# Sensitivity Analysis of Battery Aging for Model-Based PHEV Use Scenarios

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## Abstract

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## 1. Introduction

_{2}emissions of vehicles, the sales and developments of Plug-in Hybrid Electric Vehicles (PHEV) have increased drastically in the past years. In Europe, if emitting less than 50 g/km of CO

_{2}, a PHEV is placed in the category called Zero and Low Emission Vehicle regarding the European regulation on manufacturer’s fleet mean emission [1]. This leads to an increase in the all-electric range and therefore in the battery size. As batteries are not environmentally neutral, the environmental impacts of PHEVs over their lifetime could increase with the battery size.

## 2. Methodological Approach

#### 2.1. Battery Aging Model

- $q{l}_{cst}$: ${Q}_{L}$ is assumed to be a constant;
- $q{l}_{min}$: ${Q}_{L}$ refreshes its value every minute;
- $q{l}_{day}$: ${Q}_{L}$ refreshes its value every day.

#### 2.2. Use Cases Model Generation

#### 2.2.1. Vehicle Uses

- Concerning the number of trips per day, it is assumed that under a certain daily mileage ${D}_{min}$, it corresponds to a short trip (for example, a shopping trip), with a quick return, and can thus be considered as a single trip;
- Above a certain daily mileage ${D}_{max}$, it is also considered as a single trip, for example, to go on holidays or long professional displacement;
- Between these two values, the daily mileage is separated into two trips assuming home-to-work and vice versa travel. In this case, the first travel takes place at $tim{e}_{ft}$ and the second at $tim{e}_{st}$.

- if ${\mathrm{V}}_{\mathrm{trip}}<{\mathrm{V}}_{\mathrm{urban}}$,all the travel is supposed to be urban.
- if ${\mathrm{V}}_{\mathrm{urban}}<{\mathrm{V}}_{\mathrm{trip}}<{\mathrm{V}}_{\mathrm{extra}}$,$urba{n}_{part}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{extra}}-{\mathrm{V}}_{\mathrm{trip}}}{{\mathrm{V}}_{\mathrm{extra}}-{\mathrm{V}}_{\mathrm{urban}}}}$; $extr{a}_{part}=1-urba{n}_{part}$
- if ${\mathrm{V}}_{\mathrm{extra}}<{\mathrm{V}}_{\mathrm{trip}}<{\mathrm{V}}_{\mathrm{motorway}}$,$extr{a}_{part}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{motorway}}-{\mathrm{V}}_{\mathrm{trip}}}{{\mathrm{V}}_{\mathrm{motorway}}-{\mathrm{V}}_{\mathrm{extra}}}}$; $motorwa{y}_{part}=1-extr{a}_{part}$
- if ${\mathrm{V}}_{\mathrm{trip}}>{\mathrm{V}}_{\mathrm{motorway}}$,all the travel is supposed to be on motorway,

- One corresponds to the Worldwide harmonized Light-duty vehicles Test Cycles (WLTC) which is separated into three parts: urban, rural, and motorway, Figure 4;
- One is composed of the Artemis cycle which represents real driving conditions in urban, rural, and highway cases;
- One is composed of the Hyzem driving cycle [26] which has been specially developed to simulate and evaluate hybrid vehicles.

#### 2.2.2. Recharge Scenario

#### 2.3. Electrical and Thermal Model of the Battery

^{2}, and ${W}_{t}$ is the weight of the battery pack in Kg. As the parameters h and ${C}_{th}$ are really difficult to assess precisely, they are also considered as parameters in our sensitivity analysis.

- $mo{d}_{therm}$ = ${T}_{amb}$; in this case, it is assumed that the battery temperature is equal to the external temperature;
- $mo{d}_{therm}$ = $mo{d}_{1D-Rcst}$; we use the thermal model of the battery but the resistance does not depend on the temperature but only on the SOC of the battery;
- $mo{d}_{therm}$ = $mo{d}_{1D-cpl}$; we use the thermal model of the battery, and the resistance depends on both the temperature and SOC of the battery.

## 3. Model Integration and Sensitivity Analysis

## 4. Results and Discussion

#### 4.1. Use Cases

- $bat{t}_{size}$: 24 Ah is the existing battery pack of Golf GTE, an electric range of 80 km, depending on the driving conditions, can then be expected. Three values of battery sizes were studied, corresponding to 0.5, 1, and 1.5 times the reference value (24 Ah) to match the minimum and maximum autonomy of existing PHEV cars.
- ${Q}_{Lref}$: Two values of capacity loss refreshing rate are included in this study: minute or day.
- km: The mean value of km per year, corresponding to the average annual mileage of German drivers [21].
- ${c}_{rate}$ value of $1/4$ is a classical value corresponding to a full battery recharge in 4 h.
- $city$: Abu Dhabi and Reykjavik are chosen to represent extremely hot and cold climates, whereas Lyon represents average climatic conditions.
- $fa{m}_{cin}$: WLTC, Artemis, and Hyzem driving cycles are studied.
- $mont{h}_{type}$: Two cases are studied, one where the daily mileage is the same for each month of the year, this seems reasonable regarding [21]. A case with different driving behavior in summer, trying to consider holiday trips, was added.
- $So{C}_{sust}$: In the Golf GTE (and other PHEVs), the SOC strategies often discharge the batteries to a low SOC threshold and then operate the vehicle in charge-sustaining mode. From an energetic point of view, and aiming to transfer a maximum of fuel consumption to electricity, the minimum SOC threshold has to be as low as allowable by the battery. We nevertheless study two cases at 40 and 50% to assess their effect on battery aging.

