research article efficiency improvement of three-phase cascaded...

Research ArticleEfficiency Improvement of Three-Phase Cascaded H-BridgeMultilevel Inverters for Photovoltaic Systems

Nuntawat Thitichaiworakorn Nattapon Chayopitak and Natchpong Hatti

National Electronics and Computer Technology Center (NECTEC) Pathumthani 12120 Thailand

Correspondence should be addressed to Nuntawat Thitichaiworakorn nuntawatthitichaiworakornnectecorth

Received 1 March 2016 Accepted 15 May 2016

Academic Editor Santolo Meo

Copyright copy 2016 Nuntawat Thitichaiworakorn et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Medium-scale photovoltaic (PV) systems using cascaded H-bridge multilevel inverters have a capability to perform individualmaximum power point tracking (MPPT) for each PV panel or each small group of panels resulting in minimization of both powerlosses from panel mismatch and effect of partial shading They also provide high power quality modularity and possibility ofeliminating dc-dc boost stage and line-frequency transformer However each PV panel in the system is subjected to a double-line-frequency voltage ripple at the dc-link which reduces the MPPT efficiency This paper proposes a dc-link voltage ripple reductionby third-harmonic zero-sequence voltage injection for improving the MPPT efficiency Moreover a control method to achieveindividualMPPT control of each inverter cell is also presentedThe validity and effectiveness of the proposedmethods were verifiedby computer simulation

1 Introduction

Renewable energy technology has undergone a substantialdevelopment in the last three decades Photovoltaic (PV)system is promising and one of the fastest growing renewableenergy sources The worldwide cumulative installed capacityof PV systems has been increasing exponentially in the lastdecade and recently has reached a level of 178GW at theend of 2014 [1] due to the decreasing price per PV panel andgovernment policies in many countries

Generally the PV system topologies can be categorizedinto four groups (1) central inverter (2) module-integratedinverter (3) string inverter and (4) multistring inverter[2ndash5] In the central inverter topology several PV strings(PV panels connected in series) are connected in parallelwith one blocking diode per string to form a single dc-linkand are connected to the grid via a central inverter Thistopology has a simple structure a reliable control and a lowinitial cost However with only one centralized maximumpower point tracking (MPPT) control the energy yield canbe easily reduced by the effects of panel mismatch andpartial shading Dividing PV panels into smaller groups with

individual MPPT control can mitigate the problem [6 7]The module-integrated inverter topology is another side ofthe spectrum In this topology one converter operates withonly one or a few PV panels Thus the power loss frompanel mismatch can be minimized and the effect of partialshading can be mitigated However the voltage amplificationby either dc-dc boost stage or transformer is required Themodule-integrated inverter topology is intended for small PVsystems lower than 500W For the string inverter topologyone PV string is connected to the grid via one inverterTherefore the benefit of this topology is a trade-off betweencentral inverter and module-integrated inverter topologiesFinally the multistring inverter topology combines higherenergy yield of the string inverter with low initial cost ofthe central inverter topology Several PV strings each withone dedicated dc-dc converter are connected to a centralinverter Moreover in addition to the traditional two-levelinverters the use of neutral-point-clamped (NPC) inverterfor either central inverter string inverter or multistringinverter topologies has also been reported [8 9]

Recently cascaded H-bridge multilevel inverter topologyhas gained interest from many researchers for PV system

Hindawi Publishing CorporationInternational Journal of PhotoenergyVolume 2016 Article ID 2162190 10 pageshttpdxdoiorg10115520162162190

2 International Journal of Photoenergy

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

L

is

s

i1PV

i2PV

inPV

1PV

2PV

nPV

PV

PV

PV

(a)

H-

Phase a Phase b Phase c

bridgeinverter

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

L

ia1PV

ia

ib

ic

ia2PV

ianPV

a1PV

a2PV

anPV

PV

PV

PV

PV

PV

PV

PV

PV

PV

sa

sb

sc

(b)

DC AC

(c)

Figure 1 Topology of the grid-connected cascaded H-bridge PV system (a) Single-phase system (b) Three-phase system (c) H-bridgeinverter

applications [10ndash20] It is characterized by a cascade con-nection of several H-bridge inverters per one phase Thesystem topology is depicted in Figure 1 which can be either asingle-phase system in Figure 1(a) or a three-phase system inFigure 1(b) depending on the system power rating A singlePV panel can be directly connected to the dc side of each H-bridge inverter to implement the concept of one converterper one PV panel This is similar to that of the module-integrated inverter topology In this case it is possible toachieve the distributed MPPT control by each H-bridgeinverter individually [10 11] that greatly optimizes the energyyield from the PV panels The advantages of this system canbe summarized as follows

(i) The MPPT can be performed individually for eachPV panel to eliminate the effects of panel mismatchand partial shading thus maximizing the energyproduction in contrast to the system utilizing thecentralized MPPT

(ii) Due to the voltage scalability of the cascaded multi-level converter it is possible to eliminate the dc-dcboost converter stage and line-frequency transformerfor voltage elevation This can reduce the weight andcost of the PV system

(iii) The cascaded multilevel converter produces multi-level PWM output voltage waveform which has lowvoltage THD (Total Harmonic Distortion) and low

current THDTherefore EMI emission and harmonicfilter are minimized

(iv) Due to multiplicative effect of switching frequencythe system can achieve high output switching fre-quency while the device switching frequency is lowThus the switching power loss is low

(v) Themodularity of the system enables the reduction ofmanufacturing cost and easy maintenance operation

On the other hand the system can be extended to a large-capacity PV system in which the dc-link of each H-bridgeinverter can be connected to several PV strings with onededicated dc-dc converter per string that is similar to theconcept of multistring inverter topology [12 13] This kindof system can achieve a power rating of a few megawatts withthe voltage rating in the medium-voltage level However thispaper focuses on a smaller PV system with one PV panelper one H-bridge inverter for simplifying the analysis to bepresented in this paper

Since each PV panel is directly connected to a single-phase dc-ac inverter it is subjected to power fluctuation ofan angular frequency of 2120596 produced by the ac side of the H-bridge inverter The power fluctuation causes voltage rippleat the terminal of the PV panel connected to the dc-linkcapacitor The low-frequency ripple of the dc-link voltageis significant in this topology because each cell is a single-phase inverterThis is unlike in a three-phase inverter such asNPCor conventional two-level inverterwhere the three phase

International Journal of Photoenergy 3

Isc

Pmax

PPV

Voc

(Vmpp Impp)

Figure 2 Effect of dc-link voltage ripple on the MPPT [3]

legs share a common dc-link in which the ripple frequencyis higher (3120596) but much smaller in amplitude The voltageripple causes an output power ripple from the PV panelbecause the terminal voltage swings around the voltage atMPP This power ripple makes the average output powersomewhat lower than the power at MPP thus reducing theMPPT efficiency as described in Figure 2 [3] Since theMPPTalgorithm always drives the operating point to the MPP andthe center of voltage swing is always the voltage at MPPreducing the dc-link voltage ripple in each cell can increasethe average output power and the MPPT efficiency

