modeling and control of a fuel cell based z-source

/47Mechatronics Lab.

Modeling and Control of a Fuel Cell Based Z-source Converter for Distributed Generation Systems

Jin-Woo Jung, Ph. D. Student

Advisor: Prof. Ali Keyhani

October 22, 2004 (IAB’04)

Mechatronic Systems LaboratoryDepartment of Electrical and Computer Engineering

The Ohio State University

1


Publications (1)

Journal Papers

[1] J. W. Jung, H. Lee, and A. Keyhani, “Modeling, Analysis, and Control of Fuel Cell Based Distributed Generation Systems, Part I: Standalone AC Power Supply,”IEEE Transactions on Energy Conversion. (Will be submitted in November, 2004)

[2] J. W. Jung and A. Keyhani, “Modeling, Analysis, and Control of Fuel Cell Based Distributed Generation Systems, Part II: Grid-Interconnection,”IEEE Transactions on Energy Conversion. (Will be submitted in November, 2004)

[3] J. W. Jung, M. Dai, and A. Keyhani, “Modeling and Control of a Fuel Cell Based Z-Source Converter for Distributed Generation Systems ,” IEEE Transactions on Energy Conversion. (Will be submitted in October, 2004)

[4 ] J. W. Jung and A. Keyhani, “Modeling and Control of Z-Source Converter for Fuel Cell Systems,” IEEE Transactions on Energy Conversion. (Submitted in April, 2004 and being reviewed)

[5] M. N. Marwali, J. W. Jung, and A. Keyhani, “Control of Distributed Generation Systems, Part II: Load Sharing Control,” IEEE Transactions on Power Electronics, vol. 17, no. 6, pp. , Nov. 2004. (In Print)

2


Publications (2)

Conference Papers (1)[1] Jin-Woo Jung, Min Dai, and Ali Keyhani, “Modeling and Control of a Fuel Cell Based

Z-Source Converter,” IEEE Applied Power Electronics Conference (APEC’05) in Austin, Texas, USA, March 6-10, 2005. (Accepted and will be presented)

[2] Min Dai, Mohammad N. Marwali, Jin-Woo Jung, and Ali Keyhani, "A PWM rectifier control technique for three phase double-conversion UPS under unbalanced load,“IEEE Applied Power Electronics Conference (APEC’05) in Austin, Texas, USA, March 6-10, 2005. (Accepted and will be presented)

[3] Min Dai, Mohammad N. Marwali, Jin-Woo Jung, and Ali Keyhani, “Power Flow Control of a Single Distributed Generation Unit with Nonlinear Local Load,” IEEE Power SystemsConference & Exposition (PSCE’04), New York, USA, October 10-13, 2004.

[4] J. W. Jung, M. Dai, and A. Keyhani, “Optimal Control of Three-Phase PWM Inverter for UPS Systems,” IEEE Power Electronics Specialist Conference (PESC’04), pp. 2054-2059,Aachen Germany, June 20-24, 2004.

3


Publications (3)

Conference Papers (2)[5] Jin-Woo Jung and Ali Keyhani, “Design of Z-source Converter for Fuel Cells,”

Electric Supply Industry in Transition, pp. 17/62 – 17/67, AIT Thailand, 14-16 Jan. 2004.

[6] Min Dai, Ali Keyhani, Jin-Woo Jung, and A.B. Proca, "A Low Cost Fuel Cell Drive System for Electrical Vehicles," Proceedings of the 2003 Global Powertrain Congress Conference and Exposition, vol. 26, pp. 22-26, USA, Sept. 2003.

[7] A. Keyhani, M. Dai, and J. W. Jung, “Parallel Operation of Power Converters for Applications to Distributed Energy Systems,” 2nd IASTED (The International Association of Science and Technology for Development) International Conference on Power and Energy Systems, Greece, June 25-28, 2002.

