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 Manifold (SM) I st H Tank Compressor (Cp) Cooler and Humidifier W Ca,in Cathode (Ca) H O O Membrane W An,in H H O Anode (An) W Ca,out W RM,out Return Manifold (RM) W An,out A Schematic of the PEM Fuel Cell System Analyzed in the Present Work
General Parameters Used for Modeling the PEM Fuel Cell System Parameter Symbol SI Units Value Atmospheric Pressure p atm Pa.3 5 Atmospheric Temperature T atm K 98.5 Air Specific Heat Ratio γ -.4 Air Specific Heat C p J/kg/K 4 Air Density ρ a kg/m 3.3 Universal Gas Constant R J/mol/K 8.34 Air Gas Constant R a J/kg/K 86.9 Oxygen Gas Constant R O J/kg/K 59.8 Nitrogen Gas Constant R N J/kg/K 96.8 Vapor Gas Constant R v J/kg/K 46.5 Hydrogen Gas Constant R H J/kg/K 44.3 Molar Mass of Air M a kg/mol 8.97-3 Molar Mass of Oxygen M O kg/mol 3. -3 Molar Mass of Nitrogen M N kg/mol 8. -3 Molar Mass of Vapor M v kg/mol 8. -3 Molar Mass of Hydrogen M H kg/mol. -3 Faraday s Constant F A s/mol 96,487 Temperature of the Fuel Cell T fc K 353
Input Parameters Used for Modeling the PEM Fuel Cell System Parameter Symbol SI Units Value Motor Constant k t Nm/A.53 Motor Constant R CM ohm.8 Motor Constant k v V/(rad/s).53 Compressor Efficiency Compressor Motor Mechanical Efficiency η Cp -.8 η CM -.98 Number of Cells in Fuel Cell Stack n - 38 Fuel Cell Active Area A fc m 8-4 Supply Manifold Volume V SM m 3. Single Stack Cathode Volume V Ca m 3. Single Stack Anode Volume V An m 3.5 Return Manifold Volume V RM m 3.5 Supply Manifold Outlet Orifice Constant Cathode Outlet Orifice Constant Membrane Dry Density Membrane Dry Equivalent Weight k, kg/s/pa.369-5 SM out k, kg/s/pa.77-5 Ca out ρ m, dry kg/m 3 3 M, kg/mol. m dry Membrane Thickness t m m.75-4 Compressor Diameter d Cp m.86 Compressor and Motor Inertia J Cp kg m 5-5 Return Manifold Throttle Discharge Coefficient C D -.4 Return Manifold Throttle Area A T m. Average Ambient Air Relative Humidity Oxygen Mole Fraction at Cathode Inlet Hydrogen Mole Fraction at Anode Inlet φ atm -.5 x O, in -. x H, in -.
Governing Equations
Mass of Air in the Supply Manifold dm dt SM W Cp W SM, out Mass of Oxygen in the Cathode dm dt O W O W W, in O, out O, react Mass of Nitrogen in the Cathode dm dt N W W N, in N, out
Mass of Water in the Cathode dm w, Ca dt W v, Ca, in W v, Ca, out W v, Ca, gen W v, m Mass of Hydrogen in the Anode dm dt H W H W W, in H, out H, react Mass of Water in the Anode dm w, An dt W v, An, in W v, An, out W v, m
Rotational Speed of Compressor J Cp dω dt Cp ( τ τ ) CM Cp Supply Manifold Pressure dp dt SM R γ V a SM ( W T W T ) Cp Cp SM, out SM Return Manifold Pressure dp dt RM R a V T RM RM ( W W ) Ca, out RM, out
Auxiliary Equations
Supply Manifold Outlet Air Rate (Linearized Nozzle Equation) W SM, out k SM, out ( p p ) SM Ca Mass Flow Rate of Reacted Oxygen W O M, react O ni st 4F Mass Flow Rate of Reacted Hydrogen W H react M, H ni st F
Mass Flow Rate of Water Vapor Generated in the Cathode W v,, M Ca gen v ni st F Compressor Motor Torque (Static Motor Equation) τ CM η CM R k t CM ( v k ω ) CM v Cp
Steady-State Compressor Torque Compressor Air Temperature Cp atm SM Cp atm Cp P Cp W p p T C γ γ η ω τ γ γ η atm SM Cp atm atm Cp p p T T T
M. Grujicic, K. M. Chittajallu, E. H. Law and J. T. Pukrushpan, Transient Behavior of Polymer Electrolyte Membrane (PEM) Fuel Cell Systems, Submitted for Publication, June 3.