The dc-link voltage (or current) ripple reductionmethodshave been proposed for some systems and applications forexample [21] proposed a dc-link ripple current reduction forparalleled three-phase voltage-source converter (VSC) withinterleaving (this method requires two VSCs connected inparalleled) [22] proposed a dc-link voltage ripple reductionfor a transformerless modular wind generator system and[23] proposed a dc-link voltage ripple minimization methodfor a modular VSC for HVDC power transmission Thispaper presents a dc-link voltage ripple reduction in each cellof the three-phase cascaded H-bridge multilevel PV systemby injecting the third-harmonic zero-sequence voltage toimprove the MPPT efficiency The objective of the third-harmonic zero-sequence voltage injection in this paper isdifferent from that in [24] where it is used to increase the dc-link utilization and obtain higher number of voltage levelsThe method proposed in this paper does not introduce anyadditional circuit component In addition this paper alsodescribes a control method to achieve individual MPPTcontrol in each converter cellThe validity and effectiveness ofthe methods presented in this paper are verified by computersimulation

2 Circuit Configuration of the PV System

According to Figure 1 each phase of the system is a seriesconnection of multiple converter cells Each converter cellconsists of an H-bridge inverter and a PV panel as an isolateddc source The dc-link capacitor is installed in parallel witheach PV panel as a power decoupling element for absorbingthe PWM switching current produced at the dc side of each

i

C

PV iH

ia

PV HPV

Figure 3 Circuit diagram of cell a1

H-bridge inverterThe system is connected to the grid via ac-link inductor(s) 119871 Each H-bridge inverter is modulated witha unipolar PWM technique and produces three-level PWMvoltageThe frequency of triangular carrier signal for each cellis 119891119888 When several cells are connected in series a phase-shift

PWM (PS-PWM) modulation strategy is used to generatemultilevel PWM voltage waveform In this case the numberof output voltage levels becomes 2119899+1 levels (line-to-neutral)where 119899 is the number of cells per phase The phase shift ofcarrier signals of the adjacent cells is 120587119899 [25] Due to themultiplicative effect of switching frequency of the PS-PWMthe system then achieves an output switching frequency of2119899119891119888 while the device switching frequency is only at 119891

119888 This

has a positive effect on the system efficiency Moreover boththemultilevel PWMand high switching frequency character-istics have a positive effect on the harmonic performance ofthe system

3 Principle of dc-Link VoltageRipple Reduction

This paper proposes the dc-link voltage ripple reduction byinjecting the third-harmonic zero-sequence voltage compo-nent which can reduce the dc-link voltage ripple withoutintroducing any additional circuit component Since thethird-harmonic zero-sequence voltage injection techniquecan be used only for three-phase three-wire systems thispaper focuses on the three-phase cascaded H-bridge PVsystem as shown in Figure 1(b) only

Considering cell a1 depicted in Figure 3 in steady-statecondition the instantaneous power of capacitor 119901

119862has only

ac component Hence by neglecting power losses of theH-bridge inverter the instantaneous power balance can beexpressed as

119901119862= (119901PV)ac minus (119901H)ac = (VPV119894PV)ac minus (VH119894a)ac (1)

where 119901PV is the power flowing from the PV panel 119901H is thepower flowing to the dc side of H-bridge inverter and (119910)acis the ac component of 119910 In this case (VH119894a)ac is much largerthan (VPV119894PV)ac because VH and 119894a are both ac quantities whileVPV and 119894PV are both dc quantities Hence the instantaneouspower of capacitor119901

119862at the steady-state can be approximated

as

119901119862asymp minus (VH119894a)ac (2)


An approximated expression of the ac ripple component V119862of

capacitor voltage can be calculated from the dc-link capacitorpower 119901

119862as (see Appendix)

V119862=

int119901119862119889119905

119862119881119862

(3)

Hence from (2) and (3) the ac ripple component of dc-linkvoltage can be expressed as

VPV =minusint (VH119894a)ac 119889119905

119862119881PV (4)

where 119862 is the capacitance of the dc-link capacitor and 119881119862is

the nominal capacitor voltage It should be noted that119881PV and119881119862are equalVoltage VH without the third-harmonic zero-sequence

voltage injection is expressed as

VH = 119881H sin120596119905 (5)

while VH with the third-harmonic zero-sequence voltageinjection is expressed as

VH = 119881H (sin120596119905 + 1198603 sin 3120596119905) (6)

where 1198603is the relative amplitude of the third-harmonic

voltage By assuming that the system operates with powerfactor close to unity the grid current 119894a is expressed as

119894a = 119868a sin120596119905 (7)

Substituting (5) (6) and (7) into (4) the dc-link voltageripple component without the third-harmonic zero-sequencevoltage injection is expressed as

VPVorg=

119881H119868a4120596119862119881PV

sin 2120596119905 (8)

while the dc-link voltage ripplewith the third-harmonic zero-sequence voltage injection is expressed as

VPVinj=

119881H119868a4120596119862119881PV

(1 minus 1198603) sin 2120596119905 +

119881H1198603119868a8120596119862119881PV

sin 4120596119905 (9)

Figure 4 shows the plots of the estimated waveforms of dc-link voltage ripple VPVorg

and that of the dc-link voltage ripplewith the third-harmonic zero-sequence voltage injectionVPVinj

when 1198603= 04 It can be seen that injecting the third-

harmonic zero-sequence voltage can reduce the amplitudeof the dc-link voltage ripple Figure 5 shows the percentageof the dc-link voltage ripple reduction versus 119860

3obtained

from the theoretical calculation using (8) and (9) for 1198603=

0 to 08 It shows that as 1198603increases more ripple amplitude

reduction can be achieved However Figure 6 shows that theamplitude of VH is higher than 100 when 119860

3is larger than

04 (amplitude is equal to 100 when 1198603= 0) Hence the

relative amplitude1198603cannot be increased arbitrarily without

bound in order to prevent the overmodulation of H-bridgeinverter The maximum possible value of 119860

3depends on

Volta

ge

Time

(with A3 = 04)PVorg PVinj

Figure 4 Estimation of dc-link voltage ripple without and with thethird-harmonic zero-sequence voltage injection

05

101520253035404550

0 01 02 03 04 05 06 07 08 09Third-harmonic amplitude (A3)

dc-li

nk v

olta

ge ri

pple

redu

ctio

n (

)

Figure 5 Amplitude reduction of dc-link voltage ripple versus 1198603

when injecting third-harmonic zero-sequence voltage

the modulation index of the H-bridge inverter which is aratio between the voltage amplitude 119881H and the PV panelvoltage at maximum power point 119881mpp For example whenthemodulation index is equal to 085 the amplitude of VH canbe increased by the third-harmonic zero-sequence voltageinjection up to 117 (=1085 times 100) before overmodulationoccurs Thus the maximum possible value of 119860

3is 06

according to Figure 6 Hence the dc-link voltage rippleamplitude reduction can be achieved up to 39 according toFigure 5

4 Method for Individual MPPT Control

The principle of individual MPPT control is based on theindividual control of the dc-link voltage of each convertercell By considering Figure 3 the following relation can beobtained

119894H = 119894PV minus 119862119889VPV119889119905

(10)

Hence the dc-link voltage VPV can be controlled by con-trolling 119894H using the proportional-integral (PI) controller asshown in Figure 7 considering 119894PV as a disturbance In thiscase 119894H is related to 119894a as

119894H = 119889 sdot 119894a (11)

where 119889 is the duty cycle of the H-bridge inverter Thismeans that the H-bridge inverter works as a current-source


020406080

100120140160


Am

plitu

de o

fH

()

