Tania Petersen

Sébastien Rivière

Stéphanie Malbos

Yannick Chantran

Aïcha Abbas

François Chasset

Arsene Mekinian

Jérémie Sellam

Laurent Garderet

Pierre Aucouturier
email: pierre.aucouturier@inserm.fr

Arsène Mékinian

Subclasses of monoclonal (type I) immunoglobulin G cryoglobulins : 2 report on two

SUMMARY

Background : While different clinical manifestations of IgM and IgG monoclonal cryoglobulins have been demonstrated, little is known on the roles of IgG subclasses in the pathophysiology of these conditions. Methods : In two cases of myeloma-associated monoclonal (type I) cryoglobulinemia with quite distinct clinical and biological features, serum samples were analyzed using an original IgG subclass-specific immunoblotting technique.

Results : The first case had painful arthritis of hands and feet, with skin purpura and a sharp decrease of complement C4 level, and the cryoglobulin was of IgG1 subclass. The second case displayed mostly thrombotic lesions of the limb extremities, C3 and C4 serum levels were normal, and the cryoglobulin belonged to the IgG2 subclass.

Conclusion : Type I cryoglobulins of distinct IgG subclasses may result in different syndromes. In both cases, the treatment relies on eradication of the underlying plasma cell dyscrasia.

KEY WORDS

Cryoglobulin -Monoclonal immunoglobulin -IgG subclasses

INTRODUCTION

Cryoglobulins are circulating complexes of immunoglobulins (Ig) that become insoluble at low temperature. According to Brouet et al [START_REF] Brouet | Biologic and clinical significance of cryoglobulins. A report of 86 cases[END_REF], the content of cryoprecipitates defines three types based on the presence of polyclonal and/or monoclonal Ig. Type I cryoglobulins are made up of a pure monoclonal Ig that binds to itself through variable domains [START_REF] Gerber-Jenson | Molecular basis for the temperature-dependent insolubility of cryoglobulins. X. The amino acid sequence of the heavy chain variable region of McE[END_REF].

Clinical events relating to monoclonal cryoglobulinemia are quite inconstant in both form and severity: while asymptomatic cases are not uncommon, others may present with cutaneous purpuric lesions or ischemic necrosis, Raynaud phenomenon, arthritis, neurological disorders and renal complications [START_REF] Terrier | The spectrum of type I cryoglobulinemia vasculitis: new insights based on 64 cases[END_REF][START_REF] Harel | Clinico-biological characteristics and treatment of type I monoclonal cryoglobulinaemia: a study of 64 cases[END_REF].

Clinical manifestations likely relate to intrinsic properties of the cryoglobulins, including effector functions of involved monoclonal Ig. While patients with monoclonal IgM and IgG cryoglobulins seem to have distinct pathological patterns [START_REF] Harel | Clinico-biological characteristics and treatment of type I monoclonal cryoglobulinaemia: a study of 64 cases[END_REF][START_REF] Néel | Long-term outcome of monoclonal (type 1) cryoglobulinemia[END_REF], only few data are currently available on the influence of IgG subclasses.

We report on two myeloma cases with quite distinct presentations and outcome that might relate to different monoclonal cryoglobulin IgG subclasses. IgG subclass typing was performed using a homemade immunoblotting method [START_REF] Briault | Isotypy of serum monoclonal immunoglobulins in human immunodeficiency virus-infected adults[END_REF]. Briefly, serum samples were submitted to thin-layer agarose gel zone electrophoresis, transferred by single pressure onto a nitrocellulose membrane, incubated with monoclonal antibodies specific for kappa, lambda, IgG, IgG1, IgG2, IgG3 and IgG4 isotypes (clones HP6053, HP6054, HP6018, NL16, GOM2, ZG4 and RJ4, respectively), followed by alkaline phosphataseconjugated anti-mouse IgG antiserum, and then revealed with tetrazolium nitroblue / bromo-chloro-indolylphosphate.

CASE PRESENTATION Case 1 :

A 64 year-old woman with a smoldering myeloma diagnosed since two years presented with painful bilateral hand and foot arthritis combined with purpuric skin lesions of the feet (Figure 1, upper panel). She presented with anemia (Hb 8.2 g/dL) and had normal serum creatinine and calcium levels. The monoclonal IgG kappa spike had recently increased to 30.7 g/L, with concomitantly decreased serum IgA and IgM levels (0,32 g/L and 0,16 g/L, respectively). A serum cryoglobulin was then detected, and immunofixation and immunoblotting analyses of the purified cryoprecipitate demonstrated a pure monoclonal IgG of IgG1 subclass (Figure 2, upper panel).

Additional biological data showed an increased serum C-reactive protein (35.1 mg/L), a normal serum complement C3 level (1.09 g/L) and a very low complement C4 level (0.03 g/L). Serological results for HIV, HBV and HCV were negative. Proteinuria was significant (1.25 g/day) and included a predominant kappa-type monoclonal Ig free light chain (Bence-Jones proteinuria).

Imaging studies revealed osteolytic lesions of the skull. Bone marrow aspiration showed 32% dystrophic plasma cells.

A skin biopsy of a foot lesion revealed leukocytoclastic vascular lesions with microthrombi. Cutaneous and articular vasculitis was diagnosed.

An induction regimen combining Bortezomib, Cyclophosphamide and Dexamethasone (VCD) was initiated. A partial hematological remission was achieved upon four cycles.

Autologous peripheral blood stem cells were then collected, and a myeloablative chemotherapy with Melphalan 200 mg/m 2 was given followed by autologous stem cell transplantation. Three months later, she received two cycles of Bortezomib, Thalidomide and Dexamethasone (VTD) as a consolidation process. No maintenance treatment was given.

Both polyarthritis and skin purpura disappeared after the first successful cycle of VCD.

Currently, the patient is in a good general condition, with a partial remission three and a half years after the autologous peripheral blood stem cell transplantation. The serum monoclonal IgG level is stable at 2.8 g/L, and the serum cryoglobulin is currently undetectable. Serum complement fractions returned to normal levels.

Case 2 :

A 89 year-old woman presented with purpuric skin lesions and edema of the extremities.

Since 2 months she had complained about asthenia, arthralgia of metacarpal, interphalangeal, metatarsal joints and wrists. Clinical examination revealed necrotic purpuric lesions localized on the hands, legs and feet (Figure 1, lower panel). No synovitis was found.

Laboratory data showed a moderate anemia (Hb 11 g/dL), normal creatinine and calcium serum levels and no significant proteinuria or hematuria. Serum protein electrophoresis and immunofixation demonstrated a monoclonal IgG of kappa type that was assessed at 14 g/L. Serum polyclonal immunoglobulin levels were normal (IgA 1,92 g/L, IgM 0,57 g/L). An abundant serum cryoglobulin was detected, and immunofixation and immunoblotting analyses of the purified cryoprecipitate demonstrated a pure monoclonal IgG of IgG2 subclass (Figure 2, lower panel).

No bone lesions were detected on MRI and X-ray analyses. Bone marrow aspiration showed 18% dystrophic plasma cells, with a normal karyotype. There was no evidence of complement activation, as suggested by normal serum levels of C3 and C4 fractions (0,96 g/L and 0,22 g/L, respectively). Serological results for HIV, HBV and HCV were negative. The skin biopsy showed multiple thromboses of the superficial dermis capillaries, without vasculitis or inflammatory infiltrates.

Stage I myeloma with complicating cryoglobulinemia skin lesions was diagnosed and a treatment with Bortezomib and Dexamethasone was initiated. Despite this treatment, arthralgia worsened and extensive ischemic and necrotic lesions appeared on the legs (figure 1, lower panel). After a severe sepsis of urinary origin, the patient died from hemorrhagic shock related to a duodenal ulcer.

DISCUSSION

These two myeloma patients with monoclonal IgG cryoglobulins had quite distinct presentations, with predominant joint inflammatory disease in the first one versus important skin ischemic lesions in the other. In case 1, inflammation may relate at least in part to the IgG1 isotype of the cryoglobulin; indeed, this subclass is known to display pro-inflammatory effector properties, namely activation of the complement classical pathway complement and binding to Fc receptors on mononuclear cells. In case 2, normal serum complement C3 and C4 may be explained by the IgG2 isotype of the cryoglobulin. Indeed, this IgG subclass poorly activates the complement classical pathway and has little or no affinity to Fc receptors on effector cells.

Previous studies of IgG subclasses in type I cryoglobulins demonstrated an overrepresentation of IgG2 and IgG3 as compared with MGUS and myeloma, although IgG1 remained the most frequent [START_REF] Brouet | Biologic and clinical significance of cryoglobulins. A report of 86 cases[END_REF][START_REF] Virella | Heavy chain typing in IgG monoclonal gammopathies with special reference to cases of serum hyperviscosity and cryoglobulinaemia[END_REF][START_REF] Abraham | Immunological and structural properties of human monoclonal IgG cryoglobulins[END_REF]. However, these reports included no indication on the clinical and biological patterns corresponding to each IgG subclass cryoglobulin, and thus their pathological significance remains quite obscure.

To our knowledge, the only article with identified subclass of monoclonal cryoglobulins in clinically and pathologically documented patients is that from Karras et al [START_REF] Karras | Renal involvement in monoclonal (type I) cryoglobulinemia: two cases associated with IgG3 kappa cryoglobulin[END_REF] who described 2 cases with membranoproliferative glomerulonephritis due to monoclonal IgG3 kappa cryoglobulins. In both cases, kidney sub-endothelial deposits included both monotypic IgG3 kappa and complement fraction C3; distal purpuric lesions of the skin were found in one of them, with vascular deposition of IgG3 kappa and C3, and there was no evidence of cryoglobulin-related extra-renal involvement in the other reported case. This observation suggests a proneness of this IgG subclass to accumulate in the glomerulus, as also found in other monoclonal Ig deposition diseases [START_REF] Nasr | Proliferative glomerulonephritis with monoclonal IgG deposits: a distinct entity mimicking immune-complex glomerulonephritis[END_REF].

