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Lap Time Optimization ofa Sports Series Hybrid Electric Vehicle
Roberto Lot and Simos A. Evangelou
AbstractThis paper illustrates a methodology for the laptime optimization of a race series hybrid electric vehicle basedon the indirect optimal control approach. More specifically, fora vehicle with given characteristics running on a given track, theoptimal trajectory and powertrain power flow that minimize thelap time are found. The paper presents a parametric model of asports series hybrid electric vehicle, illustrates the optimizationmethod and discusses simulation results.
Index Termshybrid electric vehicle, vehicle dynamics, optimal control, lap time optimization.
I. INTRODUCTION
HYBRID electric vehicles (HEVs) are becoming morepopular due to their potential to address climate changeand the demand for a limited, but increasingly expensive,supply of fossil fuels. In addition, a new category of sportsand race HEVs is emerging [1], [2]. Although the synergybetween multiple energy sources in HEV powertrains isnormally used to bring reductions in fuel consumption andnoxious emissions, in racing the main interest is aboutperformance and lap time minimization.
In the case of road HEVs various control techniques havebeen proposed in the literature for the powertrain energymanagement, ranging from rulebased to optimisationbased[3][9]. This paper presents a global optimisationbased control approach that utilises indirect optimal control techniquesto optimise not only the powertrain energy flow but also thetrajectory of a racing HEV inside a given race circuit. Themethodology is established by utilising symbolic dynamicvehicle modelling of appropriate complexity, together withcomputationally efficient optimal control software [10].
II. MATHEMATICAL MODELThe powertrain of a Series Hybrid Electric Vehicle (S
HEV) is schematized in Figure 1 and consists of threebranches: the spark ignition engine, the battery and the transmission. In driving operating conditions, i.e. while cruisingat a constant speed or accelerating, power is request to drivethe vehicle. The spark ignition (SI) engine is mechanicallyconnected to the permanent magnet synchronous (PMS)generator, which converts the engine mechanical power Peinto the electric AC power Pg . The rectifier takes this ACpower and converts it to DC power Pr, which is providedto the DC link. In the other branch, the battery (possibly)provides additional power Pbl to the DC/DC converter, whichsteps up the battery voltage to vdc and provides power Pb tothe DC link. The overall DC link power Pr+Pb is converted
Manuscript received March 6, 2013; revised April 3, 2013.R. Lot is with Department of Industrial Engineering, University of Padova,
35100 Padova, Italy email: roberto.lot at unipd.itS. A. Evangelou is with the Departments of Electrical and Electronic,
and Mechanical Engineering, Imperial College London, London, UK email:s.evangelou at imperial.ac.uk
from DC into AC by the inverter, which supplies the electricpower Pi to the PMS motor. The latter is connected tothe wheels by means of a fixed ratio transmission, and thevehicle is finally driven with power Pt. While braking, poweris driven through the transmission to recharge the battery.Additionally, mechanical brakes extract power Ph which istransformed into heat and definitively dissipated. Optionally,the battery may be recharged by the SI engine when a powersurplus is available. In summary, the SHEV has three independent power sources, respectively battery Pb, generatorPg and brakes Ph power, that may be variously combined toobtain the desired values of vehicle speed and acceleration.In particular, the battery may conveniently provide boostpower while the vehicle is accelerating. However, the batteryenergy is limited to the value which may be recovered duringeach lap, hence it is scarse and it is important to use boosteffectively and to optimize the whole power managementprocess.
Finally, as a race driver is free to select his preferredtrajectory inside the track, to minimize the lap time it isalso necessary to optimize the trajectory and speed profile atthe same time as optimizing the power flow in the vehicle.
Details of the mathematical model of the SHEV vehicleand its powertrain, the formulation of the minimum lap timeproblem and some simulation results are discussed in thenext sections.
A. Powertrain
The first power branch (Figure 1) includes the SI engine,the PMS generator and the AC/DC converter. Modelling indetail such elements is not trivial and out of the scope of thiswork. For the purpose of lap time optimization, the essentialfeature of this power branch is that the SI engine is notmechanically connected to the vehicle transmission and soit may deliver the maximum available power regardless ofthe vehicle speed. The total power provided by this branchis found by considering that the SI engine is a source of(limited) power Pe, which is then reduced while flowingthrough the PMS generator and AC/DC converter, that haveefficiencies (assumed constant) g and r respectively:
Pr = rgPe (1)
Fig. 1. Powertrain architecture of a Series Hybrid Electric Vehicle (purplearrows indicate power losses).
