Research ArticleDynamic Sliding Mode Evolution PWM Controller for a NovelHigh-Gain Interleaved DC-DC Converter in PV System

Taizhou Bei Ping Wang Liu Yang and Zhe Zhou

Key Laboratory of Smart Grid of Ministry of Education in Tianjin University at Tianjin Tianjin 300072 China

Correspondence should be addressed to Taizhou Bei 396913440163com

Received 5 January 2014 Revised 17 April 2014 Accepted 23 April 2014 Published 21 May 2014

Academic Editor Hongjie Jia

Copyright copy 2014 Taizhou Bei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Considering the disadvantages of the traditional high-gain DC-DC converter such as big size high voltage stress of switchesand large input current ripple a novel high-gain interleaved boost converter with coupled-inductor and switched-capacitor wasproposed correspondingly and the operation principle together with the steady-state analysis of this converter was also describedBesides a new control approach-dynamic sliding mode evolution PWM controller (DSME PWM) for the novel topologicalconverter based on both dynamic evolution and sliding mode control was also presented From the simulation results andexperimental validation the proposed converter can fulfill high-gain boost low ripple of both the input current and the outputvoltage Furthermore MPPT technique can be also achieved in a short time by simulation The efficiency and stability of theconverter proposed in this paper can be improved

1 Introduction

Photovoltaic (PV) generation has an increasing influence onalleviating the energy crisis and reducing the environmen-tal pollution Researchers around the world have devotedthemselves to the performance of PV system such as stabilityefficiency and the cost as a consequence (as discussed by [1ndash3]) Generally the outputting voltage of the PV arrays rangesfrom 30V to 60V High-gain and high-efficiency DC-DCconverters are required to boost the low PV voltage to a highvoltage such as 380V for the full-bridge inverter or 760Vfor the half-bridge inverter in the 220V AC grid-connectedpower system

The good performances of any high-gain DC-DC con-verter in PV system greatly depend on the employed topologyand the corresponding control strategy So when designingthis kind of converters one should pay more attention tothese two aspects and ensure that the designed convertercan minimize the current ripple and reduce the numbers ofthe electrolytic capacitors which can reduce the circuit costand easily achieve MPPT algorithms (as discussed by [4 5])Besides power losses that existed in the converters in high-output-voltage applications should be decreased to improve

system efficiency (as discussed by [6])This paper will devoteitself to a novel scheme that can meet the requirement above

The traditional boost converters can achieve any high-gain transform by adjusting the duty cycle theoreticallyhowever the actual voltage gain is often limited by theparasitic parameters of inductances capacitances and switchdevices (as discussed by [7]) Furthermore in high-gainapplications as the duty cycle increases the input currentripple voltage or current stress of the devices and theswitching loss are all increasing as well as the diodesrsquo reverserecovery loss decreasing the efficiency of the convertersMeanwhile electromagnetic interference (EMI) is furthersevere due to the excess of 119889V119889119905 Boost converter canenhance its voltage gain via adding coupled inductances (asdiscussed by [8ndash10]) However voltage spike will occur whenturning off the switches due to the leakage inductances whichcan result in voltage stress increasing efficiency lowering andsevere EMI The aforementioned problems can be relievedby active or passive clamping methods Undoubtedly it willmake the main circuit control strategy and drive modemore complicated (as discussed by [11 12]) Another wayto improve the voltage gain of conversion is to employnonisolated cascade boost converters (as discussed by [13])

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 658674 11 pageshttpdxdoiorg1011552014658674

2 Journal of Applied Mathematics

However there are many disadvantages such as low relia-bility high costs and complexity of main circuit as well ascontrol strategy and drive methods besides the outputtingdiodesrsquo reverse recovery problem is still serious in high-gain applications Isolated boost converters can also achievehigh-gain transform by designing the ratio of transformers(as discussed by [12 14]) but the leakage inductances willcause many problems such as voltage overshoot of switchesbig voltage stress and EMI problems Meanwhile the loss oftransformers will result in lower efficiency higher costs andcomplexity of main circuit control strategy drive methodsand isolation designThe topology-combining boost convert-ers and fly-back converters can be employed in the high-step-up transform (as discussed by [15 16]) whose topology andcontrol method are fairly simple however the voltage balanceof the capacitors exists in this systemdue to series connection

The reverse-recovery loss of the output diode is one ofthe main reasons why the efficiency is limited in the highstep-up applications When the boost converter works ondiscontinuous current mode (DCM) or critical current mode(CRM) the reverse-recovery loss can be reduced but itwill increase both current stress and input current rippleTherefore it is not suitable for the high-power applicationsdue to the bulky filter circuit

Interleaved boost converters can not only develop bothpower level and power density but also average the thermalstress distribution and reduce the input current ripple (asdiscussed by [17ndash19]) which can even be eliminated whenthe duty cycle is equal to 05 theoretically However there aresome weaknesses to apply interleaved boost converter intoPV grid-connected system directly (as discussed by [7 18])The step-up ratio is not increased efficiently comparing withthe traditional single-stage boost converter as well as thevoltage stress of switches Besides it is very sensitive forordinary interleaved boost converters to mismatch the dutycycle

In allusion to all the problems mentioned above a novelhigh-gain interleaved converter with coupled inductanceand switched capacitor (hereinafter referred to as the newconverter) is proposed High-gain transform can be real-ized by designing the main circuit parameters reasonablyFurthermore the topology as well as its control method isrelatively simple Compared with conventional interleavedboost converters the new one can boost voltage on thegreat scale while reducing the voltage stress of the switchdevices significantly under the same duty cycle Moreoverthe volume of magnetic elements and the current rippleof the inductance can be reduced by introducing coupledinductance (as discussed by [8 9 14])

Control strategy of the proposed converter can easilyachieve the MPPT algorithms and minimize the currentripple to reduce the coupled inductance losses and to improvesystem efficiency and stability Moreover less computationshould also be under the consideration to ease the com-putational complexity It is worth noting that in the liter-ature some examples of high-gain DC-DC converters arepresented but they only concern the topologies while fewpapers are devoted to the controller design for such kind ofconverter although not referring to PV applications

Slidingmode control (SMC) has a high robustness againstplant uncertainty and external disturbances including thosedue to the environmental conditions in PV applicationsMany SM-based MPPT algorithms operated in differentapplications or different topologies have been proposed inpublished papers but most of those mainly focus on theSMC used in conventional boost converter Neverthelesssuch an approach has never been used in analyzing a class ofinterleaved boost converters for PV applications controlledby the SM-MPPT technique

The controller of the proposed new converter is requiredto extract the maximum energy from the PV panels andoutput the stable expectant voltage without errors Dynamicevolution control (DEC) as a new control technique hasbeen utilized in power electronics converters (not in PVapplications) DEC can ensure that the error state goes tozero in increase of time thereby improving the dynamiccharacteristic of system

A new control approach based on both SM-MPPT andDEC is proposed in this paper Since the outputting voltageand current of PV cell together with the outputting voltageof converter are involved in calculation of duty cycle allthe variation above can be compensated in the dynamicevolution

This paper is organized as follows Section 2 describes thecharacteristic of PV cells Section 3 gives the working princi-ple and performance analysis of the new converter Section 4is devoted to the controller design on basis of SM-MPPTand DEC Sections 5 and 6 afford the simulation results andexperimental validation respectively In the conclusion themain results are finally summarized

2 Characteristic of PV Cells

Formulas of the outputting current and power for the PV cellscan be seen from (1) (as discussed by [3]) and Figure 1 showsthe output characteristics of PV cells

119894pv = 119868ph minus 119868sat [exp(119902119881pv

119860119870119879) minus 1]

119875pv = 119881pv119868ph minus 119881pv119868sat [exp(119902119881pv

119860119870119879) minus 1]

(1)

where 119894pv 119881pv are the outputting current and voltage of PVcells 119868ph is the photocurrent of a single solar module 119868sat isthe reverse saturation current 119870 is Boltzmann constant 119902is electric charge 119879 is operating temperature and 119860 is P-Njunction ideal factor

When the output of the equivalent circuit of PV cellsis shorted if neglecting the reverse leakage current of thediodes we canmake an approximation that 119868ph asymp 119868sc (119868sc is theshort-cut current) which will be used in part119881 for analyzingthe inputting current in dynamic mode

Journal of Applied Mathematics 3

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

140

160

180

V (V)

P(W

)(Temp = 25∘C)

S1 = 600 (Wm2)S2 = 800 (Wm2)S3 = 1000 (Wm2)

S1

S2

S3

V-P characteristics

(a) V-P curve when temp = 25∘C different irradiance

0 10 20 30 40 50 600

50

100

150

200

250

P(W

)

V (V)

T1 = 15∘CT2 = 25∘CT3 = 35∘C

(Irradiance = 1000 Wm2)

T1T2T3

V-P characteristics

(b) V-P curve when irradiance = 1 kWm2 different temp

Figure 1 The output characteristics of PV cells

3 Principle and Steady-State Analysis

31 Principle of the New Converter The new converter withPV panels is shown in Figure 2 Capacitors 119862

1 1198622and diode

VD1 are employed to further enlarge the boost voltage underthe same duty cycle compared with any other topology

The inductances 1198711and 119871

2are closely coupled and with

the same winding orientation The coupled inductances canbe represented as a mutual inductance 119872 two equivalentleakage inductances 1198711015840

1

and 1198711015840

2

The equivalent circuit isshown in Figure 3

The relationships of the inductances are related by thefollowing equations

1198711015840

1

= 1198711minus119872

1198711015840

2

= 1198712minus119872

119872 = 119896radic11987111198712

(2)

where 119896 is coupled coefficientIn order to simplify the circuit analysis of the proposed

converter several necessary assumptions should be made asfollows

(1) All components including power switches magneticcomponents and diodes are ideal the on-state resis-tance 119877DS(ON) and all parasitic capacitances of theswitches are neglected as are the forward voltagedrops of diodes VD

1simVD3

(2) The ESR of capacitors 1198620sim 1198622and the parasitic

resistance of coupled inductances are neglected(3) 1198711= 1198712= 119871

C1

C2

VD1

VD2

VD3

S1S2 C0 R

+

+

minus

L2

ipv

PV panels

lowast

lowast

+

minus

+minus

L1

VoVpv

Figure 2 The new converter

C1

C2

VD1

VD2

VD3

S1S2C0 R

+

minus

+minusM

iL1iin

+minusiL2

L9984001

L9984002

VoVpv

Figure 3 Equivalent circuit of the new converter

(4) 1198781and 1198782work by turn and both are driven by PWM

signals

Referring to the equivalent circuits for four differentswitching states of a switching period shown in Figure 4 andthe waveforms in Figure 5 the operation of the converter canbe explained as follows

4 Journal of Applied Mathematics

C1

C2

VD1

VD2

VD3

S1S2 C0 R

M

123

+

minus

L9984001

L9984002

+minus

+minus

iL1iin

iL2

C2

C1

VoVpv

(a) Mode 1 1198781-on 1198782-off VD

1-off VD

2-on VD

3-on

C1

C2

VD1

VD2

VD3

S1S2

C0 R

M

1 2 +

minus

L9984001

L9984002

+minus

+minus

iL1iin

iL2

C2

C1

VoVpv

(b) Mode 2 1198781-on 1198782-off VD

1-off VD

2-off VD

3-off

C1

C2

VD1

VD2

VD3

S1S2 C0 R

M

12 3 +

minus

L9984001

L9984002

+minus

+minus

iL1

iL2

C2

C1

VoVpv

iin

(c) Mode 3 1198781-off 1198782-on VD

1-on VD

2-off VD

3-on

C1

C2

VD1

VD2

VD3

S1S2 C0 R

M

12

+

minus

L9984001

L9984002

+minus

+minus

iL1

iL2

C2

C1

VoVpv

iin

(d) Mode 4 1198781-off 1198782-on VD

1-off VD

2-off VD

3-off

Figure 4 Operation states and current paths of the converter

t

t

t

t

t

t

S1

S2

vc1

vc2

1 2 3 4

iL1

iL2

iin

iL1 iL2

iin = iL1 + iL2

t1 t2 t3t4t0

12057211T1205721T 12057221T 1205722T

Figure 5 Main waveforms of the new converter

Mode 1 (Figure 4(a)) At time 1199050 1198781is on while 119878

2is off the

current in the inductance 11987110158401

starts to rise linearly while 11987110158402

continues to discharge (the current in 11987110158402

was acquired in thelast switching period) Capacitor 119862

2discharges leading to its

voltage falling But the voltage of capacitor1198621rises because119862

1

is charged besides capacitor 1198620is charged by input voltage

and inductor in this mode The rates of change of 1198941198711

and 1198941198712

are approximately given by

1198891198941198711

119889119905=119881119900minus V1198621

11987110158401

1198891198941198712

119889119905= minus

119881119900minus V1198622

11987110158402

(3)

Mode 2 (Figure 4(b)) At time 1199051 1198941198712falls to zero 119894

1198711continues

to rise linearly 119877 is charged by capacitor 1198620 The rate of

change of 1198941198711

is

1198891198941198711

119889119905=119881pv

1198711

(4)

where 1198711= 1198711015840

1

+119872

Mode 3 (Figure 4(c)) At time 1199052 1198781is off while 119878

2is on 119894

1198712

starts to rise linearly from zero and 1198711starts to discharge

Besides 1198621discharges too while 119862

2starts to charge More-

over1198620is charged by input voltage and inductorThe rates of

change of 1198941198711

and 1198941198712

are approximately given by1198891198941198711

119889119905= minus

119881119900minus V1198621

11987110158401

(5)

1198891198941198712

119889119905=119881119900minus V1198622

11987110158402

(6)

V1198621+ V1198622

= 119881119900 (7)

Mode 4 (Figure 4(d)) At time 1199053 1198781is offwhile 119878

2is on 119894

1198711falls

to zero 1198941198712continues to rise linearly119877 is charged by capacitor

1198620 The rate of change of 119894

1198712is

1198891198941198712

119889119905=119881pv

1198712

(8)

where 1198712= 1198711015840

2

+119872The switching period will be repeated when 119878

1is on again

at time 1199054

Journal of Applied Mathematics 5

32 Steady-State Analysis Assume that the current wave-forms shown in Figure 5 have reached a steady state Fromthe waveform 119894

1198711as shown in Figure 5 it can be found that

119881119900minus V1198621

11987110158401

12057211119879 +

119881pv

1198711

(1205721minus 12057211) 119879 =

119881119900minus V1198621

11987110158401

(1205722minus 12057221) 119879

(9)

From the waveform 1198941198712

as shown in Figure 5 it can be foundthat

119881119900minus V1198622

11987110158402

(1205722minus 12057221) 119879 +

119881pv

1198712

12057221119879 =

119881119900minus V1198622

11987110158402

12057211119879 (10)

Also from Figure 5 the average values of 1198941198711

and 1198941198712 denoted

as 1198681and 1198682 respectively are found as

1198681=1

1198791

2

119881119900minus V1198621

11987110158401

sdot (12057211119879)2

+1

2

119881119900minus V1198621

11987110158401

sdot [(1205722minus 12057221) 119879]2

+1

2(1205721minus 12057211) 119879

sdot [2 (119881119900minus V1198621)

11987110158401

sdot 12057211119879 +

119881pv

1198711

(1205721minus 12057211) 119879]

1198682=1

1198791

2

119881119900minus V1198622

11987110158402

sdot (12057211119879)2

+1

2

119881119900minus V1198622

11987110158402

sdot [(1205722minus 12057221) 119879]2

+1

212057221119879

sdot [2 (119881119900minus V1198622)

11987110158402

sdot (1205722minus 12057221) 119879 +

119881pv

1198712

12057221119879]

(11)

Assume that the converter is lossless that is the input power119875119894is equal to the output power 119875

119900 Consider

119881pv (1198681 + 1198682) =1198812

119900

119877 (12)

where 119877 is the load resistanceIn principle (7) and (9)ndash(12) can be solved to find 119881

119900

However to simplify the calculation it is assumed that 1205721=

120572 1205722= 1 minus 120572 119871

1= 1198712= 119871 and 119871

1015840

1

= 1198711015840

2

= (1 minus 119896)119871According to some algebraic manipulations and solving aquadratic equation derived from (12) the approximate valuesof 1198811198621 1198811198622 and 119881

119900can be referred as follows

1198811198621

=3

1 minus 120572119881pv =

3

4119881119900

1198811198622

=1

1 minus 120572119881pv =

1

4119881119900

(13)

119881119900=

4

1 minus 120572119881pv (14)

The effect of R on the calculation is very little thus it isneglected when getting the approximate values as shown in(13) and (14)

A careful study of the waveforms shown in Figure 5 willreveal the following interesting facts

C1

C2

VD1

VD2

VD3

S1S2 C0 R Vo

++

minusVpv

L1

L2

+minus

+minus

Vm1

ipv

120575(Vo Vpv ipv ΔVpv Δipv )

PVpanels

PWM

DSME PWM controller

PWM 2 PWM 1

lowast

lowast

Figure 6 The control flow chart of DSME PWM controller

(1) As far as the input current 119894in is concerned theconverter appears to operate in CCM (because 119894inis continuous) Thus the peak current stress of theinductances and the input current ripple can bemaintained relatively low

(2) However since 1198941198711and 1198941198712are discontinuous the new

converter is actually operating in DCM Also sincethe rectifier diodes VD

1simVD3turn off before 119878

11198782

turns on the reverse-recovery loss of the rectifiers iseliminated

(3) Besides the current of inductance 11989411987111198941198712

has fallento zero before 119878

11198782turns on ZCS soft switching

operation during the whole switching transition isachieved

Furthermore when considering the parasitic resistancesof switches and capacitances (119903

119904and 119903119888) the switching loss due

to capacitances in the course of charging can be defined as (15)in an independent converter which contains 119899 capacitances

119882119904=119899119862

2sdot1 + 119890minus120572119879120591

1 minus 119890minus120572119879120591(Δ119881119900)2

(15)

where 119862 is the value of capacitances and 120591 = (119903119904+ 119903119888)119862

From (15) 119882119904do exist even under the ideal conditions

mainly depending on Δ119881119900 Thus in order to reduce 119882

119904

the difference between the maximum and the minimum ofvoltage in the course of charging for all capacitances shouldbe decreased

4 DSME PWM Controller

The original intention of the DSME control is to achieveMPPT as well as high-gain boost also to reduce the errorstate by forcing the error state to follow the specific pathwhich ensures the error state goes to zero in increase oftime (as discussed by [19 20]) The control flow chart ofdynamic sliding mode evolution PWM Controller can beseen in Figure 6

The modulating signal of PWM generation 1198811198981

is pro-duced by the formula 120575 combing the sliding mode theory

6 Journal of Applied Mathematics

with the dynamic evolution algorithm The specific designprocess of the DSME controller can be described as follows

41 MPPT by Sliding Mode Control When the PV cells runat the point of maximum power

119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv= 0 (16)

In SMC the state space is divided into 119904 gt 0 and 119904 lt 0 bysliding manifold (119904 = 0) The movement on 119904 = 0 can becalled sliding mode dynamics Considering 119889119875pv119889119881pv = 0the sliding manifold can be chosen as (as discussed by [2122])

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv (17)

Reconsidering the mathematical model of PV cell

119894pv = 119868ph minus 119868sat [exp(119902119881pv

119860119870119879) minus 1] (18)

Directly substituting 119894pv from (18) into (17)

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv

= minus119868sat [exp (119902

119860119870119879119881pv) sdot

119902

119860119870119879119881pv]

+ 119868ph minus 119868sat [exp(119902

119860119870119879119881pv) minus 1]

= minus119868sat (119902

119860119870119879119881pv + 1) sdot exp(

119902

119860119870119879119881pv) + (119868ph + 119868sat)

(19)

The differential form of 119904 is

119904 = minus119868sat119902

119860119870119879sdot119889119881pv

119889119905exp(

119902

119860119870119879) minus 119868sat (

119902119881pv

119860119870119879+ 1)

sdot exp(119902119881pv

119860119870119879)

119902

119860119870119879

119889119881pv

119889119905

= minus119868sat119902

119860119870119879sdot119889119881pv

119889119905exp(

119902119881pv

119860119870119879)(

119902119881pv

119860119870119879+ 2)

(20)

According to 119904 119904 lt 0 if 119904 gt 0 then 119904 lt 0 119889119881pv119889119905 gt 0119881pv willincrease when 119904 tends to sliding manifold and if 119904 lt 0 then119904 gt 0 119889119881pv119889119905 lt 0 119881pv will decrease when 119904 tends to slidingmanifoldThe dynamic accommodation of119881pv can be seen inFigure 7

Considering that the new converter is composed of twoparalleled boost converters the purpose of the additionalswitched capacitor and coupled inductance is to increase thevoltage gain Thus the drive signals for the switches can bedesigned respectively From the state space model of thesingle boost converter when the switch is on 119881pv increases

and when the switch is off 119881pv decreases (as discussed by[23])

Rearranging (17)

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv= 119881pv (

119894pv

119881pv+119889119894pv

119889119881pv) (21)

The differential form approximates to

Δ119894pv

Δ119881pvasymp119889119894pv

119889119881pv (22)

Thus

119904 = 119881pv (119894pv

119881pv+Δ119894pv

Δ119881pv) (23)

Theoretically in the case of low requirement both (17) and(24) can be chosen as the sliding surface

119904 = Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv (24)

The corresponding control law is

119906 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)] (25)

42 Evolution Path Selection The principle of determiningthe evolution path is to ensure that the error state can goto zero at any increase of time (as discussed by [24 25]) Inthis research the evolution path is an exponential function asshown in Figure 8

With this exponential evolution path the value of thedynamic characteristic of the converter will decrease expo-nentially to zero by

119867 = 1198670ℓminus120582119905

(26)

where 119867 is the dynamic characteristic of system 1198670is the

initial value of 119867 and 120582 is a dynamic evolution factor Thederivative of119867 is

119889119867

119889119905= minus120582119867

0ℓminus120582119905

= minus120582119867

119889119867

119889119905+ 120582119867 = 0

(27)

43 Analysis of Duty Cycle In order to synthesize the controllaw of the dynamic evolution controller the dynamic equa-tion of converter system has to be established and analyzedOn basis of the state space averagemodel when the converterworks on DCM the voltage and current dynamics of theconverter are given by

119881pv = 119871119889119894in119889119905

+119881119900

4(1 minus 120572) (28)

where 119871 is the inductance 119881pv is the input voltage 119894in is theinput current119881

119900is the output voltage and 120572 is the duty cycle

respectively

Journal of Applied Mathematics 7

Sliding manifold

Trajectory s chattersalong the manifoldand moves toward

VO

pv

120577 = minusΔs gt 0

s = 0

120577 = +Δ

s lt 0

Ppv

s = 0

Figure 7 The dynamic accommodation of 119881pv

119894in asymp 119894pv rearranging (28) the output voltage of convertercan be written as

119881pv = 119871119889119894pv

119889119905+119881119900

4(1 minus 120572) (29)

119881119900= 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (30)

Define a linear voltage error function as shown below

119864 = 119896119890V = 119896 (119881ref minus 119881119900) (31)

where 119896 gt 0Substituting (27) into (31) yields

119896119889119890V

119889119905+ 120582119896119890V = 0 (32)

Combining (31) with (32)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 119881

119900 (33)

Directly substituting the converter output voltage 119881119900from

(30) into (33)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (34)

The obtained duty cycle formula is given by

120572 =119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905 (35)

Formula (35) forces the state error function 119864 to satisfy thedynamic evolution function (27) Consequently the stateerror function 119864 is forced to make evolution by followingequation (35) and decrease to zero (119864 = 0) with a decreaserate 120582 The outputting voltage of converter converges to theconverters steady state

119881119900= 119881ref (36)

From the synthesis procedure it is clear that the dynamicevolution controller works on the full nonlinear system and

H

H0

0 t

H = H0eminus120582t

Figure 8 Exponential evolution path

does not need any linearization or simplification on thesystem model at all and it is interesting to note that thecontrol law in (35) consists of four distinct parts The firstpart is the feed-forward term which is calculated based onthe duty cycle at the previous sampling instant This termcompensates for variations in the input voltages The secondand third terms consist of proportional and derivative termsof the perturbations in the output voltage respectively Thelast term consists of the derivative terms of the inductancecurrent Since the inputting and outputting voltages andinductance current are involved in calculation of duty cycleit can compensate all the variation of them in the dynamicevolution

It should be noted that both119881pv and 119894pv are not the voltageand current at the MPP in (35) Consequently we expect thatthe controller can achieve MPPT and high-gain boost thusconsidering the logic relationship of control signals the finalcontrol law can be described as follows combing (25) with(35)

120575 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)]

sdot (119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905)

(37)

44 PWM Generation The PWM signals are generated bycomparing a control signal with a constant peak repetitivetriangle signal (119881cs) The frequency of the repetitive trianglesignal establishes the constant switching frequency

Figure 9 shows the PWM signals generation techniquePWM1 is produced when the control signal 119881

1198981is greater

than 119881119888119904while PWM2 is produced when 119881

1198981is less than 119881

119888119904

The values of the desired duty cycle 120572 can be gotten from(37) PWM signals are generated by (39)

1198811198981

= 120575 (38)

8 Journal of Applied Mathematics

PWM1 = 1 1198811198981

gt 119881119888119904

0 1198811198981

lt 119881119888119904

PWM2 = 1 1198811198981

lt 119881119888119904

0 1198811198981

gt 119881119888119904

(39)

5 Simulation

Some simulation has been done in order to verify theperformance of the topology and the main parameters canbe referred to in Table 1

The portion of the simulation results is given in Figure 10Figure 10(a) shows the driving signals for 119878

1and 1198782 which

are generated by the control law 120575 The outputting voltageand power of PV cell in process of MPPT can be seen inFigure 10(b) From the simulation onemay get the conclusionthat the DSME PWM controller can ensure that the PV cellruns at the maximum power point in a short time Althoughthe overshoot with a little range occurs in the process ofpower adjustment it will be eliminated soon and the stableoutputting power of 260W is achieved During the dynamicadjustment there is a stable and fleet performance withoutoscillation

In Figure 10(c) the dynamic response of the novel topol-ogy is the curve labeled overall trend This curse shows thatthe converter can raise the voltage from 36V to 400V steadilywithout a long time The curve labeled partial enlargementshows that the outputting voltage has a low voltage ripplewhen it is in stable state Considering the whole PV grid-connected system a DC-link high voltage with low rippleis convenient to choice and maintaining of the DC-linkcapacitor

As mentioned before the converter can run under fourmodes in every switching period The tendencies of 119894

1198711and

1198941198712as well as 119894in in every mode have been discussed in part III

and it is also easy to get the conclusion about 119894in = 1198941198711+ 1198941198712

based on Kirchhoff rsquos current law so the tendency of 119894in canbe obtained by the sum of 119894

