Aalborg Universitet Model-based control design of Series ...Xiongfei; Imbaquingo, Carlos Enrique; Sarrá, Eduard; Isernia, Nicola; Tonellotto, Alberto Published in: Journal of Renewable

Aalborg Universitet

Model-based control design of Series Resonant Converter based on the Discrete TimeDomain Modelling Approach for DC Wind Turbine

Chen Yu-Hsing Dincan Catalin Gabriel Kjaeligr Philip Carne Bak Claus Leth WangXiongfei Imbaquingo Carlos Enrique Sarraacute Eduard Isernia Nicola Tonellotto AlbertoPublished inJournal of Renewable Energy

DOI (link to publication from Publisher)10115520187898679

Creative Commons LicenseCC BY 40

Publication date2018

Document VersionPublishers PDF also known as Version of record

Link to publication from Aalborg University

Citation for published version (APA)Chen Y-H Dincan C G Kjaeligr P C Bak C L Wang X Imbaquingo C E Sarraacute E Isernia N ampTonellotto A (2018) Model-based control design of Series Resonant Converter based on the Discrete TimeDomain Modelling Approach for DC Wind Turbine Journal of Renewable Energy 2018 1-19httpsdoiorg10115520187898679

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Research ArticleModel-Based Control Design of Series ResonantConverter Based on the Discrete Time Domain ModellingApproach for DC Wind Turbine

Yu-Hsing Chen Catalin Gabriel Dincan Philip Kjaeligr Claus Leth Bak Xiongfei WangCarlos Enrique Imbaquingo Eduard Sarragrave Nicola Isernia and Alberto Tonellotto

Department of Energy Technology Aalborg University Aalborg Denmark

Correspondence should be addressed to Catalin Gabriel Dincan cgdetaaudk

Received 24 August 2018 Accepted 31 October 2018 Published 2 December 2018

Academic Editor Shuhui Li

Copyright copy 2018 Yu-Hsing Chen et alThis 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

This paper focuses on the modelling of the series resonant converter proposed as a DCDC converter for DC wind turbines Theclosed-loop control design based on the discrete time domain modelling technique for the converter (named SRC) operated incontinuous-conduction mode (CCM) is investigated To facilitate dynamic analysis and design of control structure the designprocess includes derivation of linearized state-space equations design of closed-loop control structure and design of gainscheduling controller The analytical results of system are verified in z-domain by comparison of circuit simulator response (inPLECS) to changes in pulse frequency and disturbances in input and output voltages and show a good agreement Furthermorethe test results also give enough supporting arguments to proposed control design

1 Introduction

MEDIUM-voltage DC (MVDC) collection of wind power isan attractive candidate to reduce overall losses and installa-tion cost especially within offshore HVDC-connected windgeneration as illustrated in Figure 1 [1] To connect DC windturbine with MVDC network (plusmn50kVDC) the series resonantconverter (SRC) serves as a step-up solid-state transformer asshown in Figure 2With the series resonant converter the DCturbine converter can take advantages of high efficiency highvoltage transformation ratio and galvanic fault isolation fordifferent ratings of turbine generator [2ndash6]

Traditional closed-loop control of SRC for the DC distri-bution system is easily implemented by detecting the zero-crossing of the resonant inductor current 119894푟 and controllingthe length of transistor and diode conduction angle 120572withoutconsidering circuit parameters of SRC [7] Additionally theoutput power flow control of SRC for DC network is achievedby controlling the phase-shift angle and frequency betweenthe two arms of H-bridge inverter [6 8 9]

Based on the discrete time domain modelling approachthe small-signal model of an improved SRC (named SRC)

is proposed [9 10] This paper continues with the small-signal plant model addressed in Section 3 and the Appendixand mainly focuses on the closed-loop control design for thesystem In the following sections the mode of operation ofSRC and small-signal plant model based on the discretetime domain modelling approach will be briefly introducedfirst The structure of closed-loop control based on theproposed small-signal plant model and the improvement inthe disturbance rejection capabilitywill be revealed To satisfythe power flow control with variable switching frequencythe gain scheduling technique will be given Finally theanalytical solution of overall system is revealed and verifiedby comparing with time-domain trace in circuit simulationmodel implemented in PLECS under different operatingpoints Furthermore the proposed control deign will bedemonstrated by a scaled-down laboratory test bench

2 Mode of Operation of SeriesResonant Converter

The mode of operation of series resonant converter (SRC)in Figure 2 is decided by the ratio between natural frequency

HindawiJournal of Renewable EnergyVolume 2018 Article ID 7898679 18 pageshttpsdoiorg10115520187898679

2 Journal of Renewable Energy

MVDC

Offshore substation platform

~

HVDC

Feeder 1

Onshore substation

AC GridFeeder 2

Generator and rectifier set

Energy dump

Turbine-side DC DC converter

~

~ ~Onshore DCAC

converter

Isolated DCDC converter MVDC connection

DC power collection systemOnshore facilities and AC

transmission networkExport cable

SubstationDCDC converterDC switchyard

Array network

GGG

LVDC

Figure 1 Generic configuration of the wind power plant with MVDC power collection