4


ORGANIZATION

I. Introduction

II. Circuit Analysis/System Modeling/PWM Implementation

A. Circuit Analysis

B. System Modeling

C. Space Vector PWM Implementation

III. Control System Design

A. Discrete-time Sliding Mode Current Controller

B. Discrete-time Optimal Voltage Controller

C. Discrete-time PI DC-link Voltage Controller

IV. Simulation Results

VI. Conclusion

5


I. INTRODUCTIONWhy are fuel cell systems increasingly used in industry?

Emerging power generation technologies

Wind turbines and photovoltaic cells (renewable technologies)

⇒ Full-grown technologies

⇒ No emission

⇒ Climate constraints: wind and sunshine

⇒ Efficiency: wind turbines (20 – 40%), photovoltaic cells (5 – 15%)

Fuel cells

⇒ Regardless of climate conditions

⇒ Hydrogen and oxygen

⇒ Products: electricity, heat, and water

⇒ Nearly zero emission

⇒ Efficiency: electricity (up to 60%), co-generation (up to 85%)

6


I. INTRODUCTIONOperating Principle of Fuel Cell Systems

Operation of the fuel cells

StacksChemical energy

toElectrical energy

Hydrogen

Oxygen

Natural gasPropane

MethanolGasoline

Reformer

Air

Power ConverterDC power AC power

Fig. 1 Fuel cell generation system.

7


I. INTRODUCTIONFeatures of Fuel Cell Systems

Types of fuel cellsPhosphoric Acid (PAFC), Solid Oxide (SOFC), Molten Carbonate (MCFC), Proton-Exchange-Membrane (PEMFC), Alkaline, Zinc-Air (ZA) fuel cells, etc.

Applications of fuel cellsResidential (domestic utility), Transportation (Fuel cell vehicle),Portable power (laptop, cell phone), Stationary (buildings, hospitals, etc),Distributed power for remote location, etc.

Characteristics of fuel cellsEnvironmental-friendly (nearly zero emission)Modular electric generationHigh efficiency (co-generation)Slow dynamic response during initial start-up and load changeBoosted for most applications due to a low DC output voltageVarying output voltage according to the load (current)

8


I. INTRODUCTIONOverview of Fuel Cell Systems (1)

Fuel Cells for stationary or distributed power applications (1)

YesYesYesYesEnvironmental – friendly (Nearly zero emission)

Natural gas, hydrogen, propane, diesel

Natural gas, hydrogen, landfill,

propane

Natural gas, hydrogen, landfill

gas, fuel oil

Natural gas, landfill gas, digester gas,

propaneFuel

3 – 250kW250kW – 10MW1kW – 10MW100 – 200kWSize

R&DR&DR&DYesCommercial Availability

WaterExcess AirExcess AirBoiling WaterCooling Medium

4,000800 – 2,0001,300 – 2,0003,000 – 3,500Installed Cost ($/kW)

200°F1,200°F1,800°F400°FOperating Temperature

PEMFCMCFCSOFCPAFCTypes

9


I. INTRODUCTIONOverview of Fuel Cell Systems (2)

Fuel Cells for stationary or distributed power applications (2)

Likely commercialization

2003/2004


2004


2004

Some commercially availableCommercial Status

Yes (80°C water)Yes (hot water, LP or HP steam)

Yes (hot water, LP or HP steam)Yes (hot water)Cogeneration

Up to 85%Up to 85%Up to 85%Up to 85%Efficiency (Cogeneration)

30 – 40%45 –55%45 – 60%36– 42%Efficiency (Electricity)

Stationary power (1997-2000),

Bus/Railroad/Automotive Propulsion

(2000-2010)

Stationary power (2000-2005)

Stationary power and Railroad Propulsion

(1998-2005)

Stationary power (1998),

Railroad Propulsion (1999)

Applications

PEMFCMCFCSOFCPAFCTypes

10


I. INTRODUCTIONWhat is the research focus?

Control of Power Electronic Interface System (i.e., Control of Power Converter)

⇒ Inexpensive, reliable, small-sized, and light-weighted

Main factors: size of L-C filter, number of power devices and sensors

⇒ Good performance:

− Nearly zero steady-state voltage (RMS) error

− Low Total Harmonic Distortion (THD)

− Fast/no-overshoot current response

− Good voltage regulation

⇒ Power generation applications (three-phase AC 208V (L-L) /60 Hz)

Research Focus

11


I. INTRODUCTIONConventional Topology (1)

Conventional: a DC-DC boost converter and a DC/AC inverter DSP controller: two controllers (DC/DC and DC/AC power converters)Power devices: ⇒ DC to DC boost converter: four power switches and four diodes⇒ DC to AC inverter: six power switchesSensors: DC input, DC output, and AC output

FuelCell(Vin)

S1 S3

S4 S2

S1 S3

S4 S6

3-phaseload

S5

S2

DC to DC boost converter DC to AC inverter

Fig. 2 Conventional system configuration.12


I. INTRODUCTIONConventional Topology (2)

Control Block Diagram

Fig. 3 Control block diagram of conventional system.