Figure 6 Amplitude of VH versus1198603with the third-harmonic zero-

sequence voltage injection (100 when 1198603= 0)

+

minusilowastHPIPV

lowastPV

Figure 7 dc-link voltage controller (output is 119894lowastH)

converter when seen from the dc side where 119894H can be madeproportional to 119894a by adjusting the duty cycle Note that 119889is an ac signal because 119894a is an ac signal while 119894H is a dcsignalTherefore the product of 119889 and 119894a becomes a dc signalAssuming that 119894a has a constant amplitude the output of PIcontroller in Figure 7 is changed to 119863 in Figure 8 where 119863is rms value of 119889 This means that the dc-link voltage canbe controlled by adjusting the duty cycle of the H-bridgeinverter However adjusting duty cycle will also affect the H-bridge inverter output voltage VH because

VH = 119889 sdot VPV (12)

Thus the duty cycle of each inverter in phase 119909 (where119909 = a b c) cannot be adjusted arbitrarily otherwise thesummation of VH in phase 119909may not be equal to Vlowast

119909 where Vlowast

119909

is the voltage command for phase 119909 obtained from the gridcurrent control In order to simultaneously control the gridcurrent and the individual dc-link voltage the output voltagecommand of each inverter must be a weighted proportion ofVlowast119909calculated by

V119909119895lowastH = Vlowast119909

119863119909119895V119909119895PV

sum119899

119895=1(119863119909119895V119909119895PV)

(13)

where 119895 = 1 minus 119899 and 119863119909119895is the rms value of the duty cycle of

cell 119909119895 obtained from the PI controller as shown in Figure 8The calculation of the voltage command Vlowast

119909for each phase

is based on the grid current control using the conventionalvoltage-oriented control with 119889119902-current decoupling [26]The 119889-axis current command 119894

lowast

119889is calculated from the

summation of error signals of all cells passing through thePI controller as shown in Figure 9 while the 119902-axis currentcommand 119894lowast

119902is set to zero

+

minusDPIPV

lowastPV

Figure 8 dc-link voltage controller (output is119863)

a1PV

a2PV

cnPV

+

minus

+

minus

+

minus

sum PI ilowastd

a1PV

cnlowastPV

a2lowastPV

Figure 9 Calculation of 119894lowast119889

The calculation of the dc-link voltage command for eachcell V119909119895lowastPV to achieve MPPT can be done by the conventionalPerturb andObserve (PampO)method the PI-based Incremen-tal Conductance method or others as described in [27]

5 Simulation Results

The simulation model of the three-phase cascaded H-bridgePV system in Figure 1(b) was createdwith four cells per phaseIn this model it is assumed that each H-bridge inverter isconnected to a PV panel and a dc-link capacitor at the dc-link network The parameters of the PV panel are based onthe PV panel model CHSM6610P-250 from Astronergy withthe nominal output power of 250W Table 1 summarizes theparameters used in the simulation model

51 Simulation Results without the Third-Harmonic Zero-Sequence Voltage Injection Figure 10 shows the steady-statesimulation results of the system in Figure 1(b) without thethird-harmonic zero-sequence voltage injection Figure 10shows that the waveforms of the three-phase voltage com-mand (Vlowasta and Vlowastb ) are sinusoidal with only fundamentalcomponentThewaveforms of the line-to-line output voltages(Vab and Vbc) are multilevel PWM voltage waveforms with 17voltage levels Therefore the waveforms of the grid currents(119894a and 119894b) are close to sinusoidal with small ripples and lowTHD The peak amplitude of the grid current in this case is


200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

2526272829303132333435

(V)

Time (ms)0 10 20

minus25minus20minus15minus10minus5

05

10152025

ia ib

(A)

Time (ms)0 10 20


050

100150200250

(V)

ab bc

Time (ms)0 10 20

minus150

minus100

minus50

0

50

100

150lowastb

lowasta

(V)

Time (ms)0 10 20

dcc1

Figure 10 Simulation results without the third-harmonic zero-sequence voltage injection

Table 1 Parameters used in the simulation model

PV panel open circuit voltage 38VPV panel short circuit current 91 APV panel maximum power voltage 30VPV panel maximum power current 83 APV panel nominal output power 250WNumber of cells per phase 4DC capacitor 3300 120583FAC inductor 1mHGrid voltage 1225 VL-L

Grid frequency 50HzSystem power rating 3 kWPWM carrier frequency 2 kHzOutput switching frequency 16 kHz

192 A The dc-link voltage waveform of cell c1 Vdcc1 containsa 100Hz ripple of 82 Vp-p and a small switching-frequency

ripple The output power of PV panel of cell c1 119875PV1 containsa 200Hz ripple of 257Wp-p and a small switching-frequencyripple with an average output power and peak output powerof 238W and 249W respectively The peak output power isthe power at MPP which is equal to a product of maximumpower voltage andmaximumpower current in Table 1 (30Vtimes83 A) The total power produced by this system is calculatedas

119901total = Vsa119894a + Vsb119894b + Vsc119894c (14)

where Vsa Vsb and Vsc are the line-to-neutral grid voltages Inthis case the calculated total power is 2856W Note that theconverter power loss is neglected in this simulation

52 Simulation Results with the Third-Harmonic Zero-Sequence Voltage Injection Figure 11 shows the steady-statesimulation results of the system in Figure 1(b) with the third-harmonic zero-sequence voltage injection (119860

3= 04) The

waveforms of three-phase voltage command (Vlowasta and Vlowastb ) are


200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

25

30

35

(V)

Time (ms)0 10 20


2575

125175225

bcab

(V)

Time (ms)0 10 20


01020304050

(V)

Time (ms)0 10 20

Third-harmonic zero-sequence voltage

minus125

minus75

minus25

25

75

125lowastb

lowasta

(V)

Time (ms)0 10 20

Fundamental component

minus125

minus75

minus25

25

75

125

Time (ms)0 10 20

(V)

ba


05

10152025

Time (ms)0 10 20

(A)

ia ibdcc1

Figure 11 Simulation results with the third-harmonic zero-sequence voltage injection (1198603= 04)

the combinations of the fundamental component and thethird-harmonic components The waveforms of the line-to-line output voltages (Vab and Vbc) are multilevel PWM voltagewaveforms similar to those in Figure 10 These waveformsdo not contain the third-harmonic component because thesystem is the three-phase three-wire system which cancelsthe zero-sequence third-harmonic components out at the

output As a result the waveforms of the grid currents (119894aand 119894b) are still close to sinusoidal with only fundamentalcomponent The peak amplitude of the grid current in thiscase is increased to 197 A The waveform of the dc-linkvoltage of cell c1 Vdcc1 contains 100Hz and 200Hz ripplecomponents as predicted by (9)The amplitude of the dc-linkvoltage ripple is decreased to 56Vp-p It can be seen that the


Table 2 Comparison of the results

wo third-harmonicinjection

wthird-harmonicinjection

ofchange

Grid currentpeak amplitude 192 A 197 A +26

dc-link voltageripple amplitude 82 Vp-p 56 Vp-p minus32

PV panel powerripple amplitude 257Wp-p 136Wp-p minus47

PV panel powerat MPP 249W 249W

PV panel averageoutput power 238W 2443W +26

Output power ofthe system 2856W 2932W +26

dc-link voltage waveform in Figure 11 and that in Figure 4are similarThis confirms the validity of dc-link voltage rippleestimation presented in Section 3

