Pablo Garcia

Luis Fernandez

Carlos A Garcia

Francisco Jurado
email: fjurado@ujaen.es

Luis M Fernández

Comparative study of PEM fuel cell models for integration in propulsion systems of urban public transport

Keywords: Vehicle Application, Theoretical Model, Control, Fuel Cell Application, Hydrogen Storage Fuel cell application, hydrogen storage, theoretical model, vehicle application. Wiley-VCH Fuel Cells

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

FC promises as the best future replacement for internal combustion engine in transportation applications, because they operate more efficiently and with low emissions. The use of this clean power source mainly focuses on public transport (buses, trains, tramways and undergrounds) and passenger vehicles, where the FC is the main power generation system and a rechargeable battery is used as auxiliary power device. Among all types, PEM FC is the best option to be used in public transport due to their high power density, specific power, low operating temperature, longevity, efficiency, good dynamic behaviour and the ability to rapidly adjust to changes in power demand [START_REF] Hoogers | Fuel Cell Technology Handbook[END_REF][START_REF] Lukic | [END_REF].

A complete model including a detailed dynamic model of the main components of PEMFC system, such as described in [START_REF] Pukrushpan | Control of fuel cell power system: Principles, modelling, analysis and feedback design[END_REF], can be used to represent its behaviour. However, reduced models have been developed in order to approximate the FC response, and thus enabling reduced computation time. A quite complete model is described in [START_REF] Caux | [END_REF], where a screw compressor is modelled, and in addition, three valves are used: a valve to control the entering hydrogen, another one controls the cathode pressure, and the third one controls the hydrogen leaving the anode. More reduced models are used by [5--8], where the concentration voltage drops are not considered and the compressor is not modelled. In [START_REF] Miotti | Proc. Vehicle Power and Propulsion (VPP'05) IEEE Conference[END_REF], a second order dynamic system represents the rotational dynamics associated with the lumped model of compressor and motor, and a pressure dependent static map is used to calculate the mass flow of the compressor. Furthermore, a reduced model for stand-alone applications is described in [START_REF] Uzunoglu | [END_REF], where the compressor is replaced by a second order function that calculates the needed hydrogen from the methanol.

This paper presents a comparative study of PEM FC models for integration in hybrid propulsion systems of urban public transport, based on a commercial FC from Nuvera, which is especially manufactured for this application [START_REF] Nuvera | Andromeda PEM fuel cell[END_REF]. A new reduced model is proposed and compared with one of most used reduced model for hybrid vehicles and portable applications. Both reduced models are evaluated when integrated in a FC-battery powered hybrid system for a real surface tramway. Furthermore, the modelling of the other components of the hybrid propulsion system and a new EMS for the hybrid tramway are presented in this paper. The reduced models are evaluated by means of simulations, where their responses are compared with that one of the complete model described in [START_REF] Pukrushpan | Control of fuel cell power system: Principles, modelling, analysis and feedback design[END_REF], which includes a detailed dynamic model of the main components of FC system. The drive cycle and the demanded load used in the dynamic simulations belong to the Metro Centro tramway that actually serves the centre of Seville, a city in Spain. It operates currently as a catenary-powered tramway in the center of Seville. One of the objectives of the Ecotrans research project, which is being carried out by a Spanish consortium and is funded by the Spanish Ministry of Science and Technology, is to retrofit the current tramway to an FCbattery hybrid tramway. This paper presents the first advances made in the modelling and control of the new hybrid system.

Modelling PEM Fuel Cell System

A commercial FC from Nuvera [START_REF] Nuvera | Andromeda PEM fuel cell[END_REF] is considered in this work for its application in the hybrid system which drives the tramway. FC selected generates a nominal power of 127 kW-235 V, so that two of these FCs are connected in parallel in order to achieve a total net power around of 200 kW 1 .

Three models of a PEM FC system are described in this section. The first one is a complete model which includes a detailed dynamic model of the main components of PEM FC system, as described in [START_REF] Pukrushpan | Control of fuel cell power system: Principles, modelling, analysis and feedback design[END_REF]. The second one corresponds to a new reduced model proposed in this paper, which is obtained from application of some simplifications to the complete model. The third one is a more reduced model, which is very used for hybrid vehicles and portable applications.

Complete Model

The complete model (CM) is composed of: • A compressor with a DC motor, which controls the incoming air (oxygen) to the cathode. The compressor is modelled by a static map which determines the air flow incoming into the cathode. • Four manifolds (two supply and two return manifolds) which keep a stable gas flow. The models are derived from application of the mass conservation principle in each manifold. • The air cooler does not address heat transfer effect and maintains the temperature of the air entering at 80 ºC. Thus, also the humidity of the gas existing is changed in the air cooler. • A humidifier which injects water into the membrane avoiding the loss of membrane properties. The model uses the thermodynamic properties of gas mixtures to calculate the condition of the flow entering the cathode. • A high pressure tank provides the hydrogen to the FC through a valve. It is controlled by a proportional control based on the pressure difference.

The PEMFC model is completed with an electrochemical model to calculate the FC output voltage, which requires the effective partial pressures of the electrodes to be determined. The partial pressures of the anode and cathode are calculated from the mass conservation law and ideal gas law. In case of the anode, it has been considered that the unused hydrogen inside the anode is returned to the inlet through a recirculation pump. On the other hand, a valve for the purging of the hydrogen is commonly used in real FCs in order to eliminate the impurities that gradually accumulate near the exhaust outlet. This purging is programmed either as a function of cell voltage or as a function of time [START_REF] Barbir | PEM Fuel Cells Theory and Practice[END_REF][START_REF] Zhu | [END_REF]. In this work, the modelling of the pump and the purging of the hydrogen has not been considered.

In addition, the FC output voltage is obtained from the sum of the Nernst's voltage, the activation voltage drop, the ohmic voltage drop and the concentration voltage drop, where all the voltage drops are function of the current density. A more detailed dynamic model of the main components of FC system is described in [START_REF] Pukrushpan | Control of fuel cell power system: Principles, modelling, analysis and feedback design[END_REF]. Figure 1 shows the configuration of the PEM FC complete model implemented in this work.

Reduced Model 1

Applying some simplifications to the previous complete model, a reduced model of PEMFC (reduced model 1, RM1) can be obtained. In this work, the following simplifications have been done:

• The gases are considered ideal.

• The FC is fed with hydrogen and air.

• The electrode channels are small enough so that the pressure drop across them is negligible. 1 For this work, the total net power is the power generated by the FC minus the power consumed by the compressor. • The exhaust of each channel is via a single orifice. The ratio of pressures between the interior and exterior of the channel is large enough to consider that the orifice is choked. • Assuming that inlet oxygen and hydrogen coming respectively from compressor and tank present a stable mass flow, the supply manifolds have been eliminated. • Return manifolds have been removed since the outlet flow of any gas is considered to be proportional to its partial pressure inside the channel. • The humidifier and air cooler are considered ideal so that the FC operates at optimum temperature (80ºC) and with constant relative humidity. • The compressor is modelled by a first order system in order to approximate its dynamic response. The FC system is composed of the FC core associated with all the auxiliary services necessary for an embedded FC system. The FC core consists of the anode and cathode blocks, which calculates the hydrogen and oxygen partial pressure, respectively, and the stack voltage model. The auxiliary services are the compressor with DC motor controlling the air flow (oxygen) to the cathode and a hydrogen valve controlling the hydrogen input flow. Figure 2 shows the configuration of this reduced model.

Anode Model

The reaction that occurs in the FC anode is the oxidation of hydrogen (supplied from a tank) by releasing electrons to the conductive electrode and on through the external load. The hydrogen partial pressure in the anode is calculated from the mass conservation law and ideal gas law.

(

)

r H out H in H an H H q q q V T R dt dp 2 2 2 2 2 - - ⋅ = (1)
where q H2 r is obtained from the Faraday`s law.

F I N q FC cell r H ⋅ ⋅ = 2 2 (2)
The hydrogen output flow q H2 out is the unused hydrogen inside the anode and it is calculated imposing the desired utilization factor (relationship between the hydrogen that reacts and the incoming hydrogen, U f ). Desired utilization factor of hydrogen has been selected from the characteristics of the commercial FC used in this work [START_REF] Nuvera | Andromeda PEM fuel cell[END_REF], where a value around of 1/3 is recommended for the FC.

        - = 1 1 2 2 f r H out H U q q (3)
where

in H r H f q q U 2 2 / = (4)
The hydrogen input flow q H2 in is calculated as the sum of the hydrogen flow that provides the tank through the hydrogen input valve q H2 valve and the hydrogen back flow (the hydrogen that does not react),

q H2 back . back H valve H in H q q q 2 2 2 + = (5)
The valve is controlled in order to minimize the difference of pressure between the anode and cathode [5,6,14]. Fig. 3 shows the control scheme of this valve. The molar flow of this valve q H2 valve is expressed as a function of the opening section S open and a constant k valve which varies depending on the difference of pressure between the anode and cathode.

valve open valve H k S q ⋅ = 2 (6)
Finally, the global utilization factor is defined as the ratio between the hydrogen flow that reacts and the hydrogen flow through the valve.

valve H r H fg q q U 2 2 / = (7)

Cathode Model

In the FC cathode, the reaction that occurs is the reduction of oxygen in air by receiving electrons. Similarly, the oxygen partial pressure in the cathode can be also obtained from the mass conservation law and ideal gas law.

