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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 7, SEPTEMBER 2011 2949

Energy-Optimal Control of Plug-in Hybrid ElectricVehicles for Real-World Driving Cycles

Stephanie Stockar, Vincenzo Marano, Marcello Canova, Giorgio Rizzoni,Fellow, IEEE, andLino Guzzella,Fellow, IEEE

AbstractPlug-in hybrid electric vehicles (PHEVs) are cur-rently recognized as a promising solution for reducing fuel con-sumption and emissions due to the ability of storing energythrough direct connection to the electric grid. Such benefits canbe achieved only with a supervisory energy management strategythat optimizes the energy utilization of the vehicle. This controlproblem is particularly challenging for PHEVs due to the possi-bility of depleting the battery during usage and the vehicle-to-gridinteraction during recharge. This paper proposes a model-basedcontrol approach for PHEV energy management that is basedon minimizing the overall CO2 emissions produceddirectly and

indirectlyfrom vehicle utilization. A supervisory energy man-ager is formulated as a global optimal control problem and thencast into a local problem by applying the Pontryagins mini-mum principle. The proposed controller is implemented in anenergy-based simulator of a prototype PHEV and validated onexperimental data. A simulation study is conducted to calibratethe control parameters and to investigate the influence of vehicleusage conditions, environmental factors, and geographic scenarioson the PHEV performance using a large database of regulatoryand real-world driving profiles.

Index TermsEnergy management, optimal control, plug-inhybrid electric vehicles.

NOMENCLATURE

Cnom Battery nominal capacity.E Energy.f Torque split factor.H Hamiltonian.Ibatt Battery current.J Cost functional.K Final state penalty term.L Lagrangian.mCO2 CO2mass flow rate.mequiv Equivalent mass flow rate.mf Fuel mass flow rate.QLHV Fuel lower heating value.

Pbatt Battery power.

Manuscript received July 4, 2010; revised January 6, 2011 and March 31,2011; accepted May 8, 2011. Date of publication June 2, 2011; date of currentversion September 19, 2011. The review of this paper was coordinated byDr. C. C. Mi.

S. Stockar, V. Marano, M. Canova, and G. Rizzoni are with the Center forAutomotive Research, The Ohio State University, Columbus, OH 43212 USA(e-mail: [email protected]).

L. Guzzella is with the Department of Mechanical and Process Engineering,Eidgenoessische Technische Hochschule Zurich, 8092 Zurich, Switzerland.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2011.2158565

R Battery internal resistance.s Equivalency factor.S Target set for the final state.t Time.T Torque.u Control law.Vbatt Battery voltage.Voc Open circuit voltage.x State variable.

Efficiency.ch Battery charger efficiency. Lagrange multiplier.i Initial condition for the Lagrange multiplier. Specific CO2content.l Scalar Lagrange multiplier.CS Fraction of driving cycle in CS mode. Angular velocity.x Feasible set for the state.AER All electric range.

BSA Belted starter alternator.

CS Charge sustaining.

CD Charge depleting.

ECMS Equivalent consumption minimization strategy.EM Electric motor.

HEV Hybrid electric vehicle.

ICE Internal combustion engine.

PHEV Plug-in HEV.

SDP Stochastic dynamic programming.

SoC State of charge.

SoE State of energy.

I. INTRODUCTION

PHEVs are today considered to be a solution to reduce fuelconsumption and CO2 emissions in the transportation sector.

Compared with conventional hybrid vehicles, the high-capacity

energy storage system of PHEVs and the ability to recharge

the battery through connection to the electric grid provide

the opportunity to control the battery depletion during vehicle

utilization, ultimately improving fuel economy.

Various studies have shown that the performance of PHEVs

depends on several factors, many of which have little or no

influence on CS hybrids and conventional vehicles [1][7]. To

name a few, the length of the driving path, the contribution of

the electricity on the overall energy consumption of the vehicle,

the cost of the electric energy, and its specific CO2

content have

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2950 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 7, SEPTEMBER 2011

been recognized as predominant factors in the assessment of

fuel economy and emissions for PHEVs.

A subject of strong interest on the part of the automotive

industry is to understand the implications of different energy

management strategies on fuel consumption, CO2 emissions,

battery life, range, and performance. To this extent, one of the

critical challenges for control design is to properly account forthe grid energy in the vehicle energy optimization problem.

Some methods have been so far proposed to design supervi-

sory controllers for PHEVs, including the minimization of an

equivalent fuel consumption, or the vehicle operating costs or

the cumulative CO2 emissions [2][4], [8], [9].

