maximumobtainableenergyharvestingpowerfrom galloping...

Research ArticleMaximum Obtainable Energy Harvesting Power fromGalloping-Based Piezoelectrics

Mohammad Yaghoub Abdollahzadeh Jamalabadi 12 Mostafa Safdari Shadloo3

and Arash Karimipour 4

1Department for Management of Science and Technology Development Ton Duc ang UniversityHo Chi Minh City 700000 Vietnam2Faculty of Civil Engineering Ton Duc ang University Ho Chi Minh City 700000 Vietnam3CORIA-UMR 6614 Normandie University CNRS-University amp INSA 76000 Rouen France4Dipartimento di Ingegneria Astronautica Elettrica ed Energetica Sapienza Universita di Roma Via Eudossiana 18Roma 00184 Italy

Correspondence should be addressed to Arash Karimipour arashkarimipourgmailcom

Received 4 March 2020 Accepted 11 May 2020 Published 7 September 2020

Academic Editor Francesco Franco

Copyright copy 2020 Mohammad Yaghoub Abdollahzadeh Jamalabadi et al (is is an open access article distributed under theCreative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium providedthe original work is properly cited

In this paper the maximum obtainable energy from a galloping cantilever beam is found (e system consists of a bluff body infront of wind which was mounted on a cantilever beam and supported by piezoelectric sheets Wind energy caused the transversevibration of the beam and the mechanical energy of vibration is transferred to electrical charge by use of piezoelectric transducer(e nonlinear motion of the EulerndashBernoulli beam and conservation of electrical energy is modeled by lumped ordinarydifferential equations (e wind forces on the bluff body are modeled by quasisteady aeroelasticity approximation where the fluidand solid corresponding dynamics are disconnected in time scales (e linearized motion of beam is limited by its yield stresswhich causes to find a limit on energy harvesting of the system(e theory founded is used to check the validity of previous resultsof researchers for the effect of wind speed tip cross-section geometry and electrical load resistance on onset speed to galloping tipdisplacement and harvested power Finally maximum obtainable average power in a standard RC circuit as a function ofdeflection limit and synchronized charge extraction is obtained

1 Introduction

Piezoelectric energy harvester uses the ambient energy andtransfers it into electric charge [1ndash7](e parametric study andthe design of piezoelectric energy harvesting from gallopingmotion is studied by Barrero-Gil et al [1] First experimentalresults are obtained by Sirohi and Mahadik [2 3] using acantilever beam exposed to air with constant velocity in a windtunnel Simulation of galloping cantilever coupled with a pi-ezoelectric transducer in an electric circuit is performed byAbdelkefi et al [4] Analytical solution of that system ofequations is presented in the work of Tan and Yan [5] As thevalues for the harvested power of Abdelkefi et al [4] and Tanand Yan [5] in some figures (watts) are beyond the order of themagnitude of experimental data of Sirohi and Mahadik [2 3]

and Jamalabadi et al [6 7] (milliwatts) this paper addressed theproblem to the linear assumption of force-deflection relationfor the EulerndashBernoulli beam (is research proposes toconsider the limitation of the yield stress of piezoelectricmaterial as the maximum point of mechanical stability as wellas energy harvesting

2 Mathematical Model

(e schematic of the system is shown in Figure 1 A bluff bodyexposed to the free stream is mounted on an EulerndashBernoullicantilever beam (e two piezoelectric wafers are attached onfree surfaces of beamwhich are in an electric circuit with electricimpedance (e y-direction galloping of the bluff body in thefirst mode of the structure is modeled by Abdelkefi et al [4] by

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 6140853 8 pageshttpsdoiorg10115520206140853

euroy (t) +(2ξω minus A) _y(t) minus Bϕ(L)2

_y(t)3

+ ω2y(t) + ϕ(L)θpV(t) 0

(1)

_V(t) minusV(t)

RCp

+θp

ϕ(L)Cp

_y(t) (2)

By assuming the following functions for motion andvoltage

y(t) ϕ(L)ymax sin(Ωt)

V(t) RC

radicΩϕ(L)ymax sin(Ωt + α)

(3)

(e integration of equations (1) and (2) term for halfperiod of motion the onset of galloping the maximumdeflection of the beam and power harvesting which are

