Reduced-order Robust Control of a Fuel Cell Air Supply System*

Proceedings of the 20th World Congress Proceedings of 20th Proceedings of the the 20th World World Congress The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World The Federation of Automatic Control The International International Federation of Congress Automatic Control Toulouse, France, July 2017 Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France, July Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 (2017) 96–101

Reduced-order Reduced-order Robust Robust Control Control ⋆of of a a Fuel Reduced-order Robust Control of a Fuel Fuel Cell Air Supply System ⋆ Cell Air Supply System ⋆ Cell Air ∗Supply System ∗∗ ∗∗∗

David David David David∗

Hern´ andez-Torres ∗ Delphine Riu ∗∗ Olivier Sename ∗∗∗ Hern´ a Hern´ andez-Torres ndez-Torres ∗ Delphine Delphine Riu Riu ∗∗ Olivier Olivier Sename Sename ∗∗∗ Hern´ andez-Torres ∗ Delphine Riu ∗∗ Olivier Sename ∗∗∗ Universit´e de Toulouse (INP, UPS; LAPLACE; CNRS); ∗ ∗ Universit´ de (INP, LAPLACE; CNRS); de Toulouse Toulouse (INP, UPS; UPS; LAPLACE; CNRS); ENSEEIHT, 2eee rue Charles Camichel, F-31071 Toulouse, France ∗ Universit´ Universit´ de Toulouse (INP, UPS; LAPLACE; CNRS); ENSEEIHT, 2 rue Charles Camichel, F-31071 Toulouse, France ENSEEIHT, 2 rue Charles Camichel, F-31071 Toulouse, France (e-mail: [email protected]). 2 [email protected]). Charles Camichel, F-31071 Toulouse, France (e-mail: ∗∗ENSEEIHT, (e-mail: [email protected]). G2ELab (UMR 5269 INPG-UJF-CNRS), 21 avenue des martyrs, ∗∗ [email protected]). ∗∗ G2ELab (e-mail: (UMR 5269 INPG-UJF-CNRS), 21 (UMR 5269 INPG-UJF-CNRS), 21 avenue avenue des des martyrs, martyrs, 38031 Grenoble, France [email protected]). ∗∗ G2ELab G2ELab (UMR 5269 (e-mail: INPG-UJF-CNRS), 21 avenue des martyrs, 38031 France (e-mail: [email protected]). ∗∗∗Grenoble, 38031 Grenoble, France (e-mail: [email protected]). GIPSA-lab, INPG, Universit´ e Grenoble Alpes, 11 rue des ∗∗∗Grenoble, France (e-mail: [email protected]). 38031 ∗∗∗ GIPSA-lab, INPG, Universit´ ee Grenoble, Grenoble 11 INPG, Universit´ Grenoble Alpes, Alpes, 11 rue rue des des math´ e matiques, 38402 France ∗∗∗ GIPSA-lab, GIPSA-lab, Universit´e Grenoble, Grenoble Alpes, 11 rue des math´ eeINPG, matiques, math´ matiques, 38402 38402 Grenoble, France France (e-mail: [email protected]). math´ ematiques, 38402 Grenoble, France (e-mail: [email protected]). (e-mail: [email protected]). (e-mail: [email protected]). Abstract: This paper presents the design of several reduced order robust control strategies Abstract: This presents the design several order Abstract: This paper paper presents thecell design of generator. several reduced reduced order robust robust control control strategies for the air supply of a hybrid fuel powerof The management the airstrategies dynamic Abstract: This paper presents thecell design of generator. several reduced order robustof control strategies for the air supply of a hybrid fuel power The management of the air dynamic for the air supply of a hybrid fuel cell power generator. The management of the air dynamic entering the fuel cell is assured by the control of the air flow of a compressor. The air supply subfor the air supply ofis hybrid by fuelthe cell powerof generator. of the air dynamic entering the fuel control the flow of compressor. The air supply subentering the fuel cell cell isaassured assured by the control of the air air flowThe of aamanagement compressor. The aircell supply subsystem is controlled to keep a desired oxygen excess ratio, thus improving the fuel dynamic entering the fuel cell is assured by the control of the air flow of a compressor. The air supply subsystem is controlled to keep a desired oxygen excess ratio, thus improving the fuel cell dynamic system is controlled to keep a desired oxygen excess ratio,these thus robust improving the fuelare cell dynamic performance. Robustness analysis studies are performed, properties contrasted system is controlled to keep a desired oxygen excess ratio,these thus robust improving the fuelare cell dynamic performance. Robustness analysis studies are performed, properties contrasted performance. Robustness analysis studies are performed, these properties are contrasted with classic control strategies, demonstrating the advantage of robust the multivariable reduced order performance. Robustness analysis studies are performed, these properties are contrasted with control strategies, demonstrating the of the reduced order with classic classic control strategies, demonstrating the advantage advantage of robust the multivariable multivariable reduced order robust methodologies. with classic control strategies, demonstrating the advantage of the multivariable reduced order robust methodologies. robust methodologies. robust © 2017, methodologies. IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Robust control, multivariable control, fuel cells, hybrid power generators, air Keywords: Robust Robust control, control, multivariable multivariable control, control, fuel fuel cells, cells, hybrid hybrid power power generators, generators, air air Keywords: compressor. Keywords: Robust control, multivariable control, fuel cells, hybrid power generators, air compressor. compressor. compressor. 