Fuel-Cell System Control Problem ( x, u w) x & f, State Equations x [ m m m p m m m p ] T O H N ω Cp SM SM w, An w, Ca RM States u v CM Controlled Variable w I st Disturbance z z z P λ net O P λ max net opt O h z ( x, u, w) Performance Variables y [ p, p ] T SM An Measurements
Linearized Model Application of Laplace Equation Yields w D u D x C z w B u B x A x zw zu z w u & W G U G Z w z u z Dynamic Feedback Controller W K U uw
Transfer Function Z ( s) () s ( G G K ) T z w zw z u W Ideal Controller Gain ideal K uw Gz ug zw uw
w z u z uw G G s s s K 3 α α α Dynamic Feedforward Controller Gain Finally, The Equation Becomes 8 8 8 3 7 4 6 5 4 6 7 8 8 8 3 7 4 6 5 4 6 3.793 7.89 4.3987 9.59 3.738 5.75 3.946.644 4.6365.7798 6.444 4.733 8.783 5.593 s s s s s s s s s s s s s K uw
Appendix Equations
Compressor Flow Rate
Mass Flow Rate of Air in the Compressor (Jensen and Kristensen Method) W Cp W cr θ Corrected Flow Rate W cr Φρ a π d 4 Cp U Cp
Ψ Ψ Φ Φ exp max max β Normalized Compressor Flow Rate where 3 3 4 4 max a M a M a M a M a Φ b M b M b β 3 3 4 4 5 5 max c M c M c M c M c M c Ψ
Dimensionless Head Parameter,,, Cp in Cp out Cp in Cp p U p p T C Ψ γ γ in Cp a Cp T R U M, γ Mach Number
Compressor Blade Tip Speed U Cp π 6 d Cp N cr Corrected Rotational Speed N cr N Cp θ Corrected Rotational Speed θ T Cp,in 88
Normalized Pressure p Cp, in p atm Regression Coefficients Regression Coefficient a i b i c i i.95-3.4449.4333 i -4.63685-5 -.34837 -.68344 Values i -5.3635-4.76567.8 i 3.7399-4 - -.4937 i 4-3.6996-5 -.58 i 5 - - -9.8755-3
Water Transport Through the Membrane
Flow Rate of Water Through the Membrane W v, m M v A fc n n d i F D w ( c c ) v, Ca t m v, An Electro-Osmotic Drag Coefficient nd.9.5 λm λm 3.4 9 Water Content 3.43 7.8ai 39.85ai 36.ai, < ai λi ( i 4.4( ai ), < ai 3 m, An, Ca)
Water-Vapor Activity a x p p ( i An Ca) v, i i v, i i, psat, i psat, i Average Water-Vapor Activity in Membrane a m a An a Ca Water Diffusion Coefficient D w D λ exp 46 33 T fc 4
Pre-Exponential Term in Above Equation D λ 6 6 6.5 ( ( λ ) ) ( 3.67( λ 3) ) 6 m m, λm <, λm 3,3 < λ < 4.5, λ m m 4.5 c Water Concentration ρm, dry λi M ( i An Ca) v, i, m, dry
Non-Linear Nozzle Flow Rate
Non-Linear Nozzle Flow Rate Equation Critical Pressure Drop ( ) ( ) ( ) > ) ( ) ( flow choked pr p p pr RT p A C flow normal pr p p pr pr pr RT p A C W crit u d u u T D crit u d u u T D γ γ γ γ γ γ γ γ γ γ γ γ crit u d crit p p pr
Fuel Cell Stack Voltage
Stack Voltage for n Fuel Cells v st nv fc Stack Voltage for Single Fuel Cell v fc E v act v ohm v conc E.9.85 3 Open Circuit Voltage 5 ( T T ) 4.385 T ln(.35 p ) ln(. p ) fc atm fc H 35 O
Activation Overpotential v.79 8.5 4.38 5 T 4 fc act a Where ( c i e ) v v v ( T T ) fc ln p Ca atm p sat.35 ( T ).73( p p ( T ) fc ln Ca.35 sat fc v a ( 5.68.68 ) O T p ( T ) p.73 p.73 ( 4 ) O ( ) ( 4.8.66 5.8.5736) T p T T fc fc sat fc sat fc fc c
Ohmic Overpotential v ohm R ohm Fuel-Cell Electrical Resistance i R ohm t m σ m Membrane Conductivity σ m ( b b ) λm exp b 33 T fc b 3 5.39 b b 35 3 3.6
