# Synthesis of Optimal Battery State-of-Charge Trajectory for Blended Regime of Plug-in Hybrid Electric Vehicles in the Presence of Low-Emission Zones and Varying Road Grades

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of PHEV Powertrain

## 3. Optimization of PHEV Control Variables

#### 3.1. Considered Driving Cycles and Scenarios

#### 3.2. Optimal Problem Formulation

#### 3.3. Optimization Results

## 4. PHEV Powertrain Control Strategy

#### 4.1. Basic Control Strategy

^{th}gear shift event), ${t}_{sh}$ is the time elapsed from the previous gear shift event, and ${t}_{th}$ is the time threshold in which the discount factor ${r}_{f}$ reaches the value of 1. The discount of the cost function is provided if the current gear ratio ${h}_{k}$ is selected to be equal to the previous gear ratio ${h}_{k-1}$, and if the elapsed time from the previous gear shift event is less than the threshold ${t}_{th}$, thus encouraging the ECMS optimization to keep the current gear ratio for somewhat prolonged period, i.e., to avoid frequent switching of gear ratio.

#### 4.2. Synthesis of Optimal SoC Reference Trajectory

#### 4.2.1. Scenario 1: Zero Road Grade and no LEZ Presence

#### 4.2.2. Scenario 2: Zero Road Grade and LEZ Presence

#### 4.2.3. Scenario 3: Variable Road Grade and no LEZ Presence

^{th}time step, and ${P}_{d,th}>0$ is the power demand threshold obtained in an iterative manner starting with the initial value conservatively set to ${P}_{d,th}=0$. The gradient $\Delta So{C}_{R,i}^{+}/\Delta {s}_{i}$ is limited with respect to SoC rate lower and upper limit values given in Eq. (23), which are obtained by feeding the maximum discharging (i.e., negative) and maximum charging (i.e., positive) battery power into the battery state Equation (6).

## 5. Simulation Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. PHEV City Bus Parameters

#### A.1. Model Parameters

^{2}; aerodynamical drag coefficient, ${C}_{d}$ = 0.70; rolling friction coefficient, ${R}_{0}$ = 0.01; empty bus weight, ${M}_{v}$ = 12635 kg, final drive ratio, ${i}_{o}$ = 4.72.

Gear no. | 1. | 2. | 3. | 4. | 5. | 6. | 7. | 8. | 9. | 10. | 11. | 12. |

Gear ratio | 14.94 | 11.73 | 9.04 | 7.09 | 5.54 | 4.35 | 3.44 | 2.70 | 2.08 | 1.63 | 1.27 | 1.00 |

#### A.2. DP Optimization Parameters

#### A.3. Control Strategy Parameters

## Appendix B. Analysis of Optimal SoC Trajectory Pattern

^{th}discrete time step, ${C}_{a}$ is the set of discrete time steps for which the additional SoC constraints are imposed, and ${K}_{SoC}$ is the penalization factor related to SoC constraint violation (set to 5·10

^{5}, herein). The normalized SoC trajectory length is defined as

^{th}discrete time step $\Delta {s}_{k}$ is normalized with respect to the total travelled distance ${s}_{f}$.

**Figure A1.**(

**a**) DP-optimal SoC trajectories of different length, (

**b**) corresponding fuel consumptions ${V}_{f}$ given versus: normalized SoC trajectory length ${L}_{SoC,norm}$, (

**c**) mean specific fuel consumption ${A}_{ek,mean}$ and (

**d**) total electric energy losses ${E}_{EL,loss}$ (3xDUB driving cycle without road grade is considered; K is the correlation index).