Proceedings of the World Congress on Engineering 2013 Vol III, WCE 2013, July 3  5, 2013, London, U.K.
ISBN: 9789881925299 ISSN: 20780958 (Print); ISSN: 20780966 (Online)
WCE 2013

The second power branch includes the battery and theDC/DC converter that steps up the low voltage on the batteryside to a high voltage on the DC link side. The battery stateof charge is described by the following differential equation:
d
dtQb = ib (2)
where Qb is the actual charge and ib the current of the battery,assumed positive during the discharge phase. Moreover, thebattery power (on the low voltage side) is:
Pbl = ibvb (3)
where vb is the closed circuit voltage of the battery, whichdepends both on the battery charge Qb and current ib. Suchdependence may be expressed in terms of the electrochemicalparameters and an equivalent electrical circuit [11], [12] asfollows:
vb = EbRib = E0+(
1 QmaxQ
)+AeB(QQmax)Rib
(4)where Eb is the open circuit voltage, Rb is the internalresistance, E0 is the nominal voltage, Qmax is the capacityof the battery, and A,B are two additional constants. TheDC/DC converter is simply modelled as a static elementhaving a constant efficiency dc. Since the converter is bidirectional, the power conversion may be described by meansof the following equation:
Pb = sign(Pb)dc Pbl (5)
where Pb is the battery power on the DC link side. Theefficiency is adjusted according to the direction of the powerflow, i.e. when the positive power flows from the battery tothe DC link Pb = dcPbl, on the contrary when the negativepower flows from the DC link to charge the battery Pbl =dcPb.
Power flow is then collected by the DC link, which drivesa bidirectional inverter. Similarly to the rectifier, the inverteris simply modelled by means of a constant efficiency factori, therefore the power balance of the DC link and inverteris described by the following equation:
Pi = sign(Pr+Pbl)i (Pr + Pbl) (6)
where once again the efficiency is adjusted according to thedirection of the power flow sign(Pr + Pbl). The invertersupplies the electric motor/generator, which is a 3phasestarconnected PMS machine. PMS machines combine anumber of attractive features when used in hybrid vehicleapplications, such as higher torquetoinertia ratio and powerdensity than ones of induction or woundrotor synchronousmachines. The dynamic electromagnetic behaviour of thePMS machine may be effectively described in the rotor dqreference frame [13] by the following nonlinear differentialequations:
Lqd
dtid = vd Rid + pLiq (7a)
Lqd
dtiq = vq Riq pLiq + (7b)
where id, vd and iq, vq are the direct and quadrature components of armature currents and terminal voltages, is therotor angular speed, while the other parameters are described
in Table I. Equations (7) model only the power losses duethe resistance R of the stator copper windings, while inreality there are other electromagnetic dissipation sources[14] such as Eddy current losses ( 2) and hysteresis losses( 2), while mechanical losses [14] include bearing losses( ) and windage losses ( 5). These additional lossesare modelled in the dynamic equation of the rotor as follows:
Jd
dt =
3
2piq + Tl + Td() (8)
where J is the rotor inertia, Tl is the mechanical torqueexchanged with the trasmission, and Td() is the dissipationtorque. The control strategy of the machine [15] uses a nulldirect current id = 0. To further simplify the model, it maybe observed that the dynamics of electromagnetic phenomenaare much faster than mechanical ones, hence transient currents may be neglected. The inertia torque J ddt is neglectedalso, as the motor inertia is much smaller than the vehicleinertia to which it is rigidly connected. These assumptionslead to the simplification of differential equations (7), (8)into a set of steadystate algebraic equations, which may beeasily solved in term of currents and voltages, leading tothe following equation for the input power Pi (exchangedwith the inverter) and output power Pm (exchanged with thetransmission):
Pi = (Tl + Td)2
3R
(Tl + Td)2
(p)2(9a)
Pm = Tl (9b)
Equations (9) are capable of describing the reversible PMSmachine both when it works as a generator, i.e. with positive power, load torque and quadrature current, or when itworks as a motor, i.e. with negative power, load torque andquadrature current. According to such conventions, the PMSefficiency is:
m =
(PmPi
)sign(Pi)(10)
and it is reported in Figure 2. The PMS machine efficiencyis very high in a wide range of operating condition, even ifit is very poor at low speeds, where the resistance losses Ri2qare predominant, and at low torques, where the mechanicallosses Td() are predominant. The figure also shows thecurrent iq , which is roughly proportional to the torque, andthe overall voltage
v2q + v
2d, which is roughly proportional
to the speed. The PMS motor is connected to the transmissionand finally to the wheels in a way that the velocity ratiobetween the motor angular speed and the vehicle forwardspeed v is constant:
=
u(11)
It is assumed that the transmission has a constant efficiencyt, the bidirectional power flow is hence modelled with thefollowing equation:
Pt = sign(Pm)t Pm (12)
i.e. when positive power flows form the PMS motor to thevehicle Pt = tPm, on the contrary when negative powerflows from the vehicle to the battery Pm = tPt.
