# On Implementing Optimal Energy Management for EREV Using Distance Constrained Adaptive Real-Time Dynamic Programming

^{*}

## Abstract

**:**

## 1. Introduction

## 2. EREV Modeling Methodology

- ${P}_{demand}$
- is the net power demand from the propulsion system.
- ${P}_{traction}$
- is the traction power required to achieve the target speed.
- ${P}_{aero}$
- is the aerodynamic resistance power.
- ${P}_{roll}$
- is the tire rolling resistance power.
- ${P}_{climb}$
- is the climbing resistance power.
- ${m}_{vehicle}$
- is the gross weight of the vehicle.
- ${m}_{pass}$
- is the net passenger weight for two passengers.
- ${v}_{target}$
- is the drive cycle generated target speed.
- ${C}_{d}$
- is the drag coefficient.
- ${\rho}_{air}$
- is the density of air at Normal Temperature and Pressure.
- ${A}_{front}$
- is the frontal area of the vehicle.
- ${C}_{rr}$
- is the rolling resistance coefficient.
- g
- is the acceleration due to gravity.
- ${\alpha}_{road}$
- is the road elevation angle.

#### 2.1. Engine Efficiency Map Estimation

#### 2.2. Transmission Efficiency Map Estimation

#### 2.3. EREV Power Flow Model Validation

## 3. Energy Management Optimization Algorithm

- ${P}_{fuel}$
- is the fuel power consumed.
- ${U}_{{P}_{fuel}}$
- is the feasible set of fuel power consumption.
- ${P}_{ESS}$
- is the ESS power consumed.
- ${U}_{{P}_{ESS}}$
- is the feasible set of ESS power consumption.
- $\lambda $
- is the co-state variable.
- ${p}_{1}\left(SoC\right)$
- is the hard penalty associated with the feasible SoC range.

#### 3.1. Distance Constrained Dynamic Programming

Algorithm 1: Compute feasible set of ESS SOC subject to driving range constraint at time step k. |

#### 3.2. Adaptive Distance Constrained Dynamic Programming

#### 3.3. Proposed Real-Time Application of Adaptive Distance Constrained Dynamic Programming

## 4. Results and Discussion

#### 4.1. Implementation of Distance Constrained Dynamic Programming

#### 4.2. Adaptive Distance Constrained Dynamic Programming

#### 4.3. Comparison of Adaptive Distance Constrained Dynamic Programming

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

API | Application Programming Interface |

APU | Auxiliary Power Unit |

BSFC | Brake Specific Fuel Consumption |

CD | Charge Depleting |

CS | Charge Sustaining |

EEC | Emissions & Energy Consumption |

EPA | Environmental Protection Agency |

EPO | Emergency Power Off |

EREV | Extended Range Electric Vehicle |

ESS | Energy Storage System |

HWFET | Highway Fuel Economy Testing |

ICE | Internal Combustion Engine |

REx | Range Extender |

SoC | State of Charge |

UDDS | Urban Dynamometer Driving Schedule |

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**Figure 3.**Battery equivalent circuit models evaluated for the power flow model. (

**a**) Zeroth, (

**b**) dual polarization.

**Figure 4.**Fuel consumption estimation model validation with (

**a**) estimated vs. measured fuel tank level comparison plot and (

**b**) the fuel tank level estimation error.

**Figure 7.**EREV research vehicle ESS SoC (State of Charge) variation comparison. Charge depleting behavior for emissions and energy consumption (EEC) cycle is shown in (

**a**–

**c**), charge sustaining behavior for the EEC cycle is shown in (

**d**–

**f**), and charge depleting/charge sustaining behavior for the urban dynamometer driving schedule (UDDS) cycle is shown in (

**g**–

**i**).

**Figure 8.**Final ESS SoC (%) achieved matrix for (

**a**) EEC, (

**b**) HWFET, (

**c**) UDDS, and (

**d**) US06 drive cycle simulations using forward propagating dynamic programming.

**Figure 9.**Net optimal energy consumption (kWh/km) matrix for (

**a**) EEC, (

**b**) HWFET, (

**c**) UDDS, and (

**d**) US06 drive cycle simulations using forward propagating dynamic programming.

**Figure 10.**Maximum achievable driving range (km) matrix for (

**a**) EEC, (

**b**) HWFET, (

**c**) UDDS, and (

**d**) US06 drive cycle simulations using forward propagation dynamic programming.

**Figure 11.**Hamiltonian in the feasible set for the (

**a**) ESS-only and (

**b**) ESS + REx operation during EEC three and nine cycle optimization taken at $t=663.0$ s, respectively.

**Figure 12.**ESS SoC and maximum driving range comparison for optimal energy consumption (ESS-only) and maximum driving range (ESS + range extender (REx) system) amongst (

**a**) EEC, (

**b**) HWFET, (

**c**) UDDS, and (

**d**) US06 drive cycle simulations using Pontryagin’s minimum principle.

**Figure 13.**Vehicle mass normalized ESS current variation with vehicle speed comparison in charge depleting (CD) mode for the UDDS cycle between research vehicle, 2014 BMW i3 Rex, and 2016 Chevrolet Volt.

