Least costly energy management for series hybrid electric ... · S. Formentin et al. / Least costly energy management for series hybrid electric vehicles As it can be easily understood,

arX

iv:1

510.

0756

6v1

[cs.

SY

] 26

Oct

201

5

Least costly energy management for series hybrid electric vehicles

Simone Formentin∗, Jacopo Guanetti, Sergio M. Savaresi

Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, P.za L. da Vinci 32, 20133 Milano, Italy.

Abstract

Energy management of plug-in Hybrid Electric Vehicles (HEVs) has different challenges from non-plug-in HEVs, due to biggerbatteries and grid recharging. Instead of tackling it to pursue energetic efficiency, an approach minimizing the driving cost incurredby the user – the combined costs of fuel, grid energy and battery degradation – is here proposed. A real-time approximation ofthe resulting optimal policy is then provided, as well as some analytic insight into its dependence on the system parameters. Theadvantages of the proposed formulation and the effectiveness of the real-time strategy are shown by means of a thorough simulationcampaign.

Keywords: Series HEV, EREV, Energy management, Dynamic Programming,Pontryagin Minimum Principle, Optimal control

1. Introduction

Hybrid Electric Vehicles (HEVs) are generally regarded toas an effective solution to improve the fuel economy and re-duce CO2 emissions with respect to Internal Combustion En-gine (ICE) vehicles. Since HEVs are usually equipped with(at least) two energy sources, a critical energy managementproblem arises, that is, a supervisory system is needed to de-termine how to generate the requested power. In the so-called“mild HEVs”, the downsized battery and the electrical motordonot allow to drive the vehicle based just on the electric power,but only to assist the ICE in low efficiency operating points.In this framework, heuristics and rule-based algorithms haveshown to provide satisfactory results. On the other hand, highlyhybridized powertrains call for more sophisticated control ap-proaches for their higher flexibility [1].

In the latter configuration, given a model of the hybrid power-train, the best performance theoretically achievable overa driv-ing schedule can be computed by means of optimization tech-niques, see,e.g., [2, 3]. A classical approach in HEVs aims atminimizing the overall fuel consumption, concurrently penal-izing excessive deviations of the battery state of charge [4, 5].Such a penalty term is very important for conventional HEVs,in which the minimization of the fuel consumptiontout courtmay lead to excessive battery charge depletion .

The above optimization approach usually yields a non-causalcontrol policy, which defines a useful upper bound in terms ofperformance for a given driving cycle. A good approxima-tion of the above optimal policy can be found using the so-called Equivalent Consumption Minimization Strategy (ECMS)- based on the Pontryagin Minimum Principle - in which theknowledge of future power requests is replaced by a cycle-dependent parameter, see [1, 6–9] for further details. Adaptive

∗Corresponding author. Tel:+39 02 2399 3498 - Fax:+39 02 2399 3412Email address:[email protected] (Simone Formentin)

variants of the ECMS have also been developed and success-fully implemented in real-time [10, 11]. Nonetheless, other realtime approaches have been explored, based,e.g., on Model Pre-dictive Control [12, 13] or Robust Control [14, 15].

The above strategies were originally conceived for conven-tional, non plug-in HEV powertrains, that is when the batterycan be recharged exclusively during vehicle operation,e.g. byregenerative braking or thermal power surplus. However, morerecent plug-in HEVs make it possible to recharge the batteryfrom the grid [16, 17]. Quite simultaneously, progresses inbat-tery technology are making big battery packs more affordable,thus extending the electric autonomy of such vehicles.

Upcoming HEVs are then more and more conceived as plug-in vehicles with a relatively large battery and a significant“all-electric range”, with a thermal unit often playing the role ofa range extender. In view of this trend, on the one hand, theneed for charge sustenance becomes less critical. On the otherhand, since the battery has a more significant impact on theoverall vehicle cost, the battery operating conditions leading tofast aging should be avoided.

Supervisory strategies have been proposed also for the en-ergy management of plug-in and series HEVs. Sticking as arelevant case to the ECMS strategies mentioned above, someimplementations for a plug-in HEV are presentede.g. in [18];quite intuitively, here the charge sustenance constraint can berelaxed, by taking into account the characteristics of the power-train and the available information on the trip to be performed.A general framework for energy optimization of plug-in HEVshas been recently introduced in [19], where the optimal data-driven tuning of the ECMS policy is also discussed. In somerecent works, battery aging is accounted for in the optimiza-tion problem. In [20] battery aging and energy consumptionare both regarded as relevant phenomena for the optimal deple-tion strategy of the battery in a plug-in HEV. In [21, 22] a simi-lar problem is tackled for HEVs with a hard charge sustenance

Preprint submitted to Elsevier October 7, 2018

http://arxiv.org/abs/1510.07566v1

S. Formentin et al./ Least costly energy management for series hybrid electric vehicles

constraint; in these works, ECMS-based strategies are devel-oped, with an additional tuning parameter affecting the weightof the aging in the cost function.

The contributions of this paper can be summarized as fol-lows. Firstly, aleast costlyformulation of the energy manage-ment is proposed, aiming to fully exploit series hybrid pow-ertrains. The underlying model also accounts for battery agingand the optimal control problem accounts for all thecost entriesrelated to both the electrical part and the thermal unit.

Secondly, by applying Dynamic Programming (DP) [23], itis shown that the resulting energy management policy does notnecessarily yield minimum fuel consumption. As a matter offact, cheap fuels like CNG (Compressed Natural Gas) can provecheaper than driving entirely on electric power, especially ifbattery purchase cost is considered; in such a scenario, a formu-lation in terms oftotal driving costis desirable from the point ofview of the user. Moreover, limited diffusion of alternative fu-els may boost the adoption of multi-fuel range-extenders [24].In the latter case, a total driving cost formulation allows to finda compromisee.g. between a relatively expensive fuel that iseasy to find, like gasoline, and a cheaper less widespread fuel,like CNG.

Unfortunately, the above DP-based solution relies upon thea-priori knowledge of the driving cycle. Therefore, as a fur-ther contribution of the paper, two causal implementationsofthe least costly energy management strategy are proposed. Theoptimal policy is first derived based on a simplified model of thepowertrain in an explicit way: although the model is less gen-eral, in this case the policy is expressed as a set of explicitrules,hence its implementation requires substantially less memoryand computational power. Furthermore it is shown that, whena more complex model of the powertrain is necessary, the op-timal policy can still be computed numerically, attaining veryclose results to the acausal benchmark. Finally, the paper in-cludes a sensitivity analysis, that investigates the performanceof the numerical policy for a broad range of model parametersand energy costs.

The remainder of the paper is as follows. A general formula-tion of the energy management problem - as well as some spe-cific formulations in terms of energy consumption minimiza-tion - is given inSection 2, where the full-fledged simulatorof the vehicle and the simulation scenarios used in the follow-ing sections are also presented. By deriving a suitable control-oriented model and an economic cost function, the least costlyenergy management approach is presented inSection 3, wherethe resulting non-causal policy is also derived by Dynamic Pro-gramming.Section 4provides the causal policies for the leastcostly energy management problem, whileSection 5discussesthe limits of applicability of such a strategy by means of a sen-sitivity study. The potential of the new approach is shown ineach section by employing both a urban and a mixed urban-motorway driving cycle. The paper is ended by some conclud-ing remarks.

2. Problem formulation and simulation setup

In this section the HEV energy management problem is pre-sented and the way it is commonly addressed in the literatureis discussed. Moreover, the simulation setup and the drivingcycles – employed in the remainder of the paper to test the pro-posed strategy – are introduced.

2.1. Problem Formulation

With “energy management problem” it is meant the problemof designing a supervisory control layer with the aim of man-aging the power dispatch between multiple sources in a HEV.More specifically, such a problem is commonly formalized asan optimal control problem over a finite time horizon. With rea-sonable knowledge of the vehicle, the speed and slope profilesof the trip can be converted into a profile of requested electricalpower in series HEVs, or mechanical power in parallel HEVs.The remainder of the paper is focused on series HEVs.

Formally, an energy management problem can be written as

minu

J = h(x(T)) +∫ T

0g(t, x(t), u(t),w(t))dt

s. t. x = f (t, x(t), u(t),w(t))

x(0) = x0

x(t) ∈ X

x(T) ∈ XT

u(t) ∈ U

(1)

whereJ is the cost function to minimize,x collects the statevariables,u is the control variable,w represents the exogenousinput variable, f denotes the state function,g is the runningcost andh is the terminal cost.x, u,w are assumed to be scalarvariables andf , g, hare assumed to be scalar, possibly nonlinearfunctions.X = [ xmin, xmax] ⊆ R is the set of admissible valuesfor the state variable; the bounds ¯xmin, xmax are assumed to bestatic.XT is the set of admissible values for the final state.U =[umin(t), umax(t)] ⊂ R × RT−1 is the set of admissible values forthe input variable; the boundsumin, umaxare assumed to possiblybe time-varying.

Many approaches proposed in the literature aim at minimiz-ing the fuel consumption for a given trip; therefore, the fuelmass flow rate is often chosen as the running cost as

g(u(t)) = mf (u(t)). (2)

The fuel mass flow rate reasonably depends on the control pol-icy u(t). The control input may be the battery current, the bat-tery power, the generated power or the ratio between batteryandgenerated power. The state variable is typically the battery stateof charge, which requires the introduction of a battery model.

