# Multiphysics modeling and optimization of the driving strategy of a light duty fuel cell vehicle

Jean-Christophe Olivier, Guillaume Wasselynck, Stéphane Chevalier, Bruno Auvity, Christophe Josset, Didier Trichet, Gaetano Squadrito, Nicolas Bernard

► To cite this version:

Jean-Christophe Olivier, Guillaume Wasselynck, Stéphane Chevalier, Bruno Auvity, Christophe Josset, et al.. Multiphysics modeling and optimization of the driving strategy of a light duty fuel cell vehicle. International Journal of Hydrogen Energy, 2017, 42 (43), pp.26943-26955. 10.1016/j.ijhydene.2017.06.243. hal-04019002v2

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Submitted on 2 Apr 2024

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# Multiphysics modeling and optimization of the driving strategy of a light duty fuel cell vehicle

Jean-Christophe Olivier<sup>a, *</sup>, Guillaume Wasselynck<sup>a</sup>, Stéphane Chevalier<sup>b</sup>, Bruno Auvity<sup>c</sup>, Christophe Josset<sup>c</sup>, Didier Trichet<sup>a</sup>, Gaetano Squadrito<sup>d</sup>, Nicolas Bernard<sup>a</sup>

## ARTICLE INFO

Article history:

Received 4 August 2016

Received in revised form

22 May 2017

Accepted 21 June 2017

Available online 21 August 2017

Keywords:

Fuel-cell powertrain

Multi-physics modeling

Optimization

Eco-marathon

## ABSTRACT

This paper presents the optimization of the driving strategy of a high efficiency fuel cell based power train. This power train is developed to equip a light duty urban-concept vehicle that runs energetic races. The objective is to go the furthest with the lowest quantity of fuel. A comprehensive dynamical model is presented, including the mechanical requirement, the thermal behavior of the fuel cell stack and the various losses and consumptions of the power train devices. This model is next integrated into a global optimization algorithm, to determine the best race strategy to be adopted. These results are validated on experimental measurements, obtained during a real race at the Shell Eco-Marathon, in 2015.

## Introduction

Given the new economic and ecological issues, car manufacturers should focus on alternative energy sources, such as electric batteries or fuel cells [8, 10, 27, 32, 33, 35, 36, 38]. Although still young and hard to handle, fuel cells, and particularly Proton Exchange Membrane Fuel Cells (PEMFCs), remain a promising way. Indeed, a hydrogen fuel cell vehicle has naturally a better performance than a diesel car (usually 30% against 22% of efficiency at the wheel). Moreover, the energy supplied by the fuel cell is electric, which allows propulsion chains more flexible with high performances [11, 37]. For

several years, a prototype vehicle with extremely low energy consumption has been developed by researchers and electrical engineering students, to run energetic races [28, 34, 39]. The energy source of this vehicle is a PEMFC. From 2010 to 2012, this vehicle has been recognized by the Shell group (through the 2010 edition of the Shell Eco-marathon competition, on the EuroSpeedway race track, Germany) as the world most energy efficient car, with an equivalent consumption of only 0.02 L of unleaded gasoline 95 per 100 km, at a minimum average speed of 30 km/h.

Since 2013, a new car with extremely low energy consumption is developed (see Fig. 1). It comes under the urban-

<footnote><sup>a</sup> Nantes University, IREENA Laboratory, Saint-Nazaire, France</footnote>

<footnote><sup>b</sup> Thermofluids for Energy and Advanced Materials Laboratory, University of Toronto, Canada</footnote>

<footnote><sup>c</sup> Nantes University, Laboratoire de Thermocinétique de Nantes, Nantes, France</footnote>

<footnote><sup>d</sup> CNR-ITAE, Messina, Italy</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: jean-christophe.olivier@univ-nantes.fr (J.-C. Olivier).</footnote>
# Multiphysics modeling and optimization of the driving strategy of a light duty fuel cell vehicle

Jean-Christophe Olivier<sup>a, *</sup>, Guillaume Wasselynck<sup>a</sup>, Stéphane Chevalier<sup>b</sup>, Bruno Auvity<sup>c</sup>, Christophe Josset<sup>c</sup>, Didier Trichet<sup>a</sup>, Gaetano Squadrito<sup>d</sup>, Nicolas Bernard<sup>a</sup>

## ARTICLE INFO

Article history:

Received 4 August 2016

Received in revised form

22 May 2017

Accepted 21 June 2017

Available online 21 August 2017

Keywords:

Fuel-cell powertrain

Multi-physics modeling

Optimization

Eco-marathon

## ABSTRACT

This paper presents the optimization of the driving strategy of a high efficiency fuel cell based power train. This power train is developed to equip a light duty urban-concept vehicle that runs energetic races. The objective is to go the furthest with the lowest quantity of fuel. A comprehensive dynamical model is presented, including the mechanical requirement, the thermal behavior of the fuel cell stack and the various losses and consumptions of the power train devices. This model is next integrated into a global optimization algorithm, to determine the best race strategy to be adopted. These results are validated on experimental measurements, obtained during a real race at the Shell Eco-Marathon, in 2015.

## Introduction

Given the new economic and ecological issues, car manufacturers should focus on alternative energy sources, such as electric batteries or fuel cells [8, 10, 27, 32, 33, 35, 36, 38]. Although still young and hard to handle, fuel cells, and particularly Proton Exchange Membrane Fuel Cells (PEMFCs), remain a promising way. Indeed, a hydrogen fuel cell vehicle has naturally a better performance than a diesel car (usually 30% against 22% of efficiency at the wheel). Moreover, the energy supplied by the fuel cell is electric, which allows propulsion chains more flexible with high performances [11, 37]. For

several years, a prototype vehicle with extremely low energy consumption has been developed by researchers and electrical engineering students, to run energetic races [28, 34, 39]. The energy source of this vehicle is a PEMFC. From 2010 to 2012, this vehicle has been recognized by the Shell group (through the 2010 edition of the Shell Eco-marathon competition, on the EuroSpeedway race track, Germany) as the world most energy efficient car, with an equivalent consumption of only 0.02 L of unleaded gasoline 95 per 100 km, at a minimum average speed of 30 km/h.

Since 2013, a new car with extremely low energy consumption is developed (see Fig. 1). It comes under the urban-

<footnote><sup>a</sup> Nantes University, IREENA Laboratory, Saint-Nazaire, France</footnote>

<footnote><sup>b</sup> Thermofluids for Energy and Advanced Materials Laboratory, University of Toronto, Canada</footnote>

<footnote><sup>c</sup> Nantes University, Laboratoire de Thermocinétique de Nantes, Nantes, France</footnote>

<footnote><sup>d</sup> CNR-ITAE, Messina, Italy</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: jean-christophe.olivier@univ-nantes.fr (J.-C. Olivier).</footnote>
<figure><img src="image_1.png" /><figcaption>Fig. 1 – The Urban-Concept vehicle Cityjoule.</figcaption></figure>

concept category, which aims to promote the development of vehicles closest to true city cars. The energy source of this vehicle is also a PEMFC. To achieve an energy consumption as low as possible, each element of the vehicle must be optimized, from the mechanical angle (drag coefficient, frontal area, slop, road-wheel friction, weight) to the electrical architecture (power converter, motor efficiencies and fuel cell peripheral elements consumption). This optimization has been done in previous works, but without taking into account the driving strategy [34]. In this paper, the optimal driving strategy that minimizes the car energy demand is presented. It is based on a multi-physics modeling of the power train, taking into account the road profile and the race constraints (average speed, stop at each turn, etc.) [1,5,7,31]. This model is then integrated into a global optimization algorithm, which lead to the best velocity profile, motor current and gearbox ratio.

The paper is structured as follows: In section *Presentation of the race and vehicle characteristics*, the vehicle specifications and the race characteristics are presented. A focus is made on power train architecture from the fuel cell stack to the mechanical needs. In section *Multi-Physics modeling of the powertrain*, a dynamical and multi-physics model is developed. In this section, each part of the power train is presented. An originality of this work is the addition of a thermal modeling of the PEMFC, in order to take into account the auxiliary stack consumption in energy balance, including the air feeding and cooling systems. The optimization results are given in Section *Results*, which allow to achieve a driving strategy minimizing the final energy consumed by the vehicle. Simulations and experimental results are compared to a real race, recorded during the previous edition of the Shell Eco-marathon, in Rotterdam, Netherlands. Conclusions are given in Section *Conclusions*.

## Presentation of the race and vehicle characteristics

The urban-concept car and its associated fuel cell power train were both designed and built to participate in the Shell Eco-Marathon energetic race (Rotterdam, Netherlands). The principle of such a competition is to drive a vehicle with one pilot

the furthest with the lowest quantity of fuel at a minimum average speed of 25 km/h. A minimum weight of 70 kg is imposed for the pilot. Each competitor will have to travel 10 laps of the race track corresponding to 16.1 km in less than 40 min. In the urban-concept category, a stop is mandatory during at each lap. To evaluate prototype car efficiency, the fuel quantity carried on board is measured before and after the attempt. Wide range of fuels can be used (Unleaded gasoline; diesel; LPG; GTL; fatty acid methyl ester; ethanol E100 and hydrogen). In the fuel cell category, an official flow meter is used to measure the volume of hydrogen consumed during the attempt. This amount of energy is converted into kilometers per kWh. It is important to notice that in the fuel cell category, battery is not allowed in the car except for safety purposes, i.e. to power the car-horn and the hydrogen sensor.

## Multi-physics modeling of the powertrain

In this section, a dynamical model of the vehicle is proposed. For a set of input parameters (e.g. motor current, road slope, ambient temperature, ...), this model calculates the evolution of the various system states, the losses and the consumption of each subsystem (see Fig. 2).

This model is organized around six main blocks: The mechanical part, the propulsion motor, the power converter, the fuel cell, the stack accessories coupled to a thermal model, and the control laws and supervision. This dynamical model is oriented towards the optimization and must be solved quickly. Thus, only the velocity $v$ and the stack temperature $T_{st}$, which are the slowest state variables, are taken into account. All other dynamics are considered fast enough to be regarded as instantaneous (see Fig. 3).

The urban-concept *Cityjoule* presented in this paper is made of two identical power trains, each supplying one of the two rear wheels. In the next sections, only one power train is studied.