## References

- Guzzella, L.; Sciaretta, A. Vehicle Propulsion Systems, 2nd ed.; Springer: Berlin, Germany, 2007. [Google Scholar]
- Škugor, B.; Cipek, M.; Deur, J. Control variables optimization and feedback control strategy design for the blended operating regime of an extended range electric vehicle. SAE Int. J. Altern. Powertrains
**2014**, 3, 152–162. [Google Scholar] [CrossRef] - Trinko, D.A.; Wendt, E.A.; Asher, Z.D.; Peyfuss, M.; Volckens, J.; Quinn, J.C.; Bradley, T.H. An Adaptive Green Zone Strategy for Hybrid Electric Vehicle Control. In Proceedings of the ITEC 2018 IEEE Transportation Electrification Conference and Expo, Long Beach, CA, USA, 13–15 June 2018; pp. 385–388. [Google Scholar]
- Soldo, J.; Skugor, B.; Deur, J. Optimal Energy Management Control of a Parallel Plug-In Hybrid Electric Vehicle in the Presence of Low-Emission Zones. SAE Technical Paper 2019-01-1215. 2019. Available online: https://www.sae.org/publications/technical-papers/content/2019-01-1215/ (accessed on 6 October 2019). [CrossRef]
- Onori, S.; Tribioli, L. Adaptive Pontryagin’s Minimum Principle supervisory controller design for the plug-in hybrid GM Chevrolet Volt. Appl. Energy
**2015**, 147, 224–234. [Google Scholar] [CrossRef] - Martinez, C.M.; Hu, X.; Cao, D.; Velenis, E.; Gao, B.; Wellers, M. Energy Management in Plug-in Hybrid Electric Vehicles: Recent Progress and a Connected Vehicles Perspective. IEEE Trans. Veh. Technol.
**2017**, 66, 4534–4549. [Google Scholar] [CrossRef] - Yu, H.; Kuang, M.; McGee, R. Trip-oriented energy management control strategy for plug-in hybrid electric vehicles. IEEE Trans. Control Syst. Technol.
**2014**, 22, 1323–1336. [Google Scholar] - Ambuhl, D.; Guzzella, L. Predictive reference signal generator for hybrid electric vehicles. IEEE Trans. Veh. Technol.
**2009**, 58, 4730–4740. [Google Scholar] [CrossRef] - Liu, Y.; Li, J.; Qin, D.; Lei, Z. Energy management of plug-in hybrid electric vehicles using road grade preview. In Proceedings of the IET International Conference on Intelligent and Connected Vehicles (ICV 2016), Chongqing, China, 22–23 September 2016. [Google Scholar]
- Gaikwad, T.; Asher, Z.; Liu, K.; Huang, M.; Kolmanovsky, I. Vehicle Velocity Prediction and Energy Management Strategy Part 2: Integration of Machine Learning Vehicle Velocity Prediction with Optimal Energy Management to Improve Fuel Economy. Available online: https://www.sae.org/publications/technical-papers/content/2019-01-1212/ (accessed on 6 October 2019).
- Xie, S.; Li, H.; Xin, Z.; Liu, T.; Wei, L. A pontryagin minimum principle-based adaptive equivalent consumption minimum strategy for a plug-in hybrid electric bus on a fixed route. Energies
**2017**, 10, 1379. [Google Scholar] [CrossRef] - Xie, S.; Hu, X.; Xin, Z.; Brighton, J. Pontryagin’s Minimum Principle based model predictive control of energy management for a plug-in hybrid electric bus. Appl. Energy
**2019**, 236, 893–905. [Google Scholar] [CrossRef] - Bouwman, K.R.; Pham, T.H.; Wilkins, S.; Hofman, T. Predictive Energy Management Strategy Including Traffic Flow Data for Hybrid Electric Vehicles. IFAC-PapersOnLine
**2017**, 50, 10046–10051. [Google Scholar] [CrossRef] - Sun, C.; Moura, S.J.; Hu, X.; Hedrick, J.K.; Sun, F. Dynamic Traffic Feedback Data Enabled Energy Management in Plug-in Hybrid Electric Vehicles. IEEE Trans. Control Syst. Technol.
**2015**, 23, 1075–1086. [Google Scholar] - Soldo, J.; Škugor, B.; Deur, J. Optimal Energy Management and Shift Scheduling Control of a Parallel Plug-in Hybrid Electric Vehicle. In Proceedings of the Powertrain Modelling and Control Conference (PMC 2018), Loughborough, UK, 10–11 September 2018. [Google Scholar]
- VOLVO 7900 ELECTRIC HYBRID SPECIFICATIONS. Available online: https://www.volvobuses.co.uk/en-gb/our-offering/buses/volvo-7900-electric-hybrid/specifications.html (accessed on 6 October 2019).