A possible strategy is theFull Electric mode,i.e. the simpleminimization of the fuel consumption, without any constraintor penalization on the final state

h(x(T)) = 0XT = X.

(3)

2


As it can be easily understood, such an approach can leadto excessive depleting of the battery charge, if the trip is suffi-ciently long or demanding. To overcome such an issue, a con-straint on the final state of charge can be set,e.g. the desiredfinal state can be enforced to be equal to the initial state suchthat perfect “charge sustenance” is accomplished. Notice thatfor the class of HEVs here considered, an arbitrary final stateof charge could be requested by the user; the charge sustainingcase is just one possible choice. Formally, this means

h(x(T)) = 0XT = x(0),

(4)

and this approach will be referred to asCharge Sustainingmode, from now on.

A hard constraint on the final state can be fulfilled when anon-causal solution of the problem is computed. When the op-timization has to be performed in real-time, soft constraints aresimpler to handle. The so-calledEquivalent Consumption Min-imization Strategy(ECMS) approach [1, 6–8] has an effectivereal-time implementation and can be stated as

h(x(T)) = ζ(x(T) − x(0))XT = X,

(5)

whereζ is a user-defined parameter usually depending on theconsidered driving cycle. Notice that ECMS strategies some-times consider a modified cost function (see e.g. [11]) that pe-nalizes not only the fuel consumption, but also any deviation ofthe state of charge from a reference valuexre f (t)

g(u(t)) = mf (u(t)) + µ

(

xre f (t) − x(t)

∆x

)2q

, (6)

whereµ is a weighting factor,q denotes the order of the penaltyterm and∆x is the largest admissible deviation of the state ofcharge.

The above formulation of the energy management problemhas been widely investigated in the literature (seee.g.[1, 6, 7]).Nonetheless, it was originally conceived for parallel HEVs, notrechargeable from the grid. Current trend in HEVs marketis oriented towards grid rechargeable vehicles with significantbattery capacity. Rather than merely increasing the efficiencyof the thermal engine, in many cases the purpose is also tohave an electric range suited for every-day urban use; a clueis the increasing commercial offer of plug-in HEVs and Ex-tended Range EVs. This peculiar aspects are considered in therecent literature when studying the energy management prob-lem for this new generation of HEVs. As a relevant and recentexample, some implementations of the aforementioned ECMSfor a plug-in HEV are presented in [18]; consistently with theremarks made above, the charge sustenance constraint is re-laxed, taking into account the significant battery capacityandthe available information on the trip to be performed. As dis-cussed in [19], the tuning of ECMS policies for plug-in HEVsis highly related to the relative weight of battery energy and fuelenergy; this weighting can be formulated to minimize, e.g.,thevehicle energy or the CO2 emissions.

In this work, a formulation of the energy management prob-lem is proposed, taking into account the phenomena with great-est impact on the overall driving cost incurred by the user. Thischoice will make the energy management a real least costly pol-icy.

The main motivation of such a choice is that, in recent ve-hicles, the electric portion of the powertrain has a very signifi-cant impact on the overall cost of the vehicle, mainly due to thebattery cost [25]. This calls for a different formulation of themanagement problem, in which also the electric costs are ex-plicitly considered in the overall minimization criterion. Sincethis study is focused on series HEVs, it is as well reasonabletoassume that a battery with significant capacity is availableonboard; in other words, for this class of HEVs, hard charge sus-tenance is not a critical need. Therefore, while the cost of bothelectric and fuel energy are relevant [19], it is also important toinclude the effect of the battery value depletion.

2.2. Simulation Setup

In order to test the proposed strategies, a full-fledged simula-tor of a series HEV was implemented using a backward facingapproach. This simulator is based on the platform developedat ETH Zurich [26] and is available on-line [27]. More detailsabout its implementation are given in Appendix A.

Backward-facing approach is generally intended for the sim-ulation of the overall energetic performance of the vehicle, overtime scales comparable to the duration of standard driving cy-cles. The subsystems are described by approximate, quasi-static or low-order dynamic models. The name of the approachrefers to the fact that the inputs to the simulator are the vehicleroad performance, for instance in terms of longitudinal speedand slope. The behavior of the vehicle is simulated proceed-ing backwards in the powertrain and computing the upstreamenergy flows. The outputs of our simulator are the fuel con-sumption, the battery state of charge and the battery state ofhealth.

A conceptual scheme of the simulator is depicted in Fig. 1,wherev is the vehicle longitudinal speed,θ is the road slope,qb

is the battery state of charge,ξb is the battery state of health, ˙mf

is the fuel consumption of the thermal generator. Notice that inFig. 1 thin lines represent electric connections, while thick linesrepresent mechanical connections.

Three main areas can be identified, a vehicle dynamics area(containing the vehicle itself, the transmission and the motor), abattery area and a thermal generation area (containing the ther-mal engine and the electric generator). The three areas are con-nected by means of an electric power link, where the sum of thebattery powerPb and of the generated powerPr has to be equalto the traction motor powerPm. Details on the parametrizationand the equations included in each block are given in the Ap-pendix.

In this work two different driving cycles are used as inputs tothe simulator. The first driving cycle, depicted in Fig. 2, isthestandard FTP urban driving cycle and is namedUrban DrivingCyclehereafter. The second driving cycle, depicted in Fig. 3, isobtained by merging the FTP urban driving cycle and the FTP

3


Transmission Motor Battery

Generator Engine

Vehiclev(t)

θ(t)

qb(t)

ξb(t)

mf (t)

Pm(t) Pb(t)

Pr(t)

Link

Figure 1: Overall composition of the Vehicle Simulator.

time (s)0 500 1000 1500

spee

d (m

/s)

0

10

20

30

Figure 2: Urban Driving Cycle.

time (s)0 1000 2000 3000 4000 5000 6000 7000

spee

d (m

/s)

0

10

20

30

Figure 3: Combined Driving Cycle.

Highway driving cycle and is namedCombined Driving Cyclehereafter.

For both driving cycles, a realistic scenario is considered,in which the vehicle has already depleted 50% of the charge,but has to accomplish another trip before having the possibil-ity to recharge the battery from the grid. Moreover, to makethe trade-off between thermal and electric power less trivial,a CNG range-extender is considered. This choice falsifies thewidespread belief that “electric is cheaper”: as a matter offact,with reasonable costs of fuel, energy and battery, the usageofa CGN engine is found to be preferable even within the rangeachievable in full-electric mode.

3. Total Cost Minimization

The proposed approach is to minimize the overall cost givenby the sum of three items: the grid energy for battery recharge,

the damage to the battery and the fuel consumed to generatepower. Since these quantities are heterogeneous, the cost func-tion is defined as the sum of the related monetary costs and it isreferred to as thedriving costhereafter.

In particular, in this section, the new formulation is intro-duced and the features of its optimal solution are discussed. Areal-time control policy will be instead object of the next sec-tion.

3.1. Mathematical Modeling

In a series HEV, the following electrical power balance holds

Pm(t) = Pr (t) + Pb(t), (7)

wherePm is the traction motor power,Pr is the generated powerandPb is the battery power, which can be computed as

Pb = vbib. (8)

From now on, positive current values will correspond to bat-tery discharging. A circuit model of the battery is needed torelate the voltagevb to the currentib. The voltage is consideredto be the sum of an open circuit term and of an ohmic term,accounting for Joule losses

vb = voc − Rbib, (9)

where the internal resistanceRb is considered to be constantand the open circuit voltagevoc to be an affine function of thebattery state of charge [28]

voc(t) = Abqb(t) + Bb, (10)

whereAb andBb are parameters fitting the real open circuit volt-age of the battery. In Figures 4 and 5 the open circuit voltageand the internal resistance of a Li-ion cell, intended for use ona pHEV and measured at different SoC levels, are illustrated.The measured values are compared with the proposed models,as defined in the rest of the section.

The state of charge of the batteryqb ∈ [0, 1] is defined ac-cording to the well established definition [28]

qb(t) = −ib(t)Qb, (11)

whereQb is the battery capacity, which slowly decreases as thebattery ages; generally, the battery is considereddeadwhen itscapacity has decreased to 80% of the nominal capacity [21].

4


qb (%)0 20 40 60 80 100

voc(V

)

0

1

2

3

4

measured

model M1

model M2

Figure 4: Open circuit voltage of a Li-ion cell: measured data vs. proposedmodels. The accuracy of (10) and (17) in describingvoc can here be appreci-ated.

qb (%)0 20 40 60 80 100

Rb(m

Ω)

25

30

35

40

measured

models M1, M2

Figure 5: Internal resistance of a Li-ion cell: measured data vs. proposed mod-els.

Since this decay is very slow,Qb can be considered constantover a driving cycle of few hours, and can be modeled as

Qb = Qnomb (1− 0.2ξb), (12)

whereQnomb is the battery nominal capacity at the beginning of

life andξb ∈ [0, 1] is the battery state of health at the beginningof the mission. The battery depth of discharge is also introducedas

db(t) = 1− qb(t). (13)

Battery aging is a rather complex phenomenon to model. Itis widely discussed in the literature and still lacks a unified ap-proach [29]. Quite intuitively, aging models are developedfordifferent purposes: manufacturers may be interested in modelsthat accurately describe the underlying electro-chemicalpro-cesses, and thus can reduce the need for expensive experimen-tal campaigns – in general, at the cost of high mathematical andcomputational complexity. On the other hand, simpler agingmodels are sought when it comes to estimate the battery agingfrom real world measurements, either to assess the current stateof health [30], or to develop high level strategies [25], as it is inour case.

