OPTIMAL CONTROL THEORY APPLIED TO HYBRID FUEL CELL POWERED VEHICLE. G. Paganelli 1, T.M. Guerra 2, S. Delprat 2 Y. Guezennec 1, G.

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1 Copyrght IFAC 15th Trennal World Congress, Barcelona, Span OPTIMAL CONTROL THEORY APPLIED TO HYBRID FUEL CELL POWERED VEHICLE G. Paganell 1, T.M. Guerra, S. Delprat Y. Guezennec 1, G. Rzzon 1 1 Center for Automotve Research - Oho State Unversty 93 Knnear Road,Columbus, Oho USA LAMIH UMR CNRS Unversté de Valencennes et du Hanaut Cambréss le Mont Houy Valencennes CEDEX 9 - France Abstract: Ths paper focuses on the energy management control strategy for a hybrd fuel cell powered vehcle. A method based on optmal control theory s presented. Ths control strategy has been appled to a representatve hybrd PEM (Proton Exchange Membrane fuel cell md-sze vehcle. Smulaton results on a pror known drvng cycle are presented and compared to a sub-optmal strategy. Copyrght IFAC Keywords: Hybrd vehcles, Energy management systems, Optmal control, Optmal power flow, Fuel cell. 1. INTRODUCTION Fuel cell powered vehcles are foreseen as an answer to the rsng concerns for the envronment. The hybrdzaton of a fuel cell powered vehcle may be requred for start-up, cold-start response or fuel cell cost, weght and volume consderatons. An other reason for hybrdzaton s the opportunty for energy effcency mprovement. Actually, the addtonal energy storage devce allows kc energy recovery and energy management optmzaton opportunty by offerng an extra degree of freedom n the power flow. Ths creates the need for a hgher level of control n the vehcle, typcally referred to as the supervsory controller or control strategy. Commonly used hybrd fuel cell control strategy are the SOCbased control, the load followng or a combnaton of both (Boettner, et al., 1, Ogburn, et al.,, Patton, et al., 1. Whle generally allowng proper operaton, those approaches do not embed any energy mnmzaton consderaton and consequently do not reach optmal fuel economy. Regardless of the topology of an hybrd powertran, the essence of the control problem s the nstantaneous management of the power flows from both power sources to acheve the overall mnmum fuel consumpton. The global nature of both the objectve (hydrogen consumpton and the constrant (accumulator state-of-charge does not lend tself to tradtonal global optmzaton technque, as the future s unknown n actual drvng crcumstances. A prevous work (Paganell & al. led to a Hybrd Fuel cell control strategy that nstantaneously optmzes the power splt between the energy converters, whle somehow accountng for the global nature of the problem. The proposed power splt algorthm, Equvalent Consumpton Mnmzaton Strategy (ECMS was based on a heurstc formulaton of the power flow wthn the electrcal accumulator. Although sutable for real tme applcaton and yeldng very good fuel economy, ths heurstc approach does not allow to reach the global optmal power flow dstrbuton. The approach presented n ths paper proposes to brdge ths gap. The objectve of ths work s thus to determne the global optmal power flow dstrbuton, n terms of energy management, wthn an hybrd fuel cell powered vehcle. It s an extenson of a prevously developed method to apply optmal control theory to a parallel hybrd electrc/ice engne vehcle (Delprat, et al., 1a.. FUEL CELL SYSTEMS Most typcally, the archtecture of a hybrd fuel cell vehcle ncludes a DC bus physcally represented by the electrcal accumulator. It s connected to the fuel cell va a DC/DC converter on one sde and to the electrc machne wth an assocated converter on the other sde, as shown n Fgure 1. Fuel Cell Power Control nput I FC +++ Fuel Cell System Power Converter DC/DC - - Power flow from Fuel Cell to DC Bus Electrc Motor Power Control nput ++++Hgh Voltage DC Bus ++++ I BATT +++ Electrcal Accumulator Hgh Voltage DC Bus Fg. 1. General Hybrd Fuel Cell System + Power Converter DC/AC - Electrc Motor Power flow between DC Bus and Electrc Motor

2 The approach presented n ths paper could easly be appled to any knd of fuel cell hybrd powertran. As a representatve example, a Proton Exchange Membrane Fuel Cell stack s assumed. The desgn and operatng parameters used are representatve of those proposed n automotve applcaton for a mdsze SUV-type vehcle, they are shown n Table 1. Table 1: Fuel Cell Stack operatng Parameters Input Parameter Value Actve Area [cm ] 4 Number of cells n seres 44 Nomnal operatng temperature [K] 353 Ar and hydrogen nlet temperature [K] 333 Ar and hydrogen nlet relatve humdty 1 [%] Anode pressure [atm] Fuel utlzaton.8 Ambent temperature [K] 73 Ar/Fuel Stochometrc Rato [kmol ar /kmol h] System Maxmum Net Power at Nomnal Operatng Temperature [kw] 86.4 Ths representatve fuel cell system has been smulated usng a hybrd fuel cell vehcle