VLVDC

Vturb

+

-frac12Lf

iturb

2Cf

2Cf

VoutRec

+

-

frac12LfioutRec

VMVDC

Lr CrN1 N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

-Vg+

-Vo+

-Vrsquog

+ＣＬ

Figure 2 Circuit topology of series resonant converter (SRC)

Output Power

vg(t)N2N1x VLVDC

-N2N1x VLVDC

t

t

t

0

0

0

10 pu

|ＣＩＯＮ２＝(Ｎ)|

Figure 3 Frequency-depended power flow control of SRC

of tank (Lr and Cr) and the switching frequency of H-bridge inverter subresonant resonant and super resonantmode In subresonant mode the switching frequency ofH-bridge inverter is lower than the natural frequency oftank The resonant operating mode is selected when theswitching frequency is equal to the natural frequency of tankIf converterrsquos switching frequency is higher than the naturalfrequency of tank the converter is operated in the superresonant mode [9]

Contrasting with the constant frequency with phase shiftcontrol which is normally applied for operation in superresonant mode to achieve ZVS at turn-on Figure 3 illustratesthe concept of frequency-depended power flow control ofSRC The converter leg of SRC consisting of switches T1

and T2 is referred to as the leading leg and the one consistingof switches T3 and T4 is referred to as the lagging leg asindicated in Figure 2 Both converter legs operate at a 50duty cycle [6 9]

Journal of Renewable Energy 3

STEP 5 STEP 6 STEP 7 STEP 8

STEP 1 STEP2 STEP 3 STEP 4

Circuit topology of DC

wind turbine converter

Clarify the resonant tank

waveform according to

the mode Of operation

Draw the equivalent

circuit Derivate the large signal

model

Linearization and derivation

of the small signal model

Define the state variables

Create state-space model of

converter

Create interesting

transfer functions

Figure 4 Flow chart of derivation of plant model of SRC

To achieve ZCS character at turn-off or minimize theturn-off current the IGBT-based SRC is designed to operateat subresonant continuous-conduction mode (subresonantCCM)This control design can drive the implemented phaseshift having the same length as the resonant pulse withoutsacrificing the advantage of linear relation to the numberof resonant pulses as depicted in Figure 3 Compared to atraditional SRCwith frequency control design in subresonantmode therefore the medium frequency transformer in theSRC addressed in this paper can be designed for a higherfrequency and avoids saturation for lower frequencies

3 Discrete Time Domain Modelling Approachfor Series Resonant Converter

Considering the efficiency subresonant mode is selected forthemode of operation of SRC for theDCwind turbine [5 6]Based on the circuit topology shown in Figure 2 Figure 5illustrates the steady-state voltage and current waveformsof SRC in subresonant mode where 120596s is the switchingfrequency (120596s=2120587 sdot fs) of SRC To apply the linear controltheory to the SRC control design deriving the plant modelof SRC with the discrete time domain modelling approachincludes the derivation of large-signal equations based onthe interesting interval shown in Figure 5 linearizationof discrete state equations and derivation of small-signaltransfer function In the derivation the voltages V푀푉퐷퐶(119905)and V퐿푉퐷퐶(119905) are assumed to be discrete in nature having theconstant values 119881표(푘) and 119881푔(푘) in interval of 119896푡ℎ event andthen switch to next states 119881표(푘+1) and 119881푔(푘+1) at the start of(119896+1)푡ℎ eventThis procedure is only valid when the variationin V표(119905) or vg(t) in the event is relatively smaller than its initialand final values [10]

With the discrete time domain modelling approach (1)gives a linearized state-space model of SRC in subresonantmode and the transfer functions between input state variablesand the defined interesting states are shown in (3) and (5)To simplify the derivation the output filter of SRC (ie Lf

and Cf) is neglected and only the DC component of outputcurrent diode rectifier ioutRec is selected as an output variableIo

To obtain the harmonic model of DC turbine converterFigure 4 gives a complete flow chart of mathematical deriva-tion of SRC plant model which describes how the SRCplant model is obtained First of all the circuit topology andmode of operation are decided as shown in Figures 2 and5 and then the equivalent circuit based on the switchingsequence of transistors is generated in Figure 6 Based onthe circuit topology shown in Figure 2 Figure 5 illustratesthe voltage and current waveforms of SRC n subresonantmode and the equivalent circuit for each event (switchinginterval) is given in Figure 6 According to Figure 6 the largesignal model of converter is created (step 4) and then theinteresting state variables (step 5) are defined to generate thesmall-signal equation and the space model of converter as insteps 6 and 7 respectively ([A] [B] [C] and [D]) Eventuallythe converter plant model (power stage of converter) isestablished based on the interesting transfer function (g1 g2and g3 in step 8) The correction of model (plant model) hasbeen confirmed and the details of derivation are given in theAppendix

[ 11990911199092

] = [119860] [12

] + [119861] [[[[

푔

표

]]]]