Therefore, the simple but effective aging model [31] is em-ployed

ξb(t) =σb

NbQnomb

|ib(t)|, (14)

meaning that the depletion of battery life is a function of thecurrent throughput.Nb is the number of charge-discharge cy-cles that the battery can stand over its entire life, in nominalconditions.σb is a weighting factor, often calledseverity fac-tor, that depends on the operating condition [31]; in our workσb depends on the battery state of charge and current

σb = σb(qb(t), ib(t)). (15)

Despite its simplicity, this model accounts for battery degra-dation in a tractable way. A similar model was used in [20],where complex electro-chemical models were reduced to beused in a supervisory controller, coming up with the integral ofa static function of current and state of charge and the integralof the current throughput. A similar approach is also used in[21]. As mentioned in the introduction, [22] also considersbat-tery aging in the energy management of an HEV. In this case, adifferent aging model is used, where instead of the severity fac-tor, an explicit function of the battery current is used; however,the aging effects due to the state of charge are not consideredwith that approach.

The thermal generation unit in a series HEV is mechanicallyunconstrained from the external world; therefore the mechani-cal operating point can be arbitrarily chosen as a function of therequested amount of power to generate. Hereafter it is assumedthat a lower level controller continuously adjusts the operatingpoint in a quasi-static manner. Given the quasi-static efficiencymaps of the thermal engine and of the electric generator, themost efficient operating points were computed as a function ofthe generated electrical power. If a thermal characterization ofthe unit was available, this lower level policy could also takeinto account the effects of thermal transients on fuel economy[32]. Therefore, under the assumption of quasi-static operation[28], the fuel power and the corresponding flow rate are

Pf (t) =Pr(t)ηr (Pr(t))

δ (Pr (t))

mf (t) =Pf (t)

λr

(16)

whereηr is the combined efficiency of the Engine-GeneratorUnit, λr is the fuel lower heating value andδ is a Dirac delta.The above equations imply that the fuel injection is active onlywhen the request of power generationPr is strictly positive. Thecombined efficiencyηr is the ratio between generated powerPr

and fuel powerPf and is therefore computed from the effciencymaps of both the engine and the generator; Figure 6 shows thecombined efficiencyηr of the Engine-Generator Unit at differ-ent power levels.

The powertrain of the series HEVs is considered to be fullydescribed by the above equations and therefore the followingdefinition is used hereafter.

Definition 1. The full control-oriented modelM1 of the pow-ertrain of a HEV is defined by the set of Equations(7)-(16).

ModelM1 is used to compute the optimal solution by DP inthe next subsection, as well as to derive a numeric approxima-

5


Pr (kW)

0 5 10 15 20 25

ηr (

%)

0

10

20

30

model M1

model M2

Figure 6: Efficiency of the Engine-Generator Unit at different power levels(data taken from the public database ADVISOR [33]).

tion of the optimal law in Section 4. By contrast, an explicitfor-mulation of the optimal law is also derived in Section 4, basedon a simpler model, with the following differences:

• The battery open circuit voltage is constant and equal tothe nominal voltage, replacing (10) with

voc(t) = vnomb , (17)

which is an approximation often used for HEVs and is rea-sonable if the battery state of charge is kept in a sufficientlynarrow range (see again Figure 4).

• The dependence of the battery severity factor from thestate of charge and current is neglected, replacing (15)with

σb = σb(qb, ib), (18)

whereqb, ib represent a nominal operating condition; thisapproximation is reasonable if the normal operating rangesof qb andib are sufficiently close to the nominal operatingconditions prescribed by the manufacturer. For instancein [25, 34] models of this kind are used and validated forplug-in HEVs; these vehicles (like series HEVs consideredhere) typically have a quite largeQb, which makes it lesslikely both to hit theqb limits and to operate at high C-rate,and thus make a simplified model with constant severityfactor reasonable.

• The thermal unit is modeled as an affine function of thegenerated power, replacing (16) with

Pf (t) = (Ar Pr (t) + Br) δ (Pr (t))

mf (t) =Pf (t)

λr

(19)

whereBr/λr represents the fuel consumption of the idlingengine andAr could be interpreted as the inverse of thegeneration efficiency. Usually, the efficiency is a func-tion of the generated power, hence the coefficients maybe found through a linear fit of the nonlinear model givenin (16). Moreover, the Willans approach [28] models theengine fuel consumption as a linear function of torque;

hence, the Willans model corresponds to (19), under themild assumption that the range engine regime is kept nar-row to maximize the efficiency.

The simple model is then formally described as follows.

Definition 2. The simplified control-oriented modelM2 of the powertrain is defined by the set of Equa-tions(7), (9), (11), (12),(13), (14), (17), (18), (19).

A formal statement of the problem defined in the previoussection is now given. Let the battery state of charge be the statevariable

x(t) = qb(t), (20)

the battery current be the input variable

u(t) = ib(t), (21)

and the traction motor power be the exogenous disturbance

w(t) = Pm(t). (22)

The state function is defined as

f (x(t), u(t)) = qb(t), (23)

and the running cost as

g(x(t), u(t)) = αvnomb Qbdb(t) + βvnom

b Qnomb ξb(t) + γPf , (24)

which sums three cost items: the grid energy used to rechargethe battery, the damage caused to the battery, the fuel consumedto generate power.α, β, γ are respectively the monetary costs of1Wh of grid energy, 1Whof battery capacity and 1Wh of fuelenergy. In the equation above, we also included the batterynominal voltagevnom

b and the grid recharging efficiencyηgrid.The first term therefore accounts for the cost of a grid chargebefore or afterthe trip at hand; the charging phase is thus de-scribed by a static model with a constant battery voltagevnom

b .Notice that the definitions ofξb and mf are different for

ModelM1 and ModelM2. Notice also that, although the mod-els of battery aging and of fuel consumption are dynamic mod-els, ξb andmf are not treated as state variables in the controlproblem statement. This is usually done withmf , but the sametreatment can be extended also toξb. As a matter of fact, this isconsistent with the approximation of consideringQb constantas in (12): the advantage of considering its dependence onξbwould be negligible when considering a driving cycle of fewhours. This is also shown in a simulation example in Subsection3.3.By contrast, it is worth considering all the resulting contri-butions ofdb, ξb andmf in the cost functionsince, for reason-able scenarios, they share the same order of magnitude. Sincethe three quantities are heterogeneous, their monetary costs canbe simply summed up, considering the unitary costsα, β, γ.

Considering the constraints,

X = [qmin, qmax] = [0.2, 0.9], (25)

represent the static bounds for the state variable, while nope-nalization function on the final state is set, that is

h(x(T)) = 0. (26)

6


Moreover, since the set of admissible final states is not re-stricted,

XT = X. (27)

The time-varying bounds for the control variableib(t) are in-stead

U = [umin(t), umax(t)], (28)

computed from the corresponding power bounds according tothe battery circuit equation

umin(t) =voc −

√

v2oc− 4RbPmin(t)

2Rb, (29)

umax(t) =voc −

√

v2oc− 4RbPmax(t)

2Rb. (30)

The time-varying power boundsPmin,Pmax ensure that staticpower bounds of the battery and of the generator are fulfilledwith the current value of the traction motor power. In view of(7), they can be computed as:

Pmin(t) = max

Pminb ,Pm(t) − Pmax

r

, (31)

Pmax(t) = min

Pmaxb ,Pm(t) − Pmin

r ,v2

oc

4Rb

, (32)

wherePminb , P

maxb are the battery power limits andPmin

r , Pmaxr are

the Engine-Generator unit power limits. All the model parame-ters and bounds defined so far are listed in Table A.8.

Then, the problem ofminimizing the total driving cost overthe given driving cycle, which will be called from now onTo-tal Cost Minimization Strategy(TCMS), can be mathematicallystated as follows:

minu

∫ T

0

(

αvnomb Qbdb(t) + βvnom

b Qnomb ξb(t) + γPf (t)

)

dt

s. t. qb(t) = −ib(t)Qb

qb(0) = q0

qb(t) ∈ [0.2, 0.9]

ib(t) ∈ [umin(t), umax(t)].

(33)

3.2. Benchmark Optimal Solution

The solution to Problem (33) can be computed off-line basedon Dynamic Programming [35]. Since such an approach doesnot directly apply to continuous time systems, a discrete-timecounterpart of the state function is considered. By BackwardEuler approach, (11) yields

qb(k+ 1) = qb(k) −Tsib(k)

Qb, k = 0, ...,N − 1, (34)

Ts being the sampling time andT = NTs being the time horizonof the mission.

