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Sliding Mode Control of Hydrogen Fuel Cell and Ultracapacitor Based Electric Power System: Electric Vehicle Application
Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of 20th World The International Federation of Congress Automatic Control Toulouse, France, July 2017 Proceedings of the the 20th9-14, World Congress Available online at www.sciencedirect.com The Federation of Automatic Control Toulouse, France, July 2017 The International International Federation of Congress Automatic Control Proceedings of the 20th9-14, World Toulouse, France, July Toulouse, France,Federation July 9-14, 9-14, 2017 2017 The International of Automatic Control Toulouse, France, July 9-14, 2017
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Sliding Mode Control ofIFAC Hydrogen Fuel Cell and Ultracapacitor Based Electric PapersOnLine 50-1 (2017)Cell 14794–14799 Sliding Mode Control of Hydrogen Fuel and Ultracapacitor Based Electric Power System: Electric Vehicle Application Based Electric Sliding Mode Control of Hydrogen Fuel Cell and Ultracapacitor Power System: Electric Vehicle Application Based Electric Sliding Mode Control of System: Hydrogen Fuel Cell and Ultracapacitor Power Electric Vehicle Application Roshini System: S. Ashok*, Yuri B. Shtessel, **, andApplication Malek Ghanes*** Power Electric Vehicle Roshini S. Ashok*, Yuri B. Shtessel, **, and Malek Ghanes*** Roshini S. Yuri **, Malek Roshini S. Ashok*, Ashok*, Yuri B. B. Shtessel, Shtessel, **, and and Malek Ghanes*** Ghanes*** *The University of Alabama in Huntsville, Huntsville,AL-35899 Roshini S. Ashok*, Yuri B. Shtessel, **, and Malek Ghanes*** *The University of Alabama in Huntsville, Huntsville,AL-35899 USA; e-mail: [email protected]). *The University University ofe-mail: Alabama in Huntsville, Huntsville, Huntsville,AL-35899 in USA;of [email protected]). The University ofAlabama Alabama in Huntsville,Huntsville,AL-35899 Huntsville, AL-35899 ***The USA; e-mail: [email protected]). *The University of Alabama in Huntsville, Huntsville,AL-35899 USA;(256)-824-6164; e-mail: [email protected]). ** The University of Alabama in Huntsville, Huntsville, AL-35899 (tel.: e-mail: [email protected]). USA; University of Alabama in Huntsville, AL-35899 ** The USA; e-mail: [email protected]). The University of Alabama in Huntsville, Huntsville, Huntsville, AL-35899 , ([email protected]** (256)-824-6164; e-mail: USA; IRCCyN) CNRS UMR 6004 ,[email protected]). Tél: +33 (0)240376913 *** Ecole Centrale de Nantes, LS2N (Ex.(tel.: (256)-824-6164; e-mail: USA; The University of Alabama in Huntsville, Huntsville, AL-35899 , ([email protected]**LS2N (tel.: (256)-824-6164; e-mail: [email protected]). USA; *** Ecole Centrale de Nantes, (Ex.(tel.: IRCCyN) CNRSnantes.fr) UMR 6004 ,[email protected]). Tél: +33 (0)240376913 IRCCyN) CNRS UMR 6004 Tél: *** (Ex. (256)-824-6164; USA; IRCCyN) CNRSnantes.fr) UMRe-mail: 6004 ,,[email protected]). Tél: +33 +33 (0)240376913 (0)240376913 ,, ([email protected]([email protected]*** Ecole Ecole Centrale Centrale de de Nantes, Nantes, LS2N LS2N (Ex.(tel.: UMR 6004 , Tél: +33 (0)240376913 , ([email protected]*** Ecole Centrale de Nantes, LS2N (Ex. IRCCyN) CNRSnantes.fr) nantes.fr) Abstract: The paper deals with controlling annantes.fr) electric power system comprised of a Hydrogen Fuel Abstract: Theboost paperand deals with controlling electric power system comprised of a Hydrogen Cell (HFC), boost/buck DC-DCan power converters and the Ultracapacitor (UC) forFuel an Abstract: The paper deals with controlling an electric power system comprised of a Hydrogen Abstract: The paper deals with controlling an electric power system comprised of a Hydrogen Fuel Cell (HFC), boost and boost/buck DC-DC power converters and the Ultracapacitor (UC)Relative forFuel an auxiliary power supply in order to control servomotor speed within a vehicle application. Cell (HFC), boost and boost/buck DC-DC power converters and the Ultracapacitor (UC) for an Abstract: The paper deals with controlling an electric power system comprised of a Hydrogen Fuel Cell (HFC), boost and boost/buck DC-DC power converters and the Ultracapacitor (UC) for an auxiliary power supply in order to control servomotor speed within a vehicle application. Relative degree approach is applied for direct control of the servomotor input voltage and speed, as well as auxiliary power supply in to control servomotor speed within a vehicle application. Cell (HFC), and boost/buck DC-DC power and the Ultracapacitor (UC) for an auxiliary power supply in order order to presence control servomotor speed within a voltage vehicle application. Relative degree approach iscurrents applied for direct control theconverters servomotor input speed, asRelative well as the HFC andboost UC in the ofof the model uncertainties. The and non-minimum phase degree approach is applied for direct control of the servomotor input voltage and speed, as well as auxiliary power supply in order to control servomotor speed within a vehicle application. Relative degree approach is applied for direct control of the servomotor input voltage and speed, as well as the HFC and UC currents in the presence of the model uncertainties. The non-minimum phase property of the DC-DC boost converter is eliminated by controlling HFC and UC currents based on the HFC and in the presence of model uncertainties. The non-minimum phase degree approach iscurrents applied for direct control the servomotor input voltage and speed,(2-ASMC) as well as the HFC and UC currents inconverter the presence ofof the the model uncertainties. Thecontrollers non-minimum phase property of theUC DC-DC boost is eliminated by controlling HFC and UC currents based on the power balance approach. The adaptive-gain second order sliding mode property of the DC-DC boost converter is eliminated by controlling HFC and UC currents based on the HFC and UC currents in the presence of the model uncertainties. The non-minimum phase property ofbalance the DC-DC boostand converter is eliminated by The controlling HFC and UC Mode currents based on power approach. The second order sliding mode controllers (2-ASMC) control the current in HFC theadaptive-gain servomotor speed. conventional Sliding Controllers the power approach. The second order sliding mode controllers (2-ASMC) property ofbalance the DC-DC boost converter is eliminated byof controlling HFC and UCload currents based on the power balance approach. The adaptive-gain second order sliding mode controllers (2-ASMC) control the current inforHFC and theadaptive-gain servomotor speed. The Sliding Mode Controllers (SMC) are designed controlling the output voltage theconventional converter and the current of the control the current in HFC and the servomotor speed. The conventional Sliding Mode Controllers the power balance approach. The adaptive-gain second order sliding mode controllers (2-ASMC) control the current in HFC and the servomotor speed. The conventional Sliding Mode Controllers (SMC) are designed for controlling the output voltage of the converter and the load current of the UC. The efficiency and robustness of the proposed SMC and 2-ASMC are confirmed via computer (SMC) designed controlling voltage of the converter and the current of control the current and infor and theofthe servomotor speed. The Sliding Mode (SMC) are designed forHFC controlling the output voltage of theconventional converter and the load load current of the the UC. Theare efficiency robustness theoutput proposed SMC and 2-ASMC are confirmed viaControllers computer simulations. UC. The efficiency the proposed SMC 2-ASMC via (SMC) designedand forrobustness controllingof voltage of and the converter andconfirmed the load current of the UC. Theare efficiency and robustness ofthe theoutput proposed SMC and 2-ASMC are are confirmed via computer computer simulations. Keywords: non-minimum phase, HFCofelectric powerControl) system, sliding mode control. simulations. UC. TheIFAC efficiency and robustness proposed SMCHosting and 2-ASMC areLtd. confirmed computer © 2017, (International Federation ofthe Automatic by Elsevier All rightsvia reserved. simulations. Keywords: non-minimum phase, HFC electric power system, sliding mode control. simulations. Keywords: non-minimum phase, HFC electric power system, sliding mode control. Keywords: non-minimum phase, HFC electric power system, sliding control. power mode supply when there is an interruption of power 1. INTRODUCTION powerHFC supply there is andemand. interruption of power Keywords: non-minimum phase, HFC electric power system, sliding mode from orcontrol. a when fast load current 1. INTRODUCTION powerHFC supply when there is an andemand. interruption of of power power power supply when there is interruption from or a fast load current The non-minimum phase property of the boost and Hydrogen Fuel Cell1. Vehicle (HFCEV) is [1]a 1. Electric INTRODUCTION INTRODUCTION from HFC or a fast load current demand. power supply when there is an interruption of from HFC or a fast loadproperty current demand. The non-minimum phase of is the boostpower and Hydrogen Fuelwhich Cell1. uses Electric Vehicle (HFCEV) is [1]aa boost/buck DC-DC power converters eliminated by hybrid vehicle, a HFC in combination with INTRODUCTION The non-minimum non-minimum phase property of is the the boost and and Hydrogen Fuelwhich Cell Electric Electric Vehicle (HFCEV) is is [1]aa from the HFC or a fast load current The phase of boost/buck DC-DC power converters eliminated by Hydrogen Fuel Vehicle (HFCEV) [1]a hybrid vehicle, a HFC in combination with controlling HFC and UC property currents,demand. which is boost one of the storage device, to Cell poweruses the electric motor. In this paper, we boost/buck DC-DC power converters is the eliminated by hybrid vehicle, which uses HFC in combination combination with The non-minimum phase of is Hydrogen Cell Electric Vehicle (HFCEV) [1]a boost/buck DC-DC power converters eliminated by controlling the HFC and UC property currents, which is boost one of and the hybrid uses aa system HFC in with storage vehicle, device, towhich power the electric motor. In this paper, weaa main contributions of the paper. control the Fuel electric power comprised ofis HFC controlling the HFC and UC converters currents, which which is one one of of the the storage device, towhich power the electric electric motor. In this this paper, paper, wea boost/buck DC-DC power is eliminated by hybrid vehicle, uses a HFC in combination with controlling the HFC and UC currents, is main contributions of the paper. storage device, to power the motor. In we control the electric power system comprised of HFC Meeting the fast load current demand is addressed by conditioned by DC-DC boost power converter, and an UC as main contributions contributions of the paper. control the byelectric electric power systemconverter, comprised of UC HFC controlling the HFC and UC currents, which is one of the storage device, to power the electric motor. In this paper, we main of the paper. Meeting the fast load current demand is addressed by control the power system comprised of HFC conditioned DC-DC boost power and an as generating two different current command profiles to follow. a storage device conditioned by boost/buck bidirectional Meeting the fast load currentcommand demand profiles is addressed addressed by conditioned byelectric DC-DC boost power power converter, and an UC as main contributions of the current paper. power system comprised of UC HFC Meeting the load current demand is by generating two different to for follow. conditioned by DC-DC boost converter, and an as acontrol storagethe conditioned by boost/buck bidirectional Specifically, afast slow current command is generated the converter, indevice order to drive the speed of the DC electric motor generating two different current command profiles to follow. aaconverter, storage device conditioned by boost/buck bidirectional Meeting current demand isis addressed by conditioned by DC-DC boost converter, and an UC as generating two different current command to for follow. Specifically, afast slowload current command is profiles generated the storage conditioned by boost/buck bidirectional order to drive thepower speed of the motor HFC, andthe the fast-current command profile generated for to its timeindevice changing command profile. In DC caseelectric of HFC, the Specifically, a fast-current slow current current command is profiles generated for the converter, indevice order to to drive the speed speed of the the DC electric motor generating two different current command to follow. acurrent conditioned by boost/buck bidirectional Specifically, a