Burak Yuksel, Orkun Karabasoglu September 18, 2011

Global Design and Control Optimization of Plug-in Hybrid Electric Vehicles for Minimum Life Cycle Cost (and Emissions)

Abstract: PHEVs have the potential to reduce fuel consumption, emissions and oil dependency in transportation sector. However the real potential can only be achieved with the right vehicle design and appropriate power management strategy. It is traditional to design the vehicle for high performance and low cost objectives and then sequentially optimize the control strategy. This approach has the potential to give suboptimal control strategy which might result in not utilizing the real potential of expensive components such as batteries. In this work we explore the design and control space of plug-in hybrid electric vehicles to find the minimum life cycle cost (and emissions). In this work a bi-level optimization strategy has been adopted since it guarantees the system level optimality [1]. The outer loop optimizes the vehicle design for minimum life cycle objectives while the inner loop optimizes the control strategy for minimum gasoline consumption by DDP algorithm which is known to give global optimum [2]. The outer loop optimizes the response surface model of life cycle cost with genetic algorithms as well as sequential quadratic programming with multiple start points. It is found that bi-level optimization gives y% less life cycle cost (and emissions) compared to sequential optimization. This approach demonstrates a useful framework to find the globally optimal vehicle design and control strategy.

Function Module

COST MODELVEHICLE

MODEL

DDP

GlobalOptimizer

x

u

CLC

CLC*

η, v

η*, v*

0

Engine_SizeX = Battery_Size

Motor_Size

Figure 1: Bi-level Design and Control Global Optimization Strategy

Function Module

ResponseSurface Fitting with Adaptive Sampling

Response Surface of Life

Cycle Cost

GAs

*x*d

*LCC

Design of Experiments Response Surface

Methodology

1) For each specific daily distance driven si from set of S=(1:1:1560) [mile/day] 2) Outer loop finds you the optimal design (di) and corresponding life cycle cost using (ui) from

inner loop3) Inner loop gives optimal control strategy (ui) for di making sure that we make the most use of

components. 4) Use Design of Experiments 5) Generate a Response Surface for Life Cycle Cost6) Optimize response surface with Genetic Algorithms7) Iterate to step 1 (account for other daily driving distances)8) Find the optimal design distribution for s=(1:1:1560). Preferably for s=1:1:100 miles. 9) Identify effects of design and control coupling10) Demonstrate the effectiveness of this algorithm by

o Optimizing only the design of a vehicle and then creating a controller sequentially. Find the life cycle cost.

o Use proposed framework to optimally design vehicle and controller. Find life cycle cost. o Compare the outcomes of two approaches.

Challenges

Different Drive cycles – Road power demand changes. Different daily distance driven – Ex: If you have a PHEV20 and you drive 10 miles a day, then you

don`t consume any gasoline, if you drive 30 miles a day, then you travel 10 miles on gasoline. Uncertainties in gasoline price and battery prices… how does optimal design change?

Mathematical Formulation

QUASI-STATIC APPROACH

In this method the input variables are the speed and acceleration. With this information, road demand is computed. With this approach it is possible to design supervisory control systems that optimize the power flows in the propulsion system without the hassle of computational burden. The drawback of the quasi-static method is its ‘backward’ formulation, i.e., the physical causality is not respected and the driving profile that has to be followed has to be known as priori. Therefore, this method is not able to handle feedback control problems or correctly deal with state events.