- Cipek, M.; Pavković, D.; Petrić, J.Š. A control-oriented simulation model of a power-split hybrid electric vehicle. Appl. Energy
**2013**, 101, 121–133. [Google Scholar] [CrossRef] - Staunton, R.H.; Ayers, C.W.; Marlino, L.D.; Chiasson, J.N.; Burress, B.A. Evaluation of 2004 Toyota Prius Hybrid Electric Drive System; Oak Ridge National Laboratory (ORNL): Oak Ridge, TN, USA, 2005. [Google Scholar]
- Yuan, Z.; Hou, S.-H.; Li, D.; Wei, G.; Hu, X. Optimal Energy Control Strategy Design for a Hybrid Electric Vehicle. Discret. Dyn. Nat. Soc.
**2013**, 132064. [Google Scholar] - EVO ELECTRIC LTD CATALOGUE, AF-230 Motor/Generator. Available online: http://www.fordmax.in.ua/wp-content/uploads/2014/12/AF-230-Spec-Sheet-V1.pdf (accessed on 6 October 2019).
- Cipek, M.; Petrić, J.; Pavković, D.; Kučinić, D. A Hydraulic Component Scalability Tool based on Willans Line Method towards the Optimal Design of Hybrid Hydraulic Vehicles. In Proceedings of the 12th Conference on Sustainable Development of Energy, Water and Environment Systems (SDEWES 2017), Dubrovnik, Croatia, 4–8 October 2017. [Google Scholar]
- Andre, D.; Meiler, M.; Steiner, K.; Wimmer, C.; Soczka-Guth, T.; Sauer, D.U. Characterization of high-power lithium-ion batteries by electrochemical impedance spectroscopy. I. Experimental investigation. J. Power Sources
**2011**, 196, 5334–5341. [Google Scholar] [CrossRef] - Bin, Y.; Li, Y.; Feng, N. Nonlinear dynamic battery model with boundary and scanning hysteresis. In Proceedings of the ASME 2009 Dynamic Systems and Control Conference, Hollywood, CA, USA, 12–14 October 2009; pp. 245–252. [Google Scholar]
- Škugor, B.; Deur, J.; Cipek, M.; Pavković, D. Design of a power-split hybrid electric vehicle control system utilizing a rule-based controller and an equivalent consumption minimization strategy. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2014**, 228, 631–648. [Google Scholar] [CrossRef] - Škugor, B.; Hrgetić, M.; Deur, J. GPS measurement-based road grade reconstruction with application to electric vehicle simulation and analysis. In Proceedings of the 11th Conference on Sustainable Development of Energy, Water and Environment Systems (SDEWES 2015), Dubrovnik, Croatia, 27 September–2 October 2015. [Google Scholar]
- Bellman, R.E.; Dreyfus, S.E. Applied Dynamic Programming; Princeton University Press: Princeton, NJ, USA, 1962. [Google Scholar]
- Cipek, M.; Škugor, B.; Čorić, M.; Kasać, J.; Deur, J. Control variable optimisation for an extended range electric vehicle. Int. J. Powertrains
**2016**, 5, 30. [Google Scholar] [CrossRef] - Škugor, B.; Soldo, J.; Deur, J. Analysis of Optimal Battery State-of-Charge Trajectory for Blended Regime of Plug-in Hybrid Electric Vehicle. In Proceedings of the International Electric Vehicle Symposium & Exhibition (EVS32), Lyon, France, 19–22 May 2019. [Google Scholar]
- Paganelli, G.; Delprat, S.; Guerra, T.M.; Rimaux, J.; Santin, J.J. Equivalent consumption minimization strategy for parallel hybrid powertrains. Proc. IEEE Veh. Technol. Conf.
**2002**, 4, 2076–2081. [Google Scholar] - Liu, K.; Asher, Z.; Gong, X.; Huang, M.; Kolmanovsky, I. Vehicle Velocity Prediction and Energy Management Strategy Part 1: Deterministic and Stochastic Vehicle Velocity Prediction Using Machine Learning. Available online: https://www.sae.org/publications/technical-papers/content/2019-01-1051/ (accessed on 6 October 2019).

