New Perspectives in Control System Design: Pseudo-Constrained to Market Responsive Control

Transcription

1 New Perspectves n Control System Desgn: Pseudo-Constraned to Market Responsve Control Donald J. Chmelewsk Department of Chemcal & Bologcal Engneerng EDOR s due to dfferent controller tunngs BOP wth less proft BOP wth more proft * * OSSOP

2 Outlne Motvatng Example Pseudo-Constraned Control Proft Control Market Responsve Control

3 Motvatng Example (Non-sothermal Reactor) F C A, T F V V r A dca F( CAn CA) VrA dt dt F( Tn T ) ( VH / C dt k( T) C A p ) r A Increase F Increased producton rate

4 Motvatng Example (Non-sothermal Reactor) F C A, T F V V r A dca F( CAn CA) VrA dt dt F( Tn T ) ( VH / C dt k( T) C A p ) r A Increase F Decrease F Increased producton rate Increase T Increase reacton rate Increase producton

5 Lmted Operatng Regon Process Lmtatons: T( t) T F( t) F (max) (max) - Catalyst protecton or onset of sde reactons - Pump lmt or lmt on downstream unt

6 Lmted Operatng Regon Process Lmtatons: T( t) T F( t) F F K c (max) ( T (max) Possble Controller: T ( sp) - Catalyst protecton or onset of sde reactons - Pump lmt or lmt on downstream unt ) F ( sp)

7 Performance n Tme Seres F T (sp) T(t) T (max) C A, T F F(t) tme F (max) F (sp) tme

8 Performance n Phase Plane T(t) * F(t)

9 Expected Dynamc Operatng Regon (EDOR) T(t) * F(t)

10 Expected Dynamc Operatng Regon (EDOR) T(t) * F(t)

11 Steady-State Relaton Controller: F K c ( T T ( sp) ) F ( sp) Steady-State Relaton: F ( sp) ( sp) f ( T )

12 Expected Dynamc Operatng Regon (EDOR) T(t) * F(t)

13 Steady-State Operatng Lne T(t) * F(t)

14 Optmal Operatng Pont T(t) * Decrease F Increase T Increase converson Increase producton F(t)

15 Optmal Operatng Pont: T(t) Another Possblty * Increase F Increased producton rate F(t)

16 Optmal Operatng Pont: Another Possblty T(t) Increase F Increased producton rate * F(t)

17 Requres Dfferent Controller Tunng T(t) * F(t)

18 Less Aggressve Tunng T(t) T (sp) T(t) T (max) * F(t) F (sp) F(t) tme F (max) tme

19 Need for Automated Tunng T(t) * * F(t)

20 Outlne Motvatng Example Pseudo-Constraned Control Proft Control Market Responsve Control

21 Process Model: Covarance Analyss (Open-Loop Case) Steady State Covarance: A x z T T x x A G wg z w(t) A x D x G w D T w (t) z(t) Plant Dx Gaussan whte nose wth covarance w

22 Expected Dynamc Operatng Regon (EDOR) z 1 EDOR defned by: 11 * z z 2

23 Closed-Loop Covarance Analyss (Full State Informaton Case) Process Model: x Ax Bu G w z u( t) Lx( t) T T ( A BL) ( A BL) G G ( D D L) ( D D L) z D x x Controller: x D u x u u D x x w Steady-State Covarance: w x u w(t) u(t) T w D w w D Plant T w L z(t) x(t)

24 Closed-Loop EDOR z 1 EDOR s from dfferent controllers * u L x 1 u L2 x z 2

25 Constraned Closed-Loop EDOR z 1 Constrants ( z z ) * z 2

26 Constraned Closed-Loop EDOR z 1 Constrants ( z z ) * z 2

27 Does there exst L such that: 1 column th Constraned Controller Exstence T w w w T u x x u x z D D D L D D L D ) ( ) ( ) ( ) ( T w T x x G G BL A BL A z T z n z 1 2