(

)

r O out O in O cat O O q q q V T R dt dp 2 2 2 2 2 - - ⋅ = (8) F o r P e e r R e v i e w F I N q FC cell r O ⋅ ⋅ = 4 2 (9)
The oxygen output flow q O2 out is calculated from the relation between the molar flow of oxygen and its partial pressure inside the channel [6,7,14]. This relation can be expressed as follows:

2 2 2 2 O O cat O O K M K p q = = (10) 2 2 2 O O out O p K q ⋅ = (11)
Another difference with respect to the CM resides in the way of calculating the total pressures in the anode and cathode. In the CM, the masses balance is performed for each element which appears in the process (nitrogen, water and oxygen in the cathode and water and hydrogen in the anode). Whereas the total pressures in the anode and cathode are obtained in this reduced model from the following expressions:

+ ⋅ + = + + = (13)
where the water partial pressure p H2O is assumed constant, and the nitrogen partial pressure p N2 is considered proportional to the oxygen partial pressure. The proportional constant has been obtained from the average value between these two pressures along the channel of the CM.

FC Voltage Model

The FC output voltage V fc is calculated as the product of the number of cells N cell and cell voltage V cell . The cell voltage can be expressed from the Nernst's instantaneous voltage E cell and the irreversible voltage or the loss voltage V irrev [START_REF] Uzunoglu | [END_REF][START_REF] Spiegel | PEM Fuel Cell Modeling and Simulation Using MATLAB[END_REF]:

cell cell fc V N V ⋅ = (14) irrev cell cell V E V - = (15)         ⋅ ⋅ ⋅ - - ⋅ - = 2 5 . 0 02 20 0 ln 2 ) ( H H g ref e cell cell p p p F T R T T k E E (16) ohm conc act irrev V V V V + + = (17)
where k e is a function of the entropy change and the Faraday`s constant.

The irreversible voltage V irrev represents the voltage drop due to several irreversible loss mechanisms, which causes the theoretical FC voltage drop. It is calculated as the sum of the activation voltage V act , concentration voltage V conc and the ohmic voltage V ohmic . A simple way to calculate them is described in [START_REF] Uzunoglu | [END_REF][START_REF] Spiegel | PEM Fuel Cell Modeling and Simulation Using MATLAB[END_REF]. The expressions used in this model are the following: [START_REF] Skvarenina | The Power Electronics Handbook[END_REF] In the CM, the irreversible voltage is calculated from more exact equations, which depend on other parameter, such as oxygen partial pressure, water saturation pressure and some constants determined empirically [START_REF] Pukrushpan | Control of fuel cell power system: Principles, modelling, analysis and feedback design[END_REF].

        ⋅ = 0 lg 2 i i F T R V den g act α (18) den fc ohmic i R V ⋅ = (19)         - ⋅ = lim 1 lg 2 i i F T R V den g conc

Compressor Model

The purpose of the compressor is to supply the inlet air mass flow to the cathode in order to maintain constant oxygen excess ratio λ O2 [START_REF] Pukrushpan | Control of fuel cell power system: Principles, modelling, analysis and feedback design[END_REF][START_REF] Hipp | [END_REF], equal to 2.5 in case of the commercial FC considered in this work. The oxygen excess ratio is defined as the ratio between the oxygen supplied and the oxygen used in the cathode.

r O in O O q q 2 2 2 = λ (21) F o r P e e r R e v i e

w

If the relative humidity is assumed to be 100%, the oxygen excess ratio is related with the air flow through the following equation:

( ) ( ) [ ] F I N M x T p p M T p M W FC O O atm sat atm a atm sat v flow air ⋅ ⋅ ⋅ ⋅ ⋅         - ⋅ ⋅ + = 4 1 1 2 2 02 λ (22)
where the saturation pressure p sat is a function of the atmospheric temperature.

In case of the CM, the air mass flow rate is determined from a compressor flow map, and a nonlinear curve method is used to model the compressor characteristics. Finally, the compressor modelling is performed in both models from the power required to drive compressor, motor torque of DC motor and compressor speed from the equilibrium torque equation or 2 nd Newton´s law, such as described in [START_REF] Pukrushpan | Control of fuel cell power system: Principles, modelling, analysis and feedback design[END_REF].

Reduced Model 2

This second reduced model (reduced model 2, RM2) is based on one of the most model used for hybrid vehicles and portable applications [5--8], which presents a more reduced order than the previous one. Figure 4 depicts the configuration of this reduced model.

In this case, the models of the anode and cathode are very similar to the previous one. In fact, the partial pressures of the hydrogen and oxygen are calculated in a similar manner.

In this model, the hydrogen output flow q H2 out is calculated from the relation between the molar flow of hydrogen and its partial pressure inside the channel [6,7,14].

2 2 2 H H out H p K q ⋅ = (23)
where K H2 is considered to be the average value of the ratio between the output hydrogen flow and the hydrogen partial pressure obtained with the RM1.

Considering Eq. ( 23) and applying the Laplace transformation to Eq. ( 1), it is obtained the following expression for the hydrogen partial pressure:

( ) r H in H H H H q q s K p 2 2 2 2 2 1 1 - + = τ (24) ) ( 2 2 T R K V g H an H = τ (25)
where τ H2 , expressed in seconds, is the value of the system pole associated with the hydrogen flow.

A similar procedure can be applied to determine the oxygen partial pressure. Considering Eq. ( 11) and applying the Laplace transformation to Eq. ( 8), the following expression can be derived for the oxygen partial pressure:

( ) r O in O O O O q q s K p 2 2 2 2 2 1 1 - + = τ (26) ) ( 2 2 T R K V g O cat O = τ (27)
where τ O2 , expressed in seconds, is the value of the system pole associated with the oxygen flow, and K O2 is the same constant used for the RM1.

In this model, it is assumed that the anode and cathode present the same constant pressure so that it is not needed to use Eqs. ( 12) and ( 13) to calculate the anode and cathode pressures, as performed in the RM1.

Apart of this, other differences with respect to the RM1 appear in the way of calculating the FC voltage and in the modelling of compressor and hydrogen input valve.

As it will be shown in the simulations, the concentration voltage drop is not very high when the FC current density is small. Assuming that the PEM FC will not work at high current density, the concentration voltage drop is ignored in this reduced model. Therefore, the FC voltage model is similar to the model described for the RM1, except for Eq. ( 20) which is not used in this model.

The other two differences are due to the ideal modelling of compressor and hydrogen input valve. In case of the compressor, it has been removed so that the power demanded by the compressor is not considered in the FC system. With this simplification, the incoming oxygen to the cathode is determined from Eq. ( 21) assuming a constant oxygen excess ratio of 2.5.

By the same way, the incoming hydrogen valve model has been removed. In this case, the hydrogen input flow is derived from Eq. ( 4) and the hydrogen valve flow is calculated from Eq. ( 7) considering a global utilisation factor near to the unit. The purposed hybrid system for the tramway is composed of: 1) PEMFC system; 2) battery; 3) dc/dc boost converter; 4) loads: tramway traction system and auxiliary services; 5) braking chopper; and 6) EMS. Figure 5 shows the configuration of hybrid system for the tramway.

The FC is the primary energy source of tramway. It is connected to a dc/dc boost converter which raises the low dc voltage delivered by the FC to the traction standard dc bus.

A rechargeable battery is used as electrical energy storage unit to supplement the output of the FC during tramway acceleration and cruise and for energy recovery during braking. It has been sized so that it store sufficient energy and provide adequate peak power for the tramway to have a specified acceleration performance and the capability to meet appropriate driving cycle.

The tramway loads supplied from the two energy sources are the auxiliary services and the traction system. The auxiliary services module represents the power consumption due to the tramway auxiliary equipment (lighting, fans, steering air conditioning systems...). The tramway traction system represents the power consumption due to the traction electric wheel motors.

Finally, the EMS determines the reference signals for the FC and the dc/dc boost converter in order to distribute accurately the power from the two electrical sources. In addition, it determines the reference signal for energy dissipation in the braking chopper when required during regenerative braking.