**Figure 1.**(

**a**) Functional scheme of considered parallel PHEV powertrain, (

**b**) idling power loss map and (

**c**) torque/power transfer efficiency map.

**Figure 2.**(

**a**) Engine specific fuel consumption map and (

**b**) M/G machine efficiency map, including corresponding maximum torque lines (blue lines).

**Figure 3.**(

**a**) Battery equivalent circuit model and (

**b**) dependencies of open-circuit voltage ${U}_{oc}$ reconstructed from and internal resistance $R$ with respect to SoC for LiFePO

_{4}battery.

**Figure 4.**Time profiles of (

**a**) vehicle velocity, (

**b**) road grade and (

**c**) bus passengers mass for city-bus recorded driving cycle DUB and (

**d**) speed time profile of Heavy Duty Urban Dynamometer Driving Schedule driving cycle (HDUDDS) assuming zero road grade and no passengers in the bus.

**Figure 5.**(

**a**) Low-emission zones (LEZ) inserted into 3xDUB driving cycle and (

**b**) 3xHDUDDS driving cycle, expressed with respect to travelled distances.

**Figure 6.**(

**a**) Sinusoidal road grade profiles for different spatial frequencies and ${s}_{f}$ = 47.3 km and (

**b**) corresponding altitude profiles.

**Figure 7.**(

**a**) Optimal SoC trajectories obtained by DP optimization and corresponding shortest-length (linear) trajectories for 3xDUB driving cycle and (

**b**) 2xHDUDDS driving cycle and for zero road grade and no LEZ presence.

**Figure 8.**(

**a**,

**c**) Optimal SoC trajectories and corresponding shortest-length (piecewise linear) trajectories in case of LEZ presence, for $So{C}_{i}$ = 90%, $So{C}_{f}$ = 30% and (

**b**,

**d**) $So{C}_{i}$= 50%, $So{C}_{f}$= 50% and for 3xDUB, 3xHDUDDS driving cycles and zero road grade.

**Figure 9.**(

**a**) Optimal SoC trajectories and corresponding shortest-length (piecewise linear) trajectories for 6xDUB driving cycle and (

**b**,

**c**) 3xDUB driving cycle in the presence of LEZs with different initial and final SoC values for recorded road grade profile from Figure 4b.

**Figure 11.**(

**a**) Optimal SoC trajectories from Figure 10 decomposed and rearranged to battery discharging (i.e., ${P}_{batt}>0$, $dSoC/dt<0$) and charging sections (i.e., ${P}_{batt}<0$, $dSoC/dt>0$) and (

**b**) SoC time derivative with respect to power demand ${P}_{d}.$

**Figure 13.**(

**a**) Synthesized SoC reference trajectories plotted along with DP-optimal trajectories for recorded road grade profile and (

**b–d**) sinusoidal profiles with different spatial frequencies.

**Figure 14.**Total fuel consumption for CD/CS and blended regimes, plotted against DP optimal results for different driving cycles and no LEZ case.

