# Electro-Thermal and Aging Lithium-Ion Cell Modelling with Application to Optimal Battery Charging

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## Abstract

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## 1. Introduction

- the design of a charging strategy that permits fast charging and a minimisation of aging resulting from battery temperature increase and large current magnitude,
- the robustness of the strategy with respect to large dynamic behaviour variations.

## 2. Cell Modelling

- an electrochemical part,
- a thermal part,
- an aging part.

#### 2.1. Description of the Electrochemical Part of the Model

_{s}is the diffusion coefficient. Resolution of the differential Equation (1) leads to the following transfer function linking the lithium concentration at the surface of the particle denoted ${C}_{s}\left(r={R}_{s}\right)$ to ${J}_{mean}^{Li}$:

_{surf}, to the cell Open Circuit Potential (OCP), denoted ${U}_{\mathrm{n}}$ and ${U}_{\mathrm{p}}$ respectively for the anode and the cathode. If the charge transfer resistance is denoted R

_{ct}(R

_{ct}can be deduced from the Butler–Volmer equation linearization), the resulting electrochemical sub-model is represented by Figure 1.

#### 2.2. Description of the Thermal Part of the Model

_{pol}(t) is the irreversible heat generation due to electrode polarization. ${R}_{f}I{\left(t\right)}^{2}$ models heat losses over the internal resistor of ${R}_{f}$.

#### 2.3. Description of the Aging Part of the Model

_{sei}) and resistance (R

_{sei}) can be computed by the following Relations [22]:

^{−1}). To obtain this coefficient, it is assumed that, after some cycles, the SEI thickness stabilizes.

_{n}, a coefficient K

_{η}is introduced in Equation (18), which thus becomes

#### 2.4. Whole Cell Model

## 3. Trajectory Planning

_{f}of twenty minutes is minimised under the following constraints:

- bounds on the charging current (Ich), in order to avoid exceeding the maximum charge current limits (3.5 C) for safety reasons,
- an increase in the SOC from 5% to 80% (but other ranges of SOC variation can be defined).

_{trans}denotes the Li-ion charges stored in (Ah), ΔC

_{loss}denotes the capacity loss (Ah), and t

_{f}(s) is the charging time. Sequential Quadratic Programming (SQP) is used to solve the optimal control problem defined by Relations (20) to (23) for various operating conditions.

_{sr}that creates aging was defined as the control signal to control the battery charging due to its high sensitivity to the charging current, as illustrated by Figure 4.

## 4. Cell Model Linearization

_{sr}) profiles.

#### 4.1. State-Space Model of the Cell Model

_{ch}(t). The state vector x(t) and the output vector y(t) are respectively given by

#### 4.2. Operating Points Definition

#### 4.3. Uncertain Linear Models Resulting from the Nonlinear Battery Model

_{sr}(s)/I

_{ch}(s) are shown in Figure 5. Among this set, one was chosen as a nominal model, with the other linear model thus defining an uncertain linear model. Figure 5 highlights a strong dependence of the frequency response of the transfer function J

_{sr}(s)/I

_{ch}(s) to the operating point. The resulting gain and phase uncertainty are taken into account in the next section for the design of a fast charging robust controller.

## 5. Design of the Fast Charging Robust Controller

#### 5.1. Closed-Loop Control

_{sr}

_{,traj}and input I

_{Ch}

_{,}

_{traj}reference signals (optimal trajectories) are known and provided by the optimal MCC trajectory planning.

_{Charge}that makes it possible to track the optimal J

_{sr}

_{,}

_{traj}is equal to the feedforward current I

_{Ch}

_{,}

_{FF}. The feedback controller thus has no effect. Conversely, if the battery behaviour differs from the behaviour of the model used for the optimisation of the charging profiles, the feedback loop adapts the charging current. It imposes the computed optimal side reaction current on the cell. As only cell temperature T, charging current I

_{Charge}and voltage U

_{cell}, are measureable signals, an aging observer using the cell model was designed to estimate the side reaction current.

#### 5.2. CRONE Control Methodology for Robust Controller Design

_{nom}(s) parameters (β

_{nom}(s) being computed for the nominal plant denoted G

_{nom}) defined by

_{l}(s) is an integer proportional integrator of order n

_{l}defined by

_{m}(s) groups several band-limited generalized templates

_{h}(s) is a low-pass filter of order n

_{h}:

_{nom}(s) = β

_{nom}(s)/(1 + β

_{nom}(s)). Order ${n}_{h}$ was chosen to ensure a bi-proper or strictly proper controller. The open-loop parameters are tuned M

_{T}, where M

_{T}denotes the resonance peak of the complementary sensitivity function T(s). Such an objective is attained by minimizing the objective function

_{Tnom}denotes a whised value for the closed-loop resonance peak computed with the nominal plant denoted G

_{nom}. The minimisation of Relation (32) is achieved under the constraints on the four closed loop sensitivity functions that follows:

^{+}= N

^{−}= 0 in Relation (29), only the independent four parameters ω

_{0}, ω

_{1}, ω

_{r}and Y

_{r}=|β(jω

_{r})|

_{dB}need to be optimized in Relation (27). In the Nichols chart, the tangency of the frequency response β

_{nom}(jω) to the desired M

_{Tnom}circle can be ensured by computing the other parameters of Relation (27), such as a

_{0}and b

_{0}. The optimal open loop transfer function being found, the controller C(s) can be obtained with the ratio of the optimal open-loop frequency responses by the nominal plant defined by

_{F}(s). Such a method has the real advantage of producing a controller whose order is rather low (classically less than 6) whatever the problem.