28 If and only f there exst X> and Y such that: 1 Constraned Controller Exstence (Convex Condton) ) ( ) ( X D Y X D D Y X D D D T T u x u x T T w w w ) ( ) ( T w T G G BY AX BY AX z n z L YX And controller s constructed as: u Lx

29 Unconstraned Controller Exstence 1 Achevable Performance Levels Unachevable 2

30 Constraned Controller Exstence 1 1 <z <z 2 2 2

31 EDOR Interpretaton <z * 1 T(t) * * 2 <z F(t)

32 such that: Pseudo-Constraned Control Y X d mn,, ) ( ) ( X D Y X D D Y X D D D T T u x u x T T w w w ) ( ) ( T w T G G BY AX BY AX z n z 1 2

33 Pareto Fronter Interpretaton 1 All Pseudo-Constraned Controllers are on the Pareto Fronter Unachevable Regon Achevable Regon 2

34 MPC Equvalence Theorem 1 (Chmelewsk & Manthanwar, 24): All controllers generated by Pseudo- Constraned Control (PCC) are concdent wth a controller generated by some Unconstraned Model Predctve Controller.

35 Inverse Optmalty Theorem 2 (Chmelewsk & Manthanwar, 24): If there exsts P > and R > such that A T P PA Q L R 1 L T PB M RL A L 1 T PB M R PB M T T T R PB P PA T L T R PB T T T then M ( L R PB) and Q L RL A P PA are such that Q M T M R R and P and L satsfy

36 Outlne Model Predctve Controller Tunng Pseudo-Constraned Control Proft Control Market Responsve Control

37 Constraned Operatng Regon CV s Steady-State Operatng Pont Constrants EDOR * MV s

38 Real-Tme Optmzaton Orgnal Nonlnear Process Model: s f ( s, m, p) q h( s, m, p) (s,m,p,q) ~ (state, mv, dst, performance) ~ (x,u,w,z)

39 Real-Tme Optmzaton ),, ( ),, ( p m s h q p m s f s Orgnal Nonlnear Process Model: max mn,, ),, ( ),, ( s.t. ) ( mn m q s q q q p m s h q p m s f q g Real-Tme Optmzaton (mnmze proft loss): (s,m,p,q) ~ (state, mv, dst, performance) ~ (x,u,w,z) RTO soluton denoted as (s ossop,m ossop,p ossop,q ossop )

40 Real-Tme Optmzaton Steady-State Operatng Pont CV s Constrants EDOR * Optmal Steady-State Operatng Pont (OSSOP) * MV s

41 Backed-off Operatng Pont (BOP) CV s Backed-off Operatng Pont (BOP) EDOR * * * MV s Optmal Steady-State Operatng Pont (OSSOP)

42 mn Steady-State BOP Selecton (Bahr, Bandon & Romagnol, 1996) Solve the followng Sem-nfnte Programmng Problem g( q) s.t. s.t. max s, m, q mn max p[ p, p ] Extensons: - Dynamc verson n Bahr, et al, (1995) max { q q ( s, m, h( s, m, - Lnearzed verson n Contreras-Dordelly & Marln (2) q f q q } p) p)

43 Devaton Varables: RTO and Controller Perspectves CV s Backed-off Operatng Pont (BOP) EDOR * * * MV s Optmal Steady-State Operatng Pont (OSSOP)

44 Two Sets of Devaton Varables max mn ),, ( ),, ( ) ( bop q q q p m s h q p m s f s q g Nonlnear

45 Two Sets of Devaton Varables max mn ),, ( ),, ( ) ( q q q p m s h q p m s f s q g Nonlnear Lnear wrt OSSOP Devaton Varables w.r.t. OSSOP: s = s bop s ossop m = m bop - m ossop p = p bop - p ossop q = q bop - q ossop max mn ' ' ' ' ' ' ' ' ' ' ' ) ( w u x q ossop q q q p D D m D s q Gp Bm As q g q g

