A Framework for the Integrated Optimization of Charging and Power Management in Plug-in Hybrid Electric Vehicles

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1

Abstract— This paper develops a dynamic programming (DP)

based framework for simultaneously optimizing the charging and

power management of a plug-in hybrid electric vehicle (PHEV).

These two optimal control problems relate to activities of the

PHEV on the electric grid (i.e., charging) and on the road (i.e.,

power management). The proposed framework solves these two

problems simultaneously in order to avoid the loss of optimality

resulting from solving them separately. The framework

furnishes optimal trajectories of the PHEV states and control

inputs over a 24-hour period. We demonstrate the framework

for 24-hour scenarios with two driving trips and different power

grid generation mixes. The results show that addressing the

above optimization problems simultaneously can elucidate

valuable insights. For example, for the chosen daily driving

scenario, grid generation mixes, and optimization objective, it is

shown that it is not always optimal to completely charge a battery

before each driving trip. In addition, the reduction in CO2

resulting from synergistic interaction of PHEVs with an electric

grid containing significant amount of wind power is studied. The

paper’s main contribution to the literature is a framework that

makes it possible to evaluate tradeoffs between charging and on-

road power management decisions.

Index Terms— Optimal Control, PHEV, Charging and Power

Management

I. INTRODUCTION

HIS HIS paper examines the problems of optimizing

the charging trajectory of a PHEV and optimally

managing PHEV power when driving. We develop an

algorithm to solve these two optimal control problems

together, thereby accounting for their interdependence. Unlike

a more traditional HEV, a PHEV is expected to exchange

significant amounts of energy with the electric grid and

provide vehicle-to-grid (V2G) services [1, 2]. In doing so,

PHEVs make it possible to replace some of the petroleum

currently being used for propulsion with other sources of

energy. To maximize this synergy, our overarching goal is to

simultaneously optimize PHEV activity on the electric grid

(i.e., charging) and on the road (i.e., power management).

This work was supported in part by the U.S. NSF under Grant #0835995.

Rakesh Patil (e-mail: [email protected]) and Zoran Filipi are with the

Department of Mechanical Engineering and Jarod Kelly is with the School of Natural Resources and Environment at the University of Michigan, Ann

Arbor,MI 48109 USA

Hosam Fathy is with the Department of Mechanical and Nuclear Engineering, Pennsylvania State University, University Park, PA 16802, USA

The literature motivates this work by highlighting several

non-trivial tradeoffs between PHEV design/control on the one

hand and charging optimization on the other. In [3], authors

reveal a tradeoff between one PHEV battery degradation

mechanism (anode-side resistive film formation) and

operating energy costs. Mitigating this tradeoff involves

charging the PHEV battery just enough to complete a given

driving trip, rather than fully charging the battery before the

trip [3]. Furthermore, Traut et al. show that the PHEV battery

size that minimizes lifecycle greenhouse gas (GHG) emissions

is not necessarily the maximum battery size [4]. The literature

also explores the use of PHEVs for V2G services such as

provision of spinning reserves and regulation. These services

can reduce GHG emissions [5], enable integration of

intermittent renewable power sources, and provide cheaper

reserves [6, 7]. Depending on the services that a PHEV

provides to the grid, it may not be possible to charge the

PHEV’s battery optimally or fully. In addition, using PHEVs

in such activities strongly influences their initial conditions for

on-road power management [8, 9], thereby motivating the

simultaneous optimization of PHEV charging and power

management.

The literature already presents significant research on

optimal PHEV charging and power management, but the

problems are considered separately. Optimal charging refers

to the problem of finding a State-of-Charge (SOC) trajectory

that minimizes some environmental or economic objective for

a given PHEV over the span of a day [9-11]. The goal is

typically to minimize PHEV costs of electricity consumption,

minimize GHG emissions, or provide V2G services to the

grid. Optimal power management refers to the problem of

delivering power to a PHEV’s wheels in the most effective

manner. Common objectives for power management are

minimization of gasoline consumption or costs associated with

a given trip [12-15]. Existing studies on optimal charging and

optimal power management typically either assume or require

the PHEV battery to be full at the onset of any given driving

trip. Our contribution is to relax this assumption and consider

the optimization of the two problems together, to enable a full

exploration of tradeoffs and synergies between these two

problems, for the first time.

Several optimal control methods have been applied to

optimal charging/power management. A review of the

methods is presented in [16]. Examples include Dynamic

A Framework for the Integrated Optimization of

Charging and Power Management in Plug-in

Hybrid Electric Vehicles

Rakesh Patil, Member, IEEE, Jarod Kelly, Zoran Filipi, Member, IEEE, Hosam Fathy, Member, IEEE

T

mailto:[email protected]





2

Programming (DP) [12, 13], Pontryagin Minimum Principle

(PMP) [14], predictive control [17], game theory [18], and

static optimization methods [3, 4]. Assuming a certain grid

mix and driving behavior, we use a Deterministic Dynamic

Programming (DDP) approach. This approach explicitly uses

Bellman’s principle for optimal control, and thus guarantees

global optimality. The output of this process is a supervisory

control trajectory that can be used to gain insights and extract

rules for subsequent implementation.

