Keywords: Energy management system, vehicular electric power system, battery and supercapacitor, Pontryagin's minimum principle, state constraints. 1

The objective of this work is to show how to control the electric power systems of a vehicle in such a manner that their power flows should be optimized in the sense of energy efficiency. As will be seen, the control problem considered in this work can be formulated as an optimization problem in the presence of several constraints. A systematic approach based on optimal control will be adopted to design the energy management strategies. Then, by means of these strategies, the electric energy will be generated and stored in the most appropriate manner so that the overall energy consumption and eventually the pollutant emissions can be minimized for a given driving cycle. To this end, both non-causal optimization method using the knowledge of the entire driving cycle and causal one are developed for two case studies with different structures of energy storage system. These strategies are then evaluated in an advanced simulation environment to point out their effectiveness.

Optimal Control Based Algorithms for Energy Management of Automotive Power Systems with Battery/Supercapacitor Storage Devices

AnhTu Nguyen*, Jimmy Lauber, Michel Dambrine Introduction Over the years, the demand of electric power consumption in conventional vehicle has become more and more important. This is due to the fact that automotive customers are more and more demanding in terms of performance, comfort and safety for their new vehicles. Hence, the number of auxiliary electric-powered devices has been constantly increased in modern vehicles, e.g. active suspension, electric brakes, catalyst heaters, etc. This increasing demand tends to double or triple the current vehicle electric load [START_REF] Soong | Investigation of Interior Permanent Magnet Offset-Coupled Automotive Integrated Starter/Alternator[END_REF]. Besides improving the efficiency of the electric components, an effective energy management strategy (EMS) is also crucial to minimize the overall energy consumption of the vehicle.

In this work, the key feature of the studied vehicle consists in the presence of an electrical supercharger (eSC) in the turbocharged air system of the spark-ignition (SI) engine. This device aims at assisting the main turbocharger to reduce the effects of "turbo lag", i.e. slow engine torque dynamics and lack of torque at low speeds. As a consequent, the drivability is significantly improved.

The energy consumption of the eSC is provided by the vehicular electric power system. To this end, the vehicle is equipped with an advanced alternator which is controlled in power. Note that this alternator is directly coupled to the vehicle primary shaft; therefore, the engine operating point can be shifted by controlling the alternator output power. This fact offers one degree of freedom for energy optimization as in the case of classical parallel hybrid electric vehicles. However, this small capacity alternator is exclusively used to generate the energy for the electric power system and cannot assist the internal combustion engine (ICE) to propel the vehicle. Note also that the considered alternator can also recover the kinetic and potential energy during the regenerative braking phases. This "free energy" is then stored in the energy storage system (ESS) and will be used later in appropriate ways.

From the above remarks, it is clear that the energy management becomes very attractive to improve the overall energy efficiency of the studied vehicle. Because of industrial specifications, the developed strategies have to satisfy several objectives. First, they can offer a global optimal solution when the driving conditions are perfectly known in advance, i.e. offline situations. Second, their adaptations for real-world driving situations (i.e. online situations) are straightforward and the resulting causal strategies behave closely as the global optimal ones. Third, the developed strategies must be simple to be implementable with limited computation and memory resources. Fourth, the strategies are based on a systematic approach so that they can be applicable to a large spectrum of component dimension without the need for extensive calibration. For all these reasons, the developed EMSs will be based on an optimal control approach using physical component models of the vehicle.

Because of the relevance to this work, a brief overview concerning the optimal based energy management issue for hybrid electric vehicle [START_REF] Khaligh | Battery, Ultracapacitor, Fuel Cell, and Hybrid Energy Storage Systems for Electric, Hybrid Electric, Fuel Cell, and Plug-In Hybrid Electric Vehicles: State of the Art[END_REF][START_REF] Çaǧatay Bayindir | A Comprehensive Overview of Hybrid Electric Vehicle: Powertrain Configurations[END_REF] will be provided. In automotive framework, there are mainly two methods based on optimal control theory which may offer a globally optimal result in offline situations: Dynamic Programming (DP) [START_REF] Bellman | Dynamic Programming[END_REF] and Pontryagin's Minimum Principle (PMP) [START_REF] Pontryagin | The Mathematical Theory of Optimal Processes[END_REF].

DP-based strategies are known to be very costly in terms of computation. Numerous efforts have been devoted to reduce the computation time [START_REF] Sciarretta | Control of Hybrid Electric Vehicles: a Survey of Optimal Energy Management Strategies[END_REF]. These strategies are often used for offline purposes (performance evaluation, component sizing) [START_REF] Sundström | Optimal Control and Design of Hybrid Electric Vehicles[END_REF]. Some adapted online versions can be found in [START_REF] Lin | Power Management Strategy for a Parallel Hybrid Electric Truck[END_REF][START_REF] Koot | Energy Management Strategies for Vehicular Electric Power Systems[END_REF][START_REF] Kessels | Energy Management for Automotive Power Nets[END_REF]. Concerning the strategies based on PMP, their optimum could not be global as in the case of DP since the PMP only provides necessary optimality conditions. However, they are much more computationally efficient and the online adaptation is more straightforward. This is the main reason why we only deal with PMP approach in this work. In the literature, many results exist on PMPbased strategies [START_REF] Delprat | Control of a Parallel Hybrid Powewtrain: Optimal Control[END_REF][START_REF] Ambühl | Explicit Optimal Control Policy and Its Practical Application for Hybrid Electric Powertrains[END_REF] or the related Equivalent Consumption Minimization Strategies (ECMS) [START_REF] Paganelli | Optimal Control Theory Applied to Hybrid Fuel Cell Powered Vehicle[END_REF][START_REF] Sciarretta | Optimal Control of Parallel Hybrid Electric Vehicles[END_REF]. Besides optimal control based approaches, some rule-based methods for energy management can also be found in literature, see [START_REF] Paladini | Super-capacitors Fuel-cell Hybrid Electric Vehicle Optimization and Control Strategy Development[END_REF][START_REF] Poursamad | Design of Genetic Fuzzy Control Strategy for Parallel Hybrid Electric Vehicles[END_REF][START_REF] Hannan | Multi-sources Model and Control Algorithm of an Energy Management System for Light Electric Vehicles[END_REF] and references therein.