Battery

Many battery systems have been experimented for hybrid vehicles (transit buses, light trucks and tramways), using Pb-Acid, Ni-Cd or Ni-MH. Practical experience shows that Ni-MH, in spite of its higher cost, is the most suitable system in terms of endurance and reliability [START_REF] Broussely | Industrial Applications of Batteries[END_REF]. Although Ni-MH batteries will be gradually replaced by Li-Ion batteries in future due to their better performance, they are currently much more expensive. For this reason, a Ni-MH battery has been selected for the tramway hybrid system developed in this work. The battery specifications are: 34 Ah, 756 V Ni-MH battery with an overall rated power of 260 kW and a peak power of 350 kW. It has been modelled by the Ni-MH battery model included in SimPowerSystems toolbox of Simulink® [18].

bat i bat bat I R E U ⋅ - = (28) ( ) ∫ ∫ ⋅ ⋅ ⋅ + ⋅ - ⋅ - = dt i B A dt i Q Q K E E p o bat exp (29)
In transportation application using Ni-MH batteries, it is preferably kept the stage-of-charge (SOC) as close to 50% as possible; or within 30 to 70%. Since at these states-of-charge, the coulombic charge efficiency is very high [START_REF] Linden | Handbook of batteries[END_REF]. These operating recommendations for the battery have been considered in EMS design. In fact, the implemented model calculates the battery SOC as [18]:

          ⋅ - ⋅ = ∫ Q dt i SOC 1 100 (%) (30)

Boost Converter

A classical dc/dc boost converter [START_REF] Skvarenina | The Power Electronics Handbook[END_REF][START_REF] Moore | [END_REF] is selected as FC power converter, which adapts the dc voltage supplied by the FC, 235V at its rated power, to the 750V traction standard dc bus.

The boost converter is composed of a high frequency inductor L 1 , an output filtering capacitor C 1 , a diode D 1 , and a main switch S 1 . Switch S 2 is a shutdown device for test security to prevent the FC stack from short circuits in case of accidental destruction of S 1 , or faulty operation of the regulator. The boost converter is based on the current control loop. An appropriate control of the fraction of time that the converter is conducting (duty cycle) allows the FC current control. Acting on the switch S 1 gate signal, it is therefore possible to determine the distribution of the power load between the two energy sources, FC and battery.

Tramway loads

As mentioned previously, the tramway loads are the auxiliary services and the traction system. In this work, both loads have been modelled as controlled dc current source connected to the dc bus, whose

Braking Chopper

In regenerative braking, some of the kinetic energy is converted into electrical energy and stored in the battery, in order to be used during the next acceleration, or whenever else needed. However, part of the regenerative braking energy must be dissipated in the braking resistor when it cannot be absorbed by the battery due to its limitations.

The braking chopper absorbs the energy produced by tramway deceleration when required. It has been modelled by a current source parallel with the dc bus. In this case, the control system determines the effective resistance to be required to dissipate the power, and thus the current is calculated from the dc bus voltage and the effective resistance.

Energy Management System

The goal of this paper is to perform a comparative study of PEMFC models and to choose the most appropriate for its integration in the modelling of the hybrid propulsion system of a tramway. It has been developed a model of all the main components of the hybrid tramway. The models are evaluated when integrated into a FC-battery powered hybrid system.

Note that the FC has a slow dynamic response due to the air compressor dynamic and the electromechanical dynamic inside the FC. In fact, the commercial FC used in this work can reach a quick power change (from 10% to 90% rated power) at less than 2 seconds. When the FC cannot reach the reference power during a quick power change (such as acceleration) due to its slow dynamic response, it is required the support of battery to provide fast, additional power and to improve the transient response. On the other hand, the battery enables the recovery and storage of regenerative braking energy, and thus the reduction of the size of the FC and the increase of the system efficiency. Furthermore, the braking chopper must operate to dissipate regenerative braking energy when it cannot be absorbed by the battery due to its limitations.

The aim of tramway EMS is to provide the power needed by optimizing the energy generated by the system and ordering the operation of the braking resistor, when necessary, during regenerative braking. Following this objective, a new control strategy is presented. It combines a state machine control strategy based on eight states and a cascade control structure composed of two control loops. It has been applied to evaluate the configuration, modelling and energy management of the power sources adopted for the tramway.

State Machine Control Strategy

State machine control strategy is based on eight states in order to decide the operating point for each component of hybrid system. It generates the reference power for FC and battery and the power to be dissipated in the braking resistor. Furthermore, this control tries to avoid continuous changes in FC reference power, since the FC dynamic response is slow. Thus, the changes in the reference power will occur when the hybrid system components reach their limitation.

In this control strategy, the following values were considered:

• The maximum and minimum powers which the FC can generate (P fc max , P fc min ), and the optimum FC power (P fc opt ), which corresponds to the maximum efficiency power of the FC. • The maximum charge power (P bat char max ) and maximum discharge power (P bat disc max ) of the battery. • The maximum and minimum battery SOC (SOC max , SOC min ). In order to avoid startup problems, an 'always on' strategy is applied to the FC [START_REF] Cho | Proc. the 8'h Workshop on Power Electronics in Transportation (WPET 2004)[END_REF][START_REF] Gao | [END_REF]. The minimum power consumed by the tramway during the driving cycle is that of the auxiliary services (61 kW), which is the only power consumed during the stops. Therefore, it was selected as the minimum power generated by the FC, P fc min . This value is close to the optimum FC power (80 kW). In addition, two operation modes were considered for the FC. The FC can operate at constant power (load levelling strategy) or by adapting the power generated to the load demanded (load following strategy).

The state machine control determines the operational state according to the demanded load, the tramway speed and the battery SOC. Three levels were considered for the battery SOC: High SOC (60%-65%), normal SOC (50%-60%) and low SOC (45%-50%). Two hysteresis cycles, as shown in Figure 6, are used for the changes among these levels. Next, the eight states considered in the control strategy are detailed. • State 1: High SOC and P load < P fc opt . The hybrid system adopts this state when the battery SOC is high and the power demanded by the tramway is less than the optimum FC power. In this state, the FC operates at minimum power, providing the auxiliary services power when the tramway is stopped. The FC is made to operate at optimum power instead of minimum power in order to improve the FC efficiency during the driving cycle. Because of the slow response of the FC, the battery must supplement the output of the FC to provide the load power, when necessary, during tramway acceleration.

• State 2: High SOC and P fc opt ≤ P load ≤ 190. In this state, the FC operates with a load following strategy, and the battery must supplement the output of the FC, when necessary. Because the maximum power demanded by the tramway is about 540 kW, the FC should provide 190 kW if the battery is made to operate at its maximum discharge power (350 kW). In this case, the FC provides an output power between its optimum power and 190 kW.

• State 3: High SOC and P load > 190. When the power demanded by the tramway is greater than 190 kW, the FC operates with a load levelling strategy, providing 190 kW. When the tramway requires more power, it is provided by the battery. In fact, it may be necessary for the battery to provide its maximum discharge power in order to supplement the FC in order to supply the maximum power demanded by the tramway. • State 4: Normal SOC and P load < P fc opt . In this state, the hybrid system works the same as in state 1.

The FC operates at minimum power, providing the auxiliary services power when the tramway is stopped. When the tramway is running, the FC generates its optimum power.

• State 5: Normal SOC and P fc opt ≤ P load ≤ P fc max . The FC operates with a load following strategy and the battery supplements the FC output, when necessary, during tramway acceleration.

• State 6: Normal SOC and P load > P fc max . In this case, the FC operates at maximum power and the battery supplements the necessary power to cover the load demand. • State 7: Low SOC and Speed=0. When the tramway is stopped and the battery SOC is low, the FC is demanded to generate its maximum power. In this state, the FC supplies the auxiliary services (61 kW) and it charges the battery (with a power of 139 kW), and in this way, the battery could rapidly achieve normal SOC. • State 8: Low SOC and Speed>0. In this state, the FC must generate the demanded load plus extra power that will charge the battery, whenever possible. The battery is required to supplement the output of the FC, when necessary, during tramway acceleration. During regenerative braking, the battery will be charged for energy recovery. However, in order to avoid excessive charging, the braking resistor dissipates energy whenever the battery reaches its operating limits.

Cascade Control Structure

This control allows FC and battery to achieve the operating points determined by the state machine control strategy according to the operating state. This cascade control is composed of two control loops, as shown in Figure 7. The outer loop controls the battery power and the inner one controls the FC generating current.

• Outer Loop Control: As mentioned previously, the control strategy generates the reference power for the battery. From the error between the battery reference power and the actual power, a PI controller generates the FC system net reference power. This power must be limited in slope and in level according to the FC system characteristic and after that with the actual FC voltage is calculated the FC reference current for the inner control loops. This current is limited according to the maximum and minimum currents that FC can generate before being compared with the actual current. • Inner Loop Control: In this control loop, a PI controller is used to determine the duty cycle of dc/dc boost through the error between the FC current reference and the actual output current. By this way, FC can generate the reference power obtained from the outer loop control.

Simulations Results

The two reduced models of PEM FC presented in this paper, the reduced model 1 (RM1) and reduced model (RM2), are evaluated in their application to the hybrid propulsion system for a real surface tramway by means of simulations, where the response of the reduced models are compared with that one of complete model (CM). With this objective, steady-state and dynamic simulations have been performed implemented in MATLAB-Simulink®.

Steady-state Simulations

From the steady-state simulations, two FC characteristic curves have been represented in order to compare the models: 1) the FC polarisation curve (voltage versus current density) in Figure 8a; and 2) the FC power curve (output power versus current density) in Figure 8b. In Figure 8b, it has been represented: 1) the output net power supplied by the two FCs, P FCT , which is calculated as the difference between the total power generated by the two FC and the power consumed by the compressors in order to impulse air to cathode; and 2) the total power generated by a FC without considering the power consumed by the compressor, P stack . As explained previously, the compressor has been considered ideal in the RM2 so that the FC output power has been calculated ignoring the power consumed by the compressor.

