# Battery Energy Management of Autonomous Electric Vehicles Using Computationally Inexpensive Model Predictive Control

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## Abstract

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## 1. Introduction

#### 1.1. Literature Review

#### 1.2. Research Contribution

#### 1.3. Paper Organization

## 2. Vehicle and Battery Dynamics

#### 2.1. Vehicle Longitudinal Dynamics

#### 2.2. Battery Dynamics

#### 2.2.1. Continuous-Time SOC Dynamics

#### 2.2.2. Motor Dynamics

#### 2.2.3. Discrete-Time Battery Dynamics

#### 2.3. Discrete-Time Model for Implementation

## 3. Optimal Control Problem Formulation

## 4. Practical Considerations for Real-Time Implementation of MPC

#### 4.1. Dynamic Programming

#### 4.2. Model Predictive Control

#### 4.2.1. Quadratic Cost Simplification

#### 4.2.2. Real-Time Computational Feasibility

## 5. Strategies for Computational Load Reduction in MPC

#### 5.1. Sampling-Time Adjustment

#### 5.2. Warmstarting

#### 5.3. Move Blocking

## 6. Simulation Results

**Simulation 1: WLTC driving cycle.**The results in urban-driving scenario obtained from the proposed method and DP are compared in Figure 10. Since the state and control trajectories using the nominal MPC are very similar to those of our approach, we omit the plots with the nominal MPC.

**Simulation 2: US06 drive cycle.**The US06 drive cycle represents the case of a highway-driving scenario. The results are shown in Figure 11.

**Discussion.**Table 4 and Figure 12 support the above claims. As expected, the speed modification that minimizes ${T}_{\mathrm{m}}^{2}$ is effective in terms of battery SOC reduction. As expected, the proposed MPC with move-blocking method does not outperform the nominal MPC but is very close to that of nominal MPC in terms of battery SOC reduction performance (see Table 4). Unfortunately, the control performances of our approach and nominal MPC shows the significant difference by comparing with DP result, but DP is not implementable in the real-time and trajectories from the DP do not consider the physical limitation of the actuator.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Increased levels of automation and connectivity through vehicle-to-everything (V2X) communication.

**Figure 3.**Variation of the open-circuit voltage ${V}_{\mathrm{oc}}$ and internal resistance ${R}_{\mathrm{b}}$ as functions of state-of-charge (SOC).

**Figure 5.**Comparison of the speed trajectories obtained by dynamic programming (DP) and human-driver maneuvers with (

**a**) WLTC and (

**b**) US06.

**Figure 6.**Comparison of the simulation results using the different cost functions with $N=10$ and ${T}_{\mathrm{s}}=1$ s. The top plot shows the battery SOC trajectories, the middle plot shows the control input evolutions, and the bottom plot shows the computation times ${t}_{\mathrm{c},k}$ at each step $k.$

**Figure 7.**(

**a**) Battery SOC reductions with respect to N, and (

**b**) computational burdens with respect to N. The computational-time constraint violation represents the ratio between the number of occurrences where ${t}_{\mathrm{c}}>{T}_{\mathrm{s}}$ and the number of samples of the entire experiment.

**Figure 8.**Distance difference between ego car and leader car with (

**a**) ${T}_{\mathrm{s}}=1$ s, (

**b**) ${T}_{\mathrm{s}}=2$ s, and (

**c**) ${T}_{\mathrm{s}}=3$ s.

**Figure 9.**Control sequence using move-blocking strategy, where ${k}_{\mathrm{b}}=3$ and $N=10.$ The first three moves are free, then blocked in consecutive blocks of three controls. Note that ${N}_{\mathrm{b}}=6,$ ${k}_{\mathrm{r}}=1,$ and the last control is within a separate block from $k=9$ to $k=10.$

**Figure 10.**Comparison between model prediction control (MPC) and DP for the WLTC driving cycle, (

**a**) distance difference between ego car and leader car, (

**b**) speed profiles, (

**c**) battery SOC trajectories, and (

**d**) control trajectories.

**Figure 11.**Comparison between MPC and DP in US06 driving cycle, (

**a**) distance difference between ego-car and leader-car, (

**b**) speed profiles, (

**c**) battery SOC trajectories, and (

**d**) control trajectories.

**Figure 12.**Comparison of the computational-time constraint violation rates obtained by nominal MPC and proposed method.

**Table 1.**Comparison of the battery state-of-charge (SOC) consumption (%) with the speed profiles obtained by dynamic programming (DP) and nominal speed.

Drive Cycle | Baseline | DP | Improvement (%) |
---|---|---|---|

WLTC | 17.55 | 14.96 | 14.76 |

US06 | 13.41 | 10.74 | 19.90 |

Without Warmstarting | With Warmstarting | |
---|---|---|

$N=15$ | 2.5336 s | 2.5210 s |

$N=20$ | 2.5646 s | 2.5512 s |

Symbol | Description | Value (Unit) |
---|---|---|

m | Vehicle total mass | 1445 (kg) |

r | Wheel radius | 0.3166 (m) |

${A}_{\mathrm{f}}$ | Vehicle frontal area | 2.06 (m${}^{2}$) |

${C}_{\mathrm{d}}$ | Aerodynamic drag coefficient | 0.312 |

$\rho $ | Air density | 1.2 (kg/${\mathrm{m}}^{3}$) |

$\theta $ | road inclination | 0 (${}^{\circ}$) |

$\gamma $ | Rolling resistance coefficient | 0.0086 |

${i}_{0}$ | Final drive ratio | 4.2 |

${v}^{\mathrm{min}},{v}^{\mathrm{max}}$ | Acceptable range of speed | (0, 150} (km/h) |

${C}_{\mathrm{b}}$ | Battery capacity | 55 (Ah) |

${\eta}_{\mathrm{b}}^{+}$ | Battery-depletion efficiency | 0.9 |

${\eta}_{\mathrm{b}}^{-}$ | Battery-recharge efficiency | 1.11 |

${\tau}_{\mathrm{min}},{\tau}_{\mathrm{max}}$ | max/min time headway | $\{1,2\}$ (s) |

N | Prediction horizon | 10 |

${T}_{\mathrm{s}}$ | Sampling time | 1 (s) |

${k}_{\mathrm{b}}$ | Control moves blocked | 3 |

**Table 4.**$\Delta \mathrm{SOC}$ obtained by different optimized speed trajectories (the values in parentheses describe the improvements compared to the baseline).

WLTC | US06 | |
---|---|---|

Baseline | 17.55 | 13.41 |

Proposed | 15.64 (10.88%) | 11.42 (14.83%) |

Nominal MPC | 15.42 (12.14%) | 11.30 (15.73%) |

DP | 14.96 (14.76%) | 10.74 (19.90%) |

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**MDPI and ACS Style**

Han, K.; Nguyen, T.W.; Nam, K.
Battery Energy Management of Autonomous Electric Vehicles Using Computationally Inexpensive Model Predictive Control. *Electronics* **2020**, *9*, 1277.
https://doi.org/10.3390/electronics9081277

**AMA Style**

Han K, Nguyen TW, Nam K.
Battery Energy Management of Autonomous Electric Vehicles Using Computationally Inexpensive Model Predictive Control. *Electronics*. 2020; 9(8):1277.
https://doi.org/10.3390/electronics9081277

**Chicago/Turabian Style**

Han, Kyoungseok, Tam W. Nguyen, and Kanghyun Nam.
2020. "Battery Energy Management of Autonomous Electric Vehicles Using Computationally Inexpensive Model Predictive Control" *Electronics* 9, no. 8: 1277.
https://doi.org/10.3390/electronics9081277