1. INTRODUCTION the thermo-dynamical sub-systems on the FC. The distur1. INTRODUCTION INTRODUCTION the thermo-dynamical sub-systems on the FC. The 1. the thermo-dynamical sub-systems on in thethis FC.article, The disturdisturbance rejection approach is considered since, 1. INTRODUCTION the thermo-dynamical sub-systems on in thethis FC.article, The disturbance rejection approach is considered since, bance rejection approach is considered in this article, since, as a general proposition, this will have an influence the Fuel cells (FC) are electro-chemical devices that convert bance rejection approach is considered in this article,on since, as a general proposition, this will have an influence on the Fuel cells (FC) are electro-chemical devices that convert as a general proposition, this will have anenergy influence on the Fuel cells energy (FC) are electro-chemical devices that convert cells life-span, avoiding non-desired high transients. chemical into electrical energy by means of an as a general proposition, this will have an influence on the Fuel cells energy (FC) are electro-chemical devices that convert cells life-span, avoiding non-desired high energy transients. chemical into electrical energy by means of an cells life-span, avoiding non-desired high energy transients. chemical pair, energy into electrical energy by means of an It should be noted that limitations on the FC performance electrode an electrolyte and a catalyzer. cells life-span, avoiding non-desired high energy transients. chemical energy into electrical energy by means of an It should beonly noted that limitations on the the FC FC performance electrode pair, pair, an an electrolyte electrolyte and and aa catalyzer. catalyzer. It be noted limitations on performance electrode areshould due not to that material characteristics, but also on the should beonly noted that limitations on the FC performance electrode and aWhen catalyzer. A fuel cellpair, may an notelectrolyte operate alone. referring to a FC It are due not to material characteristics, but also on are due not only to material characteristics, but also on the the optimization of the thermo-dynamical operating variables A fuel cell may not operate alone. When referring to a FC are due not only to material characteristics, but also on the A fuel cell may not operate alone.needed When for referring to a FC system, all the auxiliary systems, operation, are optimization of the thermo-dynamical operating variables optimization of the thermo-dynamical operating variables as, for example, the flow rate of reactant streams (Gasser, A fuel cell may not operate alone. When referring to a FC system, all allThis the auxiliary auxiliary systems, needed for aoperation, operation, are optimization of the operating variables system, the systems, needed for are included. makes the complete system rather comfor the flow of streams (Gasser, as, for example, example, the thermo-dynamical flow rate of reactant reactant streams (Gasser, 2006). As a central partrate of the FC dynamic behaviour, system, allThis the auxiliary systems, needed for aoperation, are as, included. makes the complete system rather comas, for example, the flow rate of reactant streams (Gasser, included. This makes the complete system a rather complex structure to control. The auxiliary elements can be 2006). As a central part of the FC dynamic behaviour, 2006). As a central part of the FC dynamic behaviour, control ofcentral the compressor-motor and the behaviour, supply air included. This to makes the complete system a rathercan complex structure structure to control. The auxiliary auxiliary elements can be the 2006). As a part of the FC dynamic plex control. The elements be divided into two groups: electrical and thermo-dynamical the control of the compressor-motor and the supply air the control of the compressor-motor andbethe supply air are considered. Several articles can found in the plex structure togroups: control.electrical The auxiliary elements can be flow divided into two two groups: electrical and thermo-dynamical the control of the compressor-motor and the supply air divided into and thermo-dynamical management sub-systems. The first ensures the safe conflow are considered. Several articles can be found in the flow are considered. Several