Umin 2(2ξω + C)

ρairbtipa1 ϕ2(L)Ltip + ϕ(L)ϕprime(L)L2tip + 13ϕprime2(L)L

3tip1113874 1113875

(4)

Amax ϕ(L)

Ω

(C + 2ξω minus A)

075B

1113971

(5)

Pmax C(2ξω + C minus A)

075B (6)

Before going further to derive the optimal values of thesystem parameters same as of Tan and Yan [5] it should be

noticed that the maximum bending moment at the base ofthe beam is calculated by

Mbase( _y) (23)wPZT tb2( 1113857 + tPZT( 1113857

3minus t

3PZT1113960 1113961 +(112)wbt

3b

tb2( 1113857 + tPZT( 1113857σY (7)

where the yielding stress of piezoelectric material is about312MPa in the experiment

3 Results

(e numerical (solving equations (1) and (2)) and analytical(equations (5) and (6)) solutions are calculated based on thedata provided [4] (e change in the amplitudes of the dis-placement of the tip of the cantilever beamwith the free streamvelocities at different electrical impedances from the numericaland analytical solutions are shown in Figure 2 As shown in

Figure 2 the analytical and numerical solutions are in a goodagreement In addition the limitation considered in equation(7) affected the results where soon after the onset of gallopingthe system experienced the tear in the piezoelectric sheet Tobetter assess the consequences of the constraint of equation(7) the vertical axis of Figure 2 is plotted in a log scale Asshown the maximum displacement that the beam can bear isless than the order of centimeter and for such stiffness theorder of the deflection before failure is millimeter

(e variation in the harvested power with the electricalimpedances at different free stream velocities from the

x

y

Piezoelectric sheets

R

U

Figure 1 Schematic of the galloping piezoelectric energy harvester for the case of wind direction normal to the cantilever beam

2 Mathematical Problems in Engineering

analytical and numerical solutions is revealed in Figure 3As shown in Figure 3 the results of numerical and ana-lytical methods are in a good agreement Additionally thelimitation considered in equation (7) affected the resultswhere soon after the onset of galloping the system ex-perienced the tear in the piezoelectric wafers and theharvesting of the wind energy is stopped To see the sig-nificances of the constraint of equation (7) on the systemclearer the vertical axis in Figure 3 is schemed in a log

scale As exposed the maximum harvested power in theelectric circuit is less than the order of 10minus 1 watts and forother cases the order of the harvested power before failureis milliwatts (e results are in a good agreement with theexperimental results [2 3 6 7]

By differentiating equation (6) with respect to the pa-rameter C the optimal design of the electric circuit for thegalloping system is obtained as (zPmaxzC 0⟹Co

(A minus 2ξω2))

Ro θ2p + Cp(A minus 2ξω2)

2 plusmn

θ2p + Cp(A minus 2ξω2)2

1113872 11138732

minus 4Cp(A minus 2ξω2)2

Cpω2

+ θ2p1113872 1113873

1113970

Cp(A minus 2ξω) Cpω2

+ θ2p1113872 1113873

Uonseto

4ξωρairbtipa1

1

ϕ2(L)Ltip + ϕ(L)ϕprime(L)L2tip + 13ϕprime2(L)L

3tip

Amaxo ϕ(L)

2ξω minus A

15BΩ2

1113971

Pmaxo minus(2ξω minus A)

2

3B

Ω ω

1

1 minus 05RcoCp(A minus 2ξω)

1113971

(8)

Analytical solutions numerical solutions and correctedsolutions of the amplitudes of the harvested power versus theparameter C and free stream velocity are plotted in Figure 4As shown again the limited range of parameter C is allowed

and the maximum obtainable power should be searchedthrough those values

When the value of expression under the square root ineqution (8) for Ro is negative (the velocities higher than

0 5 10 15U (ms)

δ (m

)

10ndash1

10ndash2

10ndash3

R = 105

R = 106R = 103

R = 104

Figure 2 Analytical solutions (dash lines) numerical solutions (symbols) and corrected solutions (lines) of the amplitudes of the tipdisplacement versus the electrical impedance and free stream velocity