### Fuel cell

The fuel-cell stack has a power rate of 800 W and is composed of 28 cells connected in series, each of them having an active area of 60 cm². A PEMFC is mainly described by its polarization curve. It gives the voltage level $U_{st}$ according to the supplied current $I_{st}$. The electrochemical equations give a relation

<figure><img src="image_2.png" /><figcaption>Fig. 2 – Tank-to-wheel model.</figcaption></figure>
<figure><img src="image_1.png" /><figcaption>**Fig. 1** – The Urban-Concept vehicle Cityjoule.</figcaption></figure>

concept category, which aims to promote the development of vehicles closest to true city cars. The energy source of this vehicle is also a PEMFC. To achieve an energy consumption as low as possible, each element of the vehicle must be optimized, from the mechanical angle (drag coefficient, frontal area, slop, road-wheel friction, weight) to the electrical architecture (power converter, motor efficiencies and fuel cell peripheral elements consumption). This optimization has been done in previous works, but without taking into account the driving strategy [34]. In this paper, the optimal driving strategy that minimizes the car energy demand is presented. It is based on a multi-physics modeling of the power train, taking into account the road profile and the race constraints (average speed, stop at each turn, etc.) [1,5,7,31]. This model is then integrated into a global optimization algorithm, which lead to the best velocity profile, motor current and gearbox ratio.

The paper is structured as follows: In section *Presentation of the race and vehicle characteristics*, the vehicle specifications and the race characteristics are presented. A focus is made on power train architecture from the fuel cell stack to the mechanical needs. In section *Multi-Physics modeling of the powertrain*, a dynamical and multi-physics model is developed. In this section, each part of the power train is presented. An originality of this work is the addition of a thermal modeling of the PEMFC, in order to take into account the auxiliary stack consumption in energy balance, including the air feeding and cooling systems. The optimization results are given in Section *Results*, which allow to achieve a driving strategy minimizing the final energy consumed by the vehicle. Simulations and experimental results are compared to a real race, recorded during the previous edition of the Shell Eco-marathon, in Rotterdam, Netherlands. Conclusions are given in Section *Conclusions*.

## Presentation of the race and vehicle characteristics

The urban-concept car and its associated fuel cell power train were both designed and built to participate in the Shell Eco-Marathon energetic race (Rotterdam, Netherlands). The principle of such a competition is to drive a vehicle with one pilot

the furthest with the lowest quantity of fuel at a minimum average speed of 25 km/h. A minimum weight of 70 kg is imposed for the pilot. Each competitor will have to travel 10 laps of the race track corresponding to 16.1 km in less than 40 min. In the urban-concept category, a stop is mandatory during at each lap. To evaluate prototype car efficiency, the fuel quantity carried on board is measured before and after the attempt. Wide range of fuels can be used (Unleaded gasoline; diesel; LPG; GTL; fatty acid methyl ester; ethanol E100 and hydrogen). In the fuel cell category, an official flow meter is used to measure the volume of hydrogen consumed during the attempt. This amount of energy is converted into kilometers per kWh. It is important to notice that in the fuel cell category, battery is not allowed in the car except for safety purposes, i.e. to power the car-horn and the hydrogen sensor.

## Multi-physics modeling of the powertrain

In this section, a dynamical model of the vehicle is proposed. For a set of input parameters (e.g. motor current, road slope, ambient temperature, ...), this model calculates the evolution of the various system states, the losses and the consumption of each subsystem (see Fig. 2).

This model is organized around six main blocks: The mechanical part, the propulsion motor, the power converter, the fuel cell, the stack accessories coupled to a thermal model, and the control laws and supervision. This dynamical model is oriented towards the optimization and must be solved quickly. Thus, only the velocity $v$ and the stack temperature $T_{st}$, which are the slowest state variables, are taken into account. All other dynamics are considered fast enough to be regarded as instantaneous (see Fig. 3).

The urban-concept Cityjoule presented in this paper is made of two identical power trains, each supplying one of the two rear wheels. In the next sections, only one power train is studied.

### Fuel cell

The fuel-cell stack has a power rate of 800 W and is composed of 28 cells connected in series, each of them having an active area of 60 cm². A PEMFC is mainly described by its polarization curve. It gives the voltage level $U_{st}$ according to the supplied current $I_{st}$. The electrochemical equations give a relation

<figure><img src="image_2.png" /><figcaption>**Fig. 2** – Tank-to-wheel model.</figcaption></figure>
between the output voltage $U_{st}$ (in V) and the stack efficiency $\eta_{stack}$:

$$ \eta_{stack} = \frac{1000 f U_{st}}{n_{cells} L H V_{H_2}} \qquad (1) $$

where f is the Faraday constant (96,485.34 C mol⁻¹), $n_{cells}$ the cells number and $L H V_{H_2}$ the lower heating value of hydrogen (119.96 10⁶ J kg⁻¹). Moreover, the hydrogen flow $Q_{H_2}$ (l/s), i.e. the instantaneous hydrogen consumption, is linked to the output current $I_{st}$:

$$ Q_{H_2} = \frac{I_{st} n_{cells} M_{H_2}}{f \rho_{H_2}} \qquad (2) $$

where $\rho_{H_2}$ is the hydrogen density and $M_{H_2}$ is the molar mass of hydrogen.

## Power converter

The proposed DC/DC power converter used in the power train is a classical asynchronous buck converter, located between the PEMFC and the DC-motor (see Fig. 3).

This model take into account the losses in the power stage (conduction and commutation losses of MOSFET and Schottky diode), in the smooth inductance (copper and magnetic losses) and in the small signal electronic control (voltage regulator, current sensor, ...) [6,28,29]. The proposed model allows the computation of the converter efficiency $\eta_{DCDC}$, knowing the power $P_m$ applied to the electrical motor and the losses $P_{losses}$:

$$ \eta_{DCDC} = \frac{P_m}{P_m + P_{losses}} \qquad (3) $$

with

$$ P_{losses} = P_{mosfet}^{c} + P_{diode}^{c} + P_{sw} + P_{ind}^{c} + P_{ind}^{mag} + P_{supply} \qquad (4) $$

with $P_{mosfet}^{c}$ and $P_{diode}^{c}$ the conduction losses in the power components, $P_{sw}$ the switching losses in the power stage, $P_{supply}$ the control part consumption, $P_{ind}^{c}$ and $P_{ind}^{mag}$ the copper and magnetic losses in the smooth inductance.

Conduction losses in the Schottky diode and the power MOSFET can be easily calculated by considering an ideal waveform of the motor current $i_m$, has given by Fig. 4. Assuming a conduction time $\alpha T$ for the MOSFET, its

<figure><img src="image_3.png" /><figcaption>Fig. 4 – Ideal waveform of the DC-motor current $i_m$ considered for losses calculation.</figcaption></figure>

conduction losses for an average current $I_m$, with a current ripple $\Delta I$, are given by:

$$ P_{mosfet}^{c} = \frac{1}{T} \int_{0}^{\alpha T} R_{ds(on)} i_{m}^{2}(t) dt = R_{ds(on)} \alpha \left( \frac{\Delta I^{2}}{12} + I_{m}^{2} \right) \qquad (5) $$

where $R_{ds(on)}$ is the equivalent resistance of the MOSFET and $\alpha$ the duty cycle, i.e. the ratio between the output voltage of the power stage and the voltage supplied by the fuel-cell. With the same approach, the conduction losses in the Schottky diode are given by:

$$ \begin{aligned} P_{diode}^{c} &= \frac{1}{T} \int_{0}^{T} (U_{do} + R_{d} i_{m}(t)) i_{m}(t) dt \\ &= (1 - \alpha) \left( R_{d} \frac{\Delta I^{2}}{12} + R_{d} I_{m}^{2} + U_{do} I_{m} \right) \end{aligned} \qquad (6) $$

with $U_{do}$ the equivalent drop voltage of the diode and $R_{d}$ its equivalent resistance.

For the switching losses in the power MOSFET and the Schottky diode, the considered waveforms are deliberately simplified and only allow a rough assessment of the switching losses. The used method for the MOSFET losses calculation is based on the gate charge characteristic [23,28–30]. The switch-on commutation starts once the gate voltage exceeds the threshold voltage $V_{gs(th)}$ (see Fig. 5). During fall-time of the MOSFET voltage (generally supposed greater than the current rise time which is generally neglected), the gate-to-source voltage $V_{dr}$ is nearly constant due to the Miller effect. The gate current supplied by the driver is then given by:

<figure><img src="image_4.png" /><figcaption>Fig. 3 – Diagram of the DC/DC power converter.</figcaption></figure>
between the output voltage $U_{st}$ (in V) and the stack efficiency $\eta_{stack}$:

$$ \eta_{stack} = \frac{1000 f U_{st}}{n_{cells} L H V_{H_2}} \quad (1) $$

where $f$ is the Faraday constant (96,485.34 C mol⁻¹), $n_{cells}$ the cells number and $L H V_{H_2}$ the lower heating value of hydrogen (119.96 10⁶ J kg⁻¹). Moreover, the hydrogen flow $Q_{H_2}$ (l/s), i.e. the instantaneous hydrogen consumption, is linked to the output current $I_{st}$:

$$ Q_{H_2} = \frac{I_{st} n_{cells} M_{H_2}}{f \rho_{H_2}} \quad (2) $$

where $\rho_{H_2}$ is the hydrogen density and $M_{H_2}$ is the molar mass of hydrogen.

## Power converter

The proposed DC/DC power converter used in the power train is a classical asynchronous buck converter, located between the PEMFC and the DC-motor (see Fig. 3).

This model take into account the losses in the power stage (conduction and commutation losses of MOSFET and Schottky diode), in the smooth inductance (copper and magnetic losses) and in the small signal electronic control (voltage regulator, current sensor, ...) [6,28,29]. The proposed model allows the computation of the converter efficiency $\eta_{DCDC}$, knowing the power $P_m$ applied to the electrical motor and the losses $P_{losses}$:

$$ \eta_{DCDC} = \frac{P_m}{P_m + P_{losses}} \quad (3) $$

with

$$ P_{losses} = P_{mosfet}^{c} + P_{diode}^{c} + P_{sw} + P_{ind}^{c} + P_{mag}^{ind} + P_{supply} \quad (4) $$

with $P_{mosfet}^{c}$ and $P_{diode}^{c}$ the conduction losses in the power components, $P_{sw}$ the switching losses in the power stage, $P_{supply}$ the control part consumption, $P_{ind}^{c}$ and $P_{mag}^{ind}$ the copper and magnetic losses in the smooth inductance.

