Appendix A Electric Vehicle PEM Fuel Cell Stack Parameters

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1 Appendix A Electric Vehicle PEM Fuel Cell Stack Parameters A.1 Return Manifold Polynomial Fitting Table A.1 Return manifold polynomial fitting Parameter Value Return manifold parameter p kg/s Return manifold parameter p kg/s/kpa Return manifold parameter p kg/s/kpa 2 Return manifold parameter p kg/s/kpa 3 Return manifold parameter p kg/s/kpa 4 Return manifold parameter p kg/s/kpa 5 A.2 Differential Equations Parameters For further information about the model parameters, structure and reduction, refer to [3], [2] and [1]. Table A.2 State equations parameters Parameter Expression Value B 1 n cm /R cm /J cp B 2 K v n cm /R cm /J cp 255 B 3 C p T atm φ max ρ a π/4dc 2K Ucδ/n cp θ 1/2 ef mec J cp B 4 γ 1/γ B 5 2C p T cp,in KUc B 6 1/4φ max ρ a πdc 2K Ucδ/θ 1/2 T atm γr a /V sm C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI / , Springer-Verlag London Limited

2 162 A Electric Vehicle PEM Fuel Cell Stack Parameters Table A.2 continued Parameter Expression Value B 7 V sm K sm,out /R a γr a /V sm B 8 B 7 p v,ca B 9 B 7 R N2 T st /V ca B 10 B 7 R O2 T st /V ca B 11 1/4φ max ρ a πdc 2K Ucδ/θ 1/ B 12 K sm,out B 13 K sm,out p v,ca B 14 K sm,out R N2 T st /V ca B 15 K sm,out R O2 T st /V ca B 16 G v φ des p sat,tcl K sm,out /G a B 17 B 16 p v,ca B 18 B 16 R N2 T st /V ca B 19 B 16 R O2 T st /V ca B 20 G v φ atm p sat,tatm /G a /p atm B 21 φ atm p sat,tatm /p atm B 22 G v φ ca,in p sat,tcl χ O2,ambG O2 + 1 χ O2,ambG N B 23 φ des p sat,tcl φ ca,in p sat,tcl 0 B 24 R O2 T st G O2 k ca,out /V ca B 25 B 24 p v,ca B 26 B 24 R N2 T st /V ca B 27 B 24 R O2 T st /V ca B 28 R N2 T st /V ca B 29 R O2 T st /V ca B 30 B 24 /k ca,out B 31 R O2 T st G N2 /V ca B 32 1/4G O2 n/f B 33 1 χ O2,ambG O2 /χ O2,ambG O2 + 1 χ O2,ambG N B 34 k ca,out B 35 k ca,out p v,ca B 36 k ca,out R N2 T st /V ca B 37 k ca,out R O2 T st /V ca B 38 R a T st /V rm B 39 B 38 p a1 /std 5 a B 40 B 38 p a2 /std 4 a B 41 B 38 p a3 /std 3 a B 42 B 38 p a4 /std 2 a B 43 B 38 p a5 /std a B 44 B 38 p a B 45 mean a

3 A.3 Controller Parameters 163 Table A.2 continued Parameter Expression Value B 46 B 34 B B 47 B 35 B B 48 B 36 B B 49 B 37 B B 50 B 33 B B 51 B 50 p v,ca B 52 B 50 B B 53 B 50 B B 54 B 33 B B 55 B 33 B B 56 B 33 B B 57 B 33 B B 58 G v p v,ca B 59 X O2,ca,inB B 60 X O2,ca,inB B 61 X O2,ca,inB B 62 X O2,ca,inB B 63 X O2,ca,inB B 64 X O2,ca,inB B 65 X O2,ca,inB B 66 X O2,ca,inB B 67 B 30 B B 68 B 29 G O2 B A.3 Controller Parameters Table A.3 Super-Twisting, Twisting and Sub-Optimal Parameter Value Super-Twisting algorithm parameter α 2 Super-Twisting algorithm parameter λ 3 Super-Twisting algorithm parameter ρ 0.5 Twisting algorithm parameter r Twisting algorithm parameter r Sub-Optimal algorithm parameter U 3 Sub-Optimal algorithm parameter β 0.3 Sub-Optimal algorithm parameter α 5