Due to the complexity of the control problem, heuristic

methods have often been considered [8], [10][12]. Although

rule-based methods are often successfully employed in the

industry, it is generally found that the controller design process

is cumbersome and time consuming, and its results are limited

to specific vehicle design and usage conditions. For this reason,

model-based approaches can improve on all of these drawbacks

and yield more cost-effective solutions.

SDP appears more indicated, particularly when a small num-

ber of reference driving profiles can be found as statistically

representative of the vehicle utilization [13]. Recent results

show improvements in fuel economy, operation costs, and emis-

sions [8], [14]. The SDP approach, however, requires significant

amount of data to provide a statistically relevant validation

framework. Further, the policy evaluation typically requires

large computation time, partially overcome by estimating the

control policy offline and tabulate the results in the actual

implementation.

The ECMS is a well-known approach for the online energy

management of HEVs that has recently been adapted to the su-pervisory control of PHEVs [15][20]. The proposed approach

is based on assuming that the energy expended by the vehicle

can be converted into an equivalent consumption of fuel.

The results presented lead to the conclusion that near-optimal

fuel economy can be achieved if the control algorithm depletes

the battery proportionally to the driving distance. However, this

implies that the vehicle velocity profile must be known a priori.

Such condition prevents the ECMS to be generalized, requiring

calibration of the equivalency factor for each driving profile.

Furthermore, the assumption of converting the battery energy

into an equivalent fuel mass flow rate is not formally applicable

to PHEVs since the electric energy stored from the grid dependson the energy generation mix.

This paper presents a novel supervisory energy management

strategy for charge-depleting hybrid vehicles that accounts for

the vehicle primary energy consumption, including the fuel

energy and the electric energy from the grid. The structure of

the proposed algorithm is general and adaptable to different

vehicle architectures (series, parallel, and seriesparallel) and

to any number of power splits.

The proposed approach is based on the formulation of a

global optimal control problem that minimizes the global CO2emissions produced (directly and indirectly) by vehicle use.

Pontryagins minimum principle is then applied to obtain a

local minimization problem. The control strategy is applied to aforward-oriented simulator of a seriesparallel PHEV and used

Fig. 1. Diagram of the prototype PHEV drivetrain.

TABLE IDESCRIPTION OF THE VEHICLEDRIVETRAINCOMPONENTS

to conduct vehicle performance analysis, evaluating the impact

of the control parameters for a variety of vehicle utilization andenvironmental scenarios.

This paper is organized as follows: After an overview of

the hybrid vehicle configuration and the model adopted for the

control study, a description of the energy management strategy

and its implementation into a control algorithm are given.

Simulation results are presented to evaluate the sensitivity of

the proposed control strategy to vehicle usage conditions and

environmental and geographic scenarios, also providing an

assessment of vehicle performance, fuel consumption, and CO2emissions.

II. DESCRIPTION OF THEV EHICLEAND OF THES IMULATOR

The vehicle considered in this study is a seriesparallel

prototype PHEV built on a midsize SUV platform [21], [22].

As shown in Fig. 1, the vehicle drivetrain includes a downsized

diesel engine coupled with a BSA and a six-speed automatic

transmission on the front axle and an EM on the rear axle.

Table I describes the main vehicle components. The configu-

ration chosen for this vehicle allows for a variety of operating

modes, such as pure electric drive, electric launch, engine load

shifting, motor torque assist, and regenerative braking [22].

A forward-oriented energy-based simulator was developed

and validated using a combination of driving tests and lab-oratory test data [21][24]. Fig. 2 describes the information

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Fig. 3. Block diagram of the drivetrain power flows.

to find a solution to the optimal control problem that can be

implemented on a vehicle.

Note that the approximation of converting an electrical

energy utilization into a fuel mass flow rate introduces the

equivalency factor s(t). This calibration parameter has a con-siderable impact on the battery SoC during a driving path.

For this reason, the equivalency factor must be optimized or

adapted based on the specific vehicle driving profile considered

to achieve optimal fuel economy and CS operations [27], [37].

Experimental results, however, show that the ECMS performs

close to the global optimum with modest calibration effort, with

the advantage of being implementable online [21], [31].

IV. OPTIMAL C ONTROLP ROBLEMF ORMULATION FOR

PLU G-I NH YBRID E LECTRICV EHICLEENERGYM ANAGEMENT

To define a supervisory energy management strategy for

PHEVs, an optimal control problem for charge-depleting sys-

tems is formulated here.

Compared with the energy management problem formulation

previously presented, the constraint on the final SoC defined in

(8) must be eliminated to enable charge-depleting operations.