Conduction losses in the Schottky diode and the power MOSFET can be easily calculated by considering an ideal waveform of the motor current $i_m$, has given by Fig. 4. Assuming a conduction time $\alpha T$ for the MOSFET, its

<figure><img src="image_3.png" /><figcaption>Fig. 4 – Ideal waveform of the DC-motor current $i_m$ considered for losses calculation.</figcaption></figure>

conduction losses for an average current $I_m$, with a current ripple $\Delta I$, are given by:

$$ P_{mosfet}^{c} = \frac{1}{T} \int_{0}^{\alpha T} R_{ds(on)} i_{m}^{2}(t) dt = R_{ds(on)} \alpha \left( \frac{\Delta I^{2}}{12} + I_{m}^{2} \right) \quad (5) $$

where $R_{ds(on)}$ is the equivalent resistance of the MOSFET and $\alpha$ the duty cycle, i.e. the ratio between the output voltage of the power stage and the voltage supplied by the fuel-cell. With the same approach, the conduction losses in the Schottky diode are given by:

$$ \begin{aligned} P_{diode}^{c} &= \frac{1}{T} \int_{0}^{T} (U_{do} + R_{d} i_{m}(t)) i_{m}(t) dt \\ &= (1 - \alpha) \left( R_{d} \frac{\Delta I^{2}}{12} + R_{d} I_{m}^{2} + U_{do} I_{m} \right) \end{aligned} \quad (6) $$

with $U_{do}$ the equivalent drop voltage of the diode and $R_{d}$ its equivalent resistance.

For the switching losses in the power MOSFET and the Schottky diode, the considered waveforms are deliberately simplified and only allow a rough assessment of the switching losses. The used method for the MOSFET losses calculation is based on the gate charge characteristic [23,28–30]. The switch-on commutation starts once the gate voltage exceeds the threshold voltage $V_{gs(th)}$ (see Fig. 5). During fall-time of the MOSFET voltage (generally supposed greater than the current rise time which is generally neglected), the gate-to-source voltage $V_{dr}$ is nearly constant due to the Miller effect. The gate current supplied by the driver is then given by:

<figure><img src="image_4.png" /><figcaption>Fig. 3 – Diagram of the DC/DC power converter.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 5 – Total gate charge Q<sub>sw</sub> during a commutation for the nominal current (from the IRFZ44N MOSFET datasheet).</figcaption></figure>

$$I_g = \frac{V_{dr} - V_{gs(plateau)}}{R_g} \qquad (7)$$

where R<sub>g</sub> is the total gate resistance and V<sub>gs(plateau)</sub> is the value of the gate voltage during the switch-on transient, obtained from datasheet as illustrated by Fig. 5. The total gate charge Q<sub>sw</sub> can also be deduced from this figure and finally, the switch-on time t<sub>on</sub> can be obtained with:

$$t_{on} = \frac{Q_{sw}}{I_g} = R_g \frac{Q_{sw}}{V_{dr} - V_{gs(plateau)}} \qquad (8)$$

The switch-off time t<sub>off</sub> calculation is obtained from the same Eqs. (7) And (9), but this time with a null driver voltage V<sub>d</sub>r:

$$t_{off} = R_g \frac{Q_{sw}}{V_{gs(plateau)}} \qquad (9)$$

Now, the energy losses can be calculated, considering linear variations of the voltage and the current during the rise and fall times. At the switch-on time, the load current is equal to I<sub>m</sub> + ΔI. The switch-on energy is given by:

$$E_{on} = \frac{1}{2} (I_m + \Delta I) U_{st} t_{on} \qquad (10)$$

At the switch-off time, the load current is equal to I<sub>m</sub> - ΔI and the switch-off energy is then given by:

$$E_{on} = \frac{1}{2} (I_m - \Delta I) U_{st} t_{off} \qquad (11)$$

Finally, the switching losses can be evaluated, knowing the switching frequency f<sub>sw</sub>:

$$P_{sw} = (E_{on} + E_{off}) f_{sw} \qquad (12)$$

For the coil inductance, used to smooth the current in the ironless motor, the losses occur in the aluminium winding (to reduce the weight of the coil) and in the iron core. With a current waveform as shown in Fig. 4 and with an equivalent serial resistance r<sub>L</sub>, these losses are given by:

$$P_{ind}^{c} = \frac{1}{T} \int_{0}^{T} r_{L} I_{m}^{2}(t) dt = R_{L} \left( \frac{\Delta I^{2}}{12} + I_{m}^{2} \right) \qquad (13)$$

For the core losses, hysteresis and eddy current have to be taken into account. But due to the high complexity to find analytical formulation, manufacturers typically used the Steinmetz relationship which only need three parameters and can be practically identified [26]. The conventional formulation of this relationship is obtained from the magnetic loss per unit, expressed in W/m³ and given by:

$$p_{v mag} = k_m f_{sw}^{a} \Delta I^{b} \qquad (14)$$

where k<sub>m</sub>, a and b are parameters of the material. Knowing the core volume V<sub>core</sub>, core losses are obtained by:

$$P_{ind}^{mag} = V_{core} p_{v mag} \qquad (15)$$

Finally, the small signal electronic part consumption is taken into account by a constant power value P<sub>ctrl</sub> and the Hall effect current sensor consumption P<sub>cs</sub> is taken into account with a power proportional to the motor current I<sub>m</sub>:

$$P_{supply} = P_{ctrl} + P_{cs} = k_{cs} V_{cs} |I_{m}| + P_{ctrl} \qquad (16)$$

where V<sub>cs</sub> is the supply voltage of the current sensor and k<sub>cs</sub> the ratio between the measured current and the small signal current coming from the Hall-effect sensor.

## Ironless DC-motor

The power converter supply a 200 W brushed DC motor with a nominal voltage of 24 V (MAXON RE50). The choice of this motor is linked to its high efficiency (above 94%) despite its low power rating. Moreover, the DC motor is iron-less, thus without hysteresis and eddy current losses. The losses in the DC motor are: the dry and viscous friction on the shaft, the Joule losses in the wiring and the voltages drop in the brushes. The expression of the motor efficiency can then be obtained from the following equation:

$$\eta_{motor} = \frac{k_{\phi} \omega_{mot} (I_{m} - (k_{v} \omega_{mot} + k_{d}))}{(k_{\phi} \omega_{mot} + r I_{m} + E_{b}) I_{m}} \qquad (17)$$

where k<sub>φ</sub> is the torque constant (Nm A<sup>-1</sup>), ω<sub>mot</sub> is the rotation speed (rad s<sup>-1</sup>), I<sub>m</sub> is the motor current (in A), k<sub>v</sub> is the viscous friction constant (in A rad<sup>-1</sup> s), k<sub>d</sub> is the dry friction constant (in A), r is the wiring resistance (in Ω) and E<sub>b</sub> is the voltage drop in the brushes (in V).

## Mechanical modeling

In order to obtain the dynamical behavior of the vehicle, the mechanical equations must be solved. To run, the vehicle has to overcome the efforts due to the aerodynamic F<sub>aero</sub>, the slope F<sub>slope</sub> and the rolling resistance due to friction between the wheel and the road F<sub>rr</sub>. The fundamental law of the mechanic gives the next equation to solve (see Fig. 6) [31]:

$$m \frac{dv(t)}{dt} = F_w - F_{rr} - F_{slope} - F_{aero} - F_{brake} \qquad (18)$$
<figure><img src="image_5.png" /><figcaption>Fig. 5 – Total gate charge Qsw during a commutation for the nominal current (from the IRFZ44N MOSFET datasheet).</figcaption></figure>

$$I_g = \frac{V_{dr} - V_{gs(plateau)}}{R_g} \qquad (7)$$

where R_g is the total gate resistance and Vgs(plateau) is the value of the gate voltage during the switch-on transient, obtained from datasheet as illustrated by Fig. 5. The total gate charge Qsw can also be deduced from this figure and finally, the switch-on time t_on can be obtained with:

$$t_{on} = \frac{Q_{sw}}{I_{g}} = R_{g} \frac{Q_{sw}}{V_{dr} - V_{gs(plateau)}} \qquad (8)$$

The switch-off time t_off calculation is obtained from the same Eqs. (7) And (9), but this time with a null driver voltage V_d:

$$t_{off} = R_g \frac{Q_{sw}}{V_{gs(plateau)}} \qquad (9)$$

Now, the energy losses can be calculated, considering linear variations of the voltage and the current during the rise and fall times. At the switch-on time, the load current is equal to I_m + ΔI. The switch-on energy is given by:

$$E_{on} = \frac{1}{2}(I_{m} + \Delta I)U_{st}t_{on} \qquad (10)$$

At the switch-off time, the load current is equal to I_m - ΔI and the switch-off energy is then given by:

$$E_{on} = \frac{1}{2}(I_{m} - \Delta I)U_{st}t_{off} \qquad (11)$$

Finally, the switching losses can be evaluated, knowing the switching frequency f_sw:

$$P_{sw} = (E_{on} + E_{off})f_{sw} \qquad (12)$$

For the coil inductance, used to smooth the current in the ironless motor, the losses occur in the aluminium winding (to reduce the weight of the coil) and in the iron core. With a current waveform as shown in Fig. 4 and with an equivalent serial resistance r_L, these losses are given by:

$$P_{ind}^{c} = \frac{1}{T} \int_{0}^{T} r_{L} I_{m}^{2}(t) dt = R_{L} \left( \frac{\Delta I^{2}}{12} + I_{m}^{2} \right) \qquad (13)$$

For the core losses, hysteresis and eddy current have to be taken into account. But due to the high complexity to find analytical formulation, manufacturers typically used the Steinmetz relationship which only need three parameters and can be practically identified [26]. The conventional formulation of this relationship is obtained from the magnetic loss per unit, expressed in W/m³ and given by:

$$p_{v mag} = k_{m} f_{sw}^{a} \Delta I^{b} \qquad (14)$$

where k_m, a and b are parameters of the material. Knowing the core volume V_core, core losses are obtained by:

$$P_{ind}^{mag} = V_{core} p_{v mag} \qquad (15)$$

Finally, the small signal electronic part consumption is taken into account by a constant power value PCtrl and the Hall effect current sensor consumption P_cs is taken into account with a power proportional to the motor current I_m:

$$P_{supply} = P_{ctrl} + P_{cs} = k_{cs} V_{cs} |I_{m}| + P_{ctrl} \qquad (16)$$

where V_cs is the supply voltage of the current sensor and k_cs the ratio between the measured current and the small signal current coming from the Hall-effect sensor.