**Figure 15.**DP optimal, simulated and reference SoC trajectories for case of LEZ presence, (

**a**,

**b**) 3xDUB w/ grade, (

**c**,

**d**) 3xDUB w/o grade and (

**e**,

**f**) 3xHDUDDS driving cycles and different sets of initial and final SoC values ((

**a**,

**c**,

**e**) $So{C}_{i}$ = 90%, $So{C}_{f}$ = 30%, left column and (

**b**,

**d**,

**f**) $So{C}_{i}$ = 50%, $So{C}_{f}$ = 50%, right column).

**Figure 16.**DP optimal and simulated SoC trajectories (the latter obtained for linear and more advanced, nonlinear SoC reference trajectories) for (

**a**) recorded, (

**b**) sinusoidal low frequency, (

**c**) sinusoidal mid frequency and (

**d**) sinusoidal high frequency road grade profiles, no LEZ case, and 4xDUB driving cycle.

**Figure 17.**Fuel consumption values corresponding to simulation results from Figure 16.

**Table 1.**Fuel consumption values corresponding to simulation results from Figure 15 and their relative differences in comparison with DP benchmark (given in brackets).

SoC_{i} = 90%,Target SoC_{f} = 30% | RB+ECMS vs. DP Fuel Consumption [L] | ||

Exact ΔSoC_{LEZ,}_{Σ} | Average ΔSoC_{LEZ,}_{Σ} | Doubled ΔSoC_{LEZ,}_{Σ} | |

3xDUB w/ grade | 2.83 vs. 2.80 (+0.9%) | 2.97 vs. 2.94 (+1.0%) | 3.01 vs. 2.98 (+1.1%) |

3xDUB w/o grade | 2.59 vs. 2.56 (+1.1%) | 2.60 vs. 2.57 (+1.2%) | 2.63 vs. 2.56 (+2.6%) |

3xHDUDDS | 3.15 vs. 3.08 (+2.2%) | 3.10 vs. 3.02 (+2.5%) | 3.11 vs. 3.02 (+3.0%) |

SoC_{i} = 50%,Target SoC_{f} = 50% | RB+ECMS vs. DP Fuel Consumption [L] | ||

Exact ΔSoC_{Z,}_{Σ} | Average ΔSoC_{LEZ,}_{Σ} | Doubled ΔSoC_{LEZ,}_{Σ} | |

3xDUB w/ grade | 5.02 vs. 4.99 (+0.6%) | 5.01 vs. 4.98 (+0.5%) | 5.16 vs. 5.08 (+1.5%) |

3xDUB w/o grade | 5.01 vs. 4.95 (+1.2%) | 5.10 vs. 5.04 (+1.3%) | 5.16 vs. 5.02 (+2.8%) |

3xHDUDDS | 5.72 vs. 5.66 (+1.1%) | 5.68 vs. 5.61 (+1.3%) | 5.72 vs. 5.65 (+1.2%) |

© 2019 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**

Soldo, J.; Škugor, B.; Deur, J.
Synthesis of Optimal Battery State-of-Charge Trajectory for Blended Regime of Plug-in Hybrid Electric Vehicles in the Presence of Low-Emission Zones and Varying Road Grades. *Energies* **2019**, *12*, 4296.
https://doi.org/10.3390/en12224296

**AMA Style**

Soldo J, Škugor B, Deur J.
Synthesis of Optimal Battery State-of-Charge Trajectory for Blended Regime of Plug-in Hybrid Electric Vehicles in the Presence of Low-Emission Zones and Varying Road Grades. *Energies*. 2019; 12(22):4296.
https://doi.org/10.3390/en12224296

**Chicago/Turabian Style**

Soldo, Jure, Branimir Škugor, and Joško Deur.
2019. "Synthesis of Optimal Battery State-of-Charge Trajectory for Blended Regime of Plug-in Hybrid Electric Vehicles in the Presence of Low-Emission Zones and Varying Road Grades" *Energies* 12, no. 22: 4296.
https://doi.org/10.3390/en12224296