13


I. INTRODUCTIONConventional Z-source Topology

Conventional Z-source converter: Impedance source (L-C) and a DC/AC inverterBoosted by shoot-though zero vectors (both switches turned-on)Open-loop control under only linear/heavy loadDynamic load applications like motorFuel cell modeled by a DC voltage source (battery)

Fig. 4 Conventional Z-source converter.

14


I. INTRODUCTIONProposed Z-source Topology

Proposed Z-source converter:Static load applications with fixed peak voltage/frequency (i.e., three-phase AC 208V and 60 Hz)Dynamic response of fuel cell considered System modeling/modified SVPWM implementation/closed-loop control system designGood performance under both linear load and nonlinear loadWide range of load, i.e., light load to a full load

Fig. 5 Proposed Z-source converter.15


II. Circuit Analysis/System Modeling/Space Vector PWM Implementation

Equivalent Circuit of the Fuel Cell (1)

Slow Dynamics of Reformer and Stacks

Voltage-current characteristic of a cell

16



Equivalent Circuit of the Fuel Cell (2)

Selected Equivalent Circuit Model of the Fuel CellDynamic modeling of reformer and stacks: R-C circuitVoltage-current characteristic of a cell: Region of Ohmic Polarization

Fig. 6 Equivalent circuit of the fuel cell.

17



Z-source converter configuration (1)

Z-source converter:a fuel cell, a diode, L-C impedance, a DC/AC inverter, a DSP controller, L/C filter, and a loadDiode: to prevent a reverse current that can damage the fuel cell

Fig. 7 Total system configuration with Z-source converter.

18



Z-source converter configuration (2)

Fig. 8 Total system diagram of a fuel cell based Z-source converter.

19



A. Circuit Analysis (1)

Two Operation Modes:Mode 1 (Fig. 9 (a)): Shoot-through switching mode: both switches in a leg are simultaneously turned-on

Mode 2 (Fig. 9 (b)): Non-shoot-through switching mode: basic space vectors (V0, V1, V2, V3, V4, V5, V6, V7)

(a) In the shoot-through switching mode. (b) In the non-shoot-through switching mode.

Fig. 9 Equivalent circuit of two operation modes.

20




Assumption: L1 = L2, C1 = C2

VC1 = VC2 and vL1 = vL2.

Case 1: one of shoot-through zero vectors (Fig. 9 (a)) vL1 = VC1, vf = 2VC1, and vi = 0.

Case 2: one of non-shoot-through switching vectors (Fig. 9 (b)) loop 1: vL1 = Vin – VC1 and loop 2: vi = VC1 - vL1 = 2VC1 - Vin,

where, Vin is the output voltage of the fuel cell.

( )in

z

a

z

a

inab

bC

T

z

CinbCaLLL V

TT

TT

VTT

TV

TVVTVT

dtvvVz

⋅−

−=

−=⇒=

−⋅+⋅=== ∫

21

10 1

0

11111

Average voltage of the inductors

where, Tz = Ta + Tb, Tz: switching period, Ta: total duration of shoot-through zero vectors over Tzand Tb: total duration of non-shoot-through switching vectors over Tz.