articles can be found in the literature addressing these problems. An interesting study divided into two groups: electrical and thermo-dynamical management sub-systems. The first ensures the safe conflow are considered. Several articles can be found in the management sub-systems. The first ensures the safe connection of thesub-systems. FC to the electrical and literature addressing problems. study addressing these these problems. An interesting study on the air compressor control influenceAn on interesting the FC efficiency management The firstapplication ensures the(load) safe connection of the the FC to to the theensures electrical application (load) and literature literature addressing these problems. An interesting study nection of FC electrical application (load) and the second sub-system the gas, water, air on the air compressor control influence on the FC efficiency on the air compressor control influence on the FC efficiency presented in Tekin control et al. (2006). Linear control of the nection of the FC to theensures electrical application (load) and is the second second sub-system ensures the gas, water, water, air and on the air compressor influence on the FC efficiency the sub-system the gas, air and thermal management of the FC. A complete robust control is presented in Tekin et al. (2006). Linear control of is presented in Tekin et al. of the the system on(2006). a FC isLinear studiedcontrol using modelthe second sub-system ensures gas, water, and thermo-dynamical thermal management ofby the FC. Athe complete robustair control is presented in Tekin et al. (2006). Linear control of the thermal management of the FC. A complete robust control solution was presented the authors in Hernandez-Torres thermo-dynamical system on a FC is studied using modelthermo-dynamical system on a FC is studied using modelbased control in Grujicic et al. (2004). Multivariable robust thermal management of the FC. A complete robust control solution was presented presented bythese the authors authors in Hernandez-Torres Hernandez-Torres systemet on FC is studied using modelsolution was the in et al. (2011). Of course,by sub-systems are both highly thermo-dynamical based control in Grujicic al.a(2004). (2004). Multivariable robust based control Grujicic al. H low in order PID’set controls areMultivariable proposed in robust Wang solution was presented bythese the authors in Hernandez-Torres ∞ and et al. (2011). Of course, these sub-systems are both highly based control in Grujicic et al. (2004). Multivariable robust et al. (2011). Of course, sub-systems are both highly dependent, as the gas flow in the stack directly depends on H and low order PID’s controls are proposed in Wang ∞ H and low order PID’s controls are proposed in Wang ∞ et al. (2008) and Wang and Ko (2010) respectively. An et al. (2011). Of course, these sub-systems are both highly dependent, as the gas flow in the stack directly depends on H∞al.and low order PID’sand controls are proposed in Wang dependent, as thedemand, gas flowthen in thethe stack directly depends on the load current electrical sub-system is et (2008) and Wang Ko (2010) respectively. An et al. (2008) and Wang and Ko (2010) respectively. An LPV control approach for the FC air supply was presented dependent, as the gas flow in the stack directly depends on the load current demand, then the electrical sub-system is et al. (2008) and Wang and Ko (2010) respectively. An the load current demand, then theIn electrical sub-system is LPV influenced by the flow dynamics. terms of the FC efficontrol approach for the FC air supply was presented LPV control approach for the FC air supply was presented in Hernandez-Torres et al. (2012). Non-linear control has the load current demand, then the electrical sub-system is influenced by the flow dynamics. In terms of the FC effiLPV control approach for the FC air supply was presented influenced by the flow dynamics. In terms of the FC efficiency, the by stack a betterInperformance higher in Hernandez-Torres et al. al. (2012). (2012). Non-linear control has Hernandez-Torres et Non-linear control has also been used with good using feed-back linearizainfluenced thewill flowhave dynamics. terms of theat effi- in ciency, the stack stack will have better performance atFC higher in Hernandez-Torres et al.results (2012). Non-linear control has ciency, the will have aa better performance at higher flow pressures, but this means higher compression ratios, also been used with good results using feed-back linearizaalso been used with good results using feed-back linearization and passivity in Na and Gou (2008) and Talj et al. ciency, the stackbut willthis have a better performance at ratios, higher also been used with good results using feed-back linearizaflow