1198711and 1198941198712 The first and second

one of Figure 10(d) describe the stable current of 1198941198711

and 1198941198712

and 119894in respectively over the same period in stable modeFigure 10(d) shows that the current 119894in has little ripple whichcan reduce the coupled inductance losses and improve systemefficiency

Figure 10(e) shows the voltage waves of switched capac-itors 119862

1and 119862

2 The voltage of 119862

1is three times as much as

the one of 1198622 close to three quarters of 119881

119900 this simulation

verifies the formula (13)

6 Experiments

Experiments have been finished in our laboratory just inorder to verify the validity of theoretical analysis above andthe performance of the new converter proposed in this paper

Experimental waveforms of the new converter are shownin Figure 11 The driving signals of 119878

1and 119878

2from DSP on

basis of the algorithm in this paper are given in Figure 11(a)Obviously the new converter can raise the voltage from36V to 400V when the duty cycle equals 064 (calculated

Table 1 The main parameters

Parameters Values119881ocV 443119868scA 77119881mp at MPPV 36119868mp at MPPA 72Coupling coefficient 091198711

and 1198712

mH 041198621

120583F 101198622

120583F 1001198620

120583F 22119877kΩ 1Switching frequencykHz 50

PWM1PWM2

PWM120575(Vo Vpv ipv ΔVpv Δipv )

0

Vo

Vcsipv

Vpv Vm1

Figure 9 PWM generator

from Figure 11(a)) The output voltage wave (after 10 timesattenuation) is shown in Figure 11(b) Moreover the rippleof the output voltage shown in Figure 11(b) can be limitedin 8V which can decrease the loss of switched capacitorsas shown in (15) Figure 11(c) shows the wave of the inputcurrent The little input current ripple can reduce both theloss and the volume of the coupled inductance Figures 11(d)and 11(e) show the voltage waveform of switched capacitors1198621and 119862

2 respectively (after 10 times attenuation) The

voltage of 1198621 about 300Vsim306V is three times as much as

the one of 1198622 about 92Vsim100V which is consistent with

both the analysis and the simulation results above besidesthe difference between the maximum and the minimum ofvoltage in the course of charging for each capacitance can belimited to a fairly small range which can decrease the loss ofswitched capacitors

Burrs existing in experimental waveform mainly stemfrom the effect of EMI in the experiment

7 Conclusion

This paper proposed a new high-gain interleaved converterwith coupled inductances and switched capacitors networkwhich are suited for PV application After explaining its oper-ating principle and analyzing its steady-state performancethen a control strategy for it on basis of SM-MPPT andDEC theory was designed Both simulation analysis andexperimental results showed that the proposed system hadthe following good performance

(1) Achieving MPPT and producing a high-gain voltagein an extremely short time

(2) Decreasing the ripple of the outputting voltage as wellas the inputting current and improving power density

Journal of Applied Mathematics 9

For S

1

ForS

2

(a) Driving signals for 1198781amp 1198782

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

Time (s)

Volta

ge (V

) and

pow

er (W

)

The outputting voltage and power of PV cells under MPPT

Vpv outputting voltageP outputting powerpv

(b) The output voltage and power of PV cell in process of MPPT

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

350

400

450The outputting voltage of the new topological converter

Time (s)

Volta

ge (V

)

399

3995

400

4005

401

Partial enlargement of outtputing voltage

Overall trendPartial enlargement

(c) The output voltage wave

0

5

Curr

ent (

A) The Lofwavecurrent 1 and L2 in stable mode

001 0015 002 0025 00305

1015

Time (s)

Curr

ent (

A)

iL1

iL2

The current wave of iin in the dynamic mode

65

7

75

Curr

ent (

A) The current wave of iin in stable mode

(d) The current of 1198711and 1198712and 119894in

0 001 002 003 004 005 006 007 008 009 01minus100

minus50

0

50

100

150

200

250

300

350

400

Time (s)

Volta

ge (V

)

The voltage of and

Voltage of C1

Voltage of C2

C1 C2

(e) The voltage wave of switched capacitor 1198621and 119862

2

Figure 10 Simulation results

10 Journal of Applied Mathematics

(a) Driving signals for 1198781and 1198782from DSP (b) The outputting voltage wave (X10)

(c) The wave of input current (d) The voltage wave of switched capacitor 1198621(X10)

(e) The voltage wave of switched capacitor 1198622(X10)

Figure 11 Experimental validation

(3) Reducing the volume of magnetic components andcutting the cost of system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education of China (no20120032110070)

References

[1] C Zhang and X He ldquoMaximum power point tracking by usingasymmetric fuzzy control combined with PID for photovoltaic

energy generation systemrdquo Transactions of China Electrotechni-cal Society vol 20 no 10 pp 72ndash75 2005

[2] Z Liao and X Ruan ldquoControl strategy for bi-directionalDCDC converter of a novel stand-alone photovoltaic powersystemrdquo Transactions of China Electrotechnical Society vol 23no 1 pp 97ndash103 2008

[3] M Dong J Yang K Peng and A Luo ldquoZero average incre-mental conductance maximum power point tracking controlfor photovoltaic systemrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 30 no 21 pp 48ndash53 2010

[4] J M Enrique J M Andujar and M A Bohorquez ldquoAreliable fast and low cost maximum power point tracker forphotovoltaic applicationsrdquo Solar Energy vol 84 pp 79ndash89 2010

[5] L Xie RGong and J Li ldquoAnalysis of the dynamical characteris-tics of the Interleaved boost converter inmaximumpower pointtracking for photovoltaic powerrdquo in Proceedings of the ChineseSociety for Electrical Engineering (CSEE rsquo13) vol 33 no 6 pp39ndash43 2013

Journal of Applied Mathematics 11

[6] W Li and X He ldquoReview of nonisolated high-step-up DCDCconverters in photovoltaic grid-connected applicationsrdquo IEEETransactions on Industrial Electronics vol 58 no 4 pp 1239ndash1250 2011

[7] Q LuoH Yan S Zhi C Zou and L Zhou ldquoAn interleaved highstep-up zero current transition boost converterrdquo in Proceedingsof the Chinese Society for Electrical Engineering (CSEE rsquo13) vol33 no 12 pp 18ndash21 2013

[8] W Huang and B Lehman ldquoMitigation and utilization of theinductor coupling effect in interleaved multiphase dcdc con-vertersrdquo in Proceedings of the IEEE Energy Conversion Congressand Exposition (ECCE rsquo13) pp 1822ndash1829 2013

[9] G Zhu B A McDonald and K Wang ldquoModeling and analysisof coupled inductors in power convertersrdquo IEEE Transactionson Power Electronics vol 26 no 5 pp 1355ndash1363 2011

[10] Q-B Hu B Qu and Z-Y Lu ldquoNovel step-up VRM-two-phaseinterleaved coupled-boost converterrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 26 no 9 pp 94ndash98 2006

[11] D A Grant Y Darroman and J Suter ldquoSynthesis of tapped-inductor switched-mode convertersrdquo IEEE Transactions onPower Electronics vol 22 no 5 pp 1964ndash1969 2007

[12] M Nymand and M A E Andersen ldquoHigh-efficiency isolatedboost DC-DC converter for high-power low-voltage fuel-cellapplicationsrdquo IEEE Transactions on Industrial Electronics vol57 no 2 pp 505ndash514 2010

[13] R Haroun A Cid-Pastor A El Aroudi and L Martinez-Salamero ldquoCascade connection ofDC-DC switching convertersby means of self-oscillating dc-transformersrdquo in Proceedingsof the 15th International Power Electronics and Motion ControlConference (EPEPEMC rsquo12) 2012

[14] R Guo C Wang and T Li ldquoOptimum design of couplinginductors for magnetic integration in three-phase interleavingBuck dcdc converterrdquo in Proceedings of the 8th IEEE Conferenceon Industrial Electronics and Applications (ICIEA rsquo13) pp 1029ndash1033 2013

[15] H Wu J Gu J Zhang Y Xing and G Chen ldquoHigh efficiencyhigh step-up Boost-Flyback DCDC converterrdquo Proceedings ofthe Chinese Society of Electrical Engineering vol 31 no 24 pp40ndash45 2011

[16] T J Liang andK C Tseng ldquoAnalysis of integrated boost-flybackstep-up converterrdquo IEE Proceedings Electric Power Applicationsvol 152 no 2 pp 217ndash225 2005

[17] J L Kui-Jun P Byoung-Gun K Rae-young and H Dong-seok ldquoNonisolated ZVT two-inductor boost converter with asingle resonant inductor for high step-up applicationsrdquo IEEETransactions on Power Electronics vol 27 no 4 pp 1966ndash19732012

[18] Z Lu L Zheng and Z Ma ldquoInterleaved high gain boostconverter with switched capacitor networkrdquo Transactions ofChina Electrotechnical Society vol 27 no 11 pp 154ndash156 2012

[19] A S Samosir and A H M Yatim ldquoImplementation of dynamicevolution control of bidirectional DC-DC converter for inter-facing ultracapacitor energy storage to fuel-cell systemrdquo IEEETransactions on Industrial Electronics vol 57 no 10 pp 3468ndash3473 2010

[20] A S Samosir and A H M Yatim ldquoDynamic evolution controlof bidirectionalDC-DCconverter for interfacing ultra capacitorenergy storage to fuel cell electric vehicle systemrdquo in Proceedingsof the Power Engineering Conferencee (AUPEC rsquo08) AustralasianUniversities December 2008

[21] A H ALQahtani M S Abuhamdeh Y M Alsmadi and VI Utkin ldquoPhotovoltaic power optimization using sliding mode

control with a two-axis tracking systemrdquo in Proceedings of theIEEE Energytech pp 1ndash6 2013

[22] D G Montoya C A R Paja and R Giral ldquoA new solutionof maximum power point tracking based on sliding modecontrolrdquo in Proceedings of the 39th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo13) pp 8350ndash83552013

[23] J He ldquoConnected power system based on sliding mode theoryrdquoin Study on Control Algorithm of Maximum Power Point Track-ing for Photovoltaic Grid Shandong University 2012

[24] A S Samosir and A H M Yatim ldquoDynamic evolution con-troller for single phase inverter applicationrdquo in Proceedings ofthe IEEE Symposium on Industrial Electronics and Applications(ISIEA rsquo09) vol 1 pp 530ndash535 October 2009

[25] A S Samosir M Anwari and A H M Yatim ldquoDynamicevolution control of interleaved boostDC-DC converter for fuelcell applicationrdquo in Proceedings of the 9th International Powerand Energy Conference (IPEC rsquo10) pp 869ndash874 October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

However there are many disadvantages such as low relia-bility high costs and complexity of main circuit as well ascontrol strategy and drive methods besides the outputtingdiodesrsquo reverse recovery problem is still serious in high-gain applications Isolated boost converters can also achievehigh-gain transform by designing the ratio of transformers(as discussed by [12 14]) but the leakage inductances willcause many problems such as voltage overshoot of switchesbig voltage stress and EMI problems Meanwhile the loss oftransformers will result in lower efficiency higher costs andcomplexity of main circuit control strategy drive methodsand isolation designThe topology-combining boost convert-ers and fly-back converters can be employed in the high-step-up transform (as discussed by [15 16]) whose topology andcontrol method are fairly simple however the voltage balanceof the capacitors exists in this systemdue to series connection

The reverse-recovery loss of the output diode is one ofthe main reasons why the efficiency is limited in the highstep-up applications When the boost converter works ondiscontinuous current mode (DCM) or critical current mode(CRM) the reverse-recovery loss can be reduced but itwill increase both current stress and input current rippleTherefore it is not suitable for the high-power applicationsdue to the bulky filter circuit

Interleaved boost converters can not only develop bothpower level and power density but also average the thermalstress distribution and reduce the input current ripple (asdiscussed by [17ndash19]) which can even be eliminated whenthe duty cycle is equal to 05 theoretically However there aresome weaknesses to apply interleaved boost converter intoPV grid-connected system directly (as discussed by [7 18])The step-up ratio is not increased efficiently comparing withthe traditional single-stage boost converter as well as thevoltage stress of switches Besides it is very sensitive forordinary interleaved boost converters to mismatch the dutycycle

In allusion to all the problems mentioned above a novelhigh-gain interleaved converter with coupled inductanceand switched capacitor (hereinafter referred to as the newconverter) is proposed High-gain transform can be real-ized by designing the main circuit parameters reasonablyFurthermore the topology as well as its control method isrelatively simple Compared with conventional interleavedboost converters the new one can boost voltage on thegreat scale while reducing the voltage stress of the switchdevices significantly under the same duty cycle Moreoverthe volume of magnetic elements and the current rippleof the inductance can be reduced by introducing coupledinductance (as discussed by [8 9 14])

Control strategy of the proposed converter can easilyachieve the MPPT algorithms and minimize the currentripple to reduce the coupled inductance losses and to improvesystem efficiency and stability Moreover less computationshould also be under the consideration to ease the com-putational complexity It is worth noting that in the liter-ature some examples of high-gain DC-DC converters arepresented but they only concern the topologies while fewpapers are devoted to the controller design for such kind ofconverter although not referring to PV applications

Slidingmode control (SMC) has a high robustness againstplant uncertainty and external disturbances including thosedue to the environmental conditions in PV applicationsMany SM-based MPPT algorithms operated in differentapplications or different topologies have been proposed inpublished papers but most of those mainly focus on theSMC used in conventional boost converter Neverthelesssuch an approach has never been used in analyzing a class ofinterleaved boost converters for PV applications controlledby the SM-MPPT technique

The controller of the proposed new converter is requiredto extract the maximum energy from the PV panels andoutput the stable expectant voltage without errors Dynamicevolution control (DEC) as a new control technique hasbeen utilized in power electronics converters (not in PVapplications) DEC can ensure that the error state goes tozero in increase of time thereby improving the dynamiccharacteristic of system

A new control approach based on both SM-MPPT andDEC is proposed in this paper Since the outputting voltageand current of PV cell together with the outputting voltageof converter are involved in calculation of duty cycle allthe variation above can be compensated in the dynamicevolution

This paper is organized as follows Section 2 describes thecharacteristic of PV cells Section 3 gives the working princi-ple and performance analysis of the new converter Section 4is devoted to the controller design on basis of SM-MPPTand DEC Sections 5 and 6 afford the simulation results andexperimental validation respectively In the conclusion themain results are finally summarized

2 Characteristic of PV Cells

Formulas of the outputting current and power for the PV cellscan be seen from (1) (as discussed by [3]) and Figure 1 showsthe output characteristics of PV cells

119894pv = 119868ph minus 119868sat [exp(119902119881pv

119860119870119879) minus 1]

119875pv = 119881pv119868ph minus 119881pv119868sat [exp(119902119881pv

119860119870119879) minus 1]

(1)

where 119894pv 119881pv are the outputting current and voltage of PVcells 119868ph is the photocurrent of a single solar module 119868sat isthe reverse saturation current 119870 is Boltzmann constant 119902is electric charge 119879 is operating temperature and 119860 is P-Njunction ideal factor

When the output of the equivalent circuit of PV cellsis shorted if neglecting the reverse leakage current of thediodes we canmake an approximation that 119868ph asymp 119868sc (119868sc is theshort-cut current) which will be used in part119881 for analyzingthe inputting current in dynamic mode

Journal of Applied Mathematics 3

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

140

160

180

V (V)

P(W

)(Temp = 25∘C)

S1 = 600 (Wm2)S2 = 800 (Wm2)S3 = 1000 (Wm2)

S1

S2

S3

V-P characteristics

(a) V-P curve when temp = 25∘C different irradiance

0 10 20 30 40 50 600

50

100

150

200

250

P(W

)

V (V)

T1 = 15∘CT2 = 25∘CT3 = 35∘C

(Irradiance = 1000 Wm2)

T1T2T3

V-P characteristics

(b) V-P curve when irradiance = 1 kWm2 different temp

Figure 1 The output characteristics of PV cells

3 Principle and Steady-State Analysis

31 Principle of the New Converter The new converter withPV panels is shown in Figure 2 Capacitors 119862

1 1198622and diode

VD1 are employed to further enlarge the boost voltage underthe same duty cycle compared with any other topology

The inductances 1198711and 119871

2are closely coupled and with

the same winding orientation The coupled inductances canbe represented as a mutual inductance 119872 two equivalentleakage inductances 1198711015840

1

and 1198711015840

2

The equivalent circuit isshown in Figure 3

The relationships of the inductances are related by thefollowing equations

1198711015840

1

= 1198711minus119872

1198711015840

2

= 1198712minus119872

119872 = 119896radic11987111198712

(2)

where 119896 is coupled coefficientIn order to simplify the circuit analysis of the proposed

converter several necessary assumptions should be made asfollows

(1) All components including power switches magneticcomponents and diodes are ideal the on-state resis-tance 119877DS(ON) and all parasitic capacitances of theswitches are neglected as are the forward voltagedrops of diodes VD

1simVD3

(2) The ESR of capacitors 1198620sim 1198622and the parasitic

resistance of coupled inductances are neglected(3) 1198711= 1198712= 119871

C1

C2

VD1

VD2

VD3

S1S2 C0 R

+

+

minus

L2

ipv

PV panels

lowast

lowast

+

minus

+minus

L1

VoVpv

Figure 2 The new converter

C1

C2

VD1

VD2

VD3

S1S2C0 R

+

minus

+minusM

iL1iin

+minusiL2

L9984001

L9984002

VoVpv

Figure 3 Equivalent circuit of the new converter

(4) 1198781and 1198782work by turn and both are driven by PWM

signals

Referring to the equivalent circuits for four differentswitching states of a switching period shown in Figure 4 andthe waveforms in Figure 5 the operation of the converter canbe explained as follows

4 Journal of Applied Mathematics

C1

C2

VD1

VD2

VD3

S1S2 C0 R

M

123

+

minus

L9984001

L9984002

+minus

+minus

iL1iin

iL2

C2

C1

VoVpv

(a) Mode 1 1198781-on 1198782-off VD

1-off VD

2-on VD

3-on

C1

C2

VD1

VD2

VD3

S1S2

C0 R

M

1 2 +

minus

L9984001

L9984002

+minus

+minus

iL1iin

iL2

C2

C1

VoVpv

(b) Mode 2 1198781-on 1198782-off VD

1-off VD

2-off VD

3-off

C1

C2

VD1

VD2

VD3

S1S2 C0 R

M

12 3 +

minus

L9984001

L9984002

+minus

+minus

iL1

iL2

C2

C1

VoVpv

iin

(c) Mode 3 1198781-off 1198782-on VD

1-on VD

2-off VD

3-on

C1

C2

VD1

VD2

VD3

S1S2 C0 R

M

12

+

minus

L9984001

L9984002

+minus

+minus

iL1

iL2

C2

C1

VoVpv

iin

(d) Mode 4 1198781-off 1198782-on VD

1-off VD

2-off VD

3-off

Figure 4 Operation states and current paths of the converter

t

t

t

t

t

t

S1

S2

vc1

vc2

1 2 3 4

iL1

iL2

iin

iL1 iL2

iin = iL1 + iL2

t1 t2 t3t4t0

12057211T1205721T 12057221T 1205722T

Figure 5 Main waveforms of the new converter

Mode 1 (Figure 4(a)) At time 1199050 1198781is on while 119878

2is off the

current in the inductance 11987110158401

starts to rise linearly while 11987110158402

continues to discharge (the current in 11987110158402

was acquired in thelast switching period) Capacitor 119862

2discharges leading to its

voltage falling But the voltage of capacitor1198621rises because119862

1

is charged besides capacitor 1198620is charged by input voltage

and inductor in this mode The rates of change of 1198941198711

and 1198941198712

are approximately given by

1198891198941198711

119889119905=119881119900minus V1198621

11987110158401

1198891198941198712

119889119905= minus

119881119900minus V1198622

11987110158402

(3)

Mode 2 (Figure 4(b)) At time 1199051 1198941198712falls to zero 119894

1198711continues

to rise linearly 119877 is charged by capacitor 1198620 The rate of

change of 1198941198711

is

1198891198941198711

119889119905=119881pv

1198711

(4)

where 1198711= 1198711015840

1

+119872

Mode 3 (Figure 4(c)) At time 1199052 1198781is off while 119878

2is on 119894

1198712

starts to rise linearly from zero and 1198711starts to discharge

Besides 1198621discharges too while 119862

2starts to charge More-

over1198620is charged by input voltage and inductorThe rates of

change of 1198941198711

and 1198941198712

are approximately given by1198891198941198711

119889119905= minus

119881119900minus V1198621

11987110158401

(5)

1198891198941198712

119889119905=119881119900minus V1198622

11987110158402

(6)

V1198621+ V1198622

= 119881119900 (7)

Mode 4 (Figure 4(d)) At time 1199053 1198781is offwhile 119878

2is on 119894

1198711falls

to zero 1198941198712continues to rise linearly119877 is charged by capacitor

1198620 The rate of change of 119894

1198712is

1198891198941198712

119889119905=119881pv

1198712

(8)

where 1198712= 1198711015840

2

+119872The switching period will be repeated when 119878

1is on again

at time 1199054

Journal of Applied Mathematics 5

32 Steady-State Analysis Assume that the current wave-forms shown in Figure 5 have reached a steady state Fromthe waveform 119894

1198711as shown in Figure 5 it can be found that

119881119900minus V1198621

11987110158401

12057211119879 +

119881pv

1198711

(1205721minus 12057211) 119879 =

119881119900minus V1198621

11987110158401

(1205722minus 12057221) 119879

(9)

From the waveform 1198941198712

as shown in Figure 5 it can be foundthat

119881119900minus V1198622

11987110158402

(1205722minus 12057221) 119879 +

119881pv

1198712

12057221119879 =

119881119900minus V1198622

11987110158402

12057211119879 (10)

Also from Figure 5 the average values of 1198941198711

and 1198941198712 denoted

as 1198681and 1198682 respectively are found as

1198681=1

1198791

2

119881119900minus V1198621

11987110158401

sdot (12057211119879)2

+1

2

119881119900minus V1198621

11987110158401

sdot [(1205722minus 12057221) 119879]2

+1

2(1205721minus 12057211) 119879

sdot [2 (119881119900minus V1198621)

11987110158401

sdot 12057211119879 +

119881pv

1198711

(1205721minus 12057211) 119879]

1198682=1

1198791

2

119881119900minus V1198622

11987110158402

sdot (12057211119879)2

+1

2

119881119900minus V1198622

11987110158402

sdot [(1205722minus 12057221) 119879]2

+1

212057221119879

sdot [2 (119881119900minus V1198622)

11987110158402

sdot (1205722minus 12057221) 119879 +

119881pv

1198712

12057221119879]

(11)

Assume that the converter is lossless that is the input power119875119894is equal to the output power 119875

119900 Consider

119881pv (1198681 + 1198682) =1198812

119900

119877 (12)

where 119877 is the load resistanceIn principle (7) and (9)ndash(12) can be solved to find 119881

119900

However to simplify the calculation it is assumed that 1205721=

120572 1205722= 1 minus 120572 119871

1= 1198712= 119871 and 119871

1015840

1

= 1198711015840

2

= (1 minus 119896)119871According to some algebraic manipulations and solving aquadratic equation derived from (12) the approximate valuesof 1198811198621 1198811198622 and 119881

119900can be referred as follows

1198811198621

=3

1 minus 120572119881pv =

3

4119881119900

1198811198622

=1

1 minus 120572119881pv =

1

4119881119900

(13)

119881119900=

4

1 minus 120572119881pv (14)

The effect of R on the calculation is very little thus it isneglected when getting the approximate values as shown in(13) and (14)

A careful study of the waveforms shown in Figure 5 willreveal the following interesting facts

C1

C2

VD1

VD2

VD3

S1S2 C0 R Vo

++

minusVpv

L1

L2

+minus

+minus

Vm1

ipv

120575(Vo Vpv ipv ΔVpv Δipv )

PVpanels

PWM

DSME PWM controller

PWM 2 PWM 1

lowast

lowast

Figure 6 The control flow chart of DSME PWM controller

(1) As far as the input current 119894in is concerned theconverter appears to operate in CCM (because 119894inis continuous) Thus the peak current stress of theinductances and the input current ripple can bemaintained relatively low

(2) However since 1198941198711and 1198941198712are discontinuous the new

converter is actually operating in DCM Also sincethe rectifier diodes VD

1simVD3turn off before 119878

11198782

turns on the reverse-recovery loss of the rectifiers iseliminated

(3) Besides the current of inductance 11989411987111198941198712

has fallento zero before 119878

11198782turns on ZCS soft switching

operation during the whole switching transition isachieved

Furthermore when considering the parasitic resistancesof switches and capacitances (119903

119904and 119903119888) the switching loss due

to capacitances in the course of charging can be defined as (15)in an independent converter which contains 119899 capacitances

119882119904=119899119862

2sdot1 + 119890minus120572119879120591

1 minus 119890minus120572119879120591(Δ119881119900)2

(15)

where 119862 is the value of capacitances and 120591 = (119903119904+ 119903119888)119862

From (15) 119882119904do exist even under the ideal conditions

mainly depending on Δ119881119900 Thus in order to reduce 119882

119904

the difference between the maximum and the minimum ofvoltage in the course of charging for all capacitances shouldbe decreased

4 DSME PWM Controller

The original intention of the DSME control is to achieveMPPT as well as high-gain boost also to reduce the errorstate by forcing the error state to follow the specific pathwhich ensures the error state goes to zero in increase oftime (as discussed by [19 20]) The control flow chart ofdynamic sliding mode evolution PWM Controller can beseen in Figure 6

The modulating signal of PWM generation 1198811198981

is pro-duced by the formula 120575 combing the sliding mode theory

6 Journal of Applied Mathematics

with the dynamic evolution algorithm The specific designprocess of the DSME controller can be described as follows

41 MPPT by Sliding Mode Control When the PV cells runat the point of maximum power

119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv= 0 (16)

In SMC the state space is divided into 119904 gt 0 and 119904 lt 0 bysliding manifold (119904 = 0) The movement on 119904 = 0 can becalled sliding mode dynamics Considering 119889119875pv119889119881pv = 0the sliding manifold can be chosen as (as discussed by [2122])

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv (17)

Reconsidering the mathematical model of PV cell

119894pv = 119868ph minus 119868sat [exp(119902119881pv

119860119870119879) minus 1] (18)

Directly substituting 119894pv from (18) into (17)

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv

= minus119868sat [exp (119902

119860119870119879119881pv) sdot

119902

119860119870119879119881pv]

+ 119868ph minus 119868sat [exp(119902

119860119870119879119881pv) minus 1]

= minus119868sat (119902

119860119870119879119881pv + 1) sdot exp(

119902

119860119870119879119881pv) + (119868ph + 119868sat)

(19)

The differential form of 119904 is

119904 = minus119868sat119902

119860119870119879sdot119889119881pv

119889119905exp(

119902

119860119870119879) minus 119868sat (

119902119881pv

119860119870119879+ 1)

sdot exp(119902119881pv

119860119870119879)

