a framework for the integrated optimization of charging and power management in plug-in hybrid...
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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
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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
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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
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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]
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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
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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
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Optimal Trajectories for a day with High CO2 from grid
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Cumulative Fuel [kg]
Grid Charging [kW]
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600
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C
O2 [
g/k
Wh]
Time [hr]
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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
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Optimal Trajectories with constrained charging for aday with medium CO
2 from grid
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Grid Charging [kW]
0 4 8 12 16 20 24
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g/k
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O2 [
kg]
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g/k
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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
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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
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Daily CO2 traces from different wind patterns
Average Wind
High Wind
Medium Wind
Low Wind
No Wind
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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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Based Value Proposition Analysis of Plug-In Hybrid Electric Vehicles",
0 4 8 12 16 20 240
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1
SO
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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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2 [
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10
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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
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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.