**Figure 16.**Estimated speed trace built using Google Directions application programming interface (API). (

**a**) Shows path nodes from origin to destination, (

**b**) average vehicle speed determined from current traffic conditions, and (

**c**) randomized speed traces based on the average speed.

**Figure 17.**Distance constrained dynamic programming optimization results on the research vehicle for the UDDS (

**a**–

**c**) and US06 (

**d**–

**f**) cycles.

**Figure 18.**Comparison of the vehicle’s performance with and without reoptimization using the three randomized speed traces, (

**a**–

**c**) slower than average, (

**d**–

**f**) near average, and (

**g**–

**i**) faster than average.

**Figure 19.**Net energy consumption comparison of the with and without reoptimization scenarios for the three randomized speed traces.

**Figure 20.**Comparison of the conventional and proposed algorithms for the UDDS-HWFET-UDDS-US06 cycle combinations. Subfigures (

**a**–

**c**) correspond to the single cycle, (

**d**–

**f**) correspond to the double cycle, and (

**g**–

**i**) correspond to the triple cycle.

**Figure 21.**Net energy consumption comparison of the conventional and proposed algorithms for the single, double, and triple cycles of the UDDS-HWFET-UDDS-US06 combination.

**Figure 22.**Fault insertion behavior comparison of the conventional and proposed algorithm for the UDDS-HWFET-UDDS-US06 triple cycle. (

**a**) shows the drive cycle plot, (

**b**) shows the comparison in ESS SoC, and (

**c**) shows the REx power output comparison.

System/Component | Parameter | Value | Units |
---|---|---|---|

Engine | Displacement | 0.8 | L |

Max. Power | 80.7 | kW | |

Max. Torque | 80 | Nm | |

Energy Storage System (ESS) | Max. Capacity | 18.9 | kWh |

Nominal Pack Voltage | 340 | V | |

Discharge Power Limit (Peak) | 208 | kW | |

Discharge Power Limit (Cont.) | 61.2 | kW | |

Charge Power Limit (Peak) | 102 | kW | |

Charge Power Limit (Cont.) | 20.4 | kW | |

Generator | Max. Power (Peak) | 83.8 | kW |

Max. Power (Cont.) | 58.6 | kW | |

Max. Torque (Peak) | 200 | Nm | |

Max. Torque (Cont.) | 95 | Nm | |

Traction Motor | Max. Power (Peak) | 230 | kW |

Max. Power (Cont.) | 110 | kW | |

Max. Torque (Peak) | 500 | Nm | |

Max. Torque (Cont.) | 250 | Nm | |

Gear Ratio | 4.2 | - | |

Vehicle | Gross Weight | 1875 | kg |

Passenger (x2) Weight | 160 | kg | |

Tire Rolling Radius | 0.346 | m | |

Frontal Area | 0.4 | m^{2} | |

Fuel Tank Capacity | 26.5 | L |

Substance | Molar Mass [g/mol] |
---|---|

${C}_{2.55}{H}_{7.23}O$ | 53.83 |

$C{O}_{2}$ | 44.0 |

$CO$ | 28.0 |

Driving Range | EEC Cycle | HWFET Cycle | UDDS Cycle | US06 Cycle |
---|---|---|---|---|

50 mi (≈80.5 km) | 3 | 5 | 7 | 6 |

75 mi (≈120.7 km) | 5 | 7 | 10 | 9 |

100 mi (≈160.9 km) | 7 | 10 | 13 | 12 |

125 mi (≈201.2 km) | 9 | 12 | 17 | 16 |

150 mi (≈241.4 km) | 11 | 15 | 20 | 19 |

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**MDPI and ACS Style**

Kalia, A.V.; Fabien, B.C.
On Implementing Optimal Energy Management for EREV Using Distance Constrained Adaptive Real-Time Dynamic Programming. *Electronics* **2020**, *9*, 228.
https://doi.org/10.3390/electronics9020228

**AMA Style**

Kalia AV, Fabien BC.
On Implementing Optimal Energy Management for EREV Using Distance Constrained Adaptive Real-Time Dynamic Programming. *Electronics*. 2020; 9(2):228.
https://doi.org/10.3390/electronics9020228

**Chicago/Turabian Style**

Kalia, Aman V., and Brian C. Fabien.
2020. "On Implementing Optimal Energy Management for EREV Using Distance Constrained Adaptive Real-Time Dynamic Programming" *Electronics* 9, no. 2: 228.
https://doi.org/10.3390/electronics9020228