# Computationally Efficient Energy Management in Hybrid Electric Vehicles Based on Approximate Pontryagin’s Minimum Principle

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## Abstract

**:**

## 1. Introduction

#### 1.1. Research Motivation

#### 1.2. Literature Review

#### 1.3. Main Contributions

#### 1.4. Outline

## 2. Powertrain Modelling

#### 2.1. Powertrain Configuration

#### 2.2. Modeling Processes

#### 2.2.1. Engine Model

_{e}is the engine speed, T

_{e}is the engine torque, η

_{e}is the engine efficiency, α is the engine throttle opening, T

_{emax}(n

_{e}) is the engine maximum torque at the current speed, and c, k

_{0}, k

_{1}, k

_{2}, p

_{1}, and p

_{2}are 0.002, 394.6, −198.8, −95.43, 3.788, and −72.48, respectively.

#### 2.2.2. Motor Model

_{m}is the motor speed, T

_{m}is the motor torque, η

_{m}is the motor efficiency, and P

_{b}is the required battery power (kW).

#### 2.2.3. Battery Model

_{in}is the battery resistance, V

_{oc}is the open circuit voltage, and Q

_{max}is the maximum capability.

#### 2.2.4. Transmission model

_{in}is the torque of the transmission input shaft (Nm), T

_{out}is the transmission output shaft torque (Nm), η

_{GR}is the transmission efficiency, i

_{g}(Gear) is the gear ratio at each transmission gear, Gear is the gear number, i

_{0}is the gear ratio of the final drive, w

_{in}is the angular velocity of the transmission input shaft (rad/s), and w

_{out}is the angular velocity of the transmission output shaft (rad/s).

#### 2.2.5. Vehicle Dynamics Model

_{D}is the air resistance (drag) coefficient, A is the frontal area, ${v}_{a}\left(t\right)$ is the longitudinal vehicular velocity (km/h), m is the complete vehicle curb mass, f is the rolling resistance coefficient, δ is the correction coefficient of the rotating mass, r is the wheel radius, and n

_{in}is the speed of the transmission input shaft.

## 3. The Proposed Energy Management Framework

#### 3.1. The Principle of PMP

_{e}(t) and T

_{m}(t) are the engine torque and motor torque, respectively; T

_{e}_

_{max}(n

_{e}(t)) is the engine maximum torque at the current speed, T

_{m}_

_{max}(n

_{m}(t)) is the motor maximum torque at the current speed, T

_{m}_

_{min}(n

_{m}(t)) is the motor minimum torque at the current speed, n

_{m}_

_{max}is the motor maximum speed, n

_{e}_

_{max}and n

_{e}_

_{min}are the engine maximum and minimum speed, respectively, and SOC

_{min}, SOC

_{max}are the minimum and maximum SOC, respectively.

#### 3.2. Optimization of Gearshift Command and Torque Distribution

#### 3.3. Approximate PMP

_{e}) and b(n

_{e}) are fitting coefficients related to the engine speed.

_{m}), d(n

_{m}), g(n

_{m}), f(n

_{m}) are fitting coefficients associated with the motor speed.

_{ratio}represents the motor torque divided by the total torque demand.

_{loss}is presented in Equation (32).

_{a}is the activation energy (J/mol), R

_{g}is the gas constant, T is the battery temperature in Kelvin [K], Ah is the accumulated charge throughput, i.e., the total amount of charge that can flow in and out of the battery during its operation, ${I}_{rate}$ is the current rate, z is the power law factor.

_{rate,norm}= 2.5[1/h], SOC

_{norm}= 0.35, and T

_{norm}= (273.15 + 25) K [37], and the nominal battery life is then calculated in Equation (35). In this paper, the battery temperature is assumed to be constant (25 °C).

_{a}is the ratio of battery replacement cost of 1 kg of gasoline.

