# Analysis of Optimal Battery State-of-Charge Trajectory for Blended Regime of Plug-in Hybrid Electric Vehicle

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modelling of PHEV Powertrain

## 3. Optimization of PHEV Control Variables

#### 3.1. Optimal Problem Formulation

#### 3.2. Optimisation Results

_{pass}= 0). The DUB driving cycle is repeated three times and HDUDDS two times, in order to provide battery discharging to the minimum allowable SoC level (set here to 30%) in the blended regime under the given driving conditions.

#### 3.3. Generating and Analyzing Optimal SoC Trajectories of Different Length

^{5}. The effect of extending the DP optimization with two additional SoC constraints is illustrated in Figure 6, where the optimal SoC trajectory $So{C}_{DP}$ and the cumulative cost function ${J}_{i}$ are shown with respect to travelled distance. The optimal cumulative cost function ${J}_{i}$ in the discrete time step $i$ can be calculated as:

## 4. Analysis of Optimal SoC Trajectory Patterns

#### 4.1. Simplified Case of Minimizing Solely Battery Energy Losses

^{th}route segment of length $\mathsf{\Delta}{s}_{r}$, while ${N}_{R}$ is the total number of discrete route segments. The factor ${Q}_{max}^{2}$ is omitted in Equation (23) since it is constant and does not have influence on the optimization problem solution. Under the assumption of the battery internal resistance $R$, vehicle velocity ${v}_{v,r}$, and length of all route segments $\mathsf{\Delta}{s}_{r}$ will be constant, the optimization problem can be further simplified:

^{th}route segment is obtained:

#### 4.2. More Realistic Case of Minimizing Fuel Consumption

**OP1**: power demand ${P}_{d}$ is partly satisfied by the engine and partly by the M/G machine (operating points are kept constant during the whole operation; constant $\dot{SoC}$ < 0),**OP2—Phase 1**: power demand ${P}_{d}$ is completely satisfied by the engine ($\dot{SoC}$ = 0),**Phase 2**: power demand ${P}_{d}$ is completely satisfied by the M/G machine (constant $\dot{SoC}$ < 0),**OP3—Phase 1**: power demand ${P}_{d}$ is completely satisfied by the engine which also provides additional power to recharge the battery (constant $\dot{SoC}>0$),**Phase 2**: power demand ${P}_{d}$ is completely satisfied by the M/G machine (constant $\dot{SoC}<0$).

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. PHEV City Bus Parameters

#### Appendix A.1. Model Parameters

^{2}; aerodynamical drag coefficient, ${C}_{d}$ = 0.70; rolling friction coefficient, ${R}_{0}$ = 0.01; empty bus weight, ${M}_{v}$ = 12,635 kg, final drive ratio, ${i}_{o}$ = 4.72.

**Table A1.**Transmission gear ratios [14].

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 |

#### Appendix A.2. DP Optimization Parameters

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**Figure 1.**Parallel configuration of plug-in hybrid electric vehicles (PHEV) powertrain (

**a**), transmission idle-mode power loss map (

**b**), and mechanical efficiency map (

**c**).

**Figure 2.**Engine specific fuel consumption map (

**a**), and electric machine (M/G) machine efficiency map (

**b**), given along with maximum torque lines (denoted in blue).

**Figure 3.**Battery equivalent circuit (

**a**), and dependencies of open-circuit voltage and internal battery resistance with respect to battery state-of-charge (SoC) for a considered lithium iron phosphate battery (

**b**).

**Figure 4.**City bus driving cycle including vehicle velocity (${v}_{v}$) (

**a**), road grade (${\delta}_{r}$ ) (

**b**), and passenger mass (${m}_{pass}$) (

**c**) time profiles recorded in the city of Dubrovnik (DUB); and velocity time profile for heavy-duty UDDS driving cycle (HDUDDS) which assumes a zero road (

**d**).

**Figure 5.**Optimal SoC trajectories obtained by the dynamic programing (DP) algorithm in blended regime for repetitive DUB driving cycle with varying road grade (

**a**) and zero road grade (

**b**); and for repetitive HDUDDS driving cycle (

**c**) (see Figure 4; in the case of DUB driving cycle varying passengers mass from Figure 4c is used).

**Figure 6.**Visualization of cumulative cost function ${J}_{i}$(see Equation (16)) along with the optimal SoC trajectory $So{C}_{DP}$ for the case of two additional SoC constraints (i.e., $So{C}_{constr}$ = 45% at 1/3 of total trip distance, and $So{C}_{constr}$ = 55% at 2/3 of total trip distance) and 4 x DUB driving cycle with a zero road grade.

