1

Series Hybrid Electric Vehicle Simultaneous EnergyManagement and Driving Speed Optimization

Boli Chen, Simos A. Evangelou and Roberto Lot

Abstract—The energy management (EM) and driving speedco-optimization of a series hybrid electric vehicle (S-HEV) forminimizing fuel consumption is addressed in this article on thebasis of a suitably modeled series powertrain architecture. Thepaper proposes a novel strategy that finds the optimal drivingspeed simultaneously with the energy source power split forthe drive mission specified in terms of the road geometry andtravel time. Such a combined optimization task is formulatedas an optimal control problem that is solved by an indirectoptimal control method, based on Pontryagin’s minimum prin-ciple. The optimization scheme is tested under a rural drivemission by extensive comparisons with conventional methods thatdeal with either speed optimization only or EM strategies withgiven driving cycles. The comparative results show the superiorperformance of the proposed method and provide further insightinto efficient driving.

NOTATIONi Currentl Total Length of Drive Missionm MassP PowerQ Battery CapacityR ResistanceSoC Battery State-of-ChargeSoH Battery State-of-Healths Travelled DistanceT Temperaturev Vehicle Velocityη Efficiencyρ Energy Recovery Factorτ Torqueω Angular SpeedSubscriptsb Battery i Inverterdc DC/DC Converter m Motore Engine r Rectifierg Generator t Transmissionh Mechanical Brake v Vehicle

I. INTRODUCTION

ELECTRIFICATION of personal transport is identifiedas an essential step towards addressing the climate

change threat. Ongoing efforts relate to enabling reductions

B. Chen is with the Dept. of Electronic and Electrical Engineering atUniversity College London, UK (boli.chen@ucl.ac.uk).

S. A. Evangelou is with the Dept. of Electrical and Electronic Engineering atImperial College London, UK (s.evangelou@imperial.ac.uk).

R. Lot is with the Dept. of Engineering and the Environment at theUniversity of Southampton, UK (Roberto.Lot@soton.ac.uk).

This research was supported by the EPSRC Grant EP/N022262/1.

of energy consumed by cars by shifting from conventionalfuel powertrains to more electrified ones. The end of theroad is expected to be fully electric vehicles, however thepenetration of such vehicles is hindered by the perceivedsacrifice of easy and quick refueling that average consumershave enjoyed up to now with conventional vehicles. Hybridelectric vehicles (HEVs) offer a suitable alternative designthat combines the range and refueling convenience of engineswith the cleaner electric energy of electric propulsion systems.HEV powertrains usually contain two or more energy sourcesfor propulsion, such as an internal combustion engine (ICE)and a battery, therefore they are more complex than theircounterparts in conventional vehicles. There is a variety ofarchitectures in which the sources and other components inthe powertrain can be arranged to provide propulsion to theHEV, such as series, parallel and series-parallel architectures,with further subcategories in each case. In this article, thefocus is on series architecture HEVs (S-HEVs) which includeextended-range electric vehicles, such as the BMW i3 rangeextender, the Nissan Note e-power and VIA Motors products.

In terms of powertrain operation, energy management (EM)control strategies are used to decide intelligently on how toprovide energy to the total vehicle load from the multipleenergy sources in HEV powertrains. Despite the commercialsuccess of HEVs, better fuel economy for these vehicles isbeing sought. This can be suitably achieved by new EMcontrol, which remains one of the main challenges for a HEV.

A large number of EM strategies, from rule-based tooptimization-based, have been proposed in the literature [1]–[5], among which the optimization-based approaches are usu-ally heavy in computations, even though they have reliedon predefined speed profiles upon which the optimizationis performed. Instantaneous optimization-based techniques,such as the equivalent consumption minimization strategy(ECMS), represent valid alternatives that reduce the amountof computational burden at the price of less accuracy [6],while heuristic strategies that rely on reproducing the powersplit features of optimization-based strategies [7], [8], usuallyoffer a good compromise between real-time implementabilityand optimality. However, there is no optimality guaranteefor the rule based methods, which mostly originate fromintuition. Machine learning techniques have also been recentlyadopted for HEV energy management [9], [10]. However,the performance is highly dependent upon the DP solutionsbecause the optimal EM method is learned from DP solutionsvia neural networks.

The present paper contributes a new EM strategy based onthe combined and simultaneous optimization of two important

2

Fuel tank SI EnginePMS

GeneratorRectifier

BatteryDC/DC

Converter

DC link & Inverter car

PMS Motor/

GeneratorTransmissionPrimary source

Secondary source

Propulsion Load

Pf Pe Pg Pr

Pi Pm Pt

Ph

Optimal Control Strategy

Speed profiles

Operating Constraints

Information signalMechanical & Electrical Power Flow

Route and

desired

travel time

PdcPb

Fig. 1. Block diagram of the series HEV powertrain.

aspects of a hybrid vehicle: the longitudinal speed profile andthe hybrid powertrain energy flow. The individual optimizationof the latter of these two aspects is indeed well studied in theliterature, as has been described in the previous paragraph.The individual optimization of the vehicle speed is also widelyaddressed for the purpose of minimizing fuel consumption invehicles with conventional powertrains [11]. The combinedoptimization of the two aspects is a non-trivial step thatproduces a new EM methodology [12], which to the best of theauthors’ knowledge has not been addressed sufficiently. Theproposed methodology can lead to the most efficient drivingsince both the vehicle speed and energy management are takeninto account simultaneously in the optimization, in comparisonto the methods that optimize these aspects separately andcombine them a posteriori (this investigation is provided inthis paper). The concerned problem is set up as an optimalcontrol problem (OCP) in which the global minimum fuelconsumption is sought by optimizing the speed profile andenergy management power split throughout a given vehiclemission. An indirect optimal control tool is used in this paperto find the global optimum solution.