Currently available data on IgG1 and IgG2 monoclonal cryoglobulins provide no clinical and pathological data that would tentatively explain their respective pathogenic roles. None of the present cases had objective renal involvement.

Different clinical patterns have been demonstrated in IgG and IgM monoclonal cryoglobulins, the most remarkable of which are more frequent kidney involvement and skin lesions with necrosis in IgG cryoglobulins [START_REF] Harel | Clinico-biological characteristics and treatment of type I monoclonal cryoglobulinaemia: a study of 64 cases[END_REF]. Contrary to mixed (type II) cryoglobulinemia that mostly feature leukocytoclasic vasculitis, cutaneous lesions relating to pure monoclonal cryoglobulins are most often due to thrombotic purpura with necrosis [11-13].

In our first case, the successful treatment of myeloma allowed curing arthralgia manifestations right away. On the contrary, the second patient did not respond to the initial hematological treatment and died of hemorrhagic and septic complications.

CONCLUSION

Considering the diversity of clinical expression in monoclonal IgG cryoglobulinemia, systematic identification of involved IgG subclasses could help precising the pathophysiology. The two cases reported here illustrate this point. In addition, successfully treating the underlying monoclonal plasma cell dyscrasia appears to be paramount to a successful outcome.

Figure 1 .

 1 Figure 1. Clinical symptoms : metacarpal, interphalangeal and metatarsal arthritis and limited skin purpura of a foot in patient 1 (upper panel) ; hand and foot purpuric skin lesions in patient 2 (lower panel, left) and severe ischemic and necrotic evolution (lower panel, right).

Figure 2 .

 2 Figure 2. Immunochemical studies of serum and purified cryoglobulins :

Figure 1 :Figure 2 :

 12 Figure 1 :

Sources of funding : this work was supported by Assistance Publique -Hôpitaux de Paris (AP-HP) and by Institut National de la Santé et de la Recherche Médicale (Inserm).

Declaration of interest :

The authors declare that they have no conflicts of interest regarding this study.


Djamaleddine Maamria

Kristan Gillet

Guillaume Colin

Yann Chamaillard

C Nouillant

Optimal predictive

published or not. The documents may come

I. INTRODUCTION

Nowadays, improving the energy consumption linked to transportation systems is among the major challenges facing the automotive industry. Eco-driving is considered as a major solution to face this challenge. In recent decades, engine technology and car performances have improved rapidly thanking to alternative propulsion technologies and engine downsizing, while drivers have not adapted their behavior (driving style). Eco-driving can be defined as a multi-criteria optimization problem (fuel consumption, trip duration, drivability, etc) of various tasks (navigation, guidance, stabilization) under safety constraints [START_REF] Petit | Optimal drive of electric vehicles using an inversion-based trajectory generation approach[END_REF]- [START_REF] Kim | Development of ecodriving guide system[END_REF].

The objective of eco-driving is to help the driver to improve his manner of driving in order to reduce the vehicle energy consumption. A human machine interface (HMI) integrated in the dashboard of the vehicle proposes the speed and the gear-box ratio set points to the driver. The computation of these set-points can be formulated as a dynamic optimization problem under some constraints: speed limitations, traffic information, final time and total traveled distance [START_REF] Petit | Optimal drive of electric vehicles using an inversion-based trajectory generation approach[END_REF], [START_REF] Ozatay | Analytical solution to the minimum energy consumption based velocity profile optimization problem with variable road grade ?[END_REF]- [START_REF] Mensing | Ecodriving: an economic or ecologic driving style?[END_REF]. The results reported in the literature show that the energy saving via eco-driving depends on the nature of the trip. In urban driving conditions, the energy saving may reach 20% while for highway trips, the energy saving is about 5 to 10% [START_REF] Kim | Development of ecodriving guide system[END_REF], [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF]- [START_REF] Daun | Evaluation of driving behavior and the efficacy of a predictive eco-driving assistance system for heavy commercial vehicles in a driving simulator experiment[END_REF]. Two scenarios depending on the horizon of the available information can be distinguished:

• Off-line scenario: The constraints on the speed limitations and the traffic information are fully known in advance. In this case, an estimation of the upper bound on the energy saving that can be reached through a D. Maamria, K. Gillet, G. Colin and Y. Chamaillard are with Univ. Orléans, PRISME, EA 4229, F45072, Orléans, France. djamaleddine.maamria@gmail.com b C. Nouillant is with PSA Peugeot Citroën, Direction Recherche Innovation & Technologies Avancées (DRIA), France eco-driving, can be computed through optimal control tools. In addition, heuristic driving rules to improve the vehicle fuel consumption may be extracted from the obtained optimal solution: for example, anticipate traffic flow, drive at a low engine speed and shift up early. This kind of calculation can be, for example, performed using cloud computing. This question was addressed for conventional vehicles in [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF], [START_REF] Maamria | Optimal eco-driving for conventional vehicles: simulation and experiment[END_REF], [START_REF] Sciarretta | Optimal ecodriving control: Energy-efficient driving of road vehicles as an optimal control problem[END_REF], for electric cars in [START_REF] Petit | Optimal drive of electric vehicles using an inversion-based trajectory generation approach[END_REF], [START_REF] Dib | Optimal energy management for an electric vehicle in eco-driving applications[END_REF], [START_REF] Miyatake | Theoretical study on ecodriving technique for an electric vehicle considering traffic signals[END_REF], [START_REF] Maamria | On the use of dynamic programming in eco-driving cycle computation for electric vehicles[END_REF] and for hybrid electric cars in [START_REF] Hellstrom | Design of an efficient algorithm for fuel-optimal look-ahead control[END_REF], [START_REF] Sciarretta | Optimal ecodriving control: Energy-efficient driving of road vehicles as an optimal control problem[END_REF], [START_REF] Bouvier | Determination and comparison of optimal eco-driving cycles for hybrid electric vehicles[END_REF]- [START_REF] Van Keulen | Velocity trajectory optimization in hybrid electric trucks[END_REF].

• On-line scenario: The future driving conditions are partially known for a given time or distance horizon usually called e-horizon. The information can be retrieved from many sources such as ADAS systems, sensors and navigation systems (Global Positioning System GPS unit and satellite navigation). Based on the available information, the optimizer has to find the optimal set points of the vehicle speed and the gear-box ratio in real time. The set points obtained will be sent to the driver or to a Cruise controller [START_REF] Passenberg | Combined time and fuel optimal driving of trucks based on a hybrid model[END_REF] (in the case of autonomous cars) and they have to be tracked. This question was mainly studied for trucks (conventional and hybrid) [START_REF] Passenberg | Combined time and fuel optimal driving of trucks based on a hybrid model[END_REF]- [START_REF] Neiss | Predictive speed control for a motor vehicle[END_REF] where the roadway grade is among the biggest contributors to high fuel consumption. The objective was to find the speed and gear-box ratio set-points that minimize a trade-off between the fuel consumption and the trip duration on a predefined speed range. The studies in [START_REF] Schwickart | An efficient nonlinear model-predictive eco-cruise control for electric vehicles[END_REF], [START_REF] Kamal | Eco-driving using real-time optimization[END_REF] deal with an ecodriving system using Model Predictive Control (MPC) within a given prediction horizon for passenger cars in an urban road network with complex traffic flows. The study in [START_REF] Levermore | Real-time velocity optimization to minimize energy use in passenger vehicles[END_REF] highlights the potential of real-time vehicle speed optimization as a mechanism for energy saving. A driver feedback system, designed using DP, was demonstrated in a test vehicle that uses the optimal velocity profile for the upcoming road. The prediction horizon was arbitrarily set to 1.5km.

This paper follows the same approach as the second scenario and pushes it further for passenger (conventional and electric) cars. Depending on the length of the e-horizon, the on-line scenario will lead to a sub-optimality in the energy saving with respect to the global optimum (full knowledge of the future driving conditions). We wish, by using MPC and DP, to select the value of the look-ahead horizon in order to minimize the induced sub-optimality in the energy saving while keeping the algorithm computation time reasonable. For this, various values of the e-horizon are considered and the induced sub-optimality compared to the global optimum (where the future driving conditions are known in advance) will be quantified. The choice of the look-ahead distance will be based on the optimality/computation time balance.

The paper is organized as follows. In Section II, the vehicle model is described. The computation of eco-driving cycles is detailed in Section III. Section IV presents the proposed predictive scheme. Numerical and simulation results are discussed in Section V for conventional and electric cars. In light of the results, some conclusions on the choice of the look-ahead distance are drawn.

II. VEHICLE MODELING A. Motion equations

The vehicle is modeled in a vertical plane. According to Newton's law of motion, the vehicle speed v satisfies the following differential equation

m • dv(t) dt = F t (t) -F r (t), (1)
where F t is the traction force provided by the prime mover, F r is the sum of resistance forces and m is the total vehicle mass including inertia of rotating parts. The force F r comprises the rolling resistance force, the aerodynamic drag force and a force due to the road grade. Its expression is given by

F r (t) = c 0 + c 1 • v(t) + c 2 • v(t) 2 , (2)
where c i , i = {0, 1, 2} are the coefficients of the road load equation. This model considers only the forces in the longitudinal direction.