Concentration Overpotential v conc i c i i max c 3 Where c 4 O ( 7.6 T.6) p ( T ) 3 O (.45 T.68) for p ( T ) 5 O ( 8.66 T.68) p ( T ) fc fc p.73 p.73 p.73 p.73 4 O (.6 T.54) for p ( T ) fc fc sat sat imax. c 3 fc fc sat sat fc fc < atm atm
Linearized System Matrices A -6.54 -.8897 83.8553 4.58-6.99 49.447-8.8-8.68-46.893 75.957 6.485-3.739 98.435.956.969.36835-39.9678.435 6.64 38. 4.78388-488.877-393.984 7.858-99.63699. 4.6989-5.349 B u.7569 B w -.359 -.395 -.95 -.58 D zu D zw.67856 -.4 C z.663.3688 -.896.44 -.6375 -.45555 3.897
55 5 45 8 A 6 A 4 A A Net Power, kw 4 35 3 A 8 A 6 A 4 A A 5 A.5.5 3 Oxygen Excess Ratio Variation of the Net Power With the Oxygen Excess Ratio at Different Stack-Current Levels Under Standard Operating Conditions: T fc 353 K and φ Ca
55 (a).6.55 (b) 5.5 Net Power, kw 45 4 35 3 5 max P net -.99-4 I st.7 I st.87 Oxygen Excess Ratio.45.4.35.3.5..5..5 λ -.7633-6 I opt O - 5.93-4 I st.733 st 5 5 StackCurrent, A 5 5 StackCurrent, A (a) Maximal Net Power; and (b) Optimal Oxygen Excess Ratio as a Function of Stack Current in a PEM Fuel Cell Under Standard Operating Conditions
Supply Manifold Pressure, Pa 35 35 3 75 5 5 75 5 5 5 Stack Current, A (c) opt p SM.996I st 736.4 I st 8.93 4 Compressor Motor Voltage, V 5 4 3 9 8 7 6 5 4 3 opt v CM -.36-3 I st.7 I st 4.3 5 5 StackCurrent, A (d) (c) Optimal Supply Manifold Pressure; and (d) Optimal Compressor Motor Voltage as a Function of Stack Current in a PEM Fuel Cell Under Standard Operating Conditions
(a) w I st Plant z P net P net opt λ O λ O max Static u opt v CM y p p SM An (b) w I st Plant z P net P net opt λ O λ O max Dynamic u v CM y p p SM An (a) Static and (b) Dynamic Open-Loop Feedforward Control of the PEM Fuel-Cell System
(a) 75 (b) 5 5 Static Stack Current, Amp 5 Compressor Motor Voltage, V 5 75 5 5 Dynamic 75 5 5 5 Time, s 5 5 5 5 Time, s (a) Step-Like Temporal Variation of the (Input) Stack Current, and the Corresponding: (b) Compressor Motor Voltage Optimal and Statically and Dynamically Feedforward Controlled Levels
3.5 3.5 Optimal Static (c) 6 Optimal Static (d) 3 Dynamic 5 Dynamic Oxygen Excess Ratio.75.5.5.75 Net Power, kw 4 3.5.5 5 5 5 Time, s 5 5 5 Time, s (c) Oxygen Excess Ratio; and (d) Net Power Optimal and Statically and Dynamically Feedforward Controlled Levels for the Step-Like Temporal Variation of the Stack Current
4 (a) Static (b) Static Supply Manifold/Atmospheric Pressure 3.5 3.5.5 5 5 5 Cell Voltage, V.9.8.7.6.5.4 5 5 5.5.5.5.75. Compressor Flow Rate, kg/s.3 5E-5. Current Density, A/m Temporal Responses of: (a) The Compressor and (b) The Fuel-Cell Corresponding to the Changes in the Stack Current Displayed in Figure 5(a) Under the Static Feedforward Control of the Compressor Motor Voltage. The Numbers Refer to the Time in Seconds
Supply Manifold/Atmospheric Pressure 4 3.5 3.5.5 krpm krpm 9 krpm 8 krpm 7 krpm 6 krpm 5 krpm 4 krpm 3 krpm krpm krpm.5.5.5.75. Compressor Flow Rate, kg/s Compressor Map for an Allied Signal Compressor [9]. Experimental Data are Denoted Using Triangles While the Non-Linear Curve Fitting [] Using Solid Lines