REDUCED SPLIT HEV MODEL AND EQUATIONS

Only the main difference between parallel and split model is the transmission and the torque split equations:

Figure-1 Planetary Gear

Figure-2 Planetary Gear – Quasi-static Modeling

VEHICLE

TRANSMISSION

ENGINE

MOTOR

,

,

GENERATOR

BATTERY

COST FUNCTION FOR DDP

Dynamic Programming

,

Where is the state-of-charge (SOC) and is the engine speed

,

In our problem, we are trying to find the optimal decisions which lead to optimal fuel consumption for a parallel HEV. The state and cost equations can we model like

where is the battery state of charge (SOC), is the torque split factor, is the fuel mass

consumption, is final state cost which is not defined here and is the time step. The function

represents how the SOC is changing, which is :

Where is battery current and is battery capacity. Since

and a function of and , we can represent the derivation of SOC like

The objective function to be minimized for HEV model is:

DYNAMIC PROGRAMMING RESULTS

The operating points of the engine, motor and generator after using dynamic programming in FTP-75 drive cycle:

ENGINE:

MOTOR:

GENERATOR:

The main purpose is running engine on high efficient curves when it is necessary, like meeting road demand or charging the battery.

SOC, the state variable of the inner loop, has to be kept between its minimum and maximum borders and initial SOC has to match with final SOC to calculate fuel economy more accurate:

SOC:

Abbreviations

Symbol Unit DescriptionVehicle Acceleration

Ah Battery Charge Capacity

$ Cost of Battery

$/metric ton CO2 Cost of CO2 tax

$ Daily Electricity Cost

$/kWh Electricity price per kWh

$ Cost of Engine

# Aerodynamic Coefficient

$/day Gasoline Consumption cost per day

$ Cost of Vehicle Body (Glider)

$/gallon Gasoline Price

$ Cost of Motor

$/year Life cycle cost per year

d1 kW Engine Size

d2 kW Motor Size

d3 kWh Battery Size

D # Driving days per year

kWh/day Electricity Consumption per day% Coulombic Efficiency

% Engine Efficiency

% Generator Efficiency

% Gear Box Efficiency

% Motor Efficiency

Kg/s Minimum Total Fuel Consumption for Drive Cycle

N Frictional Force

g Gallons/day Gasoline consumption per day

Ampere Battery Current

Kg Cost FunctionSec. Time StepJ/Kg Gasoline Lower Heating ValueKg Vehicle Mass

Kg/s Fuel Mass Flow

N # Total Number of Seconds

# Vehicle Gear Number

# Ring Gear Teeth Number

# Sun Gear Teeth Number

% Charge Efficiency

Mile/Gallon CS Efficiency

Mile/kWh CD Efficiency

% Charging Efficiency$/metric ton CO2 Carbon Price

Watt Battery Power

J/s Engine Chemical Power

J/s Engine Mechanical Power

Watt Motor Power

Watt Generator Power

Watt Demanded Vehicle Power

# Final Gearbox Ratio

m Wheel Radius

Charging Resistance

Discharging Resistance

Battery Inner ResistanceMiles/Day Daily Driving Distance

Miles All Electric Range Distance

SOC % State of Charge (x)

Nm Engine Torque

Nm Final Torque (Crankshaft)

Nm Generator Torque

Nm Motor Torque

Sec. Unit Time

Nm Total Torque

Nm Vehicle Torque

rd/s Engine Speed

Nm Engine Torque

Volt Battery Voltage

metric ton CO2/KWh GHGs Related to Battery Productions per KWh

metric ton CO2GHGs Related to Battery

production of “d3”size

metric ton CO2/kWh CO2 Emissions per kWh production

metric ton CO2/day Annual CO2 Emissions due to electricity consumption

metric ton CO2/gallon CO2 Emissions per gallon gasoline consumed

metric ton CO2/day Annual CO2 Emissions due to gasoline consumption

metric ton CO2/year Total life cycle GHG emissions

m/s Vehicle Linear Velocity% State Variable ( SOC )% Initial SOC

% Final SOC

rd/s Engine Angular Velocity

rd/s Final Angular Velocity

Final Angular Acceleration

rd/s Generator Angular Velocity

rd/s Motor Angular Velocity

Motor Angular Acceleration

rd/s Vehicle Angular Velocity

Vehicle Angular Acceleration

# Teeth Number Ratio (Sun/Gear)

References

[1] Fathy, Coupling between design and control

[2] Kirk, Optimal Control Theory and Introduction