B. Prime mover

In the case of a conventional vehicle, the prime mover is an Internal Combustion Engine (ICE). In this study, the ICE is a Diesel engine. This choice is not restrictive as the approach can be easily extended to gasoline engines. The fuel consumption ṁ f [g/s] is computed through a look-up table as a function of the engine torque T e [N.m] and the engine speed ω e [rpm] (see Figure 1)

ṁ f = ṁ f (ω e , T e ). (3)
In the case of an electric car, the prime mover is an electric machine. It is modeled by a quasi-static map describing the electric power (see Figure 2). This map usually includes the losses in the electric machine and the power electronic devices. The electric power P m is of the form

P m = P m (ω e , T e ). (4)

C. Battery model

The battery is represented by an equivalent circuit model comprising a voltage source U ocv in series with an electric resistance R b , both of which vary with ξ , the battery state of charge (SOC) [START_REF] Guzzella | Vehicle propulsion systems[END_REF], [START_REF] Badin | Hybrid Vehicles[END_REF]. The expression of the battery current I b is given by [START_REF] Guzzella | Vehicle propulsion systems[END_REF] - In the study [START_REF] Maamria | On the use of dynamic programming in eco-driving cycle computation for electric vehicles[END_REF], it was shown that neglecting the dynamics of ξ in the eco-driving cycle computation leads to an acceptable sub-optimality in the energy saving (less than 0.5%).

I b = 1 2R b (ξ ) U ocv (ξ ) -U 2 ocv (ξ ) -4R b (ξ ) • P m .
The inner (electrochemical) battery power is

P ech (v, T e ) = I b (v, T e ) •U ocv .
The dynamics of ξ is given by

dξ (t) dt = - I b (t) Q 0 ,
where Q 0 is the nominal battery capacity.

D. Transmission

The prime mover torque T e is related to the driver's torque demand T wh at the wheel by the relation

T wh (t) = r tire • F t (t) = η gb • R gb (t) • R t • T e (t), (5)
where R gb is the gear-box ratio, η gb is the constant gearbox efficiency and R t is the differential ratio. The torque T wh can be positive (traction) or negative (braking). Similarly, the rotational speed ω e is related to the vehicle speed v by

ω e (t) = R gb (t) • R t • v(t) r tire .
The model parameters are summarized in Table I. The numerical values are omitted for confidentiality reasons.

III. PROBLEM FORMULATION AND OFFLINE SOLUTION

For a fixed road, the idea of eco-driving is to find the best vehicle speed profile that minimizes the vehicle energy consumption. The vehicle starts from a point A at a velocity v 0 (≥ 0) and must reach a destination point B at a predefined time t f , with a velocity v 1 (≥ 0) under some constraints: speed limitations and traffic constraints [START_REF] Mensing | Ecodriving: an economic or ecologic driving style?[END_REF], [START_REF] Dib | Optimal energy management for an electric vehicle in eco-driving applications[END_REF]. This question can be solved using optimal control theory [START_REF] Petit | Optimal drive of electric vehicles using an inversion-based trajectory generation approach[END_REF], [START_REF] Mensing | Ecodriving: an economic or ecologic driving style?[END_REF].

A. OCP formulation

The cost function to be minimized over a fixed time window of duration t f is

J = t f 0 L(ω e (t), T e (t))dt.
In the case of a conventional vehicle, L is the fuel consumption ṁ f given by equation (3) while in the case of an electric car, L is the electrochemical battery power P ech given by equation ( 4).

The control variable u is composed of two components: the torque T e and the gear-box ratio R gb

u(t) = [T e (t), R gb (t)].
The considered gear-box has 6 ratios in the case of the conventional vehicle and has only one ratio for the electric vehicle. This optimization problem is carried out under the following dynamical constraints

dv(t) dt = f (v(t), u(t)), v(0) = v 0 , (6)
dx(t) dt = v(t), x(0) = 0, ( 7
)
where x is the position of the vehicle and the non-linear function f is calculated by combining [START_REF] Petit | Optimal drive of electric vehicles using an inversion-based trajectory generation approach[END_REF][START_REF] Monastyrsky | Rapid computation of optimal control for vehicles[END_REF][START_REF] Hellstrom | Design of an efficient algorithm for fuel-optimal look-ahead control[END_REF]. Due to the limitations of the vehicle speed v, of the torque T e and of the gear-box ratio R gb , the optimization must be performed under the following mixed state and input

constraints v ∈ [0, v max (x)], (8) f (v, u) ∈ [a min , a max ], (9)
T e ∈ [T min (ω e ), T max (ω e )], (10)
ω e ∈ [ω min , ω max ], (11) x(t f ) = D, (12) v(t f ) = v 1 , (13)
where v max (x) is the vehicle speed limitation at position x, D is the total traveled distance, v 1 is the desired final speed, T min and T max are the maximum and the minimum torque given by look-up tables as functions of the prime mover rotation speed ω e (see Figures 1 and2). In addition, the vehicle acceleration is constrained between its lower and upper bounds (a min , a max ) in [START_REF] Maamria | Optimal eco-driving for conventional vehicles: simulation and experiment[END_REF]. To summarize, the OCP is

(OCP) : min u t f 0 L(v, u)dt (14)
under the dynamics (6, 7), the (mixed) state and input constraints [START_REF] Dib | Optimal energy management for an electric vehicle in eco-driving applications[END_REF][START_REF] Maamria | Optimal eco-driving for conventional vehicles: simulation and experiment[END_REF][START_REF] Daun | Evaluation of driving behavior and the efficacy of a predictive eco-driving assistance system for heavy commercial vehicles in a driving simulator experiment[END_REF][START_REF] Sciarretta | Optimal ecodriving control: Energy-efficient driving of road vehicles as an optimal control problem[END_REF], and the final constraints [START_REF] Miyatake | Theoretical study on ecodriving technique for an electric vehicle considering traffic signals[END_REF][START_REF] Maamria | On the use of dynamic programming in eco-driving cycle computation for electric vehicles[END_REF]. Thus, the vehicle speed can be computed by using ( 6) once the control variable u is optimized and the initial condition of the vehicle speed is known.

B. Speed limitations

In this study, to specify the speed limitations, a certain (fixed) margin e l on an initial driving cycle speed is considered

v max (x) = v(x) + e l , v(x) > 0, 0, v(x) = 0,
where v(x) is the vehicle speed at the position x of the initial driving cycle which is used as a reference of comparison to point out the benefit of eco-driving in terms of fuel consumption reduction. Other methods to compute speed limitations can be considered [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF], [START_REF] Bouvier | Determination and comparison of optimal eco-driving cycles for hybrid electric vehicles[END_REF]. For example, the speed limits can be given by the traffic signs (legal speed limits) or provided by a cloud server depending on the vehicle position. It should be pointed out that the speed limits computation method does not impact the solving method.

C. Solving method

To solve the (OCP) defined in [START_REF] Bouvier | Determination and comparison of optimal eco-driving cycles for hybrid electric vehicles[END_REF], several methods can be used: Pontryagin Minimum Principle (PMP) or Dynamic Programming (DP). Because of the non-linear nature of the problem under consideration and the mixed state-input constraints (which are activated almost all the time), methods based on PMP are more complicated to implement [START_REF] Malisani | Dynamic control of energy in buildings using constrained optimal control by interior penalty[END_REF]. For this reason, the method proposed here is based on DP [START_REF] Bertsekas | Dynamic programming and optimal control[END_REF], [START_REF] Bryson | Applied optimal control[END_REF]. As suggested in [START_REF] Monastyrsky | Rapid computation of optimal control for vehicles[END_REF], [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF], the time-based OCP is transformed into a distance-based OCP in order to reduce the computation time. If the position space is discretized in N fixed steps of length ∆x, the time step ∆t(k), k = 1 : N is variable and is calculated from the vehicle speed v(k) and the vehicle acceleration a(k) by solving the following second order equation [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF]

∆x = 1 2 a(k) • ∆t(k) 2 + v(k) • ∆t(k).
The final constraint on the vehicle position ( 12) is fulfilled by construction (D = N • ∆x). An additional term β • ∆t(k) is added to the cost function as follows:

Js (u) = k=N ∑ k=1 [L(v(k), u(k)) + β ] ∆t(k).
The parameter β penalizes the final time t f = ∑ N k=1 ∆t(k) to obtain almost the same time duration as the initial driving cycle. The studies in [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF], [START_REF] Maamria | Which methodology is more appropriate to solve eco-driving optimal control problem for conventional vehicles?[END_REF] show that the relation between t f and β is monotone so that t f can be tuned.

IV. PREDICTIVE ECO-DRIVING CYCLE COMPUTATION

Optimal strategies obtained by DP or any other method are computed using the full knowledge of the future driving constraints (speed limitations and traffic information). In real situations, these future driving conditions are uncertain. They depend on many factors such as local traffic, infrastructure status, non-vehicle actors and weather conditions. To deal with this kind of situation, approaches based on Model Predictive Control (MPC) techniques are considered as an effective solution [START_REF] Back | Predictive control of drivetrains[END_REF]- [START_REF] Qin | A survey of industrial model predictive control technology[END_REF].