The main focus of this paper is to propose a systematic approach to design the energy management strategies that optimize the power flow of the vehicular electric power system. To this end, both online and offline strategies are considered. Thank to these optimal based EMSs, the overall energy consumption of the vehicle is minimized under all driving situations. A preliminary study on the choice of the structure of energy storage system was carried out by our industrial partners. From that, two case studies of electric power system with the same vehicle architecture will be considered. These choices are mainly due to the acceptable cost especially for the energy storage devices compared to conventional hybrid vehicles. The first case study is as in a conventional vehicle where only the battery is used to provide all onboard electric consumption and to make the electric power system more robust against peak-power demands. In this case, the EMS will exploit the freedom that the battery offers to the alternator in deciding the moment to generate electric power. This degree of freedom is generally not used [START_REF] Kessels | Energy Management for Automotive Power Nets[END_REF]. For the second case study, a hybrid storage system combining a supercapacitor [START_REF] Sharma | A Review on Electrochemical Double-layer Capacitors[END_REF] together with a battery will be used. Such a hybrid storage system has been widely used in automotive industry since it has both the high energy density of the battery and the high power density of the supercapacitor [START_REF] Fiorenti | Modeling and Experimental Validation of a Hybridized Energy Storage System for Automotive Applications[END_REF]. The supercapacitor aims at providing high currents during hard transition phases to protect the battery from fatal damages caused by overdischarge [START_REF] Khaligh | Battery, Ultracapacitor, Fuel Cell, and Hybrid Energy Storage Systems for Electric, Hybrid Electric, Fuel Cell, and Plug-In Hybrid Electric Vehicles: State of the Art[END_REF]. The supercapacitor is also used to store energy from regenerative braking and to reduce the battery size [START_REF] Fiorenti | Modeling and Experimental Validation of a Hybridized Energy Storage System for Automotive Applications[END_REF]. As will be seen, battery/supercapacitor hybrid energy storage system is more flexible in terms of optimization than the first one since it offers two degrees of freedom for EMS. However the electric structure and the control design are more complex than the first case.

In this work, we assume that the state constraints concerning the battery are not critical for optimization problem since it can be oversized. However, the supercapacitor may quickly charge and discharge due to its low specific energy compared to the battery [START_REF] Paladini | Super-capacitors Fuel-cell Hybrid Electric Vehicle Optimization and Control Strategy Development[END_REF]. Hence, the state constraints of the supercapacitor should be taken into account. To this end, a new form of penalty function will be proposed by introducing a dummy variable in the expression of the Hamiltonian. The strategies developed in this work are simple to implement, efficient in terms of fuel reduction and of computation times. They can be directly applied to parallel hybrid electric vehicles, and the formulation can be easily generalized to a large family of hybrid vehicles.

The paper is organized as follows. Section 2 first presents the studied vehicle structure with its two different electric power systems. Then, the models of some vehicle components used for control purpose are provided. In Section 3, the optimal control problems are formulated for both case studies and the Pontryagin's Minimum Principle is then applied to design the EMSs. Section 4 is devoted to the implementation issue of the developed EMSs on an advanced dynamic vehicle simulator and the analysis of the obtained results. To this end, the brief description of simulator is first given. Then, a discussion on how to use the optimal control outputs and also a simple adaptation idea to obtain causal strategies from optimal ones are presented. Next, the simulation results are performed to show the effectiveness of the developed strategies. Finally, a conclusion is given in Section 5.

Vehicle Description and Control-Based Models

The considered vehicle architecture is depicted in Figure 1. The notations are given in Appendix.

Figure 1. Representation of the studied vehicle architecture

The vehicle is equipped with a conventional powertrain with 5-speed manual transmission. The alternator is connected to the engine with a fixed gear ratio. The only difference between the two considered case studies consists in their electric power system, i.e. the "Electric System" block in Figure 1. The power flow of both case studies is described below.

Case study 1: Single storage electric power system

The power flow in this case is sketched in Figure 2. The direction of the arrows corresponds to the direction of the energy exchange between different components. The ICE produces the mechanical power ice P from chemical energy (fuel). This mechanical power ice P is divided into two parts. The first part dr P is used for vehicle propulsion. The second one , alt m P is delivered to the alternator and then converted to electrical power , alt e P . The alternator generates the power to satisfy the demand loads P of all onboard auxiliaries including the eSC. It is also used to charge the battery when necessary. The battery power bat P can be negative (when it is charged by the alternator) as well as positive (when it provides electric power for all electrical loads). It should be noticed that the eSC is controlled by engine control unit (ECU) which is out of the present work scope. However, its energy consumption profile is known and will be considered as an input of the developed EMSs.