model developed at the Oho State Unversty for ts VP- SIM vehcle performance smulator (Boettner, et al.,. It s based on a sem -emprcal pressure and temperature dependant pecewse fuel cell voltage modellng. It s assumed the auxlares consst n a varable speed compressor/expander for the ar delvery system, a pump/heat-exchanger/fan for the coolng system, a pump for hydrogen recrculaton and water pumps for humdfcaton. All the auxlares are controlled to deally track the fuel cell system power demand wth constant stochometrc ratos and are assumed to be powered by the fuel cell tself. The system power beng the remanng power avalable. The compressor s the greatest auxlary consumer wth a maxmu m power needed of 3 kw, assumng an sentropc effcency rangng from 53 to 71%. We consder here the entre system consstng of the fuel cell wth ts auxlares and we are nterested n the power out of ths system. As shown n Fgure, the maxmum model based power avalable from the fuel cell system s related to the stack temperature, t reaches a maxmum of 86.4 kw at nomnal operatng temperature (8 C or 353k. Maxmum system output power [kw] Cell temperature [degc] (nomnal temperature Fg.. VP-SIM computed Maxmum system output power avalable functon of stack temperature System Nomnal temperature (353k performances 9 Output power [kw] 8 System 7 Effcency [%] Stack current [A] Fg. 3. VP-SIM computed Fuel cell system output power and effcency functon of output current at nomnal temperature As depcted n Fgure 3 for the partcular case of nomnal temperature, the system output power ncreases along wth output current up to a certan lmt where the maxmum output power operatng pont s reached. Ths lmt s due to a more sgnfcant drop of output voltage and must be the practcal lmt of operaton of the system. Clearly above ths pont more hydrogen s consumed for less output power nducng a dramatc effcency drop. The effcency shown s computed on the bass of the system output power versus enthalpy nput to the system,.e. mass flow of hydrogen consumed tmes hydrogen LHV (Lower Heatng Value. The effcency remans hgh on the all range of operaton, ths can be manly explaned by the presence of an expander and by the compressor beng controlled to satsfy a constant ar stochometrc rato, whch s an deal case but not always practcally realsable, especally for low loads. In a pure fuel cell vehcle, the fuel cell power has to track the drver demand. Wth a hybrd confguraton, the fuel cell remans the only component to provde the energy, the buffer role of the addtonal energy storage devce allows flexblty n the choce of the fuel cell operatng pont trajectory to provde ths needed amount of energy. One can foresee that for a gven amount of power produced by the fuel cell system over a drvng cycle, dfferent fuel cell operatng pont trajectores mght lead to completely dfferent overall fuel consumpton. Ths s due to the large possble excurson of the fuel cell effcency. An other reason s the operatng pont trajectory mpacts also on the stack temperature, whch n turn sgnfcantly affects the fuel cell performance (effcency and avalable power. The purpose of the followng approach s to defne the optmal operatng pont trajectory of ths hghly coupled system to reach the mnmum overall fuel consumpton. 3. OPTIMAL CONTROL FORMULATION 3.1. Control strategy of a hybrd fuel cell vehcle The am of a hybrd control strategy s how to nstantaneously dstrbute the power on both energy sources, electrcal accumulator and fuel cell system, Practcal lmt of operaton

3 to satsfy the road load power requrement whle meetng the nstantaneous and global constrants and objectves. In our case the road load power requrement s the power requested by the tracton P em t postve or electrc motor at tme t : negatve. In a real vehcle t s contnuously gven by the drver through the gas pedal, and prorated n functon of the maxmum power avalable at the current vehcle speed. Both the battery and the fuel cell system contrbute to provde ths power: P t = P t P t (1 em batt + Where P s the power produced by the fuel cell and P batt s the power to/from the electrcal accumulator 1. Practcally, the power dstrbuton s adjusted by controllng the power produced by the fuel cell P t through the assocated power converter shown on fgure 1. The power produced (or captured by the electrcal accumulator P batt beng the remanng power needed to satsfy the total power requrement: P = P P batt The chosen control varable to adjust the fuel cell power s the current produced by the Fuel Cell I t. Ths s the avalable current output system of the system,.e., the current produced by the fuel cell less the current drawn by ts auxlares. The consdered dynamc varables of the system are the battery state of charge (SOC and the fuel cell stack temperature. They are descrbed by the followng