slow command is generated for the HFC, and the command profile is generated for converter, in order drive the of DC electric motor to storage its time changing command profile. In case of HFC, the UC. cannot vary rapidly due to slower HFC membrane HFC, and and the thea fast-current fast-current command profile is generated generated for to its time time changing command profile. In DC case ofmembrane HFC, the Specifically, slow current command is generated converter, inchanging order drive thedue speed electric HFC, command is for UC. to its command profile. In case of HFC, the current cannot vary rapidly to of slower HFC The tracking control problems areprofile addressed usingfor thethe 2dynamics should betotaken in account bythe combining themotor HFC UC. current cannot vary rapidly due to slower HFC membrane HFC, and the fast-current command profile is generated to its time changing command profile. In case of HFC, the UC. The specifically tracking control problems addressed using thefor 2current rapidly due to HFC membrane dynamics should be taken in UCs account by combining the HFC ASMC, adaptive gain are super-twisting controller with an cannot UC [4].vary The use of [2],slower [4], expedites a faster The specifically tracking control control problems problems are are addressed using using the 22dynamics should be taken taken in UCs account by combining the HFC UC. The current rapidly due to slower HFC membrane addressed the ASMC, super-twisting controller dynamics should be in account by combining the HFC with an cannot UC [4].vary The use of [2], [4], expedites a faster [4] for tracking controlling adaptive the HFC gain current through the partial varying response to the load current demand. ASMC, specifically adaptive gain super-twisting controller with an an response UC [4]. to The use of UCs [2], [4], expeditesthe faster The tracking control problems are addressed using the 2dynamics should bethe taken incurrent account by combining HFC ASMC, specifically adaptive gain super-twisting controller [4] for controlling the HFC current through the partial with UC [4]. The use of UCs [2], [4], expedites aa faster varying load demand. pressure of oxygen. The inner current loop is often [4] for for specifically controlling theThe HFCinner current through the partial varying response to theuse load current demand. ASMC, adaptive gain super-twisting controller with an response UC [4]. to The ofcurrent UCs [2], [4], expedites a faster [4] controlling the HFC current through the partial pressure of oxygen. current loop is often varying the load demand. synthetized by a classical PI controllers [5], [6]. In this work, The DC motors that are used as power drives in electric pressure ofbyoxygen. oxygen. The inner current loop is partial often [4] for theby HFC current through the varying to theare loadused current demand. pressure of The inner current loop is often synthetized a classical PI controllers [5], [6]. In this work, The DCresponse motors that asover power electric the HFCcontrolling conditioned the DC-DC boost converter is vehicles have some advantages AC drives motorsin[3]. The synthetized byoxygen. a classical classicalby PIthe controllers [5], [6]. In this work, pressure of The inner current loop is often The DC have motors that advantages are used used as asover power drives in[3]. electric synthetized by a PI controllers [5], [6]. In this work, the HFC conditioned DC-DC boost converter is The DC motors that are power drives in electric vehicles some AC motors The controlled by a conventional SMC. The nonminimum phase electric DC motors operate with lower voltage, requiring the HFC conditioned conditioned byPIthe the DC-DC boost converter is synthetized by a classical controllers [5], [6]. In this work, vehicles have some advantages over ACvoltage, motorsinrequiring [3]. The The DC motors that are used as power drives electric the HFC by DC-DC boost converter is controlled by a conventional SMC. The nonminimum phase vehicles have some advantages over AC motors [3]. The electric DC motors operate with lower property is mitigated by controlling the HFC current. The UC fewer batteries. They have higher peak torque and faster controlled by a conventional SMC. The nonminimum phase boost converter is the HFC conditioned by the DC-DC electric DC motors operate with lower voltage, requiring vehicles have some advantages over AC motors [3]. The controlled conventional SMC.athe The nonminimum phase property is by mitigated by controlling HFC current. The UC electric DC The motors with lower requiring fewer batteries. They have torque higher peak voltage, torque and faster conditioned bya the DC-DC boost/ conventional SMC also acceleration. highoperate peak enables the vehicle to be property is is by mitigated by controlling controlling the HFC current. The UC controlled aconverter conventional SMC.aathe The nonminimum phase fewer batteries. batteries. They have torque higher peak voltage, torque and faster electric DC motors operate with lower requiring property mitigated by HFC current. The UC conditioned by the DC-DC boost/ conventional SMC also fewer They have higher peak torque and faster acceleration. The high peak enables the vehicle to be controls buck to follow time-varying command more adaptable for different driving conditions. DC-DC boost/ a conventional SMC also conditioned by the property is mitigated by controlling the HFC current. The UC acceleration. The high peak torque enables the vehicle to be fewer batteries. They have higher peak torque and faster conditioned by the DC-DC boost/ a conventional SMC also controls buck converter to follow time-varying command acceleration. The high peak driving torque enables the vehicle boost to be more forin different profile generated by the HFC current command using power The adaptable challenges controlling the conditions. “HFC/UC/DC-DC controls buckbyconverter converter to follow follow time-varying command conditioned thebyDC-DC boost/ a conventional SMC also more adaptable forin different driving conditions. acceleration. The high peak driving torque enables vehiclefor to be controls buck to time-varying command profile generated the HFC current command using power more adaptable for different The boost/buck challenges controlling the conditions. “HFC/UC/DC-DC boost balance approach. Adaptive-gain twisting controller robustly and converters” based power the system an profile generated by the HFC HFC current command using power controls buck converter to follow a time-varying command The boost/buck challengesforin in controlling the conditions. “HFC/UC/DC-DC boost more adaptable different driving profile generated by the current command using power balance approach. Adaptive-gain twisting controller robustly The challenges controlling the “HFC/UC/DC-DC boost and converters” based power system for an controls the servomotor speed. electric vehicle application that are addressed in this paper are balance approach. Adaptive-gain twisting controller robustly profile by thespeed. HFC current command usingrobustly power and boost/buck converters” based power in system for an The challenges in controlling theaddressed “HFC/UC/DC-DC boost balance approach. Adaptive-gain twisting controller controlsgenerated the servomotor and boost/buck converters” power system for an electric vehicle application thatbased are this paper are as follows, 2. MATHEMATICAL MODEL OF HFC/UC/DC-DC controls the servomotor speed. balance approach. Adaptive-gain twisting controller robustly electric vehicle application application thatbased are addressed addressed in this paper paper are and boost/buck converters” power system for an controls the servomotor speed. electric vehicle that are in this are as follows, 1. The non-minimum phase property of the DC-DC boost 2. MATHEMATICAL MODEL OF HFC/UC/DC-DC CONVERTER/SERVOMOTOR controls the BOOST servomotor speed. as1.follows, follows, electric vehicle application that are addressed in DC-DC this paper are as The non-minimum phase property of the boost 2. MATHEMATICAL MATHEMATICAL MODEL OF OF HFC/UC/DC-DC HFC/UC/DC-DC and boost /buck converters is to be mitigated or 2. MODEL BOOST CONVERTER/SERVOMOTOR 1.follows, The non-minimum non-minimum phase property property of the the DC-DC boost as1. The phase of boost and boost for /buck converters is to be DC-DC mitigated or The equivalent circuit diagram of anOF electric power system BOOST CONVERTER/SERVOMOTOR eliminated tracking causal load voltage profiles in 2. MATHEMATICAL MODEL HFC/UC/DC-DC BOOST CONVERTER/SERVOMOTOR and boost for /buck converters is to to be DC-DC mitigated or 1. the Thepresence non-minimum phase property ofvoltage the boost The equivalent circuit diagram an electric system and boost /buck converters is be mitigated or eliminated causal load profiles in comprising HFC/DC-DC boost ofconverter forpower a vehicle is of tracking model uncertainties. BOOST CONVERTER/SERVOMOTOR The equivalent circuit diagram diagram ofconverter an electric electric power system eliminated for tracking causal load load voltage profiles or in and boost for /buck isdynamics tovoltage be mitigated The equivalent circuit an system comprising HFC/DC-DC boost of forpower a vehicle is eliminated causal in the presence of tracking model uncertainties. designed in Fig.1. 2. The relatively slowconverters uncertain ofprofiles the HFC comprising HFC/DC-DC boost ofconverter converter forpower vehicle is The equivalent circuit diagram an electric system the presence presence of tracking model uncertainties. eliminated for causal load profiles in comprising HFC/DC-DC boost for aa vehicle is designed in Fig.1. the of model uncertainties. 2. membrane The relatively slow uncertain dynamics thecurrent HFC challenges addressing thevoltage fast of load designed in in Fig.1. Fig.1. comprising HFC/DC-DC boost converter for a vehicle is 2. The Thepresence relatively slow uncertain uncertain dynamics of thecurrent HFC the of model uncertainties. designed 2. relatively slow the HFC membrane addressing thea backup fast of loadsource demand. A challenges controlled UC is useddynamics as of designed in Fig.1. membrane challenges addressing thea backup fast of loadsource current 2. demand. The relatively slow uncertain thecurrent HFC membrane addressing the fast load A challenges controlled UC is useddynamics as of demand. A A challenges controlled UC UC is used used as as backup source of membrane addressing theaa backup fast loadsource current demand. controlled is of Copyright © 2017 IFAC demand. A controlled UC is used as a backup source of15359 Copyright © 2017, 2017 IFAC 15359 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2017 IFAC 15359 Peer review© of International Federation of Automatic Control. Copyright ©under 2017 responsibility IFAC 15359 10.1016/j.ifacol.2017.08.2552 Copyright © 2017 IFAC 15359
Proceedings of the 20th IFAC World Congress Roshini S. Ashok et al. / IFAC PapersOnLine 50-1 (2017) 14794–14799 Toulouse, France, July 9-14, 2017
14795
the electrochemical reaction going in the HFC. Due to the insignificant values of m 1 and b1 (specifically, the value of m1
5 1 0 V ) in eq. (7) the term is typically close to m 1 3
V c o n can be neglected in eq. (6) [4]-[9],[12] 2.2 Mathematical model of HFC conditioned by DC-DC unidirectional boost power converter The dynamics of the DC-DC boost power converters are governed by the following system of differential equations [4],[11]-[13]:
Fig.1 Equivalent Circuit Diagram of HFC/UC/DC-DC boost and boost/buck converter-based electric power system for a HFCEV
2.1 Mathematical model of HFC The mathematical of HFC is derived based on the equivalent circuit in Fig. 1, the dynamics of activation over voltage, V a t , are given as [4]-[9],[12], d V at
i h fc
dt
V at
Cd
(1)
R atC d
The partial pressure of hydrogen and oxygen dynamics is presented as [15]: d dt
d dt
PH
2
Po 2
1
H
H kH
2
2
1
2
Po
o
1
PH
2
2
1
o ko 2
q
in H
2
2 k i h fc 1
(2)
d V ser
d i h fc
2
1
(3)
2
1
in
in
( qH qH )
qH
2
2
(8)
[ (1 t1 )V s e r V th fc ]
1
d V uc
2
(1 t 2 )V s e r
Luc
1
dt
in
dt
L
dt
The actuator valve dynamics are modeled as in [15], [16]: dqH
1
where t1 [ 0 ,1] is a switch (transistor T1 in Fig. 1) control function, that is transformed as, (9) t1 v 1 0 .5 , v 1 [ 0 .5 , 0 .5 ] . 2.3 Mathematical model of UC controlled by bidirectional DC-DC buck (buck/boost) converter The dynamics of the UC being charged and discharged by a bidirectional DC-DC converter are derived as in [4],[13] d iu c
q o k i h fc
[(1 t1 ) i h fc (1 t 2 ) i u c i s e r ]
Cb
dt
2
in
1
dt
C uc
V uc
(10) (11)
iu c
(4)
where t 2 v 2 0 .5 , v 2 [ 0 .5 , 0 .5 ] , and t 2 (transistor T2in Fig. 1) control function.