119902

119860119870119879

119889119881pv

119889119905

= minus119868sat119902

119860119870119879sdot119889119881pv

119889119905exp(

119902119881pv

119860119870119879)(

119902119881pv

119860119870119879+ 2)

(20)

According to 119904 119904 lt 0 if 119904 gt 0 then 119904 lt 0 119889119881pv119889119905 gt 0119881pv willincrease when 119904 tends to sliding manifold and if 119904 lt 0 then119904 gt 0 119889119881pv119889119905 lt 0 119881pv will decrease when 119904 tends to slidingmanifoldThe dynamic accommodation of119881pv can be seen inFigure 7

Considering that the new converter is composed of twoparalleled boost converters the purpose of the additionalswitched capacitor and coupled inductance is to increase thevoltage gain Thus the drive signals for the switches can bedesigned respectively From the state space model of thesingle boost converter when the switch is on 119881pv increases

and when the switch is off 119881pv decreases (as discussed by[23])

Rearranging (17)

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv= 119881pv (

119894pv

119881pv+119889119894pv

119889119881pv) (21)

The differential form approximates to

Δ119894pv

Δ119881pvasymp119889119894pv

119889119881pv (22)

Thus

119904 = 119881pv (119894pv

119881pv+Δ119894pv

Δ119881pv) (23)

Theoretically in the case of low requirement both (17) and(24) can be chosen as the sliding surface

119904 = Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv (24)

The corresponding control law is

119906 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)] (25)

42 Evolution Path Selection The principle of determiningthe evolution path is to ensure that the error state can goto zero at any increase of time (as discussed by [24 25]) Inthis research the evolution path is an exponential function asshown in Figure 8

With this exponential evolution path the value of thedynamic characteristic of the converter will decrease expo-nentially to zero by

119867 = 1198670ℓminus120582119905

(26)

where 119867 is the dynamic characteristic of system 1198670is the

initial value of 119867 and 120582 is a dynamic evolution factor Thederivative of119867 is

119889119867

119889119905= minus120582119867

0ℓminus120582119905

= minus120582119867

119889119867

119889119905+ 120582119867 = 0

(27)

43 Analysis of Duty Cycle In order to synthesize the controllaw of the dynamic evolution controller the dynamic equa-tion of converter system has to be established and analyzedOn basis of the state space averagemodel when the converterworks on DCM the voltage and current dynamics of theconverter are given by

119881pv = 119871119889119894in119889119905

+119881119900

4(1 minus 120572) (28)

where 119871 is the inductance 119881pv is the input voltage 119894in is theinput current119881

119900is the output voltage and 120572 is the duty cycle

respectively

Journal of Applied Mathematics 7

Sliding manifold

Trajectory s chattersalong the manifoldand moves toward

VO

pv

120577 = minusΔs gt 0

s = 0

120577 = +Δ

s lt 0

Ppv

s = 0

Figure 7 The dynamic accommodation of 119881pv

119894in asymp 119894pv rearranging (28) the output voltage of convertercan be written as

119881pv = 119871119889119894pv

119889119905+119881119900

4(1 minus 120572) (29)

119881119900= 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (30)

Define a linear voltage error function as shown below

119864 = 119896119890V = 119896 (119881ref minus 119881119900) (31)

where 119896 gt 0Substituting (27) into (31) yields

119896119889119890V

119889119905+ 120582119896119890V = 0 (32)

Combining (31) with (32)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 119881

119900 (33)

Directly substituting the converter output voltage 119881119900from

(30) into (33)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (34)

The obtained duty cycle formula is given by

120572 =119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905 (35)

Formula (35) forces the state error function 119864 to satisfy thedynamic evolution function (27) Consequently the stateerror function 119864 is forced to make evolution by followingequation (35) and decrease to zero (119864 = 0) with a decreaserate 120582 The outputting voltage of converter converges to theconverters steady state

119881119900= 119881ref (36)

From the synthesis procedure it is clear that the dynamicevolution controller works on the full nonlinear system and

H

H0

0 t

H = H0eminus120582t

Figure 8 Exponential evolution path

does not need any linearization or simplification on thesystem model at all and it is interesting to note that thecontrol law in (35) consists of four distinct parts The firstpart is the feed-forward term which is calculated based onthe duty cycle at the previous sampling instant This termcompensates for variations in the input voltages The secondand third terms consist of proportional and derivative termsof the perturbations in the output voltage respectively Thelast term consists of the derivative terms of the inductancecurrent Since the inputting and outputting voltages andinductance current are involved in calculation of duty cycleit can compensate all the variation of them in the dynamicevolution

It should be noted that both119881pv and 119894pv are not the voltageand current at the MPP in (35) Consequently we expect thatthe controller can achieve MPPT and high-gain boost thusconsidering the logic relationship of control signals the finalcontrol law can be described as follows combing (25) with(35)

120575 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)]

sdot (119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905)

(37)

44 PWM Generation The PWM signals are generated bycomparing a control signal with a constant peak repetitivetriangle signal (119881cs) The frequency of the repetitive trianglesignal establishes the constant switching frequency

Figure 9 shows the PWM signals generation techniquePWM1 is produced when the control signal 119881

1198981is greater

than 119881119888119904while PWM2 is produced when 119881

1198981is less than 119881

119888119904

The values of the desired duty cycle 120572 can be gotten from(37) PWM signals are generated by (39)

1198811198981

= 120575 (38)

8 Journal of Applied Mathematics

PWM1 = 1 1198811198981

gt 119881119888119904

0 1198811198981

lt 119881119888119904

PWM2 = 1 1198811198981

lt 119881119888119904

0 1198811198981

gt 119881119888119904

(39)

5 Simulation

Some simulation has been done in order to verify theperformance of the topology and the main parameters canbe referred to in Table 1

The portion of the simulation results is given in Figure 10Figure 10(a) shows the driving signals for 119878

1and 1198782 which

are generated by the control law 120575 The outputting voltageand power of PV cell in process of MPPT can be seen inFigure 10(b) From the simulation onemay get the conclusionthat the DSME PWM controller can ensure that the PV cellruns at the maximum power point in a short time Althoughthe overshoot with a little range occurs in the process ofpower adjustment it will be eliminated soon and the stableoutputting power of 260W is achieved During the dynamicadjustment there is a stable and fleet performance withoutoscillation

In Figure 10(c) the dynamic response of the novel topol-ogy is the curve labeled overall trend This curse shows thatthe converter can raise the voltage from 36V to 400V steadilywithout a long time The curve labeled partial enlargementshows that the outputting voltage has a low voltage ripplewhen it is in stable state Considering the whole PV grid-connected system a DC-link high voltage with low rippleis convenient to choice and maintaining of the DC-linkcapacitor

As mentioned before the converter can run under fourmodes in every switching period The tendencies of 119894

1198711and

1198941198712as well as 119894in in every mode have been discussed in part III

and it is also easy to get the conclusion about 119894in = 1198941198711+ 1198941198712

based on Kirchhoff rsquos current law so the tendency of 119894in canbe obtained by the sum of 119894

1198711and 1198941198712 The first and second

one of Figure 10(d) describe the stable current of 1198941198711

and 1198941198712

and 119894in respectively over the same period in stable modeFigure 10(d) shows that the current 119894in has little ripple whichcan reduce the coupled inductance losses and improve systemefficiency

Figure 10(e) shows the voltage waves of switched capac-itors 119862

1and 119862

2 The voltage of 119862

1is three times as much as

the one of 1198622 close to three quarters of 119881

119900 this simulation

verifies the formula (13)

6 Experiments

Experiments have been finished in our laboratory just inorder to verify the validity of theoretical analysis above andthe performance of the new converter proposed in this paper

Experimental waveforms of the new converter are shownin Figure 11 The driving signals of 119878

1and 119878

2from DSP on

basis of the algorithm in this paper are given in Figure 11(a)Obviously the new converter can raise the voltage from36V to 400V when the duty cycle equals 064 (calculated

Table 1 The main parameters

Parameters Values119881ocV 443119868scA 77119881mp at MPPV 36119868mp at MPPA 72Coupling coefficient 091198711

and 1198712

mH 041198621

120583F 101198622

120583F 1001198620

120583F 22119877kΩ 1Switching frequencykHz 50

PWM1PWM2

PWM120575(Vo Vpv ipv ΔVpv Δipv )

0

Vo

Vcsipv

Vpv Vm1

Figure 9 PWM generator

from Figure 11(a)) The output voltage wave (after 10 timesattenuation) is shown in Figure 11(b) Moreover the rippleof the output voltage shown in Figure 11(b) can be limitedin 8V which can decrease the loss of switched capacitorsas shown in (15) Figure 11(c) shows the wave of the inputcurrent The little input current ripple can reduce both theloss and the volume of the coupled inductance Figures 11(d)and 11(e) show the voltage waveform of switched capacitors1198621and 119862

2 respectively (after 10 times attenuation) The

voltage of 1198621 about 300Vsim306V is three times as much as

the one of 1198622 about 92Vsim100V which is consistent with

both the analysis and the simulation results above besidesthe difference between the maximum and the minimum ofvoltage in the course of charging for each capacitance can belimited to a fairly small range which can decrease the loss ofswitched capacitors

Burrs existing in experimental waveform mainly stemfrom the effect of EMI in the experiment

7 Conclusion

This paper proposed a new high-gain interleaved converterwith coupled inductances and switched capacitors networkwhich are suited for PV application After explaining its oper-ating principle and analyzing its steady-state performancethen a control strategy for it on basis of SM-MPPT andDEC theory was designed Both simulation analysis andexperimental results showed that the proposed system hadthe following good performance

(1) Achieving MPPT and producing a high-gain voltagein an extremely short time

(2) Decreasing the ripple of the outputting voltage as wellas the inputting current and improving power density

Journal of Applied Mathematics 9

For S

1

ForS

2

(a) Driving signals for 1198781amp 1198782

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

Time (s)

Volta

ge (V

) and

pow

er (W

)

The outputting voltage and power of PV cells under MPPT

Vpv outputting voltageP outputting powerpv

(b) The output voltage and power of PV cell in process of MPPT

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

350

400

450The outputting voltage of the new topological converter

Time (s)

Volta

ge (V

)

399

3995

400

4005

401

Partial enlargement of outtputing voltage

Overall trendPartial enlargement

(c) The output voltage wave

0

5

Curr

ent (

A) The Lofwavecurrent 1 and L2 in stable mode

001 0015 002 0025 00305

1015

Time (s)

Curr

ent (

A)

iL1

iL2

The current wave of iin in the dynamic mode

65

7

75

Curr

ent (

A) The current wave of iin in stable mode

(d) The current of 1198711and 1198712and 119894in

0 001 002 003 004 005 006 007 008 009 01minus100

minus50

0

50

100

150

200

250

300

350

400

Time (s)

Volta

ge (V

)

The voltage of and

Voltage of C1

Voltage of C2

C1 C2

(e) The voltage wave of switched capacitor 1198621and 119862

2

Figure 10 Simulation results

10 Journal of Applied Mathematics

(a) Driving signals for 1198781and 1198782from DSP (b) The outputting voltage wave (X10)

(c) The wave of input current (d) The voltage wave of switched capacitor 1198621(X10)

(e) The voltage wave of switched capacitor 1198622(X10)

Figure 11 Experimental validation

(3) Reducing the volume of magnetic components andcutting the cost of system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education of China (no20120032110070)

References

[1] C Zhang and X He ldquoMaximum power point tracking by usingasymmetric fuzzy control combined with PID for photovoltaic

energy generation systemrdquo Transactions of China Electrotechni-cal Society vol 20 no 10 pp 72ndash75 2005

[2] Z Liao and X Ruan ldquoControl strategy for bi-directionalDCDC converter of a novel stand-alone photovoltaic powersystemrdquo Transactions of China Electrotechnical Society vol 23no 1 pp 97ndash103 2008

[3] M Dong J Yang K Peng and A Luo ldquoZero average incre-mental conductance maximum power point tracking controlfor photovoltaic systemrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 30 no 21 pp 48ndash53 2010

[4] J M Enrique J M Andujar and M A Bohorquez ldquoAreliable fast and low cost maximum power point tracker forphotovoltaic applicationsrdquo Solar Energy vol 84 pp 79ndash89 2010

[5] L Xie RGong and J Li ldquoAnalysis of the dynamical characteris-tics of the Interleaved boost converter inmaximumpower pointtracking for photovoltaic powerrdquo in Proceedings of the ChineseSociety for Electrical Engineering (CSEE rsquo13) vol 33 no 6 pp39ndash43 2013

Journal of Applied Mathematics 11

[6] W Li and X He ldquoReview of nonisolated high-step-up DCDCconverters in photovoltaic grid-connected applicationsrdquo IEEETransactions on Industrial Electronics vol 58 no 4 pp 1239ndash1250 2011

[7] Q LuoH Yan S Zhi C Zou and L Zhou ldquoAn interleaved highstep-up zero current transition boost converterrdquo in Proceedingsof the Chinese Society for Electrical Engineering (CSEE rsquo13) vol33 no 12 pp 18ndash21 2013

[8] W Huang and B Lehman ldquoMitigation and utilization of theinductor coupling effect in interleaved multiphase dcdc con-vertersrdquo in Proceedings of the IEEE Energy Conversion Congressand Exposition (ECCE rsquo13) pp 1822ndash1829 2013

[9] G Zhu B A McDonald and K Wang ldquoModeling and analysisof coupled inductors in power convertersrdquo IEEE Transactionson Power Electronics vol 26 no 5 pp 1355ndash1363 2011

[10] Q-B Hu B Qu and Z-Y Lu ldquoNovel step-up VRM-two-phaseinterleaved coupled-boost converterrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 26 no 9 pp 94ndash98 2006

[11] D A Grant Y Darroman and J Suter ldquoSynthesis of tapped-inductor switched-mode convertersrdquo IEEE Transactions onPower Electronics vol 22 no 5 pp 1964ndash1969 2007

[12] M Nymand and M A E Andersen ldquoHigh-efficiency isolatedboost DC-DC converter for high-power low-voltage fuel-cellapplicationsrdquo IEEE Transactions on Industrial Electronics vol57 no 2 pp 505ndash514 2010

[13] R Haroun A Cid-Pastor A El Aroudi and L Martinez-Salamero ldquoCascade connection ofDC-DC switching convertersby means of self-oscillating dc-transformersrdquo in Proceedingsof the 15th International Power Electronics and Motion ControlConference (EPEPEMC rsquo12) 2012

[14] R Guo C Wang and T Li ldquoOptimum design of couplinginductors for magnetic integration in three-phase interleavingBuck dcdc converterrdquo in Proceedings of the 8th IEEE Conferenceon Industrial Electronics and Applications (ICIEA rsquo13) pp 1029ndash1033 2013

[15] H Wu J Gu J Zhang Y Xing and G Chen ldquoHigh efficiencyhigh step-up Boost-Flyback DCDC converterrdquo Proceedings ofthe Chinese Society of Electrical Engineering vol 31 no 24 pp40ndash45 2011

[16] T J Liang andK C Tseng ldquoAnalysis of integrated boost-flybackstep-up converterrdquo IEE Proceedings Electric Power Applicationsvol 152 no 2 pp 217ndash225 2005

[17] J L Kui-Jun P Byoung-Gun K Rae-young and H Dong-seok ldquoNonisolated ZVT two-inductor boost converter with asingle resonant inductor for high step-up applicationsrdquo IEEETransactions on Power Electronics vol 27 no 4 pp 1966ndash19732012

[18] Z Lu L Zheng and Z Ma ldquoInterleaved high gain boostconverter with switched capacitor networkrdquo Transactions ofChina Electrotechnical Society vol 27 no 11 pp 154ndash156 2012

[19] A S Samosir and A H M Yatim ldquoImplementation of dynamicevolution control of bidirectional DC-DC converter for inter-facing ultracapacitor energy storage to fuel-cell systemrdquo IEEETransactions on Industrial Electronics vol 57 no 10 pp 3468ndash3473 2010

[20] A S Samosir and A H M Yatim ldquoDynamic evolution controlof bidirectionalDC-DCconverter for interfacing ultra capacitorenergy storage to fuel cell electric vehicle systemrdquo in Proceedingsof the Power Engineering Conferencee (AUPEC rsquo08) AustralasianUniversities December 2008

[21] A H ALQahtani M S Abuhamdeh Y M Alsmadi and VI Utkin ldquoPhotovoltaic power optimization using sliding mode

control with a two-axis tracking systemrdquo in Proceedings of theIEEE Energytech pp 1ndash6 2013

[22] D G Montoya C A R Paja and R Giral ldquoA new solutionof maximum power point tracking based on sliding modecontrolrdquo in Proceedings of the 39th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo13) pp 8350ndash83552013

[23] J He ldquoConnected power system based on sliding mode theoryrdquoin Study on Control Algorithm of Maximum Power Point Track-ing for Photovoltaic Grid Shandong University 2012

[24] A S Samosir and A H M Yatim ldquoDynamic evolution con-troller for single phase inverter applicationrdquo in Proceedings ofthe IEEE Symposium on Industrial Electronics and Applications(ISIEA rsquo09) vol 1 pp 530ndash535 October 2009

[25] A S Samosir M Anwari and A H M Yatim ldquoDynamicevolution control of interleaved boostDC-DC converter for fuelcell applicationrdquo in Proceedings of the 9th International Powerand Energy Conference (IPEC rsquo10) pp 869ndash874 October 2010

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

140

160

180

V (V)

P(W

)(Temp = 25∘C)

S1 = 600 (Wm2)S2 = 800 (Wm2)S3 = 1000 (Wm2)

S1

S2

S3

V-P characteristics

(a) V-P curve when temp = 25∘C different irradiance

0 10 20 30 40 50 600

50

100

150

200

250

P(W

)

V (V)

T1 = 15∘CT2 = 25∘CT3 = 35∘C

(Irradiance = 1000 Wm2)

T1T2T3

V-P characteristics

(b) V-P curve when irradiance = 1 kWm2 different temp

Figure 1 The output characteristics of PV cells

3 Principle and Steady-State Analysis

31 Principle of the New Converter The new converter withPV panels is shown in Figure 2 Capacitors 119862

1 1198622and diode

VD1 are employed to further enlarge the boost voltage underthe same duty cycle compared with any other topology

The inductances 1198711and 119871

2are closely coupled and with

the same winding orientation The coupled inductances canbe represented as a mutual inductance 119872 two equivalentleakage inductances 1198711015840

1

and 1198711015840

2

The equivalent circuit isshown in Figure 3

The relationships of the inductances are related by thefollowing equations

1198711015840

1

= 1198711minus119872

1198711015840

2

= 1198712minus119872

119872 = 119896radic11987111198712

(2)

where 119896 is coupled coefficientIn order to simplify the circuit analysis of the proposed

converter several necessary assumptions should be made asfollows

(1) All components including power switches magneticcomponents and diodes are ideal the on-state resis-tance 119877DS(ON) and all parasitic capacitances of theswitches are neglected as are the forward voltagedrops of diodes VD

1simVD3

(2) The ESR of capacitors 1198620sim 1198622and the parasitic

resistance of coupled inductances are neglected(3) 1198711= 1198712= 119871

C1

C2

VD1

VD2

VD3

S1S2 C0 R

+

+

minus

L2

ipv

PV panels

lowast

lowast

+

minus

+minus

L1

VoVpv

Figure 2 The new converter

C1

C2

VD1

VD2

VD3

S1S2C0 R

+

minus

+minusM

iL1iin

+minusiL2

L9984001

L9984002

VoVpv

Figure 3 Equivalent circuit of the new converter

(4) 1198781and 1198782work by turn and both are driven by PWM

signals

Referring to the equivalent circuits for four differentswitching states of a switching period shown in Figure 4 andthe waveforms in Figure 5 the operation of the converter canbe explained as follows

4 Journal of Applied Mathematics

C1

C2

VD1

VD2

VD3

S1S2 C0 R

M

123

+

minus

L9984001

L9984002

+minus

+minus

iL1iin

iL2

C2

C1

VoVpv

(a) Mode 1 1198781-on 1198782-off VD

1-off VD

2-on VD

3-on

C1

C2

VD1

VD2

VD3

S1S2

C0 R

M

1 2 +

minus

L9984001

L9984002

+minus

+minus

iL1iin

iL2

C2

C1

VoVpv

(b) Mode 2 1198781-on 1198782-off VD

1-off VD

2-off VD

3-off

C1

C2

VD1

VD2

VD3

S1S2 C0 R

M

12 3 +

minus

L9984001

L9984002

+minus

+minus

iL1

iL2

C2

C1

VoVpv

iin

(c) Mode 3 1198781-off 1198782-on VD

1-on VD

2-off VD

3-on

C1

C2

VD1

VD2

VD3

S1S2 C0 R

M

12

+

minus

L9984001

L9984002

+minus

+minus

iL1

iL2

C2

C1

VoVpv

iin

(d) Mode 4 1198781-off 1198782-on VD

1-off VD

2-off VD

3-off

Figure 4 Operation states and current paths of the converter

t

t

t

t

t

t

S1

S2

vc1

vc2

1 2 3 4

iL1

iL2

iin

iL1 iL2

iin = iL1 + iL2

t1 t2 t3t4t0

12057211T1205721T 12057221T 1205722T

Figure 5 Main waveforms of the new converter

Mode 1 (Figure 4(a)) At time 1199050 1198781is on while 119878

2is off the

current in the inductance 11987110158401

starts to rise linearly while 11987110158402

continues to discharge (the current in 11987110158402

was acquired in thelast switching period) Capacitor 119862

2discharges leading to its

voltage falling But the voltage of capacitor1198621rises because119862

1

is charged besides capacitor 1198620is charged by input voltage

and inductor in this mode The rates of change of 1198941198711

and 1198941198712

are approximately given by

1198891198941198711

119889119905=119881119900minus V1198621

11987110158401

1198891198941198712

119889119905= minus

119881119900minus V1198622

11987110158402

(3)

Mode 2 (Figure 4(b)) At time 1199051 1198941198712falls to zero 119894

1198711continues

to rise linearly 119877 is charged by capacitor 1198620 The rate of

change of 1198941198711

is

1198891198941198711

119889119905=119881pv

1198711

(4)

where 1198711= 1198711015840

1

+119872

Mode 3 (Figure 4(c)) At time 1199052 1198781is off while 119878

2is on 119894

1198712

starts to rise linearly from zero and 1198711starts to discharge

Besides 1198621discharges too while 119862

2starts to charge More-

over1198620is charged by input voltage and inductorThe rates of

change of 1198941198711

and 1198941198712

are approximately given by1198891198941198711

119889119905= minus

119881119900minus V1198621

11987110158401

(5)

1198891198941198712

119889119905=119881119900minus V1198622

11987110158402

(6)

V1198621+ V1198622

= 119881119900 (7)

Mode 4 (Figure 4(d)) At time 1199053 1198781is offwhile 119878

2is on 119894

1198711falls

to zero 1198941198712continues to rise linearly119877 is charged by capacitor

1198620 The rate of change of 119894

1198712is

1198891198941198712

119889119905=119881pv

1198712

(8)

where 1198712= 1198711015840

2

+119872The switching period will be repeated when 119878

1is on again

at time 1199054

Journal of Applied Mathematics 5

32 Steady-State Analysis Assume that the current wave-forms shown in Figure 5 have reached a steady state Fromthe waveform 119894

1198711as shown in Figure 5 it can be found that

119881119900minus V1198621

11987110158401

12057211119879 +

119881pv

1198711

(1205721minus 12057211) 119879 =

119881119900minus V1198621

11987110158401

(1205722minus 12057221) 119879

(9)

From the waveform 1198941198712

as shown in Figure 5 it can be foundthat

119881119900minus V1198622

11987110158402

(1205722minus 12057221) 119879 +

119881pv

1198712

12057221119879 =

119881119900minus V1198622

11987110158402

12057211119879 (10)

Also from Figure 5 the average values of 1198941198711

and 1198941198712 denoted

as 1198681and 1198682 respectively are found as

1198681=1

1198791

2

119881119900minus V1198621

11987110158401

sdot (12057211119879)2

+1

2

119881119900minus V1198621

11987110158401

sdot [(1205722minus 12057221) 119879]2

+1

2(1205721minus 12057211) 119879

sdot [2 (119881119900minus V1198621)

11987110158401

sdot 12057211119879 +

119881pv

1198711

(1205721minus 12057211) 119879]

1198682=1

1198791

2

119881119900minus V1198622

11987110158402

sdot (12057211119879)2

+1

2

119881119900minus V1198622

11987110158402

sdot [(1205722minus 12057221) 119879]2

+1

212057221119879

sdot [2 (119881119900minus V1198622)

11987110158402

sdot (1205722minus 12057221) 119879 +

119881pv

1198712

12057221119879]

(11)

Assume that the converter is lossless that is the input power119875119894is equal to the output power 119875

119900 Consider

119881pv (1198681 + 1198682) =1198812

119900

119877 (12)

where 119877 is the load resistanceIn principle (7) and (9)ndash(12) can be solved to find 119881

119900

However to simplify the calculation it is assumed that 1205721=

120572 1205722= 1 minus 120572 119871

1= 1198712= 119871 and 119871

1015840

1

= 1198711015840

2

= (1 minus 119896)119871According to some algebraic manipulations and solving aquadratic equation derived from (12) the approximate valuesof 1198811198621 1198811198622 and 119881

119900can be referred as follows

1198811198621

=3

1 minus 120572119881pv =

3

4119881119900

1198811198622

=1

1 minus 120572119881pv =

1

4119881119900

(13)

119881119900=

4

1 minus 120572119881pv (14)

The effect of R on the calculation is very little thus it isneglected when getting the approximate values as shown in(13) and (14)

A careful study of the waveforms shown in Figure 5 willreveal the following interesting facts

C1

C2

VD1

VD2

VD3

S1S2 C0 R Vo

++

minusVpv

L1

L2

+minus

+minus

Vm1

ipv

120575(Vo Vpv ipv ΔVpv Δipv )

PVpanels

PWM

DSME PWM controller

PWM 2 PWM 1

lowast

lowast

Figure 6 The control flow chart of DSME PWM controller

(1) As far as the input current 119894in is concerned theconverter appears to operate in CCM (because 119894inis continuous) Thus the peak current stress of theinductances and the input current ripple can bemaintained relatively low

(2) However since 1198941198711and 1198941198712are discontinuous the new

converter is actually operating in DCM Also sincethe rectifier diodes VD

1simVD3turn off before 119878

11198782

turns on the reverse-recovery loss of the rectifiers iseliminated

(3) Besides the current of inductance 11989411987111198941198712

has fallento zero before 119878

11198782turns on ZCS soft switching

operation during the whole switching transition isachieved

Furthermore when considering the parasitic resistancesof switches and capacitances (119903

119904and 119903119888) the switching loss due

to capacitances in the course of charging can be defined as (15)in an independent converter which contains 119899 capacitances