4 164 A Electric Vehicle PEM Fuel Cell Stack Parameters Table A.4 LQR controller Parameter Value LQR controller parameter k LQR controller parameter k LQR controller parameter k LQR controller parameter k LQR controller parameter k LQR controller parameter k LQR controller parameter k LQR controller parameter k LQR controller parameter k I 0.18 References 1. Kunusch C 2006 Second order sliding mode control of a fuel cell stack using a twisting algorithm. MSc thesis, University of La Plata, Argentina in Spanish 2. Pukrushpan JT, Stefanopoulou AG, Peng H 2004 Control of fuel cell power systems. Springer, Berlin 3. Pukrushpan JT, Stefanopoulou AG, Peng H 2004 Control of fuel cell breathing. IEEE Control Syst Mag 24:30 46

5 Appendix B Laboratory FC Generation System Parameters Table B.1 General physics constants Parameter Value Dry air molar mass G a Oxygen molar mass G O2 Nitrogen molar mass G N2 Vapour molar mass G v Hydrogen molar mass G H2 Air specific constant R a Oxygen specific constant R O2 Nitrogen specific constant R N2 Vapour specific constant R v Hydrogen specific constant R H2 Faraday s constant F kg/mol kg/mol kg/mol kg/mol kg/mol Nm/kg/K Nm/kg/K Nm/kg/K Nm/kg/K Nm/kg/K C/mol Table B.2 Compressor s parameters Parameter Value Electric inductance L Electric resistance R Torque constant k φ Motor inertia J Compressor s equivalent inertia J eq Load torque coefficient A 0 Load torque coefficient A mh N m/a Nm Nm Nm Nms C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI / , Springer-Verlag London Limited

6 166 B Laboratory FC Generation System Parameters Table B.3 Polynomial fitting coefficients Parameter Value Parameter Value A 00 0 B kg/s A N m s B kg/s 2 A N m s 2 B kg/s 3 A Nm/bar B kg/s 2 /bar A Nms/bar B kg/s A Nms/bar 2 B kg/s/bar Table B.4 Humidifier and FC stack parameters Parameter Value Air manifold volume V sm m 3 Air manifold constraint K sm,out kg/s/bar Humidifier volume V hum m 3 Humidifier constraint coefficient C kg/s Humidifier constraint coefficient C kg/s/bar 2 Humidifier constraint coefficient C kg/s/bar Number of cells n 7 Cathode constraint K ca,out kg/s/bar Cathode volume V ca m 3 Membrane effective area A fc 50 cm 2 Dry membrane thickness t m cm Dry membrane density ρ m,dry kg/cm 3 Dry membrane molecular weight G m,dry 1.1 kg/mol Membrane diffusion coefficient D w cm 2 /s Water content coefficient a [H 2 O/SO 3 ] Water content coefficient a [H 2 O/SO 3 ] Water content coefficient a [H 2 O/SO 3 ] Water content coefficient a [H 2 O/SO 3 ] Electro-osmotic coefficient n [H 2 O/H + ] Electro-osmotic coefficient n [H 2 O/H + ] Electro-osmotic coefficient n [H 2 O/H + ] Charge transfer coefficient α 0.7 Exchange current density i A/cm 2 Apparent fuel cell area A app cm 2 Resistance coefficient R /cm 2 Resistance coefficient R /cm 2 Polarisation curve coefficient a 0.2 bar Polarisation curve coefficient b 0.06 V Polarisation curve coefficient m V Polarisation curve coefficient n c 15.2 cm 2 /A

7 B Laboratory FC Generation System Parameters 167 Table B.5 Operating conditions Parameter Value Humidifier temperature T hum 55 C Heating line temperature T lh 60 C Fuel cell stack temperature T st 60 C Humidifier relative humidity RH hum 0.95 Ambient relative humidity RH amb 0.5 Ambient pressure P amb 1bar Ambient temperature T amb 25 C Ambient oxygen molar fraction χ O2,amb 0.21 Hydrogen input flow W H2,an 2slpm

