# Determination of Fast Battery-Charging Profiles Using an Electrochemical Model and a Direct Optimal Control Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Battery Fast-Charging Problem

**x**= ${[{z}_{1},{z}_{2},{z}_{3}]}^{T}$, ${\mathbf{C}}_{1}$ is the row vector defined by the first row of the matrix in Equation (9), while the state ${z}_{3}={c}_{s}^{-}$ represents the bulk concentration.

## 4. Optimal Control Problem Formulation

**x**$\left(t\right)$ for the lithium-ion battery cell under the new linear model is:

- ${z}_{1}$: state 1 (mol/m${}^{3}$)
- ${z}_{2}$: state 2 (mol/m${}^{3}$)
- ${z}_{3}$: bulk concentration (mol/m${}^{3}$)

#### 4.1. Case 1: Baseline

**x**$\left(t\right)$ that maximises the bulk concentration by minimising the objective functional, J:

#### 4.2. Case 2: Balanced

**x**$\left(t\right)$, that optimises the objective functional:

#### 4.3. Case 3: Parameterisation of the Objective Functional

**x**$\left(t\right)$, that optimise the objective functional:

## 5. Results and Discussion

#### 5.1. Case 1: Baseline Scenario

#### 5.2. Case 2: Balanced Scenario

#### 5.3. Case 3: Parameterisation of the Objective Functional

#### 5.3.1. Case 3.1: $\gamma =1$

#### 5.3.2. Case 3.2: $\gamma =0$, $0\le \beta \le 1.0$ and $\kappa \in \{1,5,10\}$

#### 5.3.3. Case 3.3: $\gamma $ = 0.8, $0\le \beta \le 0.9$, $\kappa $ = 5

#### 5.4. Analysis of Energy Dissipation in the Parameterised Case

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Solution for case 1: baseline scenario. (

**a**) Optimal current profile, u. (

**b**) State of charge ${S}_{oC}$. (

**c**) Bulk concentration ${c}_{s}^{-}$. (

**d**) Surface concentration ${C}_{ss}^{-}$.

**Figure 2.**Solution for case 2: balanced scenario, with the upper bound on the final time ${T}_{f}$ = 450 s. (

**a**) Optimal current profile. (

**b**) State of charge. (

**c**) Bulk concentration ${c}_{s}^{-}$. (

**d**) Surface concentration ${C}_{ss}^{-}.$

**Figure 3.**Solution for case 2: balanced scenario, with the upper bound on the final time ${T}_{f}$ = 550 s. (

**a**) Optimal current profile. (

**b**) State of charge. (

**c**) Bulk concentration ${c}_{s}^{-}$. (

**d**) Surface concentration ${C}_{ss}^{-}.$

**Figure 4.**Solution for case 3.1: $\gamma $ = 1. (

**a**) Optimal current profile. (

**b**) State of charge. (

**c**) Bulk concentration ${c}_{s}^{-}$. (

**d**) Surface concentration ${C}_{ss}^{-}.$

**Figure 5.**Different solutions for case 3.2 ($\gamma $ = 0). Charge current profiles for $\beta $ values without the contribution of the final cost. (

**a**) Input current profiles for different values of $\beta $; (

**b**) corresponding trajectories of the state of charge for different values of $\beta $.

**Figure 6.**Solutions for case: 3.2, showing different state of charge for different values of $\beta $ with an additional penalty factor applied to the heat losses. (

**a**) Penalty factor $\kappa $ = 5. (

**b**) Penalty factor $\kappa $ = 10.

**Figure 7.**Solution for case 3.3: State of charge and optimal control trajectory considering the minimisation of the end time, the maximisation of the bulk concentration, and a penalty on the internal losses, for different values of $\beta $. (

**a**) Input current profile. (

**b**) State of charge profile.

**Figure 8.**Comparison of energy dissipation for different contributions $\kappa $ in the integrand. The baseline case is shown in orange.

**Table 1.**Summary of the different formulations of the optimal control problems. Note that the initial time ${t}_{0}=0$.

Case | Description | Path Constraints | End Cost | Integrand | Final Time |
---|---|---|---|---|---|

1 | Baseline | ${\mathbf{C}}_{1}\mathbf{x}\le {C}_{b}$ | 0 | $-{z}_{3}$ | free |

2 | Balanced | ${\mathbf{C}}_{1}\mathbf{x}\le {C}_{b}$ | 0 | $-{z}_{3}+{u}^{2}{R}_{c}$ | free |

3 | Parameterised | ${\mathbf{C}}_{1}\mathbf{x}\le {C}_{b}$ | $\gamma {t}_{f}$ | $(1-\gamma )(-\beta {z}_{3}+\kappa (1-\beta )\left({u}^{2}{R}_{c}\right))$ | free |

**Table 2.**Different objective functionals based on the values of the parameters $\beta $, $\gamma $ and $\kappa $ for case 3.