#### 4.2. Results

#### 4.3. Discussion

- The aging refreshment, which can be fixed to one day (refreshing ${Q}_{L}$ each day) and thus reduces the computational effort;
- The place of residence, at least in the manner we modeled it, i.e., a modification of the mean speed of travel (see Section 2.2);
- The family of driving cycles does not quite have an effect as this only changes the rate of discharge of the battery and does not drastically affect the SOC profile. For battery aging, it has no effect but can be sensitive in LCA as it will change the electrical (and thus fuel) consumption;
- The parameters linked to daily mileage generation ($coe{f}_{\mu}$, $coe{f}_{\sigma}$, $mont{h}_{type}$,) and the sharing between one or two trips (${D}_{min}$, ${D}_{max}$). This tends to prove that a daily mileage statistical approach is accurate enough for battery aging consideration;
- The parameters to assess the SOC profile that depends on the time of travel ($tim{e}_{ft},tim{e}_{st}$);
- The thermal model parameters ($mo{d}_{therm}$), and thus the thermal model, do not quite have an impact. It was found in the first complete sensitivity analysis that this parameter has an impact of 0.08% which means that using either a simple model or an accurate one has no effect on the battery aging in our case. Therefore, we use a simple model to reduce the computational burden and in our case, the battery temperature can be considered to be equal to the external temperature. This can be explained by the fact that in our scenarios, the car is used for a really small part of the time (less than 5% of the time for 14,000 km annual mileage). Thus, the temperatures are relatively identical whether we consider the thermal model or not. This conclusion will not be acceptable for other uses of vehicles—public transport, vehicle sharing—where the battery usage affect its internal temperature (Joules losses);
- The battery recharge rate ${c}_{rate}$ has no effect, or second-order effect, probably because the SOC profiles are not very affected by this parameter. However, fast recharge has not been considered here and the conclusion is only valid for slow recharge;
- The time of recharge $tim{e}_{rec}$ (in case of night recharge) is also non-sensitive (or its variation is too small);
- The predictive distance parameter ($dis{t}_{prev}$) does not quite have an effect either as it does not change the global SOC profile (it acts only on a few days per year for our scenario).

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Annual Mileage | 2500 | 7500 | 12,500 | 17,500 | 25,000 | 30,000 |
---|---|---|---|---|---|---|