In this paper, the integrated optimization framework is

utilized to obtain trajectories to minimize CO2 emissions. In

addition, we examine the CO2 reduction resulting from a

synergistic interaction of PHEVs with an electric grid

containing significant amounts of wind power. There is an

obvious CO2 emissions benefit of charging a PHEV from a

grid with more renewable sources. However there are two

major challenges to be addressed. First, the intermittency of

renewable sources requires more storage and regulation on the

grid [19], and using PHEV batteries for these purposes has

been studied [20, 21, 22, 23]. Second, achieving maximum

synergy requires optimized charging and driving of a PHEV

[3, 21]. In this paper we will focus on the second challenge

using our integrated optimization approach for a fair

comparison of cases rather than the insightful simulations

carried out in preliminary studies [21].

The main contributions of this paper are the DDP-based

integrated framework for optimal charging and optimal

driving and the results showcasing the value of this approach.

A quantification of total CO2 reduction through PHEVs

depending on the amount of wind on the grid is presented. The

results show that considering the dependence between the two

problems is necessary to attain total optimality. The major

challenges in this work are the integration of two problems

with different dynamics and time scales in a single optimal

control framework retaining computational tractability, and

subsequent analysis of the results. The integrated optimization

framework presented in this paper was first developed in [24],

and the incorporation of wind energy into this framework was

first explored in [25]. This paper connects these two articles

and demonstrates the flexible use and insights of using our

integrated framework.

The remainder of this paper is organized as follows.

Section II presents our PHEV and grid models. Section III

describes the optimization algorithm and framework used.

Section IV discusses the optimization results, showing the

need for our integrated combined optimization approach.

Finally, section V demonstrates the impacts of wind

penetration on reduced CO2 emissions through PHEVs.

II. MODELS

The optimal control problem is solved to obtain optimal

trajectories of the states and inputs related to PHEV charging

and on-road power management. In this section, we present

the system's states and inputs, and the constraints related to

them. First, the PHEV powertrain model is described. Then,

the interaction between the electric grid and the PHEV battery

is described. Finally, the drive cycles and CO2 traces over

which the optimal control solutions are calculated are

presented.

A. PHEV Powertrain Model

The series hybrid electric powertrain model is considered

for this study. It is schematically shown in Fig. 1 and the

component sizes are listed in Table I. The power electronics

contain a parallel bus which splits the electric current between

the generator, battery and the motor. The engine is

mechanically detached from the wheels.

The Engine, Generator and the Motor are modeled using

static maps obtained from the Powertrain System Analysis

Toolkit (PSAT). These maps provide the efficiency of each

component as a function of the component’s operating speed

and torque. The maps for battery voltage and internal

resistance are based on the Li-ion battery chemistry and are

also obtained from PSAT.

Fig. 1. Series hybrid powertrain model

TABLE I

COMPONENT SPECIFICATIONS OF THE POWERTRAIN

Component

Specification

Engine MY04 Prius 1.497 L gasoline, 57 kW, 110 Nm max

torque at 4000 RPM.

Generator Permanent magnet. 400 Nm max torque, 58kW peak

power

Battery Li-ion, 7.035Ah 75 cell SAFT model scaled by current

capacity to 13.3 kWh total energy

Motor Permanent magnet. 715 Nm max torque, 130 kW peak

power

Final Drive Ratio = 3.07

Resistance

Coefficients

f0 = 88.6 N, f1 = 0.14 N-s/m, f2 = 0.36 N-s2/m2

Equations governing the powertrain model dynamics are

represented as nonlinear ODEs. This powertrain model has 2

states: engine speed (ωe) and battery state-of-charge (SOC).

The inputs used to control the model are the fuel flow rate to

the engine ( ̇ ) and the torque demand from the generator

(τg). This model is adopted from earlier work focusing solely

on the PHEV power management problem, and is explained in

detail in [12]. To keep the model description succinct, only

the differential equations related to the states are presented

here.

For a chosen drive cycle (i.e. reference velocity trajectory),

the vehicle power demand at the wheels can be calculated.

Since the motor speed is proportional to the vehicle velocity,

the motor operating point and the subsequent motor power

Power Electronics

Engine Generator Motor Final Drive and

Wheels

Battery





3

demand at the power electronics (Pm,elec) can be calculated.

This power demand is imposed as a constraint power that the

powertrain has to satisfy at every time step. The power

flowing across the battery terminals are a result of the driving

motor and the generator powers. Thus we get the equation,

,b m elec gP P P (1)

relating the electrical power in the motor and the generator,

Pm,elec and Pg respectively, to the battery (Pb). As the battery is

represented by equivalent circuit model, the equations

describing its current (Ib) and rate of change of SOC are,

max

2

oc oc i b

b

i

b

V V 4R PI

2R

IdSOC

dt Q

(2. a-b)

where Voc and Ri are the open-circuit voltage and internal

resistance of the battery, which are functions of SOC. The

constant Qmax is the maximum charge capacity of the battery.