Case study 2: Dual storage electric power system

A sketch of the power flow in this case is depicted in Figure 3. It is worth noting that Case study 1 is nothing else than a special case of Case study 2 where the supercapacitor and the DC/DC converter are removed from the electric power system.

Figure 3. Vehicle power flow in the studied dual storage electric power system It can be observed that the consumption of onboard auxiliaries aux P can be powered either by the alternator or by the battery. The battery is also used to charge the supercapacitor through the DC/DC converter. However, the supercapacitor cannot charge the battery in this electric structure; it is exclusively used to power the eSC.

Vehicle Control-Based Model

The control models are used to develop the energy optimization algorithm. At each sampling time, the energy optimization algorithm computes the optimal control sequences that minimize the energy consumption of the vehicle. For real-time applications, the control model should have a very limited complexity. Hereafter, some control models of the components of interest for both case studies will be described. It is worth noting that for confidentiality reasons with our industrial partners, all figures

showing the data of system components are slightly scaled so that the characteristics of these components will not be revealed.

Internal combustion engine

ICE is a complex system where many physical phenomena are not easy to model, e.g. combustion process [START_REF] Heywood | Internal Combustion Engine Fundamentals[END_REF]. However, from an energetic point of view some assumptions can be considered. Here, the temperature dependency and the dynamic behavior of the ICE will be neglected. Then, ICE is characterized by a static look-up-table (LUT) giving the instantaneous fuel consumption in function of the engine torque and the engine speed, see Figure 4.

Alternator

The alternator is characterized by some static LUTs as in the case of ICE. The first one, shown in Figure 5, provides the alternator efficiency as a function of the rotary speed and the current. The second LUT needed for the alternator average model provides the maximum current that the alternator can produce as a function of the rotary speed and the current, see Figure 6. its physical limitation. Note that the alternator only works in generator mode, so its current is conventionally positive and is assumed to be measured for the optimization design problem.

Battery

For control purpose, the thermal-temperature effects of the battery are usually neglected. Then, the only state variable left in the battery is its state of charge (SOC) whose dynamics equation is given as follows:

( ) ( ) ( ) ( ) ,0 ,0 ; 0 bat bat bat bat bat bat I SOC t SOC t SOC SOC Q = - = ɺ (1)
where the initial state of charge

2

oc bat oc bat bat bat bat bat bat bat bat U SOC t U SOC t R SOC t P t I SOC t R SOC t - - = (2)
where ( ) bat P t is the battery power at the terminal voltage.

Supercapacitor

After neglecting all complex thermal-electrochemical dynamics, the only state variable left in the supercapacitor is its voltage c U which is the image of its available energy amount:

( ) ( ) ( ) ( ) ,0
; 0

sc c c c c sc I U t U t U U C = - = ɺ (3)
where the capacitance sc C and the initial voltage ,0 c U of the supercapacitor are constant and given.

Similar to the battery, the supercapacitor current expression can be also given as:

( ) ( ) ( ) ( ) ( ) 2 4 2 c c sc sc sc c sc U t U t R P t I U t R - - = (4)
where ( ) sc P t is the supercapacitor power at the terminal voltage.

DC/DC converter

The DC/DC converter is simply modeled by the following efficiency rate:

( ) ( ) , , DC o DC DC i P t P t η = (5)
where ( )

,

Optimal Control Based Energy Management

Hereafter, the optimization problem is formulated for two case studies:

• Case study 1: single storage electric power system

• Case study 2: dual storage electric power system Next, the PMP will be applied to derive the energy management strategies for these both cases.

Problem Formulation

Control objective

In this work, the goal of optimal control is to minimize the fuel consumption over the driving cycle in the time optimization horizon [ ] 0,T , then the cost function is expressed by: (

, ,

T T lhv fuel ice ice fuel ice ice Q m T t t dt P T t t dt ω ω = = ∫ ∫ ɺ J (6)
where ( ) ( ) ( )

,

Remark 1:

The pollutant emissions can be also incorporated by modifying the cost function (6) as:

( ) ( ) ( ) ( ) ( ) ( ) 1 0 , , T n fuel ice ice i i ice ice i P T t t m T t t dt ω α ω =   = +     ∑ ∫ ɺ ɶ J (7)
where i α are weighting factors provided by designers and the instantaneous pollutant emission rates

( ) ( ) ( ) , i ice ice m T t t ω ɺ (in general
x NO , CO , HC ) are given by static LUTs. However, the objective to minimize the pollutant emissions is beyond the scope of this paper; therefore only the cost function of the form (6) will be considered in the sequel.

Constraints a. Vehicle architecture constraints:

The mechanical relations between torques and speeds of the considered vehicle architecture represented in Figure 1 are given by the equations:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
w gb ice alt gb ps ice alt w T t R k t T t T t R k t T t t t t R k t R k t η ρ η ω ω ω ρ  = + =   = =   (8)
The driving cycle is usually defined by the couple ( ) ( ) ( )

, w t k t ω . Indeed, when ( ) w t ω and ( ) k t are
known, the torque requested at the wheels ( ) w T t can be easily derived for the vehicle longitudinal dynamics equation [START_REF] Sciarretta | Control of Hybrid Electric Vehicles: a Survey of Optimal Energy Management Strategies[END_REF]. In this work, engaged gear ( ) k t at each moment is chosen by the driver.