46 Two Sets of Devaton Varables Nonlnear Lnear wrt OSSOP Lnear wrt BOP g( q q mn bop ) q h( s, m, p) q' q q max g( q q' mn ossop q' D s' D m' D x q' ) g s f ( s, m, p) As' Bm' Gp' x Ax Bu Gw u q q' max w p' z D z mn x x z D u z u max D w w Devaton Varables w.r.t. OSSOP: s = s bop s ossop m = m bop - m ossop p = p bop - p ossop q = q bop - q ossop Devaton Varables w.r.t. BOP: x = s s bop u = m - m bop w = p - p bop z = q - q bop

47 Stochastc BOP Selecton (Loeblen & Perkns, 1999)

48 Assume controller L s gven and calculate : T w w w T u x x u x z D D D L D D L D ) ( ) ( ) ( ) ( T w T x x G G BL A BL A z T z n 1 Stochastc BOP Selecton (Loeblen & Perkns, 1999)

49 Assume controller L s gven and calculate : T T ( A BL) ( A BL) G G ( D D L) ( D D L) z Stochastc BOP Selecton (Loeblen & Perkns, 1999) x z T x u x x x 1n u T D Solve the followng Lnear Program: z w w w D T w mn s', m', q' g q q' s.t. As' Bm' q ' ( D s' D m') 1/ 2 q' x max q' u q mn 1/ 2 q' q' q q' max mn

50 Fxed Controller BOP Selecton Loeblen and Perkns (1999): x * EDOR * * u OSSOP Controller s fxed EDOR has fxed sze and shape

51 Varable Controller BOP Selecton Peng et al. (25): x EDOR Varable Controller * OSSOP * * u EDOR has varable sze and shape

52 Proft Control (Smultaneous BOP and Controller Selecton) EDOR s due to dfferent controller tunngs BOP wth less proft BOP wth more proft * * Max Proft Peng et al. (25)

53 Assume controller L s gven and calculate : T T ( A BL) ( A BL) G G ( D D L) ( D D L) z Stochastc BOP Selecton (Loeblen & Perkns, 1999) x z T x u x x x 1n u T D Solve the followng Lnear Program: z w w w D T w mn s', m', q' g q q' s.t. As' Bm' q ' ( D s' D m') 1/ 2 q' x max q' u q mn 1/ 2 q' q' q q' max mn

54 mn 2 1/ max 2 1/ max mn,, ' ', ', ' ' ' ' ' ') ' ( ' ' ' s.t. ' mn u x q Y X q m s q q q q q q q D m D s q Bm As q g ) ( ) ( X D Y X D D Y X D D D T T u x u x T T w w w ) ( ) ( T w T G G BY AX BY AX Proft Control (Smultaneous BOP and Controller Selecton) Peng et al. (25)

55 mn 2 1/ max 2 1/ max mn,, ' ', ', ' ' ' ' ' ') ' ( ' ' ' s.t. ' mn u x q Y X q m s q q q q q q q D m D s q Bm As q g ) ( ) ( X D Y X D D Y X D D D T T u x u x T T w w w ) ( ) ( T w T G G BL AX BY AX Peng et al. (25) Computatonal Aspects of Proft Control

56 Reverse-Convex Constrants 1 1 (z ss, +d max mn, ) 2 2 (z ss, +d max, ) 2 mn 2 1 ( q' 1 q' 1 ) 1 ( q' 1q' 1 ) z ss, q' 1 Feasble Regon

57 Global Soluton Based on Branch and Bound algorthm Regon 2 Regon 3 Regon Regon 1 Regon q' 1 z ss,

58 Proft Control Applcatons Mechancal Systems Chemcal and Reacton Systems Hybrd Vehcle Desgn Inventory Control Electrc Power System Desgn Buldng HVAC Water Resource Management