The cost functionJ to minimize is redefined in the discretetime domain as

h(x(N)) +N−1∑

k=0

g(x(k), u(k),w(k)), (35)

PSfrag replacements

Jk(x(k)) (e)

q b(%

)

time (s)

0 0.1 0.2 0.3 0.4 0.5

0 500 1000 1500

48

50

52

Figure 7: Optimal Cost-to-go functionJk(x(k)) with the TCMS in the UrbanDriving Cycle.

where

g(x(k), u(k),w(k)) = αvnomb Qb

db(k+ 1)− db(k)Ts

+ βvnomb Qnom

b

ξb(k+ 1)− ξb(k)Ts

+ γPf (k+ 1)− Pf (k)

Ts. (36)

If Backward Euler approach is used, (14),(16) yield

ξb(k+ 1) = ξb(k) + Tsσb(k)|ib(k)|

NbQnomb

(37)

Pf (k+ 1) = Pf (k) + TsPr (k)

ηr (Pr (k))δ(Pr(k))(38)

According to Dynamic Programming, the optimal cost for agiven initial stateJ∗(x(0)) is found at the last iteration of thefollowing algorithm, proceeding backwards in time fromk =N − 1 to k = 0

JN(x(N)) = h(x(N))

Jk(x(k)) = minug(x(k), u(k),w(k))

+ Jk+1( f (x(k), u(k),w(k)).

(39)

The optimal control policyπDP = µ∗0, ...µ

∗N−1 is then found as

u∗(k) = µ∗k(x(k)) = arg minug(x(k), u(k),w(k))

+ Jk+1( f (x(k), u(k),w(k)),∀k,∀x(k). (40)

Consider now the simulation environment defined in Section2.2. For the Urban Driving Cycle, the values of the optimalcost-to-go functionJk(x(k)) are depicted in Fig. 7. Intuitively,Jk(x(k)) mainly grows when moving backwards in time (i.e. indistance traveled); the dependence on the state of charge ismi-nor, because no penalty for the final state of charge is set andthe considered trip is within the electric range.

The optimal state trajectory, given an initial state of chargeqb(0) = 0.5, is highlighted with a red line. Fig. 8 shows aportion of the corresponding engine operation. As the figure

7


time (s)340 360 380 400 420 440 460

torque(N

m)/v(km/h)

0

20

40

60

80

100

120engine on

engine torque

v

Figure 8: Engine torque and vehicle speedv with the TCMS in a portion of theUrban Driving Cycle. The shaded areas indicate where the engine is on.

shows, the engine outputs power during the vehicle accelera-tions, while it is switched off when the vehicle speed (i.e. thedemandedPm) starts decreasing.

The TCMS approach is compared to the approaches pre-sented in Section 2. Analogously to the TCMS formulation,also the other approaches are implemented using DynamicProgramming; details on the formulation are in Appendix C.As noted above, the simulation scenario encompasses a CNGrange-extender, which makes the trade-off between thermal andelectric power non trivial. The trends of the battery state ofcharge under the Urban Driving Cycle are compared in Fig. 9.The corresponding final values of the total driving cost and fuelconsumption are given in Table 1.

The Full Electric strategy depletes more than 5% of thecharge, the Charge Sustaining strategy and the ECMS attain thesame final state of charge, while the TCMS performs a slightrecharge of about 2%. Notice that the ECMS was tuned on thisdriving cycle to obtain perfect charge sustenance.

The TCMS attains the minimum driving cost, the Full Elec-tric strategy is nearly 50% more expensive while the ChargeSustaining strategy and the ECMS have about 30% higher cost.Despite some deviations in the trend of the state of charge, theCharge Sustaining strategy and the ECMS attain the same costand almost the same fuel consumption. On the other hand, theTCMS uses the most fuel, while the Full Electric simply keepsthe engine off for all the cycle. Notice that different combina-tions of the cost coefficients yield a different balance betweenthermal and electrical power; for instance, it is intuitivethat,if α = β = 0, the TCMS behaves as the Full Electric strat-egy. For any combination, the TCMS attains at least the sameperformance (in terms of monetary cost) as one of the otherapproaches. The interested reader can find in Section 5 a sensi-tivity analysis of the performance of the proposed approachfora broad range of fuel prices.

The same comparison is carried out in Fig. 10 and Table 2,in the case of the Combined Driving Cycle. The main observa-tions are as follows.

First of all, since such a cycle is beyond the electric rangeof the vehicle, also the Full Electric strategy needs the ther-mal unit to complete the trip; nonetheless, it completely de-

time (s)0 500 1000 1500

q b(%

)

40

45

50

55

60TCMSECMSCSFE

Figure 9: Battery state of charge in the Urban Driving Cycle according to theconsidered approaches: TCMS, Charge Sustaining, ECMS, Full Electric.

Table 1: Driving cost and fuel consumption in the Urban Driving Cycle ac-cording to the considered approaches: TCMS, Charge Sustaining, ECMS, FullElectric.

TCMS ECMS CS FE

Driving Cost [e] 0.57 0.74 0.74 0.82Fuel Mass [g] 484.29 343.56 340.65 1.49

pletes the battery charge. In the Combined Driving Cycle, theTCMS recharges the battery by about 5%. Some features areanalogous to the Urban Driving Cycle: the Charge Sustainingstrategy and the ECMS attain the same final state of charge,although with different transients; the TCMS is the cheapeststrategy; the Charge Sustaining strategy and the ECMS attainalmost the same results and Full Electric strategy is the mostexpensive. In terms of fuel consumption, the TCMS uses themost fuel, the Charge Sustaining strategy and the ECMS havesimilar results and the Full Electric strategy attains the lowestconsumption.

From the above results, a couple of interesting facts can beinferred:

• it is a common belief that, when a driving cycle is fullyachievable in Full electric mode, this is the least costly pol-icy, according to the assumption that “electric is cheaper”.

time (s)0 1000 2000 3000 4000 5000 6000 7000

q b(%

)

20

30

40

50

60

TCMSECMSCSFE

Figure 10: Battery state of charge in the Combined Driving Cycle according tothe considered approaches: TCMS, Charge Sustaining, ECMS,Full Electric.

8


Table 2: Driving cost and fuel consumption in the Combined Driving Cycleaccording to the considered approaches: TCMS, Charge Sustaining, ECMS,Full Electric.

TCMS ECMS CS FE


Table 3: Driving cost and fuel consumption in the Urban Driving Cycle withthe TCMS approach: DP implementation with 1 and 2 state variables.

x = qb x = [qb; ξb]T

Driving Cost [e] 0.57 0.57Fuel Mass [g] 484.29 483.24

This is not always true, and in particular it is not in theconsideredscenario.

• the optimal usage of the ICE, in terms of monetary cost, isnot trivial.

As a final remark, it is reasonable to expect that in the nearfuture – thanks to technological advances – range extender willbe a more flexible component of the powertrain; notable ex-amples are multi-fuel engines and plug-in range extenders.Asa future task, the proposed approach could quite easily be ex-tended to account for cases such as (i) only expensive fuel isavailable in the first part of the trip whereas (after a gas stationis reached) it becomes cheaper in the second part, and (ii) therange extender is only available in a portion of the trip. Insce-narios like the ones above, the proposed formulation in termsof monetary cost looks more appropriate and natural than moretraditional formulations in terms of energy consumption.

3.3. State variable choice

In this subsection the choice of using only one state variable(qb) instead of two (qb andξb) is motivated by means of a sim-ulation. We show that in the scenario considered above, theeffect of consideringξb as a state variable is negligible. Thisis accomplished modifying the problem formulation for the im-plementation of the DP:

• ξb is considered as a second state variable

• the capacity in equation (11) is computed asQb =

Qnomb (1− 0.2ξb(t))

The results found with this formulation are compared withthe proposed formulation in Figure 11 and in Table 3: no sig-nificant difference is found neither in the state of charge trend,nor in the overall cost and fuel mass.

4. Real-Time Optimal Control

The results obtained in the previous section rely on thea-priori knowledge of the driving cycle. Therefore, the optimaltrajectory ofib(t) cannot be implemented in real-time.

time (s)0 500 1000 1500

q b(%

)

48

49

50

51

52

x = qb

x = [qb;ξb]T

Figure 11: Battery state of charge in the Urban Driving Cyclewith the TCMSapproach: DP implementation with 1 and 2 state variables.

In this section, an analytic and a numerical causal controlpolicy solving Problem (33) are presented. By relying on thePontryagin’s Minimum Principle [23], performance close totheoptimum are attained also in real-time, without knowledge ofthe driving cycle. At the end of this section, such policies arecompared with the benchmark results presented in Section 3.

4.1. Unconstrained Explicit Optimal Control Law

A result valid for the ModelM2 is first given discarding theconstraintsumin, umax on the control variable. In the statementwe refer to the adjoint state for the optimal control problem: adiscussion on this variable is given in Subsection 4.4.

Theorem 1. Consider ModelM2 given in Definition 2. Theoptimal battery current iob is a function of motor power Pmot

and of the adjoint state p

iob =

io,1b =

(

pQb− αvoc + β

σbvoc

Nb+ γArvoc

)

12ArRbγ

i f Pm > 0∩ Pm > P(1,4)lim (p) ∩ p < p(1,2)

lim

io,2b = 0

i f Pm > 0∩ Pm > P(2,4)lim (p) ∩ p(1,2)

lim < p < p(2,3)lim

io,3b =

(

pQb− αvoc − β

σbvoc

Nb+ γArvoc

)

12ArRbγ

i f Pm > 0∩ Pm > P(3,4)lim (p) ∩ p > p(2,3)

lim

io,1b =

(

pQb− αvoc + β

σbvoc

Nb+ γArvoc

)

12ArRbγ

i f Pm < 0∩ Pm > P(1,5)lim (p) ∩ p < p(1,2)

lim

io,4b = io,5b =

(

voc−

√

v2oc− 4RbPm

) 12Rb

else(41)

where

p(1,2)lim = Qb

(

αvoc − Arγvoc−σbβvoc

Nb

)

p(2,3)lim = Qb

(

αvoc − Arγvoc+σbβvoc

Nb

) (42)

9


and the power limits separating the different regions are givenin Equation(B.1) in Appendix B.