Figure 8a shows the FC polarisation curve obtained from each model. Note that the FC voltage depends on the voltage losses, and they vary with the current density. The polarisation curve obtained from the CM and RM1 present similar results than that shown by the commercial FC considered in this work [START_REF] Nuvera | Andromeda PEM fuel cell[END_REF]. Differences between the results of CM and RM1 are minimal from the connection of the stack to 0.95 A/cm 2 approximately, which corresponds to the common working region of the FC. After that current density, differences between both curves increase a little, due to mainly the minimal effect of the concentration drop voltage, such as observed in Figure 9b. RM2 shows higher cell voltage and higher differences, particularly for high current density. It is mainly due to that the concentration voltage drop has been ignored in this model, and this voltage drop increases with the current density.

Similar results are obtained in the FC power curve, shown in Fig. 8b. When compared with the power curves obtained by the CM, the RM1 achieves a better approximation than the RM2. Thus, the highest differences in the FC power curve obtained from the RM2 appear with high current density, because the power consumed by the compressor has not been considered in this model.

As regards with the irreversible voltage, the results obtained from each model are similar, such as shown in Figure 9a. Nevertheless, little differences between the CM and RM1 can be observed from 0.5 A/cm 2 , which are due to mainly the concentration voltage drop (Figure 9b). These differences are a little higher in case of the RM2, because they were ignored.

Dynamic Simulations

For dynamic simulations, the three models have been integrated in the described hybrid system, and the resulting models have been simulated for the real drive cycle of the Metro Centro tramway, which serves the centre of Seville, a city in Spain.

The tramway follows a symmetrical route of going and return, which last 660 s. Only 330 s have been simulated which corresponds to half of the drive cycle or to the outward route. Figure 10 shows the speed and the mechanical power required by the tramway during this route. The cycle followed by the tramway is basically composed of four trips and four stops. The first one lasts about 12 s. It consists of an acceleration and deceleration, reaching the tramway a maximum speed of 20 km/h. The following two trips are very similar. Both of them last about 60 s, and include accelerations and decelerations. However, these trips show little differences in the maximum speed and speed changes. The three first stops endure the same time, about 15 s. The fourth trip is completely different. It is the longest, about 120 s. In this case, the tramway maintains a constant speed of 15 km/h along the whole trip. Besides, the fourth stop time is higher, about 60 s.

To check the performance of the state control with normal and high battery SOC along the performed simulation, an initial battery SOC of 65% was considered. This corresponds to a high SOC. Figure 11 shows the operational states and battery SOC obtained from the models along the simulations. It can be observed that the three models operate with the same operational states during all the simulation. During the first seventy seconds, the battery SOC is high so that the system adopts the states 1, 2 and 3. From the seventy second to the end of the simulation, the system operates with the states 4, 5 and 6 because of the battery SOC is normal.

Figures 12, 13, 14 and 15 depict the internal behaviour of FC system during the driving cycle. In Figure 12, the hydrogen and oxygen partial pressures are depicted. As it can be observed, the approximation of the partial pressures is good enough in case of RM1and but only for the hydrogen partial pressure in case of RM2, because the oxygen partial pressure obtained with RM2 presents greater differences with respect to the response of CM. The anode and cathode pressures are shown in Figure 13. In this case, good results are obtained again, even though a maximum error around of 10% is obtained with the RM1, which demonstrates the right approximations performed in Eqs. ( 12) and [START_REF] Zhu | [END_REF]. For both the total FC current and output FC voltage (Fig. 14), RM2 presents higher differences with respect to CM. Finally, for a better understanding of the behaviour of each FC model, the hydrogen entering, fuel cell system efficiency and utilization factor are represented in Figure 15. The entering hydrogen is calculated from the upstream valve hydrogen flow.

dt q N M cycle valve H fc con H ∫ = 2 2 (31)
On the other hand, the FC system efficiency is obtained by the next equation [24].

∫ ∫ ⋅ = cycle H fc cycle FCT FCS dt P N dt P 2 η (32)
where and P FCT is the output net power supplied by the two FC systems considered in the hybrid tramway and P H2 is the theoretical power associated with the hydrogen, which is expressed as

stack fg therm stack stack fg therm com FC H U P U P P P η η η η ⋅ ⋅ = ⋅ ⋅ + = 2 (33)
where P FC is the output net power supplied by a FC system, P com is the power consumed by the compressor, η therm is the thermodynamic efficiency defined by the ratio of ∆G f to ∆H f , and η stack can be calculated as the relationship between the cell voltage V cell and the standard-state reversible voltage E 0 cell . Final mass hydrogen consumed by the FC system during the route is similar in case of CM and RM1, with a total mass consuming around of 0.46 kg. However, RM2 presents a lower final mass hydrogen consume, around of 0.42 kg, because, in this model, FC does not has to generate energy for any compressor. In case of FC efficiency, CM presents an average value around of 55% along the simulation. As it can be seen, RM1 achieves a better approximation of FC efficiency than RM2, which shows higher efficiency due to the removal of the compressor. As regard the utilization factor, RM1 obtains similar results to CM, an average value around of 33% along the simulation.

Figure 16 and 17 show the performance of hybrid system for driving cycle, where the powers of each one of the components of hybrid system (FC, battery, loads and braking chopper) are represented. Note that FC cannot reach the reference power during a quick power changes due to its slow dynamic response. When the tramway accelerates, it demands higher power, so that the FC tries to generate its maximum power and the battery supplements to FC producing the rest of power required by the tramway during the acceleration. In case of deceleration, it is produced regenerative braking (negative load power), so that part of regenerative power is used to charge the battery (which is limited to its maximum value) and the rest one is dissipated in the braking resistor when the battery achieves its maximum charge power.

As it can be observed in the results obtained by the two reduced models, when representing the external behaviour of FC integrated in the hybrid system of tramway (Figs. 16 and 17), are similar to those ones obtained by CM, but with an important reduction of computation time. For RM1, the reduction of computation time with respect to CM is around of 30%, whereas RM2 presents a greater reduction, around of 42%.

When compared with CM, RM1 model obtains better results than RM2 in order to represent both the internal and external behaviour of a FC integrated in hybrid system. RM2 allows approximating the power delivered by the FC to the hybrid system, and therefore, simulating the performance of tramway hybrid. However, this model presents noticeable differences in the FC operating variables, such as partial pressures, output voltage and current, and efficiency.

Conclusions

This paper has presented a comparative study of PEM FC models for integration in hybrid propulsion systems of urban public transport. A new reduced model is proposed and compared with one of most used reduced model for hybrid vehicles and portable applications. These models were evaluated in a FCbattery powered hybrid system for a surface tramway by comparing the simulated responses with that one The first reduced model (RM1), a new model proposed in this work, was derived from applying the following simplifications to the CM:

• The gases are considered ideal.

• The FC is fed with hydrogen and air.

• The electrode channels are small enough so that the pressure drop across them is negligible.

• The ratio of pressures between the inside and outside of the electrode channels is large enough to assume choked flow. • Assuming that inlet oxygen and hydrogen coming respectively from compressor and tank present a stable mass flow, the supply manifolds have been eliminated. • Return manifolds have been removed since the outlet flow of any gas is considered to be proportional to its partial pressure inside the channel. • The humidifier and air cooler are considered ideal so that the FC operates at optimum temperature (80ºC) and with constant relative humidity. • The compressor is modelled by a first order system in order to approximate its dynamic response. The second reduced model (RM2) is based on one of the most used models for hybrid vehicles and portable applications. To obtain this reduced model, the following simplifications were applied to the RM1:

• The concentration voltage drop was eliminated, and therefore it was not considered in the FC output voltage. • The compressor was considered ideal so that the power demanded by the compressor was ignored.

• The input hydrogen valve was also considered ideal so that it was removed from the model, and the incoming hydrogen flow was calculated from the global utilization factor. • It was assumed that the anode and cathode presented the same constant pressure. Steady state and dynamic simulations were performed for comparison of the models. In the steady state simulations, the FC polarisation and power curves were represented. In case of FC polarisation curve, the curve obtained from each model was compared with the real curve of the commercial FC from Nuvera considered in this work. In the dynamic simulations, the models were integrated in the tramway hybrid system and simulated for the real drive cycle of the tramway. The drive cycle and the demanded load used in the dynamic simulations correspond to the tramway that actually serves the centre of Seville, a city in Spain.

From the performed simulations, interesting conclusions were obtained. When compared the performance of the FC integrated in the tramway hybrid (powers at FC, battery, braking resistor and load), both reduced models show similar results with an important reduction of computation time with respect to the CM (reduction of 30% for RM1 and 42% for RM2). When compared the FC operating variables, RM1 model obtained better results than RM2, since RM2 shows noticeable differences in partial pressures, voltage, current and efficiency. These differences are due to that the concentration voltage drop and power consumed by compressor were ignored, and the hydrogen valve was considered ideal.