(5)
2.4 Mathematical model of a Servomotor The dynamics of a servomotor are given by the following set of equations [7] ,
2
is a switch
in
dqO
2
dt
1
qO
in
in
( qO qO ) 2
2
2
The inputs of the valves control functions, i.e.
in
qO
and
2
in
v3 q H
The output voltage
2
V h fc
in
qH
and
2
are considered as HFC in
v4 qO
2
d ise r
.
of a stack of
n
HFCs
V th fc
d ser
defined by equations V th fc n V h fc , E h fc
G1
2 Far
V h fc E h fc V o h m V c o n V a t s1 2 Far
( T s t T h fc )
R g T st 1 ln ( PH 2 ) ln ( PO 2 2 Far 2
,
V o h m i h fc R o h m ,
dt
1 J
k
i
m ser
V se r k b se r
k b ser n g T d
(12)
(13)
as the control inputs, the input-output dynamics of the system are derived based on eqs. (1) -(13): v1 , v 4 , v 2 , v 3
(7)
1
molar entropy ( J m o l K ). The universal gas constant is given as R g 8 .3 1 4 J m o l 1 K 1 . The Faraday constant is F a r 9 6 4 8 5 .3 4 1 5 s A / m o l
temperature of the HFC is given as T h fc
R a r i se r
2
)
where the Gibbs free energy G 1 ( J m o l 1 ) s 1 is the standard
described as
L in d
2.5 Mathematical model of HFC/DC-DC boost converter/UC Considering, V s e r , i h fc , iu c , PH , as the system’s outputs and
(6)
Po 2 V c o n m 1 e x p ( n i h fc ) b1 ln a1
1
1
dt
. The reference
2 9 8 .1 5 K
and the
stack temperature is T s t 3 5 3 K . The temperature ( T s t
T h fc
d V ser dt 2 d i h fc 2 dt d iu c dt d PH 2 dt
H1 H2 H3 H 4
1 i h fc Cb 1 0 0
0
Cb 0
1
15360
1 Luc
0
1 1 3 i h fc i h fc v 1 i s e r L Cb 2
0
iu c
3
2
0
0 0
V uc
1
H kH 2
where
), which is assumed to be constant during the process, is observed. The variable V c o n is dependent on the changes in the concentration of reactants as they are being consumed by
1
;
2
v1 v 4 v2 v3
(14)
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 14796 Roshini S. Ashok et al. / IFAC PapersOnLine 50-1 (2017) 14794–14799
2
1 0 .5 R g T s t L 2F
H1
1 1 i h fc i s e r 0 .5 i u c ; Cb 2
H
H
4
2
1
H
1 L
1 k P O2 O2 O2
1
PH
H kH
2
2
2
2k
,
3
1
H3
1
i h fc
1 Cb
i u c (1 v 1 )
0 .5V s e r
Luc
V uc
R g T st 2 Far
THE CONTROLLER DESIGN: RELATIVE DEGREE APPROACH: The control problem is in designing the control functions that drive the tracking errors v1 , v 4 , v 2 , v 3
,
c
dt
d iser dt d V uc dt
i h fc
1 1 PH 2 kH H2 H2 2
v
3
k
2 k i h fc
i
1 h fc
V at
Cd
command
L in d
C uc
d e1 s dt 2 d e2s 2 dt d e 3s dt d e4s
ser
dt
c
to be bounded as the feedback
control has been applied.
The first task is to design a controller
v1
1
0
0
iu c
Cb 3
2
0
1 Luc
0
0 0
V uc
1
0
H kH 2
2
v1 v 4 v2 v 3
(18)
(1 )
H 1;
H
2
i c
(2)
h fc
H 2;
H
4
P c
H
(1 )
H
2
4
,
(1 )
i H 4.2 The control problem decoupling: first step Last two equations in system (18) is completely decoupled from the first two equations: H
3. PROBLEM FORMULATION The control problem in HFC-UC/DC-DC converters based electric power system for controlling the servomotor in an electric vehicle is reduced t to the following subtasks
2
is assumed to be constant):
2
1 i h fc Cb H1 1 H 2 H 0 3 H 4 0
H 1 V ser
h fc
and
V s e r ( t ), i h fc ( t ), i u c ( t ), PH ( t )
where,
(17)
Apparently, the forced zero dynamics in eqs.(15)-(17) are stable, and the profiles V a t and i are bounded, assuming i h fc
for
tracking error dynamics,
the forced profile
2
4.1 Input-output de-coupling: Relative Degree Approach Using the relative degree approach [7], [19], [22] the inputoutput error dynamics in eq. (13) of vector relative degree r 1, 2 , 1, 1 . The system (13) is re-written in terms of
(16)
iu c
profiles
respectively ( PH
(15)
( iser R ar V ser k b ser )
1
2
c
R atC d
1
c
2
Similarly, the internal dynamics of the HFC system derived from eqs. (1), (11) and (12) as follows, d V at
c
e 2 s i h fc i h fc , e 3 s iu c iu c , e 4 s PH PH
to zero as time increases in the presence of the bounded disturbances H 1 , H 2 , H 3 , H 4 . V sce r ( t ) , i hc fc ( t ) , i ucc ( t ) , PHc are
2
1 1 PO 2 kO O2 O2 2
c
e1 s V s er V s er ,
1 1 (1 ) i h fc i s e r v 1V s e r ( i h fc ) V a t Cb 2 1 PH 2 0 .5 P O 2
4.
,
3
c
uc
3
d e3 s dt
iu c c
(1 )
d e4 s
so that
dt
P c
H
H
3
V uc Luc
(19)
v2
3
(1 )
2
1 2 k 1 PH i h fc v3 2 H kH H kH 2 2 2 2 H2
(20)
c
the output voltage follows the prescribed command profile in the presence of bounded perturbations. The second task is to design the controller v 2 so that the output current of the UC iu c follows the desired V ser V ser
command profile
c
iu c
in finite time also.
The third task is to design a controller in terms of
c
V ser
that drives s e r sce r ( t ) in accordance with the servomotor dynamics. The fourth and the fifth tasks are about designing the control functions v 3 for controlling the hydrogen partial pressure and v 4 for controlling the HFC current that allows eliminating the non-minimum phase property of the DC-DC boost converter. Specifically, HFC current is robustly controlled via the adaptive 2-ASMC (supertwisting) controller v 4 s , whose adaptive gains are not overestimated. This allows controlling the HFC current in a perturbed environment with bounded derivatives of the perturbations, whose boundaries are not known.
The error
e3 s
and
e4 s
dynamics ( iu c and
PH
2
tracking
dynamics) in (19), (20) are considered firstly for the v 2 and v 3 control functions design. The following assumption about the disturbance 3 : 3
L3
is made.
4.3. The control v 2 design The control v 2 [4], [13] is designed in terms of conventional SMC to enable the charging/discharging of the UC current in a case of a load demand at a fast rate. The “fast” command, c i u c , for the UC current, i u c , is generated in accordance with eq. (38). v 2 0 .5 s ig n e 3 s , e 3 s i u c i u c c
As soon as the UC current c
iu c
follows the command profile
in finite time by means of the SMC (21) the error becomes
e3 s 0
and, therefore,
c
iu c iu c
4.4. The control v 3 design
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iu c
(21)
in the sliding mode.
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Roshini S. Ashok et al. / IFAC PapersOnLine 50-1 (2017) 14794–14799
A PI-like controller robustly addresses the set point regulation problem by driving e 4 s PHc PH 0 in eq. (20) as time 2
2
increases v 3 k H PH 2 k i h fc k p e 4 s k i 2
2
1
1
1
e
4s
The output voltage command profile V sce r is derived in section 4.9 in accordance to [4]. The conventional SMC [4], [10], [13] v1 that represents a switching function v1 [ 0 .5, 0 .5 ]
(22)
dt
14797
v 1 0 .5 s ig n e1 s
(28)
The tuning and the design of the PI controller is understood [18] and, therefore, omitted for conciseness.
is insensitive to the bounded disturbance (
4.5 The control problem decoupling: second step
controlling the output voltage of the DC-DC boost converter by driving e1 s V sce r V s e r 0 in finite time.
The effect of DC-DC boost and boost/buck converters coupling is taken into account in eq. (18). Assuming the sliding mode is established in eq. (19), (21), first two equations in system (18) can be rewritten taking in account that v 2 is replaced by v 2 e q d e1 s dt
1 1 c iu c v 2 e q i h fc v 1 V ser H 1 C C
(23)
1
2
dt
2
i h fc c
(2)
H
2n
1 0 .5 R g T s t 2 Far L
1 k P o2 o2 O 2
v4
H
2n
H
2
1 C
iu c v 2 e q
;
v 2 eq
H 1n
d e1 s dt H 1n 2 d e2 s H 2n d t 2
1 i C h fc 0
and
c
H 1n V ser H 1n ;
H
2n
i h fc c
0
1 k P O2 O2 O2
(2)
H
placement to differential equation dynamics are derived dt
i h fc c
(2)
H
(1 )
2n
c2 e2 S
0
(25)
4.6
3
2
1 k P o2 o2 O 2
(30)
is assumed to be
with unknown boundary L 2 . The adaptive
L2
2
1/ 2
s ig n ( )
h fc
(31)
s ig n ( )
2
h s ig n h 2 , h
v4
e2 s ce2 s 0
L3
1 0 .5 R g T s t L 2 Far
h fc 2 h h ,
2n
h ,
if h fc h m
(32)
if h fc h m
h fc ( 0 ) h m , h 0
. Then the
v4 s
(33)
g1
The problem of finding real-time robust estimation of e 2 s is solved using arbitrary order robust exact differentiator [10] is similar to the eq. (43) given in section 4.9. 4.8. HFC and UC current command generator The HFC current command profile
1 0 .5 R g T s t L 2 Far
L2 ,
g1
where h , h , , h h , are arbitrary positive constants. Finally, we have
1 k P o2 o2 O 2
v4
the power balance
(27)
c
.