Brakes are simply modelled as power withdrawal, i.e asource of negative power Ph, which is converted into heatand dissipated.
Proceedings of the World Congress on Engineering 2013 Vol III, WCE 2013, July 3  5, 2013, London, U.K.
ISBN: 9789881925299 ISSN: 20780958 (Print); ISSN: 20780966 (Online)
WCE 2013

0.50.5
0.5 0.50.8
0.8
0.8
0.8
0.85
0.85
0.85
0.85
0.9
0.9
0.9
0.9
0.92 0
.92
0.92
0.92
0.94
0.94
0.94
0.95
0.95
0.95
100
0
100
200
200
400
600
speed [rpm]
torq
ue [N
]
0 1000 2000 3000 4000 5000400
300
200
100
0
100
200
Fig. 2. PMS machine efficiency (solid black), current (horizontal dotted red)and voltage (vertical dotted blue). Positive torques correspond to generator,and negative torques correspond to motor operating conditions.
By coupling and manipulating equations (5), (6), (9),(10), and (12), the vehicle power flow may be completelydescribed as a function of three independent power sources,respectively the generator Pg , battery Pb, and brakes Ph, asfollows:
Pe = 1e Pg (13a)
Pr = rPg (13b)
Pbl = signPbdc Pb (13c)
Pi = sign(rPg+Pb)i (rPg + Pb) (13d)
Pt = (imt)sign(rPg+Pb) (rPg + Pb) (13e)
The only dynamic variable of the powertrain is the batterycharge Qb, while any other variable may algebraically beexpressed as a function of Pg, Pb, Ph.
B. Vehicle and track
This section illustrates the model used to capture thevehicle gross motion on the track for the purpose of trajectoryand speed profile optimization. The vehicle is modelled asa singletrack rigid body, running on a horizontal flat track(Figure 3). The equations of motion in the longitudinal and
Fig. 3. Singletrack vehicle model.
lateral directions are respectively:
m
(d
dtu v
)= Sr + Sf cos Ff sin FD (14)
m
(d
dtv + u
)= Fr + Sf sin + Ff cos (15)
where m is the vehicle mass, u and v are respectively thelongitudinal and lateral speed, is the yaw rate, (Sf , Ff )and (Sr, Fr) are the longitudinal and lateral forces respectively on the front and rear axle, FD = 12CDAu
2 is the
aerodynamic drag resistance, and is the steering angle. Theequation for the yaw motion is:
IGd
dt = aFf cos + Sf sin bFr (16)
while IG is the yaw moment of inertia, while a and b are thedistance from the vehicle center of gravity (CoG) respectivelyof the front and rear axle.