Proof. The proof refers to Pontryagin’s Minimum Principle[23]. Define the Hamiltonian function as

H(qb, ib, p) = g(qb, ib) + pT f (qb, ib), (43)

wherep is a dynamic variable, often referred to asadjoint state,obeying to

p(t) = −∇qbH(q∗b(t), i∗b(t), p(t)), (44)

and subject to the boundary condition

p(T) = ∇qbh(q∗b(T)). (45)

The Hamiltonian reads

H(t) =

H1(t) i f Pb ≤ 0 ≤ Pm− Pb

H2(t) i f Pb = 0 ≤ Pm

H3(t) i f 0 ≤ Pb ≤ Pm

H4(t) i f Pb = Pm ≥ 0

H5(t) i f Pb = Pm ≤ 0

(46)

where

H1 = αvocib −βσbvocib

Nb+ γ

(

Ar

(

Pm− ibvoc+ Rbi2b)

+ Br

)

−pibQb

H2 = γ (Ar Pm+ Br)

H3 = αvocib +βσbvocib

Nb+ γ

(

Ar

(

Pm− ibvoc+ Rbi2b)

+ Br

)

−pibQb

H4 = αvocib +βσbvocib

Nb−

pibQb

H5 = αvocib −βσbvocib

Nb−

pibQb

(47)

By minimizing the Hamiltonian, the optimal control law isfound as

io,1b =

(

pQb− αvoc+ β

σbvoc

Nb+ γArvoc

)

12ArRbγ

io,2b = 0

io,3b =

(

pQb− αvoc− β

σbvoc

Nb+ γArvoc

)

12ArRbγ

io,4b = io,5b =

(

voc −

√

v2oc − 4RbPm

) 12Rb

(48)

where io,2b , io,4b io,5b come directly from the definition, while

io,1b , io,3b are found analyzing the first and second derivatives of

the Hamiltonian

∇uH1 = αvoc −βσbvoc

Nb− γAr (voc− 2Rbib) −

pQb

∇uH3 = αvoc +βσbvoc

Nb− γAr (voc− 2Rbib) −

pQb

∇2uH1 = 2ArRbγ > 0

∇2uH3 = 2ArRbγ > 0

(49)

The second derivatives are both positive becauseAr ,Rb, γ arepositive parameters, therefore, the Hamiltonian is convex. Theboundaries between modes 1, 2 and 3 are found studying thelimits of the above gradients whenib approaches zero

∇uH1ib→0−−−−−→ αvoc −

βσbvoc

Nb− γArvoc−

pQb

∇uH3ib→0+−−−−→ αvoc +

βσbvoc

Nb− γArvoc−

pQb

(50)

Sinceβ, σb, voc,Nb are positive andH1,H3 are convex, then∇uH3 > ∇uH1. For the same reason, the minimum amongH1,H2,H3 is found studying the sign of the two expressionsabove.H1 is minimum when∇uH1 > 0∩ ∇uH3 > 0, i.e.

p < p(1,2)lim = Qb

(

αvoc− Arγvoc −σbβvoc

Nb

)

(51)

H3 is minimum when∇uH1 < 0∩ ∇uH3 < 0, i.e.

p > p(2,3)lim = Qb

(

αvoc− Arγvoc +σbβvoc

Nb

)

(52)

H2 is minimum when∇uH1 < 0∩ ∇uH3 > 0, i.e.

p(1,2)lim < p < p(2,3)

lim (53)

Consider first the casePm > 0. In each of the three regionsjust defined, the optimal Hamiltonian amongH1,H2,H3 is com-pared toH4; the power limitsP(1,4)

lim ,P(2,4)lim ,P

(3,4)lim are the bound-

aries of the pure electric mode in each region

p < p(1,2)lim → P(1,4)

lim = Pm|H1 < H4

p(1,2)lim < p < p(2,3)

lim → P(2,4)lim = Pm|H2 < H4

p > p(2,3)lim → P(3,4)

lim = Pm|H3 < H4

(54)

Consider now the casePm < 0. Modes 2 and 3 are not fea-sible since generated power must be non negative. Therefore,whenp > p(1,2)

lim the optimal Hamiltonian isH5, otherwise bothH1 andH5 are feasible candidates. The line dividing the tworegions whereH1 andH5 are optimal is

p < p(1,2)lim → P(1,5)

lim = Pm|H1 < H5 (55)

The explicit optimal control law defines five different re-gions, each having a different expression of the optimal cur-rent in terms of the motor power and the adjoint state. Thedifferent regions are shown in Figure 12 when considering thepowetrains parameters used in our simulations.

4.2. Constrained Explicit Optimal Control Law

A result valid for ModelM2 is now given including theconstraintsumin, umax on the control variable, as defined inEquations (29), (30). These constraints reflect the bounds onboth the battery and the engine power. The map of the con-strained optimal control is depicted in Figure 13. The deriva-tion of the map follows directly from the application of the

10


p (-)-10 -5 0 5 10

Pm

(kW

)

-40

-20

0

20

40

60

iob,1 iob,2

iob,3

iob,4

iob,5

p(1,2)lim p

(2,3)lim

P(1,4)lim P

(2,4)lim

P(3,4)lim

P(1,5)lim

Figure 12: Regions of the explicit optimal control policy (unconstrained case).

p (-)-10 -5 0 5 10

Pm

(kW

)

-20

0

20

40

60

iob,1 iob,2

iob,3

iob,4

iob,5

imin(Pmin

b

)

imin(Pm − Pmaxr ) ima

x(P

max

b

)p(1)lim p

(1,2)lim p

(2,3)lim p

(3)lim

P(1,4)lim

P(2,4)lim

P(3,4)lim

P(1,5)lim

P(1,5∗)lim

P(1∗)lim

P(3∗)lim

Pmaxb

Pmaxg

Figure 13: Regions of the explicit optimal control policy (constrained case).

power bounds to the unconstrained map. Notice that in thiscase the system has 7 possible modes, instead of 5 in theunconstrained case. For this reason, the new limits (Pmin

b +

Pmaxg ),P(1,∗)

lim ,Pmaxg ,P

(3,∗)lim ,P

maxb ,P

(1,5∗)lim arise. As for the adjoint

state, the limitsp(1)lim, p

(3)lim arise as well. Notice that, forp < p(1)

lim,the boundary between thermal and pure electric modes is de-fined byP(1,5∗)

lim which is computed asP(1,5)lim but considering that

in this condition the battery power is saturatedPb = Pminb .

In the regions when constraints are active, the saturated cur-rents imin, imax are defined asumin, umax given in (29), (30).Where the battery power is saturated toPmin

b it is imin(Pminb ) =

umin(Pminb ), while the maximum power boundPmax

b is reached,it becomesimax(Pmax

b ) = umax(Pmaxb ). Instead, when the gen-

erator power is saturated toPmaxr , the optimal current becomes

imin(Pm− Pmaxr ) = umin(Pm− Pmax

r ).In the map depicted in Figure 13, different bounds on the

battery and engine power would change the constraints activein each mode, thus changing the shape of the map.

An alternative way of deriving the optimal control hasalso been proposed for an ECMS policy in [36]: theHamiltonian is evaluated for the control candidatesU =

io1, io2, i

o3, i

o4, i

o5, imin, imax and the minimization is performed on

these candidates only

iob = arg minu∈UH (56)

This approach has the advantage of using the results of the ex-plicit law for drastically reducing the number of control candi-dates, with respect to a standard numeric minimization, like theone presented in the next subsection. On the other hand, thecomplex analytic expressions of each region in terms of motorpower and adjoint state are not needed.

In the remainder of the paper we indicate withπX the controlpolicy depicted described in this subsection.

4.3. Constrained Numeric Optimal Control Law

The results presented in this section so far are referred toModelM2. We have shown that in that case, the optimal con-trol law can be expressed as an explicit function of the adjointstate and of the disturbance. As already mentioned in Section 3,ModelM2 introduces some approximation in the battery opencircuit voltage, in the battery severity factor and in the engineefficiency. In this work, the criticality of these approximationsis analyzed in the simulations at the end of this section; in ageneric application, it has to be verified based on measured dataon the system.

In this subsection, we show how to compute the optimal con-trol law when ModelM2 is not representative enough, and themore accurate ModelM1 has to be used. In such a case, the op-timal current is found by numerical minimization of the Hamil-tonian

iob = arg minu∈UH (57)

whereU is the feasible input set (considering the constraintsumin, umax) and the Hamiltonian is

H = αvnomb ib +

βσbvnomb |ib|

Nb+ γPf (Pm, ib, qb) −

pibQb

(58)

Notice that in this casevoc is an affine function of the state andσb is a function of the state and the battery current.

The policy derived with this approach is referred to as policyπN hereafter.