## Ironless DC-motor

The power converter supply a 200 W brushed DC motor with a nominal voltage of 24 V (MAXON RE50). The choice of this motor is linked to its high efficiency (above 94%) despite its low power rating. Moreover, the DC motor is iron-less, thus without hysteresis and eddy current losses. The losses in the DC motor are: the dry and viscous friction on the shaft, the Joule losses in the wiring and the voltages drop in the brushes. The expression of the motor efficiency can then be obtained from the following equation:

$$\eta_{motor} = \frac{k_{\phi} \omega_{mot} (I_{m} - (k_{v} \omega_{mot} + k_{d}))}{(k_{\phi} \omega_{mot} + r I_{m} + E_{b}) I_{m}} \qquad (17)$$

where k_φ is the torque constant (Nm A⁻¹), ω_mot is the rotation speed (rad s⁻¹), I_m is the motor current (in A), k_v is the viscous friction constant (in A rad⁻¹ s), k_d is the dry friction constant (in A), r is the wiring resistance (in Ω) and E_b is the voltage drop in the brushes (in V).

## Mechanical modeling

In order to obtain the dynamical behavior of the vehicle, the mechanical equations must be solved. To run, the vehicle has to overcome the efforts due to the aerodynamic F_aero, the slope F_slope and the rolling resistance due to friction between the wheel and the road F_r. The fundamental law of the mechanic gives the next equation to solve (see Fig. 6) [31]:

$$m \frac{dv(t)}{dt} = F_w - F_{rr} - F_{slope} - F_{aero} - F_{brake} \qquad (18)$$
<figure><img src="image_6.png" /><figcaption>Fig. 6 – Dynamical model of the mechanical part of the vehicle.</figcaption></figure>

where m is the mass of the vehicle, F<sub>w</sub> is the propulsion force applied to wheel through the electrical motor and F<sub>brake</sub> the mechanical braking force.

The aerodynamic force F<sub>aero</sub> is computed from the me-
chanical parameters of the vehicle and the wind speed v<sub>wind</sub> :

$$
F_{\text{aero}} = \frac{1}{2} \rho_{\text{air}} S C_x (v(t) + v_{\text{wind}})^2 \quad (19)
$$

with ρ<sub>air</sub> the mass density of air, C<sub>x</sub> the drag coefficient, S the
frontal area, v(t) and v<sub>wind</sub> respectively the vehicle and wind
velocities (in m s<sup>-1</sup>). The forces due to the rolling resistance
and the road inclination are calculated from the mass m of the
vehicle and the slope r<sub>slope</sub> given in %:

$$
F_{rr} = m g f_r \cos(\arctan(r_{\text{slope}})) \quad (20)
$$

$$
F_{\text{slope}} = m g \sin(\arctan(r_{\text{slope}})) \quad (21)
$$

The total drive Force F<sub>w</sub> applied through both rear wheels is
calculated from the motor torque Γ<sub>m</sub>, such as:

$$
F_w = \eta_g \frac{r_{\text{gear}}}{R} \Gamma_m \qquad (22)
$$

where R is the wheel radius, r<sub>gear</sub> the gearbox ratio of the transmission and η<sub>g</sub> its efficiency.

Thermal model of the fuel cell stack

The typical nominal power of a powertrain for an urban-
concept car is below 1 kW. For such relatively low power
need, the auxiliary devices of the system play a crucial role in
the total energy consumption. It is well known that fuel cell
stack auxiliaries cause a drop in system efficiency at low
power. It is therefore necessary to take their consumption into
account [3,4,7,15,16]. For this vehicle, the accessories of the
fuel cell are limited to an air blower and a cooling fan. The
operation and the consumption of these accessories are pre-
sented in this section. The thermal source P<sub>th src</sub> due to the
irreversibilities of the electrochemical reaction, is removed
through many different paths. In this work, it is considered
that thermal removal is achieved by the air flow rate in fuel
cell channels P<sub>th ab</sub>, the liquid-vapor phase change P<sub>th stm</sub> of the

water produced by the electrochemical reaction of the fuel
cell, the natural convection P<sub>th nat</sub> and the forced convection
assured by the cooling blower P<sub>th cb</sub>. Thus, a dynamical ther-
mal model can be expressed by next equation [19,20]:

$$
m_{fc} C_p \frac{dT_{st}}{dt} = P_{th src} - P_{th ab} - P_{th stm} - P_{th nat} - P_{th cb} \quad (23)
$$

The thermal source P<sub>th src</sub> is calculated from the total power
supplied by the PEMFC and its polarization curve:

$$
P_{\text{th src}} = (n_{\text{cells}} E_{\text{Nernst}} - U_{\text{st}}) I_{\text{st}} \quad (24)
$$

where E<sub>Nernst</sub> is the reversible voltage of a cell, calculated from
the Nernst equation [22,25] and n<sub>cells</sub> the number of cells.

The air flow in the air channel, produced by the air blower,
permits to remove a part of the heat by forced convection. The
air flow Q<sub>v ab</sub> (in l min<sup>-1</sup>), for given Stoechiometric-ratio S<sub>c</sub> and
stack current I<sub>st</sub>, is given by next equation:

$$
Q_{v ab} = 60000 \frac{5R_c(273.15 + T_{st})}{4P_a f} I_{st} n_{\text{cells}} S_c \quad (25)
$$

with R<sub>c</sub> the ideal gas constant (8.31 J mol<sup>-1</sup> K<sup>-1</sup>, T<sub>st</sub>) the stack temperature (in °C) and P<sub>a</sub> the atmospheric pressure (101,300 Pa). By assuming a perfect exchange between the air channel and the stack, the output of air temperature is equal to the mean stack temperature (T<sub>st</sub>). Moreover, the input air temperature is equal to the ambient T<sub>amb</sub>. Then the evacuated thermal power is:

$$
P_{\text{th ab}} = \frac{1}{60} \rho_{\text{air}} Q_{\text{air}} C_{\text{pair}} (T_{\text{st}} - T_{\text{amb}}) \quad (26)
$$

where ρ<sub>air</sub> is the air density (1.2 kg m<sup>-3</sup>), C<sub>pair</sub> its specific heat (1000 J K<sup>-1</sup> kg<sup>-1</sup>).

The transport of water through the different fuel cell
components (membrane, diffusion layers and channels) is still
being debated by the scientific community. However, with the
emergence of imaging techniques, such as X-ray or Neutron
radiography, the co-existence of vapor and liquid water within
the fuel cell has been emphasized by several authors
[2,9,13,14,21]. In presence of saturated gas in cathode chan-
nels, the water produced in vapor at the catalyst layer flows
through the microporous layer before to condensate in the
<figure><img src="image_6.png" /><figcaption>Fig. 6 – Dynamical model of the mechanical part of the vehicle.</figcaption></figure>

where m is the mass of the vehicle, F<sub>w</sub> is the propulsion force applied to wheel through the electrical motor and F<sub>brake</sub> the mechanical braking force.

The aerodynamic force F<sub>aero</sub> is computed from the me-
chanical parameters of the vehicle and the wind speed v<sub>wind</sub> :

$$
F_{\text{aero}} = \frac{1}{2} \rho_{\text{air}} S C_x (v(t) + v_{\text{wind}})^2 \quad (19)
$$

with ρ<sub>air</sub> the mass density of air, C<sub>x</sub> the drag coefficient, S the
frontal area, v(t) and v<sub>wind</sub> respectively the vehicle and wind
velocities (in m s<sup>-1</sup>). The forces due to the rolling resistance
and the road inclination are calculated from the mass m of the
vehicle and the slope r<sub>slope</sub> given in %:

$$
F_{rr} = m g f_r \cos(\arctan(r_{\text{slope}})) \quad (20)
$$

$$
F_{\text{slope}} = m g \sin(\arctan(r_{\text{slope}})) \quad (21)
$$

The total drive Force F<sub>w</sub> applied through both rear wheels is
calculated from the motor torque Γ<sub>m</sub>, such as:

$$
F_w = \eta_g \frac{r_{\text{gear}}}{R} \Gamma_m \qquad (22)
$$

where R is the wheel radius, r<sub>gear</sub> the gearbox ratio of the transmission and η<sub>g</sub> its efficiency.

Thermal model of the fuel cell stack

The typical nominal power of a powertrain for an urban-
concept car is below 1 kW. For such relatively low power
need, the auxiliary devices of the system play a crucial role in
the total energy consumption. It is well known that fuel cell
stack auxiliaries cause a drop in system efficiency at low
power. It is therefore necessary to take their consumption into
account [3,4,7,15,16]. For this vehicle, the accessories of the
fuel cell are limited to an air blower and a cooling fan. The
operation and the consumption of these accessories are pre-
sented in this section. The thermal source P<sub>th src</sub> due to the
irreversibilities of the electrochemical reaction, is removed
through many different paths. In this work, it is considered
that thermal removal is achieved by the air flow rate in fuel
cell channels P<sub>th ab</sub>, the liquid-vapor phase change P<sub>th stm</sub> of the

water produced by the electrochemical reaction of the fuel
cell, the natural convection P<sub>th nat</sub> and the forced convection
assured by the cooling blower P<sub>th cb</sub>. Thus, a dynamical ther-
mal model can be expressed by next equation [19,20]:

$$
m_{fc} C_p \frac{dT_{st}}{dt} = P_{th src} - P_{th ab} - P_{th stm} - P_{th nat} - P_{th cb} \quad (23)
$$

The thermal source P<sub>th src</sub> is calculated from the total power
supplied by the PEMFC and its polarization curve:

$$
P_{\text{th src}} = (n_{\text{cells}} E_{\text{Nernst}} - U_{\text{st}}) I_{\text{st}} \quad (24)
$$

where E<sub>Nernst</sub> is the reversible voltage of a cell, calculated from
the Nernst equation [22,25] and n<sub>cells</sub> the number of cells.