pressures, pressures, but this means higher compression ratios, flow means higher compression higher energy consumption of the compressor-motor subtion and passivity in Na and Gou (2008) and Talj et al. tion and passivity in Na and Gou (2008) and Talj et al. (2009) respectively. Model predictive optimal control has flow pressures, but this means higher compression ratios, higher energy energy consumption of the the compressor-motor subtion and passivity in Na and Gou (2008) and Talj et al. higher consumption of compressor-motor subsystem and a degraded overall system efficiency. An effi(2009) respectively. Model predictive optimal control has (2009) respectively. Model predictive optimal control has also been used for this type of systems in Chang and Moura higher energy consumption of the compressor-motor subsystem and a degraded overall system efficiency. An effi(2009) respectively. Model predictive optimal control has system and a degraded overall system efficiency. An efficient necessary to guarantee an optimal been also been used used for for this this type type of of systems systems in in Chang Chang and and Moura Moura (2009). systemcontrol and a is overall system efficiency. Anmaneffi- also cient control isdegraded necessary to in guarantee an avoiding optimal manalso been used for this type of systems in Chang and Moura cient control is necessary to guarantee an optimal management of hydrogen and air the system, stack (2009). (2009). cient control is necessary to in guarantee an avoiding optimal management of degradation hydrogen and air air in the system, system, stack (2009). agement of hydrogen and the stack In this work some assumptions are made to obtain simmembrane output voltageavoiding degradation) agement of degradation hydrogen and(and air in the system, avoiding stack In this this control-oriented work some some assumptions assumptions are made made to control, obtain simsimmembrane (and output voltage degradation) work are to obtain membrane degradation (and output voltage degradation) In plified models. The fuel flow the for a more reliable and efficient operation. this control-oriented work some assumptions are made to control, obtain simmembrane degradation (and output voltage degradation) In plified models. The fuel the for aa more more reliable reliable and efficient efficient operation. plified control-oriented models. The fuel flow flow control, the for and operation. air humidification and the stack temperature controls are plified control-oriented models. The fuel flow controls control, the for a more andaefficient Within thereliable context, need isoperation. identified for a robust air humidification and the stack temperature are air humidification and the stack temperature controls are assumed to be perfect. Robust control is of special interWithin the the context, context, needcontrol is identified identified fortoaa manage robust air humidification and the stack temperature controls are Within aaorder need is for robust multivariable reduced approach assumed to be perfect. Robust control is of special interassumed to be perfect. Robust control is of special interest, aiming to achieve robust stability and performance Within the context, a need is identified for a robust multivariable reduced reduced order order control control approach approach to to manage manage assumed to be perfect. Robust control is of special intermultivariable est, aiming to to achieveto robust robust stability In andthis performance ⋆ est, aiming achieve stability and performance for systems subject uncertainties. paper we multivariable approach to manage This work wasreduced possible order thankscontrol to the financial support of the est, aiming to achieveto robust stability In andthis performance ⋆ for systems subject uncertainties. paper we ⋆ This work was possible thanks to the financial support of the for systems subject to uncertainties. In this paper orwe This work was possible thanks to the financial support of the demonstrate the advantage of a multivariable reducer Grenoble Institute of Technology. Dr. Hern´ a ndez-Torres formerly ⋆ This work was possible thanks to the financial support of the for systems subject to uncertainties. In this paper we demonstrate the advantage of a multivariable reducer orGrenoble Institute of Technology. Dr. Hern´ a ndez-Torres formerly demonstrate the advantage of a multivariable reducer orGrenoble Institute of Technology. Dr. the Hern´ andez-Torres formerly with G2ELab in Grenoble, is now with University of Toulouse. demonstrate the advantage of a multivariable reducer orGrenoble Institute of Technology. Dr. the Hern´ andez-Torres formerly with in is University of with G2ELab G2ELab in Grenoble, Grenoble, is now now with with the University of Toulouse. Toulouse.

