# A Single-Degree-of-Freedom Energy Optimization Strategy for Power-Split Hybrid Electric Vehicles

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## Abstract

**:**

## 1. Introduction

^{2}and the battery’s residual energy E·SOC were chosen as the state variables, a quadratic performance index was designed to ensure the vehicle’s driving performance, sustain the battery SOC and restrain frequent and large-scale fluctuation of engine power simultaneously. The fuel economy was improved indirectly and the energy management problem was transformed into the LQR problem or the quadratic optimal tracking problem. The quadratic optimal control theory was firstly introduced by authors to deal with this kind of problem. The strategy had two control variables: the engine power and the motor power, so it was called as double-degree-of-freedom energy management strategy.

## 2. Drivetrain Architecture and Energy Management Problem Description

_{ess}. In the following discussion, only the static models of engine and motor are considered because their transient processes are relatively short and can be ignored. Firstly, a brief introduction of the battery-motor system model is given; the engine model and planetary gear model will be discussed in details in Section 4.

#### 2.1. Efficiency Model of Battery-Motor System and Its Simplification

_{OC}and the internal resistor R

_{int}are associated with the battery SOC. The battery efficiency is defined as:

#### 2.2. Energy Management Problem

## 3. Single-Degree-of-Freedom Quadratic Performance Index Strategy

#### 3.1. Extended Quadratic Optimal Control Problem and Relevant Results

#### 3.2. Derivation of Single-Degree-of-Freedom Quadratic Performance Index Strategy

#### 3.3. Analysis from the Perspective of Engineering Application

## 4. Vehicle Simulation Model

#### 4.1. Engine Model

#### 4.2. Planetary Gear Model

#### 4.3. Energy Optimization Strategy Model

## 5. Simulation Results and Comparative Analysis

#### 5.1. Test Design and the Selection of Weight Coefficient

#### 5.2. The simulation Test Results and Analysis

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Characteristics of ${V}_{OC}$ and ${R}_{int}$. (${R}_{chg}$ is the charge resistance and ${R}_{dis}$ is the discharge resistance).

**Figure 4.**Efficiency MAP of motor/generator MG1/MG2 and its controller. (

**a**) MG1 and its controller; (

**b**) MG2 and its controller.

**Figure 14.**The simulation result of single-degree-of-freedom quadratic performance index strategy (SQPIS).

**Figure 15.**The simulation result of Pontryagin’s minimum principle (PMP)-based global optimal strategy.

**Figure 16.**Engine efficiency distribution under urban dynamometer driving schedule (UDDS) driving cycle. (

**a**) rule-based EMS; (

**b**) SQPIS; and (

**c**) PMP-based strategy.

Description | Parameter | Value | Unit |
---|---|---|---|

Vehicle | Total weight | 1368 | kg |

Wheel radius | 0.287 | m | |

Frontal area | 1.746 | m^{2} | |

Aerodynamic drag coefficient | 0.3 | - | |

Rolling friction coefficient | 0.009 | - | |

Final drive ratio | 3.93 | - | |

Engine | Displacement | 1.5 | L |

Max torque | 102 @4000 rpm | Nm | |

Max power | 43 @4000 rpm | kW | |

Motor/Generator1 (MG1) and controller | Max speed | 5500 | rpm |

Max torque | 55 | Nm | |

Max power | 15 | kW | |

Motor/Generator2 (MG2) and controller | Max speed | 6000 | rpm |

Max torque | 305 | Nm | |

Max power | 31 | kW | |

Battery Package | Cell capacity | 6 | Ah |

Nominal voltage | 308 | V | |

Planetary Gear Set | Tooth number of sun gear | 30 | - |

Tooth number of ring gear | 78 | - |

Drive Cycle | Rule-Based EMS | A-ECMS | SQPIS | PMP-Based Global Optimal Strategy | ||
---|---|---|---|---|---|---|

UDDS | EFC (L/100 km) | 5.2733 | 4.0527 | 3.9984 | 3.6534 | λ = −5.0142 × 10^{−5} |

SOC(t_{f}) | 0.5505 | 0.5987 | 0.5362 | 0.5972 | ||

HWFET | EFC (L/100 km) | 4.2338 | 4.0581 | 4.0310 | 3.6445 | λ = −5.1489 × 10^{−5} |