system of equatons: em ( SOC( t+ 1 = SOC( t + P t Pem t Effbatt Te T( t+ 1 = α T( t + QI ( t, T t Te Where : SOC s the State of Charge at tme t ; ( P s the power produced by the fuel cell system at tme t ; Eff s the battery effcency dependant batt of power flow drecton ; T t s the fuel cell stack nternal temperature at tme t ; Te s the samplng tme; α represents the natural heat exchange coeffcent, t depends of the ambent temperature; Q( I _, T s the heat produced by the stack functon of current I t T t. _ and stack temperature 1 Varous effcences have been neglected n ths paper for clarty, but are ncluded n the formulaton of the control strategy actually mplemented. As can be seen n the above set of equaton, the SOC t s a electrcal accumulator state of charge pure ntegraton process. The temperature T s modelled as a smplfed frst order response type system. The chosen relevant crteron s the fuel (hydrogen consumpton on a temporal wndow of N samples. The crteron can be wrtten as : (, N 1 f (3 t = J = m I t T t Te m ( I t, T t represents the hydrogen f consumpton (n gr/s requred to produce the current T t. I ( t at the temperature In the followng sub-sectons, an approach based on optmzaton control tools (Borne, et al., 199 s proposed. 3.. Formulaton of the optmzaton problem The power produced by the fuel cell system s derved from the product of current and voltage avalable on the system output: P t = I t V I t, T t (4 To smplfy notaton, let us defne F( I, T ( t representng the voltage on the fuel cell system Eff Te multplcatve factor: output wth a ( batt (, (, F I t T t V I t T t Eff Te (5 batt It comes: P t Eff Te I t F I t, T t (6 batt The followng defnton s also used : SOC( t Pem Effbatt Te φ ( x, I + I F ( I, T α T( t + Q( I, T Te wth the state vector : T (7 xt = SOC t T t (8 The power dstrbuton on both energy sources can be wrtten as an optmzaton under constrants problem: System xt φ xt I + 1 =, (9 N 1 Crteron J = m f ( I, T Te (1 t = Constrant SOC( N SOC = (11 Introducng the lagrangan parameters vector 1 λ = λ λ mnmzaton of : N 1 t= T ( J = { m f I t, T t Te leads to the xt φ( xt I + λ + 1, } (1

4 Frst order condtons: J = λ1( t 1 λ1 = SOC J Q( I, T = λ( t 1 λ α+ T T (13 F( I, T I = T Then : t, λ λ 1 = 1 and : Q( I, T λ ( t α+ T F( I, T = λ( t 1 I T J m f ( I = I I ( t F( I, T I I ( F I, T λ (, Q I t T t T (14 (15 Computaton of soluton wth respects to (14 and (15 requres analytcal expresson of the hydrogen consumpton, the heat produce by the stack and output voltage. The VP-SIM fuel cell model allows computng of the needed varables wth a very good accuracy. Nevertheless, due to the complexty and the non-lnearty of the equatons nvolved n ths model some approxmatons have been performed Approxmaton of characterstcs Computed output voltage curves are shown on fgure 4, they are hghly related to the current wth a steeper decreasng rate at low temperature. F I t, T t s ( As defned n equaton (5, lnearly related to the fuel cell system output voltage V I t, T t. ( characterstcs Stack Voltage [V] Practcal Lmt of operaton 93k Fuel Cell Operaton Temperature Increasng 353k Stack Net current [A] Fg. 4. VP-SIM computed fuel cell system output voltage Stack Heat tranfer [kw] Fg. 5. VP-SIM computed heat transfer curves The VP-SIM model uses conservaton of energy analyss to compute the heat transfer assocated wth the fuel cell,, Q I t T t. As shown by the curves on fgure 5, the operaton of a fuel cell s essentally exothermc. Lnear pecewse approxmaton of the characterstcs Q and V provdes the requred analytcal expresson of,, F I t T t and Q I t T t to solve (14 and (15. Let n be the number of models obtaned, each of them beng defned on a segment I mn,imax,, wth of course {,..., n 1} n 1 mn max max ( T I =,I = I : (, ( ( (, ( ( F I T a T I b T = + (16 Q I T c T I d T = + (17 Accordng to the electrochemcal nner reacton of a fuel cell, the mass flow of hydrogen consumed s lnearly related to the current drawn at the stack level (18 I : m = KI f where K s a constant functon of the fuel cell actve area, the Faraday s constant and the molecular weght of hydrogen. When expressng the mass flow functon of the fuel cell current I as opposed to stack current, a slght non-lnearty appears due to the non-lnear nature of the current drawn by the auxlares. Nevertheless, as shown on fgure 6, ths non-lnearty s only relevant at hgh current and especally above the practcal lmt of operaton. Snce the operatng pont of the system must never exceed the practcal lmt of operaton, for the purpose of ths study the hydrogen mass flow s defned as a lnear functon of the stack output current: f Practcal lmt of operaton Stack Net current [A] ' m t K I t (19 93k 353k Increasng temperature The heat transfer s defned postve when t flows from the system to the envronment.