with G2ELab in Grenoble, is now with the University of Toulouse. Copyright © 2017, 2017 IFAC 98 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 98 Copyright ©under 2017 responsibility IFAC 98 Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 98 10.1016/j.ifacol.2017.08.017

Proceedings of the 20th IFAC World Congress David Hernández-Torres et al. / IFAC PapersOnLine 50-1 (2017) 96–101 Toulouse, France, July 9-14, 2017

der control approach over the classical decoupled control methodologies and the validation with complete closedloop robustness analysis.

2

VAIREX 520.08 24VDC Compressor flow map with speed contours in rpm

1.9 1.8

2. SYSTEM MODEL AND PARAMETER IDENTIFICATION

00 35

0

1.2

10

00

1.1 1

300

00 15

1.3

00

1.4

25

1.5 00

A test-bench fuel cell setup available in LEPMI laboratory (http://lepmi.grenoble-inp.fr/) is used for model identification of the fuel cell air supply system. The test-bench is composed by different elements that allow to efficiently control, operate and visualize several parameters of the FCS such as: fluid dynamics (pressure, gas flow rates, reactant stoichiometric values, etc.), thermo-dynamic parameters (stack temperature, humidification rates, etc.), or even electrical parameters (single cell voltage, stack current and power).

1.6

20

Pressure ratio

1.7

2.1 Test bench setup for system model identification

0

2

4 6 Mass flow (g/s)

8

10

Fig. 1. Compressor operation map.

prm

The stack considered in this work is a PEMFC from French c and mounted on a UBzM c frame company Paxitech with a nominal power of 475W. The arrangement is composed by a 16-cell/100 cm2 effective area stack. The air channel entering at the FC stack input is controlled by air flow and pressure regulators. For air flow control the setup is equipped with Smart series 5800S models from Brooks c , with reading/regulation capabilities using Instruments  0 − 5V control signals. For pressure reading purposes the c piezoresistive test-bench is provided by series 21 Keller  transmitters. For the pressure reading/regulation option, c was a VP50 proportional control valve from Norgren  installed for air back-pressure control.

Wcp

pcp,in =patm Tcp,in =Tatm

Compressor Map

pcp

According to the requirements of the FCS in the testbench, an air compressor was chosen to guarantee at least an air flow of 60 slpm 1 and a pressure range of 1 − 2 bars (absolute). With these characteristics and allowing an over-sizing factor for future stack expansion in the c VV-0520.08 INT model dry fixed test-bench, the Vairex vane air compressor was chosen. The compressor has an operating range of 24 − 40V and nominal power of 900 − 1200W. The air flow ranges from 0.2−10g/s at compression pressures of 1.1 − 2 bars (abs). The compressor layout and the compressor map as given by manufacturer are given in Figure 1.

Vcm

Tcp,out

ωcp

Compressor and Motor inertia

Fig. 2. Compressor model block diagram (Pukrushpan et al., 2004). air flow rate. The state in the model is the compressor speed ωcp . The inputs to the model are the air pressure patm and temperature Tatm (atmospheric), the supply manifold pressure psm (downstream pressure) and the voltage command of the compressor voltage vcm .

2.2 Air supply model Compressor modeling is somehow considered as the core of the fuel cell model. Two proposed models are presented in the following sections. First, the non-linear model proposed by Pukrushpan et al. (2004), and secondly, a linear model is proposed by Gasser (2006).

The compressor flow map can be normally found in the device manufacturer data-sheet. In Pukrushpan et al. (2004) the Jensen & Kristensen map regression method is presented to compute the compressor air flow rate, avoiding the use of look-up tables (not suitable for simulations). This method is presented with details in Pukrushpan et al. (2004) and the air flow rate Wcp is computed using a regression algorithm on the compressor map and the compressor diameter dc in m, the air density ρa in kg/m3 , and the air gas constant Ra .

The first compressor model presented here is a non-linear compressor model proposed by Pukrushpan et al. (2004). The block diagram representation of this model is shown in Figure 2. The compressor model is composed by a static compressor map, used to determinate the air flow rate through the compressor, and by a dynamic model of the system inertia, which determines the compressor speed used to find the 1

97

A look-up table is used to obtain the compressor efficiency ηcp . Then the temperature of the air leaving the compressor is computed by:

Standard liters per minute

99

Proceedings of the 20th IFAC World Congress 98 David Hernández-Torres et al. / IFAC PapersOnLine 50-1 (2017) 96–101 Toulouse, France, July 9-14, 2017

Tcp

Tatm = Tatm + ηcp

��

psm patm

� γ−1 γ

−1

�

Then the torque required to drive the compressor, is given by:

τcp

Cp Tatm = ωcp ηcp

��

psm patm

� γ−1 γ

�

− 1 Wcp

dωcp = (τcm − τcp ) dt

(3)

with Jcp the combined compressor and motor inertia in kg.m2 , ωcp the compressor speed in rad/sec, and τcm the compressor motor torque input. Substituting equation (2) into equation (3) yields to the final expression for the supply manifold dynamic. Today the use of permanent magnet DC motor is attractive due to efficiency and performance. In that case the voltage equation for the DC machine is: vcm = icm Rcm + vcm,ind

(4)

The induced voltage and torque τcm are given by: vcm,ind = kv ωcp τcm = kt icm

(5) (6)

The second compressor model presented in this section is a linear compressor model proposed in Gasser (2006). The diagram representation of this model is shown in Figure 3. icm

Rcm

vcm +_

patm

Lcm vcm,ind

M

τcm

psm

τcp

Fig. 3. Compressor model diagram (Gasser, 2006). The voltage equation of the circuit is given by: vcm = Lcm

dicm + Rcm icm + km ωcp dt

(8)

For the compressor characteristic, the relationship between the output mass flow rate Wcp and the compressor speed ωcp is assumed to be linear by a factor km,cp . A friction torque τf ric = kf ωcp is also considered in the equations. The complete linear model for the compressor-motor system is given by:

(2)

where the torque τcp is in N.m and Cp is the specific heat capacity of the air equals 1004 J.kg-1 .◦ K-1 . Both the temperature equation (1) and the torque equation (2) of the compressor are derived from turbine literature. The dynamic speed-inertia equation is given by: Jcp