Further, the equivalence between the battery energy usage

and the fuel mass flow rate shown in (9) is formally incorrect

for PHEVs, where the battery energy is mostly provided by the

grid, hence decoupled from the fuel energy. This implies that

the cost function must be redefined for PHEVs.In this paper, the cost function is defined to account for the

primary energy consumed by the vehicle during a driving path.

The most representative indicator of the well-to-wheel energy

utilization of a PHEV is given by the cumulative CO2emissions

produced by vehicle utilization, i.e.,

JPHEV(u) =

tbta

mCO2,f(t) + mCO2,e(t) dt (10)

wheremCO2,f represents the mass CO2 produced by the con-sumption of fuel (when the engine is utilized), and mCO2,e

results from the consumption of the electric energy storedonboard.

To apply the optimal control theory to the PHEV energy

management, the variables mCO2,fand mCO2,emust be relatedto vehicle system variables as follows:

mCO2,f(t) =1 Pfuel(t)

mCO2,e(t) =2 Pbatt(t)

ch(11)

where, according to Fig. 3, Pfuel is the power associated withthe fuel utilization and is determined as follows, assuming a

lower heating value(QLHV= 43MJ/kg):

Pfuel(t) = mf(t) QLHV. (12)

The term ch= 0.86 in (11) represents the battery chargerefficiency (when the vehicle is connected to the grid) [38],

and1 and2 are defined as the specific CO2 content in fueland electricity per kilowatthour. Note that 1 corresponds tothe engine brake specific CO2 (BSCO2), which can readilybe calculated from fuel consumption data. The term 2 canreasonably be estimated based on the average CO2 content

of the electricity generation mix for a specific geographic

region [39].

To account for the energy stored in the battery, the SoE is

introduced as

SoE(t) =Ebatt(t)

Enom(13)

where Enom= Cnom Voc is the nominal battery energy (inkilowatthours). Considering SoE as the new state variable in-

stead of the SoC, the state equation of the system becomes

d

dtSoE(t) = (SoC(t))

Pbatt(t)

Enom(14)

where is defined according to (5), and Pbatt is the battery

power, which is defined as positive if discharging. Note that, ifVbatt(t) =Voc, then SoE= SoC.

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STOCKARet al.: ENERGY-OPTIMAL CONTROL OF PHEVs FOR REAL-WORLD DRIVING CYCLES 2953

Based on the preceding state equation, the control variable

u(t)for the energy management problem can be defined as thevector

u(t) =

Pbatt(t);

PEM,el(t)

Pbatt(t)

(15)

where the second element represents the power split between

the rear EM and the BSA electric power outputs. Since the me-

chanical power demand to the drivetrain is known, the electric

EM and BSA power can be obtained from the efficiency maps

of the two components and simple energy balances, according

to the power flow diagram in Fig. 3.

To respect the physical limitations imposed by the drivetrain

components, the control and state variables are subject to

constraints. In particular, the battery SoE and power must be

limited to prevent abuse and aging-related issues [40], i.e.,

SoEmin

SoE(t) SoEmax

Pbatt,min Pbatt(t) Pbatt,max (16)

where, usually, SoEmin= 0.25, and SoEmax= 0.95. Furtherconstraints stem from the power limits of the drivetrain

components

PEM,min PEM(t) PEM,max

PBSA,min PBSA(t) PBSA,max (17)

where the power limits are functions of the EM and BSA speed.

V. SOLUTION OF THEP LU G-I NH YBRID E LECTRIC

VEHICLEE NERGYM ANAGEMENTP ROBLEM

The optimization problem previously defined is tackled using

Pontryagins minimum principle [35], [36], which, in principle,

allows one to obtain closed-form expressions for locally opti-

mal control signals.

In the general case, an explicit control signal can only be

found by solving a two-point boundary value problem. For the

specific problem at hand, an optimal solution can be found by

adopting the reasonable simplifications shown below.