21




Average DC-link voltage: ( )

11

0

20Cin

ab

b

z

inCbaTiii VV

TTT

TVVTT

dtvvVz

=−

=−⋅+⋅

=== ∫

Peak DC-link voltage:

ininab

zinCDCp VKV

TTT

VVV ⋅=−

=−= 1_ 2

5.00,121

1<≤≥

⋅−=

−=

z

a

z

aab

z

TT

TTTT

TK

Peak phase voltage of inverter output

where, K is called a boost factor,

22_

_inDCp

paV

KMV

MV ⋅⋅=⋅= ⇒ Control Factors: M and K

where, M denotes the modulation index, T ( )MTT

TTTT

z

b

z

abaz =+=⇒+= 1

22



B. System Modeling (1)

Simplified system circuit model

Vdc

S1 S3 S5

S4 S6 S2

+

−

AB

C

Cf

Lf

ViAB

ViBC

iiA

iiB

iiC

+−+−

VLAB

VLBC

RL

iLA

iLB

iLC

L-C Output filter Load

N

Fig. 10 Simplified system circuit model of Z-source converter.

Note that a fuel cell, a diode (D), and impedance components (L1, 2 and C1, 2) are replaced with a DC-link source (Vdc).That is, Vdc is equal to VC1 or VC2.

23




Current/voltage equations from the L-C filter (KVL and KCL)

( )

( )

( )

−−=

−−=

−−=

LALCf

iCAf

LCA

LCLBf

iBCf

LBC

LBLAf

iABf

LAB

iiC

iCdt

dV

iiC

iCdt

dV

iiC

iCdt

dV

31

31

31

31

31

31

−= 110Twhere,

−

−

101

011

iLif

if

L

CCdtd ITIV

31

31

−=

+−=

+−=

+−=

iCAf

LCAf

iCA

iBCf

LBCf

iBC

iABf

LABf

iAB

VL

VLdt

di

VL

VLdt

di

VL

VLdt

di

11

11

11

if

Lf

i

LLdtd VVI 11

+−=

where, state variables: VL (= [VLAB VLBC VLCA]T ) and Ii (= [iiAB iiBC iiCA]T = [iiA-iiB iiB-iiC iiC-iiA]T ), control input (u): Vi (= [ViAB ViBC ViCA]T ), disturbance (d): IL (= [iLA iLB iLC]T ).

24




Coordinate Transformation: three variables ⇒ two variables

c

d

q

a

abcsdq fKf =0b

where, fdq0=[fd fq f0]T, fabc=[fa fb fc]T, and f denotes either a voltage or a current variable.

−−−

=2/12/12/1

232302/12/11

32

sK

Fig. 11 Relationship between “abc” reference frame and stationary “dq” reference frame.

25




State equations in the stationary dq reference frame

idqf

Ldqf

idq

Ldqidqf

idqf

Ldq

LLdtd

CCdtd

VVI

ITIV

11

31

31

+−=

−=

−== −

31

311

23][ 2,1,,

1columnrowsisidq KTKTwhere,

1

Continuous-time state space equation of the given plant model

)()()()( tttt EdBuAXX ++=&

, ,

44

2222

2222

013

10

×

××

××

−=

IL

IC

f

fA ,

242203

1

××

−= idq

fCT

E

24

22

2210

×

×

×

= I

L f

Bwhere,

== ×

Lq

LdLdq i

i12][Id[ ]

==

×iq

ididq V

V12

Vu14×

=

idq

Ldq

IV

X , ,

26



C. Space Vector PWM Implementation (1)

Conventional Space Vector PWM:

)600,()3/sin()3/cos(

32

01

32

)sin()cos(

)(

21

2211

0

1

2

0 0

1

21

211

°≤≤

⋅⋅⋅+

⋅⋅⋅=

⋅⋅⇒

⋅+⋅=⋅∴

++= ∫∫∫ ∫+

+

αππ

αα

where

VTVTVT

VTVTVT

VdtVdtVV

dcdcrefz

refz

T

TT

TT

T

T T

ref

zz

d axis

q axis

V1(100)

V2(110)

V3(010)

V4(011)

V5(001)

V6(101)

V0(000)

V7(111)

)0,3/2(

)3/1,3/1()3/1,3/1(−

)0,3/2(−

)3/1,3/1( −− )3/1,3/1( −

T1

T2

1

2

3

4

5

6

α

refV

)()3/sin(

)sin()3/sin(

)3/sin(

210

2

1

TTTT

aTT

aTT

z

z

z

+−=

⋅⋅=

−⋅⋅=

παπ

απ

=

dc

ref

V

Vawhere

32

,

6 active vectors: V1,V2, V3, V4, V5, V62 zero vectors: V0, V7

Fig. 12 Basic space vectors and switching patterns.27




Modified Space Vector PWM Implementation

(a) Sector 1 (0°~60°) (b) Sector 2 (60°~120°)

Fig. 13 Modified SVPWM implementation.*shoot-through zero vectors (T = Ta/3)

28




Switching time calculation at each sector

Note that when the shoot-through duration (T) is equal to zero, the switching time of each power switch for the Z-source converter is exactly the same as that for the conventional one.