Controller Design for DC-DC Boost Converter
PH F C Ps e r L o a d
c
c
i h fc
is computed based on
[4], [13]:
c
i h fcV th fc i s e rV s e r ( t )
The following assumptions about the disturbances 1 , 2 , 3 : 2
g1v4 ;
v 4 s h fc
2
L1 ,
(1 ) 2
super-twisting controller is a suitable candidate [11], [17] that drives e 2 s , e 2 s 0 to a real 2-SM in finite time. The adaptive gain super- twisting controller is given below,
g1
are made as 1
h fc
0 v1 v 0 4
Equation (24) of relative degree 2 is reduced to relative degree 1 with respect to the variable, (26) e2 s c2 e2 s where c 2 0 is selected to provide a desired eigenvalue
d
obtained from eqs.(23) and (24):
1 0 .5 R g T st L 2 Far
2n
(2)
bounded
1 i u c (1 v 1 ) v 2 e q v1 Cb
0
H
The derivative of the disturbance
The decoupled input-output dynamics of the system is finally written while replacing the disturbances H 1 and H 2 from eq.(14) with
) in terms of ( v 4 )
h fc
v4 s
1 1 i h fc i s e r iu c 2
1 1 3 i h fc i h fc v 1 i s e r L Cb 2
) while
The HFC current error dynamics given in eqs. (26) and (27) are written as follows, (29) e2 s c 2 e2 s ; 2 g 1v 4 The control input ( v 4 ) for controlling the HFC current i
(24)
where, H 1n H 1
h fc
L1
must be continuous. Therefore, the controller is designed in terms of ( v 4 ). Differentiating the eq. (29) again, we obtain,
H1n
d e2
4.7 Controller Design for HFC current ( i
1
(34)
where 1 accounts for power losses (for simplicity, the power losses in the converter are neglected, and it is assumed c 1 ), i s e r is the output current command. The total voltage across n stack of HFC’s is given to be V . HFC current th fc
c
command profile i fc is generated as follows,
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Proceedings of the 20th IFAC World Congress 14798 Roshini S. Ashok et al. / IFAC PapersOnLine 50-1 (2017) 14794–14799 Toulouse, France, July 9-14, 2017
The arbitrary positive constants are given to be The output voltage 1 s , s , s , s e r m in , 1 s . command is finally derived as follows,
c
c
c
i h fc i s e r
V ser ( t )
,
(35)
V th fc c
The HFC current command profile
i h fc
is divided into two c
V ser ( t )
commands such as slow and fast commands, c
c s lo w
c fa s t
i h fc i h fc
i h fc
(36)
s lo w
where the slow command
can be generated as result of
i h fc
c low pass filtering of i h fc :
y1
5.
d i h fc
c s lo w
c
i h fc
dt
(37)
0
i h fc ,
c s lo w
Next, i fc is supposed to follow i fc
, while
iu c
will follow
s g n (
ser
)
1
s g n (
2
ser
controller
) dt
(42)
The problem lies in finding robust estimation of e s e r and its derivative in real-time. This problem is solved using a differentiator [10],
c s lo w
ser
1 s 0 .2 5;
SIMULATION STUDY
The following load voltage and current command profiles were selected in accordance with [4], [11] to ensure that the proposed HFC system can handle high power demands:
c
asymptotically the fast command profile i u c that is defined c
based on already generated
c s lo w
and
i fc
i fc
c
V th fc
i
c
V ser
c
c s lo w
i h fc
h fc
V th fc
c
V ser
c fa s t
ser
The problem in controlling the servomotor is to drive the servomotor speed s e r to follow the generated on-line reference profile [7]. To drive ( t ) , we need to design the controller in terms of V in accordance with the servomotor dynamics given in (13) c
c
ser
(43)
Fig.2 illustrates the causal tracking of the output voltage of DC-DC boost converter following the respective command profile. Fig.3 shows that the HFC’s current follows the respective HFC command profile with the help the adaptive gain super-twisting controller with the adaptive gain h f c
(38)
i h fc
4.9. Controller Design for servomotor speed
ser
c
command profiles
and a power balance condition as iu c
s e r ( t ) 1 0 0 1 0 0 s in ( 0 .2 ) t
ser
c
ser
shown in Fig.8. Fig.4 indicates the UC current ( iu c ) follows c
its respective command profile ( i u c ). The servomotor speed profile is shown in Fig.5. Fig.6 shows the HFC current control ( v 4 ) function. Fig.7 shows the partial pressure of hydrogen ( PH ) is maintained constant as time increases with 2
The error dynamics of the servo motor speed is given as.
c
ser
c
e s e r d s e r e s e r g 2V s e r ; e s e r s e r s e r
respect to its corresponding command profile(
(39)
The term in eq.(21) is considered as the differentiable disturbance, whose derivative is bounded with unknown boundary. The control input is given as V sce r in eq.(21). The
c
PH ). 2
Fig.9
indicates the adaptive gain ( s e r ) used in designing the servomotor speed controller. Bv
controller ( V sce r ) must be continuous. Therefore the control is deigned in terms of ( V ). Differentiating again, we obtain, c
ser
ser
g2,
c
g 2 y 1V s e r ;
y1 0
(40)
The term is assumed bounded L b o u n d with unknown boundary ( L b o u n d ). Then the adaptive-gain twisting controller [14] that drives e s e r , e s e r 0 to a real second order sliding mode in finite time is designed in terms of 1 s e r (s g n (
ser
1s s
1s 2
ser
)
1
s g n (
2
s g n ( s (
ser
,
ser
ser
c
V ser ( t )
,
: (41)
))
) s ),
if s e r s e r m in if s e r s e r m in
c
Fig.3 Plot of HFC command profile ( i h fc ( t ) ) and HFC current( ih fc ( t )
where, 2
2
s ( s e r , s e r ) s e r s e r 1 s s e r 2
3/2
Fig.2 Plot of the DC-DC boost converter voltage command profile ( V s ce r ( t ) ) and voltage ( V s e r )
sg n ( s e r ) s e r 4
s e r s e r s e r 0 .2 5 ser
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) (Vs.) Time
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Roshini S. Ashok et al. / IFAC PapersOnLine 50-1 (2017) 14794–14799
14799
converter was eliminated by controlling the HFC current using the adaptive gain super-twisting controller. The HFC current accurately follows the command profile without overestimating the gains in the presence of bounded perturbations with unknown bounds. The conventional sliding mode controller was implemented to follow the voltage and current causal time-varying command profiles in the DC-DC boost converter and UC current profile..
c
Fig:4 Plot of UC current command profile( i u c ( t ) ) and UC current ( iu c ( t ) ) (Vs.) Time
REFERENCES Fig.5 Plot of the servomotor speed command profile( c s e r ( t ) ) and servomotor speed( s e r ( t ) )
Fig:6 Plot of Fuel Cell current controller( v 4 )
Fig.7 Plot of partial pressure of hydrogen command profile ( c PH ) and partial pressure of hydrogen ( PH ) (Vs.)Time 2
2
Fig.8 Plot of Adaptive gain ( h f c ) (Vs.)Time
Fig.9 Plot of the Adaptive gain ( s e r )
6. CONCLUSIONS The servomotor speed within an HFC/UC/DC-DC boost and boost/buck converters based electric power system for an electric vehicle application is controlled. The adaptive gain twisting controller is used for controlling the servomotor speed. The non-minimum phase property of the DC-DC boost
[1] http://serc.berkeley.edu/fueling-the-future-hydrogen-fuel-cellvehicles-in-the-21st-century/ [2] http://www.environmentalleader.com/2012/01/10/five-reasonswhy-ultracapacitors-are-attractive-to-auto-manufacturers/ [3] http://www.engineerlive.com/content/21329 [4] R. S. Ashok; Y. B. Shtessel, “Control of Fuel Cell- based Electric Power System Using Adaptive Sliding Mode Control and Observation Techniques”, Journal of the Franklin Institute, Vol. 352, Issue 11,, November 2015, pp.4911–4934. [5] M. Hilairet, M. Ghanes, O. Béthoux, V. Tanasa, J-P. Barbotb, D. Norman-Cyrot, “A passivity-based controller for coordination of converters in a fuel cell system”, Control Engineering Practice, Vol. 21, No. 8, August 2013, pp.1097–1109. [6] M. Ghanes, M. Hilairet, J-P. Barbot , O. Bethoux, “Singular Perturbation Control for coordination of converters in a Fuel Cell System”, ELECTRIMACS, 2011, Cergy-Pontoise, France. 2011. [7] R. S. Ashok and Y. B. Shtessel, “Control of a fuel cell vehicle using adaptive sliding mode control: Servomotor application” Proceedings of ACC, 2016, pp. 2235 – 2240. [8]J. Correa, F. Farret, L. Canha, and M. Simoes ”An Electrochemical-Based Fuel- Cell Model Suitable for Electrical Engineering Automation Approach”, TIE, Vol. 51, No.5, 2004. [9] M. J. Khan, M. T. Iqbal “Dynamic Modelling and Simulation of a Fuel Cell Generator”, Fuel Cells Special Issue: Modelling of Fuel Cell Systems Volume 5, Issue 1, pages 97–104, February, 2005 [10]Y. Shtessel, C. Edwards, L. Fridman, and A. Levant, “Sliding mode control and Observation”, Birkhauser, New York, 2014. [11]J. M. Olm, X. R. Oton, and Y. Shtessel "Stable inversion-based robust tracking control in DC-DC nonminimum phase switched converters", Automatica, 47, 1, pp. 221-226, 2011. [12] C. Kunusch, P. F. Puleston, M. A. Mayosky, and J. Riera, “Sliding Mode Strategy for PEM FCs Stacks Breathing Control Using a Super-Twisting Algorithm,“ IEEE Transactions on Control System Technology, Vol. 17, no. 1, 2009, pp. 167-174. [13] R. Ashok, Y. Shtessel, and J. Smith, “Sliding mode control of electric power system comprised of fuel cells, DC-DC boost converters and UC,” Proceedings of ACC, 2013, pp. 5766 – 5771. [14] J.Kochalummoottil, Y. Shtessel, J.A Moreno, and L. Fridman, “Adaptive Twist sliding mode control: A Lyapunov Design,” in Proceedings of 50th IEEE CDC, pp. 7623-7628, 2011. [15] J.Padullésa, G.W Aultb and J.R McDonald, “An integrated SOFC plant dynamic model for power systems simulation,” Journal of Power Sources, Vol.86 , Issues 1-2, pp. 495-500. [16] K. Belmokhtar, M. H. Hammoudi, M. L. Doumbia, and K. Agbossou, “Modelling and Fuel Flow Dynamic Control of Proton Exchange Membrane Fuel Cell,” Proceedings of 4th International Conference on Power Engineering, Energy and Electrical Drives, May 2013 pp. 415-420. [17] Y. Shtessel, M. Taleb, and F. Plestan, “A novel adaptive-gain super-twisting sliding mode controller: methodology and application,” Automatica, Vol. 48, Issue 5, 2012, pp. 759-769. [18] R. Dorf, and R. Bishop, “Modern Control Systems”, 12th Edition, Prentice Hall 2010.