The starting point is a description of the system dynamics

dx(t)

dt =qf(x(t), u(t), t) (18)

with the cost functional

J(u) =

tbta

L (x(t), u(t), t) dt + K(xb, tb). (19)

The theorem introduces the Hamiltonian function

H(x(t), u(t), (t), t) =L (x(t), u(t), t)+(t)f(x(t), u(t), t)(20)

which has to be minimized at each time t to provide the optimalcontrol policy. Ifuo(t) is the optimal control policy, then thefollowing necessary conditions must be satisfied:

i) dxo(t)

dt = H|o = f(x

o(t), uo(t), t)

ii)

do(t)

dt = xH|o

iii) xo(ta) =xaiv) xo(tb) S R

n

v) H(xo(t), uo(t), o(t), t) H(xo(t), u(t), o(t), t)

If the statex(t)is bounded, namely

vi) xo(t) x(t)t [ta, tb]

x(t) = {x Rn|G(x, t) 0;G: Rnx[ta, tb] R}

whereG(x, t) defines the inequality constraints, an additionalterm is introduced in the Hamiltonian function to account for

this limitation. The corresponding Lagrange multiplier is ascalar denoted byl and subject to the KuhnTucker condition

vii) ol 0.

For the PHEV control problem, an extended Hamiltonian

function is defined based on the cost functional in (10) and the

state constraints on the battery SoE

H(x(t), u(t), (t), (t), t) =1 Pfuel(t)+

+Pbatt(t)

2ch

(t) (SoC(t))

Enom(t) (SoC(t))

Enom

(21)

with

(t) =

l, if SoE(t) SoEmaxl, if SoE(t) SoEmin0, else

(22)

where Pfuelcan be calculated from the engine fuel consumptionmaps, as in Fig. 3, and l is the scalar Lagrange multiplier forthe inequality constraints on the SoE.

The extended Hamiltonian function allows one to include the

state constraints within the same optimal control problem. Note

that (21) provides necessary conditions for optimality according

to the previously mentioned conditions. Such formulation, how-

ever, can lead to suboptimal results if the state constraints are

active. When this occurs, the optimal value for the parameter lis unknown and should be determined by applying conditions

i)vii). Since the time intervals during which the state is sliding

along the upper or lower boundary are limited in occurrence,

the value for 0l is determined here by a trial-and-error proce-dure [35].

The necessary condition for the costateo(t)is

do(t)

dt =xH|o=

x(1Pfuel(t))

x

2 Pbatt(t)

ch

+

+ x

Pbatt(t) (SoC(t))

Enom (o(t) + (t))

(23)

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with the initial condition o(t= ta) =i. Since no explicitcondition is given fori, this parameter needs to be calibrated.

The ordinary differential equation (ODE) for the costate

o(t) can further be simplified since Pfuel(t) and Pbatt(t) donot depend on the battery SoE (or SoC). However, the preceding

assumption is not valid for the battery efficiency. In fact, the bat-

tery power isPbatt(t) =Ibatt(t) Vbatt(t), whereas the maxi-mum battery power during discharging isPmax(t) =Ibatt(t) Voc(SoC(t), t). This will further penalize any operation at lowSoE when the battery efficiency is lower.

Inserting this expression in (23), the costate equation can be

rewritten as follows:

do(t)

dt =

Pbatt(t)(o(t)+(t))

Enom batt

x , ifPbatt(t)< 0

Pbatt(t)(o(t)+(t))

Enombatt(SoC(t),t)2

battx

, ifPbatt(t) 0.

(24)

According to the minimum principle, the control policy

denoted byu

o

(t)is optimal ifH(x

o

(t), u(t),

o

(t), t)presentsa global minimum with respect touo(t).As a final remark, a proof of equivalence between the ECMS

and the solution of the optimal control problem through Pon-

tryagins minimum principle was obtained for the CS HEV case

in [27] and [31]. This proof is extended here to the charge-

depleting PHEV case. In fact, the ECMS formulation presented

in (9) can be made equivalent to the Hamiltonian function

defined by (21) if the equivalency factors(t)is defined as

s(t) = 21 ch

(SoC(t), t)

Enom 1 ((t) + (t)) . (25)

VI. IMPLEMENTATION OF THEE NERGY

MANAGEMENTS TRATEGY

The solution of the optimal control problem defined by (21)

can be applied to forward-oriented models or to a vehicle

control system. Fig. 4 illustrates a procedure for the implemen-

tation of the solution into a control algorithm. Note that, al-

though the vehicle drivetrain includes three propulsion systems

(namely, an engine and two EMs), the proposed implementation

allows for the optimal torque split between an arbitrary number

of power generation elements.