τcp = kp (patm − psm )

(1)

(7)

The torque equation of the system is given by equation (3) and the DC machine torque equation is given by equation (6), as the DC machine of the compressor-motor proposed in Gasser (2006) is also a permanent magnet DC motor. The torque of the compressor is obtained by: 100

     

dicm dt dωcp dt dpsm dt





Rcm Lcm kt Jcp

−

    =     

0

1  Lcm   + 0   0 

ke 0 Lcm kd kp − − Jcp Jcp RTcp RTcp km,cp − Vsm Mair Vsm Mair kh −

0

kp Jcp RTcp Vsm Mair kh The system output is: �



� �  icm   ωcp  p  sm (9)



� �   vcm  p  atm 

� � � � icm � 0 km,cp 0 Wcp ωcp = 0 0 1 psm psm

(10)

Using the test-bench set-up preseted before and the input signals presented in Figure 4, we obtain the parameter identification results for this second model. The identification results are resumed in Table 1. The linear model proposed by Gasser (2006) was found to be well adapted to the design of the compressor system control for mainly two reasons: identification results were good and this model includes the dynamic of the compressor motor current, which will be used later to consider a true autonomous system. The model response compared to measured data are compared in Figure 5. Refer to Hernandez-Torres (2011) for complete identification results. Table 1. Identification results for the model proposed by (Gasser, 2006) Parameter km,cp kh kp ke kt kd Jcp Rcm Lcm Vsm kp (PI) ki (PI)

Identified Values 1.7002 × 10−6 6088.1 × 104 3.7477 × 10−6 0.2634 0.1936 0.0588 2.4637 × 10−5 2.1805 0.0040 0.0010 0.4912 5.9854

Units kg rad

P a.s/kg m3 s V rad N.m/A s N.m rad kg.m2 Ω H m3 – –

3. AIR SUPPLY CONTROL In this section, a H∞ control strategy for the compressor system is proposed. The control objectives for the compressor system are chosen to comply with the desired

Compressor motor speed reference (rad/s)

Proceedings of the 20th IFAC World Congress David Hernández-Torres et al. / IFAC PapersOnLine 50-1 (2017) 96–101 Toulouse, France, July 9-14, 2017

where IDC is the current delivered by the hybrid boost converters at the DC bus.

20 10

We will present now the classic strategy used to obtain a simple PI controller for this system.

0 −10 −20

3.1 Classic control approach 0

2

4

6

4

Return manifold pressure (Pa)

4

8 Time (s)

10

8 Time (s)

10

12

14

16

A very simple control design for a PI is presented. We used the function sisotool in Matlab to find a PI controller that satisfies a gain margin larger than 6dB and a phase margin larger than 45◦ . A design that presents a good trade-off between the overshoot and a stabilization time ts around 2s was chosen. The controller found satisfies GM = 12.2dB and a P M = 60◦ . The gains are given by kp = 8.33 × 104 and ki = 5.21 × 105 . This PI controller for KλO2 has the classic structure KλO2 = kp + ki /s.

x 10

2 0 −2 −4

0

2

4

6

12

14

16

Fig. 4. Inputs for compressor model identification.

A proposed strategy for oxygen excess ratio control is presented in Figure 6. In this case the speed controller Kω (s) is given by the identified PI controller with gains kp = 0.4912 and ki = 5.9854.

Compressor motor current (A)

6 Estimated response

Measured data

4 2

−2

W* cp 0

2

4

6

8 Time (s)

10

12

14

-

K λO (s)

16

ωcp

-

Vcm

Kω(s)

2

40 Compressor motor speed (rad/s)

ω*cp

Wcp

0

−4

99

Fig. 6. Compressor proposed control strategy.

20

3.2 H∞ control approach

0 −20 −40

0

2

4

6

8 Time (s)

10

12

14

16

Fig. 5. Identification results of compressor model proposed by Gasser (2006). performance of the FC. Two main objectives are considered: keep a desired oxygen excess ratio λO2 (which can be formulated as the tracking of the compressor flow Wcp ) and maximizing the net power delivered to the load. Note that we consider an autonomous system, i.e., the compressor energy is taken from the DC bus. This is not an easy requirement to meet, a trade-off has to be made because rising the supply manifold pressure increases the FC performance, but rising the pressure increases the current consumed by the compressor directly from the FC stack itself. In a simplistic solution, this is partially achieved minimizing the overshoots and properly rejecting disturbance transients on the supply manifold pressure and on the compressor speed. Note that at higher speeds more current is consumed by the compressor-motor. The compressor system dynamic is composed by the compressormotor and the FC supply manifold dynamic. The control strategies described in this section are motivated by the test-bench and the air compressor for operation of the FC under dry air conditions.