Mathematical Problems in Engineering 3

Table 1 Regular circuits

Name Output Schematic

Standard RC

Vmax (θ(1Rω)sinφ + Cp cosφ)umax

R

Vp (t)Cp

umax (Uω)

(minus 4ρbLUa1 + 8(C + Rθ2 sinφ2)3ρbLUa3)

1113969

Pave (V2max2R)

Synchronized chargeextraction

Vmax (2θCp)umax

S

D

Li CfR

umax (Uω)((minus 4ρbLUa1 + 8C + 32θ2Cp)3ρbLUa3)

1113969

Pave (CpωV2max2π)

C (1s)

101

100

10ndash2

10ndash1

10ndash3

Har

veste

d po

wer

(W)

U = 10msU = 14ms

U = 17msU = 20ms

0 05 1 2 25 3 3515 4

Figure 4 Analytical solutions (dash lines) numerical solutions (symbol) and corrected solutions (lines) of the amplitudes of the harvestedpower versus the parameter C and free stream velocity

0 5 10 15U (ms)

101

100

10ndash2

10ndash1

10ndash3

10ndash4H

arve

sted

pow

er (W

)

R = 103

R = 104R = 105

R = 106

Figure 3 Analytical solutions (dash lines) numerical solutions (symbols) and corrected solutions (lines) of the amplitudes of the harvestedpower versus the electrical impedance and free stream velocity


Tabl

e2

New

circuitinterfaces

Nam

eOutpu

tSchematic

Parallel-

synchron

ized

Vc

(2Θ

Cp(1

minusc)

+π

Rω)

um

S L i

C fR

um

(12)

Cbωπ

+(4πΘ

2 ((1

CpRω)

+(1

minusc2 2π

))C

p[(π

CpRω)

+(1

minusc)]2 )

minus(14)ρh

LUa1ω

π(316

)ρhL

Ua3ω

π((ω2

U2 )

+β2

)1113969

Pave

(V

2 maxR

)

Series-

synchron

ized

Vc

(2Θ

(1

+c)Rωπ(

1minus

c)

+2(1

+c)C

pRω)

um

SL i

C fR

um

((12)

Cbωπ

+(2πΘ

2 (1

+c)

Cp[π

(1

minusc)

+2C

pRω(

1+

c)]

)minus

(14)ρh

LUa1ω

π(316

)ρhL

Ua3ω

π((ω2

U2 )

+β2

))1113969

Pave

(4Θ

2 (1

+c)2

Rω2

[π(

1minus

c)

+2(1

+c)C

pRω]

2 )u2 m


161ms) the optimal values of the C((zCzR) 0⟹Co (θ2p2CpΩ)) is found by

Ro CpΩ1113872 1113873minus 1

Uonseto

4 ξω + θ2pCpΩρairbtipa1

1

ϕ2(L)Ltip + ϕ(L)ϕprime(L)L

2tip + 13ϕprime2(L)L

3tip

Amaxo

θ2p 2CpΩ1113872 11138731113872 1113873 + 2 ξω minus A

075BΩ2

1113971

ϕ(L)

Pmaxo θ2p 2CpΩ1113872 11138731113872 1113873 + 2 ξω minus A

15BCpΩ

Ω

ω2+

θ2p2Cp

11139741113972

(9)

x

y


CjR

Figure 5 Schematic of the galloping piezoelectric energy harvester for the case of wind direction parallel to the cantilever beam

S-SSHI EXP

P-SSHI EXPP-SSHI NUM

S-SSHI NUMStandard circuit EXPStandard cicuit NUMSCE EXPSCE NUM

25

20

15

10

5

0

Pow

er (m

W)

2 4 6 8 10 12 14 16Wind speed (ms)

Figure 6 Maximum obtainable various galloping piezoelectric energy harvester for the case of wind direction parallel to the cantileverbeam


As for the current parameters this value of the C pa-rameter is less than the allowed values of C parameter forharvested power the discussion is not necessary

Analytical solutions for galloping-based piezoelectricenergy harvesters with various interfacing circuits aresummarized in Table 1 By assuming V Vm cos(ωt + φ)

and u um cos(ωt) where tanφ (1RCpω) the maximumobtainable average power in a standard RC circuit as afunction of deflection limit is