According to Fig. 4, the variables fICE and fBSA definethe fraction of the torque demand to the drivetrain Treq thatis commanded to the engine and to the BSA, respectively. By

conducting an energy balance on the system in Fig. 3, three

matrices containing all the possible torque combinations that

satisfy the drivetrain demand are generated, i.e.,

TICE(t) = fICE Treq(t) Rnxm

TBSA(t) = fBSA(1fICE)Treq(t) Rnxm

TEM(t) =(1fBSA)(1fICE)Treq(t) Rnxm (26)

where the dimensions m and n are related to the chosenresolution for the factors fICE and fBSA. The torque request

Fig. 4. Flowchart describing the implementation of the energy managementalgorithm.

at the driveshaft Treq is evaluated using the driver acceleratorand brake commands andas follows:

Treq(t) =(t) T+max+ (t) T

max (27)

where T+maxis the maximum positive torque that the powertraincan generate combining ICE, BSA, and EM, whereas Tmaxis the maximum negative torque that can be absorbed by the

electric machines (BSA and EM), accounting for battery power

limitations.

The torque delivered by each component is then limited

according to (17). Note that the torque variables defined are

considered as mechanical and, hence, calculated at the shaft of

each component.

The electrical power provided by the battery and the power

associated with the engine fuel utilization are then computed

to evaluate the Hamiltonian function in (21). Specifically, Pfuelis determined from the engine fuel consumption, according to

(12), whereas the power of the electric machines is computed

from the efficiency maps for the BSA and EM, i.e.,

PEM,el(t) =TEM(t) EM(t) EM,el

PBSA,el(t) =TBSA(t) ICE(t) BSA,el (28)

where, for the rear EM, EM,el = 1/EM if the machine isworking as a motor, and EM,el = EM if it is working as agenerator.

For each torque split combination that satisfies the precedingconstraints, the Hamiltonian function is defined based on (21).

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In doing so, the expression(batt/SoE)in (24) is explicitlycalculated according to (5), i.e.,

dbatt(SoC(t))

dSoE =

Ibatt(t)

Voc(SoC(t))R (SoC(t))Ibatt(t)

R (SoC(t))SoC(t) R (SoC(t))Voc(SoC(t))

Voc(SoC(t))SoC(t). (29)

Note that, since the parameters Voc and R are continuouspiecewise polynomial functions [28], they can be differentiated

in the entire SoC range.

At any time step, the combination foICE and foBSA that

minimizes the Hamiltonian function matrix is chosen as the

solution of the optimization problem. It is worth observing that

the proposed algorithm, although suitable for implementation

into forward-oriented simulators or hardware-in-the-loop sys-

tems for control development and testing, cannot directly be

applied to real-time control due to the required computationand numerical optimization of the Hamiltonian function at each

time step. However, the computation effort can significantly

be decreased by precomputing the Hamiltonian function and

importing the results as maps in the vehicle control system. A

similar approach was adopted for the implementation of ECMS

to a CS HEV [21], [33], [37].

VII. RESULTS ANDA NALYSIS

The energy management algorithm was applied to the

forward-oriented PHEV simulator to conduct an evaluation of

the vehicle performance for a variety of usage conditions.The focus of the study conducted is on the effects of the con-

trol parameters on the vehicle fuel economy and CO2emissions

and the influence of driving conditions and energy generation

scenarios.

A. Vehicle Driving Scenarios

The characteristics of the driving profile have a strong impact

on the calibration of the PHEV control algorithm [10], [11],

[15], [41]. In this paper, a rich set of driving profiles was

adopted as a validation framework for the energy manage-ment control algorithm, analyzing scenarios consistent with the

driving behavior of customers and improving the generality of

the results.

The simulations were conducted on a set of regulatory and

real-world driving profiles extracted from a database of fleet

study data to statistically represent typical usage conditions of

a PHEV, including urban, extra-urban, and highway segments

with variable driving length [42].

Table V in the Appendix lists the main characteristics (veloc-

ity and energy demand at the wheel) of all the driving cycles

considered in this study. The cycles are all characterized by

a driving distance greater than the vehicle AER. This allows

for the possibility of depleting the battery, depending on thecalibration of the energy management strategy.

Fig. 5. Example of velocity profile for the controller validation (indicated asPath 3 in Table V).

Fig. 6. Summary of electricity generation mix for four countries (sources:[43][47]).

Fig. 5 shows the velocity profile of one of the nonregulatory

cycles considered. This pattern is representative of mixed-mode

driving conditions, alternating urban driving and a highway

segment.

B. Electricity Generation Scenarios

The impact of the electricity generation mix on the PHEV

utilization was evaluated by varying the specific CO2emission

coefficient 2 to encompass different energy generation sce-narios, including electricity production from both fossil fuel

and renewable sources. Four of the values considered for 2are representative of the energy generation mix for the U.S.,

Switzerland, France, and Germany, as summarized in Fig. 6.