표 = [119862] [12

] + [119863][[[[

푔

표

]]]]

(1)

where

1 = 푟2 = V푐푟

(2)


0

0

0

Kthevent

(K+1)thevent

= =

ＣＬ(ＭＮ)

６Ａ(ＭＮ)６Ｉ(ＭＮ)

ＭＮ

ＭＮ

ＭＮ

ＥＥ

Ｅ

ＥＥ

Ｅ

６Ｌ(ＭＮ)

ＣＩＯＮ２＝(ＭＮ) ＣＬ(ＭＮ)|

ＭＮ0(Ｅ) ＭＮ1(Ｅ)ＭＮ2(Ｅ)ＭＮ0(Ｅ+1)

ＭＮ1(Ｅ+1)

ＭＮ2(Ｅ+1)ＭＮ0(Ｅ+2)

|

Figure 5 Resonant inductor current and resonant capacitor voltage waveforms of SRC in subresonant CCM

-

+

-

+

vo(t) = Vo0(k)=VMVDC

ir(t)

vg(t)=Vg0(k)=

N1 N2

+ vTank(t) -

Lr Crir(t)

rsquog(t)

= N2N1 times VLVDC

(a) t0(k) le t le t1(k) (T1T4 ON)

-

+

-

+

vo(t) = Vo1(k)=-VMVDC

vg(t)=Vg1(k)=0

ir(t)N1 N2

+ vTank(t) -

Lr Crir(t)

rsquog(t)

(b) t1(k) le t le t2(k) (D1T3 ON)

-

+

-

+

vo(t) = Vo0(k+1)=-VMVDC

vg(t)=Vg0(k+1)

+ vTank(t) -

N1 N2Lr Crir(t)ir

(t)

rsquog(t)

=-

(c) t0(k+1) le t le t1(k+1) (T2 T3 ON)

vo(t) = Vo1(k+1)=VMVDC

vg(t)=Vg1(k+1)=0

N1 N2

-

++ vTank(t) -

Lr Crir(t)

-

+

ir(t)

rsquog(t)

(d) t1(k+1) le t le t2(k+1) (D2 T4 ON)

Figure 6 Equivalent circuit of SRC for large-signal analysis of conduction intervals in subresonant CCM


Compensator SRC plant model

PREF1

６－６＄

IoREFＡ＝(Ｍ)

fs

Vg

Ａ1(Ｍ)

Ａ2(Ｍ)

Ａ3(Ｍ)

Io

Vo

-

Ｌ

Figure 7 Small-signal control model of the series resonant converter SRC in subresonant CCM

and the derivation of [A] [B] [C] and [D] matrixes is shownin the Appendix Transfer functions between defined internalstate variables and input states are given by

= [ 푟

V푐푟

] = [12

] = [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(3)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(4)

and transfer functions between converter output current andinput state variables are

표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(5)

where the transfer functions 1198921(119904) 1198922(119904) and 1198923(119904) can beobtained via

[1198921 (s) 1198922 (s) 1198923 (s)] = 119862 (119878119868 minus 119860)minus1 119861 + 119863 (6)

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(7)

The transfer function 1198921(119904) describes how the output current표 is influenced by the control input variable and thetransfer functions 1198922(119904) and 1198923(119904) describe how the outputcurrent 표 is affected if any disturbance occurs in input voltage119881푔 (prop VLVDC) and the output voltage 119881표 (prop VMVDC)For example the array network (MVDC grid) containsvoltage harmonics The transfer function 1198923(119904) can be usedto evaluate the effect of voltage harmonics on the converteroutput current Detailed derivation of the above linearizedstate-space model and the expression of elements in [A] [B][C] and [D]matrix in (1) have been revealed in theAppendix

4 Model-Based Closed-Loop Control Design

Figure 7 gives an overview of small-signal control modelof SRC based on the average plant model in (5) and (6)The control design of SRC includes derivation of small-signal plant model and the design of the compensator gc Thesmall-signal transfer functions of SRC between converteroutput current and input state variables are given by (5) and(6) where the output current variation is the expression oflinear combination of the three independence inputs Therelationship between 120572 and f s in subresonant CCM in large-signal model is

120572 = ( 12119891푠

minus 12119891푟

) 2120587119891푠 = 120587 minus 120587119891푟

119891푠 (8)

where

119891푟 = 12120587radic119871푟119862푟

(9)

By substituting the perturbation terms of small-signal analy-sis into expression in (8) the small-signal expression of 120572 andf s can be obtained

120572 + = 120587 minus 120587119891푟

(119891푠 + 푠) (10)


01Hz 10Hz 10Hz 100Hz 1000Hz 10000Hz

0 dB

minus20 dB

minus40 dB

20 dB

40 dB

60 dBfp1

fz

fp2

fc

Crossover frequency

-20 dBdecade

-40 dBdecade

f (Hz)

MAG (dB)

minus60 dB

T0dB

|T(1+T)|

|T|

Figure 8 Illustration of magnitude asymptote of desired loop gain 119879(119904)|target (target curve of loop gain) [11]

where the AC component is

= minus 120587119891푟

푠 (11)