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ScienceDirect
Sliding Mode Control ofIFAC Hydrogen Fuel Cell and Ultracapacitor Based Electric PapersOnLine 50-1 (2017)Cell 14794–14799 Sliding Mode Control of Hydrogen Fuel and Ultracapacitor Based Electric Power System: Electric Vehicle Application Based Electric Sliding Mode Control of Hydrogen Fuel Cell and Ultracapacitor Power System: Electric Vehicle Application Based Electric Sliding Mode Control of System: Hydrogen Fuel Cell and Ultracapacitor Power Electric Vehicle Application Roshini System: S. Ashok*, Yuri B. Shtessel, **, andApplication Malek Ghanes*** Power Electric Vehicle Roshini S. Ashok*, Yuri B. Shtessel, **, and Malek Ghanes*** Roshini S. Yuri **, Malek Roshini S. Ashok*, Ashok*, Yuri B. B. Shtessel, Shtessel, **, and and Malek Ghanes*** Ghanes*** *The University of Alabama in Huntsville, Huntsville,AL-35899 Roshini S. Ashok*, Yuri B. Shtessel, **, and Malek Ghanes*** *The University of Alabama in Huntsville, Huntsville,AL-35899 USA; e-mail: [email protected]). *The University University ofe-mail: Alabama in Huntsville, Huntsville, Huntsville,AL-35899 in USA;of [email protected]). The University ofAlabama Alabama in Huntsville,Huntsville,AL-35899 Huntsville, AL-35899 ***The USA; e-mail: [email protected]). *The University of Alabama in Huntsville, Huntsville,AL-35899 USA;(256)-824-6164; e-mail: [email protected]). ** The University of Alabama in Huntsville, Huntsville, AL-35899 (tel.: e-mail: [email protected]). USA; University of Alabama in Huntsville, AL-35899 ** The USA; e-mail: [email protected]). The University of Alabama in Huntsville, Huntsville, Huntsville, AL-35899 , ([email protected]** (256)-824-6164; e-mail: USA; IRCCyN) CNRS UMR 6004 ,[email protected]). Tél: +33 (0)240376913 *** Ecole Centrale de Nantes, LS2N (Ex.(tel.: (256)-824-6164; e-mail: USA; The University of Alabama in Huntsville, Huntsville, AL-35899 , ([email protected]**LS2N (tel.: (256)-824-6164; e-mail: [email protected]). USA; *** Ecole Centrale de Nantes, (Ex.(tel.: IRCCyN) CNRSnantes.fr) UMR 6004 ,[email protected]). Tél: +33 (0)240376913 IRCCyN) CNRS UMR 6004 Tél: *** (Ex. (256)-824-6164; USA; IRCCyN) CNRSnantes.fr) UMRe-mail: 6004 ,,[email protected]). Tél: +33 +33 (0)240376913 (0)240376913 ,, ([email protected]([email protected]*** Ecole Ecole Centrale Centrale de de Nantes, Nantes, LS2N LS2N (Ex.(tel.: UMR 6004 , Tél: +33 (0)240376913 , ([email protected]*** Ecole Centrale de Nantes, LS2N (Ex. IRCCyN) CNRSnantes.fr) nantes.fr) Abstract: The paper deals with controlling annantes.fr) electric power system comprised of a Hydrogen Fuel Abstract: Theboost paperand deals with controlling electric power system comprised of a Hydrogen Cell (HFC), boost/buck DC-DCan power converters and the Ultracapacitor (UC) forFuel an Abstract: The paper deals with controlling an electric power system comprised of a Hydrogen Abstract: The paper deals with controlling an electric power system comprised of a Hydrogen Fuel Cell (HFC), boost and boost/buck DC-DC power converters and the Ultracapacitor (UC)Relative forFuel an auxiliary power supply in order to control servomotor speed within a vehicle application. Cell (HFC), boost and boost/buck DC-DC power converters and the Ultracapacitor (UC) for an Abstract: The paper deals with controlling an electric power system comprised of a Hydrogen Fuel Cell (HFC), boost and boost/buck DC-DC power converters and the Ultracapacitor (UC) for an auxiliary power supply in order to control servomotor speed within a vehicle application. Relative degree approach is applied for direct control of the servomotor input voltage and speed, as well as auxiliary power supply in to control servomotor speed within a vehicle application. Cell (HFC), and boost/buck DC-DC power and the Ultracapacitor (UC) for an auxiliary power supply in order order to presence control servomotor speed within a voltage vehicle application. Relative degree approach iscurrents applied for direct control theconverters servomotor input speed, asRelative well as the HFC andboost UC in the ofof the model uncertainties. The and non-minimum phase degree approach is applied for direct control of the servomotor input voltage and speed, as well as auxiliary power supply in order to control servomotor speed within a vehicle application. Relative degree approach is applied for direct control of the servomotor input voltage and speed, as well as the HFC and UC currents in the presence of the model uncertainties. The non-minimum phase property of the DC-DC boost converter is eliminated by controlling HFC and UC currents based on the HFC and in the presence of model uncertainties. The non-minimum phase degree approach iscurrents applied for direct control the servomotor input voltage and speed,(2-ASMC) as well as the HFC and UC currents inconverter the presence ofof the the model uncertainties. Thecontrollers non-minimum phase property of theUC DC-DC boost is eliminated by controlling HFC and UC currents based on the power balance approach. The adaptive-gain second order sliding mode property of the DC-DC boost converter is eliminated by controlling HFC and UC currents based on the HFC and UC currents in the presence of the model uncertainties. The non-minimum phase property ofbalance the DC-DC boostand converter is eliminated by The controlling HFC and UC Mode currents based on power approach. The second order sliding mode controllers (2-ASMC) control the current in HFC theadaptive-gain servomotor speed. conventional Sliding Controllers the power approach. The second order sliding mode controllers (2-ASMC) property ofbalance the DC-DC boost converter is eliminated byof controlling HFC and UCload currents based on the power balance approach. The adaptive-gain second order sliding mode controllers (2-ASMC) control the current inforHFC and theadaptive-gain servomotor speed. The Sliding Mode Controllers (SMC) are designed controlling the output voltage theconventional converter and the current of the control the current in HFC and the servomotor speed. The conventional Sliding Mode Controllers the power balance approach. The adaptive-gain second order sliding mode controllers (2-ASMC) control the current in HFC and the servomotor speed. The conventional Sliding Mode Controllers (SMC) are designed for controlling the output voltage of the converter and the load current of the UC. The efficiency and robustness of the proposed SMC and 2-ASMC are confirmed via computer (SMC) designed controlling voltage of the converter and the current of control the current and infor and theofthe servomotor speed. The Sliding Mode (SMC) are designed forHFC controlling the output voltage of theconventional converter and the load load current of the the UC. Theare efficiency robustness theoutput proposed SMC and 2-ASMC are confirmed viaControllers computer simulations. UC. The efficiency the proposed SMC 2-ASMC via (SMC) designedand forrobustness controllingof voltage of and the converter andconfirmed the load current of the UC. Theare efficiency and robustness ofthe theoutput proposed SMC and 2-ASMC are are confirmed via computer computer simulations. Keywords: non-minimum phase, HFCofelectric powerControl) system, sliding mode control. simulations. UC. TheIFAC efficiency and robustness proposed SMCHosting and 2-ASMC areLtd. confirmed computer © 2017, (International Federation ofthe Automatic by Elsevier All rightsvia reserved. simulations. Keywords: non-minimum phase, HFC electric power system, sliding mode control. simulations. Keywords: non-minimum phase, HFC electric power system, sliding mode control. Keywords: non-minimum phase, HFC electric power system, sliding control. power mode supply when there is an interruption of power 1. INTRODUCTION powerHFC supply there is andemand. interruption of power Keywords: non-minimum phase, HFC electric power system, sliding mode from orcontrol. a when fast load current 1. INTRODUCTION powerHFC supply when there is an andemand. interruption of of power power power supply when there is interruption from or a fast load current The non-minimum phase property of the boost and Hydrogen Fuel Cell1. Vehicle (HFCEV) is [1]a 1. Electric INTRODUCTION INTRODUCTION from HFC or a fast load current demand. power supply when there is an interruption of from HFC or a fast loadproperty current demand. The non-minimum phase of is the boostpower and Hydrogen Fuelwhich Cell1. uses Electric Vehicle (HFCEV) is [1]aa boost/buck DC-DC power converters eliminated by hybrid vehicle, a HFC in combination with INTRODUCTION The non-minimum non-minimum phase property of is the the boost and and Hydrogen Fuelwhich Cell Electric Electric Vehicle (HFCEV) is is [1]aa from the HFC or a fast load current The phase of boost/buck DC-DC power converters eliminated by Hydrogen Fuel Vehicle (HFCEV) [1]a hybrid vehicle, a HFC in combination with controlling HFC and UC property currents,demand. which is boost one of the storage device, to Cell poweruses the electric motor. In this paper, we boost/buck DC-DC power converters is the eliminated by hybrid vehicle, which uses HFC in combination combination with The non-minimum phase of is Hydrogen Cell Electric Vehicle (HFCEV) [1]a boost/buck DC-DC power converters eliminated by controlling the HFC and UC property currents, which is boost one of and the hybrid uses aa system HFC in with storage vehicle, device, towhich power the electric motor. In this paper, weaa main contributions of the paper. control the Fuel electric power comprised ofis HFC controlling the HFC and UC converters currents, which which is one one of of the the storage device, towhich power the electric electric motor. In this this paper, paper, wea boost/buck DC-DC power is eliminated by hybrid vehicle, uses a HFC in combination with controlling the HFC and UC currents, is main contributions of the paper. storage device, to power the motor. In we control the electric power system comprised of HFC Meeting the fast load current demand is addressed by conditioned by DC-DC boost power converter, and an UC as main contributions contributions of the paper. control the byelectric electric power systemconverter, comprised of UC HFC controlling the HFC and UC currents, which is one of the storage device, to power the electric motor. In this paper, we main of the paper. Meeting the fast load current demand is addressed by control the power system comprised of HFC conditioned DC-DC boost power and an as generating two different current command profiles to follow. a storage device conditioned by boost/buck bidirectional Meeting the fast load currentcommand demand profiles is addressed addressed by conditioned byelectric DC-DC boost power power converter, and an UC as main contributions of the current paper. power system comprised of UC HFC Meeting the load current demand is by generating two different to for follow. conditioned by DC-DC boost converter, and an as acontrol storagethe conditioned by boost/buck bidirectional Specifically, afast slow current command is generated the converter, indevice order to drive the speed of the DC electric motor generating two different current command profiles to follow. aaconverter, storage device conditioned by boost/buck bidirectional Meeting current demand isis addressed by conditioned by DC-DC boost converter, and an UC as generating two different current command to for follow. Specifically, afast slowload current command is profiles generated the storage conditioned by boost/buck bidirectional order to drive thepower speed of the motor HFC, andthe the fast-current command profile generated for to its timeindevice changing command profile. In DC caseelectric of HFC, the Specifically, a fast-current slow current current command is profiles generated for the converter, indevice order to to drive the speed speed of the the DC electric motor generating two different current command to follow. acurrent conditioned by boost/buck bidirectional Specifically, a slow command is generated for the HFC, and the command profile is generated for converter, in order drive the of DC electric motor to storage its time changing command profile. In case of HFC, the UC. cannot vary rapidly due to slower HFC membrane HFC, and and the thea fast-current fast-current command profile is generated generated for to its time time