In Figure 11 the ripple output power waveform of PVpanel of cell c1 is a multifrequency waveform The amplitudeof the power ripple is decreased to 136Wp-p with an averageoutput power increased to 2443W The peak power wave-form remains 249W In this case the total power produced bythis system is 2932W (calculated by (14)) which is increasedabout 26 from the previous case

53 Comparison of the Results Table 2 shows the comparisonof the simulation results of the two cases It can be seen thatinjecting the third-harmonic zero-sequence voltages with1198603= 04 can reduce the amplitudes of the dc-link voltage

ripple by 32 The PV panel power ripple is also decreasedwhile the peak of power ripple remains unchanged (equal tothe power at MPP) As a result the average output powerof each PV panel is increased and the total output poweris also increased for the same amount According to thesimulation results the total output power is increased about26 without any additional circuit component Howeverit should be noted that the percentage of the increasedpower also depends on the accuracy of the current-voltagecharacteristics of the PV panel used in the simulation

It should also be noted that the value of the dc-link voltageripple reduction of 32 in Table 2 is consistent with the valueof about 30 in Figure 5 This means that the theoreticalestimation of the dc-link voltage ripple presented in Section 3is accurate

6 Conclusion

This paper presents a dc-link voltage ripple reduction ofthe three-phase cascaded H-bridge multilevel PV systemusing the third-harmonic zero-sequence voltage injectionTherefore this method is valid only for three-phase three-wire systemsThe injection of third-harmonic zero-sequencevoltage can reduce the amplitudes of the voltage ripple andpower ripple of each PV panel As a result the average outputpower of each PV panel and the total output power are

increased According to the simulation results the dc-linkvoltage ripple reduction is 32when the relative amplitude ofthe third-harmonic voltage is 04 resulting in the total powerincrease of 26 without any additional circuit componentThis paper also presents a control method to achieve anindividual MPPT control of each converter cell

Appendix

Derivation of dc-Link Voltage Ripple Equation

The derivation presented in this section is the same as [28]and is repeated here for completeness The dc-link voltageVPV(119905) of each cell is equal to the dc-link capacitor voltageV119862(119905) Denote the dc-link capacitor instantaneous power by

119901119862(119905) and the dc-link capacitor instantaneous energy by

119882119862(119905) V119862(119905) 119901

119862(119905) and119882

119862(119905) can be related as

119882119862(119905) = int119901

119862(119905) 119889119905 =

1

2119862 (V119862(119905))2

= 1198820+ 119862(119905) (A1)

Equation (A1) shows that 119882119862(119905) can be expressed as a

summation of the dc component1198820and the ac component

119862(119905) From (A1) the dc-link capacitor voltage V

119862(119905) can be

expressed as

V119862(119905) = radic

2

119862119882119862(119905) = 119881

119862+ V119862(119905) (A2)

Equation (A2) shows that V119862(119905) can be expressed as a sum-

mation of dc mean voltage 119881119862and the ac ripple component

V119862(119905) The objective of the following section is to find V

119862(119905)

Define V119862(119905) = 119891(119909) and 119909 = (2119862)119882

119862(119905) Hence from

(A2) the following equations are obtained

119891 (119909) = radic119909

1198911015840

(119909) =1

2radic119909

(A3)

Next choose a point 119905 = 119886 for Taylor series expansion of119891(119909)which makes119882

119862(119886) = 119882

0(119862(119886) = 0) Then the following

equations are obtained

119909|119905=119886

= 1199090=21198820

119862

119891 (1199090) = radic

21198820

119862

1198911015840

(1199090) =

1

2radic21198820119862

(A4)

The function 119891(119909) can be approximated by Taylor seriesexpansion around the point 119909

0as

119891 (119909) asymp 119891 (1199090) + 1198911015840

(1199090) (119909 minus 119909

0) (A5)


By substituting (A4) into (A5) the following equation isobtained

119891 (119909) = radic21198820

119862+(2119862)119882

119862(119905) minus 2119882

0119862

2radic21198820119862

= radic21198820

119862+119862(119905)

radic21198621198820

(A6)

By comparing (A6) with (A2) the terms119881119862and V119862(119905) can be

expressed as

119881119862= radic

21198820

119862

V119862(119905) =

119862(119905)

radic21198621198820

=119862(119905)

119862119881119862

=

int119905

0

(119901119862(120591))ac 119889120591

119862119881119862

(A7)

Therefore the ac ripple component V119862(119905) can be calculated

from the ac component of the dc-link capacitor power(119901119862(119905))ac

Nomenclature

(119910)ac ac component of 1199101198603 Relative amplitude of third-harmonic voltage

119862 Capacitance of dc-link capacitor119889 Duty cycle signal of bridge cell119863 rms value of 119889119863119909119895 rms duty cycle of 119895th cell of phase 119909

119894a Grid current of phase a119868a Amplitude of 119894a119894lowast

119889 119889-axis current reference of the system

119894lowast


119894H Current of bridge cell at the dc side119894PV Current of PV panel119901119862 Instantaneous power of dc-link capacitor

119901H Instantaneous power of bridge cell at the ac side119901PV Instantaneous power of PV panelV119862 dc-link capacitor voltage

V119862 ac ripple component of dc-link capacitor

voltage119881119862 dc mean value of dc-link capacitor voltage

VH Output voltage of bridge cell at the ac side119881H Amplitude of VHVPV Voltage of PV panelVPV ac ripple component of PV panel voltage119881PV dc mean value of PV panel voltageVPVorg

Ripple component of PV panel voltage withoutthird-harmonic injection

VPVinj Ripple component of PV panel voltage with

third-harmonic injectionVlowast119909 Voltage reference of phase 119909

V119909119895lowastH Reference of VH for 119895th cell of phase 119909V119909119895PV PV panel voltage of 119895th cell of phase 119909Vsa Vsb Vsc Line-to-neutral grid voltages119882119862 Instantaneous energy of dc-link capacitor

1198820 dc component of119882

119862

119862 ac component of119882

119862

120596 Fundamental angular frequency of the system

Competing Interests

The authors declare that they have no competing interests

References

[1] SolarPower Europe Global Market Outlook for Solar Power2015ndash2019 SolarPower Europe Brussel Belgium 2015

[2] J M A Myrzik and M Calais ldquoString and module inte-grated inverters for single-phase grid connected photovoltaicsystemsmdasha reviewrdquo in Proceedings of the IEEE Bologna Pow-erTech Conference vol 2 pp 430ndash437 Bologna Italy June 2003

[3] S B Kjaer J K Pedersen and F Blaabjerg ldquoA review of single-phase grid-connected inverters for photovoltaicmodulesrdquo IEEETransactions on Industry Applications vol 41 no 5 pp 1292ndash1306 2005

[4] M Calais J Myrzik T Spooner and V G Agelidis ldquoInvert-ers for single-phase grid connected photovoltaic systemsmdashan overviewrdquo in Proceedings of the IEEE 33rd Annual PowerElectronics Specialists Conference (PESC rsquo02) vol 4 pp 1995ndash2000 June 2002

[5] Q Li and P Wolfs ldquoA review of the single phase photovoltaicmodule integrated converter topologies with three different DClink configurationsrdquo IEEE Transactions on Power Electronicsvol 23 no 3 pp 1320ndash1333 2008