For simplicity, it will be assumed that the grid energy

consumed by the PHEV has the same specific CO2 content

as the generation mix. Note that this must be considered an

approximation for the European countries, where the open

energy market may cause differences between the CO2content

of the electricity produced by each country and that consumedby the vehicle.

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C. Definition of Controller Parameters

and Performance Metrics

Based on the optimal control problem in (21), the parameters

requiring calibration are the initial condition for the Lagrange

multiplier i and the scalar Lagrange multiplier l (whichvaries the penalty on the battery SoE constraints).

The impact of the foregoing parameters will be evaluatedthrough three different metrics. First, the evolution of the

battery SoE during the driving pattern and its final value

SoEfinal(i, l)will be considered. Then, the overall CO2masscalculated with (10) and (11) will be evaluated as

mCO2 =1mfQLHV+ 21

chEnomSoE (30)

whereQLHV is the fuel lower heating value, and SoE is thedifference between initial and final SoEs.

Another variable is introduced to indicate whether the vehicle

is operating in CD or CS mode, hence identifying how fast the

control strategy depletes the battery. The variable CS definesthe fraction of the driving cycle where the vehicle operates in

CS mode at its lower SoE bound, i.e.,

CS= tCStcyc

. (31)

Specifically, tCSis calculated by considering the time duringwhich the vehicle operates within a 5% window aroundSoE= SoEmin. In the following results, the battery is assumedat SoE= SoEmax= 95%at the beginning of each cycle.

Knowledge of the fraction of the driving cycle in CS mode

is not only relevant for energy optimization but for reliability,

safety, and aging issues as well [48].

D. Analysis of Simulation Results for One Driving Cycle

and One Energy Scenario

To illustrate the results, one case study will be analyzed in

detail with reference to the driving cycle shown in Fig. 5 and the

U.S. energy generation scenario. Simulations were conducted

to evaluate the vehicle CO2 emissions, the fuel economy, and

the utilization of the battery energy in relation to the control

parameters.

Fig. 7 reports the values of the final battery SoE obtained

by varying the parameters i and l. Note that an undesiredcomplete depletion of the battery is possible for certain combi-nations of the control strategy parameters.

It is evident that l affects the ability of the controller torespect the state constraints. In particular, the SoE exceeds its

boundaries when l is below a threshold (for the consideredscenario,l

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Fig. 8. Overall vehicle CO2 emissions as a function ofi and l for thecase study (cycle Path 3, U.S. scenario). (a) Contribution from fuel energy.(b) Contribution from electric energy. (c) Combined.

Fig. 10 shows the fraction of cycle duration where the vehicle

operates in CS mode at its lower SoE bound as a functionof the control parameters. For high values ofi, the vehicle

Fig. 9. Fuel economy of the PHEV as a function ofi and l for the casestudy (cycle Path 3, U.S. scenario).

Fig. 10. Percentage of cycle in CS mode at the low SoE bound as a functionofi and l for the case study (cycle Path 3, U.S. scenario).

is operated in CD-CS mode, and the SoE reaches the lower

bound before the end of the driving path. For the driving

cycle considered, CS is slightly below 40%, meaning thatapproximately 60% of the energy requested to the drivetrain

can be satisfied with the battery.

For i> 10, the control strategy forces the vehicle to depletethe battery and, when the lower SoE bound is reached, switchesto CS mode. Conversely, as idecreases, CSdecreases steeplyto zero, and wheni 0, the control strategy is no longer ableto deplete the battery. At this condition, the final SoE is near

the same value as the initial value, hence, the control strategy

tends to operate the system in CS mode at the higher SoE

bound.

This is confirmed in Fig. 11, where the evolution of the

battery SoE during the driving cycle is represented for four

different values ofi, whereasl is set constant. Intermediatesolutions are observed for values ofi included within thetwo bounds.

In particular, a valuei= 6 allows the battery to be graduallydepleted during the cycle, reaching the lower SoE bound only at

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Fig. 11. Battery SoE profile during the driving cycle forl = 18varyingi(cycle Path 3, U.S. scenario).

Fig. 12. Optimal value of the initial conditioni as a function of vehicleenergy demand for different driving cycles (U.S. scenario).

the end of the driving pattern and avoiding any CS operations.

This operation, which is known as blended mode, allows for the

achievement of the minimum vehicle fuel consumption along a

prescribed driving cycle [17].

E. Effects of Driving Cycle Characteristics

For all the vehicle driving profiles listed in Table V and the

U.S. energy generation scenario, a full factorial design of ex-

periments was generated, varying the control parameters iandlof the supervisory controller. Then, the optimal combination(i, l)opt was determined by minimizing the CO2 emissionsproduced by the PHEV.