119882119904=119899119862

2sdot1 + 119890minus120572119879120591

1 minus 119890minus120572119879120591(Δ119881119900)2

(15)

where 119862 is the value of capacitances and 120591 = (119903119904+ 119903119888)119862

From (15) 119882119904do exist even under the ideal conditions

mainly depending on Δ119881119900 Thus in order to reduce 119882

119904

the difference between the maximum and the minimum ofvoltage in the course of charging for all capacitances shouldbe decreased

4 DSME PWM Controller

The original intention of the DSME control is to achieveMPPT as well as high-gain boost also to reduce the errorstate by forcing the error state to follow the specific pathwhich ensures the error state goes to zero in increase oftime (as discussed by [19 20]) The control flow chart ofdynamic sliding mode evolution PWM Controller can beseen in Figure 6

The modulating signal of PWM generation 1198811198981

is pro-duced by the formula 120575 combing the sliding mode theory

6 Journal of Applied Mathematics

with the dynamic evolution algorithm The specific designprocess of the DSME controller can be described as follows

41 MPPT by Sliding Mode Control When the PV cells runat the point of maximum power

119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv= 0 (16)

In SMC the state space is divided into 119904 gt 0 and 119904 lt 0 bysliding manifold (119904 = 0) The movement on 119904 = 0 can becalled sliding mode dynamics Considering 119889119875pv119889119881pv = 0the sliding manifold can be chosen as (as discussed by [2122])

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv (17)

Reconsidering the mathematical model of PV cell

119894pv = 119868ph minus 119868sat [exp(119902119881pv

119860119870119879) minus 1] (18)

Directly substituting 119894pv from (18) into (17)

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv

= minus119868sat [exp (119902

119860119870119879119881pv) sdot

119902

119860119870119879119881pv]

+ 119868ph minus 119868sat [exp(119902

119860119870119879119881pv) minus 1]

= minus119868sat (119902

119860119870119879119881pv + 1) sdot exp(

119902

119860119870119879119881pv) + (119868ph + 119868sat)

(19)

The differential form of 119904 is

119904 = minus119868sat119902

119860119870119879sdot119889119881pv

119889119905exp(

119902

119860119870119879) minus 119868sat (

119902119881pv

119860119870119879+ 1)

sdot exp(119902119881pv

119860119870119879)

119902

119860119870119879

119889119881pv

119889119905

= minus119868sat119902

119860119870119879sdot119889119881pv

119889119905exp(

119902119881pv

119860119870119879)(

119902119881pv

119860119870119879+ 2)

(20)

According to 119904 119904 lt 0 if 119904 gt 0 then 119904 lt 0 119889119881pv119889119905 gt 0119881pv willincrease when 119904 tends to sliding manifold and if 119904 lt 0 then119904 gt 0 119889119881pv119889119905 lt 0 119881pv will decrease when 119904 tends to slidingmanifoldThe dynamic accommodation of119881pv can be seen inFigure 7

Considering that the new converter is composed of twoparalleled boost converters the purpose of the additionalswitched capacitor and coupled inductance is to increase thevoltage gain Thus the drive signals for the switches can bedesigned respectively From the state space model of thesingle boost converter when the switch is on 119881pv increases

and when the switch is off 119881pv decreases (as discussed by[23])

Rearranging (17)

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv= 119881pv (

119894pv

119881pv+119889119894pv

119889119881pv) (21)

The differential form approximates to

Δ119894pv

Δ119881pvasymp119889119894pv

119889119881pv (22)

Thus

119904 = 119881pv (119894pv

119881pv+Δ119894pv

Δ119881pv) (23)

Theoretically in the case of low requirement both (17) and(24) can be chosen as the sliding surface

119904 = Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv (24)

The corresponding control law is

119906 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)] (25)

42 Evolution Path Selection The principle of determiningthe evolution path is to ensure that the error state can goto zero at any increase of time (as discussed by [24 25]) Inthis research the evolution path is an exponential function asshown in Figure 8

With this exponential evolution path the value of thedynamic characteristic of the converter will decrease expo-nentially to zero by

119867 = 1198670ℓminus120582119905

(26)

where 119867 is the dynamic characteristic of system 1198670is the

initial value of 119867 and 120582 is a dynamic evolution factor Thederivative of119867 is

119889119867

119889119905= minus120582119867

0ℓminus120582119905

= minus120582119867

119889119867

119889119905+ 120582119867 = 0

(27)

43 Analysis of Duty Cycle In order to synthesize the controllaw of the dynamic evolution controller the dynamic equa-tion of converter system has to be established and analyzedOn basis of the state space averagemodel when the converterworks on DCM the voltage and current dynamics of theconverter are given by

119881pv = 119871119889119894in119889119905

+119881119900

4(1 minus 120572) (28)

where 119871 is the inductance 119881pv is the input voltage 119894in is theinput current119881

119900is the output voltage and 120572 is the duty cycle

respectively

Journal of Applied Mathematics 7

Sliding manifold

Trajectory s chattersalong the manifoldand moves toward

VO

pv

120577 = minusΔs gt 0

s = 0

120577 = +Δ

s lt 0

Ppv

s = 0

Figure 7 The dynamic accommodation of 119881pv

119894in asymp 119894pv rearranging (28) the output voltage of convertercan be written as

119881pv = 119871119889119894pv

119889119905+119881119900

4(1 minus 120572) (29)

119881119900= 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (30)

Define a linear voltage error function as shown below

119864 = 119896119890V = 119896 (119881ref minus 119881119900) (31)

where 119896 gt 0Substituting (27) into (31) yields

119896119889119890V

119889119905+ 120582119896119890V = 0 (32)

Combining (31) with (32)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 119881

119900 (33)

Directly substituting the converter output voltage 119881119900from

(30) into (33)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (34)

The obtained duty cycle formula is given by

120572 =119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905 (35)

Formula (35) forces the state error function 119864 to satisfy thedynamic evolution function (27) Consequently the stateerror function 119864 is forced to make evolution by followingequation (35) and decrease to zero (119864 = 0) with a decreaserate 120582 The outputting voltage of converter converges to theconverters steady state

119881119900= 119881ref (36)

From the synthesis procedure it is clear that the dynamicevolution controller works on the full nonlinear system and

H

H0

0 t

H = H0eminus120582t

Figure 8 Exponential evolution path

does not need any linearization or simplification on thesystem model at all and it is interesting to note that thecontrol law in (35) consists of four distinct parts The firstpart is the feed-forward term which is calculated based onthe duty cycle at the previous sampling instant This termcompensates for variations in the input voltages The secondand third terms consist of proportional and derivative termsof the perturbations in the output voltage respectively Thelast term consists of the derivative terms of the inductancecurrent Since the inputting and outputting voltages andinductance current are involved in calculation of duty cycleit can compensate all the variation of them in the dynamicevolution

It should be noted that both119881pv and 119894pv are not the voltageand current at the MPP in (35) Consequently we expect thatthe controller can achieve MPPT and high-gain boost thusconsidering the logic relationship of control signals the finalcontrol law can be described as follows combing (25) with(35)

120575 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)]

sdot (119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905)

(37)

44 PWM Generation The PWM signals are generated bycomparing a control signal with a constant peak repetitivetriangle signal (119881cs) The frequency of the repetitive trianglesignal establishes the constant switching frequency

Figure 9 shows the PWM signals generation techniquePWM1 is produced when the control signal 119881

1198981is greater

than 119881119888119904while PWM2 is produced when 119881

1198981is less than 119881

119888119904

The values of the desired duty cycle 120572 can be gotten from(37) PWM signals are generated by (39)

1198811198981

= 120575 (38)

8 Journal of Applied Mathematics

PWM1 = 1 1198811198981

gt 119881119888119904

0 1198811198981

lt 119881119888119904

PWM2 = 1 1198811198981

lt 119881119888119904

0 1198811198981

gt 119881119888119904

(39)

5 Simulation

Some simulation has been done in order to verify theperformance of the topology and the main parameters canbe referred to in Table 1

The portion of the simulation results is given in Figure 10Figure 10(a) shows the driving signals for 119878

1and 1198782 which

are generated by the control law 120575 The outputting voltageand power of PV cell in process of MPPT can be seen inFigure 10(b) From the simulation onemay get the conclusionthat the DSME PWM controller can ensure that the PV cellruns at the maximum power point in a short time Althoughthe overshoot with a little range occurs in the process ofpower adjustment it will be eliminated soon and the stableoutputting power of 260W is achieved During the dynamicadjustment there is a stable and fleet performance withoutoscillation

In Figure 10(c) the dynamic response of the novel topol-ogy is the curve labeled overall trend This curse shows thatthe converter can raise the voltage from 36V to 400V steadilywithout a long time The curve labeled partial enlargementshows that the outputting voltage has a low voltage ripplewhen it is in stable state Considering the whole PV grid-connected system a DC-link high voltage with low rippleis convenient to choice and maintaining of the DC-linkcapacitor

As mentioned before the converter can run under fourmodes in every switching period The tendencies of 119894

1198711and

1198941198712as well as 119894in in every mode have been discussed in part III

and it is also easy to get the conclusion about 119894in = 1198941198711+ 1198941198712

based on Kirchhoff rsquos current law so the tendency of 119894in canbe obtained by the sum of 119894

1198711and 1198941198712 The first and second

one of Figure 10(d) describe the stable current of 1198941198711

and 1198941198712

and 119894in respectively over the same period in stable modeFigure 10(d) shows that the current 119894in has little ripple whichcan reduce the coupled inductance losses and improve systemefficiency

Figure 10(e) shows the voltage waves of switched capac-itors 119862

1and 119862

2 The voltage of 119862

1is three times as much as

the one of 1198622 close to three quarters of 119881

119900 this simulation

verifies the formula (13)

6 Experiments

Experiments have been finished in our laboratory just inorder to verify the validity of theoretical analysis above andthe performance of the new converter proposed in this paper

Experimental waveforms of the new converter are shownin Figure 11 The driving signals of 119878

1and 119878

2from DSP on

basis of the algorithm in this paper are given in Figure 11(a)Obviously the new converter can raise the voltage from36V to 400V when the duty cycle equals 064 (calculated

Table 1 The main parameters

Parameters Values119881ocV 443119868scA 77119881mp at MPPV 36119868mp at MPPA 72Coupling coefficient 091198711

and 1198712

mH 041198621

120583F 101198622

120583F 1001198620

120583F 22119877kΩ 1Switching frequencykHz 50

PWM1PWM2

PWM120575(Vo Vpv ipv ΔVpv Δipv )

0

Vo

Vcsipv

Vpv Vm1

Figure 9 PWM generator

from Figure 11(a)) The output voltage wave (after 10 timesattenuation) is shown in Figure 11(b) Moreover the rippleof the output voltage shown in Figure 11(b) can be limitedin 8V which can decrease the loss of switched capacitorsas shown in (15) Figure 11(c) shows the wave of the inputcurrent The little input current ripple can reduce both theloss and the volume of the coupled inductance Figures 11(d)and 11(e) show the voltage waveform of switched capacitors1198621and 119862

2 respectively (after 10 times attenuation) The

voltage of 1198621 about 300Vsim306V is three times as much as

the one of 1198622 about 92Vsim100V which is consistent with

both the analysis and the simulation results above besidesthe difference between the maximum and the minimum ofvoltage in the course of charging for each capacitance can belimited to a fairly small range which can decrease the loss ofswitched capacitors

Burrs existing in experimental waveform mainly stemfrom the effect of EMI in the experiment

7 Conclusion

This paper proposed a new high-gain interleaved converterwith coupled inductances and switched capacitors networkwhich are suited for PV application After explaining its oper-ating principle and analyzing its steady-state performancethen a control strategy for it on basis of SM-MPPT andDEC theory was designed Both simulation analysis andexperimental results showed that the proposed system hadthe following good performance

(1) Achieving MPPT and producing a high-gain voltagein an extremely short time

(2) Decreasing the ripple of the outputting voltage as wellas the inputting current and improving power density

Journal of Applied Mathematics 9

For S

1

ForS

2

(a) Driving signals for 1198781amp 1198782

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

Time (s)

Volta

ge (V

) and

pow

er (W

)

The outputting voltage and power of PV cells under MPPT

Vpv outputting voltageP outputting powerpv

(b) The output voltage and power of PV cell in process of MPPT

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

350

400

450The outputting voltage of the new topological converter

Time (s)

Volta

ge (V

)

399

3995

400

4005

401

Partial enlargement of outtputing voltage

Overall trendPartial enlargement

(c) The output voltage wave

0

5

Curr

ent (

A) The Lofwavecurrent 1 and L2 in stable mode

001 0015 002 0025 00305

1015

Time (s)

Curr

ent (

A)

iL1

iL2

The current wave of iin in the dynamic mode

65

7

75

Curr

ent (

A) The current wave of iin in stable mode

(d) The current of 1198711and 1198712and 119894in

0 001 002 003 004 005 006 007 008 009 01minus100

minus50

0

50

100

150

200

250

300

350

400

Time (s)

Volta

ge (V

)

The voltage of and

Voltage of C1

Voltage of C2

C1 C2

(e) The voltage wave of switched capacitor 1198621and 119862

2

Figure 10 Simulation results

10 Journal of Applied Mathematics

(a) Driving signals for 1198781and 1198782from DSP (b) The outputting voltage wave (X10)

(c) The wave of input current (d) The voltage wave of switched capacitor 1198621(X10)

(e) The voltage wave of switched capacitor 1198622(X10)

Figure 11 Experimental validation

(3) Reducing the volume of magnetic components andcutting the cost of system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education of China (no20120032110070)

References

[1] C Zhang and X He ldquoMaximum power point tracking by usingasymmetric fuzzy control combined with PID for photovoltaic

energy generation systemrdquo Transactions of China Electrotechni-cal Society vol 20 no 10 pp 72ndash75 2005

[2] Z Liao and X Ruan ldquoControl strategy for bi-directionalDCDC converter of a novel stand-alone photovoltaic powersystemrdquo Transactions of China Electrotechnical Society vol 23no 1 pp 97ndash103 2008

[3] M Dong J Yang K Peng and A Luo ldquoZero average incre-mental conductance maximum power point tracking controlfor photovoltaic systemrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 30 no 21 pp 48ndash53 2010

[4] J M Enrique J M Andujar and M A Bohorquez ldquoAreliable fast and low cost maximum power point tracker forphotovoltaic applicationsrdquo Solar Energy vol 84 pp 79ndash89 2010

[5] L Xie RGong and J Li ldquoAnalysis of the dynamical characteris-tics of the Interleaved boost converter inmaximumpower pointtracking for photovoltaic powerrdquo in Proceedings of the ChineseSociety for Electrical Engineering (CSEE rsquo13) vol 33 no 6 pp39ndash43 2013

Journal of Applied Mathematics 11

[6] W Li and X He ldquoReview of nonisolated high-step-up DCDCconverters in photovoltaic grid-connected applicationsrdquo IEEETransactions on Industrial Electronics vol 58 no 4 pp 1239ndash1250 2011

[7] Q LuoH Yan S Zhi C Zou and L Zhou ldquoAn interleaved highstep-up zero current transition boost converterrdquo in Proceedingsof the Chinese Society for Electrical Engineering (CSEE rsquo13) vol33 no 12 pp 18ndash21 2013

[8] W Huang and B Lehman ldquoMitigation and utilization of theinductor coupling effect in interleaved multiphase dcdc con-vertersrdquo in Proceedings of the IEEE Energy Conversion Congressand Exposition (ECCE rsquo13) pp 1822ndash1829 2013

[9] G Zhu B A McDonald and K Wang ldquoModeling and analysisof coupled inductors in power convertersrdquo IEEE Transactionson Power Electronics vol 26 no 5 pp 1355ndash1363 2011

[10] Q-B Hu B Qu and Z-Y Lu ldquoNovel step-up VRM-two-phaseinterleaved coupled-boost converterrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 26 no 9 pp 94ndash98 2006

[11] D A Grant Y Darroman and J Suter ldquoSynthesis of tapped-inductor switched-mode convertersrdquo IEEE Transactions onPower Electronics vol 22 no 5 pp 1964ndash1969 2007

[12] M Nymand and M A E Andersen ldquoHigh-efficiency isolatedboost DC-DC converter for high-power low-voltage fuel-cellapplicationsrdquo IEEE Transactions on Industrial Electronics vol57 no 2 pp 505ndash514 2010

[13] R Haroun A Cid-Pastor A El Aroudi and L Martinez-Salamero ldquoCascade connection ofDC-DC switching convertersby means of self-oscillating dc-transformersrdquo in Proceedingsof the 15th International Power Electronics and Motion ControlConference (EPEPEMC rsquo12) 2012

[14] R Guo C Wang and T Li ldquoOptimum design of couplinginductors for magnetic integration in three-phase interleavingBuck dcdc converterrdquo in Proceedings of the 8th IEEE Conferenceon Industrial Electronics and Applications (ICIEA rsquo13) pp 1029ndash1033 2013

[15] H Wu J Gu J Zhang Y Xing and G Chen ldquoHigh efficiencyhigh step-up Boost-Flyback DCDC converterrdquo Proceedings ofthe Chinese Society of Electrical Engineering vol 31 no 24 pp40ndash45 2011

[16] T J Liang andK C Tseng ldquoAnalysis of integrated boost-flybackstep-up converterrdquo IEE Proceedings Electric Power Applicationsvol 152 no 2 pp 217ndash225 2005

[17] J L Kui-Jun P Byoung-Gun K Rae-young and H Dong-seok ldquoNonisolated ZVT two-inductor boost converter with asingle resonant inductor for high step-up applicationsrdquo IEEETransactions on Power Electronics vol 27 no 4 pp 1966ndash19732012

[18] Z Lu L Zheng and Z Ma ldquoInterleaved high gain boostconverter with switched capacitor networkrdquo Transactions ofChina Electrotechnical Society vol 27 no 11 pp 154ndash156 2012

[19] A S Samosir and A H M Yatim ldquoImplementation of dynamicevolution control of bidirectional DC-DC converter for inter-facing ultracapacitor energy storage to fuel-cell systemrdquo IEEETransactions on Industrial Electronics vol 57 no 10 pp 3468ndash3473 2010

[20] A S Samosir and A H M Yatim ldquoDynamic evolution controlof bidirectionalDC-DCconverter for interfacing ultra capacitorenergy storage to fuel cell electric vehicle systemrdquo in Proceedingsof the Power Engineering Conferencee (AUPEC rsquo08) AustralasianUniversities December 2008

[21] A H ALQahtani M S Abuhamdeh Y M Alsmadi and VI Utkin ldquoPhotovoltaic power optimization using sliding mode

control with a two-axis tracking systemrdquo in Proceedings of theIEEE Energytech pp 1ndash6 2013

[22] D G Montoya C A R Paja and R Giral ldquoA new solutionof maximum power point tracking based on sliding modecontrolrdquo in Proceedings of the 39th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo13) pp 8350ndash83552013

[23] J He ldquoConnected power system based on sliding mode theoryrdquoin Study on Control Algorithm of Maximum Power Point Track-ing for Photovoltaic Grid Shandong University 2012

[24] A S Samosir and A H M Yatim ldquoDynamic evolution con-troller for single phase inverter applicationrdquo in Proceedings ofthe IEEE Symposium on Industrial Electronics and Applications(ISIEA rsquo09) vol 1 pp 530ndash535 October 2009

[25] A S Samosir M Anwari and A H M Yatim ldquoDynamicevolution control of interleaved boostDC-DC converter for fuelcell applicationrdquo in Proceedings of the 9th International Powerand Energy Conference (IPEC rsquo10) pp 869ndash874 October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

C1

C2

VD1

VD2

VD3

S1S2 C0 R

M

123

+

minus

L9984001

L9984002

+minus

+minus

iL1iin

iL2

C2

C1

VoVpv

(a) Mode 1 1198781-on 1198782-off VD

1-off VD

2-on VD

3-on

C1

C2

VD1

VD2

VD3

S1S2

C0 R

M

1 2 +

minus

L9984001

L9984002

+minus

+minus

iL1iin

iL2

C2

C1

VoVpv

(b) Mode 2 1198781-on 1198782-off VD

1-off VD

2-off VD

3-off

C1

C2

VD1

VD2

VD3

S1S2 C0 R

M

12 3 +

minus

L9984001

L9984002

+minus

+minus

iL1

iL2

C2

C1

VoVpv

iin

(c) Mode 3 1198781-off 1198782-on VD

1-on VD

2-off VD

3-on

C1

C2

VD1

VD2

VD3

S1S2 C0 R

M

12

+

minus

L9984001

L9984002

+minus

+minus

iL1

iL2

C2

C1

VoVpv

iin

(d) Mode 4 1198781-off 1198782-on VD

1-off VD

2-off VD

3-off

Figure 4 Operation states and current paths of the converter

t

t

t

t

t

t

S1

S2

vc1

vc2

1 2 3 4

iL1

iL2

iin

iL1 iL2

iin = iL1 + iL2

t1 t2 t3t4t0

12057211T1205721T 12057221T 1205722T

Figure 5 Main waveforms of the new converter

Mode 1 (Figure 4(a)) At time 1199050 1198781is on while 119878

2is off the

current in the inductance 11987110158401

starts to rise linearly while 11987110158402

continues to discharge (the current in 11987110158402

was acquired in thelast switching period) Capacitor 119862

2discharges leading to its

voltage falling But the voltage of capacitor1198621rises because119862

1

is charged besides capacitor 1198620is charged by input voltage

and inductor in this mode The rates of change of 1198941198711

and 1198941198712

are approximately given by

1198891198941198711

119889119905=119881119900minus V1198621

11987110158401

1198891198941198712

119889119905= minus

119881119900minus V1198622

11987110158402

(3)

Mode 2 (Figure 4(b)) At time 1199051 1198941198712falls to zero 119894

1198711continues

to rise linearly 119877 is charged by capacitor 1198620 The rate of

change of 1198941198711

is

1198891198941198711

119889119905=119881pv

1198711

(4)

where 1198711= 1198711015840

1

+119872

Mode 3 (Figure 4(c)) At time 1199052 1198781is off while 119878

2is on 119894

1198712

starts to rise linearly from zero and 1198711starts to discharge

Besides 1198621discharges too while 119862

2starts to charge More-

over1198620is charged by input voltage and inductorThe rates of

change of 1198941198711

and 1198941198712

are approximately given by1198891198941198711

119889119905= minus

119881119900minus V1198621

11987110158401

(5)

1198891198941198712

119889119905=119881119900minus V1198622

11987110158402

(6)

V1198621+ V1198622

= 119881119900 (7)

Mode 4 (Figure 4(d)) At time 1199053 1198781is offwhile 119878

2is on 119894

1198711falls

to zero 1198941198712continues to rise linearly119877 is charged by capacitor

1198620 The rate of change of 119894

1198712is

1198891198941198712

119889119905=119881pv

1198712

(8)

where 1198712= 1198711015840

2

+119872The switching period will be repeated when 119878

1is on again

at time 1199054

Journal of Applied Mathematics 5

32 Steady-State Analysis Assume that the current wave-forms shown in Figure 5 have reached a steady state Fromthe waveform 119894

1198711as shown in Figure 5 it can be found that

119881119900minus V1198621

11987110158401

12057211119879 +

119881pv

1198711

(1205721minus 12057211) 119879 =

119881119900minus V1198621

11987110158401

(1205722minus 12057221) 119879

(9)

From the waveform 1198941198712

as shown in Figure 5 it can be foundthat

119881119900minus V1198622

11987110158402

(1205722minus 12057221) 119879 +

119881pv

1198712

12057221119879 =

119881119900minus V1198622

11987110158402

12057211119879 (10)

Also from Figure 5 the average values of 1198941198711

and 1198941198712 denoted

as 1198681and 1198682 respectively are found as

1198681=1

1198791

2

119881119900minus V1198621

11987110158401

sdot (12057211119879)2

+1

2

119881119900minus V1198621

11987110158401

sdot [(1205722minus 12057221) 119879]2

+1

2(1205721minus 12057211) 119879

sdot [2 (119881119900minus V1198621)

11987110158401

sdot 12057211119879 +

119881pv

1198711

(1205721minus 12057211) 119879]

1198682=1

1198791

2

119881119900minus V1198622

11987110158402

sdot (12057211119879)2

+1

2

119881119900minus V1198622

11987110158402

sdot [(1205722minus 12057221) 119879]2

+1

212057221119879

sdot [2 (119881119900minus V1198622)

11987110158402

sdot (1205722minus 12057221) 119879 +

119881pv

1198712

12057221119879]

(11)

Assume that the converter is lossless that is the input power119875119894is equal to the output power 119875

119900 Consider

119881pv (1198681 + 1198682) =1198812

119900

119877 (12)

where 119877 is the load resistanceIn principle (7) and (9)ndash(12) can be solved to find 119881

119900

However to simplify the calculation it is assumed that 1205721=

120572 1205722= 1 minus 120572 119871

1= 1198712= 119871 and 119871

1015840

1

= 1198711015840

2

= (1 minus 119896)119871According to some algebraic manipulations and solving aquadratic equation derived from (12) the approximate valuesof 1198811198621 1198811198622 and 119881

119900can be referred as follows

1198811198621

=3

1 minus 120572119881pv =

3

4119881119900

1198811198622

=1

1 minus 120572119881pv =

1

4119881119900

(13)

119881119900=

4

1 minus 120572119881pv (14)

The effect of R on the calculation is very little thus it isneglected when getting the approximate values as shown in(13) and (14)

A careful study of the waveforms shown in Figure 5 willreveal the following interesting facts

C1

C2

VD1

VD2

VD3

S1S2 C0 R Vo

++

minusVpv

L1

L2

+minus

+minus

Vm1

ipv

120575(Vo Vpv ipv ΔVpv Δipv )

PVpanels

PWM

DSME PWM controller

PWM 2 PWM 1

lowast

lowast

Figure 6 The control flow chart of DSME PWM controller

(1) As far as the input current 119894in is concerned theconverter appears to operate in CCM (because 119894inis continuous) Thus the peak current stress of theinductances and the input current ripple can bemaintained relatively low

(2) However since 1198941198711and 1198941198712are discontinuous the new

converter is actually operating in DCM Also sincethe rectifier diodes VD

1simVD3turn off before 119878

11198782

turns on the reverse-recovery loss of the rectifiers iseliminated

(3) Besides the current of inductance 11989411987111198941198712

has fallento zero before 119878

11198782turns on ZCS soft switching

operation during the whole switching transition isachieved

Furthermore when considering the parasitic resistancesof switches and capacitances (119903

119904and 119903119888) the switching loss due

to capacitances in the course of charging can be defined as (15)in an independent converter which contains 119899 capacitances

119882119904=119899119862

2sdot1 + 119890minus120572119879120591

1 minus 119890minus120572119879120591(Δ119881119900)2

(15)

where 119862 is the value of capacitances and 120591 = (119903119904+ 119903119888)119862