In this section we will design full and reduced order H∞ controllers. The compressor system block diagram is presented in Figure 8 for the system proposed by Gasser (2006). Refer to Hernandez-Torres (2011) for the complete model equations. The proposed control configuration with a single performance weighting function is presented in Figure 7. This weight allows to consider the tracking objective of the compressor flow Wcp , and can be chosen as: 0.5s + 5 Wperf = (12) s + 5 × 10−8 ω Ifc Wperf ω*cp

z

Wcp -

Vst -

P

u

y K(s)

K Fig. 7. Control configuration in the P-K form for the compressor system.

Note that considering an autonomous system the following system currents relationship can be established:

The full order H∞ controller (order 5) obtained can be reduced to 1st order system, using a classic dominant order approach.

IDC = Iload + Icm

An H∞ controller was also computed using Matlab’s hinfstruct function and fixing the controller structure

(11) 101

Proceedings of the 20th IFAC World Congress 100 David Hernández-Torres et al. / IFAC PapersOnLine 50-1 (2017) 96–101 Toulouse, France, July 9-14, 2017

peaks shown in Figures 11 and 12, an improvement of the net power delivered by the FC is obtained with the MIMO H∞ control. The evolution of the operating point in the polarization curves given an idea of the degradation of cell performance after a big load current step, and the influence of the supply manifold pressure psm variation. These results show good compressor speed control.

Wcp prm

Ifc

Finally, several other dynamic results are obtained using the non-linear model, and applying a second series of load steps. A final result sequence is given in Figure 15. Note the slight oscillatory behavior of the compressor motor speed using the classic methodology. A smooth disturbance rejection is achieved on the supply manifold pressure. Figure 15 shows the good regulation of the compressor speed and the drop on the FC stack voltage after the disturbances.

psm Vst Compressor system

Fig. 8. Compressor system block diagram. to a MIMO PI. The closed-loop H∞ gain for each robust controller found are summarized in Table 2. The singular values plots of the sensitivity functions are compared in Figure 9. A slightly more robust controller is obtained with the full and reduced order H∞ controllers. Table 2. Closed-loop H∞ gain Controller Full order H∞ Reduced order H∞ Using hinfstruct Classic PI

Performance H∞ < 0.0788 H∞ < 0.1848 H∞ < 0.1865 H∞ < 0.1933

Reduced order H∞ controller Full order H∞ controller Classic PI controller

2.6 2.4 2.2

+20% Fuel cell current step

2 1.8 1.6 1.4

−20% Fuel cell current step

1.2 1

10

0

1

2

3

4

5 Time (s)

6

7

8

9

10

Fig. 10. Oxygen excess ratio simulation results using linear model.

γ/Wperf 0 −10 −20

1.9

Classic PI controller

−30

Reduced order H∞ controller

−40 Robust PI controller with function hinfstruct

−50 −60 −3 10

−2

10

−1

10

0

1

10 10 Frequency (rad/s)

2

10

3

10

Full order H∞ controller Classic PI control

+75% Fuel cell current step

1.8 Compressor oxygen excess ratio λO2

Singular values (dB)

3 2.8 Compressor oxygen excess ratio λO2

ω*cp

1.7 1.6 1.5 1.4 1.3 1.2

Fig. 9. Closed-loop singular values plot.

1.1

Using the non-linear model of the compressor system, validation of robust controller is obtained. Dynamic variations on the operating point of the FC polarization curve and the compressor flow map are presented in Figures 13 and 14 respectively after ±75% load steps. According with these results, and associating these results with the overshoot 102

10

12

14 Time (s)

16

18

20

Fig. 11. Oxygen excess ratio simulation results using nonlinear model.

Full order H∞ controller Classic PI control

500 Net power delivered by the Fuel Cell (W)

In Figures 10 and 11 the controllers regulation of the oxygen excess ratio are compared using the linear and nonlinear models (from Gasser (2006) and Pukrushpan et al. (2004)). The reduced order H∞ controller is not shown in the non-linear simulations because the results obtained are very similar to those obtained with the full order H∞ control. In the linear simulation 20% load step are applied. In the non-linear case a 75% load step is applied. The nonlinear simulation results show a clearly better disturbance rejection using the proposed H∞ control. Figure 12 shows the evolution of the net power delivered by the FC after a series of load steps. Note the several overshoots in the transient response using the classic control.