SOC(t_{f}) | 0.6082 | 0.5956 | 0.5756 | 0.6003 | ||

CSHVR | EFC (L/100 km) | 4.7662 | 3.7175 | 3.6114 | 3.5605 | λ = −5.2145 × 10^{−5} |

SOC(t_{f}) | 0.5883 | 0.6046 | 0.5515 | 0.5980 | ||

LA92 | EFC (L/100 km) | 6.4213 | 5.0392 | 4.8891 | 4.5662 | λ = −4.7331 × 10^{−5} |

SOC(t_{f}) | 0.5934 | 0.5995 | 0.5543 | 0.5989 | ||

INDIA_URBAN | EFC (L/100 km) | 4.7333 | 3.5538 | 3.4253 | 3.2982 | λ = −5.4307 × 10^{−5} |

SOC(t_{f}) | 0.5807 | 0.6043 | 0.5398 | 0.6010 | ||

INDIA_HWY | EFC (L/100 km) | 4.3588 | 3.8504 | 3.8223 | 3.6255 | λ = −4.7461 × 10^{−5} |

SOC(t_{f}) | 0.5963 | 0.5916 | 0.5637 | 0.6007 | ||

NEDC | EFC (L/100 km) | 4.6078 | 3.9271 | 3.8528 | 3.6949 | λ = −5.6184 × 10^{−5} |

SOC(t_{f}) | 0.6202 | 0.6134 | 0.6012 | 0.6001 | ||

J1015 | EFC (L/100 km) | 4.6734 | 3.7336 | 3.6696 | 3.5542 | λ = −4.9843 × 10^{−5} |

SOC(t_{f}) | 0.6074 | 0.6075 | 0.5795 | 0.5988 |

**Table 3.**Impacts of cargo mass and road slope on fuel consumption under urban dynamometer driving schedule (UDDS) driving cycle.

Cargo Mass | Rule-Based EMS | A-ECMS | SQPIS | PMP-Based Global Optimal Strategy | ||

1368 | EFC (L/100 km) | 5.2733 | 4.0527 | 3.9984 | 3.6534 | λ = −5.0142 × 10^{−5} |

SOC(t_{f}) | 0.5505 | 0.5987 | 0.5362 | 0.5972 | ||

1568 | EFC (L/100 km) | 5.8433 | 4.5030 | 4.4498 | 3.9802 | λ = −4.8372 × 10^{−5} |

SOC(t_{f}) | 0.5498 | 0.5966 | 0.5364 | 0.5988 | ||

1768 | EFC (L/100 km) | 6.4531 | 4.9280 | 4.8785 | 4.3275 | λ = −4.7621 × 10^{−5} |

SOC(t_{f}) | 0.5462 | 0.6049 | 0.5430 | 0.5981 | ||

Road Slope (0–500 m) | Rule-Based EMS | A-ECMS | SQPIS | PMP-Based Global Optimal Strategy | ||

0% | EFC (L/100 km) | 5.2733 | 4.0527 | 3.9984 | 3.6534 | λ = −5.0142 × 10^{−5} |

SOC(t_{f}) | 0.5505 | 0.5987 | 0.5362 | 0.5972 | ||

5% | EFC (L/100 km) | 5.5030 | 4.3343 | 4.2880 | 3.8715 | λ = −4.9872 × 10^{−5} |

SOC(t_{f}) | 0.5505 | 0.5983 | 0.5362 | 0.5999 | ||

10% | EFC (L/100 km) | 5.9062 | 4.6140 | 4.5462 | 4.1133 | λ = −4.8528 × 10^{−5} |

SOC(t_{f}) | 0.5505 | 0.6067 | 0.5362 | 0.5989 |

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**MDPI and ACS Style**

Xia, C.; DU, Z.; Zhang, C. A Single-Degree-of-Freedom Energy Optimization Strategy for Power-Split Hybrid Electric Vehicles. *Energies* **2017**, *10*, 896.
https://doi.org/10.3390/en10070896

**AMA Style**

Xia C, DU Z, Zhang C. A Single-Degree-of-Freedom Energy Optimization Strategy for Power-Split Hybrid Electric Vehicles. *Energies*. 2017; 10(7):896.
https://doi.org/10.3390/en10070896

**Chicago/Turabian Style**

Xia, Chaoying, Zhiming DU, and Cong Zhang. 2017. "A Single-Degree-of-Freedom Energy Optimization Strategy for Power-Split Hybrid Electric Vehicles" *Energies* 10, no. 7: 896.
https://doi.org/10.3390/en10070896