The equations describing engine operation are

( , )e e f e

e ge

flwhl

f m

d

dt J

(3. a-b)

where the engine fueling rate ( ̇ ) is an input to the model,

ωe is the engine speed, which is a state variable, τe is the

engine torque, τg is the generator torque and Jflwhl is the inertia

of the flywheel. The torque output for a given fueling rate is

obtained through the fueling rate map fe.

Several constraints affect the PHEV during its on-road

operation. The torque and power outputs of the generator,

engine and the motor cannot exceed certain speed-dependent

limits. The engine speed is a state variable and is constrained

to be a maximum of 4500 RPM. Battery SOC, is constrained

to be between 0.3 and 0.9, and the charging/discharging power

of the battery is constrained to fall within SOC-dependent

bounds. Equation (1) describes a constraint on the power flow

between the components and is implicitly implemented in the

model. These constraints form the sets X and U for driving in

Equation (4c).

B. Battery Charging and Electric Grid Dispatch Model

This section develops a model intended for optimizing the

way a single PHEV charges with grid electricity. The intent

behind this model is twofold. First, the model captures the

dynamics of battery charging, namely, Equation (2). The

model therefore has one state variable, x = {SOC} and one

input, u = {Pb} in comparison with the 2 states and 2 inputs

for the complete powertrain presented briefly in the previous

section. Second, the model allows us to quantify the grid CO2

emissions associated with PHEV charging (i.e., the emissions

associated with generating the electricity supplied to the

PHEV). To achieve this second goal, we employ an economic

grid dispatch model presented by Kelly et al. in [26]. This

dispatch model contains a list of power plants in Michigan

arranged in ascending order of generation costs ($/kWh). The

model simulates grid dispatch for a given total power demand

profile (obtained from the EPA e-grid database), and outputs

the resulting dollar costs and CO2 emissions associated with

meeting the demand. The model is also capable of

accommodating PHEV power demand.

Figure 2 illustrates a baseline scenario we use when

optimizing the power management and charging of a given

PHEV. The scenario focuses on the State of Michigan, and

assumes that the total number of PHEVs charging from the

grid at any given time is 15% of the total Michigan passenger

vehicle fleet. This percentage translates to 1.25 million

PHEVs [27], charging from the grid at the maximum possible

rate at all times during the day (which results in a grid load of

2062.5 MW). This very large number of PHEVs plugged into

the grid corresponds to a PHEV market penetration level in

the 20%-25% range. Assuming the total number of PHEVs

charging with grid electricity at every instant in time to be

constant provides a simple baseline scenario that can be easily

refined by accounting for household travel data. Our goal,

here, is to obtain baseline profiles of total grid power demand

and grid CO2 production per unit energy. The CO2 rates (Fig.

2) is in the 550-850 g/kWh range, irrespective of the grid load.

We use the profiles of CO2 production per unit energy

(described in next subsection) to optimize the charging and

power management of a single PHEV for minimal emissions.

Fig. 2 Grid Power demand and CO2 production for a year under different

PHEV penetration scenarios

A different set of constraints affect the PHEV during grid

charging than during on-road operation. The charging outlet

considered is rated at 110 V and 15 A. Thus the charging

power can be a maximum of 1.65 kW. Battery SOC is

constrained to be between 0.3 and 0.9, as it was for PHEV on-

road usage. The battery cannot discharge to the grid in the

case studies considered herein. These constraints form the sets

X and U for charging operation in Equation (4c).

C. Drive Cycle and CO2 Trajectories for Optimization

The objective in our optimal control problem is to minimize

total CO2 produced by a given PHEV’s use of fuel and

electricity. Three 24-hour CO2 trajectories were chosen from

0 2 4 6 8 10 125

10

15

20

25

30

Grid P

ow

er

Dem

and [

GW

]

0 2 4 6 8 10 12

600

700

800

Months

CO

2 [

g/k

Wh]

w /o PHEV

1 million PHEV

1.25 million PHEV





4

the CO2 per unit energy data for an entire year (in Fig. 2).

These three traces reflect days of high, medium and low CO2

(Fig. 3) relative to the CO2 traces for the entire year, covering

the entire range of CO2 rate (550-850 g/kWh). We use these

traces to account for the CO2 corresponding to PHEV

charging. Hourly grid demand data are used in the dispatch

model [26]. Hence we assume that the CO2 production

remains the same for each hour.

Fig 3. CO2 traces used for optimization

Fig 4. Drive cycle velocity graphed as a function of distance

Fig 5. A 24 hour plot of grid CO2 production and CO2 driving. (The 24 hour

period begins at 1AM, hence 8AM is the 7th hour in the graph above)

We optimize PHEV charging and power management for a

day where the PHEV performs two identical trips (23.9 miles

long) in the morning and afternoon. The total daily driving

distance of 47.8 miles necessitates either gasoline usage or

multiple battery charging events during the day, thereby

elucidating some interesting tradeoffs between battery

charging and fuel consumption. Fig. 4 shows the vehicle

velocity for each trip as a function of travel distance (obtained

from a naturalistic driving database collected at the University

of Michigan Transportation Research Institute [28]).