Pmax R

2ωθ sinφδmax( 1113857

2 (10)

and for the synchronized charge extraction

Pmax 2ωπCp

θδmax( 11138572 (11)

For the case of wind direction parallel to the cantileverbeam (see Figure 5) the governing equations are

Meurou (t) + Cb _u(t) + Ku(t) + ΘV(t) 12ρhLU2

a1_u(t)

U+ βu(t)1113888 1113889 + a3

_u(t)

U+ βu(t)1113888 1113889

3⎡⎣ ⎤⎦

I + Cp_V minus Θ _u 0

(12)

Various circuit interfaces are shown in Table 2 Maxi-mum obtainable various galloping piezoelectric energyharvester for the case of wind direction parallel to thecantilever beam is plotted in Figure 6 (e beam data isobtained from the energy harvester 2 in Table 3 of Zhao andYang [8]

4 Conclusion

In this study the nonlinear model of the galloping cantileverbeam used for piezoelectric energy harvesting is simulatednumerically with respect to the failure criteria as a limit ofthe maximum obtainable power (e ideal case of such

Table 3 Nomenclature

Symbol DescriptionA (ρairUbtipa12)(ϕ2(L)Ltip + ϕ(L)ϕprime(L)L2

tip + 13ϕprime

2(L)L3

tip)

B (ρairbtipa32U)(ϕ(L) 1113938Ltip

0 ϕ(L) + s(ϕ(L))3ds + ϕ(L) 1113938Ltip

0 s(ϕ(L)) 1113938Ltip

0 sϕprime(L)3 + ds)

C (Rθ2p(1 + C2pΩ

2R2))

Cb 2ζωnϕ2 (Lt)M 1ϕ 2 (Lt)K ωn

2ϕ2 (Lt)Θ χϕ (Lt)ρair Air densityL Length of the beamFtip (ρairU2btip2) 1113938

Ltip

0 a1( _yL + s _yprimeLU) + a3( _yL + s _yprimeLU)3ds

Mtip (ρairU2btip2) 1113938Ltip

0 a1( _yL + s _yprimeLU) + a3( _yL + s _yprimeLU)3sds

Mbase FtipL + Mtipbtip Width of the tip bodyLtip Length of the tip bodya1 a3 Aerodynamic force coefficientsR Load resistanceQ Quality factorξ Damping ratio of the structureϕ Mode shape of the structureθp Electromechanical coefficient of piezoelectric materialσY Yield strength of piezoelectric materialCp Capacity of piezoelectric layerU Wind velocityV Piezoelectric voltageβ (ϕprime(Lt)ϕ(Lt))

Ω Angular velocity of the motionχ First natural angular velocity of the cantilever beamω eminus π2Q

c Cantilever-beam displacement


system is compared with the case of maximum stress limiteddue to the yielding stress of piezoelectric material (e re-sults show that the mechanical limits of the system do notallow us to obtain the anticipated values in theory and thefeasible values are 2-3 orders of magnitude lower thanprediction values Hence the fracture limitation should beconsidered in the process of the design of galloping-basedenergy harvesters with piezoelectric materials Furthermorethe current research proposes for engineering applicationsand designing the control system for the amplitude ofgalloping is necessary as well Finally maximum obtainableaverage power in a standard RC circuit as a function ofdeflection limit and synchronized charge extraction is ob-tained In addition four electrical interfaces in galloping-based energy harvesters are assessed (e results are for afeeble coupling SCE circuit which is reasonable at higherwind while SSHI suits low wind speed (e standard circuitis suggested for strong electromechanical pairing and theSCE has the best strength against the wind and can producethe highest value of power

Data Availability

No data were used to support this study

Conflicts of Interest

(e authors declare that they have no conflicts of interest

References

[1] A Barrero-Gil G Alonso and A Sanz-Andres ldquoEnergyharvesting from transverse gallopingrdquo Journal of Sound andVibration vol 329 no 14 pp 2873ndash2883 2010

[2] J Sirohi and R Mahadik ldquoPiezoelectric wind energy harvesterfor low-power sensorsrdquo Journal of Intelligent Material Systemsand Structures vol 22 no 18 pp 2215ndash2228 2011