[6] N Femia G Lisi G Petrone G Spagnuolo and M VitellildquoDistributed maximum power point tracking of photovoltaicarrays novel approach and system analysisrdquo IEEE Transactionson Industrial Electronics vol 55 no 7 pp 2610ndash2621 2008

[7] A Bidram A Davoudi and R S Balog ldquoControl and circuittechniques to mitigate partial shading effects in photovoltaicarraysrdquo IEEE Journal of Photovoltaics vol 2 no 4 pp 532ndash5462012

[8] S Alepuz S Busquets-Monge J Bordonau J Gago DGonzalez and J Balcells ldquoInterfacing renewable energy sourcesto the utility grid using a three-level inverterrdquo IEEETransactionson Industrial Electronics vol 53 no 5 pp 1504ndash1511 2006

[9] R Gonzalez E Gubıa J Lopez and L Marroyo ldquoTransformer-less single-phase multilevel-based photovoltaic inverterrdquo IEEETransactions on Industrial Electronics vol 55 no 7 pp 2694ndash2702 2008

[10] E Villanueva P Correa J Rodriguez and M Pacas ldquoControlof a single-phase cascaded H-bridge multilevel inverter forgrid-connected photovoltaic systemsrdquo IEEE Transactions onIndustrial Electronics vol 56 no 11 pp 4399ndash4406 2009

[11] B Xiao L Hang J Mei C Riley L M Tolbert and BOzpineci ldquoModular cascaded H-bridge multilevel PV inverterwith distributed MPPT for grid-connected applicationsrdquo IEEETransactions on Industry Applications vol 51 no 2 pp 1722ndash1731 2015

[12] Y Yu G Konstantinou B Hredzak and V G Agelidis ldquoPowerbalance optimization of cascaded H-bridge multilevel convert-ers for large-scale photovoltaic integrationrdquo IEEE Transactionson Power Electronics vol 31 no 2 pp 1108ndash1120 2016

[13] Y Yu G Konstantinou B Hredzak and V G Agelidis ldquoPowerbalance of cascaded H-bridge multilevel converters for large-scale photovoltaic integrationrdquo IEEE Transactions on PowerElectronics vol 31 no 1 pp 292ndash303 2016

[14] Y Yu G Konstantinou B Hredzak and V G Agelidis ldquoOper-ation of cascaded H-bridge multilevel converters for large-scale photovoltaic power plants under bridge failuresrdquo IEEE


Transactions on Industrial Electronics vol 62 no 11 pp 7228ndash7236 2015

[15] C D Townsend Y Yu G Konstantinou and V G AgelidisldquoCascaded H-bridge multilevel PV topology for alleviation ofper-phase power imbalances and reduction of second harmonicvoltage ripplerdquo IEEE Transactions on Power Electronics vol 31no 8 pp 5574ndash5586 2016

[16] J Chavarrıa D Biel F Guinjoan C Meza and J J NegronildquoEnergy-balance control of PV cascaded multilevel grid-connected inverters under level-shifted and phase-shiftedPWMsrdquo IEEE Transactions on Industrial Electronics vol 60 no1 pp 98ndash111 2013

[17] D Sun B Ge X Yan et al ldquoModeling impedance designand efficiency analysis of quasi-Z source module in cascadedmultilevel photovoltaic power systemrdquo IEEE Transactions onIndustrial Electronics vol 61 no 11 pp 6108ndash6117 2014

[18] M Coppola F D Napoli P Guerriero D Iannuzzi S Dalientoand A D Pizzo ldquoAn FPGA-based advanced control strategyof a gridtied PV CHB inverterrdquo IEEE Transactions on PowerElectronics vol 31 no 1 pp 806ndash816 2016

[19] Y Liu B Ge H Abu-Rub and F Z Peng ldquoAn effective controlmethod for three-phase quasi-Z-source cascaded multilevelinverter based grid-tie photovoltaic power systemrdquo IEEE Trans-actions on Industrial Electronics vol 61 no 12 pp 6794ndash68022014

[20] C Cecati F Ciancetta and P Siano ldquoA multilevel inverter forphotovoltaic systems with fuzzy logic controlrdquo IEEE Transac-tions on Industrial Electronics vol 57 no 12 pp 4115ndash4125 2010

[21] D Zhang F Wang R Burgos R Lai and D BoroyevichldquoDC-link ripple current reduction for paralleled three-phasevoltage-source converters with interleavingrdquo IEEE Transactionson Power Electronics vol 26 no 6 pp 1741ndash1753 2011

[22] X B Yuan Y D Li J Y Chai and J Wang ldquoDC-link voltageripple reduction for a transformerless modular wind generatorsystemrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6Brighton UK April 2010

[23] M Tomasini R Feldman J C Clare P Wheeler D R Trainerand R S Whitehouse ldquoDC-link voltage ripple minimization inamodularmultilevel voltage source converter forHVDCpowertransmissionrdquo in Proceedings of the 14th European Conferenceon Power Electronics and Applications (EPE rsquo11) pp 1ndash10Birmingham UK September 2011

[24] S K Chattopadhyay C Chakraborty and B C Pal ldquoA hybridmultilevel inverter topology with third harmonic injection forgrid connected photovoltaic central invertersrdquo in Proceedings ofthe 21st IEEE International Symposium on Industrial Electronics(ISIE rsquo12) pp 1736ndash1741 Hangzhou China May 2012

[25] B P McGrath and D G Holmes ldquoMulticarrier PWM strategiesfor multilevel invertersrdquo IEEE Transactions on Industrial Elec-tronics vol 49 no 4 pp 858ndash867 2002

[26] L Maharjan S Inoue and H Akagi ldquoA transformerless energystorage system based on a cascade multilevel PWM converterwith star configurationrdquo IEEE Transactions on Industry Appli-cations vol 44 no 5 pp 1621ndash1630 2008

[27] T Esram andP L Chapman ldquoComparison of photovoltaic arraymaximum power point tracking techniquesrdquo IEEE Transactionson Energy Conversion vol 22 no 2 pp 439ndash449 2007

[28] H Fujita M Hagiwara and H Akagi ldquoPower flow analysis andDC-capacitor voltage regulation for the MMCC-DSCCrdquo IEEJTransactions on Industry Applications vol 132 no 6 pp 659ndash665 2012 (Japanese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

L

is

s

i1PV

i2PV

inPV

1PV

2PV

nPV

PV

PV

PV

(a)

H-

Phase a Phase b Phase c

bridgeinverter

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

H-bridge

inverter

L

ia1PV

ia

ib

ic

ia2PV

ianPV

a1PV

a2PV

anPV

PV

PV

PV

PV

PV

PV

PV

PV

PV

sa

sb

sc

(b)

DC AC

(c)

Figure 1 Topology of the grid-connected cascaded H-bridge PV system (a) Single-phase system (b) Three-phase system (c) H-bridgeinverter

applications [10ndash20] It is characterized by a cascade con-nection of several H-bridge inverters per one phase Thesystem topology is depicted in Figure 1 which can be either asingle-phase system in Figure 1(a) or a three-phase system inFigure 1(b) depending on the system power rating A singlePV panel can be directly connected to the dc side of each H-bridge inverter to implement the concept of one converterper one PV panel This is similar to that of the module-integrated inverter topology In this case it is possible toachieve the distributed MPPT control by each H-bridgeinverter individually [10 11] that greatly optimizes the energyyield from the PV panels The advantages of this system canbe summarized as follows