Despite the practical implications of following a given speedprofile, which can be enabled by advanced cruise controlsystems, the proposed methodology overall offers a number ofadvantages in comparison to existing EM strategies that takespeed profiles as known inputs: a) it removes the necessityof knowing a priori the driving cycle, which is otherwiseusually unknown in practice, b) it also removes the necessityof predicting the future driving speed, which can otherwiselead to inaccurate speed profile predictions and consequentlysuboptimal powertrain energy management, c) it can lead tothe most efficient driving as already mentioned, d) the drivemission is specified in terms of parameters than can be easilydefined by the user or measured by a navigation system, suchas the current location, the desired destination, and the pathand time to destination, and e) the proposed scheme can benaturally extended to incorporate drive mission uncertainty,such as road traffic, traffic lights and so on, by reformulatingit into a receding horizon problem, although this aspect is notstudied in the present work.

The work in the paper extends in several aspects preliminarywork on the simultaneous optimization of a series HEV speedprofile and energy management in [13] and makes furtherimportant contributions: a) while the previous work considerssolely the electrical behavior of the battery, in this work batteryhealth and thermal effects are additionally taken into accountand modeled, thus making it possible to reveal the impactof fuel efficiency optimization on the battery durability and

safety, b) in contrast to a path with a constant legal speed limitand flat surface used in the previous work, a more realisticdriving environment is created in this paper by developinga framework that takes into consideration the variable legalspeed limit and road slopes of any given path, c) real drivingspeed data is collected in this work by a recently developeddata acquisition device [14] and compared against optimizedspeed profiles for the same path to understand the natureand assess the importance of the speed optimization, d) thebenefit of the newly developed combined and simultaneousoptimization is demonstrated by comparisons with a two-step methodology that involves first conventional speed pro-file optimization followed by conventional powertrain powersharing optimization, and e) the conventional speed profileoptimization is developed further to adopt it to HEV scenariosin which there is powertrain energy recovery, which leads toadditional comparisons to those mentioned in part d).

The remainder of the paper is structured as follows. SectionII introduces the vehicle and powertrain model, including allthe powertrain component models, used in this work. TheOCP and associated operational constraints are formulated inSection III. Section IV solves the OCP for a defined vehiclemission and shows the advantages of the proposed optimiza-tion scheme by comparative results. Finally, conclusions aredrawn in Section V.

II. SERIES HYBRID ELECTRIC VEHICLE MODEL

The proposed series HEV model represents a medium-sizepassenger car, and its powertrain configuration is sketched inFig. 1. The S-HEV has three independent sources of power Pb,Pg , and Ph (corresponding respectively to the battery, genera-tor, and mechanical brakes), which can be mixed to obtain thedesired values of vehicle speed and acceleration. The hybridelectric vehicle also allows regenerative braking that conveysbraking power (negative Pt) through the transmission up tothe battery. Optimal energy management fulfils minimizationof fuel consumption by means of an appropriate power split.

The modeling of each component, which aims to capture theassociated essential physical characteristics and in particularpower losses, is presented next.

A. Propulsion Load

The propulsion load includes all the components from theinverter-driven permanent magnet synchronous (PMS) motorto the wheels, that request the load that has to be satisfied bythe two sources (primary and secondary).

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Fig. 2. Efficiency of the PMS machine (generator = positive torque, motor =negative torque) and operational bounds (blue dotted lines) for the reversiblemotor in terms of current, power, speed and voltage. The dashed horizontallines correspond to constant currents (approximately equivalent to constanttorques) and the vertical dash-dot curves correspond to constant voltages.

1) Three-phase DC to AC converter (Inverter): This workadopts a bidirectional pulse width modulated (PWM) inverter[15] in which the high frequency switching dynamics areneglected by averaging them out [16]. The efficiency ofthe energy transformation is therefore of interest, which issimply modeled as a constant efficiency factor ηi. Thus, thepower balance of the DC link and inverter is described byPi=η

sign(Pr+Pdc)i (Pr + Pdc), where ηi is adjusted according

to the direction of the power flow.2) PMS Motor/Generator: The behaviour of the PMS ma-

chine can be precisely described by a nonlinear dynamic modelin d − q reference frame [17]. This can be further simplifiedbased on the assumptions that the dynamics of electromagneticphenomena are much faster than mechanical ones and that therotor inertia torque is reasonably smaller than the load torque,leading to a set of steady-state algebraic equations in terms ofpower flow [13]:

Pm = ωmτlm, (1a)

Pi = ωm(τlm + τdm)− 2

3Rm

(τlm + τdm)2

(pmλm)2, (1b)

where Pm is the motor mechanical power and Pi is theinverter electric power, τlm is the load torque and ωm isthe rotor angular speed. τdm is the total dissipation torquethat collects a variety of electromagnetic and mechanicaldissipation, including Eddy current, hysteresis, bearing andwindage losses. In this work, τdm is assumed speed dependant,and approximated by a simple fitting technique to provide arealistic representation of the machine efficiency as comparedto experimental data [16]. The stator resistance Rm, the rotormagnetic flux λm and the number of pole pairs pm areconstants specified in Table I given at the end of Section II-D.For the reversible PMS machine, it is convenient to adopt theconvention that when it works as a generator, power (as wellas the load torque) are positive, and the power is assumednegative when it works as a motor. In view of (1), the PMSmotor/generator efficiency ηm = (Pm/Pi)

sign(Pi) may beexplicitly evaluated as a function of the load torque τlm andvelocity ωm, as shown in Fig. 2.

3) Transmission: A transmission with constant ratio gt =10 is used and hence the relation between the motor angular

Fig. 3. Single-track, non holonomic, vehicle model.

speed ωm and the vehicle forward speed v is ωm = gt v.It is assumed that the transmission has a constant efficiencyηt. The bi-directional power flow is hence modeled withPt = η

sign(Pm)t Pm, which adjusts the transmission efficiency

according to the direction of power flow.4) Brakes: Mechanical brakes are simply modeled as power

withdrawal, that is as a source of negative power Ph (Ph ≤ 0always). This power is dissipated as heat.