RM1achieves a suitable equilibrium between approximation of FC response and computational time, and therefore, it can be concluded that it results to be the most appropriate FC model for propulsion system applications from those studied in this work. water partial pressure (bar / Pa) p sat saturation pressure (bar / Pa) q H2 back hydrogen flow recovered from the anode (kg.s -1 ) q H2 in hydrogen input flow (kg.s -1 ) q H2 out hydrogen output flow (kg.s -1 ) q H2 r hydrogen flow that reacts (kg.s -1 ) q H2 valve hydrogen upstream valve flow (kg.s -1 ) q O2 in oxygen input flow (kg.s -1 ) q O2 out oxygen output flow (kg.s -1 ) q O2 r oxygen flow that reacts (kg.s -1 ) R fc FC internal resistance (Ω cm 2 ) R H2 hydrogen gas constant (J.kg -1 . K -1 ) R O2 oxygen gas constant (J.kg -1 . K -1 ) R g ideal gas constant (J.kg -

F

  function of the power consumption and the dc bus voltage.

  model (CM), which includes a detailed dynamic model of the main components of FC system.

Fig. 1

 1 Fig. 1 Configuration of the PEM FC complete model. Fig. 2 Configuration of the PEM FC reduced model 1. Fig. 3 Control scheme for hydrogen input valve Fig. 4 Configuration of the PEM FC reduced model 2. Fig. 5 Configuration of the FC-battery hybrid system for the tramway. Fig. 6 Hysteresis cycles for the battery SOC levels. Fig. 7 Cascade control structure. Fig. 8 (a) FC polarisation curve, and (b) FC power curve. Fig. 9 (a) FC total irreversible voltage, and (b) FC concentration voltage drop. Fig. 10 (a) Tramway speed, and (b) mechanical power. Fig. 11 (a) Battery SOC, and (b) operational states. Fig. 12 (a) Hydrogen partial pressure, and (b) oxygen partial pressure at FC. Fig. 13 (a) Anode pressure, and (b) cathode pressure at FC. Fig. 14 (a) Total FC current, and (b) FC output voltage. Fig. 15 (a) Hydrogen mass entering, (b) FC efficiency and, (c) FC utilization factor. Fig. 16 (a) FC output power, and (b) battery power. Fig. 17(a) Load power, and (b) power dissipated in the braking resistor.

  Fig. 1 Configuration of the PEM FC complete model. Fig. 2 Configuration of the PEM FC reduced model 1. Fig. 3 Control scheme for hydrogen input valve Fig. 4 Configuration of the PEM FC reduced model 2. Fig. 5 Configuration of the FC-battery hybrid system for the tramway. Fig. 6 Hysteresis cycles for the battery SOC levels. Fig. 7 Cascade control structure. Fig. 8 (a) FC polarisation curve, and (b) FC power curve. Fig. 9 (a) FC total irreversible voltage, and (b) FC concentration voltage drop. Fig. 10 (a) Tramway speed, and (b) mechanical power. Fig. 11 (a) Battery SOC, and (b) operational states. Fig. 12 (a) Hydrogen partial pressure, and (b) oxygen partial pressure at FC. Fig. 13 (a) Anode pressure, and (b) cathode pressure at FC. Fig. 14 (a) Total FC current, and (b) FC output voltage. Fig. 15 (a) Hydrogen mass entering, (b) FC efficiency and, (c) FC utilization factor. Fig. 16 (a) FC output power, and (b) battery power. Fig. 17(a) Load power, and (b) power dissipated in the braking resistor.

Fig. 1 Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 8 (Fig. 9 (Fig. 10 (Fig. 11 (Fig. 12 (Fig. 13 (Fig. 14 (Fig. 15 (Fig. 16 (Fig. 17

 123456891011121314151617 Fig. 1 Configuration of the PEM FC complete model. 73x28mm (600 x 600 DPI)

  Fig. 17(a) Load power, and (b) power dissipated in the braking resistor. 46x24mm (600 x 600 DPI)

Wiley-VCH Fuel Cells

Acknowledgement

This work was supported in part by Hynergreen Technologies S.A. and by the Cenit Program from the Centre for the Development of Industrial Technology, Spanish Ministry of Science and Technology, under the Ecotrans research project. A national industries consortium, leaded by CAF ('Construcciones y Auxiliar de Ferrocarriles'), and to which belongs Hynergreen Technologies, is working in this project for the development of ecological technologies for urban transport.

List of Symbols


Christian ; Kunde

Richard Hanke-Rauschenbach

Michael Mangold
email: mangold@mpi-magdeburg.mpg.de

Achim Kienle

Kai Sundmacher

Stefan Wagner

Robert Hahn

Temperature and Humidity Control of a Micro PEM Fuel Cell Stack

Keywords: Control, Dynamic Simulation, Portable, Mathematical Model, PEM Fuel Cell System, Water Management control, miniature fuel cell stack, PEM fuel cell, simulation

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

We thank the reviewer for his constructive comments. In response to the suggestions, we modified the manuscript at various points. All modifications are highlighted in blue in the revised version. In the following, we give replies to the reviewer's remarks:

1. Fig 1a is just for illustration and can be omitted.

Figure 1a is omitted in the revised version.

Explanation to Fig 1b is given subsequent to Fig The sequence of explanations should be adapted.

In the revision, the explanations to Figure 1 and Figure 2 are given according to the sequence of the figures.

A comparison to steady-state experimental data should be provided.

A new Figure 4 and the respective explanatory text have been added in the revision to include steady-state experimental results. Measurements of the high frequency impedance as a key indicator for the membrane humidification are given.

4.

The transition to a step response are calculated and measured. However no direct comparison is possible due to different setups. I suggest to adjust the parameters of the model to the experimental setup in order to provide a direct comparison.

The simulation of the step response is now done for parameters adapted to the experimental setup of Figure 6 and an explanatory text to the differences between both experimental setups is given. Since the setup according to Figure 6 is unsuitable for control due to low influence on the stack temperature, the standard parameters are still used for the rest of the paper.

5.

The developed control scheme is simple and developed from analysis of the model behaviour. The effort for an experimental test is relatively low, but no measurements with closed-loop control have been carried out. I suggest to conduct these experiments and to include experimental validation in the publication.

Actually, our current experimental setup does not yet allow the implementation of an automatic feedback control. At the moment work on an extended set-up is in progress, also motivated by the result in this manuscript. However, the design and manufacturing of a suitable circuit board with a microcontroller and the programming of the microcontroller will take some time. Our intention is to test the control scheme as soon as the extension of the experimental setup has been finished. However, this is beyond the scope of the submitted manuscript. We think that the proposed control methodology has a value in itself as it is rather general and maybe applicable to many different fuel cell stacks, not only to the one we considered in the experiments.

6. Typing error in reference [14].

Corrected.

Introduction

Fuel cells directly convert chemical energy to electrical energy. However, they are not well established yet in general applications because of their high costs, limited lifespan and high demands on operation control. With further development fuel cells are expected to play an important part in the future use of renewable energy sources.

Fuel cells have very different fields of application. Depending on the specific application, there are large differences in the mode of operation, the available manipulated variables, the limits for weight, costs and volume of the system and hence in the challenges on process control. In this work the focus is on a miniature fuel cell system for portable applications. This system should be as light, small and cheap as possible. Further, the parasitic power consumption of peripheral devices should be kept small, as it has a much stronger impact on such a low power system than on a high power system. Therefore, the intention is to cut the peripheral equipment to a minimum. Finally, the control algorithm should be implemented on a micro-controller with limited computational power. This prohibits the use of highly ambitious nonlinear control approaches. An uncontrolled open-loop operation of miniature fuel cells is also of limited value, as was shown by Shkolnikov et al. [12] F o r P e e r R e v i e w 2 who studied a 2 W power supply without extra humidification, heating or cooling. Though the system is designed for "free-breathing" operation, they conclude by experimental results that a blower is needed for optimal operation of their H 2 -PEMFC. In conclusion, control schemes are needed for micro fuel cell systems with low system complexity.

In open literature, there are hardly suitable solutions to this kind of control problem. The majority of publications on control of PEMFCs concerns classical linear control approaches [10,3,6,5,15], mainly automotive applications. Typically, a humidifier, a compressor and an independent cooling system are used to control various system states. Such a control strategy cannot be adapted to miniaturized fuel cells.

There are also some nonlinear approaches for control of PEMFCs [2,1,7,9], but those are computationally too expensive for micro fuel cell applications.

Schumacher et al. [11] propose a fuzzy algorithm for the control of the humidity in a miniature H 2 -PEMFC. The fuzzy rules use measurements of the impedance and the voltage of the fuel cell to set the system blower voltage. Test results for an experimental setup are also shown in that paper. A drawback of their solution is that they only consider the humidity while the temperature of the fuel cell has still to be kept constant via additional heating and cooling.

In the present study, a control algorithm is proposed that considers both humidity and temperature of a H 2 -PEMFC system while using only the airflow of a single blower as a manipulated variable.

The following section gives an introduction to the micro fuel cell system. In section 3 a simplistic model for this fuel cell system is derived and compared with experimental results. The model is subsequently used in section 4 for controller design and testing.