(i) The MPPT can be performed individually for eachPV panel to eliminate the effects of panel mismatchand partial shading thus maximizing the energyproduction in contrast to the system utilizing thecentralized MPPT

(ii) Due to the voltage scalability of the cascaded multi-level converter it is possible to eliminate the dc-dcboost converter stage and line-frequency transformerfor voltage elevation This can reduce the weight andcost of the PV system

(iii) The cascaded multilevel converter produces multi-level PWM output voltage waveform which has lowvoltage THD (Total Harmonic Distortion) and low

current THDTherefore EMI emission and harmonicfilter are minimized

(iv) Due to multiplicative effect of switching frequencythe system can achieve high output switching fre-quency while the device switching frequency is lowThus the switching power loss is low

(v) Themodularity of the system enables the reduction ofmanufacturing cost and easy maintenance operation

On the other hand the system can be extended to a large-capacity PV system in which the dc-link of each H-bridgeinverter can be connected to several PV strings with onededicated dc-dc converter per string that is similar to theconcept of multistring inverter topology [12 13] This kindof system can achieve a power rating of a few megawatts withthe voltage rating in the medium-voltage level However thispaper focuses on a smaller PV system with one PV panelper one H-bridge inverter for simplifying the analysis to bepresented in this paper

Since each PV panel is directly connected to a single-phase dc-ac inverter it is subjected to power fluctuation ofan angular frequency of 2120596 produced by the ac side of the H-bridge inverter The power fluctuation causes voltage rippleat the terminal of the PV panel connected to the dc-linkcapacitor The low-frequency ripple of the dc-link voltageis significant in this topology because each cell is a single-phase inverterThis is unlike in a three-phase inverter such asNPCor conventional two-level inverterwhere the three phase


Isc

Pmax

PPV

Voc

(Vmpp Impp)






i

C

PV iH

ia

PV HPV




119888 This





119862has only





as






V119862=

int119901119862119889119905

119862119881119862

(3)



119862119881PV (4)




VH = 119881H sin120596119905 (5)


VH = 119881H (sin120596119905 + 1198603 sin 3120596119905) (6)



119894a = 119868a sin120596119905 (7)


VPVorg=

119881H119868a4120596119862119881PV

sin 2120596119905 (8)


VPVinj=

119881H119868a4120596119862119881PV

(1 minus 1198603) sin 2120596119905 +

119881H1198603119868a8120596119862119881PV

sin 4120596119905 (9)





3obtained




3is larger than




3depends on

Volta

ge

Time



05

101520253035404550


dc-li

nk v

olta

ge ri

pple

redu

ctio

n (

)




3is 06




119894H = 119894PV minus 119862119889VPV119889119905

(10)


119894H = 119889 sdot 119894a (11)



020406080

100120140160


Am

plitu

de o

fH

()



+

minusilowastHPIPV

lowastPV



VH = 119889 sdot VPV (12)



119909



119863119909119895V119909119895PV

sum119899

119895=1(119863119909119895V119909119895PV)

(13)





lowast




+

minusDPIPV

lowastPV


a1PV

a2PV

cnPV

+

minus

+

minus

+

minus

sum PI ilowastd

a1PV

cnlowastPV

a2lowastPV







200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

2526272829303132333435

(V)

Time (ms)0 10 20


05

10152025

ia ib

(A)

Time (ms)0 10 20


050

100150200250

(V)

ab bc

Time (ms)0 10 20

minus150

minus100

minus50

0

50

100

150lowastb

lowasta

(V)

Time (ms)0 10 20

dcc1










3= 04) The



200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

25

30

35

(V)

Time (ms)0 10 20


2575

125175225

bcab

(V)

Time (ms)0 10 20


01020304050

(V)

Time (ms)0 10 20


minus125

minus75

minus25

25

75

125lowastb

lowasta

(V)

Time (ms)0 10 20


minus125

minus75

minus25

25

75

125

Time (ms)0 10 20

(V)

ba


05

10152025

Time (ms)0 10 20

(A)

ia ibdcc1








ofchange












6 Conclusion



Appendix




119882119862(119905) V119862(119905) 119901

119862(119905) and119882


119882119862(119905) = int119901

119862(119905) 119889119905 =

1

2119862 (V119862(119905))2

= 1198820+ 119862(119905) (A1)




119862(119905) can be

expressed as

V119862(119905) = radic

2

119862119882119862(119905) = 119881

119862+ V119862(119905) (A2)




119862(119905)

Define V119862(119905) = 119891(119909) and 119909 = (2119862)119882

119862(119905) Hence from


119891 (119909) = radic119909

1198911015840

(119909) =1

2radic119909

(A3)


119862(119886) = 119882



119909|119905=119886

= 1199090=21198820

119862

119891 (1199090) = radic

21198820

119862

1198911015840

(1199090) =

1

2radic21198820119862

(A4)


0as

119891 (119909) asymp 119891 (1199090) + 1198911015840

(1199090) (119909 minus 119909

0) (A5)



119891 (119909) = radic21198820

119862+(2119862)119882

119862(119905) minus 2119882

0119862

2radic21198820119862

= radic21198820

119862+119862(119905)

radic21198621198820

(A6)


expressed as

119881119862= radic

21198820

119862

V119862(119905) =

119862(119905)

radic21198621198820

=119862(119905)

119862119881119862

=

int119905

0

(119901119862(120591))ac 119889120591

119862119881119862

(A7)



Nomenclature





119894lowast












119862


119862


Competing Interests


References








































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Isc

Pmax

PPV

Voc

(Vmpp Impp)






i

C

PV iH

ia

PV HPV




119888 This





119862has only





as






V119862=

int119901119862119889119905

119862119881119862

(3)



119862119881PV (4)




VH = 119881H sin120596119905 (5)


VH = 119881H (sin120596119905 + 1198603 sin 3120596119905) (6)



119894a = 119868a sin120596119905 (7)


VPVorg=

119881H119868a4120596119862119881PV

sin 2120596119905 (8)


VPVinj=

119881H119868a4120596119862119881PV

(1 minus 1198603) sin 2120596119905 +

119881H1198603119868a8120596119862119881PV

sin 4120596119905 (9)





3obtained




3is larger than




3depends on

Volta

ge

Time



05

101520253035404550


dc-li

nk v

olta

ge ri

pple

redu

ctio

n (

)




3is 06




119894H = 119894PV minus 119862119889VPV119889119905

(10)


119894H = 119889 sdot 119894a (11)



020406080

100120140160


Am

plitu

de o

fH

()



+

minusilowastHPIPV

lowastPV



VH = 119889 sdot VPV (12)



119909



119863119909119895V119909119895PV

sum119899

119895=1(119863119909119895V119909119895PV)

(13)





lowast




+

minusDPIPV

lowastPV


a1PV

a2PV

cnPV

+

minus

+

minus

+

minus

sum PI ilowastd

a1PV

cnlowastPV

a2lowastPV







200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

2526272829303132333435

(V)

Time (ms)0 10 20


05

10152025

ia ib

(A)

Time (ms)0 10 20


050

100150200250

(V)

ab bc

Time (ms)0 10 20

minus150

minus100

minus50

0

50

100

150lowastb

lowasta

(V)