5 Stack H Consumpton [kg/s] x Practcal lmt of operaton Stack Net current [A] Table : Smulated Vehcle Specfcatons Mass [kg] 36 Frontal area [m ] 4 Coeffcent of drag.75 Coeffcent of rollng resstance.15 Fg. 6. VP-SIM computed stack hydrogen consumpton versus stack current at nomnal temperature The next sub-secton proposes a soluton to the optmzaton problem usng those approxmatons Towards an optmal soluton Usng (14 and (15 wth (16 and (17, and after some easy calculus we found the followng second order equaton to be solved at each samplng tme : ( ( ( ( ( a T t c T t I λ1 c ( T a ( T T t T t + ( b T t c T t I λ1 c ( T b ( T T t T t ( d T t m f I t c T t a ( T α + + T t I t T t d ( T m f ( I + α + b ( T T ( t I ( ( t c T λ 1 = ( At last, one can see that the ntal values of the lagrangan parameters,.e. 1 λ = λ λ T are enough to compute an optmal soluton wth respect to (9~(11 (Borne, et al., 199. Clearly, the fnal state of charge depends only on the ntal values of the lagrangan parameters. A smple gradent descent s used to λ whch enforce obtan 1 x( N x λ and =. 4. SIMULATION RESULTS In the followng applcaton, the fuel cell system consdered above s assocated to a 4 kw maxmum power nckel metal hydrde battery wth typcal specfc power of W/kg and specfc energy of Wh/kg. Ths confguraton s the result of a senstvty analyss presented n (Boettner, et al., 1 and s sutable for the SUV-type vehcle specfcatons presented n table. It s powered by an AC nducton electrc motor, requrng up to 16 kw. The 138 seconds long Federal Urban Drvng Schedule (FUDS was used for the smulatons conducted wth the VP-SIM smulator. The fuel cell starts at ambent temperature to represent a cold start. The ntal battery state-of-charge s.7. The process leadng to optmal results s thus: 1. Usng the VP -SIM smulator, compute the power requred by the electrc motor to meet the drvng schedule Pem ( t, t [...138] ;. Input the obtaned power needed vector nto the optmal control algorthm descrbed above to get an optmal fuel cell current dstrbuton I, t [...138] ; 3. Implement ths fuel cell current dstrbuton nto the VP-SIM smulator. Let s emphasze that, at step, the optmal current control law I t s computed usng an approxmated model accordng to equatons 16 to 19. At step 3, ths control law s then mplemented nto a more complex and accurate model, whch gves a slghtly dfferent result from the one computed wth the optmal control algorthm. Nevertheless, experences have shown that mprovement of the accuracy of the smplfed model leads to a sgnfcant ncrease of the algorthm complexty for very lttle mprovement. The sold lnes on fgure 7 show the smulated system behavour under the optmal current law for the FUDS drvng cycle. (a (b (c (d FC Net current [A] FC Net Power [kw] FC Temperature [K] Tme [s] Batery SOC [%] 8 Fg. 7. Smulated results wth optmal control on FUDS cycle

6 In order to evaluate and contrast the results of the optmal control theory approach, the well-proofed ECMS local optmzaton strategy (Paganell, et al., s used as a bass of comparson. The dashed lnes show the correspondng ECMS results. As can be seen, the current law provded by the optmal control, plot (a, s sgnfcantly smoother. Consequently, the fuel cell power and temperature, plot (b and (c, exhbt much less excurson. Nevertheless, the plot (d, shows that the battery state of charge ends at the same level n both cases. Table 3 below gves the compared fuel economy results. The fuel usage s corrected for (mnor dfferences n ntal and fnal battery state-of-charge. The correcton s based on the electrcty used (reflected by the dfference between battery fnal state of charge and ntal state of charge, usng the correspondng average fuel cell system effcency from that smulaton to convert electrcty usage to equvalent fuel consumpton. Furthermore, the