59 Proft Control Applcatons Mechancal Systems Chemcal and Reacton Systems Hybrd Vehcle Desgn Inventory Control Electrc Power System Desgn Buldng HVAC Water Resource Management

60 Fludzed Catalytc Cracker Regenerator and Separator (dynamc): Rser (pseudo steady state): (adapted from Loeblen & Perkns, 1999)

61 FCC Constrants and Economcs Process Constrants: Proft Functon: F gs F gl and F ugo are product flows (gasolne, lght gas and unconverted ol). (adapted from Loeblen & Perkns, 1999)

62 Catalyst Flow (kg/s) Inlet Ar (kg/s) Regenerator Temp (K) Cyclone Temperature (K) Proft Control vs. Fxed Controller Back-off Fxed Controller Free Controller Coke Fracton n Separator Separator Temperature (K) Fracton of Coke n Regenerator x Oxygen Mass Fracton x 1-4

63 FCC Proft Gross Proft ($/day) Dff from OSSOP ($/day) OSSOP $36,95 $. Fxed Control $34,631 - $2,274 Proft Control $35,416 - $1,489 Improves proft by 2%

64 Inventory Control Bascs Actual Inventory: stock-on-hand capable of meetng demand mmedately Inventory Poston: sum of the actual nventory and all orders placed Slver et al.(1998)

65 The (R,S) Approach wth Determnstc Demand L: Order lead tme R: Revew perod S: Order-up-to level, S ( L R) D

66 The (R,S) Approach wth Determnstc Demand a = setup cost, b = unt cost, c = holdng cost D = expected annual demand Total cost, a cdr C bd R 2 Total annual setup cost Total annual holdng cost Total annual nventory purchase cost Optmal revew perod, R 2a cd

67 The (R,S) Approach wth Determnstc Demand Example 1 a = $1/order c = $.3244/tem D = 365 tems L = 5 days 2a R cd S ( L R) D R = 15 days S = 2 tems

68 The (R,S) Approach wth Stochastc Demand Safety Stock (SS) Demand Gaussan process Average, D Standard devaton, Standard devaton of nventory at t+r+l s L R

69 Safety Stock and Servce Levels α Probablty of Stock-outs 1. 16% % % Order-up-to level, S ( L R) D SS SS ( L R) Slver et al.(1998)

70 Impact of Setup Cost Example 2 a = $1/order c = $.3244/tem D = 365 tems L = 5 days R = 15 days L < R σ = 1 tems/day^.5 α = 1 SS = 45 tems S = 245 tems Example 3 a = $.444/order c = $.3244/tem D = 365 tems L = 5 days R = 1 day L > R σ = 1 tems/day^.5 α = 1 SS = 25 tems S = 625 tems

71 Impact of Setup Cost Example 2 a = $1/order c = $.3244/tem D = 365 tems L = 5 days R = 15 days L < R σ = 1 tems/day^.5 α = 1 SS = 45 tems S = 245 tems Example 3 a = $.444/order c = $.3244/tem D = 365 tems L = 5 days R = 1 day L > R σ = 1 tems/day^.5 α = 1 SS = 25 tems S = 625 tems

72 Recursve Model I k1 I k q k d I k : actual nventory at the end of nterval k d k : total demand durng nterval k q k-θ : amount ordered at the end of nterval k-θ and arrvng at the end of nterval k θ = L/R * ponts ndcate sequence I k k Actual Inventory Tme (Days)

73 State Space Model Example 4 Convert the recursve model to state-space form, assumng θ = 3. k k k k Gw Bu Ax x 1 k u k x k u D x D z () (1) (2) 3) ( ~ ~ ~ ~ ~ k k T k k k k k q u q q q I x ~() ~ ] ~ [ k k k k k q I z d w 1, 1, G B A 1, 1 D x D u

74 LQG Approach to Inventory Control Pseudo-Constraned Control p L g x 1,, mn ) ( ) ( X BY AX BY AX G G X T T w ) ( ) ( ' X Y D X D Y D X D T T u x u x p z,..., 1, 2 Chmelewsk and Manthanwar (24)