Time (ms)0 10 20

dcc1










3= 04) The



200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

25

30

35

(V)

Time (ms)0 10 20


2575

125175225

bcab

(V)

Time (ms)0 10 20


01020304050

(V)

Time (ms)0 10 20


minus125

minus75

minus25

25

75

125lowastb

lowasta

(V)

Time (ms)0 10 20


minus125

minus75

minus25

25

75

125

Time (ms)0 10 20

(V)

ba


05

10152025

Time (ms)0 10 20

(A)

ia ibdcc1








ofchange












6 Conclusion



Appendix




119882119862(119905) V119862(119905) 119901

119862(119905) and119882


119882119862(119905) = int119901

119862(119905) 119889119905 =

1

2119862 (V119862(119905))2

= 1198820+ 119862(119905) (A1)




119862(119905) can be

expressed as

V119862(119905) = radic

2

119862119882119862(119905) = 119881

119862+ V119862(119905) (A2)




119862(119905)

Define V119862(119905) = 119891(119909) and 119909 = (2119862)119882

119862(119905) Hence from


119891 (119909) = radic119909

1198911015840

(119909) =1

2radic119909

(A3)


119862(119886) = 119882



119909|119905=119886

= 1199090=21198820

119862

119891 (1199090) = radic

21198820

119862

1198911015840

(1199090) =

1

2radic21198820119862

(A4)


0as

119891 (119909) asymp 119891 (1199090) + 1198911015840

(1199090) (119909 minus 119909

0) (A5)



119891 (119909) = radic21198820

119862+(2119862)119882

119862(119905) minus 2119882

0119862

2radic21198820119862

= radic21198820

119862+119862(119905)

radic21198621198820

(A6)


expressed as

119881119862= radic

21198820

119862

V119862(119905) =

119862(119905)

radic21198621198820

=119862(119905)

119862119881119862

=

int119905

0

(119901119862(120591))ac 119889120591

119862119881119862

(A7)



Nomenclature





119894lowast












119862


119862


Competing Interests


References








































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





V119862=

int119901119862119889119905

119862119881119862

(3)



119862119881PV (4)




VH = 119881H sin120596119905 (5)


VH = 119881H (sin120596119905 + 1198603 sin 3120596119905) (6)



119894a = 119868a sin120596119905 (7)


VPVorg=

119881H119868a4120596119862119881PV

sin 2120596119905 (8)


VPVinj=

119881H119868a4120596119862119881PV

(1 minus 1198603) sin 2120596119905 +

119881H1198603119868a8120596119862119881PV

sin 4120596119905 (9)





3obtained




3is larger than




3depends on

Volta

ge

Time



05

101520253035404550


dc-li

nk v

olta

ge ri

pple

redu

ctio

n (

)




3is 06




119894H = 119894PV minus 119862119889VPV119889119905

(10)


119894H = 119889 sdot 119894a (11)



020406080

100120140160


Am

plitu

de o

fH

()



+

minusilowastHPIPV

lowastPV



VH = 119889 sdot VPV (12)



119909



119863119909119895V119909119895PV

sum119899

119895=1(119863119909119895V119909119895PV)

(13)





lowast




+

minusDPIPV

lowastPV


a1PV

a2PV

cnPV

+

minus

+

minus

+

minus

sum PI ilowastd

a1PV

cnlowastPV

a2lowastPV







200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

2526272829303132333435

(V)

Time (ms)0 10 20


05

10152025

ia ib

(A)

Time (ms)0 10 20


050

100150200250

(V)

ab bc

Time (ms)0 10 20

minus150

minus100

minus50

0

50

100

150lowastb

lowasta

(V)

Time (ms)0 10 20

dcc1










3= 04) The



200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

25

30

35

(V)

Time (ms)0 10 20


2575

125175225

bcab

(V)

Time (ms)0 10 20


01020304050

(V)

Time (ms)0 10 20


minus125

minus75

minus25

25

75

125lowastb

lowasta

(V)

Time (ms)0 10 20


minus125

minus75

minus25

25

75

125

Time (ms)0 10 20

(V)

ba


05

10152025

Time (ms)0 10 20

(A)

ia ibdcc1








ofchange












6 Conclusion



Appendix




119882119862(119905) V119862(119905) 119901

119862(119905) and119882


119882119862(119905) = int119901

119862(119905) 119889119905 =

1

2119862 (V119862(119905))2

= 1198820+ 119862(119905) (A1)




119862(119905) can be

expressed as

V119862(119905) = radic

2

119862119882119862(119905) = 119881

119862+ V119862(119905) (A2)




119862(119905)

Define V119862(119905) = 119891(119909) and 119909 = (2119862)119882

119862(119905) Hence from


119891 (119909) = radic119909

1198911015840

(119909) =1

2radic119909

(A3)


119862(119886) = 119882



119909|119905=119886

= 1199090=21198820

119862

119891 (1199090) = radic

21198820

119862

1198911015840

(1199090) =

1

2radic21198820119862

(A4)


0as

119891 (119909) asymp 119891 (1199090) + 1198911015840

(1199090) (119909 minus 119909

0) (A5)



119891 (119909) = radic21198820

119862+(2119862)119882

119862(119905) minus 2119882

0119862

2radic21198820119862

= radic21198820

119862+119862(119905)

radic21198621198820

(A6)


expressed as

119881119862= radic

21198820

119862

V119862(119905) =

119862(119905)

radic21198621198820

=119862(119905)

119862119881119862

=

int119905

0

(119901119862(120591))ac 119889120591

119862119881119862

(A7)



Nomenclature





119894lowast












119862


119862


Competing Interests


References








































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


020406080

100120140160


Am

plitu

de o

fH

()



+

minusilowastHPIPV

lowastPV



VH = 119889 sdot VPV (12)



119909



119863119909119895V119909119895PV

sum119899

119895=1(119863119909119895V119909119895PV)

(13)





lowast




+

minusDPIPV

lowastPV


a1PV

a2PV

cnPV

+

minus

+

minus

+

minus

sum PI ilowastd

a1PV

cnlowastPV

a2lowastPV







200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

2526272829303132333435

(V)

Time (ms)0 10 20


05

10152025

ia ib

(A)

Time (ms)0 10 20


050

100150200250

(V)

ab bc

Time (ms)0 10 20

minus150

minus100

minus50

0

50

100

150lowastb

lowasta

(V)

Time (ms)0 10 20

dcc1










3= 04) The



200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

25

30

35

(V)

Time (ms)0 10 20


2575

125175225

bcab

(V)

Time (ms)0 10 20


01020304050

(V)

Time (ms)0 10 20


minus125

minus75

minus25

25

75

125lowastb

lowasta

(V)

Time (ms)0 10 20


minus125

minus75

minus25

25

75

125

Time (ms)0 10 20

(V)

ba


05

10152025

Time (ms)0 10 20

(A)

ia ibdcc1








ofchange












6 Conclusion



Appendix




119882119862(119905) V119862(119905) 119901

119862(119905) and119882


119882119862(119905) = int119901

119862(119905) 119889119905 =

1

2119862 (V119862(119905))2

= 1198820+ 119862(119905) (A1)




119862(119905) can be

expressed as

V119862(119905) = radic

2

119862119882119862(119905) = 119881

119862+ V119862(119905) (A2)




119862(119905)