Fig. 12 summarizes the results of the simulations, represent-

ing the optimal value of the Lagrange multiplier i against thevehicle energy demand at the wheel calculated for the driving

cycles considered in the study. The parameter l was set to aconstant value to ensure that the constraints on the battery SoEare always respected.

As Fig. 12 shows, the results tend to cluster within a limited

range of values for the parameter i and are almost inde-pendent on the energy demand at the wheel. A sensitivity

study was conducted to evaluate the influence on the cost

functional J(u) of errors in the optimal value of the controlparameter i. The analysis was conducted with reference to fivespecific driving patterns, representing the limit scenarios for

Fig. 12.

Table II summarizes the sensitivity results to variations in

i around the optimal value corresponding to each of the five

driving cycles considered. In all the cases, the sensitivity ofthe vehicle CO2 emissions is very limited. The results confirm

TABLE IISENSITIVITYANALYSIS OF THE COS TFUNCTIONALJ(u)

TO THEPARAMETERi (U.S. SCENARIO)

that, for the energy generation scenario considered, the control

strategy is relatively insensitive to the characteristics of the

driving pattern [26].

The behavior can be justified by considering that the pa-

rameter i is the initial condition of the costate ODE of theoptimal control problem. Therefore, its influence on the optimal

solution progressively decreases with the duration of the driving

cycle as(t)converges.In summary, the simulation results show that the vehicle CO2

emissions are relatively insensitive to the Lagrange multiplieri for the considered energy generation scenario. Furthermore,the optimal value of the control parameter, which allows the

vehicle to operate in blended mode with minimum energy

consumption, is nearly independent from the driving cycle

duration and vehicle energy demand.

Conversely, the parameter l has no impact on the vehicleperformance but ensures satisfaction of the constraints on the

battery SoE bounds. Specifically, a threshold value can be iden-

tified for l so that the state constraints are always respected,allowing one to reduce the controller calibration problem to the

sole parameteri.

This presents advantages for parameter tuning as near-optimal results can be achieved with minimal calibration effort,

in particular, without the need for information, such as the

driving length.

F. Effects of Energy Generation Scenarios

To extend the validation framework, different scenarios were

considered to evaluate the sensitivity of the control parameter

i to different values of the energy generation mix. As anexample, this analysis was initially limited to the sample driving

cycle shown in Fig. 5. Fig. 13 represents the vehicle CO2emis-

sions and engine fuel consumption against the parameter ifor the four different energy generation scenarios shown inFig. 6.

The U.S. and German energy production scenarios are rela-

tively similar, with the high specific CO2content of the electric

generation mix causing a relatively flat response of the overall

vehicle emissions to the control parameteri.Conversely, the case of Switzerland and France is rather

different, as the energy generation is predominantly composed

by renewable or low CO2primary sources. These two scenarios

offer promising opportunities for a large PHEV penetration.

Here, a higher sensitivity in the vehicle CO2 emissions can be

observed with respect to the control strategy parameter.

Fig. 14 illustrates the influence of the specific CO2content ofthe grid energy on the optimal value of the Lagrange multiplier

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Fig. 13. Impact of the energy generation mix on CO2 and fuel consumptionagainst the parameter i for different energy generation scenarios (cyclePath 3).

Fig. 14. Influence of the energy generation mix parameter2 on the optimalvalue of the parameter i (cycle Path 3).

Fig. 15. Optimal value of the initial conditioni as a function of vehicleenergy demand for different driving cycles (Swiss scenario).

i with reference to the sample driving cycle, indicating alinear correlation between 2 and i. This suggests that thecalibration of the PHEV supervisory controller could be done

when the battery is connected to the grid based on the specific

CO2 content of the electricity generation during the charging

operation.

Fig. 15 summarizes the optimal value of the Lagrange multi-

plieri against the vehicle energy demand at the wheel for allthe driving cycles considered. The low specific CO2 content of

the electric energy generation in Switzerland causes a different

behavior compared with that observed in Fig. 12 for the U.S.

scenario. Although the results are still clustered in a limited

range of i, a slightly increase dependence of the optimal

parameter value with the driving cycle energy demand can beobserved (see Table III).

TABLE IIISENSITIVITYANALYSIS OF THECOS TFUNCTIONALJ(u)

TO THEPARAMETERi ( SWISSS CENARIO)

This behavior indicates that the optimality of the control

strategy (and, consequently, the PHEV fuel consumption and

CO2 emissions) is more affected by the driving cycle charac-

teristics as the electric energy is predominantly generated from

renewable sources.