changing command profile. In DC case ofmembrane HFC, the Specifically, slow current command is generated converter, inchanging order drive thedue speed electric HFC, command is for UC. to its command profile. In case of HFC, the current cannot vary rapidly to of slower HFC The tracking control problems areprofile addressed usingfor thethe 2dynamics should betotaken in account bythe combining themotor HFC UC. current cannot vary rapidly due to slower HFC membrane HFC, and the fast-current command profile is generated to its time changing command profile. In case of HFC, the UC. The specifically tracking control problems addressed using thefor 2current rapidly due to HFC membrane dynamics should be taken in UCs account by combining the HFC ASMC, adaptive gain are super-twisting controller with an cannot UC [4].vary The use of [2],slower [4], expedites a faster The specifically tracking control control problems problems are are addressed using using the 22dynamics should be taken taken in UCs account by combining the HFC UC. The current rapidly due to slower HFC membrane addressed the ASMC, super-twisting controller dynamics should be in account by combining the HFC with an cannot UC [4].vary The use of [2], [4], expedites a faster [4] for tracking controlling adaptive the HFC gain current through the partial varying response to the load current demand. ASMC, specifically adaptive gain super-twisting controller with an an response UC [4]. to The use of UCs [2], [4], expeditesthe faster The tracking control problems are addressed using the 2dynamics should bethe taken incurrent account by combining HFC ASMC, specifically adaptive gain super-twisting controller [4] for controlling the HFC current through the partial with UC [4]. The use of UCs [2], [4], expedites aa faster varying load demand. pressure of oxygen. The inner current loop is often [4] for for specifically controlling theThe HFCinner current through the partial varying response to theuse load current demand. ASMC, adaptive gain super-twisting controller with an response UC [4]. to The ofcurrent UCs [2], [4], expedites a faster [4] controlling the HFC current through the partial pressure of oxygen. current loop is often varying the load demand. synthetized by a classical PI controllers [5], [6]. In this work, The DC motors that are used as power drives in electric pressure ofbyoxygen. oxygen. The inner current loop is partial often [4] for theby HFC current through the varying to theare loadused current demand. pressure of The inner current loop is often synthetized a classical PI controllers [5], [6]. In this work, The DCresponse motors that asover power electric the HFCcontrolling conditioned the DC-DC boost converter is vehicles have some advantages AC drives motorsin[3]. The synthetized byoxygen. a classical classicalby PIthe controllers [5], [6]. In this work, pressure of The inner current loop is often The DC have motors that advantages are used used as asover power drives in[3]. electric synthetized by a PI controllers [5], [6]. In this work, the HFC conditioned DC-DC boost converter is The DC motors that are power drives in electric vehicles some AC motors The controlled by a conventional SMC. The nonminimum phase electric DC motors operate with lower voltage, requiring the HFC conditioned conditioned byPIthe the DC-DC boost converter is synthetized by a classical controllers [5], [6]. In this work, vehicles have some advantages over ACvoltage, motorsinrequiring [3]. The The DC motors that are used as power drives electric the HFC by DC-DC boost converter is controlled by a conventional SMC. The nonminimum phase vehicles have some advantages over AC motors [3]. The electric DC motors operate with lower property is mitigated by controlling the HFC current. The UC fewer batteries. They have higher peak torque and faster controlled by a conventional SMC. The nonminimum phase boost converter is the HFC conditioned by the DC-DC electric DC motors operate with lower voltage, requiring vehicles have some advantages over AC motors [3]. The controlled conventional SMC.athe The nonminimum phase property is by mitigated by controlling HFC current. The UC electric DC The motors with lower requiring fewer batteries. They have torque higher peak voltage, torque and faster conditioned bya the DC-DC boost/ conventional SMC also acceleration. highoperate peak enables the vehicle to be property is is by mitigated by controlling controlling the HFC current. The UC controlled aconverter conventional SMC.aathe The nonminimum phase fewer batteries. batteries. They have torque higher peak voltage, torque and faster electric DC motors operate with lower requiring property mitigated by HFC current. The UC conditioned by the DC-DC boost/ conventional SMC also fewer They have higher peak torque and faster acceleration. The high peak enables the vehicle to be controls buck to follow time-varying command more adaptable for different driving conditions. DC-DC boost/ a conventional SMC also conditioned by the property is mitigated by controlling the HFC current. The UC acceleration. The high peak torque enables the vehicle to be fewer batteries. They have higher peak torque and faster conditioned by the DC-DC boost/ a conventional SMC also controls buck converter to follow time-varying command acceleration. The high peak driving torque enables the vehicle boost to be more forin different profile generated by the HFC current command using power The adaptable challenges controlling the conditions. “HFC/UC/DC-DC controls buckbyconverter converter to follow follow time-varying command conditioned thebyDC-DC boost/ a conventional SMC also more adaptable forin different driving conditions. acceleration. The high peak driving torque enables vehiclefor to be controls buck to time-varying command profile generated the HFC current command using power more adaptable for different The boost/buck challenges controlling the conditions. “HFC/UC/DC-DC boost balance approach. Adaptive-gain twisting controller robustly and converters” based power the system an profile generated by the HFC HFC current command using power controls buck converter to follow a time-varying command The boost/buck challengesforin in controlling the conditions. “HFC/UC/DC-DC boost more adaptable different driving profile generated by the current command using power balance approach. Adaptive-gain twisting controller robustly The challenges controlling the “HFC/UC/DC-DC boost and converters” based power system for an controls the servomotor speed. electric vehicle application that are addressed in this paper are balance approach. Adaptive-gain twisting controller robustly profile by thespeed. HFC current command usingrobustly power and boost/buck converters” based power in system for an The challenges in controlling theaddressed “HFC/UC/DC-DC boost balance approach. Adaptive-gain twisting controller controlsgenerated the servomotor and boost/buck converters” power system for an electric vehicle application thatbased are this paper are as follows, 2. MATHEMATICAL MODEL OF HFC/UC/DC-DC controls the servomotor speed. balance approach. Adaptive-gain twisting controller robustly electric vehicle application application thatbased are addressed addressed in this paper paper are and boost/buck converters” power system for an controls the servomotor speed. electric vehicle that are in this are as follows, 1. The non-minimum phase property of the DC-DC boost 2. MATHEMATICAL MODEL OF HFC/UC/DC-DC CONVERTER/SERVOMOTOR controls the BOOST servomotor speed. as1.follows, follows, electric vehicle application that are addressed in DC-DC this paper are as The non-minimum phase property of the boost 2. MATHEMATICAL MATHEMATICAL MODEL OF OF HFC/UC/DC-DC HFC/UC/DC-DC and boost /buck converters is to be mitigated or 2. MODEL BOOST CONVERTER/SERVOMOTOR 1.follows, The non-minimum non-minimum phase property property of the the DC-DC boost as1. The phase of boost and boost for /buck converters is to be DC-DC mitigated or The equivalent circuit diagram of anOF electric power system BOOST CONVERTER/SERVOMOTOR eliminated tracking causal load voltage profiles in 2. MATHEMATICAL MODEL HFC/UC/DC-DC BOOST CONVERTER/SERVOMOTOR and boost for /buck converters is to to be DC-DC mitigated or 1. the Thepresence non-minimum phase property ofvoltage the boost The equivalent circuit diagram an electric system and boost /buck converters is be mitigated or eliminated causal load profiles in comprising HFC/DC-DC boost ofconverter forpower a vehicle is of tracking model uncertainties. BOOST CONVERTER/SERVOMOTOR The equivalent circuit diagram diagram ofconverter an electric electric power system eliminated for tracking causal load load voltage profiles or in and boost for /buck isdynamics tovoltage be mitigated The equivalent circuit an system comprising HFC/DC-DC boost of forpower a vehicle is eliminated causal in the presence of tracking model uncertainties. designed in Fig.1. 2. The relatively slowconverters uncertain ofprofiles the HFC comprising HFC/DC-DC boost ofconverter converter forpower vehicle is The equivalent circuit diagram an electric system the presence presence of tracking model uncertainties. eliminated for causal load profiles in comprising HFC/DC-DC boost for aa vehicle is designed in Fig.1. the of model uncertainties. 2. membrane The relatively slow uncertain dynamics thecurrent HFC challenges addressing thevoltage fast of load designed in in Fig.1. Fig.1. comprising HFC/DC-DC boost converter for a vehicle is 2. The Thepresence relatively slow uncertain uncertain dynamics of thecurrent HFC the of model uncertainties. designed 2. relatively slow the HFC membrane addressing thea backup fast of loadsource demand. A challenges controlled UC is useddynamics as of designed in Fig.1. membrane challenges addressing thea backup fast of loadsource current 2. demand. The relatively slow uncertain thecurrent HFC membrane addressing the fast load A challenges controlled UC is useddynamics as of demand. A A challenges controlled UC UC is used used as as backup source of membrane addressing theaa backup fast loadsource current demand. controlled is of Copyright © 2017 IFAC demand. A controlled UC is used as a backup source of15359 Copyright © 2017, 2017 IFAC 15359 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2017 IFAC 15359 Peer review© of International Federation of Automatic Control. Copyright ©under 2017 responsibility IFAC 15359 10.1016/j.ifacol.2017.08.2552 Copyright © 2017 IFAC 15359
Proceedings of the 20th IFAC World Congress Roshini S. Ashok et al. / IFAC PapersOnLine 50-1 (2017) 14794–14799 Toulouse, France, July 9-14, 2017
14795
the electrochemical reaction going in the HFC. Due to the insignificant values of m 1 and b1 (specifically, the value of m1
5 1 0 V ) in eq. (7) the term is typically close to m 1 3
V c o n can be neglected in eq. (6) [4]-[9],[12] 2.2 Mathematical model of HFC conditioned by DC-DC unidirectional boost power converter The dynamics of the DC-DC boost power converters are governed by the following system of differential equations [4],[11]-[13]:
Fig.1 Equivalent Circuit Diagram of HFC/UC/DC-DC boost and boost/buck converter-based electric power system for a HFCEV
2.1 Mathematical model of HFC The mathematical of HFC is derived based on the equivalent circuit in Fig. 1, the dynamics of activation over voltage, V a t , are given as [4]-[9],[12], d V at
i h fc
dt
V at
Cd
(1)
R atC d
The partial pressure of hydrogen and oxygen dynamics is presented as [15]: d dt
d dt
PH
2
Po 2
1
H
H kH
2
2
1
2
Po
o
1
PH
2
2
1
o ko 2
q
in H
2
2 k i h fc 1
(2)
d V ser
d i h fc
2
1
(3)
2
1
in
in
( qH qH )
qH
2
2
(8)
[ (1 t1 )V s e r V th fc ]
1
d V uc
2
(1 t 2 )V s e r
Luc
1
dt
in
dt
L
dt
The actuator valve dynamics are modeled as in [15], [16]: dqH
1
where t1 [ 0 ,1] is a switch (transistor T1 in Fig. 1) control function, that is transformed as, (9) t1 v 1 0 .5 , v 1 [ 0 .5 , 0 .5 ] . 2.3 Mathematical model of UC controlled by bidirectional DC-DC buck (buck/boost) converter The dynamics of the UC being charged and discharged by a bidirectional DC-DC converter are derived as in [4],[13] d iu c
q o k i h fc
[(1 t1 ) i h fc (1 t 2 ) i u c i s e r ]
Cb
dt
2
in
1
dt
C uc
V uc
(10) (11)
iu c
(4)
where t 2 v 2 0 .5 , v 2 [ 0 .5 , 0 .5 ] , and t 2 (transistor T2in Fig. 1) control function.