From (15) 119882119904do exist even under the ideal conditions

mainly depending on Δ119881119900 Thus in order to reduce 119882

119904

the difference between the maximum and the minimum ofvoltage in the course of charging for all capacitances shouldbe decreased

4 DSME PWM Controller

The original intention of the DSME control is to achieveMPPT as well as high-gain boost also to reduce the errorstate by forcing the error state to follow the specific pathwhich ensures the error state goes to zero in increase oftime (as discussed by [19 20]) The control flow chart ofdynamic sliding mode evolution PWM Controller can beseen in Figure 6

The modulating signal of PWM generation 1198811198981

is pro-duced by the formula 120575 combing the sliding mode theory

6 Journal of Applied Mathematics

with the dynamic evolution algorithm The specific designprocess of the DSME controller can be described as follows

41 MPPT by Sliding Mode Control When the PV cells runat the point of maximum power

119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv= 0 (16)

In SMC the state space is divided into 119904 gt 0 and 119904 lt 0 bysliding manifold (119904 = 0) The movement on 119904 = 0 can becalled sliding mode dynamics Considering 119889119875pv119889119881pv = 0the sliding manifold can be chosen as (as discussed by [2122])

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv (17)

Reconsidering the mathematical model of PV cell

119894pv = 119868ph minus 119868sat [exp(119902119881pv

119860119870119879) minus 1] (18)

Directly substituting 119894pv from (18) into (17)

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv

= minus119868sat [exp (119902

119860119870119879119881pv) sdot

119902

119860119870119879119881pv]

+ 119868ph minus 119868sat [exp(119902

119860119870119879119881pv) minus 1]

= minus119868sat (119902

119860119870119879119881pv + 1) sdot exp(

119902

119860119870119879119881pv) + (119868ph + 119868sat)

(19)

The differential form of 119904 is

119904 = minus119868sat119902

119860119870119879sdot119889119881pv

119889119905exp(

119902

119860119870119879) minus 119868sat (

119902119881pv

119860119870119879+ 1)

sdot exp(119902119881pv

119860119870119879)

119902

119860119870119879

119889119881pv

119889119905

= minus119868sat119902

119860119870119879sdot119889119881pv

119889119905exp(

119902119881pv

119860119870119879)(

119902119881pv

119860119870119879+ 2)

(20)

According to 119904 119904 lt 0 if 119904 gt 0 then 119904 lt 0 119889119881pv119889119905 gt 0119881pv willincrease when 119904 tends to sliding manifold and if 119904 lt 0 then119904 gt 0 119889119881pv119889119905 lt 0 119881pv will decrease when 119904 tends to slidingmanifoldThe dynamic accommodation of119881pv can be seen inFigure 7

Considering that the new converter is composed of twoparalleled boost converters the purpose of the additionalswitched capacitor and coupled inductance is to increase thevoltage gain Thus the drive signals for the switches can bedesigned respectively From the state space model of thesingle boost converter when the switch is on 119881pv increases

and when the switch is off 119881pv decreases (as discussed by[23])

Rearranging (17)

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv= 119881pv (

119894pv

119881pv+119889119894pv

119889119881pv) (21)

The differential form approximates to

Δ119894pv

Δ119881pvasymp119889119894pv

119889119881pv (22)

Thus

119904 = 119881pv (119894pv

119881pv+Δ119894pv

Δ119881pv) (23)

Theoretically in the case of low requirement both (17) and(24) can be chosen as the sliding surface

119904 = Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv (24)

The corresponding control law is

119906 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)] (25)

42 Evolution Path Selection The principle of determiningthe evolution path is to ensure that the error state can goto zero at any increase of time (as discussed by [24 25]) Inthis research the evolution path is an exponential function asshown in Figure 8

With this exponential evolution path the value of thedynamic characteristic of the converter will decrease expo-nentially to zero by

119867 = 1198670ℓminus120582119905

(26)

where 119867 is the dynamic characteristic of system 1198670is the

initial value of 119867 and 120582 is a dynamic evolution factor Thederivative of119867 is

119889119867

119889119905= minus120582119867

0ℓminus120582119905

= minus120582119867

119889119867

119889119905+ 120582119867 = 0

(27)

43 Analysis of Duty Cycle In order to synthesize the controllaw of the dynamic evolution controller the dynamic equa-tion of converter system has to be established and analyzedOn basis of the state space averagemodel when the converterworks on DCM the voltage and current dynamics of theconverter are given by

119881pv = 119871119889119894in119889119905

+119881119900

4(1 minus 120572) (28)

where 119871 is the inductance 119881pv is the input voltage 119894in is theinput current119881

119900is the output voltage and 120572 is the duty cycle

respectively

Journal of Applied Mathematics 7

Sliding manifold

Trajectory s chattersalong the manifoldand moves toward

VO

pv

120577 = minusΔs gt 0

s = 0

120577 = +Δ

s lt 0

Ppv

s = 0

Figure 7 The dynamic accommodation of 119881pv

119894in asymp 119894pv rearranging (28) the output voltage of convertercan be written as

119881pv = 119871119889119894pv

119889119905+119881119900

4(1 minus 120572) (29)

119881119900= 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (30)

Define a linear voltage error function as shown below

119864 = 119896119890V = 119896 (119881ref minus 119881119900) (31)

where 119896 gt 0Substituting (27) into (31) yields

119896119889119890V

119889119905+ 120582119896119890V = 0 (32)

Combining (31) with (32)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 119881

119900 (33)

Directly substituting the converter output voltage 119881119900from

(30) into (33)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (34)

The obtained duty cycle formula is given by

120572 =119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905 (35)

Formula (35) forces the state error function 119864 to satisfy thedynamic evolution function (27) Consequently the stateerror function 119864 is forced to make evolution by followingequation (35) and decrease to zero (119864 = 0) with a decreaserate 120582 The outputting voltage of converter converges to theconverters steady state

119881119900= 119881ref (36)

From the synthesis procedure it is clear that the dynamicevolution controller works on the full nonlinear system and

H

H0

0 t

H = H0eminus120582t

Figure 8 Exponential evolution path

does not need any linearization or simplification on thesystem model at all and it is interesting to note that thecontrol law in (35) consists of four distinct parts The firstpart is the feed-forward term which is calculated based onthe duty cycle at the previous sampling instant This termcompensates for variations in the input voltages The secondand third terms consist of proportional and derivative termsof the perturbations in the output voltage respectively Thelast term consists of the derivative terms of the inductancecurrent Since the inputting and outputting voltages andinductance current are involved in calculation of duty cycleit can compensate all the variation of them in the dynamicevolution

It should be noted that both119881pv and 119894pv are not the voltageand current at the MPP in (35) Consequently we expect thatthe controller can achieve MPPT and high-gain boost thusconsidering the logic relationship of control signals the finalcontrol law can be described as follows combing (25) with(35)

120575 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)]

sdot (119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905)

(37)

44 PWM Generation The PWM signals are generated bycomparing a control signal with a constant peak repetitivetriangle signal (119881cs) The frequency of the repetitive trianglesignal establishes the constant switching frequency

Figure 9 shows the PWM signals generation techniquePWM1 is produced when the control signal 119881

1198981is greater

than 119881119888119904while PWM2 is produced when 119881

1198981is less than 119881

119888119904

The values of the desired duty cycle 120572 can be gotten from(37) PWM signals are generated by (39)

1198811198981

= 120575 (38)

8 Journal of Applied Mathematics

PWM1 = 1 1198811198981

gt 119881119888119904

0 1198811198981

lt 119881119888119904

PWM2 = 1 1198811198981

lt 119881119888119904

0 1198811198981

gt 119881119888119904

(39)

5 Simulation

Some simulation has been done in order to verify theperformance of the topology and the main parameters canbe referred to in Table 1

The portion of the simulation results is given in Figure 10Figure 10(a) shows the driving signals for 119878

1and 1198782 which

are generated by the control law 120575 The outputting voltageand power of PV cell in process of MPPT can be seen inFigure 10(b) From the simulation onemay get the conclusionthat the DSME PWM controller can ensure that the PV cellruns at the maximum power point in a short time Althoughthe overshoot with a little range occurs in the process ofpower adjustment it will be eliminated soon and the stableoutputting power of 260W is achieved During the dynamicadjustment there is a stable and fleet performance withoutoscillation

In Figure 10(c) the dynamic response of the novel topol-ogy is the curve labeled overall trend This curse shows thatthe converter can raise the voltage from 36V to 400V steadilywithout a long time The curve labeled partial enlargementshows that the outputting voltage has a low voltage ripplewhen it is in stable state Considering the whole PV grid-connected system a DC-link high voltage with low rippleis convenient to choice and maintaining of the DC-linkcapacitor

As mentioned before the converter can run under fourmodes in every switching period The tendencies of 119894

1198711and

1198941198712as well as 119894in in every mode have been discussed in part III

and it is also easy to get the conclusion about 119894in = 1198941198711+ 1198941198712

based on Kirchhoff rsquos current law so the tendency of 119894in canbe obtained by the sum of 119894

1198711and 1198941198712 The first and second

one of Figure 10(d) describe the stable current of 1198941198711

and 1198941198712

and 119894in respectively over the same period in stable modeFigure 10(d) shows that the current 119894in has little ripple whichcan reduce the coupled inductance losses and improve systemefficiency

Figure 10(e) shows the voltage waves of switched capac-itors 119862

1and 119862

2 The voltage of 119862

1is three times as much as

the one of 1198622 close to three quarters of 119881

119900 this simulation

verifies the formula (13)

6 Experiments

Experiments have been finished in our laboratory just inorder to verify the validity of theoretical analysis above andthe performance of the new converter proposed in this paper

Experimental waveforms of the new converter are shownin Figure 11 The driving signals of 119878

1and 119878

2from DSP on

basis of the algorithm in this paper are given in Figure 11(a)Obviously the new converter can raise the voltage from36V to 400V when the duty cycle equals 064 (calculated

Table 1 The main parameters

Parameters Values119881ocV 443119868scA 77119881mp at MPPV 36119868mp at MPPA 72Coupling coefficient 091198711

and 1198712

mH 041198621

120583F 101198622

120583F 1001198620

120583F 22119877kΩ 1Switching frequencykHz 50

PWM1PWM2

PWM120575(Vo Vpv ipv ΔVpv Δipv )

0

Vo

Vcsipv

Vpv Vm1

Figure 9 PWM generator

from Figure 11(a)) The output voltage wave (after 10 timesattenuation) is shown in Figure 11(b) Moreover the rippleof the output voltage shown in Figure 11(b) can be limitedin 8V which can decrease the loss of switched capacitorsas shown in (15) Figure 11(c) shows the wave of the inputcurrent The little input current ripple can reduce both theloss and the volume of the coupled inductance Figures 11(d)and 11(e) show the voltage waveform of switched capacitors1198621and 119862

2 respectively (after 10 times attenuation) The

voltage of 1198621 about 300Vsim306V is three times as much as

the one of 1198622 about 92Vsim100V which is consistent with

both the analysis and the simulation results above besidesthe difference between the maximum and the minimum ofvoltage in the course of charging for each capacitance can belimited to a fairly small range which can decrease the loss ofswitched capacitors

Burrs existing in experimental waveform mainly stemfrom the effect of EMI in the experiment

7 Conclusion

This paper proposed a new high-gain interleaved converterwith coupled inductances and switched capacitors networkwhich are suited for PV application After explaining its oper-ating principle and analyzing its steady-state performancethen a control strategy for it on basis of SM-MPPT andDEC theory was designed Both simulation analysis andexperimental results showed that the proposed system hadthe following good performance

(1) Achieving MPPT and producing a high-gain voltagein an extremely short time

(2) Decreasing the ripple of the outputting voltage as wellas the inputting current and improving power density

Journal of Applied Mathematics 9

For S

1

ForS

2

(a) Driving signals for 1198781amp 1198782

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

Time (s)

Volta

ge (V

) and

pow

er (W

)

The outputting voltage and power of PV cells under MPPT

Vpv outputting voltageP outputting powerpv

(b) The output voltage and power of PV cell in process of MPPT

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

350

400

450The outputting voltage of the new topological converter

Time (s)

Volta

ge (V

)

399

3995

400

4005

401

Partial enlargement of outtputing voltage

Overall trendPartial enlargement

(c) The output voltage wave

0

5

Curr

ent (

A) The Lofwavecurrent 1 and L2 in stable mode

001 0015 002 0025 00305

1015

Time (s)

Curr

ent (

A)

iL1

iL2

The current wave of iin in the dynamic mode

65

7

75

Curr

ent (

A) The current wave of iin in stable mode

(d) The current of 1198711and 1198712and 119894in

0 001 002 003 004 005 006 007 008 009 01minus100

minus50

0

50

100

150

200

250

300

350

400

Time (s)

Volta

ge (V

)

The voltage of and

Voltage of C1

Voltage of C2

C1 C2

(e) The voltage wave of switched capacitor 1198621and 119862

2

Figure 10 Simulation results

10 Journal of Applied Mathematics

(a) Driving signals for 1198781and 1198782from DSP (b) The outputting voltage wave (X10)

(c) The wave of input current (d) The voltage wave of switched capacitor 1198621(X10)

(e) The voltage wave of switched capacitor 1198622(X10)

Figure 11 Experimental validation

(3) Reducing the volume of magnetic components andcutting the cost of system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education of China (no20120032110070)

References

[1] C Zhang and X He ldquoMaximum power point tracking by usingasymmetric fuzzy control combined with PID for photovoltaic

energy generation systemrdquo Transactions of China Electrotechni-cal Society vol 20 no 10 pp 72ndash75 2005

[2] Z Liao and X Ruan ldquoControl strategy for bi-directionalDCDC converter of a novel stand-alone photovoltaic powersystemrdquo Transactions of China Electrotechnical Society vol 23no 1 pp 97ndash103 2008

[3] M Dong J Yang K Peng and A Luo ldquoZero average incre-mental conductance maximum power point tracking controlfor photovoltaic systemrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 30 no 21 pp 48ndash53 2010

[4] J M Enrique J M Andujar and M A Bohorquez ldquoAreliable fast and low cost maximum power point tracker forphotovoltaic applicationsrdquo Solar Energy vol 84 pp 79ndash89 2010

[5] L Xie RGong and J Li ldquoAnalysis of the dynamical characteris-tics of the Interleaved boost converter inmaximumpower pointtracking for photovoltaic powerrdquo in Proceedings of the ChineseSociety for Electrical Engineering (CSEE rsquo13) vol 33 no 6 pp39ndash43 2013

Journal of Applied Mathematics 11

[6] W Li and X He ldquoReview of nonisolated high-step-up DCDCconverters in photovoltaic grid-connected applicationsrdquo IEEETransactions on Industrial Electronics vol 58 no 4 pp 1239ndash1250 2011

[7] Q LuoH Yan S Zhi C Zou and L Zhou ldquoAn interleaved highstep-up zero current transition boost converterrdquo in Proceedingsof the Chinese Society for Electrical Engineering (CSEE rsquo13) vol33 no 12 pp 18ndash21 2013

[8] W Huang and B Lehman ldquoMitigation and utilization of theinductor coupling effect in interleaved multiphase dcdc con-vertersrdquo in Proceedings of the IEEE Energy Conversion Congressand Exposition (ECCE rsquo13) pp 1822ndash1829 2013

[9] G Zhu B A McDonald and K Wang ldquoModeling and analysisof coupled inductors in power convertersrdquo IEEE Transactionson Power Electronics vol 26 no 5 pp 1355ndash1363 2011

[10] Q-B Hu B Qu and Z-Y Lu ldquoNovel step-up VRM-two-phaseinterleaved coupled-boost converterrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 26 no 9 pp 94ndash98 2006

[11] D A Grant Y Darroman and J Suter ldquoSynthesis of tapped-inductor switched-mode convertersrdquo IEEE Transactions onPower Electronics vol 22 no 5 pp 1964ndash1969 2007

[12] M Nymand and M A E Andersen ldquoHigh-efficiency isolatedboost DC-DC converter for high-power low-voltage fuel-cellapplicationsrdquo IEEE Transactions on Industrial Electronics vol57 no 2 pp 505ndash514 2010

[13] R Haroun A Cid-Pastor A El Aroudi and L Martinez-Salamero ldquoCascade connection ofDC-DC switching convertersby means of self-oscillating dc-transformersrdquo in Proceedingsof the 15th International Power Electronics and Motion ControlConference (EPEPEMC rsquo12) 2012

[14] R Guo C Wang and T Li ldquoOptimum design of couplinginductors for magnetic integration in three-phase interleavingBuck dcdc converterrdquo in Proceedings of the 8th IEEE Conferenceon Industrial Electronics and Applications (ICIEA rsquo13) pp 1029ndash1033 2013

[15] H Wu J Gu J Zhang Y Xing and G Chen ldquoHigh efficiencyhigh step-up Boost-Flyback DCDC converterrdquo Proceedings ofthe Chinese Society of Electrical Engineering vol 31 no 24 pp40ndash45 2011

[16] T J Liang andK C Tseng ldquoAnalysis of integrated boost-flybackstep-up converterrdquo IEE Proceedings Electric Power Applicationsvol 152 no 2 pp 217ndash225 2005

[17] J L Kui-Jun P Byoung-Gun K Rae-young and H Dong-seok ldquoNonisolated ZVT two-inductor boost converter with asingle resonant inductor for high step-up applicationsrdquo IEEETransactions on Power Electronics vol 27 no 4 pp 1966ndash19732012

[18] Z Lu L Zheng and Z Ma ldquoInterleaved high gain boostconverter with switched capacitor networkrdquo Transactions ofChina Electrotechnical Society vol 27 no 11 pp 154ndash156 2012

[19] A S Samosir and A H M Yatim ldquoImplementation of dynamicevolution control of bidirectional DC-DC converter for inter-facing ultracapacitor energy storage to fuel-cell systemrdquo IEEETransactions on Industrial Electronics vol 57 no 10 pp 3468ndash3473 2010

[20] A S Samosir and A H M Yatim ldquoDynamic evolution controlof bidirectionalDC-DCconverter for interfacing ultra capacitorenergy storage to fuel cell electric vehicle systemrdquo in Proceedingsof the Power Engineering Conferencee (AUPEC rsquo08) AustralasianUniversities December 2008

[21] A H ALQahtani M S Abuhamdeh Y M Alsmadi and VI Utkin ldquoPhotovoltaic power optimization using sliding mode

control with a two-axis tracking systemrdquo in Proceedings of theIEEE Energytech pp 1ndash6 2013

[22] D G Montoya C A R Paja and R Giral ldquoA new solutionof maximum power point tracking based on sliding modecontrolrdquo in Proceedings of the 39th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo13) pp 8350ndash83552013

[23] J He ldquoConnected power system based on sliding mode theoryrdquoin Study on Control Algorithm of Maximum Power Point Track-ing for Photovoltaic Grid Shandong University 2012

[24] A S Samosir and A H M Yatim ldquoDynamic evolution con-troller for single phase inverter applicationrdquo in Proceedings ofthe IEEE Symposium on Industrial Electronics and Applications(ISIEA rsquo09) vol 1 pp 530ndash535 October 2009

[25] A S Samosir M Anwari and A H M Yatim ldquoDynamicevolution control of interleaved boostDC-DC converter for fuelcell applicationrdquo in Proceedings of the 9th International Powerand Energy Conference (IPEC rsquo10) pp 869ndash874 October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

32 Steady-State Analysis Assume that the current wave-forms shown in Figure 5 have reached a steady state Fromthe waveform 119894

1198711as shown in Figure 5 it can be found that

119881119900minus V1198621

11987110158401

12057211119879 +

119881pv

1198711

(1205721minus 12057211) 119879 =

119881119900minus V1198621

11987110158401

(1205722minus 12057221) 119879

(9)

From the waveform 1198941198712

as shown in Figure 5 it can be foundthat

119881119900minus V1198622

11987110158402

(1205722minus 12057221) 119879 +

119881pv

1198712

12057221119879 =

119881119900minus V1198622

11987110158402

12057211119879 (10)

Also from Figure 5 the average values of 1198941198711

and 1198941198712 denoted

as 1198681and 1198682 respectively are found as

1198681=1

1198791

2

119881119900minus V1198621

11987110158401

sdot (12057211119879)2

+1

2

119881119900minus V1198621

11987110158401

sdot [(1205722minus 12057221) 119879]2

+1

2(1205721minus 12057211) 119879

sdot [2 (119881119900minus V1198621)

11987110158401

sdot 12057211119879 +

119881pv

1198711

(1205721minus 12057211) 119879]

1198682=1

1198791

2

119881119900minus V1198622

11987110158402

sdot (12057211119879)2

+1

2

119881119900minus V1198622

11987110158402

sdot [(1205722minus 12057221) 119879]2

+1

212057221119879

sdot [2 (119881119900minus V1198622)

11987110158402

sdot (1205722minus 12057221) 119879 +

119881pv

1198712

12057221119879]

(11)

Assume that the converter is lossless that is the input power119875119894is equal to the output power 119875

119900 Consider

119881pv (1198681 + 1198682) =1198812

119900

119877 (12)

where 119877 is the load resistanceIn principle (7) and (9)ndash(12) can be solved to find 119881

119900

However to simplify the calculation it is assumed that 1205721=

120572 1205722= 1 minus 120572 119871

1= 1198712= 119871 and 119871

1015840

1

= 1198711015840

2

= (1 minus 119896)119871According to some algebraic manipulations and solving aquadratic equation derived from (12) the approximate valuesof 1198811198621 1198811198622 and 119881

119900can be referred as follows

1198811198621

=3

1 minus 120572119881pv =

3

4119881119900

1198811198622

=1

1 minus 120572119881pv =

1

4119881119900

(13)

119881119900=

4

1 minus 120572119881pv (14)

The effect of R on the calculation is very little thus it isneglected when getting the approximate values as shown in(13) and (14)

A careful study of the waveforms shown in Figure 5 willreveal the following interesting facts

C1

C2

VD1

VD2

VD3

S1S2 C0 R Vo

++

minusVpv

L1

L2

+minus

+minus

Vm1

ipv

120575(Vo Vpv ipv ΔVpv Δipv )

PVpanels

PWM

DSME PWM controller

PWM 2 PWM 1

lowast

lowast

Figure 6 The control flow chart of DSME PWM controller

(1) As far as the input current 119894in is concerned theconverter appears to operate in CCM (because 119894inis continuous) Thus the peak current stress of theinductances and the input current ripple can bemaintained relatively low

(2) However since 1198941198711and 1198941198712are discontinuous the new

converter is actually operating in DCM Also sincethe rectifier diodes VD

1simVD3turn off before 119878

11198782

turns on the reverse-recovery loss of the rectifiers iseliminated

(3) Besides the current of inductance 11989411987111198941198712

has fallento zero before 119878

11198782turns on ZCS soft switching

operation during the whole switching transition isachieved

Furthermore when considering the parasitic resistancesof switches and capacitances (119903

119904and 119903119888) the switching loss due

to capacitances in the course of charging can be defined as (15)in an independent converter which contains 119899 capacitances

119882119904=119899119862

2sdot1 + 119890minus120572119879120591

1 minus 119890minus120572119879120591(Δ119881119900)2

(15)

where 119862 is the value of capacitances and 120591 = (119903119904+ 119903119888)119862

From (15) 119882119904do exist even under the ideal conditions

mainly depending on Δ119881119900 Thus in order to reduce 119882

119904

the difference between the maximum and the minimum ofvoltage in the course of charging for all capacitances shouldbe decreased

4 DSME PWM Controller

The original intention of the DSME control is to achieveMPPT as well as high-gain boost also to reduce the errorstate by forcing the error state to follow the specific pathwhich ensures the error state goes to zero in increase oftime (as discussed by [19 20]) The control flow chart ofdynamic sliding mode evolution PWM Controller can beseen in Figure 6

The modulating signal of PWM generation 1198811198981

is pro-duced by the formula 120575 combing the sliding mode theory

6 Journal of Applied Mathematics

with the dynamic evolution algorithm The specific designprocess of the DSME controller can be described as follows

41 MPPT by Sliding Mode Control When the PV cells runat the point of maximum power

119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv= 0 (16)

In SMC the state space is divided into 119904 gt 0 and 119904 lt 0 bysliding manifold (119904 = 0) The movement on 119904 = 0 can becalled sliding mode dynamics Considering 119889119875pv119889119881pv = 0the sliding manifold can be chosen as (as discussed by [2122])

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv (17)

Reconsidering the mathematical model of PV cell

119894pv = 119868ph minus 119868sat [exp(119902119881pv

119860119870119879) minus 1] (18)

Directly substituting 119894pv from (18) into (17)

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv

= minus119868sat [exp (119902

119860119870119879119881pv) sdot

119902

119860119870119879119881pv]

+ 119868ph minus 119868sat [exp(119902

119860119870119879119881pv) minus 1]

= minus119868sat (119902

119860119870119879119881pv + 1) sdot exp(

119902

119860119870119879119881pv) + (119868ph + 119868sat)

(19)

The differential form of 119904 is

119904 = minus119868sat119902

119860119870119879sdot119889119881pv

119889119905exp(

119902

119860119870119879) minus 119868sat (

119902119881pv

119860119870119879+ 1)

sdot exp(119902119881pv

119860119870119879)

119902

119860119870119879

119889119881pv

119889119905

= minus119868sat119902

119860119870119879sdot119889119881pv

119889119905exp(

119902119881pv

119860119870119879)(

119902119881pv

119860119870119879+ 2)

(20)

According to 119904 119904 lt 0 if 119904 gt 0 then 119904 lt 0 119889119881pv119889119905 gt 0119881pv willincrease when 119904 tends to sliding manifold and if 119904 lt 0 then119904 gt 0 119889119881pv119889119905 lt 0 119881pv will decrease when 119904 tends to slidingmanifoldThe dynamic accommodation of119881pv can be seen inFigure 7

Considering that the new converter is composed of twoparalleled boost converters the purpose of the additionalswitched capacitor and coupled inductance is to increase thevoltage gain Thus the drive signals for the switches can bedesigned respectively From the state space model of thesingle boost converter when the switch is on 119881pv increases

and when the switch is off 119881pv decreases (as discussed by[23])

Rearranging (17)

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv= 119881pv (

119894pv

119881pv+119889119894pv

119889119881pv) (21)

The differential form approximates to

Δ119894pv

Δ119881pvasymp119889119894pv

119889119881pv (22)

Thus

119904 = 119881pv (119894pv

119881pv+Δ119894pv

Δ119881pv) (23)

Theoretically in the case of low requirement both (17) and(24) can be chosen as the sliding surface

119904 = Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv (24)

The corresponding control law is

119906 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)] (25)

42 Evolution Path Selection The principle of determiningthe evolution path is to ensure that the error state can goto zero at any increase of time (as discussed by [24 25]) Inthis research the evolution path is an exponential function asshown in Figure 8

With this exponential evolution path the value of thedynamic characteristic of the converter will decrease expo-nentially to zero by