$\mu $ | 2.57 | 2.86 | 3.11 | 3.3 | 3.54 | 3.72 |

$\sigma $ | 1.05 | 1.05 | 1.06 | 1.03 | 1 | 1.18 |

Families | Parameters | Definition |
---|---|---|

Battery | $bat{t}_{size}$ | battery size |

Aging model | ${Q}_{Lref}$ | capacity loss refreshment |

Vehicle’s use | $km$ | number of kms per year |

$PoR$ | place of residence | |

${N}_{travelmonth}$ | number of days with trips per month | |

$tim{e}_{ft}$ | time of first trip | |

$tim{e}_{st}$ | time of second trip | |

${D}_{min}$ | max distance for short trip | |

${D}_{max}$ | min distance for long trip | |

$fa{m}_{cin}$ | family of driving cycle | |

$coe{f}_{\mu}$ | mean value of daily mileage | |

$coe{f}_{\sigma}$ | standard deviation of daily mileage | |

$So{C}_{sust}$ | SOC sustaining threshold | |

$mont{h}_{type}$ | all months identical or not | |

Recharge scenarios | ${c}_{rate}$ | charging rate |

$tim{e}_{rec}$ | time of recharge | |

$mo{d}_{rech}$ | charging models | |

$rec{h}_{soc-min}$ | minimum SOC threshold | |

$dis{t}_{pred}$ | predict distance on the next day | |

Thermal model | $mo{d}_{therm}$ | thermal model |

$city$ | city of dwelling | |

h | heat transfer coefficient | |

${C}_{th}$ | specific thermal capacity |

Components | Characteristics | Values |
---|---|---|

Vehicle | weight | 1480 kg |

aerodynamic coefficient $S\xb7{C}_{x}$ | 0.0305 N·(m/s)${}^{-2}$ | |

rolling resistance ${C}_{rr}$ | 134 N | |

Battery | Capacity | 24 Ah |

nominal voltage | 345 V | |

heat transfer coefficient | 5 W/(m${}^{2}\xb7$K) | |

specific thermal capacity | 900 J/(Kg·K) | |

technology | Lithium ion | |

Engine | Max power | 150 kW @ 5000–6000 rpm |

Max Torque | 250 N.m @ 1600–3500 rpm | |

Electrical machine | Max power | 75 kW |

Maximum torque | 250 N.m | |

Auxiliary power | mean power | 611 W |

Families | Parameters | Values | Sensitivity in% |
---|---|---|---|

Battery | $bat{t}_{size}$ | 12–24–36 Ah | 32.4 |

Aging model | ${Q}_{Lref}$ | minute–days | 0.01 |

Vehicle’s use | $km$ | 7500–14,000–35,000 km | 20 |

$PoR$ | urban–extra-urban–rural | 0.08 | |

${N}_{travelmonth}$ | 20–26–29 | 2.9 | |

$tim{e}_{ft}$ | 6–8–10 a.m. | 0.03 | |

$tim{e}_{st}$ | 3–5–7 p.m. | 0.004 | |

${D}_{min}$ | 2–5–10 km | 0.002 | |

${D}_{max}$ | 80–100–200 km | 0.8 | |

$fa{m}_{cin}$ | WLTC–Artemis–Hyzem | 0.8 | |

$coe{f}_{\mu}$ | 0.8–1–1.2 | 0.8 | |

$coe{f}_{\sigma}$ | 0.8–1–1.2 | 0.6 | |

$So{C}_{sust}$ | 30–40–50% | 1.6 | |

$mont{h}_{type}$ | ident–summer | 0.02 | |

Recharge scenarios | ${c}_{rate}$ | $1/2$–$1/4$–$1/6$ | 0.8 |

$tim{e}_{rec}$ | 9–10–11 p.m. | 0.03 | |

$mo{d}_{rech}$ | trip–night–soc_trip–soc_night | 17.2 | |

$rec{h}_{soc-min}$ | 35–50–80% | 1.12 | |

$dis{t}_{pred}$ | 0–50–100 km | 0.01 | |

Thermal model | $mo{d}_{therm}$ | ${t}_{ext}$–${R}_{cst}$–${R}_{cpl}$ | 0.08 |

$city$ | Abu Dhabi–Lyon–Reykjavik | 19.6 | |

h | 2–5–8 W/(m${}^{2}\xb7$K) | 0.04 | |

${C}_{th}$ | 500–900–1500 J/(Kg·K) | 0.01 |

Families | Parameters | Values | Sensitivity in% |
---|---|---|---|

battery | $bat{t}_{size}$ | 12–24–36 Ah | 36.3 |

Vehicle’s use | $km$ | 7500–14,000–35,000 km | 15.8 |

${N}_{travelmonth}$ | 20–26–29 | 3.9 | |

$So{C}_{sust}$ | 30–40–50% | 1.7 | |

Recharge scenarios | $mo{d}_{rech}$ | tra–night–$so{c}_{tra}$–$so{c}_{night}$ | 20.3 |

$rec{h}_{soc-min}$ | 35–50–80% | 1.4 | |

Thermal model | $city$ | Abu Dhabi–Lyon–Reykjavik | 20.6 |

${\mathit{Q}}_{\mathit{L}}$ | ${\mathit{batt}}_{\mathit{size}}$ | $\mathit{km}$ | ${\mathit{N}}_{\mathit{travelmonth}}$ | ${\mathit{SoC}}_{\mathit{sust}}$ | ${\mathit{mod}}_{\mathit{rech}}$ | ${\mathit{rech}}_{\mathit{soc}-\mathit{min}}$ | $\mathit{city}$ |
---|---|---|---|---|---|---|---|

1.16% | 36 Ah | 7500 km | 20 | 30% | soc_night | 35% | Reykjavik |

5.36% | 12Ah | 35,000 km | 29 | 30% | tra | 35% | Abu Dhabi |

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## Share and Cite

**MDPI and ACS Style**

Patil, T.-D.; Vinot, E.; Ehrenberger, S.; Trigui, R.; Redondo-Iglesias, E.
Sensitivity Analysis of Battery Aging for Model-Based PHEV Use Scenarios. *Energies* **2023**, *16*, 1749.
https://doi.org/10.3390/en16041749

**AMA Style**

Patil T-D, Vinot E, Ehrenberger S, Trigui R, Redondo-Iglesias E.
Sensitivity Analysis of Battery Aging for Model-Based PHEV Use Scenarios. *Energies*. 2023; 16(4):1749.
https://doi.org/10.3390/en16041749

**Chicago/Turabian Style**

Patil, Tejas-Dilipsing, Emmanuel Vinot, Simone Ehrenberger, Rochdi Trigui, and Eduardo Redondo-Iglesias.
2023. "Sensitivity Analysis of Battery Aging for Model-Based PHEV Use Scenarios" *Energies* 16, no. 4: 1749.
https://doi.org/10.3390/en16041749