Similar to the U.S. scenario, a sensitivity study was con-

ducted on the cost functional J(u) varying the parameter ifor different driving patterns. It is possible to notice here the

increased sensitivity of the vehicle CO2 emissions to errors in

the optimal value of the control parameter.On the other hand, a considerably large error must be given

toi to detect sufficiently high variations in the cost functionalJ(u). This indicates the presence of a relatively large regionaround the sweet-spot, where the CO2 emissions and the

vehicle performance vary only marginally.

VIII. CONCLUSION

This paper has presented a novel approach to the supervisory

energy management of PHEVs. This paper has addressed the

fuel consumption and CO2emissions associated with the PHEV

use through a well-to-wheel energy balance that explicitlyaccounts for the fuel energy and grid energy utilization.

An optimal control problem was formulated by defining

a cost functional based on the cumulative CO2 produced

directly and indirectlyby the vehicle. Pontryagins minimum

principle was then applied to reduce a global optimization prob-

lem to a local minimization, allowing for the control problem

to be solved and implemented in an algorithm.

The control algorithm was then implemented on a validated

energy-based PHEV simulator. Simulations were conducted

to evaluate the sensitivity of the supervisory controller to

different vehicle utilization and energy generation scenarios.

A large database of driving profiles, including regulatory cy-cles and real-world vehicle velocity profiles extracted from

fleet studies data, was considered to provide a validation

framework.

Based on the analysis conducted, the proposed energy man-

agement strategy presents a relatively low sensitivity to the

driving profile characteristics (i.e., the energy demand at the

wheel or the driving distance). This result was achieved because

of the definition of a cost functional that formally accounts

for the different mix of primary energy forms utilized by the

PHEV, representing an improvement over the conventional

control approaches that approximate the energy utilization with

an equivalent fuel consumption metric.

In particular, the vehicle CO2emissions show the presence ofan optimal condition varying the control strategy parameter i,

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as well as a relatively large sweet-spot, where only marginal

variations from the optimal condition occur. Conversely, a

higher sensitivity to the control parameter i was observed onthe battery SoE profile and, ultimately, the vehicle operating

mode.

Furthermore, the sensitivity to vehicle usage conditions and

the tradeoff between fuel and electrical power consumptionare dependent on the specific CO2 emissions associated with

the electricity generation. In particular, a higher sensitivity was

observed for the energy generation scenarios characterized by a

low CO2content.

While this paper does not specifically address real-time

control developments, its insights are valuable when developing

energy management strategies that can lead to more readily

tunable algorithms that can address different objectives. In

particular, the analysis presented in this paper can assist in ad-

dressing differences in electricity generations between different

regions and countries, allowing for the development of energy

management strategies that can achieve, for example, minimum

CO2 emissions in the face of a different mix of electric power

generation feedstocks.

Given the increasing use of geographical information sys-

tems and navigation systems, which can lead to some de-

gree of a priori knowledge of the vehicle trajectory, the

results presented in this paper represent a step forward in

understanding of the potential of formal optimization meth-

ods in guiding the design of real-time energy management

strategies.

APPENDIX

A brief description of the driving cycle characteristics con-

sidered in the validation study is reported here. A combi-

nation of regulatory and real-world driving cycles was used

to validate the proposed supervisory energy management

strategy.

For each cycle, information on the distribution of vehicle

velocity, the driving distance, and the energy demand at the

wheel is provided. The energy demand at the wheel is computed

based on the road load equation [25]

Ewheel=M

te

ti

V

dV

dtdt+

1

2aCxAf

te

ti

V

3

dt+CrMg

te

ti

V dt

(32)

where V is the vehicle velocity, Mis the vehicle mass, ais theair density, Cxis the aerodynamic friction coefficient, Afis thevehicle frontal area,Cr is the tire rolling resistance coefficient,and g is the acceleration of gravity. The foregoing equationneglects the effects of the road grade. The vehicle parameters

are listed in Table IV.

Table V summarizes the most relevant metrics of the driving

cycles. In particular, the distribution of the distance was chosen

to ensure a driving length greater than the vehicle AER as well

as a maximum distance that is representative of typical dailycommuting trips.

TABLE IVVEHICLEPARAMETERSUSED IN(32)

TABLE VSUMMARY OF METRICS FOR THEDRIVING CYCLES

CONSIDERED IN THE STUDY

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TABLE V(Continued.)SUMMARY OFMETRICS FOR THEDRIVING

CYCLESC ONSIDERED IN THES TUDY

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