8 Appendix C Laboratory FCGS State Space Functions and Coefficients C.1 Expression of ϕx,u,t ϕx,u,t = 2B 02 m1 ut m2 x 1 x1 m 3 + A 0 + A 00 + A 10 x 2 m 5 + m 6 + A 20 x 2 m 5 + m A 01 x 1 + A 11 x 2 m 5 + m 6 x 1 + A 02 x1 2 m4 + B 01 + B 11 x 2 m 5 + m 6 + 2B 02 x 1 m1 m 2 m 3 + A 01 + A 11 x 2 m 5 + m 6 + 2A 02 x 1 m4 + B 11 m 5 B00 + B 10 x 2 m 5 + m 6 + B 20 x 2 m 5 + m B 01 x 1 + B 11 x 2 m 5 + m 6 x 1 + B 02 x1 2 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 3C3 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 C1 C 0 + B10 m 5 + 2B 20 x 2 m 5 + m 6 m 5 + B 11 m 5 x 1 B01 + B 11 x 2 m 5 + m 6 + 2B 02 x 1 m1 ut m2 x 1 x1 m 3 + A 0 + A 00 + A 10 x 2 m 5 + m 6 + A 20 x 2 m 5 + m A 01 x 1 + A 11 x 2 m 5 + m 6 x 1 + A 02 x1 2 m4 + B 11 m 5 m1 ut m2 x 1 x 1 m 3 + A 0 + A 00 + A 10 x 2 m 5 + m 6 + A 20 x 2 m 5 + m A 01 x 1 + A 11 x 2 m 5 + m 6 x 1 + A 02 x 2 1 m4 + B 01 + B 11 x 2 m 5 + m 6 + 2B 02 x 1 A10 m 5 + 2A 20 x 2 m 5 + m 6 m 5 + A 11 m 5 x 1 m4 + 2B 20 m 2 5 B00 + B 10 x 2 m 5 + m 6 + B 20 x 2 m 5 + m B 01 x 1 + B 11 x 2 m 5 + m 6 x 1 + B 02 x1 2 3C3 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 C1 C 0 + B 10 m 5 + 2B 20 x 2 m 5 + m 6 m 5 + B 11 m 5 x 1 B10 m 5 C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI / , Springer-Verlag London Limited

9 170 C Laboratory FCGS State Space Functions and Coefficients + 2B 20 x 2 m 5 + m 6 m 5 + B 11 m 5 x 1 3 2C3 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 m 5 2 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 C2 m 5 m 5 C 1 B 00 + B 10 x 2 m 5 + m 6 + B 20 x 2 m 5 + m B 01 x 1 + B 11 x 2 m 5 + m 6 x 1 + B 02 x 2 1 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 3C3 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 C1 C 0 + B 10 m 5 + 2B 20 x 2 m 5 + m 6 m 5 + B 11 m 5 x 1 3 2C3 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 R O2 m x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 C2 R O2 m 8 + R O2 m 8 C 1 3C3 m 9 x2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 + x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 + x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 C1 + C 0 G 1 m 11 1 x 2 m 5 m x 2 m 5 m C3 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 + x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 + x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 C1 + C 0 m X O2,ca,in 1 + x 2 m 5 m 10 m 14 x 3 R O2 + x 4 R N2 + x 5 R v m 8 m 12 1 K ca x3 R O2 + x 4 R N2 + x 5 R v m 8 P amb x3 R O2 G O2 x3 R O2 G O2 x 3 R O2 1 + G v x 5 R v + 1 x 3 R O2 + x 4 R N2 x 3 R O2 + x 4 R N2 a G N2 1 1 x 3 R O2 + x 4 R N2 1 x 3 R O2 + x 4 R N2 1 x3 R O2 G O2 x 3 R O2 + x 4 R N2 x 3 R 1 O2 + 1 G N2 1/4 G O 2 ni st x 3 R O2 + x 4 R N2 F + B 10 m 5 + 2B 20 x 2 m 5 + m 6 m 5 + B 11 m 5 x 1 3 2C3 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 R N2 m 8

