# Sensitivity Analysis of the Battery Model for Model Predictive Control: Implementable to a Plug-In Hybrid Electric Vehicle

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions.

## 2. Vehicle Model

#### 2.1. Overview

#### 2.2. Vehicle Dynamics Model: State Space Equations

_{m}is computed by interpolating the manufacturer look-up table over torque and speed. The electric power generated by the electric generator can be computed as follows:

_{g}is calculated by interpolating the manufacturer look-up table over torque and speed. The speed of the engine is already known as it is controlled by the MPC. Only the engine torque needs to be computed. In a discrete domain, it can be computed by Equation (14):

## 3. Model Predictive Control Strategy

#### 3.1. Overview

_{g}(t) ≤ 70 N.m

#### 3.2. Objective Function

#### 3.3. Optimization Solver

## 4. Battery Models for the Sensitivity Analysis

#### 4.1. Methodology

#### 4.2. Development of Battery Models

#### 4.3. Battery Dynamics Model Equations

_{b}is negative and is discharging when P

_{b}is positive. The battery current is then computed as follows:

#### 4.4. Aging of Battery Models

## 5. Simulation Results

- the electric generator power entirely dependent on the fuel consumption as shown in Figure 2,
- the auxiliary power defined as the same constant for the MPC model and the real vehicle,
- the electric motor power depending on its efficiency, the vehicle dynamics, and the speed profile, which are the same between the MPC model and the real vehicle.

_{oc}using “polyfit” function. Indeed, BM1 is defined as a simple constant. From BM2 to BMref, those models require the use of the Matlab function “polyval” [41] and “polyfit” [42] to compute the V

_{oc}in function of SoC. Moreover, BM4 requires a linear interpolation function of Matlab, named “interp1” [43] to calculate the internal series resistance. All of those approximation functions are computationally intense compared with other functions in the controller. More precisely, table III, providing the computational cost of each battery model, shows that “polyfit” is the most computationally demanding function.

## 6. Conclusions

_{oc}using “polyfit” function. So, increasing the battery model fidelity from BM2 to BMref does not add significant additional computation.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbol | Name | Units |

$\mathrm{t}$ | Discrete time | [s] |

$\Delta \mathrm{t}$ | Step time | [s] |

X | State variables | - |

U | Control input | - |

W | Disturbance vector | - |

${\mathrm{F}}_{\mathrm{r}}$ | Rolling resistance | [N] |

${\mathrm{F}}_{\mathrm{g}}$ | Grading resistance | [N] |

${\mathrm{F}}_{\mathrm{w}}$ | Aerodynamic drag resistance | [N] |

${\mathrm{F}}_{\mathrm{a}}$ | Acceleration ristance | [N] |

${\mathrm{F}}_{\mathrm{c}}$ | Sum of every resistant force applied to the car | [N] |

${\mathrm{P}}_{\mathrm{b}}$ | Power provided by the battery | [W] |

${\mathrm{P}}_{\mathrm{m}\_\mathrm{e}}$ | Electric power requested by the motor | [W] |

${\mathrm{P}}_{\mathrm{g}\_\mathrm{e}}$ | Power provided by the generator | [W] |

${\mathrm{P}}_{\mathrm{a}}$ | Constant power consumed by the auxiliary electric system | [W] |

${\mathrm{P}}_{\mathrm{b}}$ | Power provided by the battery | [W] |

${\mathrm{P}}_{\mathrm{c}}$ | Power requested by the car | [W] |

${\mathrm{P}}_{\mathrm{m}\_\mathrm{m}}$ | Mechanical power provided by the motor | [W] |

${\mathrm{v}}_{\mathrm{c}}$ | Car speed | [m/s] |

${\dot{\mathrm{v}}}_{\mathrm{c}}$ | Caacceleration | [m/s^{2}] |

$\mathrm{g}$ | Earth gravitational constant | [m/s^{2}] |

${\mathsf{\omega}}_{\mathrm{m}}$ | Motor speed | [rads/s] |

${\mathsf{\omega}}_{\mathrm{g}}$ | Generator speed | [rads/s] |

${\mathsf{\omega}}_{\mathrm{e}}$ | Engine speed | [rads/s] |

${\mathrm{T}}_{\mathrm{g}}$ | Generator torque | [N.m] |

${\mathrm{T}}_{\mathrm{e}}$ | Engine torque | [N.m] |

${\mathrm{T}}_{\mathrm{m}}$ | Motor torque | [N.m] |

${\mathrm{J}}_{\mathrm{f}\mathrm{l}\mathrm{w}\mathrm{h}\mathrm{l}}$ | Sum of the engine and generator flywheel moment of inertia | [kg/m^{2}] |