hydrogen usage s converted to equvalent gasolne usage based on the relatve lower heatng value of hydrogen and gasolne. Ths allows to report the results n a more ntutve form. An mprovement of more than 3% s acheved wth the optmal control approach. Table 3 FUDS smulated fuel economy Hydrogen used Gasolne equvalent fuel consumpton ECMS 67.1 g 8.14 l/1km Optmal Control (8.86 mpg 57.7 g 7.85 l/1km (9.9 mpg 5. CONCLUSION Ths paper descrbes a formulaton based on the optmal control theory for the supervsory control n charge-sustanng hybrd fuel cell vehcles on a pror known drvng cycle. Ths method allows to defne the optmal power splt between the fuel cell and the electrcal accumulator to globally mnmze the overall hydrogen consumpton. The use of optmal control theory reduces the computaton cost n comparson wth other optmzaton methods. For one specfc representatve smulated hybrd fuel cell vehcle confguraton, the proposed control strategy outperforms the fuel economy results obtaned wth other control strateges whle enforcng a chargesustanng operaton. Nevertheless, ths approach s not lmted to the battery/fuel cell confguraton presented n ths paper, t can easly be appled to other confguratons such as ultracapactor/fuel cell. Although not dramatc mprovement n fuel effcency s acheved n comparson wth the ECMS control strategy for the presented smulated drvng cycle, ths approach provdes an optmal reference result for off-lne assessment of other strateges. Implementng the result of the proposed approach for real tme control would requre the forecast of a λ, correct value of the lagrangan parameters,.e. whch wll provde an acceptable state of charge at the end of the msson. Clearly f the trp s not a pror known, t s mpossble to forecast ths value and to drectly use the proposed approach for real tme control. Nevertheless, as shown n (Delprat, et λ has a monotonc effect on state of al., 1b, charge varaton. Therefore adjustng the value of λ durng real tme operaton, usng a smple PI controller for example, allows to sustan the state of charge close to a target value. Future work wll be devoted to ths real tme adaptaton of the proposed method. REFERENCES Boettner D., G. Paganell, Y. Guezennec, G. Rzzon, M. J. Moran (. Proton Exchange Membrane (PEM fuel cell system model for automotve vehcle smulaton and control. ASME Journal of Energy Resources Technology, Volume 14, pp.1-8, March. Boettner D., G. Paganell, Y. Guezennec. M. J. Moran, G. Rzzon (1. Component Power Szng and Lmts of Operaton for Proton Exchange Membrane (PEM Fuel Cell/Battery Hybrd Automotve Applcatons. ASME - Internatonal Mechancal Engneerng Congress and Exposton IMECE1/DSC New York, November 1. Borne P., G. Dauphn-tanguy, J.P. Rchard, F. Rotella, I. Zambettaks (199. Systems control and optmzaton. Méthodes et Technques de l ngéneur, Édtons Technp (n french. Delprat S., T.M. Guerra, G. Paganell, J. Lauber, M. Delhom (1a. Control strategy optmzaton for an hybrd parallel powertran - Presented to the ACC 1, Amercan Control Conference Crystal Gateway Marrott, Arlngton, VA, USA - June 1, pp Delprat S., T.M. Guerra, J. Rmaux (1b. Optmal control of a parallel powertran : From global optmzaton to real tme control strategy, Electrc Vehcle Symposum, EVS18, Berln, October 1. Ogburn M., D.J. Nelson, W. Luttrell, B. Kng, S. Postle, R. Fahrenkrog (. Systems Integraton and Performance Issues n a Fuel Cell Hybrd Electrc Vehcle. SAE, Paper # Paganell G., Y. G. Guezennec, G. Rzzon (. Optmzng System Control Strategy for Hybrd Fuel Cell Vehcle. Paper -1-1 SP SAE World Congress, Detrot, Mchgan, USA - March 4-7,. Patton D., J. Latore, M. Ogburn, S. Gursk, P. Bryan, D. J. Nelson (1. Desgn and Development of the Vrgna Tech Fuel Cell Hybrd Electrc FutureTruck. Socety of Automotve Engneers SAE SP-1617, pp.185-.