## 4. Results and Discussion

#### 4.1. The DP-based Energy Management Formulation

#### 4.2. Extracting the Gearshift Map

#### 4.3. Optimization Performance Assessment

#### 4.3.1. The Performance Index

_{max}is the capacity of battery (Ah), N

_{cell}is the cell number of battery, SOC

_{0}, SOC

_{f}is the SOC initial and final value, respectively, U

_{ocv}_

_{discharge}is the cell voltage (V), E

_{diesel}is the energy content of diesel fuel (J/kg), ${\eta}_{diesel}$ is the efficiency of conversion from diesel engine output to the motor input, ${\eta}_{ICE}$ is the diesel engine efficiency, ${\eta}_{EM}$ is the motor efficiency, ρ

_{diesel}is the density of diesel (0.835 kg/L), d

_{cycle}is the distance traveled (m). $F{C}_{\Delta SoC\_comp}$ is the SOC compensated fuel consumption (L/100 km), FC is the actual fuel consumption without calculating SOC compensated fuel (kg).

#### 4.3.2. Optimization Performance analysis

#### 4.3.3. Battery Aging Performance Analysis

#### 4.4. Summary of Results

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclatures

Abbreviations | |

HEVs | hybrid electric vehicles |

PMP | Pontryagin’s Minimum Principle |

DP | dynamic programming |

EMSs | energy management strategies |

PHEV | plug-in HEV |

AMT | automated mechanical transmission |

SOC | state of charge |

CTCC | China typical city cycle |

FC | fuel consumption |

ECMS | Equivalent consumption minimization strategy |

Symbols | |

n_{e} | engine speed |

T_{e} | engine torque |

η_{e} | engine efficiency |

α | engine throttle opening |

T_{emax}(n_{e}) | engine maximum torque at the current speed |

n_{m} | motor speed |

T_{m} | motor torque |

η_{m} | motor efficiency |

P_{b} | required battery power |

R_{in} | battery resistance |

V_{oc} | open circuit voltage |

Q_{max} | maximum capability |

T_{out} | transmission output shaft torque |

T_{in} | torque of the transmission input shaft |

η_{GR} | transmission efficiency |

i_{g} | gear ratio of each transmission gear |

i_{0} | gear ratio of the final drive |

w_{in} | angular velocity of the transmission input shaft |

w_{out} | angular velocity of the transmission output shaft |

T_{dem}(t) | torque demand at the input shaft of the transmission |

C_{D} | air resistance (drag) coefficient |

A | frontal area |

v_{a}(t) | vehicular velocity |

m | complete vehicle curb mass |

f | rolling resistance coefficient |

δ | correction coefficient of the rotating mass |

r | wheel radius |

n_{in} | speed of the transmission input shaft |

λ (t) | co-state |

P_{b}(u_{g}(t)) | battery power |

u_{g}(t) | the optimal torque and gearshift command |

T_{m}__{min}(n_{m}(t)) | motor minimum torque at the current speed |

T_{m}__{max}(n_{m}(t) | motor maximum torque at the current speed |

T_{e}__{max}(n_{e}(t)) | engine maximum torque at the current speed |

n_{m}__{max} | motor maximum speed |