**Figure 7.**Set of DP optimal SoC trajectories of different lengths obtained by imposing an additional SoC constraint (14) (

**a**); and corresponding total fuel consumption ${V}_{f}$ shown with respect to normalized SoC trajectory length ${L}_{SoC,norm}$ (

**b**), mean engine specific fuel consumption ${A}_{ek,mean}$ (

**c**), and total electric energy losses ${E}_{EL,loss}$ (

**d**) (3 x DUB driving cycle when a zero road grade was used).

**Figure 8.**SoC trajectories obtained by constant SoC depletion rate ($So{C}_{lin}$) and by DP optimization for the constant and variable SoC-dependent battery parameters (

**a**), and corresponding SoC depletion rates (

**b**).

**Figure 9.**Fuel consumption rate ${\dot{m}}_{f}$ versus SoC depletion rate $\dot{SoC}$ (

**a**), and second derivative of ${\dot{m}}_{f}$ versus $\dot{SoC}$ curve (

**b**), given for several values of demanded power ${P}_{d}$, engine speed ${\omega}_{e}$ = 184 rad/s and $SoC$ = 50%.

**Figure 10.**Character of ${\dot{m}}_{f}$ vs. $\dot{SoC}$ dependence (convex or non-convex) for a wide range of engine speeds ${\omega}_{e}$ and driver power demands ${P}_{d}$ for the case of $SoC$ = 0.5.

**Figure 11.**Illustration of original (convex) and modified (concave) engine fuel consumption rate ${\dot{m}}_{f}$ with respect to SoC depletion rate $\dot{SoC}$ (

**a**), and the corresponding second derivatives (

**b**) for the case of $SoC$ = 50%, ${v}_{v}$ = 86 km/h, ${\omega}_{e}$ =${\omega}_{MG}$ = 184 rad/s, ${P}_{d}$ = 79.7 kW.

**Figure 12.**Illustration of three different operating scenarios through engine mean specific fuel consumption (

**a**), engine power (

**b**), total electric energy losses (

**c**), and SoC trajectory profile (

**d**) (the same operating conditions as in Figure 11: $SoC$ = 50%, ${v}_{v}$ = 86 km/h, ${\omega}_{e}$ =${\omega}_{MG}$ = 184 rad/s, ${P}_{d}$ = 79.7 kW).

**Figure 13.**Comparative cumulative fuel consumption time profiles for different operating scenarios for original (

**a**) and modified engine fuel consumption characteristics (

**b**).

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**MDPI and ACS Style**

Škugor, B.; Soldo, J.; Deur, J.
Analysis of Optimal Battery State-of-Charge Trajectory for Blended Regime of Plug-in Hybrid Electric Vehicle. *World Electr. Veh. J.* **2019**, *10*, 75.
https://doi.org/10.3390/wevj10040075

**AMA Style**

Škugor B, Soldo J, Deur J.
Analysis of Optimal Battery State-of-Charge Trajectory for Blended Regime of Plug-in Hybrid Electric Vehicle. *World Electric Vehicle Journal*. 2019; 10(4):75.
https://doi.org/10.3390/wevj10040075

**Chicago/Turabian Style**

Škugor, Branimir, Jure Soldo, and Joško Deur.
2019. "Analysis of Optimal Battery State-of-Charge Trajectory for Blended Regime of Plug-in Hybrid Electric Vehicle" *World Electric Vehicle Journal* 10, no. 4: 75.
https://doi.org/10.3390/wevj10040075