[3] J Sirohi and R Mahadik Journal of Vibration and Acousticsvol 134 Article ID 011009 2012

[4] A Abdelkefi Z Yan and M Hajj Smart Materials andStructures vol 22 2013

[5] T Tan and Z Yan ldquoAnalytical solution and optimal design forgalloping-based piezoelectric energy harvestersrdquo AppliedPhysics Letters vol 109 no 25 p 253902 2016

[6] M Y A Jamalabadi K M Kwak and S J Hwan KSNVEvol 10 p 54 2016

[7] M Y A Jamalabadi K M Kwak and S J Hwan KSNVE ASKand KSME vol 4 p 478 2017

[8] L Zhao and Y Yang ldquoComparison of four electrical interfacingcircuits in wind energy harvestingrdquo Sensors and Actuators APhysical vol 261 pp 117ndash129 2017


euroy (t) +(2ξω minus A) _y(t) minus Bϕ(L)2

_y(t)3

+ ω2y(t) + ϕ(L)θpV(t) 0

(1)

_V(t) minusV(t)

RCp

+θp

ϕ(L)Cp

_y(t) (2)

By assuming the following functions for motion andvoltage

y(t) ϕ(L)ymax sin(Ωt)

V(t) RC

radicΩϕ(L)ymax sin(Ωt + α)

(3)

(e integration of equations (1) and (2) term for halfperiod of motion the onset of galloping the maximumdeflection of the beam and power harvesting which are

Umin 2(2ξω + C)

ρairbtipa1 ϕ2(L)Ltip + ϕ(L)ϕprime(L)L2tip + 13ϕprime2(L)L

3tip1113874 1113875

(4)

Amax ϕ(L)

Ω

(C + 2ξω minus A)

075B

1113971

(5)

Pmax C(2ξω + C minus A)

075B (6)

Before going further to derive the optimal values of thesystem parameters same as of Tan and Yan [5] it should be

noticed that the maximum bending moment at the base ofthe beam is calculated by

Mbase( _y) (23)wPZT tb2( 1113857 + tPZT( 1113857

3minus t

3PZT1113960 1113961 +(112)wbt

3b

tb2( 1113857 + tPZT( 1113857σY (7)

where the yielding stress of piezoelectric material is about312MPa in the experiment

3 Results

(e numerical (solving equations (1) and (2)) and analytical(equations (5) and (6)) solutions are calculated based on thedata provided [4] (e change in the amplitudes of the dis-placement of the tip of the cantilever beamwith the free streamvelocities at different electrical impedances from the numericaland analytical solutions are shown in Figure 2 As shown in

Figure 2 the analytical and numerical solutions are in a goodagreement In addition the limitation considered in equation(7) affected the results where soon after the onset of gallopingthe system experienced the tear in the piezoelectric sheet Tobetter assess the consequences of the constraint of equation(7) the vertical axis of Figure 2 is plotted in a log scale Asshown the maximum displacement that the beam can bear isless than the order of centimeter and for such stiffness theorder of the deflection before failure is millimeter

(e variation in the harvested power with the electricalimpedances at different free stream velocities from the

x

y


R

U

Figure 1 Schematic of the galloping piezoelectric energy harvester for the case of wind direction normal to the cantilever beam





(A minus 2ξω2))


2 plusmn


1113872 11138732


Cpω2

+ θ2p1113872 1113873

1113970


+ θ2p1113872 1113873

Uonseto

4ξωρairbtipa1

1


3tip

Amaxo ϕ(L)

2ξω minus A

15BΩ2

1113971


2

3B

Ω ω

1


1113971

(8)




0 5 10 15U (ms)

δ (m

)

10ndash1

10ndash2

10ndash3

R = 105

R = 106R = 103

R = 104





Standard RC


R

Vp (t)Cp

umax (Uω)


1113969

Pave (V2max2R)


Vmax (2θCp)umax

S

D

Li CfR


1113969

Pave (CpωV2max2π)

C (1s)

101

100

10ndash2

10ndash1

10ndash3

Har

veste

d po

wer

(W)

U = 10msU = 14ms

U = 17msU = 20ms

0 05 1 2 25 3 3515 4


0 5 10 15U (ms)