5) Vehicle longitudinal dynamics: The gross motion of thevehicle is described in terms of longitudinal speed v and yawrate Ω, by using the single-track, non-holonomic vehicle modeldepicted in Figure 3. The longitudinal dynamics is describedby the following differential equation:

mvd

dtv = Fv − FT − FD − FG, (2)

where mv is the overall vehicle mass, FT = fTmvg cos θ,FD = fDv

2 and FG = mvg sin θ are resistance forcesdue to rolling drag, aerodynamics drag and road grade θrespectively. Fv is the longitudinal driving force determinedby the transmission and brakes power

Fv =Pvv

=Pt + Ph

v. (3)

The traveled distance s is calculated by integrating the longi-tudinal speed:

d

dts = v. (4)

Assuming that the wheels are non-sliding, then the vehicleturning can be modeled as: wΩ=v tan δ, where δ is thesteering angle and w the wheelbase. It is reasonable to assumethat when driving on a single lane rural road, the driverremains approximately in the middle of its lane. The roadis thus defined in terms of curvature of the road center Θcalculated from its Cartesian coordinates (x, y) as a functionof the travelled distance s:

Θ(s) =

√(d2x

ds2

)2

+

(d2y

ds2

)2

. (5)

Therefore, the vehicle yaw rate Ω is simply the product ofvehicle speed and road curvature as shown by Ω=vΘ(s).This assumption is not representative of the driver behavior atintersections, hence sharp corners are conveniently convertedinto smoother profiles by properly filtering the curvature.

B. Primary source

1) Internal Combustion Engine: A common 1.8L sparkignition naturally aspirated engine is considered. The engine

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involves very complex dynamics but these are typically muchfaster than the powertrain energy flow dynamics of interestin the present work. Hence, a characterization of the aver-age engine efficiency at steady-state operating conditions isadequate for the present purposes. The engine efficiency isderived from simulations with the industry standard engineand gas dynamics simulation software, Ricardo WAVE, whichhas been calibrated against experimental results. The efficiencyηe(ωe, τe) is expressed as a function of the crankshaft speedωe and brake torque τe at steady-state operating conditions, asshown in Fig. 4. The ICE is assumed to be “idle” for ωe ≤1000 rpm and it has a maximum efficiency ηe,max = 0.343for a brake torque of 151 Nm at 2800 rpm.

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Fig. 4. Efficiency (blue solid) and fuel mass flow rate [g/s] (red dashed) ofthe ICE with torque and speed operating points.

2) PMS Generator: The steady state equations (1) thatdescribe the reversible PMS machine are applied to describealso the PMS generator, which works only in one direction. Inthe present work, both machines are assumed to have the sameparameters. Therefore, the generator efficiency ηg = Pg/Pe isa function of generator speed and load torque, immediatelyavailable from Fig. 2.

3) Rectifier: The rectifier converts the generator AC intoDC at constant voltage Vdc. A pulse width modulated (PWM)rectifier is adopted. Similarly to the inverter, it is simply mod-eled as a constant efficiency factor ηr, such that Pr = ηrPg .

C. Secondary source

1) Battery: The Li-ion battery is modeled by a standardequivalent electrical circuit with purely ohmic impedance [4]representation, which captures the main battery dynamics bythe differential equation of the battery state-of-charge (SoC):

d

dtSoC = − ib

Qmax. (6)

where Qmax is the battery capacity, and ib is the batterycurrent, which is assumed positive during the discharge phase.Moreover, instead of having a nonlinear dependence on SoC,the battery open circuit voltage Voc is adequately approximatedas a constant for the studied charge sustaining HEV. Byconsidering the identity Pb = Vocib − Rbi

2b that is inferred

from the definitions of the battery closed circuit voltageVb = Voc − Rbib and the battery output power Pb = Vbib,battery current ib can be solved as a function of Pb. Then, the

solution is applied to reformulate (6) as

d

dtSoC = −

Voc −√V 2oc − 4PbRb

2RbQmax. (7)

a) State-of-Health (SoH): Battery SoH is a measure thatreflects the battery condition compared with a fresh battery.High fidelity SoH models, as the one utilized in a battery sizingproblem in [18], are highly nonlinear due to the associatedtemperature effects and battery capacity drops because ofSoH variations. However, in EM problems the optimizationis solved over a horizon (distance of driving mission) that is anegligible fraction of the total distance to reach battery end-of-life, hence the SoH variation is very small. Consequently, therewill be a negligible capacity drop and Qmax can be assumedto be constant. Additionally, as it will be shown, the batterytemperature is maintained within a narrow band under activecooling conditions, and therefore temperature inlfuence canalso be ignored. As such, a simple SoH model [19], capturingthe essential features [20], is adequate and is utilized in thiswork:

d

dtSoH =

−|ib|2NcycleQmax

, (8)

where Ncycle is the number of charge-discharge cycles a newbattery can take before it reaches 80% of the nominal energycapacity. By definition of the model, after this many charge-discharge cycles the SoH reaches zero.

b) Thermal model: For electric and hybrid vehicles, theprediction of the battery temperature is a vital process of thethermal management that is designed to optimize the driv-ing performance. Accurate modeling of the thermal behaviorrequires high fidelity heat transfer models that are, however,inherently complex due to the increased dimension and thedependence on battery chemistry and composition [21]. Inthis paper, a simple model that captures the main thermaleffects without significantly increasing the computation burdenis adopted. The model is given by the following equation:

d

dtTb =

Rb i2b − hcAb(Tb − Tamb)

cbmb(9)

where cb and mb are respectively the mass thermal specificcapacity and battery mass, Ab is the heat transfer area of thebattery, and hc is the heat transfer coefficient that is assumedconstant for invariant cooling air mass flow rate. Tamb = 25Cis the ambient temperature that is invariant in this study.