Furthermore, Fig. 5 shows the timing of the trips during the

day. Daily driving distance and traffic conditions substantially

affect PHEV energy usage [29]: a fact that has motivated

researchers to optimize PHEV operation over large families of

trips [13, 14]. This work, in comparison, focuses on

developing a framework to show the interplay between

optimal charging and power management. We illustrate this

framework for the trip details in Figs. 4-5, but the framework

is broadly applicable to any trip profile. For example, a day

with multiple driving trips of different lengths with their

velocity trajectories and trip start times defined can be studied

as easily as the driving conditions chosen above. This feature

of the framework is easy to understand from the description of

the framework in section III.C below.

III. OPTIMAL CONTROL FRAMEWORK

The optimal control problem is formulated as follows:

Minimize

* * * *24

2 2f dr b ch

0

CO COm t P tJ

gal kWh

subject to ( , )x f x u

(4. a-c)

,x X u U where the optimization objective, J, is the total amount of

CO2 produced during a 24-hour period. This CO2 is produced

by both driving fuel usage (the first term in the expression for

J) and grid charging (the second term). Gasoline consumption

is ̇ gallons per time step during driving ( ).

Battery charging power is given by Pb in kW and the charging

time step is . It is important to note that the

dynamics and constraints given in the above equations are

different depending on whether the PHEV is driving or

charging. These differences in the dynamics and constraints

are explained in Section II. The states and inputs of the system

are x and u respectively, and they are governed by the system

dynamics (4b). The admissible ranges of x and u are X and U

respectively. The constraints on the states (set X) and on the

inputs (set U) are also briefly described in Section II. In the

remainder of this section, the implementation of the system

model (4b) for optimization is briefly explained first. Then

the use of this model with the discrete Bellman equation for

DP is explained. Finally, the integration of the two problems

into a single optimal control algorithm is presented.

A. Backward Looking Powertrain Model

The system model represented briefly in (1-3), and in more

detail in [12], is discretized for simulation using an Euler

discretization. The model itself is simulated as a backward

looking or a-causal model. It is of the form

( ) ( ( ), ( ))u k g x k x k 1 (5)

Given the current states (at time k) and future states (at time

k+1) of the model, the inputs required for that transition can

0 4 8 12 16 20 24550

600

650

700

750

800

850

Grid C

harg

ing C

onditio

ns

CO

2 [

g/k

Wh]

Time [hr]

medium CO

2 day

high CO2 day

low CO2 day

0 5 10 15 20 250

50

100

Distance [mi]

Vehic

le

Velo

city [

mph]

0 4 8 12 16 20 24600

700

800

Grid

CO

2 [

g/k

Wh]

24 hours CO2 and vehicle velocity

0 4 8 12 16 20 240

50

100

Vehic

le

Velo

city [

mph]

Time [hr]





5

be calculated. This is in contrast with the more popular

approach of using a forward looking or causal model [13, 14,

30], which calculates the future states given the current states

and inputs,

( ) ( ( ), ( ))x k 1 f x k u k (6)

A backward looking model can be used because the

powertrain components are appropriately sized and the vehicle

model can satisfy all the power demands at the wheels.

Additionally, we verified that the model (8) has a one-to-one

correspondence (bijection) between the inputs (u(k)) and

future states (x(k+1)), for given current states.

The use of a backward looking powertrain model in our DP

method offers several advantages without which the problem

would be computationally intractable. These advantages are

explained in the next subsection and in more detail in [12].

B. Dynamic Programming

The Bellman equation is written as (11), which is the DP

algorithm used for optimal control in our work.

( , ( ))( , ( )) min { ( ( ), ( )) ( , ( ))}

i

i i j j

X k 1 x kV k x k c x k x k 1 V k 1 x k 1

(7)

where c(xi(k),x

j(k+1)) is the cost of transitioning from state

xi at time k to x

j at time k+1 and V(k,x

i(k)) is the optimal cost

to go from state xi at time k to the final time. The state space is

divided into a finite number of states at every time step that

belong to the set X(k) = {x1(k),x

2(k),…x

i(k),…x

j(k),…x

N(k)}. In

(7), the optimal cost to go from state xi at time k is calculated

as a minimization of costs associated with all reachable

transitions to states in X(k+1) (i.e. X(k+1,xi(k))). This equation

is evaluated starting from final time Tf, where the cost to go

function at the final time V(Tf,X(Tf)) is initialized for all states.

Then (7) is repeated for every state at every time step

proceeding backward in time till the initial time. Through this

process, we obtain V(1,X(1)), which gives the optimal cost to

go from every state at the initial time and is used to find the

optimal initial states and the consequent optimal trajectories.

Use of the backward looking model with the DP algorithm

shown in (7) has the following advantages compared to

previous approaches in the literature [13, 14, 30]. First, the

algorithm setup does not need interpolations to calculate the

value function as only transitions to states in X(k+1) are

considered. Second, when considering the set X(k+1) of all

possible future states, the battery charge/discharge power

limits and other such constraints on the states are used to limit

the set of allowable future states. This drastically reduces the

number of powertrain simulations. It also overcomes accuracy

issues (discussed in [31]) regarding implementing the

constraints using penalty methods. These two key advantages

make the DP problem computationally feasible in our case.