(5)
2.4 Mathematical model of a Servomotor The dynamics of a servomotor are given by the following set of equations [7] ,
2
is a switch
in
dqO
2
dt
1
qO
in
in
( qO qO ) 2
2
2
The inputs of the valves control functions, i.e.
in
qO
and
2
in
v3 q H
The output voltage
2
V h fc
in
qH
and
2
are considered as HFC in
v4 qO
2
d ise r
.
of a stack of
n
HFCs
V th fc
d ser
defined by equations V th fc n V h fc , E h fc
G1
2 Far
V h fc E h fc V o h m V c o n V a t s1 2 Far
( T s t T h fc )
R g T st 1 ln ( PH 2 ) ln ( PO 2 2 Far 2
,
V o h m i h fc R o h m ,
dt
1 J
k
i
m ser
V se r k b se r
k b ser n g T d
(12)
(13)
as the control inputs, the input-output dynamics of the system are derived based on eqs. (1) -(13): v1 , v 4 , v 2 , v 3
(7)
1
molar entropy ( J m o l K ). The universal gas constant is given as R g 8 .3 1 4 J m o l 1 K 1 . The Faraday constant is F a r 9 6 4 8 5 .3 4 1 5 s A / m o l
temperature of the HFC is given as T h fc
R a r i se r
2
)
where the Gibbs free energy G 1 ( J m o l 1 ) s 1 is the standard
described as
L in d
2.5 Mathematical model of HFC/DC-DC boost converter/UC Considering, V s e r , i h fc , iu c , PH , as the system’s outputs and
(6)
Po 2 V c o n m 1 e x p ( n i h fc ) b1 ln a1
1
1
dt
. The reference
2 9 8 .1 5 K
and the
stack temperature is T s t 3 5 3 K . The temperature ( T s t
T h fc
d V ser dt 2 d i h fc 2 dt d iu c dt d PH 2 dt
H1 H2 H3 H 4
1 i h fc Cb 1 0 0
0
Cb 0
1
15360
1 Luc
0
1 1 3 i h fc i h fc v 1 i s e r L Cb 2
0
iu c
3
2
0
0 0
V uc
1
H kH 2
where
), which is assumed to be constant during the process, is observed. The variable V c o n is dependent on the changes in the concentration of reactants as they are being consumed by
1
;
2
v1 v 4 v2 v3
(14)
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 14796 Roshini S. Ashok et al. / IFAC PapersOnLine 50-1 (2017) 14794–14799
2
1 0 .5 R g T s t L 2F
H1
1 1 i h fc i s e r 0 .5 i u c ; Cb 2
H
H
4
2
1
H
1 L
1 k P O2 O2 O2
1
PH
H kH
2
2
2
2k
,
3
1
H3
1
i h fc
1 Cb
i u c (1 v 1 )
0 .5V s e r
Luc
V uc
R g T st 2 Far
THE CONTROLLER DESIGN: RELATIVE DEGREE APPROACH: The control problem is in designing the control functions that drive the tracking errors v1 , v 4 , v 2 , v 3
,
c
dt
d iser dt d V uc dt
i h fc
1 1 PH 2 kH H2 H2 2
v
3
k
2 k i h fc
i
1 h fc
V at
Cd
command
L in d
C uc
d e1 s dt 2 d e2s 2 dt d e 3s dt d e4s
ser
dt
c
to be bounded as the feedback
control has been applied.
The first task is to design a controller
v1
1
0
0
iu c
Cb 3
2
0
1 Luc
0
0 0
V uc
1
0
H kH 2
2
v1 v 4 v2 v 3
(18)
(1 )
H 1;
H
2
i c
(2)
h fc
H 2;
H
4
P c
H
(1 )
H
2
4
,
(1 )
i H 4.2 The control problem decoupling: first step Last two equations in system (18) is completely decoupled from the first two equations: H
3. PROBLEM FORMULATION The control problem in HFC-UC/DC-DC converters based electric power system for controlling the servomotor in an electric vehicle is reduced t to the following subtasks
2
is assumed to be constant):
2
1 i h fc Cb H1 1 H 2 H 0 3 H 4 0
H 1 V ser
h fc
and
V s e r ( t ), i h fc ( t ), i u c ( t ), PH ( t )
where,
(17)
Apparently, the forced zero dynamics in eqs.(15)-(17) are stable, and the profiles V a t and i are bounded, assuming i h fc
for
tracking error dynamics,
the forced profile
2
4.1 Input-output de-coupling: Relative Degree Approach Using the relative degree approach [7], [19], [22] the inputoutput error dynamics in eq. (13) of vector relative degree r 1, 2 , 1, 1 . The system (13) is re-written in terms of
(16)
iu c
profiles
respectively ( PH
(15)
( iser R ar V ser k b ser )
1
2
c
R atC d
1
c
2
Similarly, the internal dynamics of the HFC system derived from eqs. (1), (11) and (12) as follows, d V at
c
e 2 s i h fc i h fc , e 3 s iu c iu c , e 4 s PH PH
to zero as time increases in the presence of the bounded disturbances H 1 , H 2 , H 3 , H 4 . V sce r ( t ) , i hc fc ( t ) , i ucc ( t ) , PHc are
2
1 1 PO 2 kO O2 O2 2
c
e1 s V s er V s er ,
1 1 (1 ) i h fc i s e r v 1V s e r ( i h fc ) V a t Cb 2 1 PH 2 0 .5 P O 2
4.
,
3
c
uc
3
d e3 s dt
iu c c
(1 )
d e4 s
so that
dt
P c
H
H
3
V uc Luc
(19)
v2
3
(1 )
2
1 2 k 1 PH i h fc v3 2 H kH H kH 2 2 2 2 H2
(20)
c
the output voltage follows the prescribed command profile in the presence of bounded perturbations. The second task is to design the controller v 2 so that the output current of the UC iu c follows the desired V ser V ser
command profile
c
iu c
in finite time also.
The third task is to design a controller in terms of
c
V ser
that drives s e r sce r ( t ) in accordance with the servomotor dynamics. The fourth and the fifth tasks are about designing the control functions v 3 for controlling the hydrogen partial pressure and v 4 for controlling the HFC current that allows eliminating the non-minimum phase property of the DC-DC boost converter. Specifically, HFC current is robustly controlled via the adaptive 2-ASMC (supertwisting) controller v 4 s , whose adaptive gains are not overestimated. This allows controlling the HFC current in a perturbed environment with bounded derivatives of the perturbations, whose boundaries are not known.
The error
e3 s
and
e4 s
dynamics ( iu c and
PH
2
tracking
dynamics) in (19), (20) are considered firstly for the v 2 and v 3 control functions design. The following assumption about the disturbance 3 : 3
L3
is made.
4.3. The control v 2 design The control v 2 [4], [13] is designed in terms of conventional SMC to enable the charging/discharging of the UC current in a case of a load demand at a fast rate. The “fast” command, c i u c , for the UC current, i u c , is generated in accordance with eq. (38). v 2 0 .5 s ig n e 3 s , e 3 s i u c i u c c
As soon as the UC current c
iu c
follows the command profile
in finite time by means of the SMC (21) the error becomes
e3 s 0
and, therefore,
c
iu c iu c
4.4. The control v 3 design
15361
iu c
(21)
in the sliding mode.
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Roshini S. Ashok et al. / IFAC PapersOnLine 50-1 (2017) 14794–14799
A PI-like controller robustly addresses the set point regulation problem by driving e 4 s PHc PH 0 in eq. (20) as time 2
2
increases v 3 k H PH 2 k i h fc k p e 4 s k i 2
2
1
1
1
e
4s
The output voltage command profile V sce r is derived in section 4.9 in accordance to [4]. The conventional SMC [4], [10], [13] v1 that represents a switching function v1 [ 0 .5, 0 .5 ]
(22)
dt
14797
v 1 0 .5 s ig n e1 s
(28)
The tuning and the design of the PI controller is understood [18] and, therefore, omitted for conciseness.
is insensitive to the bounded disturbance (
4.5 The control problem decoupling: second step
controlling the output voltage of the DC-DC boost converter by driving e1 s V sce r V s e r 0 in finite time.
The effect of DC-DC boost and boost/buck converters coupling is taken into account in eq. (18). Assuming the sliding mode is established in eq. (19), (21), first two equations in system (18) can be rewritten taking in account that v 2 is replaced by v 2 e q d e1 s dt
1 1 c iu c v 2 e q i h fc v 1 V ser H 1 C C
(23)
1
2
dt
2
i h fc c
(2)
H
2n
1 0 .5 R g T s t 2 Far L
1 k P o2 o2 O 2
v4
H
2n
H
2
1 C
iu c v 2 e q
;
v 2 eq
H 1n
d e1 s dt H 1n 2 d e2 s H 2n d t 2
1 i C h fc 0
and
c
H 1n V ser H 1n ;
H
2n
i h fc c
0
1 k P O2 O2 O2
(2)
H
placement to differential equation dynamics are derived dt
i h fc c
(2)
H
(1 )
2n
c2 e2 S
0
(25)
4.6
3
2
1 k P o2 o2 O 2
(30)
is assumed to be
with unknown boundary L 2 . The adaptive
L2
2
1/ 2
s ig n ( )
h fc
(31)
s ig n ( )
2
h s ig n h 2 , h
v4
e2 s ce2 s 0
L3
1 0 .5 R g T s t L 2 Far
h fc 2 h h ,
2n
h ,
if h fc h m
(32)
if h fc h m
h fc ( 0 ) h m , h 0
. Then the
v4 s
(33)
g1
The problem of finding real-time robust estimation of e 2 s is solved using arbitrary order robust exact differentiator [10] is similar to the eq. (43) given in section 4.9. 4.8. HFC and UC current command generator The HFC current command profile
1 0 .5 R g T s t L 2 Far
L2 ,
g1
where h , h , , h h , are arbitrary positive constants. Finally, we have
1 k P o2 o2 O 2
v4
the power balance
(27)
c
.