The characteristic parameters of the battery are chosen toemulate a realistic battery for a non plug-in HEV. The batteryenergy capacity is 1.5kWh (as in the Nissan note e-power).It corresponds to a small-sized battery and as such it mustbe capable of delivering continuous currents of 10C or morein practice for sufficient propulsion power [22]. Therefore,the maximum charging/discharging C-rates are set to 10 Cand 20 C respectively. However, repetitive charge-dischargecycles under such C-rate levels would rapidly make the batterywear out with deep discharges. Therefore, the battery SoC islimited in this work within the narrow range of 50%-80%.The thermal parameters used are mainly obtained from [23],[24]. In particular, as mentioned in [23], the heat transfercoefficient hc may practically vary between 5 W/m2/K (natural

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convection) and 65 W/m2/K (forced-air convective cooling)depending on the cooling intensity. In this paper, hc is setto 25 W/m2/K, by which an environment with intermediate aircooling is assumed. The thermal specific capacity cb is set to800J/kg/K, which is a validated value for a Li-ion battery cell[23]. Furthermore, following the work in [24], it is assumedthat the battery pack used in this work consists of 72 prismaticpouch cells (dimension: 16 × 65 × 151 mm), formed in twoarrays, with coolant passages (width of the coolant passages is3 mm) between each two cells to allow fresh air for circulation(see the configuration of the active air cooled battery systemproposed in [24]). As such, the heat transfer area Ab can benumerically determined. The remaining battery parameters arereported in Table I, provided in Section II-D.

2) DC/DC converter: The battery voltage is amplified bya high-efficiency bidirectional DC/DC converter. Similarly tothe rectifier the converter is modeled as a static elementhaving constant efficiency ηdc [25]. The bidirectional powerconversion is thus described by Pdc = η

sign(Pb)dc Pb, in which Pb

is the battery power on the low voltage side, Pdc is the batterypower on the DC link side, and the efficiency is adjustedaccording to the direction of the power flow.

D. Component integration

In this Section a fuel consumption model is developedanalytically by the efficiency analysis of the engine-generatorsystem. Thereafter, all the individual components are assem-bled together to form the overall vehicle model, in which theaccuracy of transient behavior and complexity are balanced.

1) Fuel consumption model: The mechanical integrationof the ICE and the PMS generator enables the calculationof the efficiency ηf of the transformation of fuel chemicalpower into electric power, which is simply the product of theengine and generator efficiencies: ηf = ηeηg . The mechanicalseparation from the wheels allows a requested engine power tobe supplied by freely choosing among different combinationsof torque and speed, which optimize the engine-generatorefficiency. As shown in Fig. 5, the relationship between the

Fig. 5. Fuel mass rate with generator power, when the most efficient enginetorque-speed operating point is followed at each power value.

engine fuel mass rate mf and Pg is approximately linear,provided the engine-generator unit is continuously operatedalong the locus of the most efficient torque-speed operatingpoints. The fuel mass dynamic equation is therefore given by:

d

dtmf = mf0 +

PgqHV αf

(10)

where mf0 = 0.12 g/s is the fuel mass rate when the engine isidle, qHV = 44MJ/kg is the gasoline lower heating value andαf = 0.34 is the equivalent factor of power transformation.

2) Overall powertrain model: In view of the power flowdescribed already for each component, the vehicle power flowmay be described as functions of the three independent powersources Pb, Pg , Ph, with the particular power output of thetransmission given by:

Pt=(ηiηmηt)sign

(ηrPg+η

sign(Pb)

dc Pb

)(ηrPg + η

sign(Pb)dc Pb

).

From a practical perspective, the three powers are not con-trolled directly in this work, but via the time-derivativesjg, jb, jh of the associated forces Fg, Fb, Fh, respec-tively, which have the dimensions of jerk, to ensuresmooth controls and to avoid unrealistic jerky manoeuvres.Therefore, the three independent power sources are cal-culated by Pg=Fgv, Pb=Fbv, Ph=Fhv. Considering statevariables x,[mf , SoC, v, s, Fg, Fb, Fh]T and control inputsu,[jg, jb, jh]T , the following system is obtained by collecting(2), (4), (7), (10) and the dynamics of Fg, Fb and Fh:

d

dt

mf

SoCvsFgFbFh

=

mf0 + Pg/(qHV αf )

−(Voc −√V 2oc − 4PbRb)/(2RbQmax)

(Pt + Ph)/(mv v)− (FT + FD + FG)/mv

vmv jgmv jbmv jh

.

(11)Any other time-varying variables of the system, such as

electric currents and voltages, torques and the vehicle powerflow can be explicitly calculated by algebraic equations withrespect to x and u. In addition, battery state of health andtemperature are calculated respectively from equations (8)and (9) for monitoring purposes. The main characteristicparameters of the vehicle model are summarized in Table I.

TABLE IMAIN VEHICLE MODEL PARAMETERS

symbol value descriptionmv 1500 kg vehicle massfT 0.01 tyre rolling resistance coefficientfD 0.47 aerodynamics drag coefficientQmax 5 Ah battery capacityRb 0.2056 Ω battery internal resistanceVoc 300 V battery open circuit voltageNcycle 2000 battery life cyclemb 36 kg battery masscb 800 J/kg/K battery mass specific heat capacityAb 1.56 m2 battery heat transfer areahc 25 W/m2/K battery heat transfer coefficientηr, ηi, ηdc 0.96 efficiency of converters & invertersRm 90 Ω PMS machine stator resistanceλm 0.21 Wb PMS machine rotor magnetic fluxpm 6 PMS machine number of polesηt 0.96 efficiency of the transmissiongt 10 transmission ratio

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III. OPTIMAL CONTROL PROBLEM FORMULATION

Conventional energy management strategies are designed tofind proper power component combinations that minimize fuelconsumption for given vehicle speed profiles. However, thefuture driving speed is not known during real world driving.The idea proposed in this paper is based on a different concept,as shown in Fig. 1. The OCP is formulated by specifyingthe vehicle mission subjected to performance and operatingconstraints that must be satisfied. Instead of in terms of vehiclespeed and acceleration, the drive mission is defined in termsof route and a specified traveling time. Therefore, accordingto the new methodology (OCP-Joint), the powertrain energyflow and speed profile along the route are jointly provided bythe solution of the optimization.