Eventually the system transfer function in Figure 5 can beexpressed as

표 (s) = (1119881푀푉퐷퐶) 119892푐 (minus120587119891푟) 11989211 + 119892푐 (minus120587119891푟) 1198921

푅퐸퐹

+ 11989221 + 119892푐 (minus120587119891푟) 1198921

푔

+ 11989231 + 119892푐 (minus120587119891푟) 1198921

표

(12)

Equation (12) can be further expressed as the following

표 (s) = 1119881푀푉퐷퐶

1198791 + 119879푅퐸퐹 + 11989221 + 119879푔 + 11989231 + 119879표 (13)

with a loop gain

119879 (s) = 119892푐

minus120587119891푟

1198921 (14)

where the loop gain is defined by the product of gains aroundforward and feedback paths [11]

5 Disturbance Rejection Capability

The closed-loop control design of SRC is implemented viathe compensator gc which is applied to shape the loop gainof the system (ie 119879(119904)) Considering the transfer function ofoutput current given in (13) the relationship between 표 and푔 is shaped by closed-loop control as

표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉0=0

= 11989221 + 119879 (15)

The variation in output current Io caused by 푔 can bealleviated by increasing the magnitude of the loop gain 119879(119904)when the closed-loop control design is integrated with theSRC plant model The system transfer functions in (13) also

show that the variation reduction of Io due to variation inMVDC network will benefit from a high loop gain 119879(119904)

표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

= 11989231 + 119879 (16)

Furthermore consider the tracking performance of outputcurrent control in (17)

표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0

= 1119881푀푉퐷퐶

1198791 + 119879 (17)

Assume that a constant power reference 119875푅퐸퐹 is applied to thecontrol loop with a constant MVDC source and a constantLVDC source A large loop gain |119879(119904)| (ie |119879(119904)| ≫ 1) canalso make sure of a good DC current tracking performanceas shown in (18)

119868표119875푅퐸퐹

asymp 1119881푀푉퐷퐶

(18)

Therefore the objective of the compensator gc is togovern the system with a desired loop gain (ie 119879(119904) =119879(119904)|target) where the deviation of desired loop transferfunction 119879(119904)|target can be found by simply evaluating themagnitude asymptote in Figure 8

119879 (s) = 119892푐

minus120587119891푟

1198921

= 1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

(19)

Considering the desired loop gain 119879(119904)|target illustrated inFigure 8 the disturbance rejection capability of the outputcurrent for a frequency range below the crossover frequency(119891푐) can be improved with closed-loop control For exampleat the low frequency range (119891 lt 119891푐) the output current 119868표

is almost in direct proportion to the power reference signal119875푅퐸퐹

1198791 + 119879 asymp 1 for 119891 lt 119891푐 (|119879| ≫ 1)119879 for 119891 gt 119891푐 (|119879| ≪ 1) 997904rArr

표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0


(20)


Furthermore a high loop gain provides a good disturbancereduction to the variation on input voltage 119881푔 and outputvoltage 119881표 by the factor 1|119879|


표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0푉0=0

asymp 1198922119879 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

asymp 1198923119879

(21)

Typically the crossover frequency f c should be less thanapproximately 10 of switching frequency of SRC (119891푐 lt01119891푠) to limit the harmonics caused by PWM switching [11]Based on (19) therefore compensator gc|OP under a certainoperating point (OP) can be expressed by

119892푐1003816100381610038161003816푂푃 = 119879|푂푃target(minus120587119891푟) 1198921

1003816100381610038161003816푂푃

= 119891푟minus120587sdot 11198921

1003816100381610038161003816푂푃

1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃target

(22)

Equations (23)ndash(27) summarize the parameters (ie Q120596푝1 120596푝2 120596푧 120596푐 and 120579) which are used to shape the loop gain119879(119904) via the compensator 119892푐 The crossover frequency f c andthe low-frequency pole at 119891푝1 are defined as

119891푐 = 01119891푠 (23)

119891푝1 = 145119891푐 (24)

The low-frequency zero at 119891푧 and high-frequency pole at119891푝2 can be chosen according to crossover frequency 119891푐 andrequired phase margin 120579 as follows

119891푧 = 119891푐radic1 minus sin (120579)1 + sin (120579) (25)

119891푝2 = 119891푐radic1 + sin (120579)1 minus sin (120579) (26)

where the angle 120579 is a phase lead angle of compensator at f cThe DC gain of target loop gain 119879(119904)|target is

119879표 = ( 119891푐119891푝1

)2 times radic 119891푧119891푝2

(27)

TheQ-factor is used to characterize the transient responseof closed-loop system Using a high Q-factor can increasethe dynamic response during transient but it can also causeovershoot and ringing on power devices In practical appli-cation the Q-factor must be sufficiently low to keep enough