C. Integrated Optimization Framework

The two optimal control problems - of charging and power

management, have the same objective (i.e. to reduce CO2

emissions), but number of states, the system dynamics,

constraints and the time steps involved are different.

Furthermore, as illustrated in Fig. 5, the charging and driving

events occur though out the day and the optimal

trajectory/actions taken during one event affects the other.

To tackle these differences, but still use the Bellman

equation (7) effectively, ensuring optimality, a complete

optimal framework is used. The constraints on the state at the

boundaries of the charging and driving events are used to

integrate these two problems. Since the engine speed state (ωe)

has to be zero at the beginning of a driving trip and ωe can be

any value at the end of the trip, we use these constraints to

connect the two problems. This framework proceeds

backwards in time, starting at the end of the 24 hour period

under consideration (see Fig 5). The following steps (and Fig.

6) explain the method used:

Step 1 - Optimal charging is the problem under

consideration at the final time step, so the value function is

initialized for every discretized value of SOC only.

Step 2 - Equation (7) is then applied proceeding backwards

in time until the end of the second driving trip.

Step 3 - At this time step the value function is modified

from a 1D array (V1) with entries for each discretization of

SOC only to a 2D array (V2) with entries for each

discretization of SOC and ωe. This is done by concatenating or

repeating the same value function for every discretization of

ωe.

Step 4 - Equation 7 is then applied proceeding backwards in

time until the beginning of the second driving trip with (V2).

Step 5 - At this time step the value function is modified

back from a 2D array (V2) to a 1D array (V1) by using the

value function entries associated with only the zero engine

speed state as that is the only allowable engine speed at the

beginning of a driving trip.

Step 6 - Follow steps 2 through 5 until the beginning of the

first driving trip.

Step 7 - Follow step 2 until the initial time step to get the

value function (cost-to-go) for all SOC discretizations.

Essentially the loop in Fig. 6 is followed twice starting with

the initialization block.

Fig 6. Complete Optimal Framework

Execute Bellman Equation

with grid charging model

Execute Bellman Equation with

vehicle model

Modify V by

concatenation. Same

function for all ωe

V = Rm

cost to go

for each SOC

V = RnxRm

cost to go for

each SOC and ωe

m = 4500 SOC grid points

n = 30 ωe grid points

V - Value function

..

..

1

2

1

V

V n

V

( , )x f x u

{ , }ex SOC

{ , }g fu m

Modify V by

extraction. Only ωe =0

is allowed at start of

trip ( )1 2V V 1

V = RnxRm

cost to go for

each SOC and ωe

V = Rm

cost to go

for each SOC

{ }x SOC { }bu P

Initialize V

for each SOC





6

During step 2, the system dynamics have one state and input

with constraints related to grid charging, while during step 4

the states, system dynamics and constraints are those related to

PHEV on-road usage. This framework is computationally

feasible due to the use of a backward looking model and the

interpretation of the Bellman equation as presented in (7). This

framework can also be used for solving a generic combination

of optimal control problems which are linked through a state

variable (SOC in our case). Furthermore, it can be extended to

link multiple optimal control problems with well-defined

transition of states from one problem to the other.

IV. RESULTS

The framework presented in the previous section is used to

obtain optimal state and control input trajectories for a 24 hour

period. The driving trips occur at the 7th and 16th hour of the

24 hour period respectively. The initial and final SOC are the

same to ensure no undue advantage of using electricity. For

each optimization case, the optimal trajectories are obtained

for initial and final SOC of 0.3 through 0.9 in increments of

0.1 (seven runs). Three optimization cases are examined, for

the three CO2 cases of high, medium and low (Fig. 3). It is

important to note while analyzing the results, that these CO2

trajectories are obtained on different days from the Michigan

grid mix, which is coal intensive. The CO2 produced by

gasoline combustion is 8.78 kg/gal [32].

Previous studies regarding grid integration of PHEVs

assume PHEV population numbers and complete control over

all PHEV batteries when not being used on the road [20, 22,

23]. These studies are extremely relevant to understand the

total load and controllability aspects of the load. However, we

wish to study the impacts of PHEV grid charging on its

power-management and vice versa. Thus, our studies focus on

a single PHEV’s 24 hour charging opportunities. In other

words, this study focuses on the optimal control strategies that

result in the least CO2 impact for that individual PHEV owner

only.

Fig 7. Optimal trajectories for a day with high CO2 from the grid

Three important observations can be inferred from the

results. First, the framework produces optimal trajectories that

factor in electric grid conditions while making power

management decisions and vice versa. For example, on a day

when grid CO2 rate is higher (Fig. 7) than gasoline CO2 rate,

the solutions do not encourage charging, but instead use a

significant amount of gasoline while driving. For the day

when grid CO2 rate is lower (Fig. 8), the battery is charged

sufficiently to complete the first trip in all-electric and in

excess before the second trip. For the medium CO2 day (Fig.