119867 = 1198670ℓminus120582119905

(26)

where 119867 is the dynamic characteristic of system 1198670is the

initial value of 119867 and 120582 is a dynamic evolution factor Thederivative of119867 is

119889119867

119889119905= minus120582119867

0ℓminus120582119905

= minus120582119867

119889119867

119889119905+ 120582119867 = 0

(27)

43 Analysis of Duty Cycle In order to synthesize the controllaw of the dynamic evolution controller the dynamic equa-tion of converter system has to be established and analyzedOn basis of the state space averagemodel when the converterworks on DCM the voltage and current dynamics of theconverter are given by

119881pv = 119871119889119894in119889119905

+119881119900

4(1 minus 120572) (28)

where 119871 is the inductance 119881pv is the input voltage 119894in is theinput current119881

119900is the output voltage and 120572 is the duty cycle

respectively

Journal of Applied Mathematics 7

Sliding manifold

Trajectory s chattersalong the manifoldand moves toward

VO

pv

120577 = minusΔs gt 0

s = 0

120577 = +Δ

s lt 0

Ppv

s = 0

Figure 7 The dynamic accommodation of 119881pv

119894in asymp 119894pv rearranging (28) the output voltage of convertercan be written as

119881pv = 119871119889119894pv

119889119905+119881119900

4(1 minus 120572) (29)

119881119900= 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (30)

Define a linear voltage error function as shown below

119864 = 119896119890V = 119896 (119881ref minus 119881119900) (31)

where 119896 gt 0Substituting (27) into (31) yields

119896119889119890V

119889119905+ 120582119896119890V = 0 (32)

Combining (31) with (32)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 119881

119900 (33)

Directly substituting the converter output voltage 119881119900from

(30) into (33)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (34)

The obtained duty cycle formula is given by

120572 =119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905 (35)

Formula (35) forces the state error function 119864 to satisfy thedynamic evolution function (27) Consequently the stateerror function 119864 is forced to make evolution by followingequation (35) and decrease to zero (119864 = 0) with a decreaserate 120582 The outputting voltage of converter converges to theconverters steady state

119881119900= 119881ref (36)

From the synthesis procedure it is clear that the dynamicevolution controller works on the full nonlinear system and

H

H0

0 t

H = H0eminus120582t

Figure 8 Exponential evolution path

does not need any linearization or simplification on thesystem model at all and it is interesting to note that thecontrol law in (35) consists of four distinct parts The firstpart is the feed-forward term which is calculated based onthe duty cycle at the previous sampling instant This termcompensates for variations in the input voltages The secondand third terms consist of proportional and derivative termsof the perturbations in the output voltage respectively Thelast term consists of the derivative terms of the inductancecurrent Since the inputting and outputting voltages andinductance current are involved in calculation of duty cycleit can compensate all the variation of them in the dynamicevolution

It should be noted that both119881pv and 119894pv are not the voltageand current at the MPP in (35) Consequently we expect thatthe controller can achieve MPPT and high-gain boost thusconsidering the logic relationship of control signals the finalcontrol law can be described as follows combing (25) with(35)

120575 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)]

sdot (119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905)

(37)

44 PWM Generation The PWM signals are generated bycomparing a control signal with a constant peak repetitivetriangle signal (119881cs) The frequency of the repetitive trianglesignal establishes the constant switching frequency

Figure 9 shows the PWM signals generation techniquePWM1 is produced when the control signal 119881

1198981is greater

than 119881119888119904while PWM2 is produced when 119881

1198981is less than 119881

119888119904

The values of the desired duty cycle 120572 can be gotten from(37) PWM signals are generated by (39)

1198811198981

= 120575 (38)

8 Journal of Applied Mathematics

PWM1 = 1 1198811198981

gt 119881119888119904

0 1198811198981

lt 119881119888119904

PWM2 = 1 1198811198981

lt 119881119888119904

0 1198811198981

gt 119881119888119904

(39)

5 Simulation

Some simulation has been done in order to verify theperformance of the topology and the main parameters canbe referred to in Table 1

The portion of the simulation results is given in Figure 10Figure 10(a) shows the driving signals for 119878

1and 1198782 which

are generated by the control law 120575 The outputting voltageand power of PV cell in process of MPPT can be seen inFigure 10(b) From the simulation onemay get the conclusionthat the DSME PWM controller can ensure that the PV cellruns at the maximum power point in a short time Althoughthe overshoot with a little range occurs in the process ofpower adjustment it will be eliminated soon and the stableoutputting power of 260W is achieved During the dynamicadjustment there is a stable and fleet performance withoutoscillation

In Figure 10(c) the dynamic response of the novel topol-ogy is the curve labeled overall trend This curse shows thatthe converter can raise the voltage from 36V to 400V steadilywithout a long time The curve labeled partial enlargementshows that the outputting voltage has a low voltage ripplewhen it is in stable state Considering the whole PV grid-connected system a DC-link high voltage with low rippleis convenient to choice and maintaining of the DC-linkcapacitor

As mentioned before the converter can run under fourmodes in every switching period The tendencies of 119894

1198711and

1198941198712as well as 119894in in every mode have been discussed in part III

and it is also easy to get the conclusion about 119894in = 1198941198711+ 1198941198712

based on Kirchhoff rsquos current law so the tendency of 119894in canbe obtained by the sum of 119894

1198711and 1198941198712 The first and second

one of Figure 10(d) describe the stable current of 1198941198711

and 1198941198712

and 119894in respectively over the same period in stable modeFigure 10(d) shows that the current 119894in has little ripple whichcan reduce the coupled inductance losses and improve systemefficiency

Figure 10(e) shows the voltage waves of switched capac-itors 119862

1and 119862

2 The voltage of 119862

1is three times as much as

the one of 1198622 close to three quarters of 119881

119900 this simulation

verifies the formula (13)

6 Experiments

Experiments have been finished in our laboratory just inorder to verify the validity of theoretical analysis above andthe performance of the new converter proposed in this paper

Experimental waveforms of the new converter are shownin Figure 11 The driving signals of 119878

1and 119878

2from DSP on

basis of the algorithm in this paper are given in Figure 11(a)Obviously the new converter can raise the voltage from36V to 400V when the duty cycle equals 064 (calculated

Table 1 The main parameters

Parameters Values119881ocV 443119868scA 77119881mp at MPPV 36119868mp at MPPA 72Coupling coefficient 091198711

and 1198712

mH 041198621

120583F 101198622

120583F 1001198620

120583F 22119877kΩ 1Switching frequencykHz 50

PWM1PWM2

PWM120575(Vo Vpv ipv ΔVpv Δipv )

0

Vo

Vcsipv

Vpv Vm1

Figure 9 PWM generator

from Figure 11(a)) The output voltage wave (after 10 timesattenuation) is shown in Figure 11(b) Moreover the rippleof the output voltage shown in Figure 11(b) can be limitedin 8V which can decrease the loss of switched capacitorsas shown in (15) Figure 11(c) shows the wave of the inputcurrent The little input current ripple can reduce both theloss and the volume of the coupled inductance Figures 11(d)and 11(e) show the voltage waveform of switched capacitors1198621and 119862

2 respectively (after 10 times attenuation) The

voltage of 1198621 about 300Vsim306V is three times as much as

the one of 1198622 about 92Vsim100V which is consistent with

both the analysis and the simulation results above besidesthe difference between the maximum and the minimum ofvoltage in the course of charging for each capacitance can belimited to a fairly small range which can decrease the loss ofswitched capacitors

Burrs existing in experimental waveform mainly stemfrom the effect of EMI in the experiment

7 Conclusion

This paper proposed a new high-gain interleaved converterwith coupled inductances and switched capacitors networkwhich are suited for PV application After explaining its oper-ating principle and analyzing its steady-state performancethen a control strategy for it on basis of SM-MPPT andDEC theory was designed Both simulation analysis andexperimental results showed that the proposed system hadthe following good performance

(1) Achieving MPPT and producing a high-gain voltagein an extremely short time

(2) Decreasing the ripple of the outputting voltage as wellas the inputting current and improving power density

Journal of Applied Mathematics 9

For S

1

ForS

2

(a) Driving signals for 1198781amp 1198782

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

Time (s)

Volta

ge (V

) and

pow

er (W

)

The outputting voltage and power of PV cells under MPPT

Vpv outputting voltageP outputting powerpv

(b) The output voltage and power of PV cell in process of MPPT

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

350

400

450The outputting voltage of the new topological converter

Time (s)

Volta

ge (V

)

399

3995

400

4005

401

Partial enlargement of outtputing voltage

Overall trendPartial enlargement

(c) The output voltage wave

0

5

Curr

ent (

A) The Lofwavecurrent 1 and L2 in stable mode

001 0015 002 0025 00305

1015

Time (s)

Curr

ent (

A)

iL1

iL2

The current wave of iin in the dynamic mode

65

7

75

Curr

ent (

A) The current wave of iin in stable mode

(d) The current of 1198711and 1198712and 119894in

0 001 002 003 004 005 006 007 008 009 01minus100

minus50

0

50

100

150

200

250

300

350

400

Time (s)

Volta

ge (V

)

The voltage of and

Voltage of C1

Voltage of C2

C1 C2

(e) The voltage wave of switched capacitor 1198621and 119862

2

Figure 10 Simulation results

10 Journal of Applied Mathematics

(a) Driving signals for 1198781and 1198782from DSP (b) The outputting voltage wave (X10)

(c) The wave of input current (d) The voltage wave of switched capacitor 1198621(X10)

(e) The voltage wave of switched capacitor 1198622(X10)

Figure 11 Experimental validation

(3) Reducing the volume of magnetic components andcutting the cost of system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education of China (no20120032110070)

References

[1] C Zhang and X He ldquoMaximum power point tracking by usingasymmetric fuzzy control combined with PID for photovoltaic

energy generation systemrdquo Transactions of China Electrotechni-cal Society vol 20 no 10 pp 72ndash75 2005

[2] Z Liao and X Ruan ldquoControl strategy for bi-directionalDCDC converter of a novel stand-alone photovoltaic powersystemrdquo Transactions of China Electrotechnical Society vol 23no 1 pp 97ndash103 2008

[3] M Dong J Yang K Peng and A Luo ldquoZero average incre-mental conductance maximum power point tracking controlfor photovoltaic systemrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 30 no 21 pp 48ndash53 2010

[4] J M Enrique J M Andujar and M A Bohorquez ldquoAreliable fast and low cost maximum power point tracker forphotovoltaic applicationsrdquo Solar Energy vol 84 pp 79ndash89 2010

[5] L Xie RGong and J Li ldquoAnalysis of the dynamical characteris-tics of the Interleaved boost converter inmaximumpower pointtracking for photovoltaic powerrdquo in Proceedings of the ChineseSociety for Electrical Engineering (CSEE rsquo13) vol 33 no 6 pp39ndash43 2013

Journal of Applied Mathematics 11

[6] W Li and X He ldquoReview of nonisolated high-step-up DCDCconverters in photovoltaic grid-connected applicationsrdquo IEEETransactions on Industrial Electronics vol 58 no 4 pp 1239ndash1250 2011

[7] Q LuoH Yan S Zhi C Zou and L Zhou ldquoAn interleaved highstep-up zero current transition boost converterrdquo in Proceedingsof the Chinese Society for Electrical Engineering (CSEE rsquo13) vol33 no 12 pp 18ndash21 2013

[8] W Huang and B Lehman ldquoMitigation and utilization of theinductor coupling effect in interleaved multiphase dcdc con-vertersrdquo in Proceedings of the IEEE Energy Conversion Congressand Exposition (ECCE rsquo13) pp 1822ndash1829 2013

[9] G Zhu B A McDonald and K Wang ldquoModeling and analysisof coupled inductors in power convertersrdquo IEEE Transactionson Power Electronics vol 26 no 5 pp 1355ndash1363 2011

[10] Q-B Hu B Qu and Z-Y Lu ldquoNovel step-up VRM-two-phaseinterleaved coupled-boost converterrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 26 no 9 pp 94ndash98 2006

[11] D A Grant Y Darroman and J Suter ldquoSynthesis of tapped-inductor switched-mode convertersrdquo IEEE Transactions onPower Electronics vol 22 no 5 pp 1964ndash1969 2007

[12] M Nymand and M A E Andersen ldquoHigh-efficiency isolatedboost DC-DC converter for high-power low-voltage fuel-cellapplicationsrdquo IEEE Transactions on Industrial Electronics vol57 no 2 pp 505ndash514 2010

[13] R Haroun A Cid-Pastor A El Aroudi and L Martinez-Salamero ldquoCascade connection ofDC-DC switching convertersby means of self-oscillating dc-transformersrdquo in Proceedingsof the 15th International Power Electronics and Motion ControlConference (EPEPEMC rsquo12) 2012

[14] R Guo C Wang and T Li ldquoOptimum design of couplinginductors for magnetic integration in three-phase interleavingBuck dcdc converterrdquo in Proceedings of the 8th IEEE Conferenceon Industrial Electronics and Applications (ICIEA rsquo13) pp 1029ndash1033 2013

[15] H Wu J Gu J Zhang Y Xing and G Chen ldquoHigh efficiencyhigh step-up Boost-Flyback DCDC converterrdquo Proceedings ofthe Chinese Society of Electrical Engineering vol 31 no 24 pp40ndash45 2011

[16] T J Liang andK C Tseng ldquoAnalysis of integrated boost-flybackstep-up converterrdquo IEE Proceedings Electric Power Applicationsvol 152 no 2 pp 217ndash225 2005

[17] J L Kui-Jun P Byoung-Gun K Rae-young and H Dong-seok ldquoNonisolated ZVT two-inductor boost converter with asingle resonant inductor for high step-up applicationsrdquo IEEETransactions on Power Electronics vol 27 no 4 pp 1966ndash19732012

[18] Z Lu L Zheng and Z Ma ldquoInterleaved high gain boostconverter with switched capacitor networkrdquo Transactions ofChina Electrotechnical Society vol 27 no 11 pp 154ndash156 2012

[19] A S Samosir and A H M Yatim ldquoImplementation of dynamicevolution control of bidirectional DC-DC converter for inter-facing ultracapacitor energy storage to fuel-cell systemrdquo IEEETransactions on Industrial Electronics vol 57 no 10 pp 3468ndash3473 2010

[20] A S Samosir and A H M Yatim ldquoDynamic evolution controlof bidirectionalDC-DCconverter for interfacing ultra capacitorenergy storage to fuel cell electric vehicle systemrdquo in Proceedingsof the Power Engineering Conferencee (AUPEC rsquo08) AustralasianUniversities December 2008

[21] A H ALQahtani M S Abuhamdeh Y M Alsmadi and VI Utkin ldquoPhotovoltaic power optimization using sliding mode

control with a two-axis tracking systemrdquo in Proceedings of theIEEE Energytech pp 1ndash6 2013

[22] D G Montoya C A R Paja and R Giral ldquoA new solutionof maximum power point tracking based on sliding modecontrolrdquo in Proceedings of the 39th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo13) pp 8350ndash83552013

[23] J He ldquoConnected power system based on sliding mode theoryrdquoin Study on Control Algorithm of Maximum Power Point Track-ing for Photovoltaic Grid Shandong University 2012

[24] A S Samosir and A H M Yatim ldquoDynamic evolution con-troller for single phase inverter applicationrdquo in Proceedings ofthe IEEE Symposium on Industrial Electronics and Applications(ISIEA rsquo09) vol 1 pp 530ndash535 October 2009

[25] A S Samosir M Anwari and A H M Yatim ldquoDynamicevolution control of interleaved boostDC-DC converter for fuelcell applicationrdquo in Proceedings of the 9th International Powerand Energy Conference (IPEC rsquo10) pp 869ndash874 October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

with the dynamic evolution algorithm The specific designprocess of the DSME controller can be described as follows

41 MPPT by Sliding Mode Control When the PV cells runat the point of maximum power

119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv= 0 (16)

In SMC the state space is divided into 119904 gt 0 and 119904 lt 0 bysliding manifold (119904 = 0) The movement on 119904 = 0 can becalled sliding mode dynamics Considering 119889119875pv119889119881pv = 0the sliding manifold can be chosen as (as discussed by [2122])

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv (17)

Reconsidering the mathematical model of PV cell

119894pv = 119868ph minus 119868sat [exp(119902119881pv

119860119870119879) minus 1] (18)

Directly substituting 119894pv from (18) into (17)

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv

= minus119868sat [exp (119902

119860119870119879119881pv) sdot

119902

119860119870119879119881pv]

+ 119868ph minus 119868sat [exp(119902

119860119870119879119881pv) minus 1]

= minus119868sat (119902

119860119870119879119881pv + 1) sdot exp(

119902

119860119870119879119881pv) + (119868ph + 119868sat)

(19)

The differential form of 119904 is

119904 = minus119868sat119902

119860119870119879sdot119889119881pv

119889119905exp(

119902

119860119870119879) minus 119868sat (

119902119881pv

119860119870119879+ 1)

sdot exp(119902119881pv

119860119870119879)

119902

119860119870119879

119889119881pv

119889119905

= minus119868sat119902

119860119870119879sdot119889119881pv

119889119905exp(

119902119881pv

119860119870119879)(

119902119881pv

119860119870119879+ 2)

(20)

According to 119904 119904 lt 0 if 119904 gt 0 then 119904 lt 0 119889119881pv119889119905 gt 0119881pv willincrease when 119904 tends to sliding manifold and if 119904 lt 0 then119904 gt 0 119889119881pv119889119905 lt 0 119881pv will decrease when 119904 tends to slidingmanifoldThe dynamic accommodation of119881pv can be seen inFigure 7

Considering that the new converter is composed of twoparalleled boost converters the purpose of the additionalswitched capacitor and coupled inductance is to increase thevoltage gain Thus the drive signals for the switches can bedesigned respectively From the state space model of thesingle boost converter when the switch is on 119881pv increases

and when the switch is off 119881pv decreases (as discussed by[23])

Rearranging (17)

119904 =119889119875pv

119889119881pv= 119894pv + 119881pv

119889119894pv

119889119881pv= 119881pv (

119894pv

119881pv+119889119894pv

119889119881pv) (21)

The differential form approximates to

Δ119894pv

Δ119881pvasymp119889119894pv

119889119881pv (22)

Thus

119904 = 119881pv (119894pv

119881pv+Δ119894pv

Δ119881pv) (23)

Theoretically in the case of low requirement both (17) and(24) can be chosen as the sliding surface

119904 = Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv (24)

The corresponding control law is

119906 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)] (25)

42 Evolution Path Selection The principle of determiningthe evolution path is to ensure that the error state can goto zero at any increase of time (as discussed by [24 25]) Inthis research the evolution path is an exponential function asshown in Figure 8

With this exponential evolution path the value of thedynamic characteristic of the converter will decrease expo-nentially to zero by

119867 = 1198670ℓminus120582119905

(26)

where 119867 is the dynamic characteristic of system 1198670is the

initial value of 119867 and 120582 is a dynamic evolution factor Thederivative of119867 is

119889119867

119889119905= minus120582119867

0ℓminus120582119905

= minus120582119867

119889119867

119889119905+ 120582119867 = 0

(27)

43 Analysis of Duty Cycle In order to synthesize the controllaw of the dynamic evolution controller the dynamic equa-tion of converter system has to be established and analyzedOn basis of the state space averagemodel when the converterworks on DCM the voltage and current dynamics of theconverter are given by

119881pv = 119871119889119894in119889119905

+119881119900

4(1 minus 120572) (28)

where 119871 is the inductance 119881pv is the input voltage 119894in is theinput current119881

119900is the output voltage and 120572 is the duty cycle

respectively

Journal of Applied Mathematics 7

Sliding manifold

Trajectory s chattersalong the manifoldand moves toward

VO

pv

120577 = minusΔs gt 0

s = 0

120577 = +Δ

s lt 0

Ppv

s = 0

Figure 7 The dynamic accommodation of 119881pv

119894in asymp 119894pv rearranging (28) the output voltage of convertercan be written as

119881pv = 119871119889119894pv

119889119905+119881119900

4(1 minus 120572) (29)

119881119900= 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (30)

Define a linear voltage error function as shown below

119864 = 119896119890V = 119896 (119881ref minus 119881119900) (31)

where 119896 gt 0Substituting (27) into (31) yields

119896119889119890V

119889119905+ 120582119896119890V = 0 (32)

Combining (31) with (32)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 119881

119900 (33)

Directly substituting the converter output voltage 119881119900from

(30) into (33)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (34)

The obtained duty cycle formula is given by

120572 =119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905 (35)

Formula (35) forces the state error function 119864 to satisfy thedynamic evolution function (27) Consequently the stateerror function 119864 is forced to make evolution by followingequation (35) and decrease to zero (119864 = 0) with a decreaserate 120582 The outputting voltage of converter converges to theconverters steady state

119881119900= 119881ref (36)

From the synthesis procedure it is clear that the dynamicevolution controller works on the full nonlinear system and

H

H0

0 t

H = H0eminus120582t

Figure 8 Exponential evolution path

does not need any linearization or simplification on thesystem model at all and it is interesting to note that thecontrol law in (35) consists of four distinct parts The firstpart is the feed-forward term which is calculated based onthe duty cycle at the previous sampling instant This termcompensates for variations in the input voltages The secondand third terms consist of proportional and derivative termsof the perturbations in the output voltage respectively Thelast term consists of the derivative terms of the inductancecurrent Since the inputting and outputting voltages andinductance current are involved in calculation of duty cycleit can compensate all the variation of them in the dynamicevolution

It should be noted that both119881pv and 119894pv are not the voltageand current at the MPP in (35) Consequently we expect thatthe controller can achieve MPPT and high-gain boost thusconsidering the logic relationship of control signals the finalcontrol law can be described as follows combing (25) with(35)

120575 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)]

sdot (119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905)

(37)

44 PWM Generation The PWM signals are generated bycomparing a control signal with a constant peak repetitivetriangle signal (119881cs) The frequency of the repetitive trianglesignal establishes the constant switching frequency

Figure 9 shows the PWM signals generation techniquePWM1 is produced when the control signal 119881

1198981is greater

than 119881119888119904while PWM2 is produced when 119881

1198981is less than 119881

119888119904

The values of the desired duty cycle 120572 can be gotten from(37) PWM signals are generated by (39)

1198811198981

= 120575 (38)

8 Journal of Applied Mathematics

PWM1 = 1 1198811198981

gt 119881119888119904

0 1198811198981

lt 119881119888119904

PWM2 = 1 1198811198981

lt 119881119888119904

0 1198811198981

gt 119881119888119904

(39)

5 Simulation

Some simulation has been done in order to verify theperformance of the topology and the main parameters canbe referred to in Table 1

The portion of the simulation results is given in Figure 10Figure 10(a) shows the driving signals for 119878

1and 1198782 which

are generated by the control law 120575 The outputting voltageand power of PV cell in process of MPPT can be seen inFigure 10(b) From the simulation onemay get the conclusionthat the DSME PWM controller can ensure that the PV cellruns at the maximum power point in a short time Althoughthe overshoot with a little range occurs in the process ofpower adjustment it will be eliminated soon and the stableoutputting power of 260W is achieved During the dynamicadjustment there is a stable and fleet performance withoutoscillation

In Figure 10(c) the dynamic response of the novel topol-ogy is the curve labeled overall trend This curse shows thatthe converter can raise the voltage from 36V to 400V steadilywithout a long time The curve labeled partial enlargementshows that the outputting voltage has a low voltage ripplewhen it is in stable state Considering the whole PV grid-connected system a DC-link high voltage with low rippleis convenient to choice and maintaining of the DC-linkcapacitor

As mentioned before the converter can run under fourmodes in every switching period The tendencies of 119894

1198711and

1198941198712as well as 119894in in every mode have been discussed in part III

and it is also easy to get the conclusion about 119894in = 1198941198711+ 1198941198712

based on Kirchhoff rsquos current law so the tendency of 119894in canbe obtained by the sum of 119894

1198711and 1198941198712 The first and second

one of Figure 10(d) describe the stable current of 1198941198711

and 1198941198712

and 119894in respectively over the same period in stable modeFigure 10(d) shows that the current 119894in has little ripple whichcan reduce the coupled inductance losses and improve systemefficiency

Figure 10(e) shows the voltage waves of switched capac-itors 119862

1and 119862

2 The voltage of 119862

1is three times as much as

the one of 1198622 close to three quarters of 119881

119900 this simulation

verifies the formula (13)

6 Experiments

Experiments have been finished in our laboratory just inorder to verify the validity of theoretical analysis above andthe performance of the new converter proposed in this paper

Experimental waveforms of the new converter are shownin Figure 11 The driving signals of 119878

1and 119878

2from DSP on

basis of the algorithm in this paper are given in Figure 11(a)Obviously the new converter can raise the voltage from36V to 400V when the duty cycle equals 064 (calculated

Table 1 The main parameters

Parameters Values119881ocV 443119868scA 77119881mp at MPPV 36119868mp at MPPA 72Coupling coefficient 091198711

and 1198712

mH 041198621

120583F 101198622

120583F 1001198620

120583F 22119877kΩ 1Switching frequencykHz 50

PWM1PWM2

PWM120575(Vo Vpv ipv ΔVpv Δipv )

0

Vo

Vcsipv

Vpv Vm1

Figure 9 PWM generator

from Figure 11(a)) The output voltage wave (after 10 timesattenuation) is shown in Figure 11(b) Moreover the rippleof the output voltage shown in Figure 11(b) can be limitedin 8V which can decrease the loss of switched capacitorsas shown in (15) Figure 11(c) shows the wave of the inputcurrent The little input current ripple can reduce both theloss and the volume of the coupled inductance Figures 11(d)and 11(e) show the voltage waveform of switched capacitors1198621and 119862

2 respectively (after 10 times attenuation) The

voltage of 1198621 about 300Vsim306V is three times as much as

the one of 1198622 about 92Vsim100V which is consistent with

both the analysis and the simulation results above besidesthe difference between the maximum and the minimum ofvoltage in the course of charging for each capacitance can belimited to a fairly small range which can decrease the loss ofswitched capacitors

Burrs existing in experimental waveform mainly stemfrom the effect of EMI in the experiment

7 Conclusion

This paper proposed a new high-gain interleaved converterwith coupled inductances and switched capacitors networkwhich are suited for PV application After explaining its oper-ating principle and analyzing its steady-state performancethen a control strategy for it on basis of SM-MPPT andDEC theory was designed Both simulation analysis andexperimental results showed that the proposed system hadthe following good performance

(1) Achieving MPPT and producing a high-gain voltagein an extremely short time

(2) Decreasing the ripple of the outputting voltage as wellas the inputting current and improving power density

Journal of Applied Mathematics 9

For S

1

ForS

2

(a) Driving signals for 1198781amp 1198782

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

Time (s)

Volta

ge (V

) and

pow

er (W

)