29


III. Control System DesignEntire Control-loop Structure

Fig. 14 Total control system block diagram.

where, Three controllers1. Discrete-time Optimal Voltage Controller2. Discrete-time Sliding Mode Current Controller (DSMC) 3. Discrete-time PI DC-link Voltage Controller

One observer1. Asymptotic Observer for estimation of load currents

30


III. Control System DesignA. Discrete-time Sliding Mode Current Controller (1)

Block diagram of Current Controller

Fig. 15 Discrete-time current controller using DSMC.

31



Asymptotic observer design for estimation of load currents

State equation for asymptotic observer

( )LdqLdqLdq

LdqLdqidqf

idqf

L

LCCdt

d

VVI

VVTIV

−=

−⋅−=

ˆˆ3

13

1ˆ

1

1( )Ldq ˆ

Continuous-time state-space representation for asymptotic observer

( ) ( )aaLdqLdqLdq

aabbaaa

LLtt

tttt

XXVVId

XAXAXAX

−=−==

−+=ˆˆ)(ˆ)(ˆ

)()()(ˆ)(ˆ

11

&

= ×223

1 IC f

bA ,

−= idq

fa C

LTA

31 ,where, ]ˆ[ˆ

Ldqa VX = , ][ idqb IX = , ][ Ldqa VX = ,

L1 = a constant observer gain.

32



Discrete-time state-space model for asymptotic observer

( )

−==

−+=+

)()(ˆ)(ˆ)(ˆ)()()(ˆ)1(ˆ

1

****

kkLkk

kkkk

aaLdq

aabbaaa

XXId

XAXAXAX

∫ −= zza

T

aT de

0

)(** ττ AA A∫ −= zza

T

bT

b de0

)(* ττ AA A,where, A zaT

a e A=*,

Block diagram of asymptotic observer

Fig. 16 Block diagram of asymptotic observer.

33



Continuous-time state space equation

−==

++=

)()()()()(

)(ˆ)()()(

1

11

ttttt

tttt

refidq yyeXCy

dEBuAXX&

,

44

2222

2222

013

0

×

××

××

−=

IL

IC

f

fA

1

,24

22

2210

×

×

×

= I

L f

B , 24220

31

××

−= idq

fCT

E14×

=

idq

Ldq

IV

X , where,

=

10000100

1C[ ]

==

×iq

ididq V

V12

Vu , , [ ]

==

iq

ididq I

IIy1 ,

==

Lq

LdLdq i

iˆˆ

]ˆ[ˆ Id

Discrete-time state space equation

−==

++=+

)()()()()(

)(ˆ)()()1(

1

11

***

kkkkk

kkkk

refidq yyeXCy

dEuBXAX

∫ −= zz

T T de0

)(* ττ EE A, ∫ −= zz

T T de0

)(* ττ BB Awhere, zTe AA =* ,

34



Sliding-mode manifold

)()()()()( _11_11 kkkkk refref yXCyys −=−=

Equivalent control law

0)1()(ˆ)()(

)1()1()1(

_1*

1*

1*

1

_11

=+−++=

+−+=+

kkkk

kkk

ref

ref

ydECuBCXAC

yys

( ) ( ))(ˆ)()()( *1

*1

*1*1 kkkk idqeq dECXACIBCu −−=∴

−

Control limit

>

≤=

00

0

)(for )()(

)(for )()( ukk

ku

ukkk

eqeqeq

eqeq

uuu

uuu dcVu

32

0 =where,

35


III. Control System DesignB. Discrete-time Optimal Voltage Controller (1)

Block diagram of Voltage Controller

Robust Servomechanism Controller (RSC)

⇒ Servo compensator: internal model principle

⇒ Stabilizing compensator: optimal control

Fig. 17 Discrete-time optimal voltage controller using RSC.