9), the grid CO2 rate before the first trip is not as favorable to

encourage charging which results in a small amount of

gasoline usage for the first trip. Such tradeoffs between grid

charging and power management are specific to minimizing

CO2, the chosen grid mix, the driving distance and trip times

during the day. However, the framework allows us to expose

these tradeoffs depending on the choice of the objectives and

conditions.

Fig 8. Optimal trajectories for a day with low CO2 from the grid

Fig 9. Optimal trajectories for a day with medium CO2 from the grid

Second, the framework can be used to assess the impacts of

specific grid activities on a PHEVs power management and its

ability to achieve a chosen objective. For example, we enact a

constraint to restrict PHEV charging during the peak load

hours of the grid (hours 12-18). Fig. 10 shows that this results

in lower amount of charging before the second trip resulting in

0 4 8 12 16 20 240

0.5

1

SO

C

Optimal Trajectories for a day with High CO2 from grid

0 4 8 12 16 20 240

1

2

3

Charg

ing P

ow

er

[kW

]

Fuel [k

g]

0 4 8 12 16 20 24

600

700

800

Grid

C

O2 [

g/k

Wh]

Time [hr]

Cumulative Fuel [kg]

Grid Charging [kW]

0 4 8 12 16 20 240

0.5

1

SO

C

Optimal Trajectories for a day with Low CO2 from grid

0 4 8 12 16 20 240

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ing P

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[kW

]

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g]


Grid Charging [kW]

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CO

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g/k

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Optimal Trajectories for day with Medium CO2 from grid

0 4 8 12 16 20 240

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Time [hr]





7

higher fuel usage for the second trip compared to the

unconstrained case. However, due to a significant section of

the day having low CO2 grid rate (hours 8-22 in Fig. 10), most

of the charging still occurs during favorable grid CO2 periods.

Thus, the constraint results in only a 1% increase in the total

CO2, but the framework shows how on-road power

management is affected.

Fig 10. Optimal trajectories with no charging constraint at peak load (hours

12-18) for a day with medium CO2 from the grid

Fig 11. Optimal trajectories for a day with medium CO2 from the grid for different initial SOC

Finally, we study the impact of different initial SOC

conditions. In the previous cases (Figs. 7-10), the initial SOC

was chosen as 0.7. In Fig. 11, we observe that the optimal

trajectories are heavily influenced by the initial SOC. The total

CO2 produced in the case for 0.3, 0.7 and 0.9 initial SOC are

8.02 kg, 7.64 kg and 7.61 kg respectively. The 0.7 and 0.9

initial SOC cases have sufficient battery energy due to

previous evening's charging (indicated by the initial charge in

the battery) and hence require little or no charging before the

first trip. For the initial SOC = 0.3 case, if we imagine that the

consumer could not charge the battery in the previous evening,

then it is not optimal from a CO2 perspective to charge the

PHEV overnight completely. Rather, it is optimal to drive the

first trip on gas and charge the battery for the second trip. This

non-intuitive solution is possible to obtain only due to the

consideration of both the charging and power management in

a single framework.

V. IMPACTS OF WIND PENETRATION

This section investigates the reduction in total CO2

emissions resulting from an individual PHEV's charging from

a grid with significant wind penetration. We use the integrated

optimal control framework presented in section III to study the

CO2 reduction impact of a cleaner grid through PHEVs.

Recent forecasts suggest that the electric grid will include a

significant amount of renewable power sources [2, 33].

Studies suggest a significant reduction in CO2 emissions

resulting from a grid with higher penetration of renewable

resources [2, 33]. A PHEV is considered a key enabler in

clean transportation through charging from a cleaner grid. In

this paper we specifically analyze this CO2 emissions impact

of using PHEVs with a cleaner grid.

A. Wind Power and Grid Dispatch and CO2

The wind power data used in our studies are obtained from

the Eastern Wind Integration and Transmission Study

(EWITS) [34]. AWS-Truewind created this dataset with

assistance from NREL/DOE. This data consists of three years

(2004-2006) of 10-minute time step wind speeds. The study

evaluated potential sites for wind power development and

forecasted wind power output at sites with high potential. In

our work, projected wind power from offshore wind in the

state of Michigan is used.

Fig 12. Wind Power trace summed over all sites and averaged over all days of a year (Yearly Averaged Traces).

Due to significant variability in wind speeds over time and

space, obtaining optimal charging and power management

trajectories for a specific chosen wind power trace is of little

use. Fig. 12 shows a graph of wind power output, which is

summed up over all offshore locations, and averaged over all

days of the year. The datasets show that the wind output has

significant temporal variability, but averaging over all days of

the year produces identical 24-hour wind traces for the 3

0 4 8 12 16 20 240

0.5

1

SO

C

Optimal Trajectories with constrained charging for aday with medium CO

2 from grid

0 4 8 12 16 20 240

1

2

3

Charg

ing P

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[kW

]

Fuel [k

g]


Grid Charging [kW]

0 4 8 12 16 20 24

600

700

800

Grid

CO

2 [

g/k

Wh]

Time [hr]

No Charging

0 4 8 12 16 20 240

0.5

1

SO

C

Optimal trajectories for day with medium CO2 from grid

0 4 8 12 16 20 240

5

10

Cum

ula

tive

C

O2 [

kg]

0 4 8 12 16 20 24600

700

800

Grid

CO

2 [

g/k

Wh]

Time [hr]

SOC = 0.7

SOC = 0.9

SOC = 0.3

0 5 10 15 20 251000

1200

1400

1600

1800

Time [hr]

Pow

er

[MW

]

One Year Averaged Power from all sites

2004

2005

2006





8

years. We call these the Yearly Averaged Traces. In our study,

we only use data for 2004, and will focus on that particular

yearly averaged trace.