An alternative methodology proposed in this work to solvethe complete optimization problem is a two-step approach(OCP-2-step) that successively combines two conventionalmethods: 1) driving speed optimization (OCP-S), and 2)energy management optimization (OCP-EM), as shown Fig. 6.The OCP-2-step is related to conventional methods in which,

Route & travel

time

Full HEV

model

OCP-Joint

Driving

speed

Energy

management

OCP-S OCP-EMSpeed

HEV powertrain

model

Car motion

model only

OCP-2-step

Fig. 6. Optimal control problems formulated for the series HEV.

for example, the power split is optimized for a given speedprofile (that may or may not be an optimal speed profile), orthe speed profile alone is optimized for a conventional (non-hybrid) vehicle. The present work compares the mentionedoptimization approaches and exposes the benefits in vehiclefuel efficiency of conducting the combined optimization OCP-Joint, which takes into account both the characteristics of thedrive mission as well as the powertrain, as demonstrated laterin Section IV. The general formulation of the OCPs studiedin this paper is as follows:

minu

J(x,u) =

∫ tf

0

L(x,u, t) dt+ φ(x(tf )) (12a)

subject to:d

dtx = f (x,u, t) (12b)

ψ (x,u, t) ≤ 0 (12c)b (x(0),x(tf )) = 0, (12d)

where J(x,u) is the main objective. The vector x representsthe system state vector, which evolves according to the differ-ential equation x=f(x,u, t), and the vector u represents thecontrol input. The inequality constraints and the boundary con-ditions are taken into account by (12c) and (12d). Finally, tfdenotes the total traveling time. In each OCP described below,the respective J , x, u, constraints and boundary conditions aredifferent, and are specified.

a) Vehicle speed optimization (OCP-S): Driving speedoptimization in terms of fuel efficiency has been well studiedin the literature for conventional vehicles [11]. In this section,this conventional approach is developed further and extendedto HEVs by considering energy recovery during braking. Afixed energy recovery factor is defined as the ratio betweenregenerative braking power and the total braking power:

ρ = Pt/Pv, ∀Pv < 0 (13)

with ρ ∈ [0, 1]. The boundary case ρ=0 corresponds to aconventional powertrain, since in the present work the fuelmass rate is linearly dependent upon the generator outputpower (see Fig. 5), while ρ = 1 is associated with recoveringall braking energy as in a highly hybridized or fully electricvehicle. The total driving/braking power is incorporated intothe dynamic model of the longitudinal dynamics of the vehicleas the total driving/braking force Fv (see (3)), as follows:

d

dt

vsFv

=

Fv/mv − (FT + FD + FG)/mv

vmv jv

(14)

with x , [v, s, Fv]T . The OCP is formulated to find jerk input

u = jv , associated with the total driving/braking power andconsequently the speed v, which minimizes the net energyfrom the powertrain:

J =

∫ tf

0

Pt dt =

∫ tf

0

max(ρFv, Fv)v dt. (15)

The inequality constraints of this OCP are as follows. In-equality constraints are imposed on the speed and accelerationfor driving safety and comfort. Specifically,

0 ≤ v ≤ vmax, (16)

where vmax is the legal speed limit1. Instead of the commonlyused ellipse of adherence of tires, the longitudinal and lateralacceleration are constrained within an acceleration diamond,as everyday drivers use accelerations remarkably smaller thanadherence limits [26]. The constraint is described as follows:∣∣∣∣Fv/mv

ax,max

∣∣∣∣+

∣∣∣∣ vΩ

ay,max

∣∣∣∣ ≤ 1 . (17)

where Fv/mv and vΩ are respectively the longitudinal andthe lateral acceleration applied by the driver, while ax,maxand ay,max are their maximum allowed values. The total inputpower is limited according to

vFv ≤ Pv,max , (18)

where Pv,max is determined according to the sum of theindividual power limits of the engine and the battery branches.Finally, jv is bounded within ±1m/s3.

The following boundary conditions are used to define theinitial and terminal states:

s(0) = 0 , s(tf ) = l,v(0) = v(tf ) = vmin, Fv(0) = Fv(tf ) = 0,

(19)

1To avoid the issue of singularity when the power is divided by v, a non-zero lower bound is imposed, such that v ≥ vmin with vmin a small constant.

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where l is the length of the path and tf is expressed in termsof l and an average speed for the mission.

b) Powertrain power-split optimization (OCP-EM): Thegeneral problem of optimizing the EM of an HEV while itfollows a predefined speed, for example according to a givendrive cycle, is standard, while it has also been specificallyformulated in numerous prior studies as an OCP; see forexample [3], [27]. Here such an EM OCP formulation isadapted and described for the proposed vehicle model.

For the given value of the vehicle speed and accelera-tion, the necessary driving force Fv is uniquely determinedby (2) whereas the power split is freely assigned. Basedon the developed series HEV model, the dynamic modelused for EM is part of (11), comprising only dynamics ofx=[mf , SoC, Fg, Fb, Fh]T (and SoH and Tb for monitoring),subject to an algebraic constraint among Fb, Fg and Fh givenby (3). The OCP is formulated to find u=[jb, jg, jh]T thatminimizes J=mf (tf ), subject to multiple constraints that keepthe operating conditions of the powertrain components insidetheir admissible range, as follows. The generator power islimited according to:

0 ≤ Pg ≤ Pg,max, (20)

with Pg,max = 86kW. The battery is constrained in terms ofSoC and current:

SoCmin < SoC < SoCmax, ibc < ib < ibd, (21)

where SoCmin=0.5, SoCmax=0.8, and ibd=100A and ibc=−50A are determined by the discharging and charging C-rate limits respectively. The PMS motor (and indirectly theinverter) is constrained by the operational limits as shown inFig 2. Braking power is constrained to be negative:

Ph ≤ 0 , (22)

and jg, jb, jh are bounded within ±1m/s3.The following boundary conditions are imposed to complete

the formulation:

SoC(0) = SoC(tf ) = 0.65, mf (0) = 0, (23)Fb(0) = Fb(tf ) = Fg(0) = Fg(tf ) = Fh(0) = Fh(tf ) = 0.