The outputting voltage and power of PV cells under MPPT

Vpv outputting voltageP outputting powerpv

(b) The output voltage and power of PV cell in process of MPPT

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

350

400

450The outputting voltage of the new topological converter

Time (s)

Volta

ge (V

)

399

3995

400

4005

401

Partial enlargement of outtputing voltage

Overall trendPartial enlargement

(c) The output voltage wave

0

5

Curr

ent (

A) The Lofwavecurrent 1 and L2 in stable mode

001 0015 002 0025 00305

1015

Time (s)

Curr

ent (

A)

iL1

iL2

The current wave of iin in the dynamic mode

65

7

75

Curr

ent (

A) The current wave of iin in stable mode

(d) The current of 1198711and 1198712and 119894in

0 001 002 003 004 005 006 007 008 009 01minus100

minus50

0

50

100

150

200

250

300

350

400

Time (s)

Volta

ge (V

)

The voltage of and

Voltage of C1

Voltage of C2

C1 C2

(e) The voltage wave of switched capacitor 1198621and 119862

2

Figure 10 Simulation results

10 Journal of Applied Mathematics

(a) Driving signals for 1198781and 1198782from DSP (b) The outputting voltage wave (X10)

(c) The wave of input current (d) The voltage wave of switched capacitor 1198621(X10)

(e) The voltage wave of switched capacitor 1198622(X10)

Figure 11 Experimental validation

(3) Reducing the volume of magnetic components andcutting the cost of system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education of China (no20120032110070)

References

[1] C Zhang and X He ldquoMaximum power point tracking by usingasymmetric fuzzy control combined with PID for photovoltaic

energy generation systemrdquo Transactions of China Electrotechni-cal Society vol 20 no 10 pp 72ndash75 2005

[2] Z Liao and X Ruan ldquoControl strategy for bi-directionalDCDC converter of a novel stand-alone photovoltaic powersystemrdquo Transactions of China Electrotechnical Society vol 23no 1 pp 97ndash103 2008

[3] M Dong J Yang K Peng and A Luo ldquoZero average incre-mental conductance maximum power point tracking controlfor photovoltaic systemrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 30 no 21 pp 48ndash53 2010

[4] J M Enrique J M Andujar and M A Bohorquez ldquoAreliable fast and low cost maximum power point tracker forphotovoltaic applicationsrdquo Solar Energy vol 84 pp 79ndash89 2010

[5] L Xie RGong and J Li ldquoAnalysis of the dynamical characteris-tics of the Interleaved boost converter inmaximumpower pointtracking for photovoltaic powerrdquo in Proceedings of the ChineseSociety for Electrical Engineering (CSEE rsquo13) vol 33 no 6 pp39ndash43 2013

Journal of Applied Mathematics 11

[6] W Li and X He ldquoReview of nonisolated high-step-up DCDCconverters in photovoltaic grid-connected applicationsrdquo IEEETransactions on Industrial Electronics vol 58 no 4 pp 1239ndash1250 2011

[7] Q LuoH Yan S Zhi C Zou and L Zhou ldquoAn interleaved highstep-up zero current transition boost converterrdquo in Proceedingsof the Chinese Society for Electrical Engineering (CSEE rsquo13) vol33 no 12 pp 18ndash21 2013

[8] W Huang and B Lehman ldquoMitigation and utilization of theinductor coupling effect in interleaved multiphase dcdc con-vertersrdquo in Proceedings of the IEEE Energy Conversion Congressand Exposition (ECCE rsquo13) pp 1822ndash1829 2013

[9] G Zhu B A McDonald and K Wang ldquoModeling and analysisof coupled inductors in power convertersrdquo IEEE Transactionson Power Electronics vol 26 no 5 pp 1355ndash1363 2011

[10] Q-B Hu B Qu and Z-Y Lu ldquoNovel step-up VRM-two-phaseinterleaved coupled-boost converterrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 26 no 9 pp 94ndash98 2006

[11] D A Grant Y Darroman and J Suter ldquoSynthesis of tapped-inductor switched-mode convertersrdquo IEEE Transactions onPower Electronics vol 22 no 5 pp 1964ndash1969 2007

[12] M Nymand and M A E Andersen ldquoHigh-efficiency isolatedboost DC-DC converter for high-power low-voltage fuel-cellapplicationsrdquo IEEE Transactions on Industrial Electronics vol57 no 2 pp 505ndash514 2010

[13] R Haroun A Cid-Pastor A El Aroudi and L Martinez-Salamero ldquoCascade connection ofDC-DC switching convertersby means of self-oscillating dc-transformersrdquo in Proceedingsof the 15th International Power Electronics and Motion ControlConference (EPEPEMC rsquo12) 2012

[14] R Guo C Wang and T Li ldquoOptimum design of couplinginductors for magnetic integration in three-phase interleavingBuck dcdc converterrdquo in Proceedings of the 8th IEEE Conferenceon Industrial Electronics and Applications (ICIEA rsquo13) pp 1029ndash1033 2013

[15] H Wu J Gu J Zhang Y Xing and G Chen ldquoHigh efficiencyhigh step-up Boost-Flyback DCDC converterrdquo Proceedings ofthe Chinese Society of Electrical Engineering vol 31 no 24 pp40ndash45 2011

[16] T J Liang andK C Tseng ldquoAnalysis of integrated boost-flybackstep-up converterrdquo IEE Proceedings Electric Power Applicationsvol 152 no 2 pp 217ndash225 2005

[17] J L Kui-Jun P Byoung-Gun K Rae-young and H Dong-seok ldquoNonisolated ZVT two-inductor boost converter with asingle resonant inductor for high step-up applicationsrdquo IEEETransactions on Power Electronics vol 27 no 4 pp 1966ndash19732012

[18] Z Lu L Zheng and Z Ma ldquoInterleaved high gain boostconverter with switched capacitor networkrdquo Transactions ofChina Electrotechnical Society vol 27 no 11 pp 154ndash156 2012

[19] A S Samosir and A H M Yatim ldquoImplementation of dynamicevolution control of bidirectional DC-DC converter for inter-facing ultracapacitor energy storage to fuel-cell systemrdquo IEEETransactions on Industrial Electronics vol 57 no 10 pp 3468ndash3473 2010

[20] A S Samosir and A H M Yatim ldquoDynamic evolution controlof bidirectionalDC-DCconverter for interfacing ultra capacitorenergy storage to fuel cell electric vehicle systemrdquo in Proceedingsof the Power Engineering Conferencee (AUPEC rsquo08) AustralasianUniversities December 2008

[21] A H ALQahtani M S Abuhamdeh Y M Alsmadi and VI Utkin ldquoPhotovoltaic power optimization using sliding mode

control with a two-axis tracking systemrdquo in Proceedings of theIEEE Energytech pp 1ndash6 2013

[22] D G Montoya C A R Paja and R Giral ldquoA new solutionof maximum power point tracking based on sliding modecontrolrdquo in Proceedings of the 39th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo13) pp 8350ndash83552013

[23] J He ldquoConnected power system based on sliding mode theoryrdquoin Study on Control Algorithm of Maximum Power Point Track-ing for Photovoltaic Grid Shandong University 2012

[24] A S Samosir and A H M Yatim ldquoDynamic evolution con-troller for single phase inverter applicationrdquo in Proceedings ofthe IEEE Symposium on Industrial Electronics and Applications(ISIEA rsquo09) vol 1 pp 530ndash535 October 2009

[25] A S Samosir M Anwari and A H M Yatim ldquoDynamicevolution control of interleaved boostDC-DC converter for fuelcell applicationrdquo in Proceedings of the 9th International Powerand Energy Conference (IPEC rsquo10) pp 869ndash874 October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

Sliding manifold

Trajectory s chattersalong the manifoldand moves toward

VO

pv

120577 = minusΔs gt 0

s = 0

120577 = +Δ

s lt 0

Ppv

s = 0

Figure 7 The dynamic accommodation of 119881pv

119894in asymp 119894pv rearranging (28) the output voltage of convertercan be written as

119881pv = 119871119889119894pv

119889119905+119881119900

4(1 minus 120572) (29)

119881119900= 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (30)

Define a linear voltage error function as shown below

119864 = 119896119890V = 119896 (119881ref minus 119881119900) (31)

where 119896 gt 0Substituting (27) into (31) yields

119896119889119890V

119889119905+ 120582119896119890V = 0 (32)

Combining (31) with (32)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 119881

119900 (33)

Directly substituting the converter output voltage 119881119900from

(30) into (33)

119896119889119890V

119889119905+ (120582119896 minus 1) 119890V + 119881ref = 4119881pv + 120572119881119900 minus 4119871

119889119894pv

119889119905 (34)

The obtained duty cycle formula is given by

120572 =119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905 (35)

Formula (35) forces the state error function 119864 to satisfy thedynamic evolution function (27) Consequently the stateerror function 119864 is forced to make evolution by followingequation (35) and decrease to zero (119864 = 0) with a decreaserate 120582 The outputting voltage of converter converges to theconverters steady state

119881119900= 119881ref (36)

From the synthesis procedure it is clear that the dynamicevolution controller works on the full nonlinear system and

H

H0

0 t

H = H0eminus120582t

Figure 8 Exponential evolution path

does not need any linearization or simplification on thesystem model at all and it is interesting to note that thecontrol law in (35) consists of four distinct parts The firstpart is the feed-forward term which is calculated based onthe duty cycle at the previous sampling instant This termcompensates for variations in the input voltages The secondand third terms consist of proportional and derivative termsof the perturbations in the output voltage respectively Thelast term consists of the derivative terms of the inductancecurrent Since the inputting and outputting voltages andinductance current are involved in calculation of duty cycleit can compensate all the variation of them in the dynamicevolution

It should be noted that both119881pv and 119894pv are not the voltageand current at the MPP in (35) Consequently we expect thatthe controller can achieve MPPT and high-gain boost thusconsidering the logic relationship of control signals the finalcontrol law can be described as follows combing (25) with(35)

120575 =1

2[1 + sign (Δ119881pv sdot 119894pv + Δ119894pv sdot 119881pv)]

sdot (119881ref minus 4119881pv

119881119900

+(120582119896 minus 1) 119890V

119881119900

+119896

119881119900

119889119890V

119889119905+4119871

119881119900

119889119894pv

119889119905)

(37)

44 PWM Generation The PWM signals are generated bycomparing a control signal with a constant peak repetitivetriangle signal (119881cs) The frequency of the repetitive trianglesignal establishes the constant switching frequency

Figure 9 shows the PWM signals generation techniquePWM1 is produced when the control signal 119881

1198981is greater

than 119881119888119904while PWM2 is produced when 119881

1198981is less than 119881

119888119904

The values of the desired duty cycle 120572 can be gotten from(37) PWM signals are generated by (39)

1198811198981

= 120575 (38)

8 Journal of Applied Mathematics

PWM1 = 1 1198811198981

gt 119881119888119904

0 1198811198981

lt 119881119888119904

PWM2 = 1 1198811198981

lt 119881119888119904

0 1198811198981

gt 119881119888119904

(39)

5 Simulation

Some simulation has been done in order to verify theperformance of the topology and the main parameters canbe referred to in Table 1

The portion of the simulation results is given in Figure 10Figure 10(a) shows the driving signals for 119878

1and 1198782 which

are generated by the control law 120575 The outputting voltageand power of PV cell in process of MPPT can be seen inFigure 10(b) From the simulation onemay get the conclusionthat the DSME PWM controller can ensure that the PV cellruns at the maximum power point in a short time Althoughthe overshoot with a little range occurs in the process ofpower adjustment it will be eliminated soon and the stableoutputting power of 260W is achieved During the dynamicadjustment there is a stable and fleet performance withoutoscillation

In Figure 10(c) the dynamic response of the novel topol-ogy is the curve labeled overall trend This curse shows thatthe converter can raise the voltage from 36V to 400V steadilywithout a long time The curve labeled partial enlargementshows that the outputting voltage has a low voltage ripplewhen it is in stable state Considering the whole PV grid-connected system a DC-link high voltage with low rippleis convenient to choice and maintaining of the DC-linkcapacitor

As mentioned before the converter can run under fourmodes in every switching period The tendencies of 119894

1198711and

1198941198712as well as 119894in in every mode have been discussed in part III

and it is also easy to get the conclusion about 119894in = 1198941198711+ 1198941198712

based on Kirchhoff rsquos current law so the tendency of 119894in canbe obtained by the sum of 119894

1198711and 1198941198712 The first and second

one of Figure 10(d) describe the stable current of 1198941198711

and 1198941198712

and 119894in respectively over the same period in stable modeFigure 10(d) shows that the current 119894in has little ripple whichcan reduce the coupled inductance losses and improve systemefficiency

Figure 10(e) shows the voltage waves of switched capac-itors 119862

1and 119862

2 The voltage of 119862

1is three times as much as

the one of 1198622 close to three quarters of 119881

119900 this simulation

verifies the formula (13)

6 Experiments

Experiments have been finished in our laboratory just inorder to verify the validity of theoretical analysis above andthe performance of the new converter proposed in this paper

Experimental waveforms of the new converter are shownin Figure 11 The driving signals of 119878

1and 119878

2from DSP on

basis of the algorithm in this paper are given in Figure 11(a)Obviously the new converter can raise the voltage from36V to 400V when the duty cycle equals 064 (calculated

Table 1 The main parameters

Parameters Values119881ocV 443119868scA 77119881mp at MPPV 36119868mp at MPPA 72Coupling coefficient 091198711

and 1198712

mH 041198621

120583F 101198622

120583F 1001198620

120583F 22119877kΩ 1Switching frequencykHz 50

PWM1PWM2

PWM120575(Vo Vpv ipv ΔVpv Δipv )

0

Vo

Vcsipv

Vpv Vm1

Figure 9 PWM generator

from Figure 11(a)) The output voltage wave (after 10 timesattenuation) is shown in Figure 11(b) Moreover the rippleof the output voltage shown in Figure 11(b) can be limitedin 8V which can decrease the loss of switched capacitorsas shown in (15) Figure 11(c) shows the wave of the inputcurrent The little input current ripple can reduce both theloss and the volume of the coupled inductance Figures 11(d)and 11(e) show the voltage waveform of switched capacitors1198621and 119862

2 respectively (after 10 times attenuation) The

voltage of 1198621 about 300Vsim306V is three times as much as

the one of 1198622 about 92Vsim100V which is consistent with

both the analysis and the simulation results above besidesthe difference between the maximum and the minimum ofvoltage in the course of charging for each capacitance can belimited to a fairly small range which can decrease the loss ofswitched capacitors

Burrs existing in experimental waveform mainly stemfrom the effect of EMI in the experiment

7 Conclusion

This paper proposed a new high-gain interleaved converterwith coupled inductances and switched capacitors networkwhich are suited for PV application After explaining its oper-ating principle and analyzing its steady-state performancethen a control strategy for it on basis of SM-MPPT andDEC theory was designed Both simulation analysis andexperimental results showed that the proposed system hadthe following good performance

(1) Achieving MPPT and producing a high-gain voltagein an extremely short time

(2) Decreasing the ripple of the outputting voltage as wellas the inputting current and improving power density

Journal of Applied Mathematics 9

For S

1

ForS

2

(a) Driving signals for 1198781amp 1198782

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

Time (s)

Volta

ge (V

) and

pow

er (W

)

The outputting voltage and power of PV cells under MPPT

Vpv outputting voltageP outputting powerpv

(b) The output voltage and power of PV cell in process of MPPT

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

350

400

450The outputting voltage of the new topological converter

Time (s)

Volta

ge (V

)

399

3995

400

4005

401

Partial enlargement of outtputing voltage

Overall trendPartial enlargement

(c) The output voltage wave

0

5

Curr

ent (

A) The Lofwavecurrent 1 and L2 in stable mode

001 0015 002 0025 00305

1015

Time (s)

Curr

ent (

A)

iL1

iL2

The current wave of iin in the dynamic mode

65

7

75

Curr

ent (

A) The current wave of iin in stable mode

(d) The current of 1198711and 1198712and 119894in

0 001 002 003 004 005 006 007 008 009 01minus100

minus50

0

50

100

150

200

250

300

350

400

Time (s)

Volta

ge (V

)

The voltage of and

Voltage of C1

Voltage of C2

C1 C2

(e) The voltage wave of switched capacitor 1198621and 119862

2

Figure 10 Simulation results

10 Journal of Applied Mathematics

(a) Driving signals for 1198781and 1198782from DSP (b) The outputting voltage wave (X10)

(c) The wave of input current (d) The voltage wave of switched capacitor 1198621(X10)

(e) The voltage wave of switched capacitor 1198622(X10)

Figure 11 Experimental validation

(3) Reducing the volume of magnetic components andcutting the cost of system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education of China (no20120032110070)

References

[1] C Zhang and X He ldquoMaximum power point tracking by usingasymmetric fuzzy control combined with PID for photovoltaic

energy generation systemrdquo Transactions of China Electrotechni-cal Society vol 20 no 10 pp 72ndash75 2005

[2] Z Liao and X Ruan ldquoControl strategy for bi-directionalDCDC converter of a novel stand-alone photovoltaic powersystemrdquo Transactions of China Electrotechnical Society vol 23no 1 pp 97ndash103 2008

[3] M Dong J Yang K Peng and A Luo ldquoZero average incre-mental conductance maximum power point tracking controlfor photovoltaic systemrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 30 no 21 pp 48ndash53 2010

[4] J M Enrique J M Andujar and M A Bohorquez ldquoAreliable fast and low cost maximum power point tracker forphotovoltaic applicationsrdquo Solar Energy vol 84 pp 79ndash89 2010

[5] L Xie RGong and J Li ldquoAnalysis of the dynamical characteris-tics of the Interleaved boost converter inmaximumpower pointtracking for photovoltaic powerrdquo in Proceedings of the ChineseSociety for Electrical Engineering (CSEE rsquo13) vol 33 no 6 pp39ndash43 2013

Journal of Applied Mathematics 11

[6] W Li and X He ldquoReview of nonisolated high-step-up DCDCconverters in photovoltaic grid-connected applicationsrdquo IEEETransactions on Industrial Electronics vol 58 no 4 pp 1239ndash1250 2011

[7] Q LuoH Yan S Zhi C Zou and L Zhou ldquoAn interleaved highstep-up zero current transition boost converterrdquo in Proceedingsof the Chinese Society for Electrical Engineering (CSEE rsquo13) vol33 no 12 pp 18ndash21 2013

[8] W Huang and B Lehman ldquoMitigation and utilization of theinductor coupling effect in interleaved multiphase dcdc con-vertersrdquo in Proceedings of the IEEE Energy Conversion Congressand Exposition (ECCE rsquo13) pp 1822ndash1829 2013

[9] G Zhu B A McDonald and K Wang ldquoModeling and analysisof coupled inductors in power convertersrdquo IEEE Transactionson Power Electronics vol 26 no 5 pp 1355ndash1363 2011

[10] Q-B Hu B Qu and Z-Y Lu ldquoNovel step-up VRM-two-phaseinterleaved coupled-boost converterrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 26 no 9 pp 94ndash98 2006

[11] D A Grant Y Darroman and J Suter ldquoSynthesis of tapped-inductor switched-mode convertersrdquo IEEE Transactions onPower Electronics vol 22 no 5 pp 1964ndash1969 2007

[12] M Nymand and M A E Andersen ldquoHigh-efficiency isolatedboost DC-DC converter for high-power low-voltage fuel-cellapplicationsrdquo IEEE Transactions on Industrial Electronics vol57 no 2 pp 505ndash514 2010

[13] R Haroun A Cid-Pastor A El Aroudi and L Martinez-Salamero ldquoCascade connection ofDC-DC switching convertersby means of self-oscillating dc-transformersrdquo in Proceedingsof the 15th International Power Electronics and Motion ControlConference (EPEPEMC rsquo12) 2012

[14] R Guo C Wang and T Li ldquoOptimum design of couplinginductors for magnetic integration in three-phase interleavingBuck dcdc converterrdquo in Proceedings of the 8th IEEE Conferenceon Industrial Electronics and Applications (ICIEA rsquo13) pp 1029ndash1033 2013

[15] H Wu J Gu J Zhang Y Xing and G Chen ldquoHigh efficiencyhigh step-up Boost-Flyback DCDC converterrdquo Proceedings ofthe Chinese Society of Electrical Engineering vol 31 no 24 pp40ndash45 2011

[16] T J Liang andK C Tseng ldquoAnalysis of integrated boost-flybackstep-up converterrdquo IEE Proceedings Electric Power Applicationsvol 152 no 2 pp 217ndash225 2005

[17] J L Kui-Jun P Byoung-Gun K Rae-young and H Dong-seok ldquoNonisolated ZVT two-inductor boost converter with asingle resonant inductor for high step-up applicationsrdquo IEEETransactions on Power Electronics vol 27 no 4 pp 1966ndash19732012

[18] Z Lu L Zheng and Z Ma ldquoInterleaved high gain boostconverter with switched capacitor networkrdquo Transactions ofChina Electrotechnical Society vol 27 no 11 pp 154ndash156 2012

[19] A S Samosir and A H M Yatim ldquoImplementation of dynamicevolution control of bidirectional DC-DC converter for inter-facing ultracapacitor energy storage to fuel-cell systemrdquo IEEETransactions on Industrial Electronics vol 57 no 10 pp 3468ndash3473 2010

[20] A S Samosir and A H M Yatim ldquoDynamic evolution controlof bidirectionalDC-DCconverter for interfacing ultra capacitorenergy storage to fuel cell electric vehicle systemrdquo in Proceedingsof the Power Engineering Conferencee (AUPEC rsquo08) AustralasianUniversities December 2008

[21] A H ALQahtani M S Abuhamdeh Y M Alsmadi and VI Utkin ldquoPhotovoltaic power optimization using sliding mode

control with a two-axis tracking systemrdquo in Proceedings of theIEEE Energytech pp 1ndash6 2013

[22] D G Montoya C A R Paja and R Giral ldquoA new solutionof maximum power point tracking based on sliding modecontrolrdquo in Proceedings of the 39th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo13) pp 8350ndash83552013

[23] J He ldquoConnected power system based on sliding mode theoryrdquoin Study on Control Algorithm of Maximum Power Point Track-ing for Photovoltaic Grid Shandong University 2012

[24] A S Samosir and A H M Yatim ldquoDynamic evolution con-troller for single phase inverter applicationrdquo in Proceedings ofthe IEEE Symposium on Industrial Electronics and Applications(ISIEA rsquo09) vol 1 pp 530ndash535 October 2009

[25] A S Samosir M Anwari and A H M Yatim ldquoDynamicevolution control of interleaved boostDC-DC converter for fuelcell applicationrdquo in Proceedings of the 9th International Powerand Energy Conference (IPEC rsquo10) pp 869ndash874 October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

PWM1 = 1 1198811198981

gt 119881119888119904

0 1198811198981

lt 119881119888119904

PWM2 = 1 1198811198981

lt 119881119888119904

0 1198811198981

gt 119881119888119904

(39)

5 Simulation

Some simulation has been done in order to verify theperformance of the topology and the main parameters canbe referred to in Table 1

The portion of the simulation results is given in Figure 10Figure 10(a) shows the driving signals for 119878

1and 1198782 which

are generated by the control law 120575 The outputting voltageand power of PV cell in process of MPPT can be seen inFigure 10(b) From the simulation onemay get the conclusionthat the DSME PWM controller can ensure that the PV cellruns at the maximum power point in a short time Althoughthe overshoot with a little range occurs in the process ofpower adjustment it will be eliminated soon and the stableoutputting power of 260W is achieved During the dynamicadjustment there is a stable and fleet performance withoutoscillation

In Figure 10(c) the dynamic response of the novel topol-ogy is the curve labeled overall trend This curse shows thatthe converter can raise the voltage from 36V to 400V steadilywithout a long time The curve labeled partial enlargementshows that the outputting voltage has a low voltage ripplewhen it is in stable state Considering the whole PV grid-connected system a DC-link high voltage with low rippleis convenient to choice and maintaining of the DC-linkcapacitor

As mentioned before the converter can run under fourmodes in every switching period The tendencies of 119894

1198711and

1198941198712as well as 119894in in every mode have been discussed in part III

and it is also easy to get the conclusion about 119894in = 1198941198711+ 1198941198712

based on Kirchhoff rsquos current law so the tendency of 119894in canbe obtained by the sum of 119894

1198711and 1198941198712 The first and second

one of Figure 10(d) describe the stable current of 1198941198711

and 1198941198712

and 119894in respectively over the same period in stable modeFigure 10(d) shows that the current 119894in has little ripple whichcan reduce the coupled inductance losses and improve systemefficiency

Figure 10(e) shows the voltage waves of switched capac-itors 119862

1and 119862

2 The voltage of 119862

1is three times as much as

the one of 1198622 close to three quarters of 119881

119900 this simulation

verifies the formula (13)

6 Experiments

Experiments have been finished in our laboratory just inorder to verify the validity of theoretical analysis above andthe performance of the new converter proposed in this paper

Experimental waveforms of the new converter are shownin Figure 11 The driving signals of 119878

1and 119878

2from DSP on

basis of the algorithm in this paper are given in Figure 11(a)Obviously the new converter can raise the voltage from36V to 400V when the duty cycle equals 064 (calculated

Table 1 The main parameters

Parameters Values119881ocV 443119868scA 77119881mp at MPPV 36119868mp at MPPA 72Coupling coefficient 091198711

and 1198712

mH 041198621

120583F 101198622

120583F 1001198620

120583F 22119877kΩ 1Switching frequencykHz 50

PWM1PWM2

PWM120575(Vo Vpv ipv ΔVpv Δipv )

0

Vo

Vcsipv

Vpv Vm1

Figure 9 PWM generator

from Figure 11(a)) The output voltage wave (after 10 timesattenuation) is shown in Figure 11(b) Moreover the rippleof the output voltage shown in Figure 11(b) can be limitedin 8V which can decrease the loss of switched capacitorsas shown in (15) Figure 11(c) shows the wave of the inputcurrent The little input current ripple can reduce both theloss and the volume of the coupled inductance Figures 11(d)and 11(e) show the voltage waveform of switched capacitors1198621and 119862

2 respectively (after 10 times attenuation) The

voltage of 1198621 about 300Vsim306V is three times as much as

the one of 1198622 about 92Vsim100V which is consistent with

both the analysis and the simulation results above besidesthe difference between the maximum and the minimum ofvoltage in the course of charging for each capacitance can belimited to a fairly small range which can decrease the loss ofswitched capacitors

Burrs existing in experimental waveform mainly stemfrom the effect of EMI in the experiment

7 Conclusion

This paper proposed a new high-gain interleaved converterwith coupled inductances and switched capacitors networkwhich are suited for PV application After explaining its oper-ating principle and analyzing its steady-state performancethen a control strategy for it on basis of SM-MPPT andDEC theory was designed Both simulation analysis andexperimental results showed that the proposed system hadthe following good performance