10 C.1 Expression of ϕx,u,t x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 C2 R N2 m 8 + R N2 m 8 C 1 m 9 x2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 3C3 + x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 + x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 C1 + C 0 m 11 1 G 1 a x 2m 5 m x 2 m 5 m C3 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 + x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 + x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 C1 + C 0 m x 2 m 5 m 10 G v m Ga c a i nx 3 R O2 + x 4 R N2 + x 5 R v m 8 m 12 1 X O2,ca,in 1 x 3 R O2 m 8 G O2 x 3 R O2 m 8 + x 4 R N2 m 8 1 x 3 R O2 m 8 G O2 x 3 R O2 m 8 + x 4 R N2 m x 3 R O2 m 8 G N2 1 x 3 R O2 m 8 + x 4 R N2 m 8 K ca x3 R O2 + x 4 R N2 + x 5 R v m 8 P amb 1 + G v x 5 R v m 8 x 3 R O2 m 8 + x 4 R N2 m 8 1 x 3 R O2 m 8 G O2 x 3 R O2 m 8 + x 4 R N2 m 8 x 3 R O2 m G N2 x 3 R O2 m 8 + x 4 R N2 m 8 + B 10 m 5 + 2B 20 x 2 m 5 + m 6 m 5 + B 11 m 5 x 1 3 2C3 x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 R v m x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 C2 R v m 8 + R v m 8 C 1 3C3 G v m 12 x2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 + x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 + x 2 m 5 x 3 R O2 + x 4 R N2 + x 5 R v m 8 C1 + C 0 G 1 1 x 2 m 5 m G v m 10 + G a x 2 m 5 m 10 + x 2 m 5 b 2 3 C 3 + x 2 m 5 b 2 2 C 2 + x 2 m 5 b 2 C 1 + C G v m 10 G a x 2 m 5 m 10 1 G v m 12 x2 m 5 b 2 3 C 3 a

11 172 C Laboratory FCGS State Space Functions and Coefficients + x 2 m 5 b 2 2 C 2 + x 2 m 5 b 2 C 1 + C 0 G 1 a 1 x 2 m 5 m G v m 10 + G a x 2 m 5 m 10 + x 2 m 5 b 2 3 C 3 + x 2 m 5 b 2 2 C 2 + x 2 m 5 b 2 C 1 + C G v m 10 G a x 2 m 5 m 10 1 G v m 12 Gax 3 R O2 + x 4 R N2 + x 5 R v m 8 m 12 K ca x3 R O2 + x 4 R N2 + x 5 R v m 8 P amb 1 + K ca x3 R O2 + x 4 R N2 + x 5 R v m 8 P amb 1 + G v x 5 R v m 8 x 3 R O2 m 8 + x 4 R N2 m 8 1 x 3 R O2 m 8 G O2 x 3 R O2 m 8 + x 4 R N2 m 8 x 3 R O2 m A fc F G N /2 G vni st F x 3 R O2 m 8 + x 4 R N2 m 8 n 0 + n 1 a 0 + a 1 2 x 7m 15 + x 5 m 16 + a 2 4 x 7m 15 + x 5 m a 3 8 x 7m 15 + x 5 m n 2 a 0 + a 1 2 x 7m 15 + x 5 m 16 + a 2 4 x 7m 15 + x 5 m a x 7m 15 + x 5 m 16 3 I st a0 + a 1 x 5 m 16 + a 2 x5 2 m a 3x5 3m3 16 ρ m,dry G m,dry a 0 + a 1 x 7 m 15 + a 2 x 2 7 m a 3x 3 7 m3 15 ρ m,dry G m,dry Dw t m G v A fc n C.2 Model Coefficients m 1 = K φ /R, m 2 = K φ 30/π, m 3 = A 1 30/π, m 4 = π/30/j eq m 5 = T sm R a /V hum, m 6 = P sat T sm RH amb + RH hum,ca P sat T hum,ca m 8 = T st /V ca, m 9 = G v RH hum,ca P sat T hum,ca, m 10 = P sat T sm RH amb m 11 = G v P sat T sm RH amb /G a, m 12 = RH hum,ca P sat T hum,ca m 13 = R O2 T st G O2, m 14 = G v RH hum,ca P sat T hum,ca /G a m 15 = T st R v /V an /P sat T lh,an, m 16 = R v T st /V ca /P sat T lh,ca

12 C.3 Feedforward Control Action 173 m 17 = RH an,in P sat T lh,an, m 18 = T st /V an, m 19 = T st R v /V an m 20 = T st R H2 /V an C.3 Feedforward Control Action u ff = W 6 air,ref W 5 air,ref W 4 air,ref W 3 air,ref W 2 air,ref W air,ref