75 Std. Dev. Inventory Inventory LQG Approach to Inventory Control Example 5: Solve scenaro n example 3 usng state-space model x mn, L, p C 1 g B Std. Dev. Starts = varance of nventory = varance of starts Case A: g 1 = 1, g 2 = Case B: g 1 = 1, g 2 = 5 Case C: g 1 = 1, g 2 = 1 A Starts C B A

76 Std. Dev. Inventory Inventory Connecton to (R,S) Approach Case A gves a controller dentcal to the (R,S) approach p mn g x, L, 1 Case A: g 1 = 1, g 2 = Case B: g 1 = 1, g 2 = 5 Case C: g 1 = 1, g 2 = 1 C B Std. Dev. Starts A Starts C B A

77 Extenson to Two-Echelon System Decentralzed controllers

78 Extenson to Two-Echelon System Centralzed controllers

79 Std. Dev. Inventory Tank 2 Extenson to Two-Echelon System Centralzed B Centralzed A Decentralzed Std. Dev. Inventory Tank 1 Decentralzed: (R,S) approach appled to both tanks Centralzed A: CMV problem wth g = [1 1 ] Centralzed B: Centralzed A case wth more weght placed on nventory of the frst tank

80 Inventory Inventory Extenson to Two-Echelon System 2 Tank 1 3 Tank Decentralzed Centralzed A 2 Decentralzed Centralzed A Starts Starts Small safety stock ncrease at I 2, leads to large safety stock decrease at I 1

81 Mult-Echelon System Delvery tme = 2 Retal 1 Demand 1 Delvery tme = 4 Warehouse 1 Delvery tme = 1 Delvery tme = 6 Delvery tme = 6 Retal 2 Demand 2 Plant Delvery tme = 6 Delvery tme = 4 Delvery tme = 6 Delvery tme = 2 Warehouse 2 Delvery tme = 2 Retal 3 Retal 4 Demand 3 Demand 4

82 Mult-Echelon System

83 Closed-loop Smulaton: Inventory at Retal 1

84 Closed-loop Smulaton: Inventory at Warehouse 1

85 Mult-Echelon System Delvery tme = 2 Retal 1 Demand 1 Delvery tme = 4 Warehouse 1 Delvery tme = 1 Delvery tme = 6 Delvery tme = 6 Retal 2 Demand 2 Plant Delvery tme = 6 Delvery tme = 4 Delvery tme = 6 Delvery tme = 2 Warehouse 2 Delvery tme = 2 Retal 3 Retal 4 Demand 3 Demand 4

86 Outlne Model Predctve Controller Tunng Pseudo-Constraned Control Proft Control Market Responsve Control

87 Electrc Power Management Power Produced Equals Power Consumed

88 Power Management wth Renewable Power Power Produced Power Consumed

89 MW MW Power Management Wth Renewable Power Dspatchable Renewable Load

90 Motvaton Structure Utlty Perspectve Merchant Perspectve Drven by Consumers Relablty Requrements Focused on Captal Costs Drven by Opportunty Attenton to Market Prces Focused on Revenue

91 Cents per kw hr Electrcty Spot Prce Merchant Perspectve 4 3 RTP Electrcty Forecasted Data Tme (days) Drven by Opportunty Attenton to Market Prces Focused on Revenue

92 Integrated Gasfcaton Combned Cycle (IGCC)

93 Dspatchable IGCC Synthess Gas Storage Compressed Ar Storage

94 Cryogenc Ar Separaton Unt (CASU) Compressor Work Compressed Ar N 2 Rch Vapor Lqud N 2 Ar Pretreatment GOX GN2 Low Pressure Column Ar Heat Exchanger GOX Expander Expander Hgh Pressure Column Crude Lqud Oxygen