101

100

10ndash2

10ndash1

10ndash3

10ndash4H

arve

sted

pow

er (W

)

R = 103

R = 104R = 105

R = 106



Tabl

e2

New

circuitinterfaces

Nam

eOutpu

tSchematic

Parallel-

synchron

ized

Vc

(2Θ

Cp(1

minusc)

+π

Rω)

um

S L i

C fR

um

(12)

Cbωπ

+(4πΘ

2 ((1

CpRω)

+(1

minusc2 2π

))C

p[(π

CpRω)

+(1

minusc)]2 )

minus(14)ρh

LUa1ω

π(316

)ρhL

Ua3ω

π((ω2

U2 )

+β2

)1113969

Pave

(V

2 maxR

)

Series-

synchron

ized

Vc

(2Θ

(1

+c)Rωπ(

1minus

c)

+2(1

+c)C

pRω)

um

SL i

C fR

um

((12)

Cbωπ

+(2πΘ

2 (1

+c)

Cp[π

(1

minusc)

+2C

pRω(

1+

c)]

)minus

(14)ρh

LUa1ω

π(316

)ρhL

Ua3ω

π((ω2

U2 )

+β2

))1113969

Pave

(4Θ

2 (1

+c)2

Rω2

[π(

1minus

c)

+2(1

+c)C

pRω]

2 )u2 m



Ro CpΩ1113872 1113873minus 1

Uonseto


1



3tip

Amaxo

θ2p 2CpΩ1113872 11138731113872 1113873 + 2 ξω minus A

075BΩ2

1113971

ϕ(L)


15BCpΩ

Ω

ω2+

θ2p2Cp

11139741113972

(9)

x

y


CjR


S-SSHI EXP



25

20

15

10

5

0

Pow

er (m

W)

2 4 6 8 10 12 14 16Wind speed (ms)






Pmax R


2 (10)


Pmax 2ωπCp

θδmax( 11138572 (11)



a1_u(t)

U+ βu(t)1113888 1113889 + a3

_u(t)

U+ βu(t)1113888 1113889

3⎡⎣ ⎤⎦


(12)


4 Conclusion




tip + 13ϕprime

2(L)L3

tip)


0 ϕ(L) + s(ϕ(L))3ds + ϕ(L) 1113938Ltip

0 s(ϕ(L)) 1113938Ltip


C (Rθ2p(1 + C2pΩ

2R2))



Ltip









Data Availability




References













(A minus 2ξω2))


2 plusmn


1113872 11138732


Cpω2

+ θ2p1113872 1113873

1113970


+ θ2p1113872 1113873

Uonseto

4ξωρairbtipa1

1


3tip

Amaxo ϕ(L)

2ξω minus A

15BΩ2

1113971


2

3B

Ω ω

1


1113971

(8)




0 5 10 15U (ms)

δ (m

)

10ndash1

10ndash2

10ndash3

R = 105

R = 106R = 103

R = 104





Standard RC


R

Vp (t)Cp

umax (Uω)


1113969

Pave (V2max2R)


Vmax (2θCp)umax

S

D

Li CfR


1113969

Pave (CpωV2max2π)

C (1s)

101

100

10ndash2

10ndash1

10ndash3

Har

veste

d po

wer

(W)

U = 10msU = 14ms

U = 17msU = 20ms

0 05 1 2 25 3 3515 4


0 5 10 15U (ms)

101

100

10ndash2

10ndash1

10ndash3

10ndash4H

arve

sted

pow

er (W

)

R = 103

R = 104R = 105

R = 106



Tabl

e2

New

circuitinterfaces

Nam

eOutpu

tSchematic

Parallel-

synchron

ized

Vc

(2Θ

Cp(1

minusc)

+π

Rω)

um

S L i

C fR

um

(12)

Cbωπ

+(4πΘ

2 ((1

CpRω)

+(1

minusc2 2π

))C

p[(π

CpRω)

+(1

minusc)]2 )

minus(14)ρh

LUa1ω

π(316

)ρhL

Ua3ω

π((ω2

U2 )

+β2

)1113969

Pave

(V

2 maxR

)

Series-

synchron

ized

Vc

(2Θ

(1

+c)Rωπ(

1minus

c)