The air flow in the air channel, produced by the air blower,
permits to remove a part of the heat by forced convection. The
air flow Q<sub>v ab</sub> (in l min<sup>-1</sup>), for given Stoechiometric-ratio S<sub>c</sub> and
stack current I<sub>st</sub>, is given by next equation:

$$
Q_{v ab} = 60000 \frac{5R_c(273.15 + T_{st})}{4P_a f} I_{st} n_{\text{cells}} S_c \quad (25)
$$

with R<sub>c</sub> the ideal gas constant (8.31 J mol<sup>-1</sup> K<sup>-1</sup>, T<sub>st</sub>) the stack
temperature (in °C) and P<sub>a</sub> the atmospheric pressure
(101,300 Pa). By assuming a perfect exchange between the air
channel and the stack, the output of air temperature is equal
to the mean stack temperature (T<sub>st</sub>). Moreover, the input air
temperature is equal to the ambient T<sub>amb</sub>. Then the evacuated
thermal power is:

$$
P_{\text{th ab}} = \frac{1}{60} \rho_{\text{air}} Q_{\text{air}} C_{\text{pair}} (T_{\text{st}} - T_{\text{amb}}) \quad (26)
$$

where ρ<sub>air</sub> is the air density (1.2 kg m<sup>-3</sup>), C<sub>pair</sub> its specific heat (1000 J K<sup>-1</sup> kg<sup>-1</sup>).

The transport of water through the different fuel cell
components (membrane, diffusion layers and channels) is still
being debated by the scientific community. However, with the
emergence of imaging techniques, such as X-ray or Neutron
radiography, the co-existence of vapor and liquid water within
the fuel cell has been emphasized by several authors
[2,9,13,14,21]. In presence of saturated gas in cathode chan-
nels, the water produced in vapor at the catalyst layer flows
through the microporous layer before to condensate in the
macro-porous layer. The liquid water is then transported under capillary fingering regime and removed toward the channels in an eruptive regime [14,21]. Nevertheless, the particular fuel cell operating conditions used in this study, i.e. low current density, high air stoichiometry, low relative humidity (ambient air) and dead-ended anode, may lead to significantly different transport regime. As a result, Lee et al. [18] reported an absence of liquid water in the channels of a fuel cell operating in similar conditions. According to this, the present model assumed that the liquid water presents in the GDL is evaporated and drained through the channel in vapor form. Thus, the power drained by the liquid-vapor phase change is given by:

$$P_{\text{th stm}} = \dot{m}_{H_2O} L_{H_2O} \qquad (27)$$

where $\dot{m}_{H_2O}$ is the water mass flow-rate (g s⁻¹) evacuated through the air channel and $L_{H_2O}$ the latent heat of vaporization (2500 kJ kg⁻¹). The amount of water produced by the reaction depends only on the stack current $I_{st}$ provided by the fuel cell. The flow-rate $\dot{m}_{H_2O}$ is then given by the following equation:

$$\dot{m}_{H_2O} = n_{\text{cells}} \frac{R_c(273.15 + T_{st})}{4P_a F} I_{st} \qquad (28)$$

The natural convection exchange coefficient is deduced from experimental measurements and expressed as:

$$P_{\text{th nat}} = k_{\text{nat}}(T_{\text{st}} - T_{\text{amb}}) \qquad (29)$$

The last term of Eq. (23) is the forced convection by the cooling blower. For the studied fuel cell stack, each cell is separated from the other by a corrugated iron sheet. Knowing the air flow-rate $\dot{m}_{cb}$ (attention, $Q_{v cb}$ après) and considering a perfect exchange with the fuel cell, the thermal power removed by the cooling blower is given by:

$$P_{\text{th cb}} = \rho_{\text{air}} Q_{v cb} C_p \text{ air} (T_{\text{st}} - T_{\text{amb}}) \qquad (30)$$

Now that each part of the thermal model was described, its coupling with the electrical model of the fuel cell stack and accessories must be done. From the stack current $I_{st}$ and temperature $T_{st}$, Eq. (26), (27) and (29) can be directly calculated. For Eq. (30), the air flow rate is an input parameter for the model. In this work, it is considered that the cooling blower is controlled through its voltage $U_{cb}$. The same assumption is made for the control of the air blower ($U_{ab}$).

The air flow rate of the cooling blower and air feeding fan are measured on a fuel-cell stack made of 28 cells (31 cm² of active area) manufactured by the society MES [24]. The cooling system is a 12 V radial blower, Micronel U97 and the air feeding fan is a 12 V axial blower, Micronel D484.

Measurements on this fuel cell assembly allow to deduce the airflow and power consumption of the auxiliaries, as shown in Fig. 7.

For both blower, a least squares regression is used to fit the experimental data:

$$Q_{v ab} = 0.75 U_{ab} \text{ and } P_{ab} = 4.0 U_{ab}^3 \qquad (31)$$

$$Q_{v cb} = 4.00 U_{cb} \text{ and } P_{cb} = 7.5 U_{cb}^3 \qquad (32)$$

## Driving and auxiliary management laws

The last part of the model to be detailed is the supervision with the control laws. These functions are performed by a microcontroller ATMEGA 128. This system receives and processes various information such as the setting motor current, stack temperature, fuel cell current, vehicle velocity, etc.

For the cooling regulation, an hysteresis controller is used. The cooling blower is turned on with a given voltage value, if the temperature exceed a threshold value $T_{max}$ and while it remains greater than a lower value $T_{min}$. Even if this control law is not optimal, it has the advantage of being extremely robust with respect to changes in operating conditions (stack losses, ambient temperature, ...). Further, to smooth the power supplied by the stack, the cooling blower is supplied only during the free-wheel phase (i.e. deceleration phase), when the electrical motor is turned off and the vehicle consumption is minimum (see Fig. 8).

For the driving strategy, each lap of the race can be decomposed in three sequences. The first one is a starting phase from 0 km/h to an intermediate velocity $v_{min}$, with a constant motor current $I_{start}$. During this phase, the acceleration should be high enough so as not to lose too much time for the rest of the lap. In contrast, for a too high torque applied to the wheel, the power train efficiency could be greatly reduced.

The second phase is a constant speed strategy. The objective is to maintain an average speed as constant as possible, with the greatest possible efficiency. In Ref. [34], it is shown that the optimum efficiency is obtained by performing a series of acceleration and free wheel. The acceleration phases are carried out with the optimum torque, which maximizing the energy efficiency of the power chain. The parameters to be optimized are the motor current $I_{acc}$ applied during the acceleration phases and speed hysteresis thresholds $v_{min}$ and $v_{max}$. The optimization of these parameters must improve the global energetic performance of the vehicle.

## Formulation of the optimization problem

For the optimization of this problem, a standard formulation is used. The objective function can be expressed has the minimization of the fuel consumption on a full race (10 laps), under constraint of a minimum average speed of 25 km/h. The tank-to-wheel model illustrated by Fig. 2 is implemented in Matlab. The state-space variables taken into account are the fuel consumption $m_{H_2}$, the velocity $v(t)$ and the stack temperature $T_{st}(t)$, according to Eqs. (18) and (23). Both equations are computed using a 4th order Runge-Kutta solver for one lap and using a relaxation method to reach a thermal equilibrium. Indeed, it is easier and safer to rely on a nominal thermal operating point. The problem is then formulated as an ordinary differential equation (ODE) such as:

$$\frac{dx}{dt} = f(x, t), \text{ with } x(0) = x_0 \qquad (33)$$

where the state space vector $x(t) = [v, T_{st}, m_{H_2}]^T$. The relaxation method intends to reinject the final temperature value of the simulation (one lap here) as the initial value of the next calculation. To accelerate the convergence, the initial temperature is chosen equal to the lower bound of the
macro-porous layer. The liquid water is then transported under capillary fingering regime and removed toward the channels in an eruptive regime [14,21]. Nevertheless, the particular fuel cell operating conditions used in this study, i.e. low current density, high air stoichiometry, low relative humidity (ambient air) and dead-ended anode, may lead to significantly different transport regime. As a result, Lee et al. [18] reported an absence of liquid water in the channels of a fuel cell operating in similar conditions. According to this, the present model assumed that the liquid water presents in the GDL is evaporated and drained through the channel in vapor form. Thus, the power drained by the liquid-vapor phase change is given by:

$$P_{\text{th stm}} = \dot{m}_{H_2O} L_{H_2O} \qquad (27)$$

where $\dot{m}_{H_2O}$ is the water mass flow-rate (g s⁻¹) evacuated through the air channel and $L_{H_2O}$ the latent heat of vaporization (2500 kJ kg⁻¹). The amount of water produced by the reaction depends only on the stack current $I_{st}$ provided by the fuel cell. The flow-rate $\dot{m}_{H_2O}$ is then given by the following equation:

$$\dot{m}_{H_2O} = n_{\text{cells}} \frac{R_c(273.15 + T_{\text{st}})}{4P_a F} I_{\text{st}} \qquad (28)$$

The natural convection exchange coefficient is deduced from experimental measurements and expressed as:

$$P_{\text{th nat}} = k_{\text{nat}}(T_{\text{st}} - T_{\text{amb}}) \qquad (29)$$

The last term of Eq. (23) is the forced convection by the cooling blower. For the studied fuel cell stack, each cell is separated from the other by a corrugated iron sheet. Knowing the air flow-rate $\dot{m}_{cb}$ (attention, $Q_{v cb}$ après) and considering a perfect exchange with the fuel cell, the thermal power removed by the cooling blower is given by:

$$P_{\text{th cb}} = \rho_{\text{air}} Q_{v cb} C_p \text{ air} (T_{\text{st}} - T_{\text{amb}}) \qquad (30)$$

Now that each part of the thermal model was described, its coupling with the electrical model of the fuel cell stack and accessories must be done. From the stack current $I_{st}$ and temperature $T_{st}$, Eq. (26), (27) and (29) can be directly calculated. For Eq. (30), the air flow rate is an input parameter for the model. In this work, it is considered that the cooling blower is controlled through its voltage $U_{cb}$. The same assumption is made for the control of the air blower ($U_{ab}$).