Case | $\mathit{\gamma}$ | $\mathit{\beta}$ | $\mathit{\kappa}$ | End Cost | Integrand |
---|---|---|---|---|---|

3.1 | 1 | n/a | n/a | ${t}_{f}$ | 0 |

3.2 | 0 | $0.1\le \beta \le 0.9$ | $\{1,5,10\}$ | 0 | $-\beta {z}_{3}+\kappa (1-\beta ){u}^{2}{R}_{c}$ |

3.3 | $\gamma =0.8$ | $0.1\le \beta \le 0.9$ | $\left\{5\right\}$ | $\gamma {t}_{f}$ | $(1-\gamma )(-\beta {z}_{3}+\kappa (1-\beta )\left({u}^{2}{R}_{c}\right))$ |

**Table 3.**Comparison of the results from [3] with the results obtained in this work, in particular for case 1. The switching time refers to the point in time at which the surface concentration reaches its upper limit.

Parameter | Results from [3] | This Work | $\mathbf{\Delta}$ |
---|---|---|---|

Final Time, ${t}_{f}$ (s) | 450 (fixed) | 450 | 0% |

Switching Time (s) | 317 | 311 | 1.89% |

${S}_{oC}$ | 0.5 | 0.5 | 0% |

${c}_{s}^{-}\left({t}_{f}\right)$ (mol/m${}^{3}$) | 15,000 | 15,000 | 0% |

$u\left({t}_{f}\right)$ | 0C | −0.012C | 1.2% ^{1} |

^{1}Based on the absolute error.

Final Time, ${\mathit{t}}_{\mathit{f}}$ (s) | Cost | Max Relative Local Error |
---|---|---|

450 | 3.44 × ${10}^{6}$ | 8.38 × ${10}^{-6}$ |

550 | 6.43 × ${10}^{6}$ | 1.73 × ${10}^{-5}$ |

**Table 5.**Estimation of the energy loss in KWh during battery charging for different contributions of lithium ($\beta $) and charging current ($\kappa $) in the objective functional.

$\mathit{\kappa}$ | $\mathit{\beta}$ = 0.3 | $\mathit{\beta}$ = 0.6 | $\mathit{\beta}$ = 0.8 | $\mathit{\beta}$ = 0.9 | $\mathit{\beta}$ = 1.0 |
---|---|---|---|---|---|

1 | 0.2131 | 0.2444 | 0.2451 | 0.2456 | 0.2473 |

2 | 0.1852 | 0.2358 | 0.2451 | 0.2455 | 0.2473 |

3 | 0.1756 | 0.2209 | 0.2436 | 0.2456 | 0.2473 |

4 | 0.1721 | 0.2066 | 0.2409 | 0.2455 | 0.2473 |

5 | 0.1705 | 0.1964 | 0.2373 | 0.2451 | 0.2473 |

6 | 0.1697 | 0.1896 | 0.2328 | 0.2444 | 0.2473 |

7 | 0.1692 | 0.1852 | 0.2237 | 0.2434 | 0.2473 |

8 | 0.1689 | 0.1812 | 0.2067 | 0.2423 | 0.2473 |

9 | 0.1686 | 0.1784 | 0.1898 | 0.2409 | 0.2473 |

10 | 0.1684 | 0.1764 | 0.1729 | 0.2394 | 0.2473 |

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**MDPI and ACS Style**

Gonzalez-Saenz, J.; Becerra, V.
Determination of Fast Battery-Charging Profiles Using an Electrochemical Model and a Direct Optimal Control Approach. *Batteries* **2024**, *10*, 2.
https://doi.org/10.3390/batteries10010002

**AMA Style**

Gonzalez-Saenz J, Becerra V.
Determination of Fast Battery-Charging Profiles Using an Electrochemical Model and a Direct Optimal Control Approach. *Batteries*. 2024; 10(1):2.
https://doi.org/10.3390/batteries10010002

**Chicago/Turabian Style**

Gonzalez-Saenz, Julio, and Victor Becerra.
2024. "Determination of Fast Battery-Charging Profiles Using an Electrochemical Model and a Direct Optimal Control Approach" *Batteries* 10, no. 1: 2.
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