{\rule{4pt}{0ex}}{\mathrm{ms}}^{-1}$ |

${k}_{-}$ | $7.3\xb7{10}^{-5}\phantom{\rule{4pt}{0ex}}{\mathrm{ms}}^{-1}$ |

D | $3.1\xb7{10}^{-12}\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ |

${V}_{off}$ (charge) | $0.032\phantom{\rule{4pt}{0ex}}\mathrm{V}$ |

${V}_{off}$ (discharge) | $-0.191\phantom{\rule{4pt}{0ex}}\mathrm{V}$ |

Parameter | Value (Aged Diffusion Cell) |
---|---|

${c}_{1a}$ | 919 mol/m${}^{3}$ |

${c}_{1c}$ | 807 mol/m${}^{3}$ |

${R}_{stack}$ | $1.47\phantom{\rule{4pt}{0ex}}\mathrm{\Omega}$ |

${k}_{+}$ | $3.9\xb7{10}^{-1}\phantom{\rule{4pt}{0ex}}{\mathrm{ms}}^{-1}$ |

${k}_{-}$ | $4.7\xb7{10}^{-5}\phantom{\rule{4pt}{0ex}}{\mathrm{ms}}^{-1}$ |

D | $7.4\xb7{10}^{-12}\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ |

${V}_{off}$ (charge) | $0.028\phantom{\rule{4pt}{0ex}}\mathrm{V}$ |

${V}_{off}$ (discharge) | $-0.162\phantom{\rule{4pt}{0ex}}\mathrm{V}$ |

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## Share and Cite

**MDPI and ACS Style**

Badenhorst, W.; Jensen, C.M.; Jakobsen, U.; Esfahani, Z.; Murtomäki, L.
Control-Oriented Electrochemical Model and Parameter Estimation for an All-Copper Redox Flow Battery. *Batteries* **2023**, *9*, 272.
https://doi.org/10.3390/batteries9050272

**AMA Style**

Badenhorst W, Jensen CM, Jakobsen U, Esfahani Z, Murtomäki L.
Control-Oriented Electrochemical Model and Parameter Estimation for an All-Copper Redox Flow Battery. *Batteries*. 2023; 9(5):272.
https://doi.org/10.3390/batteries9050272

**Chicago/Turabian Style**

Badenhorst, Wouter, Christian M. Jensen, Uffe Jakobsen, Zahra Esfahani, and Lasse Murtomäki.
2023. "Control-Oriented Electrochemical Model and Parameter Estimation for an All-Copper Redox Flow Battery" *Batteries* 9, no. 5: 272.
https://doi.org/10.3390/batteries9050272