13 Index A Air compressor, 75, 106 humidifier, 77, 113 supply, 6, 24 supply manifold, 76, 112 Algorithm sub-optimal, 56, 67, 94, 139, 140 super twisting, 56, 63, 86, 88, 139, 144 twisting, 56, 93, 139, 140 Alkaline fuel cell, 4 Anode channels, 78, 120 reaction, 3 Armature current, 107 voltage, 107 B Back-diffusion, 121 Bipolar plate, 22 C Catalyst, 13, 20 Catalytic oxidation, 3, 13 reduction, 3, 13 Cathode channels, 77, 117 reaction, 3 starvation, 30 Charge transfer coefficient, 16 Chattering, 36, 47 Compressor, 24 Control affine system, 37 Current collector, 16 D DC motor, 75 DC/AC converter, 25 DC/DC converter, 25 Degradation, 30 Diaphragm vacuum pump, 106, 111 Diffeomorphism, 37 Differential inclusion, 49, 56, 86, 139 Direct methanol fuel cell, 5 Discontinuous control action, 39, 47 E Efficiency, 17 Electro-osmotic drag, 19, 79, 121 Electrochemical potential, 123 Electrode, 20 Electron, 14 Energy conversion, 17 Enthalpy, 17 Entropy change, 15 Equivalent control, 40, 42 Exchange current density, 16 F Faraday s constant, 14, 79 Feedforward control, 86, 87, 141 Filippov differential inclusion, 50 method, 45 sense, 51 solution, 51 Finite time convergence, 64 Fuel cell, 3 active area, 79 apparent area, 124 stack, 5, 117 C. Kunusch et al., Sliding-Mode Control of PEM Fuel Cells, Advances in Industrial Control, DOI / , Springer-Verlag London Limited

14 176 Index G Gas diffusion layer, 21 Gibbs free energy, 14 H Heat management, 24 Higher order sliding mode, 48 Humidification, 26 Humidifier, 26 Humidity ratio, 118 Hydrogen, 1 Hydrogen supply, 24 I Ideal sliding, 40 Invariance conditions, 41 K Kalman observer, 98 L Lie derivative, 37, 41, 137 Line heater, 27 Load torque, 75, 107 Losses activation, 15, 123 concentration, see diffusion diffusion, 16, 123 ohmic, 16, 123 LQR, 30, 97, 100 Lyapunov function, 40 M Majorant curve, 62, 65 Mass mole fraction, 118 Matching condition, 44 MEA, 21, 26, 125 Membrane conductivity, 16 dry thickness, 16 electrode assembly, see MEA water content, 16, 125 Modulation factor, 68 Molten carbonate fuel cell, 4 Motor torque, 75, 107 N Nernst voltage, 15 Net power, 82, 90, 134 Normal form, 52 Nozzle, 112 O Oxygen, 3 control, 30 excess ratio, see stoichiometry starvation, 73, 83, 84, 135 stoichiometry, 73, 83, 90, 135, 145 supply, 24 P Parasitic load, 24 Peak detector, 68 PEM fuel cell, 5, 10, 13, 17, 30 Phosphoric fuel cell, 4 PID, 30 Platinum, 13, 20 Polarisation curve, 17, 123, 126 Polymer electrolyte membrane fuel cell, see PEM fuel cell Polymeric membrane, 19, 79 Potential difference, 14 Power conversion, 17 Proton, 14 Proton exchange membrane fuel cell, see PEM fuel cell Protonic conductivity, 14 R Regularity condition, 51 Relative degree, 37, 52, 54, 55 Return manifold, 80 Reversible voltage, 14 S Scalar field, 37 Sealing gasket, 22 Second order sliding mode, 54 Shaft angular speed, 107 Sliding manifold, see surface mode control, 35 surface, 37, 39, 82, 136 variable, 137 Solid oxide fuel cell, 5 Stack current, 89 voltage, 90 State space model, 130 T Tafel equation, 16 Thermal management, 6 Torque disturbance, 88 Transversality condition, 42 V Vector field, 37 Voltage drop, 3

15 Index 177 W Water diffusion, 19 management, 6, 24 transport, 121 Windup, 147

Control of Proton Electrolyte Membrane Fuel Cell Systems. Dr. M. Grujicic Department of Mechanical Engineering

Control of Proton Electrolyte Membrane Fuel Cell Systems Dr. M. Grujicic 4 Department of Mechanical Engineering OUTLINE. Feedforward Control, Fuel Cell System. Feedback Control, Fuel Cell System W Cp Supply