4.4. Adjoint State

For policy πX (both unconstrained and constrained), i.e.when ModelM2 is considered, the adjoint state is constant

p = 0 (59)

while for policy πN, i.e. when ModelM1 is considered, theadjoint state is subject to the dynamic equation

p(t) = −∇qbg(qb(t), ib(t)) − p(t)′∇qb f (qb(t), ib(t))

= −βvnom

b

Nb|ib(t)|∇qbσb(t) − γ∇qb Pf (t).

(60)

11


time (s)0 500 1000 1500

q b(%

)

48

49

50

51

52

πDP

πX

πN

Figure 14: Battery state of charge for the Urban Driving Cycle: benchmark so-lution (πDP), analytical real-time solution (πX) and numerical real-time solution(πN) of Problem (33).

Table 4: Driving cost and fuel consumption for the Urban Driving Cycle:benchmark solution (πDP), analytical real-time solution (πX) and numericalreal-time solution (πN) of Problem (33).

πDP πX πN

Driving Cost [e] 0.566 0.568 0.566Fuel Mass [g] 484.29 500.75 484.30

The proof of these statements comes directly from the appli-cation of Pontryagin’s Minimum Principle.

When the final state constraint is not active, the boundarycondition for the adjoint state is simply

p(T) = ∇qbh = 0 (61)

Hereafter, we take the approximation ˙p ≈ 0 to compute theapproximated policyπN in a causal framework.

4.5. Simulation ResultsThe two proposed real-time implementationsπX, πN of the

TCMS were compared to the non-causal implementationπDP

presented in Section 3. The trends of the battery state of chargeunder the Urban Driving Cycle are compared in Fig. 14. Thecorresponding final values of the total driving cost and fuelcon-sumption are given in Table 4. Apparently policyπN approxi-mates with high accuracy the results of the Dynamic Program-ming. Also the explicit policyπX gives very close results, withnegligible difference in the final cost and an increase in fuelconsumption of about 3%; this is reflected in a slightly higher(about 0.13%) state of charge at the end of the cycle.

The same comparison is carried out in Fig. 15 and Table 5, inthe case of the Combined Driving Cycle. Similar comments ap-ply: the monetary cost attained by policyπX is not significantlydifferent from policiesπN, πDP; on the other hand, the fuel con-sumption is about 3.5% higher. Also in this case this is reflectedin a higher final state of charge: in this case the deviation fromthe other two policies is more relevant (about 1%).

5. Sensitivity Analysis

In this section it is compared how policyπN proposed in Sec-tion 4 performs with respect to the benchmark policyπDP pro-

time (s)0 1000 2000 3000 4000 5000 6000 7000

q b(%

)

40

45

50

55

60

πDP

πX

πN

Figure 15: Battery state of charge for the Combined Driving Cycle: bench-mark solution (πDP), analytical real-time solution (πX) and numerical real-timesolution (πN) of Problem (33).

Table 5: Driving cost and fuel consumption for the Combined Driving Cy-cle: benchmark solution (πDP), analytical real-time solution (πX) and numericalreal-time solution (πN) of Problem (33).

πDP πX πN

Driving Cost [e] 3.070 3.073 3.070Fuel Mass [g] 2723.4 2813.3 2742.9

posed in Section 3, when some parameters are different fromthe nominal case at hand. Our goal is to empirically evaluatetherange of validity of the real-time numeric solution, bearing inmind that it is found with the approximation ˙p(t) ≈ 0. Clearly,the approximation is expected to be more critical in those con-ditions where ˙p is significantly different from zero during themission. From (58), the approximated policyπN is expected tobe good whenp(t)≪ αvnom

b Qb,∀t.Equation (60) suggests that the derivative of the adjoint state

depends on parametersβ, γ and on the gradients∇xσb, ∇xPf ;sincePf is a function ofPg = Pm − Pb, it is affected by batteryparameters and by the engine efficiency. For the sake of sim-plicity, the remainder of the paper focuses on the sensitivity toγ andAb. The cost coefficientγ is proportional to the fuel cost,which can significantly change over the time and from countryto country. The parameterAb is the gradient of the open circuitvoltage with respect to the state of charge, which can signif-icantly change from battery to battery, and possibly also overthe life of the battery itself. Nonetheless, it should be noted thatthe range considered for both the parameters is much larger thana realistic variability and is taken into account only for scien-tific analysis. This sensitivity study can clearly be extended toother parameters and to the analytical policyπX. The study ishere necessarily limited and leaves possible extensions tofutureresearch.

The sensitivity of the state of charge variationqb(0)− qb(T)to γ andAb is shown in Fig. 16; the corresponding sensitivity ofthe adjoint state variationp(0)− p(T) is given in Fig. 17. Thevariation of state of charge depends primarily onγ, while thedependence onAb is minor. Notice that this result also showshow the performance summarized in Table 1 is affected by theseparameters: in particular, high values ofγ make the electric

12


Ab = 0

Ab = 35

Ab = 70

Ab = 105

Ab = 140

PSfrag replacements

γ (e/kWh)

q b(0

)−q b

(T)

(%)

0.05 0.1 0.15 0.2−4

−2

0

2

4

6

Figure 16: Battery state of charge variation over the Urban Driving Cycle fordifferent values of the fuel costγ and the battery parameterAb. The gray dashedline indicates the state of charge variation with the Full Electric strategy.

Ab = 0

Ab = 35

Ab = 70

Ab = 105

Ab = 140

PSfrag replacements

γ (e/kWh)

p(0

)−p(

T)

(-)

0.05 0.1 0.15 0.2−0.1

0

0.1

0.2

0.3

0.4

Figure 17: Adjoint state variation over the Urban Driving Cycle for differentvalues of the fuel costγ and the battery parameterAb.

power more convenient. In Section 3, the state of charge vari-ation corresponding to the Full Electric strategy is found to be0.055 (shown in Fig. 17 with a gray dashed line), which is alsothe variation attained by the TCMS when the fuel is expensiveenough. The variation of the adjoint state is negligible forsmallvalues ofγ, which correspond to policies that tend to increasethe state of charge; it tends to increase significantly when bothγandAb increase, and the maximum is attained for their extremevalues (γ = 0.23e/kWh, Ab = 140).

The benchmark solution and the numeric real-time solutionare compared in Fig. 18 and in Table 6, for the Urban DrivingCycle and for two different choices ofγ andAb:

(a) γ = 0.08e/kWh, Ab = 70, i.e., the nominal parametriza-tion employed in the previous section;

(b) γ = 0.23e/kWh, Ab = 140,i.e., the parametrization whichattains the maximum variation of the adjoint statep(0) −p(T).

The case (a) was already commented in Section 4. As for thecase (b), although the costate variation is significant, theeffect

time (s)0 500 1000 1500

q b(%

)

40

45

50

55

60πDP (a)

πN (a)

πDP (b)

πN (b)

Figure 18: Sensitivity analysis for the state of charge in the Urban DrivingCycle: benchmark policyπDP and numerical policyπN in cases (a) and (b).

Table 6: Sensitivity analysis for the driving cost and the fuel consumption inthe Urban Driving Cycle: benchmark policyπDP and numerical policyπN incases (a) and (b).

πDP (a) πN (a) πDP (b) πN (b)

Driving Cost [e] 0.566 0.566 0.8120 0.8121Fuel Mass [g] 484.28 484.30 0 0

on the policy is negligible; the policyπN gives the same resultsas the benchmark policyπDP in terms of cost, state of chargeand fuel consumption.

The same study was carried out also for the Combined Driv-ing Cycle. The sensitivity of the state of charge variationqb(0) − qb(T) to γ and Ab is shown in Fig. 19; the corre-sponding sensitivity of the adjoint state variationp(0) − p(T)is given in Fig. 20. The dependence of bothqb(0)− qb(T) andp(0) − p(T) on γ andAb is almost the same observed for theUrban Driving Cycle. The most remarkable difference is theamplitude of the variations. More in detail, notice that ifγ ishigh enough, the optimal policy completely depletes the batteryandqb(0)− qb(T) = 0.3. Also in this case, that this result alsoshows how the performance summarized in Table 2 is affectedby Ab andγ. As for the variation of the adjoint state, it is maxi-mum for the extreme values (γ = 0.23e/kWh, Ab = 140) as inthe previous case.

The benchmark policyπDP is compared to policyπN inFig. 21 and in Table. 7, for the Combined Driving Cycle andfor the same choices ofγ andAb as before. The first case wasalready commented in Section 4. As for the second case, a sig-nificant difference between the two policies is observed. Theincrease of both the final cost and the final fuel consumption isof about 3.5%, whereas the difference between the trends of thestate of charge is more relevant: the benchmark policy reachesthe lower bound only at the end of the cycle, while policyπN

leads to a longer part of the mission spent at the lower boundfor the state of charge. This deterioration of performance can beattributed to the underlying approximation ˙p ≈ 0, that is con-tradicted by the large variation of the adjoint state observed inFig. 20.

In conclusion, the proposed policyπN shows good perfor-mance in almost all the tested situations. The only criticalsitua-

13


Ab = 0

Ab = 35

Ab = 70

Ab = 105

Ab = 140

PSfrag replacements

γ (e/kWh)

q b(0

)−q b

(T)

(%)

0.05 0.1 0.15 0.2−10

0

10

20

30

Figure 19: Battery state of charge variation over the Combined Driving Cyclefor different values of the fuel costγ and the battery parameterAb. The graydashed line indicates the state of charge variation with theFull Electric strategy.