(1) Achieving MPPT and producing a high-gain voltagein an extremely short time

(2) Decreasing the ripple of the outputting voltage as wellas the inputting current and improving power density

Journal of Applied Mathematics 9

For S

1

ForS

2

(a) Driving signals for 1198781amp 1198782

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

Time (s)

Volta

ge (V

) and

pow

er (W

)

The outputting voltage and power of PV cells under MPPT

Vpv outputting voltageP outputting powerpv

(b) The output voltage and power of PV cell in process of MPPT

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

350

400

450The outputting voltage of the new topological converter

Time (s)

Volta

ge (V

)

399

3995

400

4005

401

Partial enlargement of outtputing voltage

Overall trendPartial enlargement

(c) The output voltage wave

0

5

Curr

ent (

A) The Lofwavecurrent 1 and L2 in stable mode

001 0015 002 0025 00305

1015

Time (s)

Curr

ent (

A)

iL1

iL2

The current wave of iin in the dynamic mode

65

7

75

Curr

ent (

A) The current wave of iin in stable mode

(d) The current of 1198711and 1198712and 119894in

0 001 002 003 004 005 006 007 008 009 01minus100

minus50

0

50

100

150

200

250

300

350

400

Time (s)

Volta

ge (V

)

The voltage of and

Voltage of C1

Voltage of C2

C1 C2

(e) The voltage wave of switched capacitor 1198621and 119862

2

Figure 10 Simulation results

10 Journal of Applied Mathematics

(a) Driving signals for 1198781and 1198782from DSP (b) The outputting voltage wave (X10)

(c) The wave of input current (d) The voltage wave of switched capacitor 1198621(X10)

(e) The voltage wave of switched capacitor 1198622(X10)

Figure 11 Experimental validation

(3) Reducing the volume of magnetic components andcutting the cost of system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education of China (no20120032110070)

References

[1] C Zhang and X He ldquoMaximum power point tracking by usingasymmetric fuzzy control combined with PID for photovoltaic

energy generation systemrdquo Transactions of China Electrotechni-cal Society vol 20 no 10 pp 72ndash75 2005

[2] Z Liao and X Ruan ldquoControl strategy for bi-directionalDCDC converter of a novel stand-alone photovoltaic powersystemrdquo Transactions of China Electrotechnical Society vol 23no 1 pp 97ndash103 2008

[3] M Dong J Yang K Peng and A Luo ldquoZero average incre-mental conductance maximum power point tracking controlfor photovoltaic systemrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 30 no 21 pp 48ndash53 2010

[4] J M Enrique J M Andujar and M A Bohorquez ldquoAreliable fast and low cost maximum power point tracker forphotovoltaic applicationsrdquo Solar Energy vol 84 pp 79ndash89 2010

[5] L Xie RGong and J Li ldquoAnalysis of the dynamical characteris-tics of the Interleaved boost converter inmaximumpower pointtracking for photovoltaic powerrdquo in Proceedings of the ChineseSociety for Electrical Engineering (CSEE rsquo13) vol 33 no 6 pp39ndash43 2013

Journal of Applied Mathematics 11

[6] W Li and X He ldquoReview of nonisolated high-step-up DCDCconverters in photovoltaic grid-connected applicationsrdquo IEEETransactions on Industrial Electronics vol 58 no 4 pp 1239ndash1250 2011

[7] Q LuoH Yan S Zhi C Zou and L Zhou ldquoAn interleaved highstep-up zero current transition boost converterrdquo in Proceedingsof the Chinese Society for Electrical Engineering (CSEE rsquo13) vol33 no 12 pp 18ndash21 2013

[8] W Huang and B Lehman ldquoMitigation and utilization of theinductor coupling effect in interleaved multiphase dcdc con-vertersrdquo in Proceedings of the IEEE Energy Conversion Congressand Exposition (ECCE rsquo13) pp 1822ndash1829 2013

[9] G Zhu B A McDonald and K Wang ldquoModeling and analysisof coupled inductors in power convertersrdquo IEEE Transactionson Power Electronics vol 26 no 5 pp 1355ndash1363 2011

[10] Q-B Hu B Qu and Z-Y Lu ldquoNovel step-up VRM-two-phaseinterleaved coupled-boost converterrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 26 no 9 pp 94ndash98 2006

[11] D A Grant Y Darroman and J Suter ldquoSynthesis of tapped-inductor switched-mode convertersrdquo IEEE Transactions onPower Electronics vol 22 no 5 pp 1964ndash1969 2007

[12] M Nymand and M A E Andersen ldquoHigh-efficiency isolatedboost DC-DC converter for high-power low-voltage fuel-cellapplicationsrdquo IEEE Transactions on Industrial Electronics vol57 no 2 pp 505ndash514 2010

[13] R Haroun A Cid-Pastor A El Aroudi and L Martinez-Salamero ldquoCascade connection ofDC-DC switching convertersby means of self-oscillating dc-transformersrdquo in Proceedingsof the 15th International Power Electronics and Motion ControlConference (EPEPEMC rsquo12) 2012

[14] R Guo C Wang and T Li ldquoOptimum design of couplinginductors for magnetic integration in three-phase interleavingBuck dcdc converterrdquo in Proceedings of the 8th IEEE Conferenceon Industrial Electronics and Applications (ICIEA rsquo13) pp 1029ndash1033 2013

[15] H Wu J Gu J Zhang Y Xing and G Chen ldquoHigh efficiencyhigh step-up Boost-Flyback DCDC converterrdquo Proceedings ofthe Chinese Society of Electrical Engineering vol 31 no 24 pp40ndash45 2011

[16] T J Liang andK C Tseng ldquoAnalysis of integrated boost-flybackstep-up converterrdquo IEE Proceedings Electric Power Applicationsvol 152 no 2 pp 217ndash225 2005

[17] J L Kui-Jun P Byoung-Gun K Rae-young and H Dong-seok ldquoNonisolated ZVT two-inductor boost converter with asingle resonant inductor for high step-up applicationsrdquo IEEETransactions on Power Electronics vol 27 no 4 pp 1966ndash19732012

[18] Z Lu L Zheng and Z Ma ldquoInterleaved high gain boostconverter with switched capacitor networkrdquo Transactions ofChina Electrotechnical Society vol 27 no 11 pp 154ndash156 2012

[19] A S Samosir and A H M Yatim ldquoImplementation of dynamicevolution control of bidirectional DC-DC converter for inter-facing ultracapacitor energy storage to fuel-cell systemrdquo IEEETransactions on Industrial Electronics vol 57 no 10 pp 3468ndash3473 2010

[20] A S Samosir and A H M Yatim ldquoDynamic evolution controlof bidirectionalDC-DCconverter for interfacing ultra capacitorenergy storage to fuel cell electric vehicle systemrdquo in Proceedingsof the Power Engineering Conferencee (AUPEC rsquo08) AustralasianUniversities December 2008

[21] A H ALQahtani M S Abuhamdeh Y M Alsmadi and VI Utkin ldquoPhotovoltaic power optimization using sliding mode

control with a two-axis tracking systemrdquo in Proceedings of theIEEE Energytech pp 1ndash6 2013

[22] D G Montoya C A R Paja and R Giral ldquoA new solutionof maximum power point tracking based on sliding modecontrolrdquo in Proceedings of the 39th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo13) pp 8350ndash83552013

[23] J He ldquoConnected power system based on sliding mode theoryrdquoin Study on Control Algorithm of Maximum Power Point Track-ing for Photovoltaic Grid Shandong University 2012

[24] A S Samosir and A H M Yatim ldquoDynamic evolution con-troller for single phase inverter applicationrdquo in Proceedings ofthe IEEE Symposium on Industrial Electronics and Applications(ISIEA rsquo09) vol 1 pp 530ndash535 October 2009

[25] A S Samosir M Anwari and A H M Yatim ldquoDynamicevolution control of interleaved boostDC-DC converter for fuelcell applicationrdquo in Proceedings of the 9th International Powerand Energy Conference (IPEC rsquo10) pp 869ndash874 October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

For S

1

ForS

2

(a) Driving signals for 1198781amp 1198782

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

Time (s)

Volta

ge (V

) and

pow

er (W

)

The outputting voltage and power of PV cells under MPPT

Vpv outputting voltageP outputting powerpv

(b) The output voltage and power of PV cell in process of MPPT

0 001 002 003 004 005 006 007 008 009 010

50

100

150

200

250

300

350

400

450The outputting voltage of the new topological converter

Time (s)

Volta

ge (V

)

399

3995

400

4005

401

Partial enlargement of outtputing voltage

Overall trendPartial enlargement

(c) The output voltage wave

0

5

Curr

ent (

A) The Lofwavecurrent 1 and L2 in stable mode

001 0015 002 0025 00305

1015

Time (s)

Curr

ent (

A)

iL1

iL2

The current wave of iin in the dynamic mode

65

7

75

Curr

ent (

A) The current wave of iin in stable mode

(d) The current of 1198711and 1198712and 119894in

0 001 002 003 004 005 006 007 008 009 01minus100

minus50

0

50

100

150

200

250

300

350

400

Time (s)

Volta

ge (V

)

The voltage of and

Voltage of C1

Voltage of C2

C1 C2

(e) The voltage wave of switched capacitor 1198621and 119862

2

Figure 10 Simulation results

10 Journal of Applied Mathematics

(a) Driving signals for 1198781and 1198782from DSP (b) The outputting voltage wave (X10)

(c) The wave of input current (d) The voltage wave of switched capacitor 1198621(X10)

(e) The voltage wave of switched capacitor 1198622(X10)

Figure 11 Experimental validation

(3) Reducing the volume of magnetic components andcutting the cost of system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education of China (no20120032110070)

References

[1] C Zhang and X He ldquoMaximum power point tracking by usingasymmetric fuzzy control combined with PID for photovoltaic

energy generation systemrdquo Transactions of China Electrotechni-cal Society vol 20 no 10 pp 72ndash75 2005

[2] Z Liao and X Ruan ldquoControl strategy for bi-directionalDCDC converter of a novel stand-alone photovoltaic powersystemrdquo Transactions of China Electrotechnical Society vol 23no 1 pp 97ndash103 2008

[3] M Dong J Yang K Peng and A Luo ldquoZero average incre-mental conductance maximum power point tracking controlfor photovoltaic systemrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 30 no 21 pp 48ndash53 2010

[4] J M Enrique J M Andujar and M A Bohorquez ldquoAreliable fast and low cost maximum power point tracker forphotovoltaic applicationsrdquo Solar Energy vol 84 pp 79ndash89 2010

[5] L Xie RGong and J Li ldquoAnalysis of the dynamical characteris-tics of the Interleaved boost converter inmaximumpower pointtracking for photovoltaic powerrdquo in Proceedings of the ChineseSociety for Electrical Engineering (CSEE rsquo13) vol 33 no 6 pp39ndash43 2013

Journal of Applied Mathematics 11

[6] W Li and X He ldquoReview of nonisolated high-step-up DCDCconverters in photovoltaic grid-connected applicationsrdquo IEEETransactions on Industrial Electronics vol 58 no 4 pp 1239ndash1250 2011

[7] Q LuoH Yan S Zhi C Zou and L Zhou ldquoAn interleaved highstep-up zero current transition boost converterrdquo in Proceedingsof the Chinese Society for Electrical Engineering (CSEE rsquo13) vol33 no 12 pp 18ndash21 2013

[8] W Huang and B Lehman ldquoMitigation and utilization of theinductor coupling effect in interleaved multiphase dcdc con-vertersrdquo in Proceedings of the IEEE Energy Conversion Congressand Exposition (ECCE rsquo13) pp 1822ndash1829 2013

[9] G Zhu B A McDonald and K Wang ldquoModeling and analysisof coupled inductors in power convertersrdquo IEEE Transactionson Power Electronics vol 26 no 5 pp 1355ndash1363 2011

[10] Q-B Hu B Qu and Z-Y Lu ldquoNovel step-up VRM-two-phaseinterleaved coupled-boost converterrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 26 no 9 pp 94ndash98 2006

[11] D A Grant Y Darroman and J Suter ldquoSynthesis of tapped-inductor switched-mode convertersrdquo IEEE Transactions onPower Electronics vol 22 no 5 pp 1964ndash1969 2007

[12] M Nymand and M A E Andersen ldquoHigh-efficiency isolatedboost DC-DC converter for high-power low-voltage fuel-cellapplicationsrdquo IEEE Transactions on Industrial Electronics vol57 no 2 pp 505ndash514 2010

[13] R Haroun A Cid-Pastor A El Aroudi and L Martinez-Salamero ldquoCascade connection ofDC-DC switching convertersby means of self-oscillating dc-transformersrdquo in Proceedingsof the 15th International Power Electronics and Motion ControlConference (EPEPEMC rsquo12) 2012

[14] R Guo C Wang and T Li ldquoOptimum design of couplinginductors for magnetic integration in three-phase interleavingBuck dcdc converterrdquo in Proceedings of the 8th IEEE Conferenceon Industrial Electronics and Applications (ICIEA rsquo13) pp 1029ndash1033 2013

[15] H Wu J Gu J Zhang Y Xing and G Chen ldquoHigh efficiencyhigh step-up Boost-Flyback DCDC converterrdquo Proceedings ofthe Chinese Society of Electrical Engineering vol 31 no 24 pp40ndash45 2011

[16] T J Liang andK C Tseng ldquoAnalysis of integrated boost-flybackstep-up converterrdquo IEE Proceedings Electric Power Applicationsvol 152 no 2 pp 217ndash225 2005

[17] J L Kui-Jun P Byoung-Gun K Rae-young and H Dong-seok ldquoNonisolated ZVT two-inductor boost converter with asingle resonant inductor for high step-up applicationsrdquo IEEETransactions on Power Electronics vol 27 no 4 pp 1966ndash19732012

[18] Z Lu L Zheng and Z Ma ldquoInterleaved high gain boostconverter with switched capacitor networkrdquo Transactions ofChina Electrotechnical Society vol 27 no 11 pp 154ndash156 2012

[19] A S Samosir and A H M Yatim ldquoImplementation of dynamicevolution control of bidirectional DC-DC converter for inter-facing ultracapacitor energy storage to fuel-cell systemrdquo IEEETransactions on Industrial Electronics vol 57 no 10 pp 3468ndash3473 2010

[20] A S Samosir and A H M Yatim ldquoDynamic evolution controlof bidirectionalDC-DCconverter for interfacing ultra capacitorenergy storage to fuel cell electric vehicle systemrdquo in Proceedingsof the Power Engineering Conferencee (AUPEC rsquo08) AustralasianUniversities December 2008

[21] A H ALQahtani M S Abuhamdeh Y M Alsmadi and VI Utkin ldquoPhotovoltaic power optimization using sliding mode

control with a two-axis tracking systemrdquo in Proceedings of theIEEE Energytech pp 1ndash6 2013

[22] D G Montoya C A R Paja and R Giral ldquoA new solutionof maximum power point tracking based on sliding modecontrolrdquo in Proceedings of the 39th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo13) pp 8350ndash83552013

[23] J He ldquoConnected power system based on sliding mode theoryrdquoin Study on Control Algorithm of Maximum Power Point Track-ing for Photovoltaic Grid Shandong University 2012

[24] A S Samosir and A H M Yatim ldquoDynamic evolution con-troller for single phase inverter applicationrdquo in Proceedings ofthe IEEE Symposium on Industrial Electronics and Applications(ISIEA rsquo09) vol 1 pp 530ndash535 October 2009

[25] A S Samosir M Anwari and A H M Yatim ldquoDynamicevolution control of interleaved boostDC-DC converter for fuelcell applicationrdquo in Proceedings of the 9th International Powerand Energy Conference (IPEC rsquo10) pp 869ndash874 October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

(a) Driving signals for 1198781and 1198782from DSP (b) The outputting voltage wave (X10)

(c) The wave of input current (d) The voltage wave of switched capacitor 1198621(X10)

(e) The voltage wave of switched capacitor 1198622(X10)

Figure 11 Experimental validation

(3) Reducing the volume of magnetic components andcutting the cost of system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education of China (no20120032110070)

References

[1] C Zhang and X He ldquoMaximum power point tracking by usingasymmetric fuzzy control combined with PID for photovoltaic

energy generation systemrdquo Transactions of China Electrotechni-cal Society vol 20 no 10 pp 72ndash75 2005

[2] Z Liao and X Ruan ldquoControl strategy for bi-directionalDCDC converter of a novel stand-alone photovoltaic powersystemrdquo Transactions of China Electrotechnical Society vol 23no 1 pp 97ndash103 2008

[3] M Dong J Yang K Peng and A Luo ldquoZero average incre-mental conductance maximum power point tracking controlfor photovoltaic systemrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 30 no 21 pp 48ndash53 2010

[4] J M Enrique J M Andujar and M A Bohorquez ldquoAreliable fast and low cost maximum power point tracker forphotovoltaic applicationsrdquo Solar Energy vol 84 pp 79ndash89 2010

[5] L Xie RGong and J Li ldquoAnalysis of the dynamical characteris-tics of the Interleaved boost converter inmaximumpower pointtracking for photovoltaic powerrdquo in Proceedings of the ChineseSociety for Electrical Engineering (CSEE rsquo13) vol 33 no 6 pp39ndash43 2013

Journal of Applied Mathematics 11

[6] W Li and X He ldquoReview of nonisolated high-step-up DCDCconverters in photovoltaic grid-connected applicationsrdquo IEEETransactions on Industrial Electronics vol 58 no 4 pp 1239ndash1250 2011

[7] Q LuoH Yan S Zhi C Zou and L Zhou ldquoAn interleaved highstep-up zero current transition boost converterrdquo in Proceedingsof the Chinese Society for Electrical Engineering (CSEE rsquo13) vol33 no 12 pp 18ndash21 2013

[8] W Huang and B Lehman ldquoMitigation and utilization of theinductor coupling effect in interleaved multiphase dcdc con-vertersrdquo in Proceedings of the IEEE Energy Conversion Congressand Exposition (ECCE rsquo13) pp 1822ndash1829 2013

[9] G Zhu B A McDonald and K Wang ldquoModeling and analysisof coupled inductors in power convertersrdquo IEEE Transactionson Power Electronics vol 26 no 5 pp 1355ndash1363 2011

[10] Q-B Hu B Qu and Z-Y Lu ldquoNovel step-up VRM-two-phaseinterleaved coupled-boost converterrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 26 no 9 pp 94ndash98 2006

[11] D A Grant Y Darroman and J Suter ldquoSynthesis of tapped-inductor switched-mode convertersrdquo IEEE Transactions onPower Electronics vol 22 no 5 pp 1964ndash1969 2007

[12] M Nymand and M A E Andersen ldquoHigh-efficiency isolatedboost DC-DC converter for high-power low-voltage fuel-cellapplicationsrdquo IEEE Transactions on Industrial Electronics vol57 no 2 pp 505ndash514 2010

[13] R Haroun A Cid-Pastor A El Aroudi and L Martinez-Salamero ldquoCascade connection ofDC-DC switching convertersby means of self-oscillating dc-transformersrdquo in Proceedingsof the 15th International Power Electronics and Motion ControlConference (EPEPEMC rsquo12) 2012

[14] R Guo C Wang and T Li ldquoOptimum design of couplinginductors for magnetic integration in three-phase interleavingBuck dcdc converterrdquo in Proceedings of the 8th IEEE Conferenceon Industrial Electronics and Applications (ICIEA rsquo13) pp 1029ndash1033 2013

[15] H Wu J Gu J Zhang Y Xing and G Chen ldquoHigh efficiencyhigh step-up Boost-Flyback DCDC converterrdquo Proceedings ofthe Chinese Society of Electrical Engineering vol 31 no 24 pp40ndash45 2011

[16] T J Liang andK C Tseng ldquoAnalysis of integrated boost-flybackstep-up converterrdquo IEE Proceedings Electric Power Applicationsvol 152 no 2 pp 217ndash225 2005

[17] J L Kui-Jun P Byoung-Gun K Rae-young and H Dong-seok ldquoNonisolated ZVT two-inductor boost converter with asingle resonant inductor for high step-up applicationsrdquo IEEETransactions on Power Electronics vol 27 no 4 pp 1966ndash19732012

[18] Z Lu L Zheng and Z Ma ldquoInterleaved high gain boostconverter with switched capacitor networkrdquo Transactions ofChina Electrotechnical Society vol 27 no 11 pp 154ndash156 2012

[19] A S Samosir and A H M Yatim ldquoImplementation of dynamicevolution control of bidirectional DC-DC converter for inter-facing ultracapacitor energy storage to fuel-cell systemrdquo IEEETransactions on Industrial Electronics vol 57 no 10 pp 3468ndash3473 2010

[20] A S Samosir and A H M Yatim ldquoDynamic evolution controlof bidirectionalDC-DCconverter for interfacing ultra capacitorenergy storage to fuel cell electric vehicle systemrdquo in Proceedingsof the Power Engineering Conferencee (AUPEC rsquo08) AustralasianUniversities December 2008

[21] A H ALQahtani M S Abuhamdeh Y M Alsmadi and VI Utkin ldquoPhotovoltaic power optimization using sliding mode

control with a two-axis tracking systemrdquo in Proceedings of theIEEE Energytech pp 1ndash6 2013

[22] D G Montoya C A R Paja and R Giral ldquoA new solutionof maximum power point tracking based on sliding modecontrolrdquo in Proceedings of the 39th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo13) pp 8350ndash83552013

[23] J He ldquoConnected power system based on sliding mode theoryrdquoin Study on Control Algorithm of Maximum Power Point Track-ing for Photovoltaic Grid Shandong University 2012

[24] A S Samosir and A H M Yatim ldquoDynamic evolution con-troller for single phase inverter applicationrdquo in Proceedings ofthe IEEE Symposium on Industrial Electronics and Applications(ISIEA rsquo09) vol 1 pp 530ndash535 October 2009

[25] A S Samosir M Anwari and A H M Yatim ldquoDynamicevolution control of interleaved boostDC-DC converter for fuelcell applicationrdquo in Proceedings of the 9th International Powerand Energy Conference (IPEC rsquo10) pp 869ndash874 October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

[6] W Li and X He ldquoReview of nonisolated high-step-up DCDCconverters in photovoltaic grid-connected applicationsrdquo IEEETransactions on Industrial Electronics vol 58 no 4 pp 1239ndash1250 2011

[7] Q LuoH Yan S Zhi C Zou and L Zhou ldquoAn interleaved highstep-up zero current transition boost converterrdquo in Proceedingsof the Chinese Society for Electrical Engineering (CSEE rsquo13) vol33 no 12 pp 18ndash21 2013

[8] W Huang and B Lehman ldquoMitigation and utilization of theinductor coupling effect in interleaved multiphase dcdc con-vertersrdquo in Proceedings of the IEEE Energy Conversion Congressand Exposition (ECCE rsquo13) pp 1822ndash1829 2013

[9] G Zhu B A McDonald and K Wang ldquoModeling and analysisof coupled inductors in power convertersrdquo IEEE Transactionson Power Electronics vol 26 no 5 pp 1355ndash1363 2011

[10] Q-B Hu B Qu and Z-Y Lu ldquoNovel step-up VRM-two-phaseinterleaved coupled-boost converterrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 26 no 9 pp 94ndash98 2006

[11] D A Grant Y Darroman and J Suter ldquoSynthesis of tapped-inductor switched-mode convertersrdquo IEEE Transactions onPower Electronics vol 22 no 5 pp 1964ndash1969 2007

[12] M Nymand and M A E Andersen ldquoHigh-efficiency isolatedboost DC-DC converter for high-power low-voltage fuel-cellapplicationsrdquo IEEE Transactions on Industrial Electronics vol57 no 2 pp 505ndash514 2010

[13] R Haroun A Cid-Pastor A El Aroudi and L Martinez-Salamero ldquoCascade connection ofDC-DC switching convertersby means of self-oscillating dc-transformersrdquo in Proceedingsof the 15th International Power Electronics and Motion ControlConference (EPEPEMC rsquo12) 2012

[14] R Guo C Wang and T Li ldquoOptimum design of couplinginductors for magnetic integration in three-phase interleavingBuck dcdc converterrdquo in Proceedings of the 8th IEEE Conferenceon Industrial Electronics and Applications (ICIEA rsquo13) pp 1029ndash1033 2013

[15] H Wu J Gu J Zhang Y Xing and G Chen ldquoHigh efficiencyhigh step-up Boost-Flyback DCDC converterrdquo Proceedings ofthe Chinese Society of Electrical Engineering vol 31 no 24 pp40ndash45 2011

[16] T J Liang andK C Tseng ldquoAnalysis of integrated boost-flybackstep-up converterrdquo IEE Proceedings Electric Power Applicationsvol 152 no 2 pp 217ndash225 2005

[17] J L Kui-Jun P Byoung-Gun K Rae-young and H Dong-seok ldquoNonisolated ZVT two-inductor boost converter with asingle resonant inductor for high step-up applicationsrdquo IEEETransactions on Power Electronics vol 27 no 4 pp 1966ndash19732012

[18] Z Lu L Zheng and Z Ma ldquoInterleaved high gain boostconverter with switched capacitor networkrdquo Transactions ofChina Electrotechnical Society vol 27 no 11 pp 154ndash156 2012

[19] A S Samosir and A H M Yatim ldquoImplementation of dynamicevolution control of bidirectional DC-DC converter for inter-facing ultracapacitor energy storage to fuel-cell systemrdquo IEEETransactions on Industrial Electronics vol 57 no 10 pp 3468ndash3473 2010

[20] A S Samosir and A H M Yatim ldquoDynamic evolution controlof bidirectionalDC-DCconverter for interfacing ultra capacitorenergy storage to fuel cell electric vehicle systemrdquo in Proceedingsof the Power Engineering Conferencee (AUPEC rsquo08) AustralasianUniversities December 2008

[21] A H ALQahtani M S Abuhamdeh Y M Alsmadi and VI Utkin ldquoPhotovoltaic power optimization using sliding mode

control with a two-axis tracking systemrdquo in Proceedings of theIEEE Energytech pp 1ndash6 2013

[22] D G Montoya C A R Paja and R Giral ldquoA new solutionof maximum power point tracking based on sliding modecontrolrdquo in Proceedings of the 39th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo13) pp 8350ndash83552013

[23] J He ldquoConnected power system based on sliding mode theoryrdquoin Study on Control Algorithm of Maximum Power Point Track-ing for Photovoltaic Grid Shandong University 2012

[24] A S Samosir and A H M Yatim ldquoDynamic evolution con-troller for single phase inverter applicationrdquo in Proceedings ofthe IEEE Symposium on Industrial Electronics and Applications(ISIEA rsquo09) vol 1 pp 530ndash535 October 2009

[25] A S Samosir M Anwari and A H M Yatim ldquoDynamicevolution control of interleaved boostDC-DC converter for fuelcell applicationrdquo in Proceedings of the 9th International Powerand Energy Conference (IPEC rsquo10) pp 869ndash874 October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of