Controller Design for DC-DC Boost Converter
PH F C Ps e r L o a d
c
c
i h fc
is computed based on
[4], [13]:
c
i h fcV th fc i s e rV s e r ( t )
The following assumptions about the disturbances 1 , 2 , 3 : 2
g1v4 ;
v 4 s h fc
2
L1 ,
(1 ) 2
super-twisting controller is a suitable candidate [11], [17] that drives e 2 s , e 2 s 0 to a real 2-SM in finite time. The adaptive gain super- twisting controller is given below,
g1
are made as 1
h fc
0 v1 v 0 4
Equation (24) of relative degree 2 is reduced to relative degree 1 with respect to the variable, (26) e2 s c2 e2 s where c 2 0 is selected to provide a desired eigenvalue
d
obtained from eqs.(23) and (24):
1 0 .5 R g T st L 2 Far
2n
(2)
bounded
1 i u c (1 v 1 ) v 2 e q v1 Cb
0
H
The derivative of the disturbance
The decoupled input-output dynamics of the system is finally written while replacing the disturbances H 1 and H 2 from eq.(14) with
) in terms of ( v 4 )
h fc
v4 s
1 1 i h fc i s e r iu c 2
1 1 3 i h fc i h fc v 1 i s e r L Cb 2
) while
The HFC current error dynamics given in eqs. (26) and (27) are written as follows, (29) e2 s c 2 e2 s ; 2 g 1v 4 The control input ( v 4 ) for controlling the HFC current i
(24)
where, H 1n H 1
h fc
L1
must be continuous. Therefore, the controller is designed in terms of ( v 4 ). Differentiating the eq. (29) again, we obtain,
H1n
d e2
4.7 Controller Design for HFC current ( i
1
(34)
where 1 accounts for power losses (for simplicity, the power losses in the converter are neglected, and it is assumed c 1 ), i s e r is the output current command. The total voltage across n stack of HFC’s is given to be V . HFC current th fc
c
command profile i fc is generated as follows,
15362
Proceedings of the 20th IFAC World Congress 14798 Roshini S. Ashok et al. / IFAC PapersOnLine 50-1 (2017) 14794–14799 Toulouse, France, July 9-14, 2017
The arbitrary positive constants are given to be The output voltage 1 s , s , s , s e r m in , 1 s . command is finally derived as follows,
c
c
c
i h fc i s e r
V ser ( t )
,
(35)
V th fc c
The HFC current command profile
i h fc
is divided into two c
V ser ( t )
commands such as slow and fast commands, c
c s lo w
c fa s t
i h fc i h fc
i h fc
(36)
s lo w
where the slow command
can be generated as result of
i h fc
c low pass filtering of i h fc :
y1
5.
d i h fc
c s lo w
c
i h fc
dt
(37)
0
i h fc ,
c s lo w
Next, i fc is supposed to follow i fc
, while
iu c
will follow
s g n (
ser
)
1
s g n (
2
ser
controller
) dt
(42)
The problem lies in finding robust estimation of e s e r and its derivative in real-time. This problem is solved using a differentiator [10],
c s lo w
ser
1 s 0 .2 5;
SIMULATION STUDY
The following load voltage and current command profiles were selected in accordance with [4], [11] to ensure that the proposed HFC system can handle high power demands:
c
asymptotically the fast command profile i u c that is defined c
based on already generated
c s lo w
and
i fc
i fc
c
V th fc
i
c
V ser
c
c s lo w
i h fc
h fc
V th fc
c
V ser
c fa s t
ser
The problem in controlling the servomotor is to drive the servomotor speed s e r to follow the generated on-line reference profile [7]. To drive ( t ) , we need to design the controller in terms of V in accordance with the servomotor dynamics given in (13) c
c
ser
(43)
Fig.2 illustrates the causal tracking of the output voltage of DC-DC boost converter following the respective command profile. Fig.3 shows that the HFC’s current follows the respective HFC command profile with the help the adaptive gain super-twisting controller with the adaptive gain h f c
(38)
i h fc
4.9. Controller Design for servomotor speed
ser
c
command profiles
and a power balance condition as iu c
s e r ( t ) 1 0 0 1 0 0 s in ( 0 .2 ) t
ser
c
ser
shown in Fig.8. Fig.4 indicates the UC current ( iu c ) follows c
its respective command profile ( i u c ). The servomotor speed profile is shown in Fig.5. Fig.6 shows the HFC current control ( v 4 ) function. Fig.7 shows the partial pressure of hydrogen ( PH ) is maintained constant as time increases with 2
The error dynamics of the servo motor speed is given as.
c
ser
c
e s e r d s e r e s e r g 2V s e r ; e s e r s e r s e r
respect to its corresponding command profile(
(39)
The term in eq.(21) is considered as the differentiable disturbance, whose derivative is bounded with unknown boundary. The control input is given as V sce r in eq.(21). The
c
PH ). 2
Fig.9
indicates the adaptive gain ( s e r ) used in designing the servomotor speed controller. Bv
controller ( V sce r ) must be continuous. Therefore the control is deigned in terms of ( V ). Differentiating again, we obtain, c
ser
ser
g2,
c
g 2 y 1V s e r ;
y1 0
(40)
The term is assumed bounded L b o u n d with unknown boundary ( L b o u n d ). Then the adaptive-gain twisting controller [14] that drives e s e r , e s e r 0 to a real second order sliding mode in finite time is designed in terms of 1 s e r (s g n (
ser
1s s
1s 2
ser
)
1
s g n (
2
s g n ( s (
ser
,
ser
ser
c
V ser ( t )
,
: (41)
))
) s ),
if s e r s e r m in if s e r s e r m in
c
Fig.3 Plot of HFC command profile ( i h fc ( t ) ) and HFC current( ih fc ( t )
where, 2
2
s ( s e r , s e r ) s e r s e r 1 s s e r 2
3/2
Fig.2 Plot of the DC-DC boost converter voltage command profile ( V s ce r ( t ) ) and voltage ( V s e r )
sg n ( s e r ) s e r 4
s e r s e r s e r 0 .2 5 ser
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) (Vs.) Time
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Roshini S. Ashok et al. / IFAC PapersOnLine 50-1 (2017) 14794–14799
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converter was eliminated by controlling the HFC current using the adaptive gain super-twisting controller. The HFC current accurately follows the command profile without overestimating the gains in the presence of bounded perturbations with unknown bounds. The conventional sliding mode controller was implemented to follow the voltage and current causal time-varying command profiles in the DC-DC boost converter and UC current profile..
c
Fig:4 Plot of UC current command profile( i u c ( t ) ) and UC current ( iu c ( t ) ) (Vs.) Time
REFERENCES Fig.5 Plot of the servomotor speed command profile( c s e r ( t ) ) and servomotor speed( s e r ( t ) )
Fig:6 Plot of Fuel Cell current controller( v 4 )
Fig.7 Plot of partial pressure of hydrogen command profile ( c PH ) and partial pressure of hydrogen ( PH ) (Vs.)Time 2
2
Fig.8 Plot of Adaptive gain ( h f c ) (Vs.)Time
Fig.9 Plot of the Adaptive gain ( s e r )
6. CONCLUSIONS The servomotor speed within an HFC/UC/DC-DC boost and boost/buck converters based electric power system for an electric vehicle application is controlled. The adaptive gain twisting controller is used for controlling the servomotor speed. The non-minimum phase property of the DC-DC boost
[1] http://serc.berkeley.edu/fueling-the-future-hydrogen-fuel-cellvehicles-in-the-21st-century/ [2] http://www.environmentalleader.com/2012/01/10/five-reasonswhy-ultracapacitors-are-attractive-to-auto-manufacturers/ [3] http://www.engineerlive.com/content/21329 [4] R. S. Ashok; Y. B. Shtessel, “Control of Fuel Cell- based Electric Power System Using Adaptive Sliding Mode Control and Observation Techniques”, Journal of the Franklin Institute, Vol. 352, Issue 11,, November 2015, pp.4911–4934. [5] M. Hilairet, M. Ghanes, O. Béthoux, V. Tanasa, J-P. Barbotb, D. Norman-Cyrot, “A passivity-based controller for coordination of converters in a fuel cell system”, Control Engineering Practice, Vol. 21, No. 8, August 2013, pp.1097–1109. [6] M. Ghanes, M. Hilairet, J-P. Barbot , O. Bethoux, “Singular Perturbation Control for coordination of converters in a Fuel Cell System”, ELECTRIMACS, 2011, Cergy-Pontoise, France. 2011. [7] R. S. Ashok and Y. B. Shtessel, “Control of a fuel cell vehicle using adaptive sliding mode control: Servomotor application” Proceedings of ACC, 2016, pp. 2235 – 2240. [8]J. Correa, F. Farret, L. Canha, and M. Simoes ”An Electrochemical-Based Fuel- Cell Model Suitable for Electrical Engineering Automation Approach”, TIE, Vol. 51, No.5, 2004. [9] M. J. Khan, M. T. Iqbal “Dynamic Modelling and Simulation of a Fuel Cell Generator”, Fuel Cells Special Issue: Modelling of Fuel Cell Systems Volume 5, Issue 1, pages 97–104, February, 2005 [10]Y. Shtessel, C. Edwards, L. Fridman, and A. Levant, “Sliding mode control and Observation”, Birkhauser, New York, 2014. [11]J. M. Olm, X. R. Oton, and Y. Shtessel "Stable inversion-based robust tracking control in DC-DC nonminimum phase switched converters", Automatica, 47, 1, pp. 221-226, 2011. [12] C. Kunusch, P. F. Puleston, M. A. Mayosky, and J. Riera, “Sliding Mode Strategy for PEM FCs Stacks Breathing Control Using a Super-Twisting Algorithm,“ IEEE Transactions on Control System Technology, Vol. 17, no. 1, 2009, pp. 167-174. [13] R. Ashok, Y. Shtessel, and J. Smith, “Sliding mode control of electric power system comprised of fuel cells, DC-DC boost converters and UC,” Proceedings of ACC, 2013, pp. 5766 – 5771. [14] J.Kochalummoottil, Y. Shtessel, J.A Moreno, and L. Fridman, “Adaptive Twist sliding mode control: A Lyapunov Design,” in Proceedings of 50th IEEE CDC, pp. 7623-7628, 2011. [15] J.Padullésa, G.W Aultb and J.R McDonald, “An integrated SOFC plant dynamic model for power systems simulation,” Journal of Power Sources, Vol.86 , Issues 1-2, pp. 495-500. [16] K. Belmokhtar, M. H. Hammoudi, M. L. Doumbia, and K. Agbossou, “Modelling and Fuel Flow Dynamic Control of Proton Exchange Membrane Fuel Cell,” Proceedings of 4th International Conference on Power Engineering, Energy and Electrical Drives, May 2013 pp. 415-420. [17] Y. Shtessel, M. Taleb, and F. Plestan, “A novel adaptive-gain super-twisting sliding mode controller: methodology and application,” Automatica, Vol. 48, Issue 5, 2012, pp. 759-769. [18] R. Dorf, and R. Bishop, “Modern Control Systems”, 12th Edition, Prentice Hall 2010.
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