Then, it can be noticed from ( 8) that, with a given driving cycle, neither the engine speed nor the alternator speed can be chosen by the energy management strategy, the only degree of freedom of the studied architecture is the alternator torque (or ICE torque).

b. Mechanical constraints: Due to the physical limitations of the ICE and the alternator, their speeds and torques are subject to the following constraints:

( ) ( ) ,min ,max ,min ,max ice ice ice alt alt alt t t ω ω ω ω ω ω ≤ ≤    ≤ ≤   (9) and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
T t T t T t T t T t T t ω ω ω ω  ≤ ≤   ≤ ≤   (10)
By considering the physical alternator torque limits, the engine torque limits at each instant t are given as, where the primary shaft torque

( ) ( ) ( ) ps ice alt T t T t T t ρ = +
is derived from the driving cycle:

( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ,min ,max : ice ice ice ice ice ice ice ice T t T t T t T t T t ω ω ≤ ≤ = ∈ T (11) where ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) {
 = -   = -   (12)
c. Electric power system constraints: As mentioned above, the only difference between the two considered case studies consists in their energy storage systems. Case study 1 deals only with one battery, see Figure 7. The optimization problem formulation in this case is similar to the one of conventional parallel hybrid electric vehicles with only one system state of the battery (1). From the electric structure in Figure 7, the battery current can be computed as:

( ) ( ) ( ) bat alt loads I t I t I t = - (13)
where the electric load current

, bat alt aux DC i I t I t I t I t = - - ( ) ( ) ( ) ( )
, sc DC o eSC I t I t I t = - (14) ( ) ( ) ( )
The DC/DC output current is constrained by:

( ) ( ) ( ) { } , , , , , , : DC o

I

are respectively the minimum and maximum output current of the converter.

For safe operation and cycle life extension, the battery SOC and battery current are both limited:

( )

,min ,max bat bat bat SOC SOC t SOC ≤ ≤ (17) ( )
,min ,max bat bat bat

I I t I ≤ ≤ (18)
For the supercapacitor, the voltage c U and the current sc I are subject to the following constraints:

( )

,min ,max c c c U U t U ≤ ≤ (19) ( )
,min ,max sc sc sc

I I t I ≤ ≤ (20)
Both battery and supercapacitor are considered as energy buffer systems. Therefore, the charge sustaining condition should be fulfilled by EMS for both of them. Concretely, one should have ( ) ( )

0 bat bat bat SOC SOC T SOC ∆ - ≜ (21)
( ) ( )

0 c c c U U T U ∆ - ≜ (22)

Application of Pontryagin's Minimum Principle

Next, the PMP will be applied to the two cases. Only offline optimal solutions will be considered in this subsection. Thereafter, for simplicity, the explicit time-dependence of the variables is omitted except for confusing situations.

Case study 1: Single storage electric power system

Taking into account ( 1) and ( 6), the Hamiltonian in this case is defined as:

( ) ( ) ( ) 1 1 ,0 , , , bat ice fuel ice bat bat ice bat SOC Q C T P SO I T λ ω λ = - H (23)
where the battery current

( ) b bat at SOC I
is computed by [START_REF] Paganelli | Optimal Control Theory Applied to Hybrid Fuel Cell Powered Vehicle[END_REF]. The necessary optimality conditions provided by PMP in this case are given as:

( ) ( ) ( ) * * * ,0 1 ,0 ; 0 bat bat bat bat bat bat I SOC SOC SOC SOC Q λ ∂ ⋅ = = - = ∂ ɺ H (24) ( ) ( )
* * * 1 1 ,0 bat bat bat bat bat I SOC SOC Q SOC λ λ ∂ ∂ ⋅ = - = ∂ ∂ ɺ H (25)
( )

* ,0 bat bat SOC T SOC ≈ ( 26
) * ,min ,max bat bat bat SOC SOC SOC ≤ ≤ (27) ( ) ( ) [ ] * * * 1 * * 1 , , , , ; 0,
; i bat ba ce ice ice t ice SOC T SOC T t T T λ λ ∀ ∈ ∀ ≤ ∈ H H T (28)
Several comments can be made regarding these optimality conditions. First, the conditions ( 24) and [START_REF] Koot | Energy Management for Vehicular Electric Power Systems[END_REF] provide respectively the dynamics of the system state and its associated co-state. However, neither an initial condition nor a final condition on the co-state is available. Second, it is important to emphasize that when the charge sustaining condition guaranteed by ( 26) is required for the EMS, the battery usually operates only in a small range of SOC [START_REF] Kim | Optimal Control of Hybrid Electric Vehicles Based on Pontryagin's Minimum Principle[END_REF]. As a consequence, the open circuit voltage and the internal resistance of the battery may not vary so much in this range. Therefore, from

(2), it can be concluded that the battery current

( ) b bat at SOC I
is not significantly affected by the variation of battery SOC. Combining this fact with condition [START_REF] Koot | Energy Management for Vehicular Electric Power Systems[END_REF], it follows that:

1 1 10 0 λ λ λ ≈ ⇒ = ɺ ( 29
)
where the constant 10 λ has to be determined. This assumption has been exploited in many other previous works [START_REF] Delprat | Control of a Parallel Hybrid Powewtrain: Optimal Control[END_REF][START_REF] Sciarretta | Control of Hybrid Electric Vehicles: a Survey of Optimal Energy Management Strategies[END_REF][START_REF] Kim | Optimal Control of Hybrid Electric Vehicles Based on Pontryagin's Minimum Principle[END_REF]. Third, our studied battery has an important nominal capacity ,0 bat Q , then, the state constraints (27) will never be violated. Fourth, at each instant t , the optimal control * ice T minimizing the Hamiltonian can be exhaustively searched in the torque admissible set ice T defined in [START_REF] Delprat | Control of a Parallel Hybrid Powewtrain: Optimal Control[END_REF]. This can be numerically done by testing all torque possibilities of the set ice T at each time step.