36



Discrete-time state space plant model

)(ˆ)()()1( *** kkkk dEuBXAX ++=+

Augmented system model including DSMC

=++=+

)()()(ˆ)()()1( 1

kkkkkk

dd

ddd

XCydEuBXAX

( ) 1*1

* −= BCBB d( ) *

11*

1** ACBCBAA −

−=d

[ ]2222 0 ××= IdC )()( ,1 kk idqcmdIu =, ,

where, , , ( ) *1

1*1

** ECBCBEE −−=d ,

[ ]

==

Lq

LdLdqd V

VVy

37



Continuous-time servo-compensator

Sinusoidal Tracking/disturbance model

LdqLdqvdqvdqcc VVeeBηAη −=+= *,&

4422222

2222

00

×××

××

⋅−

=I

I

ici ω

A2422

220

××

×

=

IciB

121232244

44244

44441

000000

×××

××

××

=

c

c

c

c

AA

A

2123

2

1

×

=

c

c

c

cBBBB

, , , , where, A

Ac is the given tracking/disturbance poles, ωi (i = 1, 2, 3), ω1 = ω, ω2 = 5⋅ω, ω3 = 7⋅ω.

ω = 2πf [rad/sec], f = 60 Hz.

Discrete-time servo-compensator

)()()1( ** kkk vdqcc eBηAη +=+

∫ −= zzc

T

cT

c de0

)(* ττ BB Awhere, zcTc eAA =* ,

38



Augmented system combining both the plant and the servo-compensator

)(ˆ)(ˆˆ)(ˆ)(ˆˆ)1(ˆd_ref211 kkkkk yEdEuBXAX +++=+

Linear quadratic performance index

)()()(ˆ)(ˆ0

kkkkJk

TT∑ +=∞

=uuXQX εε

where, Q is a symmetrical positive-definite matrix and ε>0 is a small number

[ ] )()()()(

)(ˆ)( 21211 kkkk

kk ηKXKηX

KKXKu −−=

−=−=

Control input

=

)()(

)(ˆkk

kηX

X , ,

−

= **

0ˆcdc

d

ACBA

A

=

0ˆ dBB ,

=

0ˆ

1dE

E ,

= *2

0ˆcB

E , where,

)()( ,1 kk idqcmdIu = , , )()(ˆ kk LdqId = )()( *_ kk Ldqrefd Vy =

39


III. Control System DesignC. Discrete-time PI DC-link Voltage Controller (1)

Discrete PI DC-link voltage controller equation

( ) ( )

−+⋅⋅+−−⋅+−=

−=

)1()(2

)1()()1()(

C22*

kekeTKkekeKkTkT zip

C )()()( kkke VV

Block diagram of DC-link Voltage Controller

Fig. 18 Discrete-time PI controller to regulate the DC-link average voltage.

40


III. Control System DesignC. Discrete-time PI DC-link Voltage Controller (2)

Duration (Tcal = Ta/3) of shoot-through zero vectors

Desired capacitor voltage: VC1 = VC2 = 340 V

( )( )

( )( )

=⋅=

=⋅=⇒=∴

⋅−−

=−−

=⇒⋅−

−=

−=

VVwhereTT

VVwhereTTTT

TVV

VVVVTV

TT

TT

VTT

TV

ina

inaacal

zin

in

inC

inCain

z

a

z

a

inab

bC

300,,0.1050

130,,0.3814

3

680340

221

1

zmin_

zmax_

2

22

41


IV. Simulation ResultsSystem Parameters for Simulations

Table II. System Parameters for Simulations

Rr = 0.05 Ω, Cr = 42.8 mF, Rs = 0.02 Ω, Cs = 4.2 mFTime Constant of Reformer and Stacks