n_{e}__{max} | engine maximum speed |

n_{e}__{min} | engine minimum speed |

SOC_{min} | minimum SOC |

SOC_{max} | maximum SOC |

sh(t) | gearshift command |

g(t) | optimal gear number |

u(t) | engine torque and motor torque |

T_{e}__{opt} | engine torque for the flexible torque demand |

T_{m}__{opt} | motor torque for the flexible torque demand |

E_{△SOC} | energy produced by SOC deviation |

SOC_{f} | SOC final value |

SOC_{0} | SOC initial value |

N_{cell} | cell number of battery |

FC_{△SOC}__{comp} | SOC compensated fuel consumption |

U_{ocv}__{discharge} | cell voltage |

E_{diesel} | energy content of diesel fuel |

η_{diesel} | efficiency of conversion from diesel energy to electricity |

η_{ICE} | diesel engine efficiency |

η_{EM} | motor efficiency |

ρ_{diesel} | density of diesel |

d_{cycle} | distance traveled |

Q_{loss} | battery capacity loss |

α β $\eta $ | fitting coefficients related to SOC |

E_{a} | activation energy |

R_{g} | gas constant |

T | battery temperature in Kelvin |

Ah | accumulated charge throughput |

I_{rate} | current rate |

z | power law factor |

R_{ratio} | motor torque divided by the total torque demand |

I_{b}(t) | current during the trip |

K | weight factor for battery aging cost |

C_{a} | ratio of battery replacement cost of 1 kg of gasoline |

${\dot{m}}_{f}\left(u\left(k\right)\right)$ | fuel consumption at k instant |

x(k) | state variable at time instant k |

SOC(k) | battery state of charge at current instant |

g(k) | gear number at time instant k |

u(k) | control variable |

r(k) | ratio of the motor torque to the total torque demand |

d(k) | gearshift command |

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**Figure 1.**Powertrain topology of a parallel hybrid electric vehicle (HEV) [30].

**Figure 2.**Engine efficiency map [31].

**Figure 3.**Motor efficiency map [31].

**Figure 13.**Current of A-PMP including the battery aging. ($\left|\overline{{I}_{b}}\right|\left(t\right)=11.4569\left(K=100\right)$, $\left|\overline{{I}_{b}}\right|\left(t\right)=11.1624\left(K=200\right)$).

n_{e} (r/min) | a | b | R^{2} |
---|---|---|---|

800 | 0.005 | 0.0761 | 0.9851 |

900 | 0.0055 | 0.0753 | 0.9921 |

1000 | 0.0063 | −0.0053 | 0.9976 |

1100 | 0.0065 | 0.0892 | 0.9961 |

1200 | 0.0068 | 0.1229 | 0.9971 |

1300 | 0.0072 | 0.1326 | 0.9974 |

1400 | 0.0077 | 0.1691 | 0.9969 |

1500 | 0.0081 | 0.2020 | 0.9959 |

1600 | 0.0087 | 0.2261 | 0.9958 |

1700 | 0.0092 | 0.2699 | 0.9950 |

1800 | 0.0098 | 0.2177 | 0.9960 |

1900 | 0.0104 | 0.2660 | 0.9952 |

2000 | 0.0110 | 0.3217 | 0.9942 |

2100 | 0.0118 | 0.3575 | 0.9934 |

2200 | 0.0124 | 0.4377 | 0.9915 |

2300 | 0.0128 | 0.4923 | 0.9903 |

2400 | 0.0134 | 0.5848 | 0.9878 |

2500 | 0.0141 | 0.6348 | 0.9858 |

2600 | 0.0147 | 0.6877 | 0.9837 |

n_{m} (r/min) | c (×10^{−6}) | d (×10^{−6}) | R^{2} | g (×10^{−6}) | f (×10^{−6}) | R^{2} |
---|---|---|---|---|---|---|