Micro Fuel Cell Stack

In this paper a miniaturized fuel cell system for power output below 20 W is investigated. The system has been developed by the Fraunhofer IZM Berlin. It has to be light-weighted and small to fulfill the demands of applications in e.g. portable electronics, small unmanned vehicles and autonomous sensors. Parasitic power consumption of peripherals should also be considerably smaller than the total power output. Any peripherals have to be kept at a minimum for these reasons.

Consequently, a single blower is used for the cooling and the air supply of this fuel cell system. The airflow of this blower is splitted in one fraction that provides the cathode with oxygen and removes product water vapor from the system. The second fraction passes over external cooling fins to remove heat from the system. This concept is shown in Figure 1. The ratio between external and internal airflow is determined by the geometry of the construction. It influences the general humidification of the fuel cell and is therefore an important design parameter. H 2 -PEMFCs are used in the investigated system. They can be operated sufficiently well at low temperatures near ambient conditions and involve a less complex chemical reaction than other competitors in the low temperature range. The core of the system is a stack consisting of ten fuel cells connected electrically in series with an active reaction area of 10 cm 2 each.

In the fuel cells, hydrogen is oxidized to water according to the following overall reaction.

g 2 2 2 O H O 2 1 H → +
The free reaction enthalpy is partly converted to electrical energy. The rest is dissipated and released as heat. The basic configuration of a single fuel cell is illustrated in Figure 2. The protonic conductivity of the membrane strongly depends on its humidification. Since humidifiers are not applicable to the system, dry hydrogen is fed to the fuel cell. Additionally, the osmotic drag drives water through the membrane to the cathode side and water is only produced in the cathodic electrode. Therefore humidification of the anode side of the membrane only works by back-diffusion of water from the cathode side to the anode side. To reduce local dehydration, the membrane is chosen as thin as possible.

To further reduce peripherals there is no recirculation of hydrogen. Instead the anodic gas channels are operated in the so called dead-end mode, i. e. the outlet of the anodic gas channels is closed. Since impurities can accumulate in the anodic gas channels due to membrane crossover and impurities in the hydrogen feed, the anodic gas channels have to be purged periodically by briefly opening the anode outlet.

There are several requirements for the operation of the fuel cell system. The membrane has to be kept well humidified for good proton conductivity. The humidity also has to be kept low enough that water does not condensate in the fuel cell, because it would block the gas transport to the reaction sites. The temperature of the fuel must not exceed upper boundaries given by the materials used in the system. The reaction kinetics on the opposite require high temperatures. Additionally, the strong coupling between the temperature and the relative humidity has to be taken into account.

These requirements have to be met by using only the airflow of the blower as a manipulated variable and with a control algorithm that can be easily implemented and run on a microcontroller. In the following, a control-oriented model for a H 2 -PEMFC is derived, that focuses on heat and water management of the system. The notation used in the subsequent sections is given in Table 1 andTable 2.

Model Structure and Basic Model Assumptions

The model is a lumped system that describes the dynamics of the fuel cell stack in terms of cell temperature and water vapor concentration in the cathodic gas channels. The following main model assumptions are made:

• Spatial gradients are neglected in the whole system.

• The water content of the membrane is in equilibrium with the cathodic gas channels.

• Local drying of the membrane is not described by the model. The fuel cell performance will therefore be overestimated for higher current densities.

• Water is only present in the membrane and in gaseous form. Liquid water is not accounted for.

• The chemical reaction in the fuel cell produces water in gaseous form.

• The mass storage capacities of the cathodic gas channels and the membrane are dominant. Nitrogen, water vapor and oxygen is present in the cathodic gas channels, the membrane only stores water.

• The heat capacity of the solid parts of the cell is the dominant energy storage capacity.

• The dynamics of the fuel cell are dominated by the water balance and the energy balance. Faster dynamics (e.g. charge balances) are assumed to be instantaneous.

• Voltage losses are caused by the cathodic overvoltage, the protonic resistance of the membrane and the constant ohmic resistance of the electrodes and current collectors. The system is assumed to be connected to an electronic device that sets the output current of the fuel cell stack to specified values.

• The ideal gas law is valid in all gaseous phases. • The pressure is constant in the whole system.

• Loss of water and heat due to purging of the anodic gas channels is neglected.

• The anodic feed is assumed to be dry, pure hydrogen. The cathodic feed is assumed to be ambient air.

∆ - ∆ - + = φ φ η M C 0 cell cell U U (1)
The open circuit voltage is computed using the Nernst equation, where θ G r ∆ is the standard free reaction enthalpy for the oxidation of hydrogen with the reaction product being water vapor. For the anode side pure hydrogen at approximately standard pressure is assumed. Therefore hydrogen does not appear in the open circuit voltage.

(

)

        ⋅ - ∆ - = 2 1 O cell 0 cell / 1 ln 2 p p zF RT zF G U r θ (2)
The relation between current density cell i and cathodic overvoltage C

( ) ( )      - ⋅ - ⋅ =                 - - -         ⋅ - = - - 1 ref 1 cell a ref O CC O ref C, 0, C 0, C cell C C cell C C 0, cell 2 2 1 exp exp T T R E c c i i RT zF RT zF i i η α η α (3) )
The transport of protons through the membrane is driven by the potential difference M φ ∆ . The protonic conductivity M κ of the membrane depends on the temperature and on the water content of the membrane λ . The according relations are taken from [8].

M M cell M h i ⋅ = ∆ κ φ (4) ( ) ( ) ( ) ( ) ( ) ( ) 1 - 2 3 353K M, 1 1 cell 353K M, M m S 2658 . 0 0298 . 0 0013 . 0 K K 353 1183 6 . 0 exp 2640 exp λ λ λ κ λ κ κ + + = - ⋅ + - ⋅ - ⋅ = - - F o r P e e r R e v i e w 7
The ohmic resistances of the electrodes and the current collectors are comprised in the cell resistance cell

R . cell cell cell R A i ⋅ ⋅ = ∆ Ω φ (5)

Mass Balances

First the oxygen balance should be derived. The dynamics of the oxygen concentration in the cathodic gas channels is assumed to be fast in comparison with the water dynamics and the temperature dynamics. The quasi-stationary material balance reads:

zF i nA F RT p c F cell cell out cell CC CC O CC in, in O 2 1 / x 0 2 2 - - = (6)
The equation given above can be converted into an explicit expression for the oxygen concentration in the cathode channels CC O 2 c . The unknown cathodic molar outflow out F will be looked at the end of this subsection.

Next the water balance is developed. An approach according to [4] is followed here. The integral water balance around the whole stack reads:

zF i nA F x F x t n cell cell out CC O H CC in, in O H O H 2 2 2 d d + - = (7)
In the balance given above the anodic inflow does not appear as the hydrogen feed enters the cell dry.

The water is assumed to distribute homogeneously in the membrane and the cathode gas compartment. Due to its small volume the anodic gas compartment was neglected.

M O H M CC O H CC O H 2 2 2 c V c V n + = (8)
As mentioned before the cathodic gas compartment and the membrane are assumed to be in thermodynamic equilibrium. The membrane water concentration can therefore be expressed as a function of the channel water concentration and the temperature.

( )

cell CC O H M O H M O H , 2 2 2 T c c c = (9)
The expression given above is established by the following set of constitutive equations [8], forming the algebraic sub-system (10)-( 16).

ρ λ X X X + = (11) dry O H dry M O H dry O H dry M O H 2 2 2 2 1 X M X M M λ ρ ρ λ ρ ρ ρ + + = (12)
The water phase equilibrium relation reads or the channel relative humidity rH defined as ( )

( ) ( ) K 303 K 353 K 303 cell K 50 K 303 λ λ λ λ - ⋅ - + = cell T T (13) 3 O H 2 O H O H K
cell sat cell CC O H O H 2 2 T p RT c rH a = = (16)
For numerical treatment and the subsequent controller synthesis it is convenient to express the water balance (7) in terms of

t n d d O H 2
. Time derivation of Equations ( 8) and ( 9)

gives t T T c V t c c c V t c V t n d d d d d d d d cell cell M O H M CC O H CC O H M O H M CC O H CC O H 2 2 2 2 2 2 ⋅ ∂ ∂ + ⋅ ∂ ∂ + ⋅ = (17) Substitution of t n d d O H 2
in Equation ( 7) by the expression given above yields

zF i nA F x F x t T T c V t c c c V V cell cell out CC O H CC in, in O H cell cell M O H M CC O H CC O H M O H M CC 2 2 2 2 2 2 d d d d + - = ⋅ ∂ ∂ + ⋅         ∂ ∂ + (18)
Finally, a relation for the cathodic molar outflow out F should be derived.

Starting point is the total material balance for the cathode channel.