Table 1 Parameters of SRC plant model

Low voltage DC (VLVDC) 404 (kVDC)Medium voltage DC (VMVDC) 1000 (kVDC)Transformer winding voltage ratio (N1 N2) 1 25Rated output power 119875표푢푡 10 (MW)Resonant inductor 119871푟 781 (mH)Resonant capacitor 119862푟 025 (uF)

phase margins and alleviate voltage and current stress onpower devices [11] Additionally since the power flow controlof SRC depends on the control of switching frequencyf s the parameters of target curve and the coefficient oftransfer function gc have to be changed according to differentoperating points (different output powers) To make surethat the compensator gc can match with different outputpower requirements therefore a gain scheduling approach isproposed which will be revealed in the next section

6 Design of Digital GainScheduling Controller

Gain scheduling controller is designed to access the param-eter of compensator 119892푐 in real time and then adjust itbased on the different operating points Figure 9 gives acomplete digital controller of SRC based on the small-signalcontrol model and the bilinear transformation The digitalcontroller of SRC consists of a small-signal controller a gainscheduling controller a feedforward control loop and a DCcomponent calculator (119868표 calculator) The controller is imple-mented in z-domain with a variable interrupt frequency 119891푖푛푡

(119891푖푛푡 prop switching frequency 119891푠) With the bilinear transformthe general form of the discrete-time representation of thecompensator 119892푐 can be expressed as

119892푐 (z) = 11988651199115 + 11988641199114 + 11988631199113 + 11988621199112 + 1198861119911 + 119886011988751199115 + 11988741199114 + 11988731199113 + 11988721199112 + 1198871119911 + 1198870

(28)

where coefficients 119886푛 and 119887푛 (n=0sim5) are used to specify thecoefficients of numerator and denominator

To design the gain scheduling controller coefficientsan and bn in (28) are evaluated under different operatingpoints (ie different output power) with (22)ndash(27) A trendin the variation of each coefficient (ie PREF vs an andbn) is recorded and then is formulated via the polynomialapproximation as shown in Figures 12 and 13 which willbe discussed in the next section (Section 7) Eventually thecoefficient of gc(z) for SRC in the subresonant CCM can beadjusted by a continuous function such as 119886푛 = 119891(119875푅퐸퐹) and119887푛 = 119891(119875푅퐸퐹) in real time to avoid any potential turbulencescaused by gain-changing

7 Verification of Closed-Loop Control Design

With the SRC topology in Figure 2 and the controller shownin Figure 9 Tables 1 and 2 give the parameters used inthe state-space model and circuit simulation models (tools)


sf

Switching model

Small-signal controller

Gain Scheduling controller

Feedforward control

PREFIoREF

gC(z)

Vg Vo

1

６－６＄

1

６－６＄

fs (prop PREF prop IoREF)

fs

PREF

Io = Io + Io

Io

Io Calculatorlowast lowastIo = io0PLECS

Io (prop PREF prop IoREF)

ioPLECS = io0PLECS+ioℎPLECS

Figure 9 Control block of the series resonant converter SRC in z-domain

Table 2 Specifications of digital controller

Switching frequency 119891푠10k (Hz) (full load)

Interrupt frequency ofdigital controller 119891푖푛푡 (ie119891푖푛푡 = 2119891푠)

20k (Hz) (full load)

Q-factor 10Phase margin 120579 52∘

Power reference signal119875푅퐸퐹

10MW (full load)

Sampling rate of ADconversion

1M (Hz)

Duty cycle 50

for verifying the validation of overall system in z-domainThe control model in the subresonant CCM is verified toidentify the accuracy of proposed small-signal model andthen the results of coefficient assessment of 119892푐(119911) with thegain scheduling controller are integrated with control loopand are tested by a ramp-power reference

By applying a +05 stepping perturbation to all inputstate variables Figures 10 and 11 give the analytical solutionsof small-signal model of SRC and the results obtained fromthe time-domain switching model implemented in PLECSThe SRC with closed-loop control is commanded to deliveraround 90MW DC power and 75MW DC power to MVDCnetwork respectively Figures 10 and 11 show that both thesteady state and transient state in the analytical model matchwith the results generated by switching model Thereforedynamics of SRC switching model can be predictable andcontrolled with the proposed small-signal model

Figures 12 and 13 give the result of coefficient assessmentof 119892푐(119911) for the design of the gain scheduling controller Basedon (28) the trend in the variation of coefficients 119886푛 and 119887푛

in subresonant CCM from 575MW to 10MW (05MWstep)is identified and then the variation of each coefficient isapproximated with a 3rd polynomial (ie an(PREF)|PolyFit andbn(PREF)|PolyFit) According to the variation in output powerreference 119875푅퐸퐹 the gain scheduling controller accesses thepolynomial 119892푐(119911) to regulate its coefficient in real time To

Table 3 Specifications of laboratory test bench

Low voltage DC source(VLVDC)

216 (VDC)

DC component ofmedium voltage source(VMVDC0)

400 (VDC)

Transformer windingvoltage ratio (N1 N2)

1 2

Rated output power 119875표푢푡550 (W)