Ab = 0

Ab = 35

Ab = 70

Ab = 105

Ab = 140

PSfrag replacements

γ (e/kWh)

p(0

)−p(

T)

(-)

0.05 0.1 0.15 0.2−1

−0.5

0

0.5

1

1.5

Figure 20: Adjoint state variation over the Combined Driving Cycle for differ-ent values of the fuel costγ and the battery parameterAb.

tion involves a long driving cycle and a combination of high fuelcost and high dependence of battery voltage on state of charge,which can falsify the approximations on the adjoint state andconsequently make policyπN suboptimal.

While such a combination of parameters is quite extreme atpresent, notice that the performance degradation could be lim-ited by estimating online the optimal value of the adjoint state,in a similar way to what is done for the ECMS strategies; thisamounts to dropping the approximationp(t) ≈ 0 and introduc-ing an estimatep(t) ≈ p(t). Notice that the accuracy of the es-timate p would reasonably depend on the availability of someinformation on the trip (like elevation and traffic). This task is,however, outside the scope of this paper and is left for futureresearch.

6. Conclusions and Remarks

In this paper, the Total Cost Minimization Strategy (TCMS)was proposed as a suitable approach for least costly energy

time (s)0 1000 2000 3000 4000 5000 6000 7000

q b(%

)

20

40

60

80πDP (a)

πN (a)

πDP (b)

πN (b)

Figure 21: Sensitivity analysis for the state of charge in the Combined DrivingCycle: benchmark policyπDP and numerical policyπN in cases (a) and (b).

Table 7: Sensitivity analysis for the driving cost and the fuel consumption inthe Combined Driving Cycle: benchmark policyπDP and numerical policyπN

in cases (a) and (b).

πDP (a) πN (a) πDP (b) πN (b)


management in extended range EVs and, more generally, forseries HEVs.

Specifically, the optimization goal was formulated as theoverall cost given by the cost of the grid energy, the batterylifeand the fuel consumption over a given trip. This choice appearsmore suitable than standard (constrained) fuel minimization,according to the future scenario where more and more HEVswill be plug-in and characterized by high-capacity batteries.Moreover, the use of a “monetary cost” instead of the standard“energy consumption” allows to sum up heterogeneous termswithout the need of tuning some weighting coefficients. To-gether with the optimization problem, the model of the power-train was modified accordingly, in order to take into accounttheeffect of battery aging.

By means of simulations on a full-fledged model of the ve-hicle, the least costly policy was compared with other policiesminimizing different objective functions, when a CNG rangeextender is available. Here, it was shown that, with currenten-ergy costs, the least costly policy does not lead to a full-electricpolicy even when the driving cycle is within the all-electricrange.

Since the benchmark optimum computed using DynamicProgramming cannot be implemented without thea-prioriknowledge of the driving cycle, a real-time solution of the prob-lem based on the Pontryagin Minimum Principle was studied.In this framework, some analytical guidelines and the numer-ical solution were provided. It was finally shown that such areal-time strategy is an excellent approximation of the bench-mark result for any reasonable combination of model parame-ters/energy costs.

Future works will be devoted to the implementation of theproposed TCMS approach on a real-world vehicle setup.

14


Acknowledgements

This work was partially supported by “Ricerca sul SistemaEnergetico” RSE S.p.a., Milano, Italy.

References

References

[1] A. Sciarretta, L. Guzzella, Control of hybrid electric vehicles, IEEE Con-trol Systems 27 (2) (2007) 60 –70.

[2] S. Delprat, J. Lauber, T. Guerra, J. Rimaux, Control of a parallel hybridpowertrain: optimal control, IEEE Transactions on Vehicular Technology53 (3) (2004) 872–881.

[3] S. Barsali, C. Miulli, A. Possenti, A control strategy tominimize fuel con-sumption of series hybrid electric vehicles, IEEE Transactions on EnergyConversion 19 (1) (2004) 187–195.

[4] J. Won, R. Langari, M. Ehsani, An energy management and charge sus-taining strategy for a parallel hybrid vehicle with cvt, IEEE Transactionson Control Systems Technology 13 (2) (2005) 313–320.

[5] J. Won, R. Langari, Intelligent energy management agentfor a parallel hy-brid vehicle-part ii: torque distribution, charge sustenance strategies, andperformance results, IEEE Transactions on Vehicular Technology 54 (3)(2005) 935–953.

[6] A. Sciarretta, M. Back, L. Guzzella, Optimal control of parallel hy-brid electric vehicles, IEEE Transactions on Control Systems Technology12 (3) (2004) 352–363.

[7] L. Serrao, S. Onori, G. Rizzoni, ECMS as a realization of pontryagin’sminimum principle for HEV control, in: Proceedings of the AmericanControl Conference, 2009, pp. 3964–3969.

[8] G. Paganelli, T. M. Guerra, S. Delprat, J.-J. Santin, M. Delhom,E. Combes, Simulation and assessment of power control strategies fora parallel hybrid car, Proceedings of the Institution of Mechanical Engi-neers, Part D: Journal of Automobile Engineering 214 (7) (2000) 705–717. doi:10.1243/0954407001527583 .

[9] N. Kim, S. Cha, H. Peng, Optimal control of hybrid electric ve-hicles based on Pontryagin’s minimum principle, IEEE Transac-tions on Control Systems Technology 19 (5) (2011) 1279–1287.doi:10.1109/TCST.2010.2061232 .

[10] C. Musardo, G. Rizzoni, Y. Guezennec, B. Staccia, A-ECMS: An adap-tive algorithm for hybrid electric vehicle energy management, EuropeanJournal of Control 11 (4) (2005) 509–524.

[11] D. Ambuhl, L. Guzzella, S. Member, Predictive Reference Signal Gener-ator for Hybrid Electric Vehicles, IEEE Transactions on Vehicular Tech-nology 58 (9) (2009) 4730–4740.

[12] H. Borhan, A. Vahidi, A. Phillips, M. Kuang, I. Kolmanovsky,S. Di Cairano, Mpc-based energy management of a power-splithy-brid electric vehicle, IEEE Transactions on Control Systems Technology20 (3) (2012) 593–603.

[13] P. Poramapojana, Predictive control of hybrid vehiclepowertrain for intel-ligent energy management, M.Sc. Thesis, Michigan Technological Uni-versity, 2012.

[14] P. Pisu, G. Rizzoni, A comparative study of supervisorycontrol strate-gies for hybrid electric vehicles, IEEE Transactions on Control SystemsTechnology 15 (3) (2007) 506–518.

[15] P. Pisu, E. Silani, G. Rizzoni, S. Savaresi, A LMI-basedsupervisory ro-bust control for hybrid vehicles, in: Proceedings of the American ControlConference, 2003, pp. 4681 – 4686.

[16] T. Bradley, A. Frank, Design, demonstrations and sustainability impactassessments for plug-in hybrid electric vehicles, Renewable and Sustain-able Energy Reviews 13 (1) (2009) 115–128.

[17] J. Axsen, K. Kurani, Hybrid, plug-in hybrid, or electric – What do carbuyers want?, Energy Policy 61 (2013) 532–543.

[18] A. Sciarretta, L. Serrao, P. Dewangan, P. Tona, E. Bergshoeff, C. Bordons,L. Charmpa, P. Elbert, L. Eriksson, T. Hofman, M. Hubacher, P. Iseneg-ger, F. Lacandia, A. Laveau, H. Li, D. Marcos, T. Nuesch, S. Onori,P. Pisu, J. Rios, E. Silvas, M. Sivertsson, L. Tribioli, A.-J. van der Hoeven,M. Wu, A control benchmark on the energy management of a plug-in hy-brid electric vehicle, Control Engineering Practice 29 (2014) 287–298.doi:10.1016/j.conengprac.2013.11.020 .

[19] C. Guardiola, B. Pla, S. Onori, G. Rizzoni, Insight intothe hev/phev opti-mal control solution based on a new tuning method, Control EngineeringPractice 29 (2014) 247–256.

[20] S. J. Moura, J. L. Stein, H. K. Fathy, Battery-health conscious power man-agement in plug-in hybrid electric vehicles via electrochemical modelingand stochastic control, Control Systems Technology, IEEE Transactionson 21 (3) (2013) 679–694.

[21] L. Serrao, S. Onori, A. Sciarretta, Y. Guezennec, G. Rizzoni, Optimalenergy management of hybrid electric vehicles including battery aging,American Control Conference (ACC), 2011 (3) (2011) 2125–2130.

[22] S. Ebbesen, P. Elbert, L. Guzzella, Battery state-of-health per-ceptive energy management for hybrid electric vehicles, IEEETransactions on Vehicular Technology 61 (7) (2012) 2893–2900.doi:10.1109/TVT.2012.2203836 .

[23] D. Bertsekas, Dynamic programming and optimal control, Athena Scien-tific Belmont, 1995.

[24] Fuerex - Homepage.URL http://www.fuerex.eu/

[25] C.-S. N. Shiau, N. Kaushal, C. T. Hendrickson, S. B. Peterson, J. F.Whitacre, J. J. Michalek, Optimal plug-in hybrid electric vehicle designand allocation for minimum life cycle cost, petroleum consumption, andgreenhouse gas emissions, Journal of Mechanical Design 132(9) (2010)091013.