#### 5.3. Design of a CRONE Controller for Fast Charging

- a sensitivity function S(s) resonance peak lower than 6dB to reach a good stability degree;
- a nominal resonance peak ${M}_{Tnom}$ of function T(s) equal to 1.7 dB for a small overshoot of the nominal response to a step of the reference signal of ${J}_{sr}$;
- a closed loop bandwidth close to 0.2 rad/s;
- a control effort sensitivity less than 10 A (${I}_{ch}$) for a variation of ${J}_{sr}$ of 10 μA in high-frequency.

#### 5.4. Analysis of the Control Loop Performance

_{sr}diagram in the figure shows a tracking error close to 2 (μA·m

^{−3}). In spite of plant uncertainties, the controller tracking performance is thus guaranteed.

#### 5.5. Improvement of the Control Strategy

_{Charge}and side reaction current J

_{sr},

_{traj}optimal trajectories against SOC.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Description of the implementation of the fast charging functions in a battery management system (BMS).

**Figure 3.**An example of a Multi-stage Constant Current (MCC) profile with 6 slots ($N=6$). Each value I

_{ch}(k) has to be optimised to find the current optimal profile.

**Figure 4.**Trajectories computed for a fixed charging time of 20 min, at T = 35 °C, for a cell at the Beginning of Life (BOL) in a State of Charge (SOC) range of 5–80%.

**Figure 5.**Tansfer function J

_{sr}(s)/I

_{ch}(s) frequency responses for the considered operating points.

**Figure 7.**Nichols chart of the open loop for the uncertainty domains (

**green**) associated with the nominal plant (

**blue**) and Nichols abacus (

**red**).

**Figure 8.**Gain diagram of the four sensitivity functions (solid line, green for the nominal behaviour, red and blue for two extreme behaviours of the plant) and user defined constraints (dotted lines).

Symbol | Parameter | Unit |
---|---|---|

$m$ | Mass of the cell | kg |

${C}_{p}$ | Specific heat capacity | J·kg^{−1}·K^{−1} |

${U}_{pol}$ | Polarization voltage | V |

${R}_{f}$ | High frequency resistance | Ω |

I | Input current | A |

$\alpha $ | Heat transfer coefficient | W·m^{−2}·K |

$A$ | Cell surface area | m^{2} |

${T}_{amb}$ | Ambient temperature | K |

Symbol | Parameter | Unit |
---|---|---|

${a}_{s}^{n}$ | Specific surface area | m^{−1} |

${j}_{0,sei}$ | Exchange current density | A·m^{−2} |

${\alpha}_{n}$ | Symmetry factor | - |

n | Number of transferred electrons | - |

${E}_{a}$ | Activation Energy | kJ·mol^{−1} |

${\delta}_{sei}$ | SEI layer thickness | m |

${M}_{sei}$ | Molar mass of SEI layer | kg·mol^{−1} |

$\rho $ | Density of SEI layer | kg·m^{−3} |

${\kappa}_{sei}$ | SEI layer conductivity | S·m^{−1} |

${R}_{sei,init}$ | Initial resistance of SEI layer | Ω·m^{2} |

${R}_{p}$ | Resistance of side reaction product | Ω·m^{2} |

${I}_{ch}$ | Charging current | A |

**Table 3.**Variables in the state and output vectors of the nonlinear cell model * I = p (cathode), n (anode).

Symbol | Parameter | Unit |
---|---|---|

$\mathsf{\Delta}{C}_{loss}$ | Capacity loss | mAh |

${\delta}_{SEI}$ | Thickness of SEI layer | m |

${C}_{avg,{i}^{*}}$ | Electrode average concentration | Ah |

${C}_{1}^{part,i{\text{}}^{*}}$ | Electrode partial concentration | Ah |

${U}_{bat}$ | Battery terminal voltage | V |

${\varphi}_{en}$ | Anode potential | V |

$OCV$ | Open Circuit Voltage | V |

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**MDPI and ACS Style**

Mohajer, S.; Sabatier, J.; Lanusse, P.; Cois, O.
Electro-Thermal and Aging Lithium-Ion Cell Modelling with Application to Optimal Battery Charging. *Appl. Sci.* **2020**, *10*, 4038.
https://doi.org/10.3390/app10114038

**AMA Style**

Mohajer S, Sabatier J, Lanusse P, Cois O.
Electro-Thermal and Aging Lithium-Ion Cell Modelling with Application to Optimal Battery Charging. *Applied Sciences*. 2020; 10(11):4038.
https://doi.org/10.3390/app10114038

**Chicago/Turabian Style**

Mohajer, Sara, Jocelyn Sabatier, Patrick Lanusse, and Olivier Cois.
2020. "Electro-Thermal and Aging Lithium-Ion Cell Modelling with Application to Optimal Battery Charging" *Applied Sciences* 10, no. 11: 4038.
https://doi.org/10.3390/app10114038