The idea of MPC is the following: the future driving conditions are only known for a look-ahead distance D 0 ≤ D and the speed trajectory will be recalculated every traveled distance D f called optimization frequency where D f < D 0 . The total traveled distance D is divided into n intervals D i , i = 1 : n where n is the ratio between D and the D f . The previous OCP described in [START_REF] Bouvier | Determination and comparison of optimal eco-driving cycles for hybrid electric vehicles[END_REF] will be solved n times

(OCP i ) : min u t i+1 t i L(v, u)dt ( 15
)
under the dynamics (6, 7), the (mixed) state and input constraints [START_REF] Dib | Optimal energy management for an electric vehicle in eco-driving applications[END_REF][START_REF] Maamria | Optimal eco-driving for conventional vehicles: simulation and experiment[END_REF][START_REF] Daun | Evaluation of driving behavior and the efficacy of a predictive eco-driving assistance system for heavy commercial vehicles in a driving simulator experiment[END_REF][START_REF] Sciarretta | Optimal ecodriving control: Energy-efficient driving of road vehicles as an optimal control problem[END_REF]. The time t i corresponds to the moment when the vehicle reaches the position (i • D f ) while the time t i+1 corresponds to the moment when the vehicle reaches the position (i • D f + D 0 ). Thus, the constraints on the vehicle position are

x(t i ) = i • D f , x(t i+1 ) = i • D f + D 0 .
The initial vehicle speed at time t i is given by the final speed reached at the end of the previous segment. The final speed value is free at the end of each segment except for the last segment:

v(t n ) = v 1 ,
where v 1 is the desired final speed at the end of journey. The optimization scheme is illustrated in Figure 3. The global problem for a traveled distance of D is divided into n sub-problems solved successively. At each traveled distance D f , the speed limitations and the traffic information are updated. A new vehicle speed trajectory is calculated for a distance horizon D 0 by solving the OCP (15) using the method described in Section (III-C). The computed speed trajectory has to be followed by the driver. To avoid the effects due to human interaction, the vehicle is assumed to follow the speed set points precisely (in simulation). Figure 4 shows the inputs, the outputs and the parameters of the algorithm. The inputs are the speed limitations, traffic information and road grade. The outputs of the algorithm are the vehicle speed and the gear-box ratio set points to the low level controllers. The parameter D f controls the launch of the algorithm: its value has to be chosen to find a good trade-off between the optimality of the solution and the computation time. The algorithm can also be launched in the case of sudden events: change in the traffic constraints, appearance of any other barriers (persons crossing the road), difference between real speed and set point. By this methodology, a feedback is introduced.

V. NUMERICAL RESULTS

Five driving cycles were selected: ECE-15 (Urban Driving Cycle UDC), EUDC (the Extra-urban driving cycle), WLTC (the Worldwide harmonized Light vehicles Test Cycle), the Artemis Urban and Artemis Rural driving cycle [START_REF] Michel | The ARTEMIS European driving cycles for measuring car pollutant emissions[END_REF]. The duration t f without stop phases, the total traveled distance D, the final SOC (electric case) and the fuel consumption (conventional case) for each initial driving cycle are given in Table II. The value of e l for the speed limitation was arbitrarily fixed to 2km/h.

A standard computer equipped with an Intel Core i5-4590, CPU 3.30GHz with 8GB of RAM was used. The look-ahead distance D 0 for the six scenarios tested is given in Table III.

Scenario I is considered as a reference of comparison (global optimum) in this study as the look-ahead distance D 0 equals the total traveled distance: full knowledge of the

D f =    D, for Scenario I, D 0 2 , otherwise. ( 16
)
For the DP parameters, a distance step of ∆x = 10m was used in the case of the ECE and the A. Urban and ∆x = 20m for the other driving cycles. This value of ∆x was chosen so as to get a time step approximately around 1s [START_REF] Maamria | Which methodology is more appropriate to solve eco-driving optimal control problem for conventional vehicles?[END_REF]. The value of β is tuned iteratively such that all the eco-driving cycles and the initial driving cycles have almost the same duration (with an error ≤ 0.5% on the final time t f ). This constraint on the trip duration allows us to make a fair comparison between the initial and the computed eco-driving cycles.

The objective in this study is to find the most suitable choice of D 0 for which the sub-optimality induced by the receding horizon strategy with respect to the global solution is reduced while keeping the computation time reasonable. The final aim is an embedded algorithm to compute the optimal vehicle speed set-points. Two cases are studied: electric and conventional.

A. Case of an electric car

The cost function L is the electrochemical battery power P ech . The control variable is the electric machine torque T e . The following mesh parameters were used: 0.02m/s for the vehicle speed and 2N.m for the electric machine torque. The initial value of the SOC was 90%.

For scenario I, the results in terms of energy consumption, energy saving through eco-driving with respect to the initial driving cycle and computation time α are given in Table IV. Figure 5 summarizes the sub-optimality in [%] of the tested scenarios with respect to scenario I. The extra-energy consumption with respect to scenario I increases when the look ahead distance D 0 decreases, and its maximum value is less than 1.4%. It should be pointed out that the energy saving of scenario I with respect to the initial driving cycle is at least 12.3% (see Table IV): in the worst case, the energy saving with respect to the initial driving cycle is 10.9%. Two cases are distinguished:

• Driving cycles with low mean speed value (ECE-15 and A. Urban): the maximum sub-optimality is less than 0.6% for scenario V I. When D 0 = 0.3km and D f = 0.15km (case not presented in Table III), the suboptimality reaches 0.9% for the ECE-15. • Driving cycles with high mean speed value (A. Rural, WLTC, EUDC): scenario V where the sub-optimality is less than 0.3% can be considered as a good choice.

The vehicle speed trajectories for the Artemis Rural cycle are given in Figure 6. When D 0 ≥ 1km, the eco-driving speed trajectories are similar. When D 0 < 1km, a problem of vehicle speed drops appears because the vehicle speed is free at the end of each segment. The value of D f has to be decreased in order to minimize this phenomenon. The mean computation time for each optimization and the number of optimizations n for the various scenarios are given in Table V. These mean time values represent the time needed to get the new speed trajectory at each iteration. The number of optimizations represents the load of the CPU and the electric energy consumption associated to this use during the trip. As can be seen from this table, the mean computation time is low and approximately constant for each scenario (in the same column): its value for scenarios V (EUDC, A. Rural and WLTC cases) and V I (ECE-15 and A. Urban cases) is 0.5s in the worst case (the time needed to travel 7.5m if vehicle speed is 50km/h and 15m if vehicle speed is 100km/h). This time is always lower than the time needed to travel a distance ∆x (the distance step in the DP).

B. Case of a Conventional vehicle

The cost function L is the fuel consumption ṁ f . The control variables are the engine torque T e and the gear-box ratio R gb . The following mesh parameters are chosen based on the study [START_REF] Maamria | Which methodology is more appropriate to solve eco-driving optimal control problem for conventional vehicles?[END_REF]: ∆v = 0.1m/s for the vehicle speed and ∆T e = 2N.m for the engine torque.

For scenario I, the results in terms of fuel consumption, fuel saving through eco-driving with respect to the initial driving cycle and computation time α are given in Table VI. The induced sub-optimality in [%] with respect to scenario I is given in Figure 7. The extra-fuel consumption increases when D 0 decreases, and its maximum value is less than 3.2%. In the case of the ECE-15 and the Artemis Urban, the sub-optimality is less than 0.2% for all the considered scenarios. In the case of the Artemis Rural, the WLTC and the EUDC, scenario V , where the sub-optimality is less than 0.7%, is considered as a reasonable choice. It should be pointed out that the fuel consumption reduction of scenario I with respect to the initial driving cycle is at least 17% (see Table VI): in the worst case, the fuel saving with respect to the initial driving cycle is reduced to 14%.

The vehicle speed trajectories for the EUDC case are given in Figure 8. The speed trajectories are similar for D 0 ≥ 2km. When D 0 is lower than 2km, a problem of vehicle speed drops appears as in the electric case. To reduce this phenomenon, the value of D f must be decreased when D 0 <= 1km as illustrated in Figure 8 where D 0 = 1km and D f = 0.26km. The sub-optimality is also reduced: for D f = 0.5km, the sub-optimality is 0.7% and for D f = 0.26km, the sub-optimality is 0.5%. On the other hand, the number of iterations n increases from 14 for D f = 0.5km to 27 for D f = 0.26km.

The number of optimizations n and the mean computation time for each optimization for the various scenarios are given in Table VII. For scenarios V (case of the EUDC, the A. Rural and the WLTC) and V I (case of the ECE-15 and the A. Urban), the computation time is about 1s in the worst case: the time needed to travel 28m if the vehicle speed is 100km/h. This time is always lower than the time needed to travel a distance of 2∆x (the distance step in the DP). This result is promising to embed the proposed algorithm.

VI. CONCLUSION

The computation of eco-driving cycles for electric and conventional vehicles using receding horizon control was addressed and formulated as an optimal control problem. The objective was to study the impact of the look-ahead distance on the optimality of the solution. The result is that, for the two systems considered, in the case of urban driving cycles (mean speed values less than 35km/h), a look-ahead distance of 300m to 500km leads to a sub-optimality less than 0.6% in the energy consumption compared to the global solution (where the future driving conditions are known). This result is interesting to size sensors that predict future driving conditions. For highway driving cycles, a look-ahead distance of 1km to 2km can be used as the induced suboptimality in the energy consumption is less than 0.7%. Moreover, the computation times are relatively small (1s in the worst case): lower than the time needed to travel a distance of 30m. This result is a first promising step for a real-time computation using an Engine Control Unit (ECU).

To embed the algorithm, two parameters have to be tuned: the first parameter is used to reach almost the same final time as the initial driving cycle. The second parameter is the frequency with which the optimization is launched so as to find a trade-off between optimality and computation time. Optimizing the tuning of these two parameters is the subject of current investigations and the first results are promising. It is also planned to extend this study to hybrid electric cars.

Fig. 1 .