F o r P e e r R e v i e w 9 t c V zF i nA F F t c V d d 2 1 d d M O H M cell cell out CC in, CC t CC 2 - + - = (19)
The last term in the equation given above considers the amount of water to be exchanged with the membrane instantaneously in order to obey the thermodynamic equilibrium (9). Time derivation of Equation ( 9) and multiplication with M V gives:

t T T c V t c c c V t c V d d d d d d cell cell M O H M CC O H CC O H M O H M M O H M 2 2 2 2 2 ⋅ ∂ ∂ + ⋅ ∂ ∂ = (20)
Using the ideal gas law and considering the assumption of constant pressure Equation ( 19) can be finally converted to

t T RT p V t T T c V t c c c V zF i nA F F d d d d d d 2 1 cell 2 cel CC CC cell cell M O H M CC O H CC O H M O H M cell cell CC in, out 2 2 2 2 ⋅ + ⋅ ∂ ∂ - ⋅ ∂ ∂ - + = (21)

Energy Balance

The energy balance for the system reads with H & being enthalpy flows and el P and th Q being the electrical power and the heat flow drawn from the system:

th el out in d d Q P H H t E - - - = & & (22)
By incorporating the model assumptions and assuming temperature independent heat capacities the energy balance can be transformed as follows:

( ) ( ) th el cell cell cell amb gas , in cell solid , d d Q P H zF i nA T T c F t T C r p p - - ∆ - ⋅ + - ⋅ =
The heat flow th Q leaving the stack is modelled to be linearly dependent on the temperature difference between fuel cell and environment.

( ) Using those relations and substituting the electrical power in the energy balance gives the following compact dynamical equation for the cell temperature:

amb cell th T T k Q env - ⋅ =
( ) ( ) ( ) cell th cell cell env gas , in cell amb cell solid , d d U U i nA k c F T T t T C p p - ⋅ + + ⋅ - = (25)
Equation ( 25) is used in the model, where in F is the total molar flow entering the system.

zF i nA F F cell cell CC in, in + = (26)
The heat transport coefficient env k is dependent on ext F and fitted to experimental data.

( ) ( )

1 1 1 1 ext 1 env K W 05 . 0 K W 301 . 0 s mol 078 . 0 - - - - - + + ⋅ = F k (27)
Additionally, the heat capacity of the gas gas , p c does not vary much with composition and is assumed to be constant.

Resulting Fuel Cell Model

The resulting fuel cell model consists of two differential equations, the equation for the water content (18) and the equation for the cell temperature (25). For the design of the controller, these equations have to be linked to the power of the air blower that will be used as the manipulated variable. To do this, it is assumed that the The model is simulated in MATLAB using the parameters given in Table 3.

Simulation and Comparison to Experimental Data

The stationary solutions for a range of the current density are shown in Figure 3. The other parameters are kept constant. Both the water vapor concentration in the cathode channels and the cell temperature rise roughly linear with the current density. This leads to a single maximum for the relative humidity in the channels (Figure 3d). Depending on the parameters this maximum may be significantly higher than the environmental humidity. Only a well humidified membrane has a good conductivity for protons. The effect of self-humidification is therefore necessary for operation of the system without additional humidification of the inlet gas streams.

A similar behavior of the humidity can be observed when the airflow is varied at a constant cell current (not shown here). Temperature and water concentration increase approximately linearly with an increase of the reciprocal of CC in,

F

. Again, this leads to a single maximum in the steady state humidity as a function of CC in, F . To validate the stationary behaviour of the model, impedance measurements were taken at a voltage frequency of 1 kHz. At such a rather high frequency the fuel cell's impedance can be seen as a measure for its ohmic resistance and hence for the humidification of the membrane. Figure 4 shows a comparison between the simulated ohmic resistance and the measured impedance. The qualitative agreement is good. Both curves show a distinct minimum that is due to a maximum of humidity as described above. The absolute values of simulated resistance and measured impedance deviate, because the simulated value includes the membrane resistance only, whereas the measured impedance additionally contains other ohmic resistances that are independent of the current density. Further, there is a slight offset between the simulated and the measured minimum of the impedance. This can be attributed to unmodelled loss mechanisms such as local drying of the membrane due to osmotic drag. These losses lead to an increased heat production and therefore a lower humidification of the membrane than the model would predict. (left) simulation, membrane impedance only, (right) measurement of the whole system. Measurement setup as described in [13]. The airflow is kept constant. The minimum impedance corresponds to a maximum of humidification as shown in Figure 3d.

The validation of the dynamic behavior of the model was done based on a preliminary fuel cell stack design that differs slightly from the final design used for the control studies. The reason is that the preliminary stack design allowed humidity measurements in the stack, which are much harder in the final layout. The preliminary stack differs from the final design in the following points relevant for the model: six instead of ten fuel cells in the stack; no active external cooling, i.e. the heat transfer coefficient env k is constant; higher total heat capacity

C

had to be adapted for comparison with experiments, as indicated in Table 3.

The transient behavior of the fuel cell model shown in Figure 5 reflects the different time constants for the dynamics of temperature and water vapor concentration. Steps of the current density were applied to the system while keeping the other parameters constant (Figure 5a). Changes of the temperature are found to be slower than those of the water vapor concentration. This results in an overshoot of the relative humidity (Figure 5d), because the saturation pressure also changes more slowly than the partial pressure of water vapor. On the rising slope of the relative humidity water is bound by the membrane and partially released on the falling slope. As a consequence there is also an overshoot of the water vapor concentration. Its maximum is shifted towards the falling slope of the relative humidity.

After a step to a higher current density the cell voltage drops instantaneously according to the actual ohmic resistance of the cell. The voltage then reaches temporarily higher values due to better humidification and lower membrane resistance. The maximum of the cell voltage therefore correlates with the maximum of the relative humidity.

A comparison with the experimental results in Figure 6 indicates a reasonable accuracy of the model. One difference is that the simulated temperature approximately follows a linear first-order step response, whereas the measured temperature rises very slowly in the immediate vicinity of the steady-state (Figure 6b). This may be explained by the slow heating of the endplates, which affect the measured temperature in the middle of the stack. To include this effect in the model, the assumption of a homogeneous stack temperature would have to be dropped.

Development of a Control Scheme

In this section a global control algorithm is developed based on analysis of the phase portraits of the two-dimensional system (18), (25). Because there is only one manipulated variable -the airflow -, but two control variables -temperature and water content -, it is not possible to control both control variables simultaneously. Instead, the objective is to formulate a control algorithm that takes the system in some suboptimal way to the desired set point while avoiding critical system states. In the present case, critical system state mainly means a state where the humidity exceeds The control algorithm is developed in three steps. First, the dynamic behavior of the system is analyzed by generalized phase portraits. This gives a deeper insight on the global effect of different airflows. In the second step, a suitable steady state set point is chosen. Again due to the single manipulated variable, the set point cannot be chosen arbitrarily, but has to be feasible in the sense that there must be some constant airflow leading to this steady state. The third step consists in the design of a controller. The control algorithm should be simple enough for implementation on a microcontroller. Further, it should be robust against model errors, as it is based on a simple qualitative model. These two points are more important than an excellent performance of the closed-loop system.

Analysis of the Airflow Influence by Generalized Phase Portraits

As the fuel cell model is of second order, its dynamic behavior can be visualized conveniently by phase portraits. However, a classical phase portrait constructed for a single value of the airflow is not very useful for controller design. Instead one is interested to know how choosing a specific airflow value determines the timederivatives of the system states and thus the "moving direction" of the system in the state space. If one evaluates the system dynamics at one point in the state space for all possible airflow values, one gets all directions the system can be forced to move, starting from this system state. Such an analysis is depicted in Figure 7. Figure 7 shows the angles of direction in which the system can move from different positions in the state space. One should distinguish the ideal case of an unlimited airflow between zero and infinity and the realistic case of an airflow in a certain interval. The lower bound of this interval is defined by the fuel cell's oxygen demand. Oxygen depletion reduces the efficiency of the fuel cell. Therefore the airflow is chosen to always be greater than is necessary for an oxygen stochiometric factor of three. The upper bound is given by the fact that the blower can only produce a limited maximal airflow. The influence of the restricted and the unrestricted choice of the airflow on the system is indicated in Figure 7 by dark grey and light grey respectively. Figure 7 can be used to choose feasible steady state operation points and to construct trajectories that lead to these desired operation points without entering the forbidden shaded area where condensation occurs. For example, it can be seen thatindependent of the chosen control algorithm -a limited airflow is not able to prevent condensation for certain states at low temperatures below 295 K. This is because trajectories starting at points close to the shaded forbidden area always point into that area, no matter what airflow is chosen. If one wants to avoid condensation at low temperatures, one can do that by increasing the maximum power of the blower. On the other hand, if the cell temperature is sufficiently high, above about 305 K, then it will always be possible to avoid condensation by suitable control. Figure 8 shows the possible moving directions of the system analogously to Figure 7 but with the additional information of the speed of change for all possible directions. The speed of change is computed by

2 cell scal cell, 2 CC O H CC scal O, H d d 1 d d 1 d d 2 2         +         = = t T T t c c t x v (28)
and deforms the circular area segments of Figure 7. It can be seen in Figure 8 that changes of the water vapor concentration are much faster than changes of the temperature. This can be explained by the bigger storage capacities for heat than for water in the system. For practical purposes it is often sufficient to keep the system states in a region around the operation point. This operation window is indicated by a light grey rectangular area in the following Figures.