Resonant inductor 119871푟200 (mH)

Resonant capacitor 119862푟10 (uF)

Switching frequency 119891푠 800 (Hz)Output filter inductor 119871푓 25 (mH)Output filter capacitor119862푓

10 (mF)

Resistive load 119877푙표푎푑125 (Ω)

Interrupt frequency 119891푖푛푡

(ie 119891푖푛푡 = 2119891푠)16k (Hz)


1M (Hz)

Duty cycle 50

evaluate the adequacy of control design of overall systemfinally the time-trace simulation of output power flow controlis given in Figure 14 with a ramp-power reference PREF from01MW to 10MW and vice versa The results show that theoutput currentpower (Io) of the series resonant converter canbe well controlled whenmagnitude output powers referencesare changed

8 Laboratory Test Results

To verify the control design first the circuit simulation iscarried out with circuit simulation tool of PLECS and thenthe controller is implemented in a scaled-down laboratorytest bench The circuit configuration of test bench andthe corresponding parameters are shown in Figure 15 andTable 3 respectively where the MVDC network is simulated


Step-changing in PREF(z) GclosedIoPREF(z)=Io(z)PREF(z) Step-changing in Vg(z) GclosedIoVg(z)=Io(z)Vg(z)

Step-changing in Vo(z) GclosedIoVo(z)=Io(z)Vo(z)

(A)

96

955

95

945

94

97

96

95

94

93

96

95

94

93

92

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1

Figure 10 Dynamics of output current Io generated by both the switching model and derived state-space model with the closed-loopcontroller when +05 of step-changing is applied in PREF Vg and Vo respectively (119875푅퐸퐹 90MW 997888rarr 9045MW Vg 10101kVDC 997888rarr101515kVDC Vo 1000kVDC 997888rarr 1005kVDC blue circle dynamic of state-space model in z-domain red line dynamic of electrical signalin PLECS circuit model the interrupt time of digital controller Tint=1(2x119891푠|op) =1(2x900Hz) sec)

Step-changing in Vo(z) GclosedIoVo(z)=Io(z)Vo(z)76

755

75

745

74

Step-changing in PREF(z) GclosedIoPREF(z)=Io(z)PREF(z)765

76

755

75

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1

Step-changing in Vg(z) GclosedIoVg(z)=Io(z)Vg(z)

775

77

765

755

75

76

78

(A)

t (Sec)093 094 095 096 097 098 099 1

Figure 11 Dynamics of output current Io generatedbyboth the switchingmodel andderived state-spacemodelwith the closed-loop controllerwhen +05 of step-changing is applied in PREF Vg and Vo respectively (PREF 75MW 997888rarr 75375MW Vg 10101kVDC 997888rarr 101515kVDC Vo1000kVDC 997888rarr 1005kVDC blue circle dynamic of state-space model in z-domain red line dynamic of electrical signal in PLECS circuitmodel the interrupt time of digital controller Tint=1(2x119891푠|op) =1(2x750Hz) sec)


6 7 8 9 10x 106

0

minus01

minus02

Coefficient a0 (PREF) and its polynomial approximation

PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

05

04

03

02

minus02

minus06

0

minus04

06

02

minus02

04

0

08

minus08

minus1minus09

minus11

1

06

08

04

a0(PREF)

a0(PREF)|PolyFit

a2(PREF)

a2(PREF)|PolyFit

a4(PREF)

a4(PREF)|PolyFit

a1(PREF)

a1(PREF)|PolyFit

a3(PREF)

a3(PREF)|PolyFit

a5(PREF)

a5(PREF)|PolyFit

0

1

2 3

4

5

Figure 12 Design of gain scheduling controller piecewise continuous functions of numerator of 119892푐(119911) and its polynomial approximation(3rd) in subresonant CCM

6 7 8 9 10x 106

0

minus005

minus01

Coefficient b0 (PREF) and its polynomial approximation

PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

minus3

minus2

minus1

minus35

minus3

minus25

b0(PREF)

b0(PREF)|PolyFit

b2(PREF)

b2(PREF)|PolyFit

b4(PREF)

b4(PREF)|PolyFit

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

1

05

0

5

3

4

2

1

05

0

b1(PREF)

b1(PREF)|PolyFit

b3(PREF)

b3(PREF)|PolyFit

b5(PREF)

b5(PREF)|PolyFit

b0

b1

b2

b3

b4

b5

Figure 13 Design of gain scheduling controller piecewise continuous functions of denominator of 119892푐(119911) and its polynomial approximation(3rd) in subresonant CCM


Io

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(A)

(a) Ramp-up power PREF=Pout=01MW 㨀rarr 10MWslop=495MWsec

Io(A)

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(b) Ramp-down power PREF=Pout=10MW 㨀rarr 01MW slop=-495MWsec

Figure 14 Output current (Io) of the series resonant converter with a ramp-power reference PREF to verify the design of gain schedulingcontroller and demonstrate the start-up process of DC wind converter

SRC (Fig 2)