 1 Fig. 1. Specific fuel consumption SFC [g/kWh] of the ICE as a function of the engine torque and engine speed. For confidentiality reasons, the data are normalized.

Fig. 2 .

 2 Fig. 2. Electric Machine Power [kW] as a function of the electric machine torque and electric machine rotational speed.

Fig. 3 .

 3 Fig. 3. Predictive algorithm scheme for eco-driving.

Fig. 4 .

 4 Fig. 4. Algorithm inputs and outputs.

Fig. 5 .

 5 Fig. 5. Sub-optimality in the energy consumption [%] versus scenarios of Table III: Electric case.

Fig. 6 .

 6 Fig. 6. Vehicle Speed [km/h] for the Artemis Rural cycle: Electric case.

Fig. 7 .

 7 Fig. 7. Extra-fuel consumption [%] versus scenarios of Table III: Conventional case.

Fig. 8 .

 8 Fig. 8. Vehicle Speed [km/h] for the EUDC cycle: Conventional case.

TABLE VII MEAN

 VII COMPUTATION TIME [S] FOR EACH ITERATION AND THE NUMBER OF OPTIMIZATIONS n (BETWEEN BRACKETS) FOR THE VARIOUS SCENARIOS: CONVENTIONAL CASE

		II	III	IV	V	V I
	ECE-15	2.9 (3)	3 (3)	2 (5)	1.2 (9)	0.7 (17)
	A. Urban	3.6 (3)	4 (3)	2.5 (5)	1.4 (9)	0.7 (18)
	A. Rural	4.6 (9)	3.4 (12) 2.1 (18)	1 (35)	0.5 (67)
	WLTC	4.6 (12) 3.3 (16) 2.1 (23) 0.9 (46) 0.5 (88)
	EUDC	4.2 (4)	3.1 (5)	2 (7)	0.9 (14) 0.4 (27)


Djamaleddine Maamria
email: djamaleddine.maamria@gmail.com

Kristan Gillet
email: kristan.gillet@univ-orleans.fr

Guillaume Colin
email: guillaume.colin@univ-orleans.fr

Yann Chamaillard
email: yann.chamaillard@univ-orleans.fr

Cédric Nouillant
email: cedric.nouillant@mpsa.com

Computation of eco-driving cycles for Hybrid Electric Vehicles: Comparative analysis

Keywords: Eco-driving, Hybrid Electric Vehicle (HEV), Dynamic Programming (DP), Energy management, Complexity analysis, Model reduction

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Introduction

Spurred by environmental requirements, economic factors and energysaving interests, eco-driving has attracted much attention from the scientific community in the last decade. It is now considered as a major solution to reduce the energy consumption linked to transportation. It can be seen as a multi-criteria optimization (fuel consumption, duration, drivability, etc) of various tasks (navigation, guidance, stabilization) under safety constraints.

In other words, the idea of eco-driving is to calculate the vehicle velocity trajectory that minimizes the vehicle energy consumption under constraints: speed limitations, final time and total traveled distance. This question can be solved using optimal control tools.

For conventional vehicles, fuel consumption, engine emissions or any combination of both over a fixed time window is the cost function to be minimized [START_REF] Mensing | Vehicle trajectory optimization for application in eco-driving[END_REF][START_REF] Mensing | Ecodriving: an economic or ecologic driving style?[END_REF]. For full electric cars, the cost function to be minimized is the electric power requested by the electric machine [START_REF] Dib | Optimal energy management for an electric vehicle in eco-driving applications[END_REF][START_REF] Petit | Optimal drive of electric vehicles using an inversion-based trajectory generation approach[END_REF][START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF][START_REF] Sciarretta | Optimal ecodriving control: Energy-efficient driving of road vehicles as an optimal control problem[END_REF][START_REF] Miyatake | Theoretical study on ecodriving technique for an electric vehicle considering traffic signals[END_REF]. The duration of the trip can be considered as an additional degree of freedom in the optimization. A trade-off between the fuel consumption and the duration can be found. Two dynamics are usually considered: the position and the speed of the vehicle. For these two architectures (conventional and electric), two control variables are used: the engine or the electric machine torque and the gear-box ratio while the main constraints bear on speed limitations, vehicle stops and total traveled distance (F. [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF][START_REF] Sciarretta | Optimal ecodriving control: Energy-efficient driving of road vehicles as an optimal control problem[END_REF] However, having an additional energy source increases the complexity of the models and thus the algorithms used to calculate eco-driving cycles as mentioned in [START_REF] Van Keulen | Velocity trajectory optimization in hybrid electric trucks[END_REF]. In the case of hybrid electric vehicles, additional state and control variables have to be considered in the optimization: the battery State Of Charge (SOC) with a constraint on its final value and the electric machine torque.

The work in [START_REF] Kim | Model predictive control of velocity and torque split in a parallel hybrid vehicle[END_REF] presents a strategy that optimizes both the speed profile and the torque split between the electric machine and the engine using a Gradient method. More recently, the algorithms in (F. [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF][START_REF] Ngo | An optimal control-based algorithm for hybrid electric vehicle using preview route information[END_REF] combine dynamic programming with the Energy Management System (EMS) design for a Hybrid Electric Vehicle (HEV) to calculate eco-driving cycles. A bi-level approach that reduces computation time was suggested in [START_REF] Ngo | An optimal control-based algorithm for hybrid electric vehicle using preview route information[END_REF]. The optimal control strategy is calculated by decoupling the optimization of the control variables. In a first step (an outer loop) the speed trajectory is optimized assuming that the vehicle is propelled only by the internal combustion engine or the electric machine. In a second step, the power split between the engine and the electric machine is optimized in an inner loop for a given vehicle speed, gear-box ratio and wheel torque. The missing point is the quantification of the sub-optimality induced by the method used. A similar approach was used in [START_REF] Sciarretta | Optimal ecodriving control: Energy-efficient driving of road vehicles as an optimal control problem[END_REF] where an overview of eco-driving problems for various architectures (electric, conventional and hybrid electric cars) was given. Analytical solutions were suggested in the case where the gear-box ratios are not optimized.

Later, in [START_REF] Heppeler | Fuel efficiency analysis for simultaneous optimization of the velocity trajectory and the energy manage-ment in hybrid electric vehicles[END_REF], the authors worked on the direct optimization of the EMS for an HEV with a small deviation from the given desired vehicle velocity as an additional degree of freedom. It was shown that the additional degree of freedom for the velocity decreases fuel consumption by about 6.8% compared to a real-time power split strategy and by about 4.3% compared to an off-line power split algorithm with a fixed velocity trajectory. The work in [START_REF] Bouvier | Determination and comparison of optimal eco-driving cycles for hybrid electric vehicles[END_REF] compared two approaches to calculate eco-driving cycles for a parallel HEV in terms of fuel saving. The study concluded that in order to generate the best speed trajectory in terms of fuel consumption, it is necessary to consider that the vehicle is an HEV: this consideration saves up to 3%. However, the comparison of the computation time of the two methods was not investigated. This paper follows the path described above and pursues the analysis further. A parallel HEV equipped with a Diesel engine is considered. This choice is not restrictive, as the methodology presented here could be easily transposed to other cases of interest. The objective is to calculate, within a reasonable time, an eco-driving cycle for a HEV under final time, distance and SOC constraints while fulfilling the speed limits. We wish to find a trade-off between the accuracy of the DP solution and the complexity of the algorithms used to obtain this solution (an accuracy/complexity balance).

For this purpose, four methods to calculate eco-driving cycles are considered:

• The first method is based on solving directly the optimal control problem associated to eco-driving for HEVs.

• The second method is based on reducing the number of state variables by introducing a tuning parameter to satisfy the SOC final constraint.

• The third method is based on decoupling the optimization of the control variables into two steps. In a first step, an eco-driving cycle is calculated assuming that the vehicle is propelled only by the engine.

In the second step, to follow the calculated eco-driving cycle, an off-line energy management strategy is designed to optimize the torque split and the gear-box ratios.

• The last method is similar to the previous one where only the torque split is optimized in the second step.

These methods are compared in terms of fuel consumption, state trajectories, computation time of the DP and memory (RAM) use. Based on the numerical results, a conclusion about the chosen trade-off between accuracy/complexity is drawn.

The paper is organized as follows. In Section II, the vehicle model is described. The calculation of eco-driving cycles is detailed in Section III.

Section IV details the proposed numerical methods to calculate eco-driving cycles for an HEV. Numerical and simulation results are discussed in Section V. In light of the results, some conclusions on the most convenient method to be used are drawn based on a trade-off between optimality/complexity.

Vehicle Modeling

The system considered here is a dual shaft parallel mild hybrid with an electric machine (EM ) connected to the engine by a belt (Figure 1).

The gearbox is between the power-train and the wheel. This architecture allows regenerative braking (the electric machine works as a generator during braking phases), hybrid and zero-emission vehicle (ZEV) modes. Due to the architecture choice, during the ZEV mode, the engine injection is cut off and the electric machine produces power, keeping the engine rotating. This system was used in [START_REF] Michel | Optimizing fuel consumption and pollutant emissions of gasoline-hev with catalytic converter[END_REF][START_REF] Simon | Gasolinehev equivalent consumption and pollutant minimization strategy[END_REF].

Motion equations

The vehicle is modeled in a vertical plane. According to Newton's law of motion, the vehicle speed v satisfies the following differential equation

m • dv(t) dt = F t (t) -F r (t), (1)
where F t is the traction force to be provided by the engine, F r is the sum of resistance forces and m is the total vehicle mass including the rotating parts. The force F r comprises the rolling resistance force, the aerodynamic drag force. Its expression is given by

F r (t) = c 0 + c 1 • v(t) + c 2 • v(t) 2 , (2)
where c i , i = {0, 1, 2} are the constant coefficients of the road load equation.