. % Humidity 5% Humidity 4. 5 Pa 3.5 5 Pa Cell Voltage, V.8.6.4 3. 5 Pa.5 5 Pa. 5 Pa.5 5 Pa.. 5 Pa 5E-5..5. Current Density, A/m Polarization Curves for a Single PEM Fuel Cell at 353K and at Different Pressures of the Fully- Humidified (Solid Lines) and 5% Relative Humidified (Dashed Lines) Air in the Cathode
M. Grujicic, K. M. Chittajallu and J. T. Pukrushpan, Control of the Transient Behavior of Polymer Electrolyte Membrane (PEM) Fuel Cell Systems, Submitted for Publication, July 3.
Fuel-Cell System Control Problem ( x, u w) x & f, State Equations x [ m m m p m m m p ] T O H N ω Cp SM SM w, An w, Ca RM States u v CM Controlled Variable w I st Disturbance z z z P λ net O P λ max net opt O h z ( x, u, w) Performance Variables [ W, p v ] T y, Cp SM st Measurements
Linearized Model Integral State Variable w D u D x C y w D u D x C z w B u B x A x yw yu y zw zu z w u & opt W Cp W Cp q &
Feedback Control of the Control Variable ( ) q K x x K u I d P Where o d d x x x dt u R u q Q q z Q z J T I T z T Cost Function dt u R u q Q q x Q C C x J T I T z z T z T Cost Function Can be Redefined as
Weighing Function Matrix Q [ ] T Q x Q I Where Q C x T z Q z C z K Optimal Gain [ ] T T K K R B P P I T T Where PA A P Q PBR B P K P This Procedure Yields 3 3 [ 3.6 3. 47.45 7.853 4.7.64.5 68.57] K I.857
Modal Canonical Form x c T x Resulting Matrices in Canonical Coordinate System A c TAT [ ] B T c B w B u C c C T y
Partitioning of the Matrices A c A cu A cd B c B B cu cd C c [ C C ] cu cd
Reduced Order Observer Gain (Linear Quadratic Guassian Method) L u T SCcuWy Where S is the Solution of This Equation SA T cu A cu S V x SC T cu W y C cu S Positive Definite Weighting Matrices V x [ ] T.. αb B diag cu cu W y 6 diag [ ]
Observer Gain [ ] T L T L u The Resulting L is L 7.748 3.669 3.346 645. 33.486 76.97 9.78.835.594.55 83.78 447.48 6.8 48.83 9.455 5.8859 93.4 6.477 349..7 4.6 34.87 69.945 66.99
The Resulting Linearized Equations Voltage ( ) w D u D x C y w D u D x C z y y L w B u B x A x yw yu y zw zu z w u ˆ ˆ ˆ ˆ ˆ ˆ& ( ) q K x x K u I d P ˆ
Linearized System Matrices A -6.54 -.8897 83.855 4.58-6.99 49.447-8.8-8.68-46.893 75.96 6.485-3.74 98.433.956.969.3684-39.968.435 6.64 38. 4.7839-488.88-393.984 7.85-99.637. 4.6989-5.349 B u.7569 B w -.359 -.395.68436 -.95 -.6898 D zu D zw D yu D yw -.4 -.4 -.3468 C z.663.3688 -.896.44 -.6375 -.45555 3.897 C y 4.7839-5.977 3.974.6643 -.5747
Eigen Values, Eigen Vectors and Observability Eigenvalues ε -.8-9.9-46.55-4.454-4.94-3.53 -.9536 -.55 Eigenvectors x.9e-6 -.747 -.8597.7E-6.57.3 -.954 -.6 x -.98.4 -.33 -.4855.355 -.36 -.75 -.578 x3-9.e-7 -.74689 -.88-4.5E-7.6936 -.499.5873 -.454 x4 -.94E-6 -.388 -.3857-5.7E-6.9776 -.83 -.39835 -.33 x5.88e-7.485.657 9.E-7.98 -.49 -.3 -.8 x6-5.39e-7.6447.8538 8.73E-6.7 -.366 -.4 -.9378 x7 -.9567.876.36736.9889 -.389.44.9.33 x8 8.5E-8.9788 -.37 -.5E-6.466 -.389 -. -.546 Observability rank(εi-a; Cy) 8 8 8 8 8 8 8 8 cond(εi-a; Cy).43 48.6 8.7 54.38 3.6 898.35 374.8 86.