+2(1

+c)C

pRω)

um

SL i

C fR

um

((12)

Cbωπ

+(2πΘ

2 (1

+c)

Cp[π

(1

minusc)

+2C

pRω(

1+

c)]

)minus

(14)ρh

LUa1ω

π(316

)ρhL

Ua3ω

π((ω2

U2 )

+β2

))1113969

Pave

(4Θ

2 (1

+c)2

Rω2

[π(

1minus

c)

+2(1

+c)C

pRω]

2 )u2 m



Ro CpΩ1113872 1113873minus 1

Uonseto


1



3tip

Amaxo

θ2p 2CpΩ1113872 11138731113872 1113873 + 2 ξω minus A

075BΩ2

1113971

ϕ(L)


15BCpΩ

Ω

ω2+

θ2p2Cp

11139741113972

(9)

x

y


CjR


S-SSHI EXP



25

20

15

10

5

0

Pow

er (m

W)

2 4 6 8 10 12 14 16Wind speed (ms)






Pmax R


2 (10)


Pmax 2ωπCp

θδmax( 11138572 (11)



a1_u(t)

U+ βu(t)1113888 1113889 + a3

_u(t)

U+ βu(t)1113888 1113889

3⎡⎣ ⎤⎦


(12)


4 Conclusion




tip + 13ϕprime

2(L)L3

tip)


0 ϕ(L) + s(ϕ(L))3ds + ϕ(L) 1113938Ltip

0 s(ϕ(L)) 1113938Ltip


C (Rθ2p(1 + C2pΩ

2R2))



Ltip









Data Availability




References












Standard RC


R

Vp (t)Cp

umax (Uω)


1113969

Pave (V2max2R)


Vmax (2θCp)umax

S

D

Li CfR


1113969

Pave (CpωV2max2π)

C (1s)

101

100

10ndash2

10ndash1

10ndash3

Har

veste

d po

wer

(W)

U = 10msU = 14ms

U = 17msU = 20ms

0 05 1 2 25 3 3515 4


0 5 10 15U (ms)

101

100

10ndash2

10ndash1

10ndash3

10ndash4H

arve

sted

pow

er (W

)

R = 103

R = 104R = 105

R = 106



Tabl

e2

New

circuitinterfaces

Nam

eOutpu

tSchematic

Parallel-

synchron

ized

Vc

(2Θ

Cp(1

minusc)

+π

Rω)

um

S L i

C fR

um

(12)

Cbωπ

+(4πΘ

2 ((1

CpRω)

+(1

minusc2 2π

))C

p[(π

CpRω)

+(1

minusc)]2 )

minus(14)ρh

LUa1ω

π(316

)ρhL

Ua3ω

π((ω2

U2 )

+β2

)1113969

Pave

(V

2 maxR

)

Series-

synchron

ized

Vc

(2Θ

(1

+c)Rωπ(

1minus

c)

+2(1

+c)C

pRω)

um

SL i

C fR

um

((12)

Cbωπ

+(2πΘ

2 (1

+c)

Cp[π

(1

minusc)

+2C

pRω(

1+

c)]

)minus

(14)ρh

LUa1ω

π(316

)ρhL

Ua3ω

π((ω2

U2 )

+β2

))1113969

Pave

(4Θ

2 (1

+c)2

Rω2

[π(

1minus

c)

+2(1

+c)C

pRω]

2 )u2 m



Ro CpΩ1113872 1113873minus 1

Uonseto


1



3tip

Amaxo

θ2p 2CpΩ1113872 11138731113872 1113873 + 2 ξω minus A

075BΩ2

1113971

ϕ(L)


15BCpΩ

Ω

ω2+

θ2p2Cp

11139741113972

(9)

x

y


CjR


S-SSHI EXP



25

20

15

10

5

0

Pow

er (m

W)

2 4 6 8 10 12 14 16Wind speed (ms)






Pmax R


2 (10)


Pmax 2ωπCp

θδmax( 11138572 (11)



a1_u(t)

U+ βu(t)1113888 1113889 + a3

_u(t)

U+ βu(t)1113888 1113889

3⎡⎣ ⎤⎦


(12)


4 Conclusion




tip + 13ϕprime

2(L)L3

tip)