200 | −0.202 | 6.049 | 0.989 | −0.075 | 2.727 | 0.987 |

600 | −0.620 | 19.82 | 0.988 | −0.223 | 8.319 | 0.986 |

1500 | −1.231 | 24.75 | 0.995 | −0.712 | 13.33 | 0.996 |

1800 | −1.294 | 10.41 | 0.999 | −0.968 | 5.577 | 0.999 |

2100 | −1.751 | 33.34 | 0.996 | −0.995 | 15.64 | 0.996 |

2300 | −1.911 | 31.76 | 0.997 | −1.099 | 13.42 | 0.997 |

2500 | −2.306 | 54.03 | 0.996 | −1.097 | 20.25 | 0.999 |

2600 | −2.382 | 58.75 | 0.994 | −1.152 | 22.20 | 0.997 |

**Table 3.**Specifications of the vehicle [29].

Item | Parameter | Value |
---|---|---|

Engine | Maximum Power (kW) | 125 |

Maximum Torque (Nm) | 600 | |

Maximum speed (r/min) | 2600 | |

Motor | Maximum Power (kW) | 120 |

Maximum Torque (Nm) | 650 | |

Maximum speed (r/min) | 2600 | |

Battery | Cell open circuit voltage (V) | 3.8 |

Capacity (Ah) | 70 | |

Voltage (V) | 650 | |

Transmission | AMT gear ratio | [3.583 2.22 1.36 1 0.74] |

Final gear ratio | 6.17 | |

Vehicle | Vehicle mass (kg) | 18,000 |

Roll coefficient | 0.015 | |

C_{d} | 0.65 | |

A | 6.73 | |

Radius (m) | 0.5715 | |

δ | 1.04 |

Variables | Upper Limit | Lower Limit | Grid Number |
---|---|---|---|

SOC(k) | 0.8 | 0.2 | 61 |

g(k) | 4 | 0 | 5 |

r(k) | −1 | 1 | 21 |

d(k) | −1 | 1 | 3 |

Methods | Cycles | Strategies | Fuel (L/100 km) | Final_SOC | Fuel Changes (%) | Gearshift Events |
---|---|---|---|---|---|---|

DP | CTCC | Without drivability | 24.36 | 0.5954 | 0 | 85 |

With drivability | 24.36 | 0.5940 | 0 | 59 | ||

PMP | CTCC | Without drivability | 24.77 | 0.5996 | 0 | 64 |

With drivability | 26.23 | 0.5989 | 5.8 | 54 |

Methods | Cycles | Strategies | Fuel (L/100 km) | Final_SOC | Fuel Changes (%) | Gearshift Events | Computation Time (s) |
---|---|---|---|---|---|---|---|

DP | CTCC | Without drivability | 24.36 | 0.5954 | 0 | 85 | 26.88 |

With drivability | 24.36 | 0.5940 | 0 | 59 | |||

PMP | CTCC | Without drivability | 24.77 | 0.5996 | 1.7 | 64 | 7.55 |

With drivability | 26.23 | 0.5989 | 0 | 54 | |||

A-PMP | CTCC | Without drivability | 24.82 | 0.5994 | 0 | 212 | 3.98 |

With drivability | 25.31 | 0.5995 | 1.9 | 54 |

Parameter | Value |
---|---|

α | 2694.5 |

β | 6022.2 |

η | 152.5 |

R_{g} | 8.314 J/(mol·K) |

E_{a} | 31,500 J/mol |

z | 0.56 |

C_{a} | 950 |

K | Fuel (L/100 km) | SOC Final | Ah_{eff} |
---|---|---|---|

0 | 25.27 | 0.5995 | - |

100 | 25.82 | 0.6029 | 4.0973 |

150 | 25.98 | 0.6039 | 4.0331 |

200 | 26.12 | 0.6046 | 3.9885 |

250 | 26.25 | 0.6056 | 3.9241 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, F.; Wang, L.; Coskun, S.; Cui, Y.; Pang, H.
Computationally Efficient Energy Management in Hybrid Electric Vehicles Based on Approximate Pontryagin’s Minimum Principle. *World Electr. Veh. J.* **2020**, *11*, 65.
https://doi.org/10.3390/wevj11040065

**AMA Style**

Zhang F, Wang L, Coskun S, Cui Y, Pang H.
Computationally Efficient Energy Management in Hybrid Electric Vehicles Based on Approximate Pontryagin’s Minimum Principle. *World Electric Vehicle Journal*. 2020; 11(4):65.
https://doi.org/10.3390/wevj11040065

**Chicago/Turabian Style**

Zhang, Fengqi, Lihua Wang, Serdar Coskun, Yahui Cui, and Hui Pang.
2020. "Computationally Efficient Energy Management in Hybrid Electric Vehicles Based on Approximate Pontryagin’s Minimum Principle" *World Electric Vehicle Journal* 11, no. 4: 65.
https://doi.org/10.3390/wevj11040065