Compressor Motor Voltage, V 5 4 3 9 8 7 6 5 4 3 opt v CM -.36-3 I st.7 I st 4.3 5 5 StackCurrent, A (a) Compressor Flow Rate, kg/s.8.7.6.5.4 W - 4. -7 I Cp.3 5 5 StackCurrent, A st (b) 4.3-4 I st -.5 (a) Optimal Compressor Motor Voltage; and (b) Optimal Compressor Flow Rate as a Function of Stack Current in a PEM Fuel Cell Under Standard Operating Conditions
.6.55 (c) 55 (d).5 5 Oxygen Excess Ratio.45.4.35.3.5..5..5 opt λ O -.7633-6 I st - 5.93-4 I st.733 Net Power, kw 45 4 35 3 5 max P net -.99-4 I st.7 I st.87 5 5 StackCurrent, A 5 5 StackCurrent, A (c) Optimal Oxygen Excess Ratio; and (d) Maximal Net Power as a Function of Stack Current in a PEM Fuel Cell Under Standard Operating Conditions
I st w I st z P λo net P λ max net opt O ( I ) st ( I ) st v W opt CM opt CM ( I ) st ( I ) st - Feedforward Controller Fuel Cell Stack u v CM y W p v st Cp SM xˆ Observer Integral Feedback Controller Observer-Based Feedback Controller for a PEM Fuel-Cell System
Stack Current, Amp 5 5 (a) Compressor Motor Voltage, V 3 5 5 5 (b) Static Feedforward Feedback 5 5 5 Time, s 5 5 5 Time, s (a) Step-Like Temporal Variation of the (Input) Stack Current, and the Corresponding: (b) Compressor Motor Voltage for Static Feedforward and Observer-Based Feedback Controlled Levels
Change in Oxygen Excess Ratio.75.5.5 -.5 -.5 -.75 (c) Static Feedforward Feedback Net Power Change, kw 4 - -4-6 (d) Static Feedforward Feedback - 5 5 5 Time, s -8 5 5 5 Time, s (c) Change in Oxygen Excess Ratio; and (d) Change in Net Power for Static Feedforward and Observer-Based Feedback Controlled Levels for Step-Like Temporal Variation in Stack Current
Supply Manifold/Atmospheric Pressure 4 3.5 3.5.5 (a) krpm krpm 5 5.5.5.5.75. 3 4 5 6 krpm krpm krpm krpm Observer-based Feedback Compressor Flow Rate, kg/s 7 krpm 5 8 krpm 9 krpm krpm krpm Cell Voltage, V.9.8.7.6.5.4 (b) 5.3 5E-5. 5 Current Density, A/m Observer-based Feedback 5 kpa 5 kpa kpa 4 kpa 3 kpa 35 kpa 5 kpa Temporal Responses of: (a) The Compressor and (b) The Fuel-Cell Corresponding to the Changes in the Stack Current Displayed in Figure 4(a) Under The Observer- Based Feedback Control of the Compressor Motor Voltage. The Numbers Associated With Arrowed Lines Refer to the Time in Seconds
Sensitivity Magnitude - - -3 - - 3 Frequency, rad/s Frequency Dependence of the Input-Sensitivity Magnitude for the PEM Fuel-Cell System With an Observer-Based Feedback Controller