0 ϕ(L) + s(ϕ(L))3ds + ϕ(L) 1113938Ltip

0 s(ϕ(L)) 1113938Ltip


C (Rθ2p(1 + C2pΩ

2R2))



Ltip









Data Availability




References










Tabl

e2

New

circuitinterfaces

Nam

eOutpu

tSchematic

Parallel-

synchron

ized

Vc

(2Θ

Cp(1

minusc)

+π

Rω)

um

S L i

C fR

um

(12)

Cbωπ

+(4πΘ

2 ((1

CpRω)

+(1

minusc2 2π

))C

p[(π

CpRω)

+(1

minusc)]2 )

minus(14)ρh

LUa1ω

π(316

)ρhL

Ua3ω

π((ω2

U2 )

+β2

)1113969

Pave

(V

2 maxR

)

Series-

synchron

ized

Vc

(2Θ

(1

+c)Rωπ(

1minus

c)

+2(1

+c)C

pRω)

um

SL i

C fR

um

((12)

Cbωπ

+(2πΘ

2 (1

+c)

Cp[π

(1

minusc)

+2C

pRω(

1+

c)]

)minus

(14)ρh

LUa1ω

π(316

)ρhL

Ua3ω

π((ω2

U2 )

+β2

))1113969

Pave

(4Θ

2 (1

+c)2

Rω2

[π(

1minus

c)

+2(1

+c)C

pRω]

2 )u2 m



Ro CpΩ1113872 1113873minus 1

Uonseto


1



3tip

Amaxo

θ2p 2CpΩ1113872 11138731113872 1113873 + 2 ξω minus A

075BΩ2

1113971

ϕ(L)


15BCpΩ

Ω

ω2+

θ2p2Cp

11139741113972

(9)

x

y


CjR


S-SSHI EXP



25

20

15

10

5

0

Pow

er (m

W)

2 4 6 8 10 12 14 16Wind speed (ms)






Pmax R


2 (10)


Pmax 2ωπCp

θδmax( 11138572 (11)



a1_u(t)

U+ βu(t)1113888 1113889 + a3

_u(t)

U+ βu(t)1113888 1113889

3⎡⎣ ⎤⎦


(12)


4 Conclusion




tip + 13ϕprime

2(L)L3

tip)


0 ϕ(L) + s(ϕ(L))3ds + ϕ(L) 1113938Ltip

0 s(ϕ(L)) 1113938Ltip


C (Rθ2p(1 + C2pΩ

2R2))



Ltip









Data Availability




References











Ro CpΩ1113872 1113873minus 1

Uonseto


1



3tip

Amaxo

θ2p 2CpΩ1113872 11138731113872 1113873 + 2 ξω minus A

075BΩ2

1113971

ϕ(L)


15BCpΩ

Ω

ω2+

θ2p2Cp

11139741113972

(9)

x

y


CjR


S-SSHI EXP



25

20

15

10

5

0

Pow

er (m

W)

2 4 6 8 10 12 14 16Wind speed (ms)






Pmax R


2 (10)


Pmax 2ωπCp

θδmax( 11138572 (11)



a1_u(t)

U+ βu(t)1113888 1113889 + a3

_u(t)

U+ βu(t)1113888 1113889

3⎡⎣ ⎤⎦


(12)


4 Conclusion




tip + 13ϕprime

2(L)L3

tip)


0 ϕ(L) + s(ϕ(L))3ds + ϕ(L) 1113938Ltip

0 s(ϕ(L)) 1113938Ltip


C (Rθ2p(1 + C2pΩ

2R2))



Ltip









Data Availability




References













Pmax R


2 (10)


Pmax 2ωπCp

θδmax( 11138572 (11)



a1_u(t)

U+ βu(t)1113888 1113889 + a3

_u(t)

U+ βu(t)1113888 1113889

3⎡⎣ ⎤⎦


(12)


4 Conclusion




tip + 13ϕprime

2(L)L3

tip)


0 ϕ(L) + s(ϕ(L))3ds + ϕ(L) 1113938Ltip

0 s(ϕ(L)) 1113938Ltip


C (Rθ2p(1 + C2pΩ

2R2))



Ltip









Data Availability




References











Data Availability




References










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