95 Why Not O2 Storage? Work Compressor Compressed Ar N 2 Rch Vapor Lqud N 2 Ar Pretreatment GOX GN2 Low Pressure Column Ar Heat Exchanger GOX Expander Expander Hgh Pressure Column Crude Lqud Oxygen

96 Why Not O2 Storage? Compressor Work N 2 Rch Vapor Ar Pretreatment Compressed Ar Ar GOX GN2 Low Pressure Column Lqud N 2 Cryogenc dstllaton has very large response tme Heat Exchanger GOX Expander Expander Hgh Pressure Column Typcally the slowest unt of the whole IGCC process Crude Lqud Oxygen

97 Why Compressed Ar Storage? Work Compressed Ar Storage Compressed Ar Compressor Compressed Ar N2 Rch Vapor Low Pressure Column Lqud N2 95% of CASU power s used by the Man Ar Compressor. Ar Pretreatment Ar GOX GN2 Heat Exchanger GOX Expander Expander Hgh Pressure Column Crude Lqud Oxygen Man Ar Compressor can respond quckly. Dstllaton Unt can stll be run at constant throughput.

98 Dspatchablty from IGCC C G F 6, P to CASU C 4 η 4 F 5, P e F 4, P e F, P C η F 2, P s F, P s C N to grd Compressed Ar Storage P s, T s, V s

99 Cents per kw hr Electrcty Spot Prce Merchant Perspectve 4 3 RTP Electrcty Forecasted Data Tme (days) Drven by Opportunty Attenton to Market Prces Focused on Revenue

100 Response to Market Changes EDOR s due to dfferent controller tunngs BOP wth less proft BOP wth more proft * * OSSOP Peng et al. (25)

101 Electrc Prce Model Whte Nose Input Shapng Flter Sequence wth Electrcty Prce Characterstcs

102 Electrc Prce Model Whte Nose Input Shapng Flter Sequence wth Electrcty Prce Characterstcs Measured Electrcty Prce State Estmator and/or Predctor Predcton of Electrcty Prce

103 Cents per kw hr Electrc Prce Model Whte Nose Input Shapng Flter Sequence wth Electrcty Prce Characterstcs Measured Electrcty Prce State Estmator and/or Predctor Predcton of Electrcty Prce 4 3 RTP Electrcty Forecasted Data Tme (days)

104 Model Predctve Control T max pe( t)* vp ( t) dt v p ( t) where p ( t) ~ the predcted prce (or value) v e p ( t) ~ the producton rate

105 Model Predctve Control T max pe( t)* v p ( t) dt v p ( t) where p ( t) ~ the predcted prce (or value) and Constrants nclude : v ( t) S( t) e p v p ( t) ~ the producton rate ~ amount n storage v max p and S( t) S max

106 Model Predctve Control T max pe( t)* v p ( t) dt E[ pe * v p ] Ce v p ( t) where p ( t) ~ the predcted prce (or value) and Constrants nclude : v e p ( t) S( t) v p ( t) ~ the producton rate ~ amount n storage v max p and S( t) S max

107 System Desgn max v p ( t) T pe( t)* vp( t) dt E * p e vp Ce ( v p ( t) max v max p and max How does v and S mpact C p e S( t) S max )?

108 System Desgn max v p ( t) T pe( t)* vp( t) dt E * p e vp Ce ( v p ( t) max v max p and max How does v and S mpact C p e S( t) S max )? MPCcannot answer ths queston!