Hence, the constraints on the control variable ice T are "naturally" considered. Fifth, it is clear that the optimal solution at each instant t depends on the initial conditions of the system state ,0 bat SOC and the co-state 10 λ . The former initial condition is given, however, the latter on is not known a priori.

The value of 10 λ depends on boundary condition of the terminal cost at final time T , i.e. the future information of the driving conditions. Indeed, the determination/estimation of this value is crucial to reach the optimal solution as close as possible. For offline situations where driving cycles are given in advance, the value of 10 λ can be iteratively computed by a "root finding algorithm" [START_REF] Delprat | Control of a Parallel Hybrid Powewtrain: Optimal Control[END_REF].

Case study 2: Dual storage electric power system

In this case, two dynamical systems (1) and ( 3) are available for the energy storage system. As previously highlighted, the state constraints [START_REF] Fiorenti | Modeling and Experimental Validation of a Hybridized Energy Storage System for Automotive Applications[END_REF] of the supercapacitor should be taken into account.

To this end, a new dummy variable has been introduced whose dynamics is defined as [START_REF] Kirk | Optimal Control Theory: An Introduction[END_REF]:

( ) d c X T U ɺ ≜ (30)
where the function ( ) c T U in (30) is defined as:

( ) ( ) ( ) 2 2
,min ,min ,max ,max

c c c c c c c c c T U U U sg U U U U sg U U     = - - + - -     (31)
and the function ( ) sg ⋅ in (31) is given as:

( ) Taking into account the dynamics (1), ( 3) and (30), the augmented Hamiltonian for the Case study 2 is defined as follows:

0, 0 1, 0 x sg x x <   ≥  ≜ (
( ) Then, the necessary optimality conditions are given by ( 24)-( 27) together with the following ones: ( )

( ) ( ) ( ) ( ) , 1 2 1 ,0 , 2 , , , , , , , bat bat
( ) ( ) ( ) * * ,0 2 * , * , ; 0 DC o a sc c c c c sc I U U I U U C λ ∂ ⋅ = = - = ∂ ɺ H (35) ( ) * ,0 c c U T U ≈ (36) * ,min ,max c c c U U U ≤ ≤ (37) ( ) ( ) ( ) ( ) 2 2 * * * * * ,min ,min ,max ,max * 0 0 d c c c c c c c d d a c X U U sg U U U U sg U U X λ ∂ ⋅     = = - - + - -     ∂ = ɺ H (38) ( ) ( ) ( ) ( ) ( ) * * * * * 1 2
U U Q U C U U U sg U U U U sg I U U λ λ λ λ λ ∂ ∂ ∂ ⋅ = - = + ∂ ∂ ∂     - - - - - -     ɺ ⋯ H (39) ( ) * a d d X λ ∂ ⋅ = -∂ ɺ H (40) ( ) ( ) [ ] ( ) * * * * * * * * * * 1 * * 2 , 1 2 , , , , , , , , , , , , , , , 0, ;
* * 0 0 a d d d d X λ λ λ ∂ ⋅ = - = ⇒ = ∂ ɺ H ( 42
)
where 0 d λ is the constant to be determined. Note that if the supercapacitor has an important capacitance sc C , its state constraints [START_REF] Fiorenti | Modeling and Experimental Validation of a Hybridized Energy Storage System for Automotive Applications[END_REF] will be then trivial. In this case, 0 d λ can be set equal to 0, which means that the supercapacitor state constraints are not taken into account.

From the supercapacitor current ( ) 4), it follows that:

sc c I U expression in (
( ) ( ) 2 4 sc c sc c c c sc sc I U I U U U R P ∂ = - ∂ - (43)
Then, the condition (39) can be rewritten as:

( ) ( ) ( ) ( ) * * 2 * 2 2 * * * * * 0 ,min ,min 0 ,max ,max 4 2 2 sc c sc c sc sc d c c c c d c c c c I U C U R P U U sg U U U U sg U U λ λ λ λ - = -     - - - - - -     ɺ ⋯ (44)
The trajectory * (36) are satisfied. However, using a "root finding algorithm" as in Case study 1 to iteratively compute these three values would not be appropriate due to excessive computation times. A simple method, which is more effective in terms of computation times, will be proposed latter. Although this method only offers sub-optimal control sequences of * , it can be directly used for online implementation.

Physical interpretation of Hamiltonian

This subsection aims at pointing out the physical meaning of the Hamiltonians and the co-states in the previous definitions [START_REF] Borutzky | Bond Graph Methodology: Development and Analysis of Multidisciplinary Dynamic System Models[END_REF] and (34). To this end, only Hamiltonian of Case study 2 is considered since it is of a more general form than the one in [START_REF] Borutzky | Bond Graph Methodology: Development and Analysis of Multidisciplinary Dynamic System Models[END_REF].