The air flow rate of the cooling blower and air feeding fan are measured on a fuel-cell stack made of 28 cells (31 cm² of active area) manufactured by the society MES [24]. The cooling system is a 12 V radial blower, Micronel U97 and the air feeding fan is a 12 V axial blower, Micronel D484.

Measurements on this fuel cell assembly allow to deduce the airflow and power consumption of the auxiliaries, as shown in Fig. 7.

For both blower, a least squares regression is used to fit the experimental data:

$$Q_{v ab} = 0.75 U_{ab} \text{ and } P_{ab} = 4.0 U_{ab}^3 \qquad (31)$$

$$Q_{v cb} = 4.00 U_{cb} \text{ and } P_{cb} = 7.5 U_{cb}^3 \qquad (32)$$

## Driving and auxiliary management laws

The last part of the model to be detailed is the supervision with the control laws. These functions are performed by a microcontroller ATMEGA 128. This system receives and processes various information such as the setting motor current, stack temperature, fuel cell current, vehicle velocity, etc.

For the cooling regulation, an hysteresis controller is used. The cooling blower is turned on with a given voltage value, if the temperature exceed a threshold value $T_{max}$ and while it remains greater than a lower value $T_{min}$. Even if this control law is not optimal, it has the advantage of being extremely robust with respect to changes in operating conditions (stack losses, ambient temperature, ...). Further, to smooth the power supplied by the stack, the cooling blower is supplied only during the free-wheel phase (i.e. deceleration phase), when the electrical motor is turned off and the vehicle consumption is minimum (see Fig. 8).

For the driving strategy, each lap of the race can be decomposed in three sequences. The first one is a starting phase from 0 km/h to an intermediate velocity $v_{min}$, with a constant motor current $I_{start}$. During this phase, the acceleration should be high enough so as not to lose too much time for the rest of the lap. In contrast, for a too high torque applied to the wheel, the power train efficiency could be greatly reduced.

The second phase is a constant speed strategy. The objective is to maintain an average speed as constant as possible, with the greatest possible efficiency. In Ref. [34], it is shown that the optimum efficiency is obtained by performing a series of acceleration and free wheel. The acceleration phases are carried out with the optimum torque, which maximizing the energy efficiency of the power chain. The parameters to be optimized are the motor current $I_{acc}$ applied during the acceleration phases and speed hysteresis thresholds $v_{min}$ and $v_{max}$. The optimization of these parameters must improve the global energetic performance of the vehicle.

## Formulation of the optimization problem

For the optimization of this problem, a standard formulation is used. The objective function can be expressed has the minimization of the fuel consumption on a full race (10 laps), under constraint of a minimum average speed of 25 km/h. The tank-to-wheel model illustrated by Fig. 2 is implemented in Matlab. The state-space variables taken into account are the fuel consumption $m_{H_2}$, the velocity $v(t)$ and the stack temperature $T_{st}(t)$, according to Eqs. (18) and (23). Both equations are computed using a 4th order Runge-Kutta solver for one lap and using a relaxation method to reach a thermal equilibrium. Indeed, it is easier and safer to rely on a nominal thermal operating point. The problem is then formulated as an ordinary differential equation (ODE) such as:

$$\frac{dx}{dt} = f(x, t), \text{ with } x(0) = x_0 \qquad (33)$$

where the state space vector $x(t) = [v, T_{st}, m_{H_2}]^T$. The relaxation method intends to reinject the final temperature value of the simulation (one lap here) as the initial value of the next calculation. To accelerate the convergence, the initial temperature is chosen equal to the lower bound of the
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7 – Characteristics of the cooling and air blowers.</figcaption></figure>

regulator (around 40 °C). The steady state is achieved in just a few iterations.

For the optimization, a particle swarm algorithm is used with five input parameters (see Fig. 8), which are the motor currents $I_{start}$ and $I_{acc}$, the maximum speed $v_{max}$, the speed ripple $\Delta V$ which is the difference between $v_{max}$ and $v_{min}$, and the gearbox ratio $r_{gear}$ between the electric motor and the wheel. The tuning of the particle swarm algorithm is classical, with a population of 50 particles and a maximum number of iterations of 200 [12]. To sum up, the optimization problem can be formulated such as:

Minimize: $g(p) = (\text{fuel consumption})$

with: $p = \{I_{start}; I_{acc}; v_{max}; \Delta V; r_{gear}\}$

subject to: $\text{mean}(v(t)) \ge 25\text{km/h}$ and $\text{max}(T_{st}(t)) < 45^{\circ}\text{C}$

Both constraints are evaluated at the end of each simulation. If a constraint is not respected, the solution is degraded by the addition of a weight on the objective function:

$$g(p) \leftarrow g(p) + 2^{\frac{25-\text{mean}(v)}{f_v}} + 2^{\frac{\text{max}(T_{st})-45}{f_T}} \quad (34)$$

where $f_v$ and $f_T$ are the setting weights on the velocity and temperatures respectively.

## Results

The optimization results for the runs of the shell eco marathon in Rotterdam is summarized by Table 1. This optimization takes into account the slope of the road and is obtained for an ambient temperature of 25°C. The list of the vehicle parameters, used for the optimization, is given by Table 3. As shown in Fig. 9, the vehicle is launch in 30 secondes (0–25 km/
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7 – Characteristics of the cooling and air blowers.</figcaption></figure>

regulator (around 40 °C). The steady state is achieved in just a few iterations.

For the optimization, a particle swarm algorithm is used with five input parameters (see Fig. 8), which are the motor currents $I_{start}$ and $I_{acc}$, the maximum speed $v_{max}$, the speed ripple $\Delta V$ which is the difference between $v_{max}$ and $v_{min}$, and the gearbox ratio $r_{gear}$ between the electric motor and the wheel. The tuning of the particle swarm algorithm is classical, with a population of 50 particles and a maximum number of iterations of 200 [12]. To sum up, the optimization problem can be formulated such as:

Minimize: $g(\mathbf{p}) = (\text{fuel consumption})$

with: $\mathbf{p} = \{I_{start}; I_{acc}; v_{max}; \Delta V; r_{gear}\}$

subject to: $\text{mean}(v(t)) \ge 25\text{km/h}$ and $\text{max}(T_{st}(t)) < 45^{\circ}\text{C}$

Both constraints are evaluated at the end of each simulation. If a constraint is not respected, the solution is degraded by the addition of a weight on the objective function:

$$g(\mathbf{p}) \leftarrow g(\mathbf{p}) + 2^{\frac{25-\text{mean}(v)}{v}} + 2^{\frac{\text{max}(T_{st})-45}{T}} \quad (34)$$

where $f_v$ and $f_T$ are the setting weights on the velocity and temperatures respectively.

## Results

The optimization results for the runs of the shell eco marathon in Rotterdam is summarized by Table 1. This optimization takes into account the slope of the road and is obtained for an ambient temperature of 25°C. The list of the vehicle parameters, used for the optimization, is given by Table 3. As shown in Fig. 9, the vehicle is launch in 30 secondes (0–25 km/
<figure><img src="image_9.png" /><figcaption>Fig. 8 – Typical speed, current and temperature profiles for each lap.</figcaption></figure>

<table><caption>Table 1 – Optimization result.</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>I<sub>start</sub></td><td>10.8 A</td></tr><tr><td>I<sub>acc</sub></td><td>3.1 A</td></tr><tr><td>v<sub>max</sub></td><td>27.8 km/h</td></tr><tr><td>ΔV</td><td>1.6 km/h</td></tr><tr><td>r<sub>gear</sub></td><td>200/14</td></tr><tr><td>Average velocity</td><td>25.0 km/h</td></tr><tr><td>Fuel cell losses</td><td>3.04 Wh/lap</td></tr><tr><td>Motor losses</td><td>0.39 Wh/lap</td></tr><tr><td>DC-DC converter losses</td><td>0.23 Wh/lap</td></tr><tr><td>Mechanical transmission losses</td><td>0.11 Wh/lap</td></tr><tr><td>Supervision system consumption</td><td>0.37 Wh/lap</td></tr><tr><td>Mechanical energy</td><td>3.7 Wh/lap</td></tr><tr><td>Hydrogen consumption</td><td>7.96 Wh/lap</td></tr><tr><td>Tank-to-wheel efficiency</td><td>46.20%</td></tr><tr><td>Performance</td><td>4.94 Wh/km<br>1855.75 km/L<sub>SP95</sub></td></tr></tbody></table>

<table><caption>Table 3 – Parameters of the vehicle.</caption><thead><tr><th>Parameter</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>mass</td><td>M</td><td>150 kg</td></tr><tr><td>drag coefficient</td><td>C<sub>x</sub></td><td>0.1</td></tr><tr><td>frontal area</td><td>S</td><td>0.88 m<sup>2</sup></td></tr><tr><td>air density</td><td>ρ<sub>air</sub></td><td>1.2 kg m<sup>-3</sup></td></tr><tr><td>rolling resistance coefficient</td><td>f<sub>r</sub></td><td>2.10<sup>-3</sup></td></tr><tr><td>transmission efficiency</td><td>η<sub>g</sub></td><td>0.95</td></tr><tr><td>number of cells</td><td>n<sub>cells</sub></td><td>28</td></tr><tr><td>mass of the fuel cell</td><td>m<sub>fc</sub></td><td>1 kg</td></tr><tr><td>heat capacity of the fuel cell</td><td>C<sub>p</sub></td><td>750 J kg<sup>-1</sup> K<sup>-1</sup></td></tr><tr><td>heat capacity of the air</td><td>C<sub>p,air</sub></td><td>1000 J kg<sup>-1</sup> K<sup>-1</sup></td></tr><tr><td>latent heat of vaporization</td><td>L<sub>H2O</sub></td><td>2500 kJ kg<sup>-1</sup></td></tr><tr><td>natural convection coefficient</td><td>k<sub>nat</sub></td><td>0.25 W K<sup>-1</sup></td></tr><tr><td>ambient temperature</td><td>T<sub>amb</sub></td><td>25°C</td></tr><tr><td>high threshold temperature</td><td>T<sub>max</sub></td><td>44°C</td></tr><tr><td>low threshold temperature</td><td>T<sub>min</sub></td><td>41°C</td></tr><tr><td>oxygen stoichiometric-ratio</td><td>S<sub>c</sub></td><td>5</td></tr></tbody></table>