Note that the charge sustainability is addressed in the presentwork by employing SoC boundary conditions, rather than in-troducing an additional term in the optimization cost function.This condition enables to make a fair comparison of fuelconsumption between the solutions of different optimizationmethods, without having to refer to “equivalent fuel” quan-tification caused by unequal boundary SoC values betweenthe different solutions, which may be inaccurate. Finally,Tb(0) = Tamb = 25 C and SoH(0) = 1 are introduced.

c) Combined optimization of both vehicle speed andpower split (OCP-Joint): In this framework, the OCP for-mulation includes the optimization of the power split (as inOCP-EM), however, the driving speed is unspecified, whichleads to a formulation by which both EM and forward speedare simultaneously optimized. By this new formulation, the op-timization problem becomes more complex and powerful thanany of the individual optimization problems and it represents

new knowledge. The state space model in this case is alreadyspecified in (11) with x , [mf , SoC, v, s, Fg, Fb, Fh]T , wherev is now included as a state according to (2), and (3) does notrepresent an algebraic constraint as in OCP-EM. The objectiveof the OCP-Joint is to minimize J = mf (tf ) analogouslyto the OCP-EM by finding three inputs u = [jb, jg, jh]T ,however the speed profile v now also results from the opti-mization, as a consequence of the jerk inputs u. The inequalityconstraints and the boundary conditions in this case collect allthe conditions (except (18)) imposed in OCP-S and OCP-EM.

d) Two-step optimization of vehicle speed and power split(OCP-2-step): The (OCP-2-step) simply obtains the speedprofile by solving OCP-S and combining that a posteriori withOCP-EM. OCP-2-step will be used in Section IV to bench-mark OCP-Joint (for ρ=0, OCP-2-step is precisely related toexisting optimization schemes in the literature).

The foregoing OCPs may be solved by various methodsin different categories, such as dynamic programming, directmethods and indirect methods. In this work, these OCPsare solved by an indirect method embedded OCP solver(named PINS) that is based on C++ [28]. This method solvesthe optimization problem by converting it into a boundaryvalue problem resulting from Pontryagin’s minimum principle(PMP). Path constraints (for example, battery current limits,acceleration diamond and energy source power limits) are notexplicitly included, instead, the inequalities are approximatedand included in the Lagrange term

∫ tf0L(x,u, t) dt by us-

ing smooth barrier functions. Non-smooth functions, such assign(·) are smoothed by approximations. The smoothness ofthe various approximation functions is freely tunable, subjectto a well-known trade-off between accuracy and robustness(convergence to a solution) of the solver. It has been shownin [29] that this indirect solver offers comparable performanceto solvers pertaining to other categories in terms of accuracy,robustness and computational speed.

IV. SIMULATION RESULTS

In this Section the OCPs defined in Section III are solved forthe specific vehicle mission shown in Fig. 7 that correspondsto a rural route with scarce traffic. This route has beendeliberately chosen to minimize the traffic influence, which isnot taken into account in this paper. Road geometry including

Fig. 7. 18.9km rural route selected for the vehicle missionhttps://goo.gl/ytrVUM.

8

0 2 4 6 8 10 12 14 16 18

Travelled distance [km]

0

20

40

60

80S

peed [km

/h]

Experimental

Filtered Experimental

OCP-S with = 0

OCP-Joint

average

A

BC

D

E

F

G

H I

J

legal limit

Fig. 8. Real and optimized speed profiles with respect to traveled distance.

the actual gradient is exported from Google MyMaps andit is then converted into the curvature model. The variablespeed limit imposed throughout the journey captures the realconditions of the chosen path. Since traffic is not considered,the speed is not constrained to the behavior of other vehicles.

A. Driving speed optimization assessment against real driving

With the aim of showing the importance of driving speedoptimization in terms of fuel saving and investigating theassociated impact on the battery temperature and SoH, anunoptimized human-driver speed profile of a petrol passengercar for the route in Fig. 7 is utilized for comparison with theoptimal solutions. OCP-S is deployed with ρ = 0 to simulatea conventional vehicle, while the main idea OCP-Joint is alsoused for benchmarking. The comparative results are shown inFigure 8, where the trip is requested to be accomplished in1326s (identical to the real case), which corresponds to thevehicle being driven at an average speed of 51.3 km/h withidentical initial and terminal speeds to the real case. Notethat the sudden dips (highlighted by dashed ellipses) in thereal profile are caused by isolated traffic, therefore for faircomparisons they are not taken into account; the ‘modifiedreal’ speed profile, also shown in Fig. 8, is used instead, whichis obtained by passing the real profile though a median filterto remove such very sharp changes in speed.

Compared to the real driving speed, the optimal profileslead to more efficient vehicle operation (as will be quantifiedin Table II) by using the information of the upcoming roadslopes and speed limit. For example, at the beginning ofthe journey, the human driver accelerates to the speed limit,and upon reaching it, decelerates dramatically because of theapproaching junction at 1 km. Conversely, the optimal profilesstop increasing at a much lower speed, and they are thenfollowed by a milder deceleration to pass through the firstcorner. It is also noticed that at 7 km, the human driver exceedsthe legal speed limit by view of the sign ahead indicating thehigher limit, whereas the optimized results do not suffer fromsuch a human ‘error’ and strictly obey the legal speed limit atall times.

To further understand the nature of optimal as comparedto real driving, two different cases of OCP-EM are solved byinjecting: a) the optimal speed from OCP-S (OCP-2-step), andb) the modified real speed profile; and comparing these withthe solutions of OCP-Joint in terms of fuel economy as well

as battery temperature and SoH. In the latter OCP-EM case itmay be argued that the real driving speed is collected from aconventional vehicle, which is not consistent with the hybridpowertrain architecture deployed in OCP-EM. However, thehuman driver may not significantly change his/her driving stylewhen the conventional car is replaced by a hybrid one withsimilar engine power, as the driving style of a particular driveris less influenced by the powertrain architecture but highlydepends on the maximum power that can be delivered.