Tz = 1/(5.4 kHz) = 185.2 µsecSwitching/Sampling Period

VL, RMS = 208 V (L-L), f = 60 HzAC Output Voltage

Lf = 1000 µH, Cf = 200 µFInverter Output Filters

L1 = L2 = 200 µH, C1 = C2 = 1000 µFImpedance Components

Pout = 10 kVAOutput Rated Power

VC2 = 340 VDesired Average DC-link Voltage

Vin = 130 ∼ 300 VFuel Cell Output Voltage

where, ⇒ Heavy load (10 kW): the output voltage of fuel cell is 130 V ⇒ Light load (0.5 kW): that of fuel cell is 300 V ⇒ Linear load: a resistor and an inductor⇒ Nonlinear load: an three-phase inductor (2 mH), a three-phase diode bridge,

a DC-link capacitor (800 µF), and a resistor (7 Ω)⇒ Load change: 5 kW (250V) to 10 kW (130V), vice versa

42


IV. Simulation ResultsA. Linear Load (R)

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10

200

400

V in, V

C2 [V

] VinVC2

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-400

-200

0

200

400

V LAB, V

LBC

, VLC

A

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-50

0

50

Time [sec]

i LA, i

LB, i

LC

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10

200

400

V in, V

C2 [V

] VinVC2

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-400

-200

0

200

400

V LAB, V

LBC

, VLC

A

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1

-2

0

2

Time [sec]

i LA, i

LB, i

LC(a) 130V/10kW (Heavy load) (b) 300V/0.5kW (Light load)

Fig. 19 Simulation waveforms under a linear load.

(1: Fuel cell output voltage (Vin) and capacitor voltage (VC2),2: line to line voltages (VLAB, VLBC, VLCA),

3: load phase currents (iLA, iLB, iLC).)

43


IV. Simulation ResultsB. Nonlinear Load (Rectifier load)

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10

200

400

V in, V

C2 [V

] VinVC2

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-400

-200

0

200

400

V LAB, V

LBC

, VLC

A

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-50

0

50

Time [sec]

i LA, i

LB, i

LC [A

]

Fig. 20 Simulation waveforms under a nonlinear load (130V/7Ω).



44


IV. Simulation ResultsC. Load Change

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10

5

10

15

P [k

W]

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10

200

400

V in, V

C2 [V

]

VinVC2

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-400-200

0200400

V LAB, V

LBC

, VLC

A

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-50

0

50

Time [sec]

i LA, i

LB, i

LC

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10

5

10

15

P [k

W]

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10

200

400

V in, V

C2 [V

]

VinVC2

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-400-200

0200400

V LAB, V

LBC

, VLC

A

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-50

0

50

Time [sec]

i LA, i

LB, i

LC(a) Load increase (b) Load decrease

Fig. 21 Simulation waveforms under a load change at 70msec.



45


IV. Simulation Resultsd. Six gating signals and estimated load current

0.0167 0.0168 0.0168 0.0169 0.0169 0.017 0.017 0.0170

1

2PWM Signals for Six Power Switches

S1

0.0167 0.0168 0.0168 0.0169 0.0169 0.017 0.017 0.0170

1

2

S4

0.0167 0.0168 0.0168 0.0169 0.0169 0.017 0.017 0.0170

1

2

S3

0.0167 0.0168 0.0168 0.0169 0.0169 0.017 0.017 0.0170

1

2

S6

0.0167 0.0168 0.0168 0.0169 0.0169 0.017 0.017 0.0170

1

2

S5

0.0167 0.0168 0.0168 0.0169 0.0169 0.017 0.017 0.0170

1

2

S2

Time [s ec]

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-50

0

50

Load current (iLA )and estimated load current (i*LA)

i LA a

nd i*

LA [A

]

iLAi*LA

0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-50

0

50

i LA a

nd i*

LA [A

]

iLAi*LA

(a) Gating signals for six power switches. (b) Estimated load currents under heavy load.

Fig. 22 Six gating signals and estimated load current.

(S1 and S4: phase A, S3 and S6: phase B, S5 and S2: phase C,Upper switches: S1, S3, and S5,Lower switches: S4, S6, and S2.)

46


VI. Conclusions

L-C Impedance components instead of a DC to DC boost converter

Inexpensive, reliable, small-sized, and light-weighted

Slow dynamic response of fuel cell

Discrete-time state-space system model

Modified PWM Technique using Shoot-through zero vectors

Feedback controller design under both linear and nonlinear load

Good performance: Nearly zero steady-state voltage (RMS) error Low Total Harmonic Distortion (THD)Fast/no-overshoot current responseGood voltage regulation

47

modeling and control of a fuel cell based z-source

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