Tire longitudinal forces are strictly related to the powerflow (13), in particular the propulsive power is transferredto the real axle only, while braking power is distributed onboth axle with constant ratio rb:
Sf =rb min(Pt + Ph, 0)
u(17a)
Sr =Pt + Ph rb min(Pt + Ph, 0)
u(17b)
where suffixes f and r refer to the front and rear axle.Vehicle directionality is controlled by means of the steer
ing angle . Tire lateral forces are assumed to be proportionalto the cornering stiffness K, (small) sideslip angle and tirevertical load N as follows:
Ff = kff ' Kf( a + v
u
)Nf (18a)
Fr = krr ' Krb vu
Nr (18b)
Tire loads are calculated taking into account the load transferfrom the front to the rear axle during traction (and vice versaduring braking), according to the (approximate) expressions:
Nr =a
a+ bmg +
h
a+ b(Sr + Sf ) (19a)
Nf =b
a+ bmg h
a+ b(Sr + Sf ) (19b)
where h is the vehicle center of gravity height. The trackis assumed to be flat and to lay on the horizontal planexy. The curvature of the track reference line may becalculated from its cartesian coordinates (x, y) as a functionof the travelled space s as follows:
(s) =
(d2x
ds2
)2+
(d2y
ds2
)2(20)
To add the second dimension of the striplike track model itis sufficient to specify the distance from the borders, whichpossibly depends on the position s. To track the positionand orientation of the vehicle, it is very convenient to usethe triple of curvilinear coordinates (s, n, ), where s and nare respectively the longitudinal and lateral position on theroad strip and is the vehicle heading relative to the road.As depicted in Figure 3, such coordinates are related to thevehicle speeds by the following differential relations:
d
dts =
u cos v sin1 n(s)
(21a)
d
dtn = u sin+ v cos (21b)
d
dt = u cos v sin
1 n(s)(s) (21c)
In conclusion, sections IIA and IIB describe the powertrain flow and the vehicle gross motion as a function of 7variables x(t) = {Qb, u, v,, s, n, }T and 4 control inputsu(t) = {Pg, Pb, Ph, }T .
Proceedings of the World Congress on Engineering 2013 Vol III, WCE 2013, July 3  5, 2013, London, U.K.
ISBN: 9789881925299 ISSN: 20780958 (Print); ISSN: 20780966 (Online)
WCE 2013

TABLE IMODEL PARAMETERS
Symbol Value Parameter
m 1050 kg vehicle massh 0.450 m center of gravity (CoG) heighta 1.150 m CoG to front axleb 1.000 m CoG to rear axle distanceeIz 800 kg m2 yaw moment of inertia
12CDA 0.378 kg/m aerodynamic drag factorKr,Kf 20 rad1 tires nondimensional sideslip stiffnessr, f 1.05 tires friction coefficientrb 0.60 braking torque ratio on the front axle
Pe,max 125 kW SI engine max powerE0 200 V battery nominal voltage
Qmax 10 Ah battery capacityib,d 100 A battery max discharging currentib,c 50 A battery max recharging currentRb 0.200 Battery internal resistancevdc 1000 V DC link voltagep 6 number of pole pairs of the PMS motorR 0.040 PMS stator resistance
Ld, Lq 450 mH stator inductances of PMS motor 0.20 Wb rotor magnetic flux of PMS motor
im,max 250 A max current of PMS motor 9 transmission ratior 0.96 rectifier efficiencydc 0.96 DC/DC converter efficiencyi 0.96 inverter efficiencyt 0.85 transmission efficiency (tires included)
III. THE MINIMUM LAP TIME PROBLEM
The minimum lap time problem consists in finding thevehicle control inputs that minimize the time T necessary tomove the vehicle from the starting line to the finish one of thegiven track, by satisfying the mechanical equations of motionas well as inequality constraints such as tires adherence, maxpower, track width, etc. As the final value T of the timevariable t is clearly undefined, while the curvilinear abscissas varies from between fixed initial point s = 0 and endpoint s = L, it is convenient to mathematically formulatethe problem in terms of the independent variable s insteadof t. Such optimal control problem (OCP) may be formulatedas follows:
find: minuU
T (22a)
subject to:d
dsx = f (x,u, s) (22b)
(x,u, s) 0 (22c)b (x(0),x(L)) = 0 (22d)
where x and u are respectively the state variables and inputsvector, (22b) is the state space model in the s domain,(22c) are algebraic inequalities that may bound both the statevariables and control inputs and (22d) is the set of boundaryconditions used to (partially) specify the vehicle state at thebeginning and at the end of the maneuver.