The results in terms of battery lifetime, and average fuelconsumption are shown in Table II, while Fig. 9 shows thebattery current and temperature profiles for the three optimiza-tion cases. The available distance the vehicle can travel before

TABLE IICOMPARISON AMONG DIFFERENT OPTIMIZATION APPROACHES

Battery life Fuel economy[km] [L/100km]

OCP-Joint 157500 3.84OCP-2-step 127700 4.08Exp. speed+OCP-EM 89573 5.92

0 2 4 6 8 10 12 14 16 18

Travelled distance [km]

-50

0

50

100

Cu

rre

nt

[A]

experimental +OCP-EM

OCP-2-step

OCP-Joint

0 2 4 6 8 10 12 14 16 18Travelled distance [km]

25

25.5

26

26.5

27

Te

mp

era

ture

[°C

] Experimental + OCP-EM

OCP-2-step

OCP-Joint

Fig. 9. Battery current and temperature profiles with Tamb = 25C.

the battery reaches end-of-life is calculated according to thedegradation suggested by the end of the present mission. Itis shown that the OCP-Joint beyond substantially improvedfuel economy, offers enhanced battery durability comparedto the other two cases, as OCP-Joint is the most boundedthroughout the simulation in terms of positive and negative

9

current maximum values. The beneficial behavior of OCP-Joint in terms of battery current and consequently in terms ofSoH, together with the relevance of temperature variations, isexplained as follows.

From the battery thermal model (9), Tb is affected by ib andthe convective cooling (subject to the temperature differencefrom ambient temperature). As such, it may be the casethat up to a certain amount of battery current there is novisible change in battery temperature because the resistive heatRb i

2b is suppressed by the cooling effect, while only even

higher operating currents will increase Tb. As illustrated inFig. 9, a large current spike always results in a significantstep change in temperature. For example, the main temperatureincrease of OCP-2-step occurs at the very beginning when alarge discharging current is requested. Comparing to the othercases, the OCP-Joint solution has overall smaller ib, but theduration of its larger charging currents appearing at 5.5 kmand 11.5 km is much longer. Hence, the terminal temperaturein all optimization cases ends up being very similar.

Battery current on the other hand represents the only vari-able that influences SoH, which is particularly proportionalto the integration of |ib| (see (8)). Therefore, it is clear that asignificant positive jump in the temperature value (for examplethe one appears at 2.5 km) is certainly an indication of a largeib that will also lead to a dramatic SoH drop. Since all thethree optimization cases are charge sustained, the integrationof their current profiles over [0, tf ] are all zero. This meansthat any large positive variations in the current (even in theform of spikes) will have to be compensated by negativevariations, thus overall increasing the average value of |ib| andhence the SoH drop. A real driver tends to drive the vehiclewith higher values of acceleration and deceleration than theoptimal ones. Hence more power will be requested from thepowewtrain, and most likely the battery itself, implying largecurrent spikes, in the case of the experimental + OCP-EMcase. Also, when the powertrain characteristics are not takeninto account in the optimization, as in the case of OCP-2-step, the speed profile solution tends to have the PnG form, inwhich the initial part of each PnG segment is large accelerationand therefore large battery current. In contrast, in OCP-Jointthe optimization takes into account that large battery poweramounts are less efficient than low power values, therefore itrestricts the battery usage to lower current values.

B. Assessment of driving speed combined optimization

To gain more insight into the behavior of different opti-mization schemes and to expose the benefits of combinedoptimization, the next example is carried out to explicitlycompare the simultaneously optimized results of OCP-Jointand the solutions of OCP-2-step. The same vehicle missionas defined in the last example is used with the same averagespeed and boundary speed values.

As it can be noticed in Fig. 10, the consideration of roadelevation in the optimization mainly gives rise to an additiveperturbation to the optimal speed solution in comparison tothe flat road. In order to focus on and compare the mainmechanisms that produce the nature of the optimized speed

profiles, the elevation of the route is omitted for the subsequentanalysis in this section. In Fig. 10, the speed profile solved bythe joint optimization is compared with the solutions from thespeed optimization by OCP-S for different vehicle topologies.The results show that the optimized speed profile is dependentupon the selected optimization strategy and also varies as ρchanges when OCP-S is deployed. It can be seen that theoptimal driving speed for the case ρ = 0 (conventional vehiclewith no braking energy recovery) follows a pattern namedpulse-and-glide (PnG), which consists of rapid accelerationuntil a maximum velocity is reached, followed by a periodof coasting to pass through the next corner. Such a speedpattern has been proven to be the most fuel efficient strat-egy in conventional vehicles [11]. This phenomenon of PnGfades away in the optimal solutions as the powertrain shiftstowards the electric architecture that allows more regenerativebrakes by increasing ρ. Eventually, it is hardly observed inthe optimal profile when ρ=1 (hybrid vehicle with brakingenergy recovery), because the power amounts requested foracceleration and deceleration are balanced to take advantageof the fully recoverable braking energy. Moreover, the speedsolution produced by the conventional powertrain formulationis closer to the simultaneously optimized speed (OCP-Joint)than that of the electric powertrain case in terms of root meansquare (RMS) difference. The reason is that in the proposedarchitecture the battery is only a minor energy source, givingrise to a hybrid powertrain closer to the conventional type.

To complete the comparison, the speed profiles deter-mined by the OCP-S speed optimization respectively forρ = 0, 0.2, 0.4, 0.6, 0.8, and 1 are fed to OCP-EM (leading tofive cases of OCP-2-step), the solutions of which including thepower split and the corresponding average fuel consumptionare compared with their counterpart found by the combinedoptimization scheme OCP-Joint. The comparative results arereported in Fig. 11. It can be seen that the proposed OCP-Jointstrategy benefits the fuel economy by at least 1.3% due to theinclusion of the actual hybrid powertrain in the optimization.Although the hybrid powertrain is neglected in OCP-S, theoptimality of the solution in that case can be enhanced bymanipulating ρ to allow a suitable amount of regenerativebraking. In the present case, ρ = 0.4 gives the closest solutionto OCP-Joint in terms of fuel consumption.