400

Overshoot Pnet = Pfc − Pcm

300 200 100 0 −100 0

5

10

15

20

Time (s)

Fig. 12. Net power delivered by the FC.

25

Proceedings of the 20th IFAC World Congress David Hernández-Torres et al. / IFAC PapersOnLine 50-1 (2017) 96–101 Toulouse, France, July 9-14, 2017

4. CONCLUSION

1 Data-sheet polarization curve for pca = 1.5 bar and λO2 = 2

0.95 0.9 Cell voltage (V)

0.85

A reduced order robust control with H∞ performance is presented in this paper. The results obtained were satifatory when compared to a classical controller using PI control in terms of robustness and dynamics. The proposed control is multivariable with the advantage of implementation on a real system by using a reduced order controller. The obtained controller performance was validated using the non-linear model of the FC air supply system. A near future perspective in this work is the control validation on the real-time test-bench set-up.

Simulated polarization curves from identified model for λO2 = 1.6

+75% Fuel cell current step at t=10s

0.8 0.75 0.7

−75% Fuel cell current step at t=20s

0.65 0.6 0.55

0

0.1

0.2 0.3 0.4 Fuel Cell current density (A/cm2 )

0.5

0.6

REFERENCES

Fig. 13. Polarization curve.

0

120

1.55 Compressor pressure ratio

Chang, Y.A. and Moura, S.J. (2009). Air Flow Control in Fuel Cell Systems: An Extremum Seeking Approach . In American Control Conference, ACC 2009, 4299–4304. Gasser, F. (2006). An Analytical, Control-Oriented State Space Model for a PEM Fuel Cell System. Ph.D. thesis, ´ Ecole Polytechnique F´ed´erale de Laussane. Grujicic, M., Chittajallu, K.M., Law, E.H., and Pukrushpan, J.T. (2004). Model-based control strategies in the dynamic interaction of air supply and fuel cell. Proc. Instn Mech. Engrs., Part A: J. Power and Energy, 218. Hernandez-Torres, D. (2011). Robust control of hybrid electro-chemical generators. Ph.D. thesis. Hernandez-Torres, D., Riu, D., and Sename, O. (2012). An LPV Control Approach for a Fuel Cell Power Generator Air Supply System. In American Control Conference, ACC 2012, 4299–4304. Hernandez-Torres, D., Riu, D., Sename, O., and Druart, F. (2011). A robust multivariable approach for hybrid fuel cell supercapacitor power generation system. European Physical Journal of Applied Physics, 54(2). Na, W.K. and Gou, B. (2008). Feedback-LinearizationBased Nonlinear Control for PEM Fuel Cells. IEEE Transactions on Energy Conversion, 23(1), 179–190. Pukrushpan, J.T., Stefanopoulou, A.G., and Peng, H. (2004). Control of Fuel Cell Power Systems. Principles, Modeling, Analysis and Feedback Design. Springer. Talj, R., Ortega, R., and Hilairet, M. (2009). A controller tuning methodology for the air supply system of a PEM fuel-cell system with guaranteed stability properties. International Journal of Control, 82(9), 1706–1719. Tekin, M., Hissel, D., Pera, M., and Kauffmann, J. (2006). Energy consumption reduction of a PEM fuel cell motorcompressor group thanks to efficient control laws. Journal of Power Sources, 156, 57–63. Wang, F., Chen, H., Yang, Y., and Yen, J. (2008). Multivariable robust control of a proton exchange membrane fuel cell system. Journal of Power Sources, 177, 393– 403. Wang, F. and Ko, C. (2010). Multivariable robust PID control for a PEMFC system. International Journal of Hydrogen Energy, 35, 10437–10445.

Compressor map with speed contours in rad/s 14

1 1.620

+75% Fuel cell current step at t=10s

1.5 100

12

0

1.45

100

1.4

−75% Fuel cell current step at t=20s 120

0

10

80 1.35 80

1.3

0

0.5

1

1.5

2 2.5 3 3.5 Compressor flow (Kg/s)

4

4.5

5 −4

x 10

Fig. 14. Compressor map.

−4

Wcp (Kg/s)

x 10

psm (Pa) ωcp (rad/s)

4 2 10 5 x 10

15

20

25

30

10

15

20

25

30

10

15

20

25

30

10

15

20

25

30

10

15

25

30

1.6 1.4

150

100

vst (V)

12 11 10 40 ifc (A)

101

30 20

20 Time (s)

Fig. 15. Simulation results using non-linear model (reduced order controller in continuous line and classic PI control in dotted line). 103