<table><caption>Table 2 – Sensitivity of the optimization results against parameter variations.</caption><thead><tr><th></th><th>Reference</th><th>I<sub>start</sub></th><th>I<sub>acc</sub></th><th>v<sub>max</sub></th><th>ΔV</th><th>r<sub>gear</sub></th></tr></thead><tbody><tr><td>optimization</td><td></td><td>10.8 A</td><td>3.1 A</td><td>27.8 km/h</td><td>1.6 km/h</td><td>200/14</td></tr><tr><td>variations</td><td></td><td>+0.1 A</td><td>+0.1 A</td><td>+1.0 km/h</td><td>+0.1 km/h</td><td>—</td></tr><tr><td>consumption</td><td>4.9 Wh/km</td><td>+0.90%</td><td>+0.08%</td><td>+2.59%</td><td>+3.14%</td><td>—</td></tr><tr><td>average speed</td><td>25.0 km/h</td><td>+0.21%</td><td>+0.04%</td><td>+1.23%</td><td>+0.18%</td><td>—</td></tr><tr><td>Stack temperature</td><td>42.90°C</td><td>+0.05%</td><td>+0.02%</td><td>+0.32%</td><td>+0.15%</td><td>—</td></tr></tbody></table>

h). An average speed of 27 km/h is then maintained with a small velocity ripple of ΔV = 1.6 km/h, due to the successions of acceleration and free wheel phases. At the end the vehicle is able to brake very quickly and efficiently. During the lap, and by considering an initial temperature of 41°C, the cooling

system is activated during the free wheel phases to maintain a stack temperature between 41 and 44°C.

With this set of parameters, the global tank-to-wheel efficiency for one lap of the race is greater than 45% and an energy performance of about 500 Wh/100 km (i.e. an equivalent
<figure><img src="image_9.png" /><figcaption>Fig. 8 – Typical speed, current and temperature profiles for each lap.</figcaption></figure>

<table><caption>Table 1 – Optimization result.</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>I<sub>start</sub></td><td>10.8 A</td></tr><tr><td>I<sub>acc</sub></td><td>3.1 A</td></tr><tr><td>v<sub>max</sub></td><td>27.8 km/h</td></tr><tr><td>ΔV</td><td>1.6 km/h</td></tr><tr><td>r<sub>gear</sub></td><td>200/14</td></tr><tr><td>Average velocity</td><td>25.0 km/h</td></tr><tr><td>Fuel cell losses</td><td>3.04 Wh/lap</td></tr><tr><td>Motor losses</td><td>0.39 Wh/lap</td></tr><tr><td>DC-DC converter losses</td><td>0.23 Wh/lap</td></tr><tr><td>Mechanical transmission losses</td><td>0.11 Wh/lap</td></tr><tr><td>Supervision system consumption</td><td>0.37 Wh/lap</td></tr><tr><td>Mechanical energy</td><td>3.7 Wh/lap</td></tr><tr><td>Hydrogen consumption</td><td>7.96 Wh/lap</td></tr><tr><td>Tank-to-wheel efficiency</td><td>46.20%</td></tr><tr><td>Performance</td><td>4.94 Wh/km<br>1855.75 km/L<sub>SP95</sub></td></tr></tbody></table>

<table><caption>Table 3 – Parameters of the vehicle.</caption><thead><tr><th>Parameter</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>mass</td><td>M</td><td>150 kg</td></tr><tr><td>drag coefficient</td><td>C<sub>x</sub></td><td>0.1</td></tr><tr><td>frontal area</td><td>S</td><td>0.88 m<sup>2</sup></td></tr><tr><td>air density</td><td>ρ<sub>air</sub></td><td>1.2 kg m<sup>-3</sup></td></tr><tr><td>rolling resistance coefficient</td><td>f<sub>r</sub></td><td>2.10<sup>-3</sup></td></tr><tr><td>transmission efficiency</td><td>η<sub>g</sub></td><td>0.95</td></tr><tr><td>number of cells</td><td>n<sub>cells</sub></td><td>28</td></tr><tr><td>mass of the fuel cell</td><td>m<sub>fc</sub></td><td>1 kg</td></tr><tr><td>heat capacity of the fuel cell</td><td>C<sub>p</sub></td><td>750 J kg<sup>-1</sup> K<sup>-1</sup></td></tr><tr><td>heat capacity of the air</td><td>C<sub>p air</sub></td><td>1000 J kg<sup>-1</sup> K<sup>-1</sup></td></tr><tr><td>latent heat of vaporization</td><td>L<sub>H2O</sub></td><td>2500 kJ kg<sup>-1</sup></td></tr><tr><td>natural convection coefficient</td><td>k<sub>nat</sub></td><td>0.25 W K<sup>-1</sup></td></tr><tr><td>ambient temperature</td><td>T<sub>amb</sub></td><td>25°C</td></tr><tr><td>high threshold temperature</td><td>T<sub>max</sub></td><td>44°C</td></tr><tr><td>low threshold temperature</td><td>T<sub>min</sub></td><td>41°C</td></tr><tr><td>oxygen stoichiometric-ratio</td><td>S<sub>c</sub></td><td>5</td></tr></tbody></table>

<table><caption>Table 2 – Sensitivity of the optimization results against parameter variations.</caption><thead><tr><th></th><th>Reference</th><th>I<sub>start</sub></th><th>I<sub>acc</sub></th><th>v<sub>max</sub></th><th>ΔV</th><th>r<sub>gear</sub></th></tr></thead><tbody><tr><td>optimization</td><td></td><td>10.8 A</td><td>3.1 A</td><td>27.8 km/h</td><td>1.6 km/h</td><td>200/14</td></tr><tr><td>variations</td><td></td><td>+0.1 A</td><td>+0.1 A</td><td>+1.0 km/h</td><td>+0.1 km/h</td><td>—</td></tr><tr><td>consumption</td><td>4.9 Wh/km</td><td>+0.90%</td><td>+0.08%</td><td>+2.59%</td><td>+3.14%</td><td>—</td></tr><tr><td>average speed</td><td>25.0 km/h</td><td>+0.21%</td><td>+0.04%</td><td>+1.23%</td><td>+0.18%</td><td>—</td></tr><tr><td>Stack temperature</td><td>42.90°C</td><td>+0.05%</td><td>+0.02%</td><td>+0.32%</td><td>+0.15%</td><td>—</td></tr></tbody></table>

h). An average speed of 27 km/h is then maintained with a small velocity ripple of ΔV = 1.6 km/h, due to the successions of acceleration and free wheel phases. At the end the vehicle is able to brake very quickly and efficiently. During the lap, and by considering an initial temperature of 41°C, the cooling

system is activated during the free wheel phases to maintain a stack temperature between 41 and 44°C.

With this set of parameters, the global tank-to-wheel efficiency for one lap of the race is greater than 45% and an energy performance of about 500 Wh/100 km (i.e. an equivalent
<figure><img src="image_10.png" /><figcaption>Fig. 9 – Optimal profiles for one lap of the Cityjoule vehicle, on the Rotterdam race.</figcaption></figure>

consumption of 1 L<sub>SP95</sub> for 1800 km) is achieved. This simulation result is obtained for ideal climatic conditions without wind. This performance can thus be viewed as being the maximum value to be reached by the vehicle.

Considering the thermal behavior of the fuel cell, the proposed dynamical model permits to estimate the repartition of the heat through the various possible paths (natural convection, air feeding, liquid to vapor phase, cooling system). Fig. 10 gives the thermal power repartition in steady state, to maintain an average temperature of 43°C, as shown on Fig. 9.

Heat is thus mainly evacuated by the cooling system and the liquid to steam phase change of the water produced by the reaction. It must be noted that the assumption of a complete conversion to steam of the water produced in liquid form is still debated by the scientific community [14,17,18,21]. That is why in future works, experimental measurements must be done to confirm this hypothesis.

Fig. 11 presents the Sankey diagram of the powertrain for a lap. This energy diagram reveals that the main source of energy loss is the PEMFC stack: 38% of the primary hydrogen
<figure><img src="image_10.png" /><figcaption>Fig. 9 – Optimal profiles for one lap of the Cityjoule vehicle, on the Rotterdam race.</figcaption></figure>

consumption of 1 L<sub>SP95</sub> for 1800 km) is achieved. This simulation result is obtained for ideal climatic conditions without wind. This performance can thus be viewed as being the maximum value to be reached by the vehicle.

Considering the thermal behavior of the fuel cell, the proposed dynamical model permits to estimate the repartition of the heat through the various possible paths (natural convection, air feeding, liquid to vapor phase, cooling system). Fig. 10 gives the thermal power repartition in steady state, to maintain an average temperature of 43°C, as shown on Fig. 9. Heat

is thus mainly evacuated by the cooling system and the liquid to steam phase change of the water produced by the reaction. It must be noted that the assumption of a complete conversion to steam of the water produced in liquid form is still debated by the scientific community [14,17,18,21]. That is why in future works, experimental measurements must be done to confirm this hypothesis.

Fig. 11 presents the Sankey diagram of the powertrain for a lap. This energy diagram reveals that the main source of energy loss is the PEMFC stack: 38% of the primary hydrogen
<figure><img src="image_11.png" /><figcaption>**Fig. 10 – Heat evacuation paths for an average temperature of 43°C.**</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 11 – Sankey diagram for one lap.**</figcaption></figure>

energy is lost in the electrochemical process. A total of 4.6% of the primary energy is consumed by the fuel cell stack auxiliaries, constituted by the air blower and the cooling fan. Finally, 46% of the primary hydrogen energy is available to the wheel.

Finally, a sensitivity analysis is conducted on the optimization results (see Table 2). From the data set obtained by optimization, each variable is slightly modified to assess its influence on different criteria. Thus, the starting current and the acceleration current are increased by 0.1 A, the maximum speed by 1 km/h and the speed ripple by 0.1 km/h. Because the gear ratio is not considered as a strategy variable, it is not modified. The sensitivity is computed on 3 different parameters, which are the performance, the average speed and the average value of the stack temperature along 1 lap.