Apart from the fuel economy, the simultaneous optimizationalso improves the battery thermal behavior and durability,which can be inferred from the battery operating current,analogously to the previous example in Section IV-A. InFig. 12, the current profile of the OCP-Joint is comparedwith two cases pertaining to the OCP-2-step respectively withρ = 0 and ρ = 1. Due to the removal of the elevations,the relationship between battery current, and consequentlytemperature and SoH can be more clearly observed in thiscase. It is clear that the OCP-Joint produces the most boundedib, with also the fewest fluctuations throughout the mission,thereby the solution of OCP-Joint tends to deliver enhancedbattery thermal and SoH performance. It can be seen in Fig. 11that the solution of OCP-Joint is able to restrain the increaseof battery temperature by about 12% as compared to the mostfuel efficient OCP-2-step case (ρ = 0.4), and the benefit is

10

0 2 4 6 8 10 12 14 16 18Travelled distance [km]

0

20

40

60

80S

pee

d [km

/h]

OCP-Joint with road slope

OCP-Joint

OCP-S with = 0

OCP-S with = 1

legal limit

Fig. 10. Optimal speed profile under different scenarios. The values of RMS difference between the profiles obtained by OCP-Joint and OCP-S respectivelywith ρ = 0 and ρ = 1 are 3.83 and 4.14 when road elevation is ignored.

0 0.2 0.4 0.6 0.8 10

1

2

3

4

fuel usage L

/100km

incre

ase [%

]

3.247 3.244 3.2403.244

3.257

3.317

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

battery

dura

bili

ty im

pro

vem

ent [%

]

battery terminal temperature increase

battery life decrease

Fig. 11. Benefit of OCP-Joint compared to OCP-2-step with different ρ interms of average fuel consumption (fuel economy is shown on top of each barin terms of L/100km) and battery durability when road elevation is ignored.

0 2 4 6 8 10 12 14 16 18

Travelled distance [km]

-10

0

10

20

Cu

rre

nt

[A]

OCP-2-step with =0

OCP-2-step with =1

OCP-Joint

peak current is 45A

Fig. 12. Battery current profiles when road elevation is ignored.

increased when ρ diverges from 0.4. The depleted SoH isalmost steady for different choices of ρ, which amounts tothe battery life being saved by approximately 40% by thecombined optimization.

Finally, the best behaved OCP-2-step case with ρ = 0.4 isselected for further comparison with the simultaneously opti-mized solution of OCP-Joint. In this case the RMS differencebetween the two optimized speed profiles is further reducedto 3.46, which is lower than the differences calculated forother values of ρ. In Fig. 13, the battery SoC and power splitobtained in both scenarios are compared. The main differencebetween the two optimal solutions is that the individuallyoptimized speed has a higher acceleration after passing acorner, therefore in the acceleration phase it requests higherpropulsive power, which is mainly provided by the ICE, whilethe battery provides an additional, small amount of power. Byobserving the SoC, the battery is depleted more for the OCP-2-step at the beginning of the mission to fulfill the powerdemand requested by the associated speed profile. To remainstrictly charge sustaining, less battery is used for the OCP-2-step case at the end of the mission while both scenarios havesimilar behavior in between.

0 2 4 6 8 10 12 14 16 18Travelled distance [km]

-5

0

5

10

15

Pow

er

[kW

]

engine

battery

brakes

solid: OCP-Joint

dotted: OCP-2-step with =0.4

0 2 4 6 8 10 12 14 16 18

Travelled distance [km]

62

63

64

65

66

SoC

[%

]

OCP-Joint

OCP-2-step with =0.4

Fig. 13. Comparison of OCP-Joint and OCP-2-step with ρ = 0.4 in termsof (Top) power flow of engine Pg , battery Pb, mechanical brakes Ph and(Bottom) battery SoC when road elevation is ignored.

V. CONCLUSIONS

The energy management control and the driving speedoptimization of a series HEV in the context of fuel economyare addressed in this paper by an optimal control methodology.The novelty of the proposed method is that it solves the opti-mal speed profile and the power split simultaneously, with thedrive mission including only the path, the specified travelingtime and constraints that address safety and comfort. Theless complex sole optimization scheme of the vehicle speedand the energy management respectively are also presentedand combined, forming a two-step scheme with the sameoverall objective as the proposed simultaneous optimizationscheme, to benchmark against the proposed strategy in termsof fuel efficiency and battery durability. It is shown that thesimultaneous optimization is solvable and it performs better interms of fuel economy while also improving battery lifetime.Nevertheless, the optimality of the two-step methodology canbe improved by suitably tuning the objective function for thespeed optimization via a parameter that is associated with theregenerative braking capability of the powertrain.

From a technological perspective, the presented methodol-ogy works in real time as implemented in the PINS suite andrequires as input only basic information on the vehicle stateand route characteristics that are easily available on a naviga-tion system. It can also be used for benchmarking purposes fora system where the computation power is rigorously limited.

11

Future research efforts will be devoted to the extension of themethodology to the case where traffic conditions and othermission uncertainties are taken into consideration.

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Boli Chen (M’16) received the B. Eng. in Elec-trical and Electronic Engineering in 2010 fromNorthumbria University, UK. In 2011 and 2015,he respectively received the MSc and the Ph.D.in Control Systems from Imperial College London,UK. Currently, he is a Lecturer in the Departmentof Electronic and Electrical Engineering, UniversityCollege London, U.K. His research focuses on con-trol, optimization, estimation and identification of arange of complex dynamical systems, mainly fromautomotive and power electronics areas.

Simos A. Evangelou (M’05-SM’17) received theB.A./M.Eng. degree in electrical and informationsciences from the University of Cambridge, Cam-bridge, U.K., and the Ph.D. degree in control en-gineering from Imperial College London, London,U.K.. He is currently a Reader with the Departmentof Electrical and Electronic Engineering, ImperialCollege London, London, U.K. Dr. Evangelou is aMember of IFAC Technical Committee AutomotiveControl and on the editorial board of internationaljournals and conferences, including the IEEE Con-

trol Systems Society Conference Editorial Board.

Roberto Lot is Professor of Automotive Engineer-ing at the University of Southampton (UK) since2014. He received a PhD in Mechanics of Machinesin 1998 and a Master Degree cum laude in Me-chanical Engineering in 1994 from the Universityof Padova (Italy). His research interests includedynamics and control of road and race vehicles,contributing to make our vehicles safer, faster, andeco-friendlier. He has directed several national andinternational research projects and published morethan 100 scientific papers.