Define V119862(119905) = 119891(119909) and 119909 = (2119862)119882

119862(119905) Hence from


119891 (119909) = radic119909

1198911015840

(119909) =1

2radic119909

(A3)


119862(119886) = 119882



119909|119905=119886

= 1199090=21198820

119862

119891 (1199090) = radic

21198820

119862

1198911015840

(1199090) =

1

2radic21198820119862

(A4)


0as

119891 (119909) asymp 119891 (1199090) + 1198911015840

(1199090) (119909 minus 119909

0) (A5)



119891 (119909) = radic21198820

119862+(2119862)119882

119862(119905) minus 2119882

0119862

2radic21198820119862

= radic21198820

119862+119862(119905)

radic21198621198820

(A6)


expressed as

119881119862= radic

21198820

119862

V119862(119905) =

119862(119905)

radic21198621198820

=119862(119905)

119862119881119862

=

int119905

0

(119901119862(120591))ac 119889120591

119862119881119862

(A7)



Nomenclature





119894lowast












119862


119862


Competing Interests


References








































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

2526272829303132333435

(V)

Time (ms)0 10 20


05

10152025

ia ib

(A)

Time (ms)0 10 20


050

100150200250

(V)

ab bc

Time (ms)0 10 20

minus150

minus100

minus50

0

50

100

150lowastb

lowasta

(V)

Time (ms)0 10 20

dcc1










3= 04) The



200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

25

30

35

(V)

Time (ms)0 10 20


2575

125175225

bcab

(V)

Time (ms)0 10 20


01020304050

(V)

Time (ms)0 10 20


minus125

minus75

minus25

25

75

125lowastb

lowasta

(V)

Time (ms)0 10 20


minus125

minus75

minus25

25

75

125

Time (ms)0 10 20

(V)

ba


05

10152025

Time (ms)0 10 20

(A)

ia ibdcc1








ofchange












6 Conclusion



Appendix




119882119862(119905) V119862(119905) 119901

119862(119905) and119882


119882119862(119905) = int119901

119862(119905) 119889119905 =

1

2119862 (V119862(119905))2

= 1198820+ 119862(119905) (A1)




119862(119905) can be

expressed as

V119862(119905) = radic

2

119862119882119862(119905) = 119881

119862+ V119862(119905) (A2)




119862(119905)

Define V119862(119905) = 119891(119909) and 119909 = (2119862)119882

119862(119905) Hence from


119891 (119909) = radic119909

1198911015840

(119909) =1

2radic119909

(A3)


119862(119886) = 119882



119909|119905=119886

= 1199090=21198820

119862

119891 (1199090) = radic

21198820

119862

1198911015840

(1199090) =

1

2radic21198820119862

(A4)


0as

119891 (119909) asymp 119891 (1199090) + 1198911015840

(1199090) (119909 minus 119909

0) (A5)



119891 (119909) = radic21198820

119862+(2119862)119882

119862(119905) minus 2119882

0119862

2radic21198820119862

= radic21198820

119862+119862(119905)

radic21198621198820

(A6)


expressed as

119881119862= radic

21198820

119862

V119862(119905) =

119862(119905)

radic21198621198820

=119862(119905)

119862119881119862

=

int119905

0

(119901119862(120591))ac 119889120591

119862119881119862

(A7)



Nomenclature





119894lowast












119862


119862


Competing Interests


References








































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


200205210215220225230235240245250

PPV1

(W)

Time (ms)0 10 20

25

30

35

(V)

Time (ms)0 10 20


2575

125175225

bcab

(V)

Time (ms)0 10 20


01020304050

(V)

Time (ms)0 10 20


minus125

minus75

minus25

25

75

125lowastb

lowasta

(V)

Time (ms)0 10 20


minus125

minus75

minus25

25

75

125

Time (ms)0 10 20

(V)

ba


05

10152025

Time (ms)0 10 20

(A)

ia ibdcc1








ofchange












6 Conclusion



Appendix




119882119862(119905) V119862(119905) 119901

119862(119905) and119882


119882119862(119905) = int119901

119862(119905) 119889119905 =

1

2119862 (V119862(119905))2

= 1198820+ 119862(119905) (A1)




119862(119905) can be

expressed as

V119862(119905) = radic

2

119862119882119862(119905) = 119881

119862+ V119862(119905) (A2)




119862(119905)

Define V119862(119905) = 119891(119909) and 119909 = (2119862)119882

119862(119905) Hence from


119891 (119909) = radic119909

1198911015840

(119909) =1

2radic119909

(A3)


119862(119886) = 119882



119909|119905=119886

= 1199090=21198820

119862

119891 (1199090) = radic

21198820

119862

1198911015840

(1199090) =

1

2radic21198820119862

(A4)


0as

119891 (119909) asymp 119891 (1199090) + 1198911015840

(1199090) (119909 minus 119909

0) (A5)



119891 (119909) = radic21198820

119862+(2119862)119882

119862(119905) minus 2119882

0119862

2radic21198820119862

= radic21198820

119862+119862(119905)

radic21198621198820

(A6)


expressed as

119881119862= radic

21198820

119862

V119862(119905) =

119862(119905)

radic21198621198820

=119862(119905)

119862119881119862

=

int119905

0

(119901119862(120591))ac 119889120591

119862119881119862

(A7)



Nomenclature





119894lowast












119862


119862


Competing Interests


References








































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





ofchange












6 Conclusion



Appendix




119882119862(119905) V119862(119905) 119901

119862(119905) and119882


119882119862(119905) = int119901

119862(119905) 119889119905 =

1

2119862 (V119862(119905))2

= 1198820+ 119862(119905) (A1)




119862(119905) can be

expressed as

V119862(119905) = radic

2

119862119882119862(119905) = 119881

119862+ V119862(119905) (A2)




119862(119905)

Define V119862(119905) = 119891(119909) and 119909 = (2119862)119882

119862(119905) Hence from


119891 (119909) = radic119909

1198911015840

(119909) =1

2radic119909

(A3)


119862(119886) = 119882



119909|119905=119886

= 1199090=21198820

119862

119891 (1199090) = radic

21198820

119862

1198911015840

(1199090) =

1

2radic21198820119862

(A4)


0as

119891 (119909) asymp 119891 (1199090) + 1198911015840

(1199090) (119909 minus 119909

0) (A5)



119891 (119909) = radic21198820

119862+(2119862)119882

119862(119905) minus 2119882

0119862

2radic21198820119862

= radic21198820

119862+119862(119905)

radic21198621198820

(A6)


expressed as

119881119862= radic

21198820

119862

V119862(119905) =

119862(119905)

radic21198621198820

=119862(119905)

119862119881119862

=

int119905

0

(119901119862(120591))ac 119889120591

119862119881119862

(A7)



Nomenclature





119894lowast












119862


119862


Competing Interests


References








































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



119891 (119909) = radic21198820

119862+(2119862)119882

119862(119905) minus 2119882

0119862

2radic21198820119862

= radic21198820

119862+119862(119905)

radic21198621198820

(A6)


expressed as

119881119862= radic

21198820

119862

V119862(119905) =

119862(119905)

radic21198621198820

=119862(119905)

119862119881119862

=

int119905

0

(119901119862(120591))ac 119889120591

119862119881119862

(A7)



Nomenclature





119894lowast












119862


119862


Competing Interests


References








































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

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