Let us define the following variables:

( )

1 , 2 1 2 0 ; b oc ba c t t sc a U S Q C s O U C s λ λ - - ≜ ≜ (45)
Then, the expression of the Hamiltonian in (34) can be rewritten as: clearer with (46). Indeed, this is the sum of the weighted powers of all energy sources available in the vehicle. In other words, the Hamiltonian represents an equivalent fuel power; and the variables 1 s and 2 s are used to converts the inner battery power and the inner supercapacitor power into the equivalent quantities of fuel power. That is why these variables are usually referred to equivalence factors [START_REF] Paganelli | Optimal Control Theory Applied to Hybrid Fuel Cell Powered Vehicle[END_REF][START_REF] Sciarretta | Optimal Control of Parallel Hybrid Electric Vehicles[END_REF]. The more important these variables are, the more expensive the electric energy is. So, it is more beneficial to recover the energy by regenerative braking. On the contrary, the lower these variables are, the cheaper the electric energy is also. As a consequent, it is more beneficial to use the electric machine to generate the energy (for the case of hybrid vehicles).

( ) ( ) ( ) ( ) ( ) , 1 2 1 , 2 , , , , , , , , ,
In the cases where the state constraints are present, the dynamics d X ɺ indicates these constraints are whether or not violated. Then, the term

0 d d d d X X λ λ = ɺ
ɺ is incorporated into the Hamiltonian as a penalty function. The constant 0 d λ should be selected to be very high such that the supercapacitor state lies in its bound limits in very short time. Since there is no penalty if the state remains between its upper and lower limits, the energy management strategies can make full use of the supercapacitor over the allowable range.

Implementation and Results Analysis

Dynamic Vehicle Simulator

Before analyzing the results, it should be noticed that all developed strategies in this work are validated with a dynamic vehicle simulator calibrated with real vehicle data. This simulator consists of all appropriate models of different elements constituting the studied vehicle, including the driver and the energy management strategy. An overview of the simulation environment in this work is illustrated in Figure 9. As can be seen, the considered simulation environment has two separated parts: the vehicle model part and the energy management system part. In what follows, some issues directly related to the implementation of the developed energy management strategies into the simulator are first discussed. Next, a simple idea to derive a causal EMS for real-time applications is presented. Then, to show the performance of the developed strategies in terms of energy consumption efficiency, they will be also compared to a baseline strategy where the energy storage system (ESS: battery and/or supercapacitor) will be practically never charged or discharged. As a consequent, the alternator will be always activated to generate all energy needed for onboard electric demand. Note also that these baseline strategies, provided by our industrial partners, will not be detailed for confidentiality reason.

Implementation

How to use the optimal control sequences?

The developed EMSs provide the engine torque and also the alternator torque in both case studies.

They are often used as reference signals to control the ICE and the alternator, respectively. However, in this project, the ECU is designed by another industrial partner and only optimal alternator torque * alt T will be used to control the reference voltage of the alternator. The control scheme of the alternator is illustrated in Figure 10. It is important to remark that the approach used in our work does not require any modifications of the vehicle structure (drive train and electric power system). The only simple task for implementation is to replace the existing controller(s) of the baseline strategy with those developed in this paper.

Online adaptation

As previously emphasized, it is possible to obtain the optimal solutions only when all information of the entire driving cycle is available a priori. In subsection 3.2, we also showed that the optimization problems consist finally in determining the constant 10 λ for Case study 1 and constants 10 λ and 20 λ for Case study 2 with a "root finding algorithm". However, these strategies are not causal and cannot be applicable for real-world applications. Therefore, an adaptation of these strategies for online implementation is necessary. Over the years, a great deal of efforts has been investigated to cope with online strategies based on non-causal optimal ones [START_REF] Sciarretta | Control of Hybrid Electric Vehicles: a Survey of Optimal Energy Management Strategies[END_REF]. The crucial point of this problem is to find out an appropriate way to adapt the co-state(s) in such a manner that the behavior of causal strategies is as close as possible to the corresponding optimal solution. For simplicity and for computation efficiency, the so-called " λ -control" method is adopted in this work [START_REF] Delprat | Optimal Control of a Parallel Powertrain: from Global Optimization to Real Time Control Strategy[END_REF][START_REF] Koot | Energy Management for Vehicular Electric Power Systems[END_REF][START_REF] Kessels | Energy Management for Automotive Power Nets[END_REF]. This method is based on a feedback control which is easy to implement, see Figure 11. The expression of the estimated λ ɶ is given as:

( ) ( ) 0 0 t p sp i sp K SOC SOC K SOC SOC λ λ = + - + - ∫ ɶ ( 47
)
where sp SOC is the SOC set point of the considered energy system storage. This value is given as ( )

0 sp SOC SOC =
if the charge sustaining condition is considered. p K , i K are the gains of the PI controller and 0 λ is the initial guess. It can be noticed that the idea of the " λ -control" method is simply to keep the SOC of each energy storage system in a reference range of variation defined by sp SOC . In other words, the feedback " λ -control" aims at preventing the overcharge or depletion of the considered ESS in long term, however, its SOC may "freely" vary in short term. For this reason, the PI controller gains should be selected rather low. A detail on this discussion can be found in [START_REF] Koot | Energy Management for Vehicular Electric Power Systems[END_REF].