+

VMVDC

iturbVMVDCh

VMVDC0

Perturbation

DC power sourceRload

DAux

+Vturb

minus

minus

Figure 15 Configuration of laboratory test bench (VMVDC0 DC component of medium voltage power source VMVDCh perturbation source)

by a unidirectional power flow DC power source with acontrollable perturbation

Figures 16(a) and 16(b) depict the system response when apositive and a negative step perturbation (001pu) in MVDCnetwork are applied respectively The control design exhibitsa close behavior in either simulation or experimental testThere is some small tracking error during the transientbetween the simulation and test resultsThis usually is causedby the estimated error of components and stray inductancewhich is not considered in simulation model Figure 16(c)represents how the output current behaves when a step-change (026pu) is applied in the power reference signalPREF Under the proposed control law for SRC both thesimulation and test result show that the DC component ofDC turbine output current (119894turb) tracking performance canbe guaranteed However a small oscillation (asymp40Hz) duringthe transient of step-change of power reference signal inthe experimental test is observed due to the series diodeDAux (in Figure 15) which is reverse-biased at this testoccasion

9 Conclusion

A model-based control design of SRC for DC wind powerplant based on small-signal plant model in the discrete time-domain modelling is revealed This paper continues withthe modelling of SRC given in the Appendix and mainlyaddresses the closed-loop control design for the system Thecontrol design process contains the derivation of state-spaceplant model design of closed-loop control structure anddesign of gain scheduling controller Compared with thetraditional frequency-depended power flow control whichrelied on open-loop structure the SRC with the closed-loopstructure can gain a better disturbance rejection capabilityfor the output power control The verification of proposeddigital controller including plant model is addressed in boththe analytical model and the time-domain circuit simulationimplemented in PLECS in Section 7 by evaluating the SRCwith the stepping-perturbation under the subresonant CCMFurthermore gain scheduling approach is implemented bythe polynomial approximation and tested under different


Time [s]

Test ResultSimulation

098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]

(a) 40VDC step-up disturbance (001pu) in theMVDCvoltage (VMVDC)

Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]

(b) 40VDC step-down disturbance (001pu) in the MVDC voltage(VMVDC)

1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]

(c) Step-up disturbance (026pu) in the power reference signal PREF(equivalent of current reference signal 137A㨀rarr173A)

Figure 16 Dynamic response of output current of DC wind turbine converter (iturb) when a step-up-down disturbance is injected in systemat t=10 [s]operating points (different output powers) Integrating thegain scheduling controller with closed-loop structure enablesthe system to automatically adjust parameters of controllerin real time to satisfy different output power requirementswithout sacrificing the control performance Finally Sec-tion 8 shows that all the test results give enough supportingarguments to the proposed control design

Appendix

The objective of the study is to understand the harmonicsdistribution of offshore DC wind farm and how the DC windturbines are affected by harmonics from MVDC gird Thissection summarizes the derivation of plant model of DCwind turbine based on the discrete time domain modellingapproach (discrete time domain modelling approach [10]steps 1-8) which can help the reader to reach the plant modelof DC wind turbine (SRC) and then conduct control deignofDCwind turbineThe following discussion will give a com-plete derivation process including the corresponding flowchart of the derivation of SRC plant model given in Figure 4

Steps 1 -3 Decide the Circuit Topology of DC Turbine Con-verter ResonantTankWaveform andEquivalentCircuit Steps

1-3 describe the circuit topology of SRC (DC wind turbineconverter) and mode of operation which is operated insubresonant CCM as in Figures 2 and 5 The correspondingequivalent circuit for the SRC in subresonant CCM is givenin Figure 6 where the waveform is divided by different timezone (different switching sequence) based on the discretetime domainmodelling approach proposed by King R J [10]Those figures (Figures 2 5 and 6) are used to generate thelarge signal model of SRC

Step 4 Large Signal Model Based on Figure 6 the objective ofderivation of large-signal model is to express the final value ofinteresting state variables in each switching interval with theinitial values The procedure is only valid when the variationin output voltage V표(119905) (MVDC grid voltage) or input voltageV푔(119905) (LVDC voltage) in the event (switching) is relativelysmaller than its initial and final values [10] Equations (A1)to (A16) give the derivation of large-signal model of resonantinductor current ir(t) and resonant capacitor voltage vCr(t)and their end values at 119896푡ℎ event in terms of initial values of119896푡ℎ event

For t0(k) le t le t1(k) (T1 T4 ON)

V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)

The resonant inductor current 119894푟(t) and resonant capacitorvoltage vCr(t) can be obtained by solving (A1)

119894푟 = 1119885푟

(119881푔0(푘) minus 119881표0(푘) minus 119881퐶푟0(k)) sin (120596푟t)+ 119868푟0(푘)cos (120596푟t)

(A3)

V퐶푟 = 119881푔0(푘) minus (119881푔0(푘) minus 119881표0(푘) minus 119881퐶푟0(k)) cos (120596푟t)+ 119868푟0(푘)119885푟 sin (120596푟t) minus 119881표0(푘)