Fig 13. Wind Power traces on different days compared to the yearly averaged

trace.

Fig 14. Amount of power and energy supplied by wind in different cases to

satisfy grid power demand

Fig 15. CO2 traces produced by a grid mix with different amounts of wind

To assess the performance of an average optimal solution

on different days, three days with different levels of wind

power are considered. A day with 85 percentile wind energy

content was selected as a "high wind" day. The "medium day"

has 50 percentile and the "low day" is chosen as a day with 15

percentile wind energy content. Fig. 13 shows the wind traces

corresponding to these days along with the yearly averaged

trace.

The actual amount of power and energy supplied by wind

power to satisfy the grid demand is shown in Fig. 14. It is

observed that that on the high wind day, up to 18% of the grid

energy is from wind, compared to only 4% on the low wind

day. Fig. 14 also shows that even on the day with lowest wind,

at some point during the day, wind power supplied close to

15% of the demand. Hence all these scenarios describe

significant wind penetration levels compared to current levels.

Finally, Fig. 15 shows the CO2 traces resulting from the

different amounts of wind supplied during a particular day.

These traces impact the optimization algorithm's decision to

charge from the grid as explained in the previous results

section.

B. Results

In each case 7 optimization problems are solved

corresponding to an initial and final SOC ranging from 0.3 to

0.9 in steps of 0.1. The results are highly sensitive to the

driving behavior and the grid mix assumptions. However, our

goal is to analyze each optimal solution in detail and learn

from these 'best case' scenarios.

The results we present will be the optimal trajectories for an

individual PHEV which interacts with the grid and follows

driving pattern as described in section II.C. The following are

the highlights of the results

1. Total CO2 emissions impact of a PHEV decreases due

to inclusion of wind power (even in the case of lowest

wind power)

2. Due to our integrated charging and power management

framework, we capture the optimal benefits of wind

power on minimizing propulsion CO2. It is shown that

the optimal trajectories vary significantly depending on

the wind penetration.

3. The loss of optimality resulting from wind variability

in Fig. 13 is quantified by providing a comparison of

the optimal result for each case and the 'yearly average'

result.

Next, we explain each of these results in more detail. It is

obvious that adding large quantities of wind will result in

reduced CO2 per unit energy produced on the grid (Fig. 15). It

is less straightforward to understand the impact of increased

wind power on reduction in CO2 produced from

transportation. Comparing the cases without any wind power

on the grid and with a low amount of wind power on the grid

(low amount of wind power as shown in Fig. 13), we observe

a reduction in total CO2 impact of a PHEV. Fig. 16 shows the

resulting optimal trajectories and CO2 impact (2% lower with

wind) over 24 hours. The difference in the charging

trajectories is observed in the first few hours, when the CO2

output of the grid is lower due to wind.

A comparison of optimal trajectories for the three days with

different levels of wind penetration is shown in Fig. 17. The

differences are due to - 1) the tradeoffs between the CO2

produced by gasoline fuel and electricity and 2) due to varying

0 5 10 15 20 250

500

1000

1500

2000

2500

3000

3500

4000Wind Power on chosen days

Time [hr]

Win

d P

ow

er

[MW

]

Yearly Average

High Day

Medium Day

Low Day

0

5

10

15

20

25

30

High Med Low Avg. Yr.

Max(%)

Min(%)

Energy(%)

Wind Power and Energy Penetration (%)

0 4 8 12 16 20 24550

600

650

700

750

800

850

900

Time [hr]

Grid C

O2 [

g/k

Wh]

Daily CO2 traces from different wind patterns

Average Wind

High Wind

Medium Wind

Low Wind

No Wind





9

amounts of wind throughout the day, resulting in different

CO2 from the grid at different times. Thanks to an integrated

approach, we can capture these differences. For example, in

the medium wind case, we observe that higher grid CO2 (due

to low wind) during hours 10-16 results in the optimal solution

using some gasoline for the second trip rather than charging

before the second trip. Such tradeoffs between gasoline usage

and charging are seen in all the cases depending on the CO2

produced by the chosen grid mix.

Fig 16. Optimal Trajectories for days with no and low wind

Fig 17. Optimal Trajectories for days of high, low and medium wind

As mentioned in section V.A, it is not practical to

implement optimal solutions for every single day due to wind

variability. So we obtain an optimal solution for the yearly

average wind case. This average optimal solution is simulated

for different wind power days and its performance is

compared to the respective optimal CO2 on these days in Fig.