Simulation Results

In this work, the Artemis Road cycle [START_REF] André | The ARTEMIS European Driving Cycles for Measuring car Pollutant Emissions[END_REF] will be exclusively considered to validate the different energy management strategies. This driving cycle represents real driving conditions and allows achieving a realistic evaluation of the obtained results. The vehicle speed and gear position of this real-world driving cycle are shown in Figure 12. For all simulations presented hereafter, the tracking performance of the vehicle with respect to the speed reference of the considered driving cycle is always perfectly guaranteed.

Figure 12. Artemis Road cycle: vehicle speed (up) and imposed gearbox ratio (bottom)

Case study 1

The following strategies are implemented and their results will be compared:

• BL1: Baseline strategy for Case study 1 where the battery is not practically used, so the alternator power is almost equal to the required electric load.

• PMP1: PMP-based optimal strategy for Case study 1 with a given driving cycle.

• RT1: Real-time strategy for Case study 1 with 1 λ ɶ estimated by (47). The desired and the realized alternator torques obtained from PMP1 strategy are presented in Figure 13. It can be noticed that the alternator toque, which is indirectly imposed by the voltage reference of the electric power system, globally tracks the optimal alternator torque provided by the PMP1 strategy. However, the alternator has its own dynamics. Hence, the realized torque tends to 0 after a certain time for each alternator activation. This problem, which is unavoidable, will degrade the fuel saving performance of the PMP1 strategy. Indeed, the alternator can take some electric energy which is unscheduled by PMP1 strategy. Figure 13. Optimal alternator toque provided by PMP1 strategy and real alternator provided by simulator for all the driving cycle (up) and their zooms (bottom) Figure 14 shows the comparison of realized alternator torques between PMP1 and BL1 strategies. As previously stated, BL1 tries to maintain a constant alternator voltage; such that the battery is practically not used and all onboard electric load energy is directly supplied by the alternator. Hence, the alternator will be always activated for this strategy and this can increase the fuel consumption.

Concerning PMP1 strategy, it schedules the alternator activation at appropriate moments (deceleration phases, battery charging) and with appropriate quantities of torque. In such a manner, PMP1 strategy can help to recover a certain amount of "free energy" coming from regenerative braking. In addition, with this optimal strategy, the alternator can be also used to shift the operating From Figure 15, it can be observed that if the co-state 1 λ ɶ is fine tuned then RT1 strategy has the same behaviors as PMP1 strategy although it does not need any information on the future of driving cycle. As previously stated, searching offline optimal solution with "root finding algorithm" can be too expensive in terms of simulation time. Hence, this will be not presented here and only two strategies are implemented and compared in this case:

• BL2: Baseline strategy for Case study 2 uses only the supercapacitor. The DC/DC converter is controlled by a heuristic strategy of industrial partner. This strategy aims at guaranteeing that the voltage in the supercapacitor side is always superior to the one in the battery side and the supercapacitor energy is always kept between certain levels.

• RT2: Real-time strategy for Case study 2 with 1 λ ɶ and 2 λ ɶ estimated by (47).

The results of BL2 and RT2 strategies are compared in Figure 16. The same comments on the alternator activation can be done as in Case study 1, i.e. the alternator is mostly activated by RT2 strategy to recover regenerative braking energy. Moreover, RT2 strategy also activates the DC/DC converter more often than BL2 strategy to charge the supercapacitor at appropriate moments. As a consequent, both energy storage systems (ESS) are better exploited in RT2 strategy than in BL2 strategy as shown in Figure 17. Figure 18 shows that supercapacitor voltage always remains in its operating range, i.e. it is always higher than the battery voltage as imposed by the electric power system and lower than the supercapacitor voltage maximal value (16.2V). Thanks to the penalty function

Figure 2 .

 2 Figure 2. Power flow of the studied vehicle with single storage electric power system

Figure 4 .

 4 Figure 4. Representation of the instantaneous fuel consumption of the studied engine

Figure 5 .Figure 6 .

 56 Figure 5. Representation of the alternator efficiency

  are given. The expression of the battery current can be given as[START_REF] Lin | Power Management Strategy for a Parallel Hybrid Electric Truck[END_REF]:

  converter output and input powers. The efficiency of the DC/DC converter is given.

Figure 7 .

 7 Figure 7. Sketch of the electric structure of Case study 1

  and represents all onboard auxiliary demand including the consumption of eSC. The current delivered by the alternator ( ) alt I t can be easily derived from the optimal alternator torque at each time step. It is noticed from (13) that if the alternator is optimally controlled, then, the battery use is also indirectly optimized in the sense of energy efficiency.For Case study 2, both battery and supercapacitor are considered. They are linked by a DC/DC converter. The electric structure of this case is depicted in Figure8. In this case, the dynamics of the supercapacitor (3) should be considered together with[START_REF] Soong | Investigation of Interior Permanent Magnet Offset-Coupled Automotive Integrated Starter/Alternator[END_REF] for optimization problem to fully take advantage of all electric structure potential.

Figure 8 .