A. State space model
The vehicle model described in Sections IIA and IIB hasto be slightly modified to take into account that in the OCP
formulation (22) it is allowed to bound the controls u, but itis not possible to guarantee that such controls remain smoothand to avoid unrealistic jerky maneuvers. For this reason,powers and steering angle are not controlled directly, but viatheir (bounded) time derivative, as follows:
d
dt = (23a)
d
dtPg = mujg (23b)
d
dtPh = mujb (23c)
d
dtPw = mujh (23d)
where power controls j have dimensions of jerk [ms3].To be consistent with the OCP formulation (22b), the inde
pendent variable t must be replaced with the new independentvariable s in all model equations, according to the followingderivation rule:
x =dx
ds=dx
dt
dt
ds=dx
dtt (24)
where the time variation with respect to the track length issimply calculated by inverting the relation (21a) as follows:
t =1 n(s)
u cos v sin(25)
At this point the sdomain state space model has 11 statevariables:
x(s) = {t, Qb, u, v,, n, , Pg, Pb, Ph, }T (26)
and 4 inputs:
u(s) = {, jg, jb, jh}T (27)
B. Inequality constraints and boundary conditions
Inequalities (22c) are used to keep the vehicle operatingconditions inside their admissible range. Power train constraints include the limitation of the SI engine power:
0 Pe Pe,max (28)
The battery is constrained in terms of charge and current:
Qb,min Qb Qb,max (29a)ib,d ib ib,c (29b)
The PMS motor/generator is constrained in terms of voltageand current:
v2d + v2q
v2dc3
(30a)
iq im,max (30b)
Braking power is constrained to be negative:
Ph 0 (31)
The vehicle must remain inside the track borders:
(bl w) n br w (32)
while w is the vehicle width and bl, br are the left and rightborders distance.Tires adherence is constrained inside their traction ellipse:
F 2r + S2r (rNr)
2 (33a)
F 2f + S2f (fNf )
2 (33b)
Proceedings of the World Congress on Engineering 2013 Vol III, WCE 2013, July 3  5, 2013, London, U.K.
ISBN: 9789881925299 ISSN: 20780958 (Print); ISSN: 20780966 (Online)
WCE 2013

where r, r are the friction coefficients respectively for therear and front tire.
To complete the problem definition it is necessary tospecify boundary conditions (22d). As the optimization ismade on a closed loop track, it is natural to impose cyclicboundary conditions for all state variables x(s), except forthe time t  which is obviously acyclic.
C. Solution of the Minimum Lap Time problem
The optimal control problem defined in equations (22) maybe solved by using various methods [16], such as non linearprogramming, dynamic programming or indirect methods.The latter approach has been used in this work. Summarizing,the inequality constraints (22c) have been replaced with some(approximately) equivalent terms in the penalty function(22a) by means of a wall function W which is null when theconstraint is satisfied and becomes very large as the boundaryis approached and possibly exceeded. Moreover, the presenceof equality constraints (22b) is managed with the Lagrangesmultiplier methods. In this manner, the constrained OCPproblem (22) is converted into the equivalent, unconstrainedminimization of the following functional:
J(x,u, s) = t(L)+j
L0
W (j)ds+i
L0
i(xi fi)ds
(34)To minimize J(), the first variation principle is used and theminimization problem is finally converted into a differentialboundary value problem (BVP). More details are given in[10], [17]. Since this process heavily requires the manipulation of the equations of motion at symbolically level, thewhole problem formulation has been carried out in Maple[18]. In particular the mathematical model of the vehiclehas been modelled by using the MBSymba package [19],then the OCP problem has been formulated by using theXOptima package, which also automatically generates C++code ready to be compiled. Finally, the numerical integrationof the BVP problem is performed by using the specializedsolver described in [10].
IV. SIMULATION RESULTS
Simulations have been carried out for the exemplary HEVvehicle, whose characteristics are summarized in table I,and the Mugello circuit. Figure 4 shows the circuit andalso the calculated optimal trajectory, speed and acceleration. In order to better understand the significance of thebattery power boost and its optimal utilization, the vehicleperformances with and without the battery are compared. Thefigure highlights that the boost (blue line) slightly increasesacceleration and top speed, leading to a difference in thelap time of 189.19 s versus 191.24 s (i.e. +2.05 s) when thebattery is disabled. Secondly, the figure shows that speedand acceleration are equal both at the beginning and at theend of the lap, according to the imposed cyclic boundaryconditions. While acceleration is limited by engine batterypower, deceleration is limited by tire adherence. The latter isdepicted in Figure 5 in terms of the lateral force to tire loadratio vs longitudinal force to tire load ratio, for both tires.In particular this picture shows that the optimal maneuvermakes large utilization of combined longitudinal and lateral
200 0 200 400 600 800 1000
300
200
1000
100
0 1000 2000 3000 4000 5000 6000 70000
20
40
60
u [m
/s]
curvilinear abscissa [m]
0 1000 2000 3000 4000 5000 6000 700010
5
0
5
a x [m
/s2 ]
curvilinear abscissa [m]
Fig. 4. Lap time optimization at the Mugello circuit: optimal trajectory,speed and longitudinal acceleration (solid blue: reference HEV vehicle,dashed red: battery disabled).