To take the road grade α into account, the coefficient c 0 will be not constant and its expression will be

c 0 = c a 0 + m • g • sin(α), ( 3
)
where g is the acceleration of gravity, c a 0 is the road load coefficient. This model considers only the forces in the longitudinal direction. In this study, the road grade is null.

Internal Combustion Engine (ICE)

The ICE under consideration is a Diesel engine. The fuel consumption ṁf (g/s) is computed through a look-up table as a function of the engine rotational speed (ω eng ) and the effective engine torque (T eng ) (see Figure 2) ṁf = ṁf (ω eng , T eng ).

(4)

Electric machine model

The electric machine is modeled by a quasi-static map describing its electric power. The electric power P m consumed (in traction mode) or supplied to the battery (in recuperation mode) is of the form

P m = P m (ω el , T el ), (5)
Normalized Engine Torque Normalized Engine Speed where T el is the electric machine torque and w el is the electric machine rotational speed. This map includes the losses in the electric machine and the power electronic devices. The electric machine torque is limited by speeddependent upper and lower bounds of the form (bold blue and black lines in

Specific Fuel Consumption [g/kWh] 0,2 0,4 0,6 0,8 1 0 0,2 0,4 0,6 0,8 1 SFC Maximum Torque
Figure 3)

T elmin (ω el ) ≤ T el ≤ T elmax (ω el ). (6)

Battery model

The battery of Li-ion type is represented by an equivalent circuit model comprising a voltage source U ocv in series with an electric resistance R bat , both of which vary with ξ, the battery state of charge (SOC) (Guzzella and , 2013;[START_REF] Badin | Hybrid Vehicles[END_REF]. The expression of the battery current I b is

Sciarretta

I b = 1 2R bat (ξ) U ocv (ξ) -U 2 ocv (ξ) -4R bat (ξ) • P b , (7)
where U ocv and R bat are given by look-up tables as functions of ξ (see Figures 4 and5) and P b is the power requested from the battery given by

P b = P m . (8)
In reality, U ocv and R bat depend also on the battery mode (discharging or charging). These dependencies are neglected in this study as they will not 3.9

4

SOC [%] U ocv [V] Figure 5: U ocv [V] dξ(t) dt = - I b (t) Q 0 , (9)
where Q 0 is the nominal battery capacity. In order to simplify the notation, the dynamics of ξ considering a given initial condition ξ 0 is written as

dξ(t) dt = g(v(t), ξ(t), T el (t)), ξ(0) = ξ 0 . ( 10
)
The inner (electrochemical) battery power is defined by

P ech (v, ξ, T e ) = I b (v, ξ, T el ) • U ocv (ξ). (11)

Transmission

The engine torque T eng and the electric machine torque T el are related to the torque required at the wheel T wh by

T wh (t) = η gb • R gb (t) • R t • [T eng (t) + R el • T el (t)] , (12)
where R gb is the gear-box ratio, r tire is the wheel radius, η gb is the gearbox efficiency (assumed to be constant), R el is the constant motor-to-wheel transmission ratio and R t is the differential ratio.

In the case of a conventional vehicle (with only an ICE), T wh can be calculated using the formula

T wh (t) = η gb • R gb (t) • R t • T eng (t). ( 13
)
During braking phases, the maximum energy allowed by the electric machine is recovered and is used to recharge the battery. The remaining part is dissipated by the braking system.

Similarly, the rotational speed ω eng of the ICE and ω el of the electric machine are related to the vehicle speed v by

ω el (t) = R el • ω eng (t) = R gb (t) • R t • R el • v(t) r tire . (14)
The model parameters are summarized in Table 1. The coefficients of the road load equations are omitted for confidentiality reasons.

Problem formulation

For a fixed road, the eco-driving methodology consists of finding the best speed profile minimizing the vehicle power consumption knowing that the vehicle starts from a point A at a given speed v 0 (≥ 0) and must reach a destination point B at time t f , with a velocity v 1 (≥ 0). For an HEV, an additional constraint on the final SOC value is introduced. This constraint allows the comparison of many solutions by guaranteeing that they reach the same level of battery energy at the end of the driving cycle [START_REF] Guzzella | Vehicle propulsion systems[END_REF][START_REF] Sciarretta | Optimal control of parallel hybrid electric vehicles[END_REF]. This problem can be solved using optimal control tools [START_REF] Mensing | Ecodriving: an economic or ecologic driving style?[END_REF][START_REF] Petit | Optimal drive of electric vehicles using an inversion-based trajectory generation approach[END_REF]. The cost function to be minimized is the fuel consumption over a fixed time window of duration t f

J = t f 0 ṁf (ω eng (t), T eng (t))dt. ( 15
)
The control variable u is composed of three components: the engine torque T eng , the electric machine torque T el and the gear-box ratio R gb

u(t) = [T eng (t), T el (t), R gb (t)]. ( 16
)
This optimization is carried out under the following dynamical constraints

dx(t) dt = v(t), x(0) = 0, ( 17
)
dv(t) dt = f (v(t), u(t)), v(0) = v 0 , (18)
dξ(t) dt = g(v(t), ξ(t), u(t)), ξ(0) = ξ 0 , ( 19
)
where x is the vehicle position and f is calculated by combining (1, 2, 12)

f = 1 m [-c 0 -c 1 • v -c 2 • v 2 + η gb r tire • R gb • R t • (T eng + R el • T el )]. (20)
Since the vehicle speed, the battery state of charge, the engine torque and speed, the electric machine torque and speed and the gear-box ratio are limited, and the final position, final speed and the final value of ξ are fixed, the optimization must be performed under the following constraints

v(t) ∈ [0, v max (x)], (21)
f (v, u) ∈ [a min , a max ], (22)
ξ(t) ∈ [ξ min , ξ max ], (23)
T eng (t) ∈ [T min (ω eng (t)), T max (ω eng (t))], (24)
ω eng (t) ∈ [ω min , ω max ], (25)
T el (t) ∈ [T elmin (ω el (t)), T elmax (ω el (t))], (26)
ω el (t) ∈ [ω elmin , ω elmax ], (27)
x(t f ) = D, (28)
v(t f ) = v 1 , (29)
ξ(t f ) = ξ t , (30)
where D is the total traveled distance, ξ t is the desired final SOC, ξ min and ξ max are fixed values. The limitations T min , T max , T elmin and T elmax are given by look-up tables as a function of the engine speed ω eng and the electric machine speed ω el .

The speed limitations v max in ( 21) are given as a function of the vehicle position and not of time [START_REF] Mensing | Vehicle trajectory optimization for application in eco-driving[END_REF][START_REF] Dib | Optimal energy management for an electric vehicle in eco-driving applications[END_REF]. In this study, the considered initial and final values of the vehicle speed are zero

v 0 = v 1 = 0. ( 31
)
Equation ( 22) limits the vehicle acceleration between its maximum and minimum values. The acceleration is an algebraic function of the vehicle speed and the control variables. This function can be evaluated for all the possible choices of the vehicle speed and the control variables. The values not satisfying the constraint on the acceleration are excluded.

The constraints on the engine torque in (24) and the electric machine torque in ( 26) are mixed input-state constraints, they depend on the vehicle speed v and the gear-box ratio. The constraints on the rotational speeds in ( 25) and ( 27) are mixed input-state constraints, they depend on the vehicle speed and the gear-box ratio.

For the battery, the current I b is limited between its maximum and minimum values in the case of battery charging and discharging. This constraint is not considered in the problem formulation. It will be checked a posteriori.

To summarize, the OCP considered in this paper is

(OCP ) : min u t f 0 ṁf (v, u)dt (32)
under the dynamics (17,18,19), the state and input constraints (21,22,23,24,25,26,27), and the final constraints (28, 29, 30).

Speed limitations

To compute an eco-driving cycle from a given driving cycle, the following constraints [START_REF] Mensing | Ecodriving: an economic or ecologic driving style?[END_REF] have to be included:

• the same final distance x(t f ), the same number of stops and the same duration t f as the initial driving cycle,

• the vehicle speed limitations depending on the vehicle position x.

In this study, to specify the speed limits, a certain (fixed) margin e l on the initial driving cycle speed is considered

v max (x) =    v(x) + e l , v(x) > 0, 0, v(x) = 0, (33)
where v(x) is the speed value of the initial cycle at the position x. Other types of limits can be considered as in (F. [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF][START_REF] Bouvier | Determination and comparison of optimal eco-driving cycles for hybrid electric vehicles[END_REF], where a set of legal speed limits v lim were used. The process of identifying the speed limit for a given position x can be described in two steps:

1. find index j for which v lim (j -1) + e l < v(x) and v lim (j) + e l ≥ v(x).

2. v max (x) = v lim (j).

The choice of speed limits does not impact the solving method. Then, the objective is to find a new speed trajectory that takes these constraints into account and leads to a lower fuel consumption.

Solving Methods

The solution considered here is based on Dynamic Programming (DP) [START_REF] Bertsekas | Dynamic programming and optimal control[END_REF]. It is well-known that the number of the state and the control variables greatly impacts the numerical methods. Considering additional state and control variables increases the level of complexity and the computational burden. It may also jeopardize the robustness of numerical methods employed to compute the optimal trajectories.