 8 Figure 8. Sketch of the electric structure of Case study 2

2 λ 20 λ and 0 dλ

 2200 is obtained by integrating both sides of (44) which of Case study 2 is now reduced to the choice of the three values 10 λ , in such a manner that both boundary conditions (charge sustaining conditions)[START_REF] André | The ARTEMIS European Driving Cycles for Measuring car Pollutant Emissions[END_REF] and

  the fuel power, the inner battery power and the inner supercapacitor power. The physical meaning of the Hamiltonian becomes

Figure 9 .

 9 Figure 9. Simulation environment in the thesis framework

Figure 10 .

 10 Figure 10. Control scheme of the alternator

Figure 11 .

 11 Figure 11. Online estimation of the co-state λ ɶ

  point of the ICE to other regions that require relative less fuel. Moreover, the battery will be better exploited as shown in Figure15.

Figure 14 .Figure 15 .

 1415 Figure 14. Comparison of realized alternator torques between BL1 and PMP1 strategies

Figure 16 .

 16 Figure 16. Comparison of alternator torque and converter current between BL2 and RT2 strategies

Figure 17 .

 17 Figure 17. Comparison of ESS state of charges between BL2 and RT2 strategies

  Hamiltonian, the supercapacitor voltage only touches its upper limit for a very short time (around 100s) of the driving cycle.

Figure 18 .

 18 Figure 18. Voltages of energy storage systems for RT2 strategy

  }

	ice T	,min	ice ω	t	max	ice T	,min	ice ω	, t T t ps	ρ	alt T	,max	ω	alt	t
	ice T	,max	ice ω	t	min	ice T	,max	ice ω	, t T t ps	ρ	alt T	,min	ω	alt	t

Table 1

 1 summarizes the energy consumption of the considered strategies and the fuel saving of PMP1

	Vehicle Speed (km/h)	0 50 100 150	0	100	200	300	400	500	600 Reference Realized	700	800	900	1000
									Time (s)
	Alternator Torque (Nm)	0 5 10	0	100	200	300	400	500	600	700	800	900 BL1	1000
									Time (s)
	Alternator Torque (Nm)	0 5 10 15	0	100	200	300	400	500	600	700	800	900 OPTI1	1000
									Time (s)
			81
		80.5
	Battery SOC (%)	79.5 80 79			BL1 PMP1 RT1
		0 78.5	100	200	300	400	500	600	700	800	900	1000
									Time (s)
			0.6
			0.4
			0 0.2	100	200	300	400	500	600	700	800	900	1000

0.8 1 1.2 1.4 1.6 Time (s) λ (-) Estimated λ and

  RT1 strategies with respect to BL1 strategy for Artemis Road cycle. This table shows that the proposed strategies (OPTI1 and RT1) are not only effective to reduce the fuel consumption but also can guarantee the sustaining charge condition of the battery.

	Strategy Fuel Use [g] Fuel Saving [%]	∆	bat SOC	[%]
	BL1	700.072	0		0.361
	OPTI1	685. 597	2.068		0.056
	RT1	685.776	2.042		0.111

Table 1 .

 1 Summary of energy consumption for different strategies of Case study 1 4.3.2. Case study 2

Table 2

 2 summarizes the energy consumption of both strategies, and the fuel saving of RT2 strategy with respect to BL2 strategy for Artemis Road cycle. From the results, it can be concluded that the RT2 strategy is effective for fuel consumption reduction.

	Strategy Fuel Use [g] Fuel Saving [%]	∆	bat SOC	[%]	∆	SOC	sc	[%]
	BL2	701.819	0		0.019			9.394
	RT2	690.322	1.634		0.076			0.054

Table 2 .

 2 Summary of energy consumption for different strategies of Case study 2As can be seen, the results obtained on this unique cycle are very promising. However, several tests with other driving cycles would be necessary to show that the extra energy consumption of eSC

	12.5
	0 12	100	200	300	400	500	600	700	800	900	1000

13 13.5 14 14.5 15 15.5 16 16.5 Time (s) ESS Voltage (V) Supercapacitor Battery

Acknowledgements

This research is sponsored by the International Campus on Safety and Intermodality in Transportation the Nord-Pas-de-Calais Region, the European Community, the Regional Delegation for Research and Technology, the Ministry of Higher Education and Research, and the French National Center for Scientific Research. This work is also sponsored by the FUI and VALEO Group through the SURAL'HY project.

would be compensated with some effective energy management strategies (note that the eSC is practically only activated in low load region).

Concluding Remarks

In this work, PMP-based strategies are developed to "optimally" control the vehicular electric power systems. We have shown that this approach has several advantages for real-time implementation. First, causal strategies, which can mimic the behavior of optimal solution, can be easily obtained with a simple feedback control scheme. Second, it is very efficient in terms of time computation because it is based on the instantaneous minimization of the Hamiltonian. The effectiveness of the approach is pointed out through several simulation results for both case studies.

Most of fuel saving comes from regenerative braking which is "free" energy. Despite the additional cost for hardware investments, the dual storage electric power system offers a limited fuel saving performance compared to single storage electric power system for the given parameter values in the simulator AMESim. However, this electric structure may be used to reduce the capacity of the battery since it is not practically used or it would be more interesting for electric hybrid vehicles with "stop and start" operation.

The energy management problem considered in this work can be directly applied to parallel hybrid electric vehicles. They are also easily generalized to others hybrid architectures with some slight modifications.

Appendix. Some notations of vehicle variables used in this paper

Variable