109 Expected Proft Whte Nose Input Shapng Flter p e (t) E[p e* v p ] Manpulated Varables (Controller s u=lx) Process Model v p (t)

110 Re-Scalng of Prce w(t) Shapng Flter p' e (t) ( p' e pe) E[p' e* v p ] Manpulated Varables Process Model v p (t) (Controller s u=lx)

111 Correlatng Prce and Usage If E 2 ( ' ) p e v p and p' e p e then v p ( t) p e ( t)

112 Correlatng Prce and Usage ) ( ) ( then t p t v e p ) ' ( If 2 p e v p E e e p p ' and ] [ ] [ 2 ] [ Also, p p e e v E v p E p E ] [ ] [ ] [ p p e e v E v p E p E e p e e C v p E p E ] [ ] [ 2

113 Maxmum Proft L, C e max c R 2 R p e ( c E[ ]) E E E 2 ( ' ) p e v p 2 max 2 v p v ) ( p 2 max 2 S (S )

114 Maxmum Levelzed Proft L,, v max max p, S max c R c L,1 v max p c L,2 S max E E E 2 ( ' ) p e v p 2 max 2 v p v ) ( p 2 max 2 S (S )

115 Maxmum Levelzed Proft L,, v max max p, S max E c R c L,1 v max p 2 ( ' ) p e v p c L,2 S max E E 2 max 2 v p v ) ( p 2 max 2 S (S ) Non-Convex Problem (but global soluton from branch and bound)

116 Captal Costs c Compressor CASU Storage cost: Water Reservor $.54/m 3 of workng volume. Compressed Ar Inventory Compressor cost: $16/kW

117 Dspatchablty from IGCC C G F 6, P to CASU C 4 η 4 F 5, P e F 4, P e F, P C η F 2, P s F, P s C N to grd Compressed Ar Storage P s, T s, V s

118 MW $/MWh MW Dspatchable Operaton CASU Man Compressor 8 a Stroage Man Compressor 8 b 6 4 Mm 3 Volume of Storage c Prce of Electrcty 2 d tme, day

119 Revenue, mllon $/day Changes n Revenue b wth storage no storage tme, day

120 Levelzed Annual Revenue Compressor Costs Storage Costs Levelzed Revenue Wthout Storage $96M - $368M/yr Wth Storage $192M $.2M $377M/yr 2.5% mprovement n levelzed revenue

121 Dspatchable IGCC Synthess Gas Storage Compressed Ar Storage

122 Acknowledgements Current Students: Syed K. Amed, Ben Omell and Davd Mendoza-Serrano Former Students: Amt Manthanwar (MS), Mchael Peng (PhD), Mng-We Yang (PhD) and Wa-Kt Ong (UG) Collaborators Professor Durango-Cohen (Stuart Busness School, IIT) Professor Abbason (Chem Eng, IIT) Professor Muehleson (Arch Eng, IIT) Professor Moschandreas (Env Eng, IIT) Fundng: Natonal Scence Foundaton (CBET 96796) IIT Graduate College and Armour College og Engneerng IIT Chemcal & Bologcal Engneerng Department

123 Conclusons Relatonshp between controller performance and plant proft quantfed. Enables proft guded controller and closed-loop system desgn. Applcable to a broad set of problems from a varety of dscplnes. Lnear controller can be used for market responsveness. Non-convex, but global soluton methods used.

124 Hybrd Vehcle Desgn Power Bus arm fc bat scap R arm Fuel Cell E fc R bat E bat R scap E scap E arm L arm w arm

125 DC-DC Converters Power Bus arm fc afc bat abat scap ascap R arm Fuel Cell E fc DC-DC Converter R bat E bat DC-DC Converter R scap E scap DC-DC Converter E arm L arm w arm k fc k bat k scap

126 Servo-Loops wth PI Controllers PI (sp) P fc + - PI (sp) P bat + - PI (sp) P scap + - k fc P fc k bat P bat k scap P scap Vehcle Power System

127 Supervsory Control Hgh Level Controller (sp) P mot PI (sp) P fc + - PI (sp) P bat + - PI (sp) P scap + - k fc P fc k bat P bat k scap P scap Vehcle Power System