In order to reduce the computation time, the method suggested in (F. [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF][START_REF] Bouvier | Determination and comparison of optimal eco-driving cycles for hybrid electric vehicles[END_REF][START_REF] Monastyrsky | Rapid computation of optimal control for vehicles[END_REF]) is used, where the time-based OCP (with a fixed time step ∆t) is transformed into a space-based OCP (with a fixed distance step ∆x) using

d dt = v • d dx . ( 34
)
In the space-based OCP, the stop phases are removed from the driving cycle.

A comparison between the time-based and the space-based OCP solutions for a conventional vehicle is given in [START_REF] Maamria | Which methodology is more appropriate to solve eco-driving optimal control problem for conventional vehicles[END_REF]. If the position space is discretized in N points with a fixed step ∆x, the time step ∆t(k), k = 1 : N is variable and is implicitly calculated from the vehicle speed v(k) and the vehicle acceleration a(k) by solving the second order equation

∆x = 1 2 a(k) • ∆t(k) 2 + v(k) • ∆t(k). (35)
The x is omitted from the OCP (32). To calculate the right value of β, a rootfinding method can be used to drive the final time error to zero as done in [START_REF] Mensing | Vehicle trajectory optimization for application in eco-driving[END_REF][START_REF] Sciarretta | Optimal ecodriving control: Energy-efficient driving of road vehicles as an optimal control problem[END_REF].

Based on this simplification, four methods to calculate eco-driving cycles for HEVs are defined in a complexity order as follows:

Method 1

In this method, the optimal control problem to be solved is the OCP described in (32) with the introduction of the tunable term β • ∆t(k) in the cost function as a terminal cost. The OCP has two state variables (v, ξ) and three control variables (T eng , T el , R gb ). The cost function to be minimized is

k=N k=1 [ ṁf (v(k), u(k)) + β] • ∆t(k) (36)
under the dynamics (18, 19), the state and input constraints (21,22,23,24,25,26,27), and the final constraints (29, 30). The parameter β controls the duration of the trip t f : it has been shown that the relations between β and t f is monotonic. This method is denoted by (M 1 ) and it is considered as a reference of comparison in this study.

Method 2

The dynamic of ξ has been considered in the OCP (32) because of the final constraint ( 30). Moreover, it has been shown, by using the Pontryagin Minimum Principle (PMP) [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF], in the energy management system design for HEVs that neglecting the dependance of U ocv and R bat in ξ leads to a quasi-optimal fuel consumption [START_REF] Serrao | Optimal energy management of hybrid electric vehicles including battery aging[END_REF][START_REF] Sciarretta | Optimal control of parallel hybrid electric vehicles[END_REF][START_REF] Kim | Optimal control of hybrid electric vehicles based on pontryagins minimum principle[END_REF]. Thus, to reduce the number of state variables from 2 (v, ξ) to 1 (v), a new tunable quantity is added to the cost function (36) as follows

k=N k=1 ṁf (v(k), u(k)) + β + µ H lhv P ech (v(k), ξ, u(k)) • ∆t(k), ( 37
)
where H lhv is the lower heating value of the fuel and ξ is a fixed value of ξ used to calculate mean (constant) values of R b and U ocv in the expression of the electrochemical power P ech defined in equation ( 11). The parameter µ is used to bring the final SOC to its target value ξ t : for a fixed value of β, the relation between the final SOC and µ is monotone. Because of the absence of information about ξ in the backward loop, the instantaneous constraint ( 23) cannot be handled. This method was suggested in (F. [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF][START_REF] Bouvier | Determination and comparison of optimal eco-driving cycles for hybrid electric vehicles[END_REF] and it is denoted, in what follows, by (M 2 ).

Method 3

The third method (M 3 ) is based on decoupling the optimization of the control variables. The method involves two steps:

1. Step1 : An eco-driving cycle is calculated assuming that the vehicle is propelled only by the ICE (vehicle parameters such as the weight and the road load coefficients do not change). The cost function to be minimized is the fuel consumption. The state variable is the vehicle speed v and the control variables are the engine torque T eng and the gear-box ratio R gb . The torque required at the wheel T wh is related to T eng and R gb by the relation ( 13). The associated OCP is min

(Teng,R gb ) k=N k=1 [ ṁf (ω eng (k), T eng (k)) + β] • ∆t(k) (38)
under the dynamics (18), the state and input constraints (21,22,24,25,26,27), and the final constraints (29). The tunable parameter β penalizes t f to obtain almost the same duration as the initial driving cycle. The expression of f in the dynamics of v, for this case, is

f = 1 m [-c 0 -c 1 • v -c 2 • v 2 + η gb r tire • R gb • R t • T eng ]. ( 39
)
This problem was studied in (F. [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF][START_REF] Sciarretta | Optimal ecodriving control: Energy-efficient driving of road vehicles as an optimal control problem[END_REF][START_REF] Maamria | Which methodology is more appropriate to solve eco-driving optimal control problem for conventional vehicles[END_REF][START_REF] Hooker | Optimal driving for single-vehicle fuel economy[END_REF][START_REF] Ozatay | Analytical solution to the minimum energy consumption based velocity profile optimization problem with variable road grade[END_REF][START_REF] Kamal | On board ecodriving system for varying road-traffic environments using model predictive control[END_REF].

2.

Step2 : Using the DP, an energy management strategy is calculated for the eco-driving cycle obtained in Step1. For this purpose, the vehicle is assumed to follow the eco-driving cycle (the torque at the wheel T wh is imposed). The objective is to determine the torque split between the electric machine and the ICE and the gear-box ratio in order to minimize the fuel consumption. The state variable is ξ with its final constraint (30). The control variables are the engine torque T eng and the gear-box ratio R gb . As the torque T wh is imposed, the electric machine torque T el is calculated from the torque balance in (12). On the other hand, the engine and the electric machine speeds are free as the gearbox ratio is considered as a control variable. They are calculated using formula ( 14) where only the vehicle speed v is known. The associated OCP can be written min

(Teng,R gb ) k=N k=1 ṁf (w eng (k), T eng (k)) • ∆t(k) (40)
under the dynamics (19), state and input constraints (23,24,25,26,27), and the final constraints (30). The final time t f is fixed (the duration of the eco-driving cycle in Step1). The PMP can not be used to solve the OCP (40) because the cost function is not convex with respect to the discrete control variable R gb [START_REF] Nesch | Convex optimization for the energy management of hybrid electric vehicles considering engine start and gearshift costs[END_REF].

Method 4

This last method (M 4 ) is also based on decoupling the optimization of the control variables into two steps:

1. Step1 : This step is the same as Step1 in Section 4.3. 20 2. Step2 : Using the DP, an energy management strategy is calculated for the eco-driving cycle obtained in Step1. This step is similar to Step 2 in Section 4.3 where the gear-box ratio is fixed [START_REF] Guzzella | Vehicle propulsion systems[END_REF][START_REF] Serrao | Optimal energy management of hybrid electric vehicles including battery aging[END_REF][START_REF] Sciarretta | Optimal control of parallel hybrid electric vehicles[END_REF][START_REF] Kim | Optimal control of hybrid electric vehicles based on pontryagins minimum principle[END_REF]. The control variable is the engine torque T eng . As the torque T wh is imposed, the electric machine torque T el is calculated from the torque balance in ( 12). On the other hand, the engine and the electric machine speeds are imposed as the gear-box ratio is fixed. They are calculated using formula ( 14) where the vehicle speed v and the gear-box ratio R gb are known. The associated OCP can be written min

Teng k=N k=1 ṁf (ω eng (k), T eng (k)) • ∆t(k) (41)
under the dynamics (19), the state and input constraints (23,24,26), and the final constraints (30).

A similar approach was suggested in [START_REF] Sciarretta | Optimal ecodriving control: Energy-efficient driving of road vehicles as an optimal control problem[END_REF][START_REF] Ngo | An optimal control-based algorithm for hybrid electric vehicle using preview route information[END_REF] where the second step was performed using the PMP [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF]. Convex optimization can also be used for the second step [START_REF] Nesch | Convex optimization for the energy management of hybrid electric vehicles considering engine start and gearshift costs[END_REF].

The number of state and control variables and the parameters to be tuned for each method are summarized in Table 2. The objective of this study is to find a trade-off between the optimality of the solution (fuel consumption), the number of parameters to be tuned and the computation time of the DP(s).

States variables Control variables

Tuning parameters for each cycle are given in Table 3. The stop phases are removed from the driving cycles. The optimization is performed only when the vehicle moves.

M 1 2(v, ξ) 3(T eng , T el , R gb ) 1 (β) M 2 1(v) 3(T eng , T el , R gb ) 2 (β, µ) M 3 1(v) + 1(ξ) 2(T eng , R gb )+2(T eng , R gb ) 1 (β) M 4 1(v) + 1(ξ) 2(T eng , R gb )+1(T eng ) 1 (β)

Numerical Results

Six

The impact of the discretization step size on the optimality is not investigated. The study [START_REF] Maamria | Which methodology is more appropriate to solve eco-driving optimal control problem for conventional vehicles[END_REF] For the SOC, a step of ∆ξ = 0.25% is used in the method (M 1 ) and of ) and (M 4 ). The initial value of the SOC is ξ(0) = 60%. The gear-box considered has 6 ratios.

The constraints on the battery current I b are not considered in the problem formulation but checked in the forward computation (a posteriori).

For this study, a computer equipped with an Intel Core i7 2.30 GHz with 128 GB of RAM was used. The solution given by the method (M 1 ) is considered as a reference of comparison. However, this method is the most expensive in terms of Random-Access Memory (RAM) use and it is limited to three driving cycles: ECE, LWLTC and MWLTC (RAM saturation).

Optimality and computation time

The four methods are compared in terms of fuel consumption (Table 4), state trajectories, final time t f and the desired final SOC for all the methods (Table 5), computation time for each iteration (Table 6) and the RAM use during the backward loop (Table 7).