1 0.5 0 0.5 1
1
0.5
0
0.5
1
long
itudi
nal a
dher
ence
lateral adherence
rear tire
1 0.5 0 0.5 1
1
0.5
0
0.5
1
long
itudi
nal a
dher
ence
lateral adherence
front tire
Fig. 5. Adherence of the rear and front tires (solid blue: reference HEVvehicle, dashed red: battery disabled).
forces, as only race drivers do. Figure 6 highlights the vehicle power flow from the spark ignition engine and battery tothe PMS motor/generator. During acceleration, the SI engineis capable of providing the maximum power regardless ofthe vehicle speed, indeed the engine may easily operate atthe speed of maximum power thanks to the absence of anymechanical coupling between the engine and the tractionaxle. The battery presents a more evident on/off operationalmode, approximately switching from the positive maximumof delivered power to the negative maximum of rechargepower. As it is required that the battery state of charge(SoC) at the end of the lap should be equal to the beginning(as depicted in figure 7), the integral of battery power isslightly negative to counterbalance power train losses. Itmay be observed that the battery is recharged not only byrecovering energy during braking, but also by using the SIengine power when there is some surplus (e.g. at s=3000 m)and even during acceleration (e.g. 1250 . s . 1500,6500 . s . 6800, etc.). This effect is more evident byanalyzing the PMS power: the maximum battery power isused to boost the acceleration from s = 400 m (blue line vsred line), at s = 1020 m the battery boost is not more usedbecause the energy stored in the battery is not more sufficient.From s = 1180 m, the reference case highlights that the fullavailable SI engine power is not provided to the PMS motor
Proceedings of the World Congress on Engineering 2013 Vol III, WCE 2013, July 3  5, 2013, London, U.K.
ISBN: 9789881925299 ISSN: 20780958 (Print); ISSN: 20780966 (Online)
WCE 2013

0 1000 2000 3000 4000 5000 6000 70000
50
100
150
Pen
gine
[kW
]
0 1000 2000 3000 4000 5000 6000 700010
0
10
20
Pba
ttery
[kW
]
0 1000 2000 3000 4000 5000 6000 70000
50
100
150
PP
MS [k
W]
curvilinear abscissa [m]
Fig. 6. Power of the SI engine, battery and PMS motor/generator (solidblue: reference HEV vehicle, dashed red: battery disabled).
0 1000 2000 3000 4000 5000 6000 700075
80
85
90
SoC
[%]
curvilinear abscissa [m]
Fig. 7. Battery State of Charge (solid blue: reference HEV vehicle, dashedred: battery disabled).
(as in the case of disabled battery), but it is split betweenthe motor and the battery. In other words, to minimize thelap time, it is convenient to use the battery to boost duringthe initial phase of the acceleration, even if it is necessary torelinquish some propulsive power in the successive phase torecharge the battery. The proper timing of battery boost andrecharging (shown in figure 7) is hence crucial for the laptime minimization. This task is excellently carried out withthe global optimization approach adopted, while it would bevery difficult to optimally manage the power flow by meansof some predeterminate control strategy.
V. CONCLUSIONS
This paper illustrates a methodology for the lap timeoptimization of a race series hybrid electric vehicle basedon the indirect optimal control approach. This method isvery powerful because it does not require the definitionof a specific control architecture or strategy in advance,while system inputs are given as a result of the optimizationprocess. Second, the optimization is performed globally, i.ealong the whole track. This is essential in this kind ofproblem, where the battery introduces a strong couplingamong the different sections of the circuit: battery boostmay be used while exiting from a curve only if the battery
has been recharged while braking in another section, notnecessarily close to the first one.
Conversely, the indirect method is not so popular due tothe difficulties associated with the problem implementationas well as the numerical solution. The transformation ofthe constrained optimization problem into an equivalentboundary value problem (BVP) requires the definition of anoptimizationtailored model and the symbolic manipulationof the model equations, that in the present work havebeen performed by using computer symbolic algebra tools.This approach also leads to very computationally efficientapplications, with the optimizations in this paper carried outin times much faster than real time, highlighting that thisapproach may be applied effectively to more complex vehiclemodels as well.
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Proceedings of the World Congress on Engineering 2013 Vol III, WCE 2013, July 3  5, 2013, London, U.K.
ISBN: 9789881925299 ISSN: 20780958 (Print); ISSN: 20780966 (Online)
WCE 2013