128 Power to Motor (kw) Speed (mph) Drve Cycle Data and Modelng tme (sec) tme (sec)

129 Hgh Level Battery Model E bat P bat P (loss) bat

130 Hgh Level Battery Model E bat P bat P (loss) bat E mn bat E mn E bat bat E max bat E max bat eˆ bat m bat

131 Hgh Level Battery Model E bat P bat P (loss) bat E mn bat E mn E bat bat E max bat E max bat eˆ bat m bat P P P mn bat mn bat P mn bat P max bat bat rate d Cˆ, bat eˆ batmbat ˆ rate, c bat eˆ batmbat C

132 Power and Energy Constrants E bat P C E max rate,c max bat bat bat E eˆ m max bat bat bat E mn bat P bat P C E mn rate,d max bat bat bat

133 Constrants a Functon of Mass E bat P C E max rate,c max bat bat bat E eˆ m max bat bat bat P bat

134 Aspect Rato a Functon of C-Rate E bat P C E max rate,c max bat bat bat E eˆ m max bat bat bat P bat

135 Power Losses E bat P bat P (loss) bat E mn bat E mn E bat bat E max bat E max bat eˆ bat m bat P ( loss) bat lˆ P bat 2 bat m bat P P P mn bat mn bat P mn bat P max bat bat rate d Cˆ, bat eˆ batmbat ˆ rate, c bat eˆ batmbat C

136 Operatng Regon wth Power Losses E bat P bat

137 Hgh Level Super Cap Model E sc P sc P (loss) sc E mn sc E mn E sc sc E max sc E max sc eˆ sc m sc P ( loss) sc 2 Psc lˆ m sc sc P P P mn sc mn sc mn sc P P max sc sc rate d Cˆ, sc eˆ scmsc ˆ rate, c sc eˆ scmsc C

138 Detaled Fuel Cell Models Anode Sold Materal Current Collector In Cathode (H 2, H 2 O) H Ar n 2 O 2 H + H + H 2 O H + H + H + N 2 Anode Exhaust H + H + H + H 2 O Cathode Exhaust MEA

139 Hgh Level Fuel Cell Model P P fc fc P mn fc P P mn fc P max pˆ fc fc fc m P fc max fc P P P mn fc mn fc max fc P fc C C P rate, d fc rate, c fc pˆ max fc pˆ fc fc m m fc fc

140 Desgn Problem Formulaton mn{ cˆ m cˆ m cˆ m } st.. fc fc bat bat sc sc P P P P fc o bat sc m m m m m v o fc bat sc Pbat bat Pbat bat lbatmbat Pbat lbatm bat P P sc sc sc l m sc sc sc sc sc sc P l m T AX BY AX BY Go mvg 1 1 Go mvg1 w x u T T x u D X D X D Y X D Y P E E 2 max fc max bat max sc P P P max fc max bat max sc P fc E E bat sc P 1 2 E 3 4 P 5 6 fc E bat sc P P P P bat sc fc bat P sc P fc mn fc E E mn bat mn sc P P P mn bat mn sc mn fc

141 Case Study Technology Lthum Battery Super- Capactor PEM Fuel Cell Cost $59/kg $93/kg $3/kg C-Rate.5 hr hr -1 1 hr -1 Power Densty 1 W/kg 11, W/kg 1 W/kg Appetecch & Prosn (25) Portet, Taberna, Smon, Flahaut, & Laberty-Robert (25) Murphy, Csar & Clarke (1998)

142 Case Study Soluton Technology Lthum Battery Super- Capactor PEM Fuel Cell Mass 3 kg 2.8 kg 3.5 kg Nomnal Power.1 kw 1.5 kw 1.8 kw Total Captal Cost Cost $177 $26 $15 $3,8

143 P fc (kw) Fuel Cell Operatng Regon 3.5 Fuel Cell 2.1 Fuel Cell Power(kW) P fc (kw/s) x tme(hr)

144 E sc (kj) Super Cap Operatng Regon 7 Super Capactor 15 SuperCap Power, kw P sc (kw) tme(hr)

145 E bat (kj) Battery Operatng Regon 1 8 Battery Battery Power, kw P bat (kw) tme(hr)