In Table 4, the second and the third row (Init-conv and Init-HEV) give the From Table 5, the error on the final time for the tested methods is less than 0.7% with the same final SOC (ξ(t f )). From Table 4 given the fuel consumption, it can be said that: • Methods (M 2 ) and (M 1 ) are close in terms of fuel consumption: suboptimality less than 1% while computation time is divided by at least 85. This small difference is not surprising as the formulation of the method (M 2 ) is similar to a PMP transformation with a tunable parameter µ (for the EMS design using PMP, µ is the equivalence factor).

• The method (M 1 ) overloads the RAM, it uses at least 76GB.

ECE EUDC NEDC LWLTC MWLTC WLTC M 4 ≤ 1 ≤ 1 ≤ 1 ≤ 1 ≤ 1 ≤ 1 M 3 ≤ 1 ≤ 1 ≤ 1 ≤ 1 ≤ 1 ≤ 2 M 2 ≤ 2 ≤ 2 ≤ 3 ≤ 2 ≤ 2 ≤ 3 M 1 76 - - 85 95 -
• The sub-optimality induced by using (M 3 ) compared to (M 2 ) is less than 2% while the computation times are close for short-driving cycles (ECE, LWLTC, MWLTC) and (M 3 ) is faster otherwise.

• The induced sub-optimality by using (M 4 ) compared to (M 2 ) is less than 3.5% except for the ECE case (where the error is 5%) while the computation time is divided by at least 8. The advantage of methods (M 3 ) and (M 4 ) compared to (M 2 ) is that only one parameter (β)

has to be tuned. In the studies (F. [START_REF] Mensing | Optimal energy utilization in conventional, electric and hybrid vehicles and its application to eco-driving[END_REF][START_REF] Monastyrsky | Rapid computation of optimal control for vehicles[END_REF]Maamria et al., 2016a), it was shown that, for conventional and electric vehicles, the relation between the final time t f and β is monotone: t f decreases when β increases.

• Fuel consumption reduction depends on the nature of the driving cycle.

The fuel saving can be partially correlated to the mean speed v of the initial driving cycle: it increases when v decreases. It is the lowest for the EUDC while it is the highest for the ECE and the LWLTC.

5.2. Tuning of µ and β for the method (M 2 )

For the method (M 2 ), two parameters have to be tuned, and thus at least two iterations are needed. Figures 6 and7 show the sensitivity of t f and the final SOC to β and µ for the LWLTC:

1. For a fixed value of β, the variation of t f is small while the relation between the final SOC and µ is monotone: increasing the value of µ increases the final value of the SOC.

2. For a fixed value of µ, the variation of final SOC is small while the relation between the final time and β is monotone: increasing the value of β decreases t f .

A similar analysis was conducted for other fixed values of β and µ and for the other driving cycles. These relations make the search for the good (M 3 ), the induced sub-optimality is reduced to 2% while the computation time is still reasonable.

Conclusion

The calculation of eco-driving cycles for HEVs was studied. Four methods to solve the associated OCP were compared in terms of fuel consumption saving, state trajectories, computation time and memory (RAM) use with the objective of finding a trade-off between complexity and optimality. The result is that, for the parallel HEV under consideration, the simplest method, based on decoupling the optimization of the control variables into two steps, among all possible choices is accurate enough to guarantee a near optimal fuel saving (a sub-optimality of 3%) while ensuring a reasonable computa-tion time. By considering the gear-box ratio as a degree of freedom in the energy management system optimization, the induced-sub-optimality is reduced to 1.5% while the computation time is multiplied by 10. The reference method (the most complicated) overloads the RAM (at least 70GB) and has an inconvenient computation time (compared to the simplest method). An intermediate method, based on reducing the number of state variable was also tested. Its drawback is the introduction of an additional tuning parameter.

Figure 1 :

 1 Figure 1: Parallel mild-hybrid architecture

Figure 2 :

 2 Figure 2: Specific fuel consumption SFC (g/kWh) of the ICE as a function of engine rotational speed and engine torque. For confidentiality reasons, the data are normalized.

Figure 3 :

 3 Figure 3: Electric Machine Efficiency as a function of the electric machine rotational speed and torque. For confidentiality reasons, the data are normalized.

  impact the conclusion of the conducted analysis. The same battery model is used for what follows. The battery current I b is also limited by its minimum value in the case of charging operation and its maximum value in the case of discharging phases. The dynamics of ξ is given by

Figure 4 :

 4 Figure 4: Internal resistance R bat [mΩ]

  acceleration a(k) is calculated from the vehicle speed v(k) and the control variables u(k). The final constraint on the vehicle position (28) is fulfilled by construction (D = N • ∆x). An additional tunable term β • ∆t(k) is added to the cost function as a terminal cost: β penalizes the final time to obtain almost the same time duration as the initial driving cycle. The state variable

  normalized driving cycles are considered: ECE-15 (the urban driving cycle), EUDC (the Extra-urban driving cycle), NEDC (the New European driving cycle), WLTC (the Worldwide harmonized Light vehicles Test Cycle), the low (LWLTC) and medium (MWLTC) phases of the WLTC. The duration without stop phases, the total traveled distance, mean speed value v and e l

  addressed this question for conventional vehicles. Based on the results obtained, the step sizes for the vehicle speed, position and engine torque are chosen as follows: ∆v = 0.1m/s for the vehicle speed, ∆x = 10m for the distance in the case of ECE and LWLTC and ∆x = 20m for the other driving cycles. For the control inputs u, steps of 2N.m for the engine and the electric machine torques are used.

Figure 12 :

 12 Figure 12: Summary: Methods comparison in terms of fuel consumption reduction [%],normalized computation time (with respect to (M 4 )) and the RAM use.

Table 1 :

 1 Vehicle model parameters

		Description	Value Unit
	m	Vehicle mass	1930	kg
	r tire	Wheel radius	0.34	m
	η gb	Gear-box efficiency	0.87	-
	R gb	Gearbox ratios	6	-
	R t	Differential ratio	3.53	-
	R el	Motor-to-wheel ratio	2.5	-
	ω idle	Engine idle speed	750	rpm
	ω min	min engine speed	750	rpm
	ω max	max engine speed	4000 rpm
	ω elmax max electric machine speed 10000 rpm
	ξ min	SOC min value	20	%
	ξ max	SOC max value	90	%
	a min	Acceleration min value	-2	m/s 2
	a max	Acceleration max value	1.5	m/s 2
	3.1. Optimal Control Problem (OCP) formulation

Table 2 :

 2 Difference between tested methods: state, control variables and tuning parameters.

Table 3 :

 3 Driving cycle parameters

	Cycle Name Time [s] Distance [km] e l [km/h] v [km/h]
	ECE-15	135	1.01	2	26.6
	EUDC	360	6.9	4	69
	MWLTC	386	5	3	46
	LWLTC	445	2.98	1	24.1
	NEDC	900	10.95	3	43.8
	WLTC	1574	22.72	3	52
	∆ξ = 0.02% in the second step of the methods (M 3

Table 4 :

 4 Fuel consumption [g]

		ECE EUDC NEDC LWLTC MWLTC WLTC
	Init-conv 50.6 299.5	501.7	123.2	206.7	973.3
	Init-HEV 33.2 283.8	414.8	91.1	178.4	829.3
	(M 4 )	21.6 229.3	317.3	69.1	122.5	665
	(M 3 )	20.7	227	310.9	67.4	120.8	655.9
	(M 2 )	20.5 226.6	310.3	66	118.3	649.1
	(M 1 )	20.3	-	-	65.9	118.2	-

fuel consumption for the initial driving cycles in the cases of a conventional and a hybrid vehicles (torque split only), respectively. They are used to assess the fuel saving of eco-driving for the considered methods.

Table 5 :

 5 Final time [s] and the desired final SOC [%]

	t f	ECE EUDC NEDC LWLTC MWLTC WLTC
	M 4	134.9 360.8	900.3	434.4	385.1	1575.3
	M 3	134.9 360.8	900.3	434.4	385.1	1575.3
	M 2	134.3 360.2	900.3	434.4	385.8	1575.2
	M 1	134.7	-	-	434.4	385.7	-
	ξ(t f )	60	60.21	60.39	60	59.97	60.21

Table 6 :

 6 Computation Time [s] for each iteration

		ECE EUDC NEDC LWLTC MWLTC WLTC
	M 4	6	29.7	47.8	17.3	17	104.6
	M 3 63.6 228.1	415	171	159.1	707.5
	M 2	50	368	596	144.5	171.4	1332
	M 1 4300	-	-	11000	14600	-

Table 7 :

 7 RAM use during backward loop[GB]

SOC(t f ) [%]

Variation of µ for a fixed β Variation of β for a fixed µ values of β and µ easier. First, one can tune β to get a final time t f near its target value. After, µ is tuned to bring the final SOC to its desired value.

The values of β and µ are adjusted iteratively.

State and control variables trajectories

The speed and SOC trajectories versus the distance for the LWLTC cycle are shown in Figures 8 and9.

Summary of the study

The main objective of this study is to calculate, within a reasonable time, an eco-driving cycle for HEVs. The comparison of the tested methods in terms of fuel consumption reduction (with respect to fuel consumption of conventional vehicles), the normalized computation time (with respect to the computation time of the method (M 4 )) and the RAM use is given in Figure 12. Based on these results, the methods (M 4 ) and (M 3 ) are the most suitable: the method (M 4 ) gives a first good estimation of the fuel saving with a reasonable computation time. By considering the gear-box optimization in