${\mathrm{M}}_{\mathrm{i}}$ | Inertia mass due to all rotating parts | [kg] |

$\mathrm{M}$ | Car mass | [kg] |

${\mathsf{\eta}}_{\mathrm{b}}$ | Battery coulomb efficiency | - |

${\mathsf{\eta}}_{\mathrm{g}}$ | Generator efficiency | - |

${\mathsf{\eta}}_{\mathrm{g}\mathrm{e}\mathrm{a}\mathrm{r}}$ | Motor gear ratio efficiency | - |

${\mathsf{\eta}}_{\mathrm{m}}$ | Motor efficiency | - |

${\mathsf{\eta}}_{\mathrm{e}}$ | Enge efficiency | - |

${\mathrm{V}}_{\mathrm{B}}$ | Battery voltage | [V] |

$\mathrm{V}\mathrm{o}\mathrm{c}$ | Battery open circuit voltage | [V] |

R_{S} | Series resistance | [Ω] |

R_{T_S} | Short transient voltage response resistance | [Ω] |

C_{T_S} | Short transient voltage response capacitance | [F] |

R_{T_L} | Long transient voltage response resistance | [Ω] |

C_{T_L} | Long transient voltage response capacity | [F] |

${\mathrm{I}}_{\mathrm{B}}$ | Current provided/received by the battery | [A] |

$\mathrm{S}\mathrm{o}\mathrm{C}$ | Battery State of Charge | - |

${\mathrm{C}}_{\mathrm{b}}$ | Nominal battery capacity | [A.h] |

${\mathrm{Q}}_{\mathrm{l}}$ | Battery capacity fades | - |

ΔSoC | Equivalent cumulative SoC variation for a given C_{rate} | - |

${\mathrm{E}}_{\mathrm{b}}$ | Nominal energy capacity of the battery | [J] |

$\mathrm{E}\mathrm{O}{\mathrm{L}}_{\mathrm{e}\mathrm{v}}$ | End of life percentage of a battery for EVs | - |

$\mathrm{E}\mathrm{O}{\mathrm{L}}_{\mathrm{s}\mathrm{t}}$ | End of life percentage of a battery for stationary applications | |

${\mathsf{\epsilon}}_{\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{e}}$ | Cradle-to-gate embodied primary energy per unit of electrical energy capacity for lithium battery | - |

${\mathrm{e}}_{\mathrm{d}\_\mathrm{f}}$ | Energy density of the fuel | [J/kg] |

$\mathrm{J}$ | Energy cost function | [J] |

${\dot{\mathrm{m}}}_{\mathrm{f}}$ | Instant fuel consumption | [kg/s] |

${\mathrm{C}}_{\mathrm{r}0}$ | Car static rolling coefficient | - |

${\mathrm{C}}_{\mathrm{r}1}$ | Car dynamic rolling coefficient | - |

${\mathrm{C}}_{\mathrm{D}}$ | Car drag coefficient | - |

${\mathsf{\epsilon}}_{0}$ | Gear reduction | - |

$\mathrm{r}\mathrm{d}$ | Wheels radius | [m] |

${\mathrm{A}}_{\mathrm{f}}$ | Car front size surface | [m^{2}] |

${\mathsf{\rho}}_{\mathrm{a}}$ | Air density | [kg/m^{3}] |

R | Perfect gas constant | [J/mol·K] |

E_{a} | Activation energy | J/mol |

z | Power law factor | - |

B | Pre-exponent factor | - |

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**Figure 2.**Series PHEV block diagram of the Subaru BRZ 2015 Urban Dynamometer Driving Schedule (UDDS).