Choice of Steady State Set Points

Choosing the Locally Best Airflow

As first step towards the final control algorithm, a rather simple approach is tested. The idea is always to choose the airflow that provides the fastest approach to the set point in the current state. The speed of approach is defined here as the negative change of the squared distance to the center of the operation window and computed as follows.

( ) ( )

setcenter cell cell cell setcenter CC, O H CC O H CC O H cell 2 scal cell, cell CC O H 2 CC scal O, H CC O H 2 scal cell, cell 2 CC scal O, H CC O H , d d d d d d 2 1 2 2 2 2 2 2 2 2 T T T c c c t T T T t c c c T T c c t v ∆ - = ∆ - = ∆ ∆ - ∆ - =                   ∆ +         ∆ - = (29)
At each point in the state space, the optimal airflow is chosen that maximizes v in (29). The result is given in Figure 9. One can see several things from this diagram. First, the simple approach is able to take the system from nearly arbitrary initial states to the desired operation window marked by the grey box. The only exceptions are initial states with a low temperature and high humidity, where the blower is too weak to prevent condensation. The second point is that the air blower nearly always operates on minimum or maximum power. If the temperature is too high, the airflow is set to a maximum to cool down the system. If the temperature is too low, the minimum airflow is chosen in order to accumulate reaction heat. This rule is broken only at the border to the condensation area, where a high airflow is needed to blow out redundant water. If the temperature is on the desired level one can increase humidity by a small airflow or decrease it by a high airflow. The qualitative behavior shown in Figure 9 does hardly change for different parameter sets and operation windows. These deviations inspire an even simpler control algorithm presented in the following.

Formulation of the Control Algorithm

Based on the results of the previous section, a simple control algorithm is proposed for the miniature fuel cell stack:

• Rule 1: If the system is "above" or "right" of the operation window, apply the maximal airflow.

• Rule 2: If the system is "below" or "left" of the operation window, apply the minimal airflow.

• Rule 3: If the relative humidity violates an upper threshold, apply the maximal airflow. This rule overrides the second rule.

For implementation the regions have to be described in terms of cell temperature and water vapor concentration. Less than the maximal airflow may be sufficient to prevent the fuel cell from flooding, but since quantitative values are not known the maximum is the safest value to choose. Additionally, linear transition zones are defined between regions with different airflows to prevent chattering. Rapid changes of the airflow cannot be realized because of the inertia of the blower and the air itself. This problem is reduced by continuously changing the airflow.

The control algorithm can be expressed in terms of cell temperature and water vapor concentration by the following relations. The linear transition zones are left out here for the sake of readability. The operation window is characterized by the

F F F rH rH F rH rH F F F c c T T T T F c c T T T T F F nom =      < ≥ =            < ∧ < ∨ < > ∧ > ∨ > = (30)
A graphical representation of the control algorithm is shown in Figure 10c. The resulting trajectories of the closed-loop system (Figure 10d) lead to the operation window and are composed of trajectories relating to the maximal and minimal airflow (Figures 10a and10b) in the respective regions as well as trajectory parts that lie in the region boundaries. Inside the operation window a nominal airflow is applied that keeps the system at the operation point. For practical application this value can be obtained from steady-state measurements. In summary, the problem of two control variables and one manipulated variable is solved by focusing on the temperature in most cases. Only when the humidity in the cell becomes critical or when the temperature has the nominal value, the airflow is used to control the water concentration. The reason for the success of this strategy lies in the different time constants of the water and the energy balance. As the dynamics of the water concentration are very fast, one can quickly adjust the water concentration without disturbing the temperature strongly.

The proposed control algorithm is very simple. The rules depend only on the current states of the system. The implementation on a microcontroller should therefore pose no problem. The demands on sensor equipment are also modest, as only a temperature and a humidity measurement are required. 5

Conclusions

In the present work a controller has been designed that adjusts temperature and humidity of a miniaturized fuel cell system by using the airflow of a blower as a single manipulated variable. The controller design is based on a phase plane analysis of a qualitative fuel cell model. The resulting controller is simple and easy to implement. It only needs measurement information that is readily available in a miniature fuel cell stack. The control law relies mainly on the qualitative properties of the model, and therefore should be robust against model errors.

It has been shown that the controller is able to take the system from a rather large region of attraction to the desired operation window. Limitations exist for low fuel cell temperatures, where it may be hard to heat up the stack and to prevent condensation simultaneously. These difficulties may be overcome by modifications of the system design like a stronger air-blower. Another alternative might be to use the cell current as an additional manipulated variable, if permitted by the actual application. simulation, membrane impedance only, (right) measurement of the whole system. Measurement setup as described in [13]. The airflow is kept constant. The minimum impedance corresponds to a maximum of humidification as shown in Figure 3d.

c 3 m mol 1 - - gas , p c 1 1 mol K J 30 - - - solid , p C * ) K J 64 ( K J 60 1 1 - - - CC in, ext / F F * ) 0 ( 25 - max F 1 s mol 0.01 - - M h m 10 18 6 - ⋅ Fraunhofer IZM Berlin cell i 2 m A 4000 - - env k * K W 08 . 0 1 - - n * ) 6 ( 10

3 Figure 1 :

 31 Figure 1: Scheme of the airflow in the fuel cell system. The light grey area in the center represents the fuel cell stack.

Figure 2 :

 2 Figure 2: Basic configuration of a H 2 -PEMFC with a) gas channels b) gas diffusion layers c) electrodes and d) the membrane. The anode side is operated in dead-end mode here.

  The computation of a single cell voltage includes the open-circuit voltage 0 cell U and the voltage loss mechanisms considered in this model, i.e. the cathode activation overpotential, resistance to proton transport in the membrane and resistance to electron transport in the electrodes and the current collectors.

Ω

  and oxygen concentration. Parameter values for the reaction kinetics are taken from[14].

(

  by the reaction (second term on right-hand side) can be rewritten using the heating value voltage th U :

F

  generated by the blower is equal to the gas flow at the cathode inlet To simplify this study, it is assumed that ext F can be set directly by the controller.

Figure 3 :Figure 4 :

 34 Figure 3: Stationary solutions for the variation of the current density. The airflow is kept constant at 1 CC in, s mol 0023 . 0 -

  are used. Therefore the three model parameters n , env k , solid p,

Figure 5 :

 5 Figure 5: System response to a current density step. The airflow is kept constant at 1 CC in, s mol 0024 . 0 -

Figure 6 :

 6 Figure 6: Measurement of the system response to a current density step. Measurement setup as described in [13]. The airflow is kept constant; (a) current density (b) cell voltage (c) temperature (d) relative humidity (e) water vapor concentration. The presented data is for an experimental setup without external cooling.

Figure 7 :.Figure 8 :

 78 Figure 7: Possible directions of trajectories for an unlimited maximal inlet airflow CC in, 1 s mol 0016 . 0 F ≤ -

Figure 7

 7 Figure 7 also indicates feasible steady state set points. These are the points on the dash-dotted line where the opening of the grey segments flips from top to bottom.One can in principle choose any point on this line as set point. As Figure7is valid only for one fixed cell current, the line of feasible set points will shift if the electric

Figure 9 :F

 9 Figure 9: Direction of the maximal speed of change towards the operation point. The respective inlet airflow CC in, F is indicated by the color spanning from dark grey for minimal airflow to light grey for maximal airflow.

c

  which both describe intervals with maximal and minimal acceptable values.

Figure 10

 10 Figure 10: a) and b) show the phase portraits for the maximal and the minimal airflow respectively. The light grey area marks the operation window. c) Graphical representation of the control algorithm. The applied airflow depends only on the system states. d) The trajectories of the closed-loop system. The current density is fixed at 2 cell m A 4000 -

Figure 1 : 3 Figure 2 : 4 Figure 3 : 12 Figure 4 :

 13243124 Figure 1: Scheme of the airflow in the fuel cell system. The light grey area in the center represents the fuel cell stack.3

13

 13

Figure 5 : 14 Figure 6 : 15 Figure 7 :.

 5146157 Figure 5: System response to a current density step. The airflow is kept constant at 1 CC in, s mol 0024 . 0 -

  17

Figure 8 :Page

 8 Figure 8: Possible directions of trajectories (see Figure 7) rated with the respective speed of change. The values at each point (+) are normalized to the

Figure 9 :FFigure 10 :Figure 1 :

 9101 Figure 10: a) and b) show the phase portraits for the maximal and the minimal airflow respectively. The light grey area marks the operation window. c) Graphical representation of the control algorithm. The applied airflow depends only on the system states. d) The trajectories of the closed-loop system. The current density is fixed at 2 cell m A 4000 -

Figure 2 :Figure 3 :Figure 6 :Figure 7 :Figure 8 :Figure 9 :Figure 10 Figure 10

 2367891010 Figure 2: Basic configuration of a PEMFC 93x131mm (600 x 600 DPI)

Table 2

 2

	Page 25 of 42	Fuel Cells
	: Indices
	Symbol	Description
	A	anode
	AC	anode gas channels
	amb	ambient
	C	cathode
	CC	cathode gas channels
	ext	outside the fuel cell stack
	gas	gas phase
	in	inlet
	M	membrane
		Wiley-VCH

Wiley-VCH Fuel Cells