These results show that the sensitivity of the driving strategy on the performance is quite high and especially on the velocity profile $v_{max}$ and $\Delta V$. A variation of only 1 km/h on the maximum speed lead to a consumption increase of more than 2%. This sensitivity is mainly due to the higher aerodynamic drag when the velocity increases. A quite high sensitivity on the performance is also obtained for the starting and acceleration currents, which is due this time to a global lower efficiency of the powertrain. An important criterion of a race is to consider the minimum average speed of 25 km/h. It is interesting to see that the increase in the motor current (i.e. torque) does not significantly increase the average speed vehicle, while it increases the vehicle consumption. It seems then more interesting to increase the maximum speed, while keeping the same velocity ripple and motor current. With this strategy, the powertrain efficiency remains optimum, as well as the average number of starts per lap. Finally, a sensitivity analysis is done on the average stack Temperature, which is representative of the mean value of the stack losses. Once again, the higher sensitivity is on the velocity profile, due to an higher mechanical need when the average speed increase. As a conclusion, the velocity ripple, starting current and acceleration current must be as close as there values obtained by optimization. To adapt the speed of the car during a race, only the maximum velocity has to be changed.

## Experimental results on the Rotterdam track

During the 2015 edition of the Shell Eco-marathon, which took place in May in the city center of Rotterdam, Netherlands, a full run has been recorded. The measurements include the instantaneous speed (via an in-vehicle GPS), the voltage and current of both fuel cell stacks, and the stack temperature (through a PT1000 sensor). The total distance covered is of 16 km which is performed in 10 laps. With an average speed of 25 km/h, the duration of a run must not be more than 38 min (i.e. 2300 s). Fig. 12 gives a comparison between experimental results and simulation results obtained in the same conditions, ie an ambient temperature of 27°C (which is the temperature inside the fuel cell compartment) and an equivalent average wind speed of 15 km/h.

Finally, the measured results are very close to simulations. The power supplied by each fuel cell stack reached in both cases up to 250 W, as well as the average power which is about 110 W. As explain before, the driving strategy bases on changing only the maximum speed during the race, to adapt the average speed per lap. Concerning the stack temperature, the low and high thresholds are 5°C higher than in the previous section of this paper. This is due to the fact that the temperature measurement is placed at the hottest point, which is about 4–5°C above the average stack temperature. With this slight adjustment, a good agreement is obtained. The recorded performance for this run is 796 Wh/100 km, by considering a lower heating value of 33 kWh/kg and a volumic density of 0.08988 kg m⁻³. The simulation results give a consumption of 820 Wh/100 km, that is to say an error of only 3%. The difference with the performance of 500 Wh/100 km obtained with ideal climatic conditions (see the previous section)
<figure><img src="image_11.png" /><figcaption>**Fig. 10 – Heat evacuation paths for an average temperature of 43°C.**</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 11 – Sankey diagram for one lap.**</figcaption></figure>

energy is lost in the electrochemical process. A total of 4.6% of the primary energy is consumed by the fuel cell stack auxiliaries, constituted by the air blower and the cooling fan. Finally, 46% of the primary hydrogen energy is available to the wheel.

Finally, a sensitivity analysis is conducted on the optimization results (see Table 2). From the data set obtained by optimization, each variable is slightly modified to assess its influence on different criteria. Thus, the starting current and the acceleration current are increased by 0.1 A, the maximum speed by 1 km/h and the speed ripple by 0.1 km/h. Because the gear ratio is not considered as a strategy variable, it is not modified. The sensitivity is computed on 3 different parameters, which are the performance, the average speed and the average value of the stack temperature along 1 lap.

These results show that the sensitivity of the driving strategy on the performance is quite high and especially on the velocity profile $v_{max}$ and $\Delta V$. A variation of only 1 km/h on the maximum speed lead to a consumption increase of more than 2%. This sensitivity is mainly due to the higher aerodynamic drag when the velocity increases. A quite high sensitivity on the performance is also obtained for the starting and acceleration currents, which is due this time to a global lower efficiency of the powertrain. An important criterion of a race is to consider the minimum average speed of 25 km/h. It is interesting to see that the increase in the motor current (i.e. torque) does not significantly increase the average speed vehicle, while it increases the vehicle consumption. It seems then more interesting to increase the maximum speed, while keeping the same velocity ripple and motor current. With this strategy, the powertrain efficiency remains optimum, as well as the average number of starts per lap. Finally, a sensitivity analysis is done on the average stack Temperature, which is representative of the mean value of the stack losses. Once again, the higher sensitivity is on the velocity profile, due to an higher mechanical need when the average speed increase. As a conclusion, the velocity ripple, starting current and acceleration current must be as close as there values obtained by optimization. To adapt the speed of the car during a race, only the maximum velocity has to be changed.

## Experimental results on the Rotterdam track

During the 2015 edition of the Shell Eco-marathon, which took place in May in the city center of Rotterdam, Netherlands, a full run has been recorded. The measurements include the instantaneous speed (via an in-vehicle GPS), the voltage and current of both fuel cell stacks, and the stack temperature (through a PT1000 sensor). The total distance covered is of 16 km which is performed in 10 laps. With an average speed of 25 km/h, the duration of a run must not be more than 38 min (i.e. 2300 s). Fig. 12 gives a comparison between experimental results and simulation results obtained in the same conditions, ie an ambient temperature of 27°C (which is the temperature inside the fuel cell compartment) and an equivalent average wind speed of 15 km/h.

Finally, the measured results are very close to simulations. The power supplied by each fuel cell stack reached in both cases up to 250 W, as well as the average power which is about 110 W. As explain before, the driving strategy bases on changing only the maximum speed during the race, to adapt the average speed per lap. Concerning the stack temperature, the low and high thresholds are 5°C higher than in the previous section of this paper. This is due to the fact that the temperature measurement is placed at the hottest point, which is about 4–5°C above the average stack temperature. With this slight adjustment, a good agreement is obtained. The recorded performance for this run is 796 Wh/100 km, by considering a lower heating value of 33 kWh/kg and a volumic density of 0.08988 kg m⁻³. The simulation results give a consumption of 820 Wh/100 km, that is to say an error of only 3%. The difference with the performance of 500 Wh/100 km obtained with ideal climatic conditions (see the previous section)
<figure><img src="image_13.png" /><figcaption>Fig. 12 – Experimental (left) and simulation (right) results of a race on the Rotterdam track during the Shell Eco-Marathon 2015.</figcaption></figure>

and this measured result is mainly explained by the meteorological conditions, and especially wind.

## Conclusions

In this paper, an analytical tank-to-wheel model of an Urban-Concept fuel cell vehicle is presented. It takes into account the mechanical needs, the losses of each stage of the power train (fuel cell and auxiliaries, power converter, motor, transmission) and the dynamical thermal behavior of the fuel cell to account accurately the consumption of the fuel cell stack auxiliaries. Using a set of five driving parameters (speed thresholds, gear ratio and motor currents), an optimal driving strategy is obtained using a global optimization algorithm. This multiphysical and dynamic model is based on the consumption and the main interactions between the various elements of the car. Therefore, the optimal driving strategy is obtained using a global optimization algorithm which accounts for race constraints. This optimization leads to a maximum performance of 500 Wh/100 km for this urban vehicle, with an average speed of 25 km/h. A sensitivity analysis is done and shows that to adjust the average speed of the vehicle during a real race, it is recommended to change the maximum value of the speed profile without changing the motor currents or the velocity ripple. Today, the powertrain of this urban concept vehicle does not includes storage elements for the braking energy recovering. This is mainly explained by the ultra lightweight of the vehicle. Indeed, with a total weight of 150 kg, the

total recoverable kinetic energy at 25 km/h is only 1 Wh. Since each lap requires about 8 Wh of hydrogen energy, the potential gain from such an energy recovery system remains below than 15%. Nevertheless, in the context of such energy competitions, any gain of consumption is valuable. This is why the integration of supercapacitors in the vehicle is planed.

## Acknowledgment

The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreement ETRERA 2020 num. [609543].

## REFERENCES

[1] Ettihir Khalid, Boulon Loic, Agbossou Kodjo. Energy management strategy for a fuel cell hybrid vehicle based on maximum efficiency and maximum power identification. IET Electr Syst Transp December 2016;6(4):261–8.

[2] Antonacci P, Chevalier S, Lee J, Yip R, Ge N, Bazylak A. Feasibility of combining electrochemical impedance spectroscopy and synchrotron x-ray radiography for determining the influence of liquid water on polymer electrolyte membrane fuel cell performance. Int J Hydrogen Energy 2015;40(46):16494–502. http://www.sciencedirect.com/science/article/pii/S0360319915300562.
<figure><img src="image_13.png" /><figcaption>Fig. 12 – Experimental (left) and simulation (right) results of a race on the Rotterdam track during the Shell Eco-Marathon 2015.</figcaption></figure>

and this measured result is mainly explained by the meteorological conditions, and especially wind.

## Conclusions

In this paper, an analytical tank-to-wheel model of an Urban-Concept fuel cell vehicle is presented. It takes into account the mechanical needs, the losses of each stage of the power train (fuel cell and auxiliaries, power converter, motor, transmission) and the dynamical thermal behavior of the fuel cell to account accurately the consumption of the fuel cell stack auxiliaries. Using a set of five driving parameters (speed thresholds, gear ratio and motor currents), an optimal driving strategy is obtained using a global optimization algorithm. This multiphysical and dynamic model is based on the consumption and the main interactions between the various elements of the car. Therefore, the optimal driving strategy is obtained using a global optimization algorithm which accounts for race constraints. This optimization leads to a maximum performance of 500 Wh/100 km for this urban vehicle, with an average speed of 25 km/h. A sensitivity analysis is done and shows that to adjust the average speed of the vehicle during a real race, it is recommended to change the maximum value of the speed profile without changing the motor currents or the velocity ripple. Today, the powertrain of this urban concept vehicle does not includes storage elements for the braking energy recovering. This is mainly explained by the ultra lightweight of the vehicle. Indeed, with a total weight of 150 kg, the

total recoverable kinetic energy at 25 km/h is only 1 Wh. Since each lap requires about 8 Wh of hydrogen energy, the potential gain from such an energy recovery system remains below than 15%. Nevertheless, in the context of such energy competitions, any gain of consumption is valuable. This is why the integration of supercapacitors in the vehicle is planed.

## Acknowledgment

The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreement ETRERA 2020 num. [609543].

## REFERENCES

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