**Figure 7.**Error between the SoC computed by the model predictive control (MPC) model and the reference vehicle during two Urba n Dynamometer Driving Schedule (UDDS,

**up**) and Highway Fuel Economy Test (HWFET,

**down**) drive cycles.

**Figure 8.**Error between the fuel consumption computed by the MPC model and the reference vehicle during two UDDS (

**up**) and HWFET (

**down**) drive cycles.

**Figure 9.**Error between the battery capacity fade computed by the MPC model and the reference vehicle during 2 UDDS (

**up**) and HWFET (

**down**) drive cycles.

**Figure 10.**Error between the objective function computed by the MPC model and the reference vehicle during two UDDS (

**up**) and HWFET (

**down**) drive cycles.

**Figure 11.**Fuel consumption comparison at the end of the two UDDS (

**up**) and HWFET (

**down**) drive cycles.

**Figure 13.**Objective function comparison at the end of the two UDDS (

**up**) and HWFET (

**down**) drive cycles.

Power-Train Components | Name | Characteristics |
---|---|---|

Energy Storage System (ESS) | Lithium iron phosphate (LFP) prismatic cells from A123 | Capacity = 39.2 Ah; nominal voltage = 340 V; nominal energy = 13.3 kWh; configuration: 7×15s2p. |

Internal Combustion Engine (ICE) | Model MPE850 from Weber | 41 kW, 2 cylinders, 850 cc. |

Electric Generator | Model YASA-400 | 93 kW, axial flux permanent magnet. |

Electric Motors Unit | Model GVK210-100L6 from Linamar | 2 × 80 kW, unit ratio = 8.49. |

Vehicle dynamics | 2015 Subaru BRZ Limited | Drag coefficient = 0.28; frontal area = 1.9695 m^{2}; PHEV mass = 1300 kg; wheel radius = 0.3 m. |

Name | Notation | Value |
---|---|---|

Algorithm type | Algorithm | Interior-point |

Step tolerance | TolX | 1 |

Function tolerance | TolFun | 0.01 |

Maximum Iteration | MaxIter | 1000 |

Maximum Function Evaluation | MaxFunEval | 1000 |

Constraint Tolerance | TolCon | 0.1 |

Name | BM1 | BM2 | BM3 | BM4 | BM5 | BMref |
---|---|---|---|---|---|---|

V_{oc} (V) | 350 | g(SoC) | ||||

R_{S} (Ω) | 0 | 0 | 0.15 | f(SoC) | 0.095 | 0.1094 |

R_{T_S} (Ω) | 0 | 0 | 0 | 0 | 0.0118 | 0.1111 |

C_{T_S} (F) | 0 | 0 | 0 | 0 | 287.8 | 422.7 |

R_{T_L} (Ω) | 0 | 0 | 0 | 0 | 0 | 0.1115 |

C_{T_L} (F) | 0 | 0 | 0 | 0 | 0 | 10,196 |

Function | BM1 | BM2 | BM3 | BM4 | BM5 | BMref | |
---|---|---|---|---|---|---|---|

I_{b} | - | 1 | 1 | 9 | 9 | 1 | 1 |

V_{oc} | “Polyfit” | - | 936 | 936 | 936 | 936 | 936 |

“Polyval” | - | 31 | 31 | 31 | 31 | 31 | |

V_{b} | - | - | - | 2 | 2 | 3 | 4 |

R_{s} | “Interp1” | - | - | - | 10 | - | - |

V_{T_S} | - | - | - | - | - | 4 | 4 |

V_{T_L} | - | - | - | - | - | - | 4 |

Total Computation Cost | 1 | 968 | 978 | 988 | 975 | 980 | |

Computation time | 512 s | 919 s | 921 s | 977 s | 903 s | 909 s | |

Simulation time | 2740 s | 2740 s | 2740 s | 2740 s | 2740 s | 2740 s | |

Battery model computation time | 0.0097 s | 0.0170 s | 0.0170 s | 0.0180 s | 0.0168 s | 0.0169 s |

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## Share and Cite

**MDPI and ACS Style**

Sockeel, N.; Shi, J.; Shahverdi, M.; Mazzola, M.
Sensitivity Analysis of the Battery Model for Model Predictive Control: Implementable to a Plug-In Hybrid Electric Vehicle. *World Electr. Veh. J.* **2018**, *9*, 45.
https://doi.org/10.3390/wevj9040045

**AMA Style**

Sockeel N, Shi J, Shahverdi M, Mazzola M.
Sensitivity Analysis of the Battery Model for Model Predictive Control: Implementable to a Plug-In Hybrid Electric Vehicle. *World Electric Vehicle Journal*. 2018; 9(4):45.
https://doi.org/10.3390/wevj9040045

**Chicago/Turabian Style**

Sockeel, Nicolas, Jian Shi, Masood Shahverdi, and Michael Mazzola.
2018. "Sensitivity Analysis of the Battery Model for Model Predictive Control: Implementable to a Plug-In Hybrid Electric Vehicle" *World Electric Vehicle Journal* 9, no. 4: 45.
https://doi.org/10.3390/wevj9040045