18. In this manner we quantify the loss of optimality resulting

from wind variability. In particular, there is a maximum loss

of ~10% amongst the cases considered. Looking at the

variation in wind power in Fig. 13 and in the wind penetration

in Fig.14, it can be observed that despite such variations, the

final CO2 loss (of optimality) by applying the average optimal

solution is 10% in the worst case. This is due to the narrower

range of variations in the resulting CO2 produced by the grid

as seen in Fig. 15. Quantifying the optimality lost in reducing

CO2 emissions through PHEVs due to variability of wind

power on the grid is a unique finding of this study.

Fig 18. Comparison of loss of CO2 reduction for different cases

VI. CONCLUSION

An integrated optimal framework which considers the

dependence between optimal charging and optimal on-road

power management is developed. By exploiting the constraints

on the engine speed state at the boundaries of the charging and

power management problems, these two optimal control

problems with different number of states, different dynamics

and time scales are integrated. The steps of the framework are

detailed and applied to 24 hour scenarios of charging and

driving. Three cases with days of high, low and medium CO2

are studied along with two naturalistic driving trips.

The results show that it is necessary to consider an

integrated approach to optimally charge and drive a PHEV.

Utilizing our novel framework, for the objective of

minimizing CO2 under the driving and grid conditions

assumed in this paper, it is shown that it is not optimal to fully

charge the battery before every driving trip as assumed in the

literature. The framework was also applied to assess the

optimal CO2 reduction benefits for PHEVs achievable from an

increased wind power scenario on the grid. The results show

that our integrated approach produces synergistic results with

a 2% CO2 reduction even on a day with low wind. Since it is

not practical to implement optimal solutions for every day due

to wind variability, the loss of optimality resulting from using

optimal solutions for an average case is presented.

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0 4 8 12 16 20 240

0.5

1

SO

C

Comparison of Optimal Trajectories for days with no wind and low wind

0 4 8 12 16 20 240

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cum

ula

tive

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kg]

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Rakesh M. Patil (M’12) received

his B.Tech. degree in Mechanical

Engineering from Indian Institute of

Technology, Kharargpur, in 2007,

where he received the Institute

Silver Medal. He received his M.S.

and Ph.D. degrees in Mechanical Engineering from the

University of Michigan, Ann Arbor, in 2009 and 2012

respectively. He has worked as a technical consultant for the X

Prize foundation and is currently a Postdoctoral Researcher at

NEC Labs America.

Jarod C. Kelly is an assistant

research scientist in the School of

Natural Resources and Environment

at the University of Michigan. He

received his PhD (2008) and MS

(2005) in mechanical engineering

from the University of Michigan, and

his BS (2003) in mechanical

engineering from the University of Oklahoma.

Zoran Filipi received the M.S. and

PhD. degrees in Mechanical Engineering

from the University of Belgrade,

Belgrade, Serbia, in 1987 and 1992

respectively.

He is currently a Professor and

Timken Endowed Chair in Vehicle





11

System Design at the Clemson University, International

Center for Automotive Research (ICAR). He was with the

University of Michigan between 1994 and 2011, where he

rose to the rank of Research Professor, and served as the

director of the Center for Engineering Excellence through

Hybrid Technology and the deputy director of the Automotive

Research Center, a U.S. Army Center of Excellence for

modeling and simulation of ground vehicles.

He is the Editor-in-Chief of the SAE International Journal

for Alternative Powertrain, and the Associate Editor of the

ASME Journal of Engineering for Gas Turbines and Power.

His primary research interests are advanced engine concepts,

alternative powertrains, and energy for transportation. In

particular, Filipi’s efforts on simulation and analysis of hybrid

propulsion systems focus on architecture, design and control

of electric and hydraulic hybrid powertrains, and powertrain-

in-the-loop integration. He has published more than 150

papers in international journals and refereed conference

proceeding papers.

Dr. Filipi is a Fellow of the Society of Automotive

Engineers (SAE) and a member of the SAE Executive

Committee for Powertrains. He is the recipient of the SAE

Forest R. McFarland Award, the IMechE Donald Julius Groen

Award, IMechE Journal of Automobile Engineering Best

Paper Award, and the University of Michigan Research

Faculty Achievement Award.

Hosam K. Fathy earned his B.Sc.,

M.S., and Ph.D. degrees – all in

Mechanical Engineering – from the

American University in Cairo in 1997,

Kansas State University in 1999, and

University of Michigan in 2003,

respectively. He was a Control Systems

Engineer at Emmeskay, Inc. (now part

of LMS International) in 2003/2004. He

served as a postdoctoral Research Fellow and Assistant

Research Scientist at The University of Michigan from 2004-

2006, and from 2006-2010, respectively. Since 2010, he has

served as an Assistant Professor at Penn State University. His

Control Optimization Laboratory (COOL) currently employs

12 graduate, undergraduate, and postdoctoral researchers, and

focuses on battery health modeling and control, as well as the

optimal control of energy systems.

Documents

A Framework for the Integrated Optimization of Charging and Power Management in Plug-in Hybrid Electric Vehicles