[26] L. Guzzella, A. Amstutz, The QuasiStatic Simulation (QSS) Tool-box, http://www.idsc.ethz.ch/Downloads/DownloadFiles/qss(2005).

[27] J. Guanetti, S. Formentin, S. Savaresi, The series HEV simulator SHE,http://home.deib.polimi.it/formentin/SHE.html (2014).

[28] L. Guzzella, A. Sciarretta, Vehicle Propulsion Systems: Introduction toModeling and Optimization, Springer, 2007.

[29] V. Ramadesigan, P. W. Northrop, S. De, S. Santhanagopalan, R. D. Braatz,V. R. Subramanian, Modeling and simulation of lithium-ion batteriesfrom a systems engineering perspective, Journal of The ElectrochemicalSociety 159 (3) (2012) R31–R45.

[30] F. Todeschini, S. Onori, G. Rizzoni, An experimentallyvalidated capacitydegradation model for li-ion batteries in phevs applications, Proceedingsof the 8th IFAC Symposium on Fault Detection, Supervision and Safetyof Technical Processes 8 (1) (2012) 456–461.

[31] V. Marano, S. Onori, Y. Guezennec, G. Rizzoni, N. Madella, Lithium-ionbatteries life estimation for plug-in hybrid electric vehicles, in: VehiclePower and Propulsion Conference, VPPC, IEEE, 2009, pp. 536–543.

[32] I. Arsie, G. Rizzo, M. Sorrentino, Effects of engine thermal transients onthe energy management of series hybrid solar vehicles, Control Engineer-ing Practice 18 (11) (2010) 1231–1238.

[33] K. Wipke, M. Cuddy, S. Burch, Advisor 2.1: a user-friendly advancedpowertrain simulation using a combined backward/forward approach, Ve-hicular Technology, IEEE Transactions on 48 (6) (1999) 1751–1761.

[34] Lithium-ion battery cell degradation resulting from realistic vehicle andvehicle-to-grid utilization, Journal of Power Sources 195(8) (2010)2385–2392.

[35] M. Sniedovich, Dynamic programming: Foundations and principles,CRC press, 2010.

[36] D. Ambuhl, O. Sundstrom, A. Sciarretta, L. Guzzella,Explicit op-timal control policy and its practical application for hybrid electricpowertrains, Control Engineering Practice 18 (12) (2010) 1429–1439.doi:10.1016/j.conengprac.2010.08.003 .

Appendix A. Simulation Oriented Model Details

As described in Section 2, the simulator is made of a vehicledynamics part, a battery-related part and a thermal generationarea.

The vehicle dynamics area is represented in Fig 1 by theblocks related toVehicle, TransmissionandMotor. The blockVehicledescribes the relationships among the vehicle’s longitu-dinal speedv, the slopeθ and the wheel’s rotational speedωw

15

http://dx.doi.org/10.1243/0954407001527583

http://dx.doi.org/10.1109/TCST.2010.2061232

http://dx.doi.org/10.1016/j.conengprac.2013.11.020

http://dx.doi.org/10.1109/TVT.2012.2203836

http://www.fuerex.eu/

http://www.fuerex.eu/

http://www.idsc.ethz.ch/Downloads/DownloadFiles/qss

http://home.deib.polimi.it/formentin/SHE.html

http://dx.doi.org/10.1016/j.conengprac.2010.08.003


Table A.8: Simulator and control oriented model parameters

Ab (V) 70 Bb (V) 320 vnomb (V) 355

Qnomb (Ah) 65 Nb (−) 2000 Rb (mΩ) 500

Ar (−) 3.43 Br (kW) 5.61λr (kJ/g) 47 ρ f (kg/l) 0.2α ( ekWh) 0.2 qmin (−) 0.2 qmax(−) 0.9β ( ekWh) 500 Pb,min (kW) -50 Pb,max(kW) 50γ ( ekWh) 0.077 Pr,min (kW) 0 Pr,max(kW) 25M (kg) 1500 Cx (−) 0.22 ρ (kg/m3) 1.18Rw (m) 0.3 Cv (kg/s) 0 r (−) 3.5A (m2) 2 Cr (−) 0.008 ηt (−) 0.98

and torqueTw as

Mv =Tw

Rw− Fb − F f

ωw =v

Rw

whereM is the vehicle mass,Rw is the wheel radius,Fb is thebraking force. The friction termF f can be detailed as

F f = −Mgsinθ −Cr Mgcosθ −Cvv−12ρACxv

2

whereg is the gravitational acceleration,Cr ,Cv,Cx are respec-tively the vehicle’s roll, viscous and drag coefficients,ρ is theair density,A is the vehicle’s reference area.

The blockTransmissiondefines the relation that links thewheel’s speedωw and wheel torqueTw to the traction motorspeedωm and the motor torqueTm as

Tm =η−sign(Tw)t

rTw

ωm = rωw

wherer is the transmission ratio andη is the transmission effi-ciency.

The blockMotor models the traction motor power as

Pel = Tmωmηm(Tm, ωm)−sign(Pm)

whereηm is the motor efficiency, depending on the mechanicaloperating point.

As for the battery and thermal generation ar-eas, the underlying model is mainly described by(7), (9), (10), (11), (14), (15), (16) presented in Section 3.Equation (12) is replaced by

Qb(t) = Qnomb (1− 0.2ξb(t)).

The numerical values of parameters used in the Simulatorand in the Control Oriented Models are given in Table A.8.

Appendix B. Power Limits of the Explicit Control Law

The power limits separating the regions of the optimal con-trol law of Theorem 1 are defined as

P(1,4)lim = −(Ψ13 − 4N1.5

b Q0.5b pΨ1 + Ψ7 + Ψ8 + 4N1.5

b Q1.5b αvocΨ1

−Ψ11 −Ψ9 + 6NbQ2bσbαβv

2oc+ Ψ10 − 6NbQbσbβpvoc

+ N0.5b Q1.5

b σbβvocΨ14− 4Ar NbQ2bσbβγv

2oc)/Ψ2

P(2,4)lim = −(Ψ13 + NbpΨ6 + Ψ7 + Ψ8 − NbQbαvocΨ6 − QbσbβvocΨ6

−Ψ11 + Ψ12 + 2Ar BrN2bQ2

bRbγ2 − Ψ14 − Ar N

2bQ2

bαγv2oc

+ Ar N2bQbγpvoc− Ar NbQ2

bσbβγv2oc)/(2A2

r N2bQ2

bRbγ2)

P(3,4)lim = −(Ψ13 + Ψ7 + Ψ8 −Ψ11 − Ψ9 + Ψ12 + Ψ10 −Ψ14

+ Ψ5 −Ψ3 −Ψ4)/Ψ2

P(1,5)lim = −(Ψ13 + Ψ7 + Ψ8 −Ψ11 − Ψ9 − Ψ12 + Ψ10 + Ψ14

−Ψ5 + Ψ3 −Ψ4)/Ψ2

(B.1)

where

Ψ1 =

√

Qbσbαβv2oc− σbβpvoc+ Ar BrNbQbRbγ2 − Ar Qbσbβγv2

oc,

Ψ2 = 4A2r N2

bQ2bRbγ

2, Ψ3 = 4√

Ar BrRbN2bQ2

bαγvoc,

Ψ4 = 4√

Ar BrRbNbQ2bσbβγvoc, Ψ5 = 4

√

Ar BrRbN2bQbγp,

Ψ6 = (Ψ13 + Ψ7 + Ψ8 − Ψ11 + Ψ9 + Ψ12 + Ψ10 − Ψ14

− 2Ar N2bQ2

bαγv2oc+ 2ArN

2bQbγpvoc− 2Ar NbQ2

bσbβγv2oc)

0.5,

Ψ7 = N2bQ2

bα2v2

oc, Ψ8 = Q2bσ

2bβ

2v2oc,

Ψ9 = A2r N2

bQ2bγ

2v2oc, Ψ10 = 4Ar BrN

2bQ2

bRbγ2,

Ψ11 = 2N2bQbαpvoc, Ψ12 = 2NbQ2

bσbαβv2oc,

Ψ13 = N2b p2, Ψ14 = 2NbQbσbβpvoc.

Appendix C. DP formulation for fuel-based cost functions

The ECMS, CS, FE formulations defined in Section 2 arecompared with the TCMS formulation in Section 3; to ensure afair comparison, all approaches are implemented with DP, usingthe same underlying model (i.e. the discretized equations (34),(38)). Using the same notation of Section 3, the discretizedrunning cost for ECMS, CS, FE is

g(x(k), u(k),w(k)) =mf (k+ 1)−mf (k)

Ts. (C.1)

The constraints on the inputu(k) ∈ U and on the state of chargex(k) ∈ X are defined in (28) and (25), respectively. The terminalcost is simplyh(x(k), u(k),w(k)) = 0 for FE and CS, while forthe ECMS it is

h(x(k), u(k),w(k)) = ζ(qb(N) − qb(0)). (C.2)

The final state constraint for CS isx(N) ∈ x(0), while forECMS and CS it is simplyx(N) ∈ X.

16

Documents

Least costly energy management for series hybrid electric ... · S. Formentin et al. / Least costly energy management for series hybrid electric vehicles As it can be easily understood,