(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)

At time t = t1(k) the tank current ir makes a zero crossingcommutating T1 and T4 off and turning on D1 and T3Therefore

119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟

(119881푔0(푘) minus 119881표0(푘) minus 119881퐶푟0(k)) sin (120596푟t1(푘))+ 119868푟0(푘) cos (120596푟t1(푘)) = 0

(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)

tan (120596푟푠120573퐾) = minus119868푟0(푘)119885푟(119881푔0(푘) minus 119881표0(푘) minus 119881퐶푟0(k)) (A8)

0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)

119881퐶푟1(푘) = V퐶푟 (t1(푘))= 119881푔0(푘) minus (119881푔0(푘) minus 119881표0(푘) minus 119881퐶푟0(k))

sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟

sdot sin (120596푟푠120573퐾) minus 119881표0(푘)

(A10)

For t1(k) le t le t2(k) (D1 T3 ON)

119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where

t耠 = t minus t1(k) (A12)

V퐶푟 (t耠) = (119881표1(푘) + 119881퐶푟1(k)) cos (120596푟t耠) minus 119881표1(푘) (A13)

where

119868푟1(푘) = 0119881푔1(푘) = 0119881표1(푘) = minus119881푀푉퐷퐶 = minus119881표0(푘)

(A14)

Eventually the inductor current ir(t) and capacitor voltagevCr(t) at time t=t2(k) can be represented by

119868푟2(푘) = [minussin (120596푟푠120573퐾) sdot sin (120596푟푠120572퐾)] sdot 119868푟0(푘)

+ [minus 1119885푟

sdot cos (120596푟푠120573퐾) sdot sin (120596푟푠120572퐾)] sdot 119881퐶푟0(k)

+ [ 2119885푟

sdot sin (120596푟푠120572퐾) + minus1119885푟

sdot cos (120596푟푠120573퐾)sdot sin (120596푟푠120572퐾)] sdot 119881표0(푘) + [minus1119885푟

sdot sin (120596푟푠120572퐾) + 1119885푟

sdot cos (120596푟푠120573퐾) sdot sin (120596푟푠120572퐾)] sdot 119881푔0(푘)

(A15)

119881퐶푟2(푘) = [119885푟 sdot sin (120596푟푠120573퐾) sdot cos (120596푟푠120572퐾)] sdot 119868푟0(푘)

+ [cos (120596푟푠120573퐾) sdot cos (120596푟푠120572퐾)] sdot 119881퐶푟0(k) + [minus2sdot cos (120596푟푠120572퐾) + cos (120596푟푠120573퐾) sdot cos (120596푟푠120572퐾) + 1]sdot 119881표0(푘) + [cos (120596푟푠120572퐾) minus cos (120596푟푠120573퐾)sdot cos (120596푟푠120572퐾)] sdot 119881푔0(푘)

(A16)

where

120596푠 sdot (t2(푘) minus t1(푘)) = 120572퐾 (A17)

The large-signal expression of resonant inductor current ir(t)and resonant capacitor voltage vCr(t) in (119896 + 1)푡ℎ event(t0(k+1) le t le t1(k+1) and t1(k+1) le t le t2(k+1)) can be obtainedwith the same process as derivation of equations as in (A1)-(A16)

Steady-State Solution of Large-Signal Model Equation (A18)gives the conditions for calculating steady-state solution(operating points) of discrete state equation

119868푟2(푘) = minus119868푟0(푘)119881퐶푟2(푘) = minus119881퐶푟0(k)

(A18)


By substituting (A18) into (A15) and (A16) the steady-statesolution of 119868푟0(푘) and 119881퐶푟0(푘) can be expressed in terms of119881표0(푘) 119881푔0(푘) 120573k and 120572k

119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)

where the overbar is used to indicated the steady-state valueof interesting state variables

To simplify the derivation the output filter of SRC (ieLf and Cf) is neglected due to very slow dynamics in voltageand current compared with the resonant inductor currentand resonant capacitor and only the DC component ofoutput current diode rectifier ioutRec is selected as an outputvariable io Therefore during theKth event the output currentequation delivered by the SRC is expressed as

119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠

sin (120596푟푠120573퐾)sdot [1 minus cos (120596푟푠120572푘)] sdot 119868푟0(푘) + 1120574푘

sdot minus 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟

cos (120596푟푠120573퐾) sdot [1 minus cos (120596푟푠120572푘)]sdot 119881퐶푟0(k) + 1120574푘


1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟

(cos (120596푟푠120573퐾) minus 2) sdot [1 minus cos (120596푟푠120572푘)]sdot 119881표0(푘) + 1120574푘

sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟

(1 minus cos (120596푟푠120573퐾)) sdot [1 minus cos (120596푟푠120572푘)]sdot 119881푔0(푘)

(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)

and 119868푟0(푘) is the initial value of Inductor current 119881퐶푟0(푘) isthe initial value of capacitor voltage 119881표0(푘) is the initial valueof rectifier output voltage and 119881푔0(푘) is then initial valueof input voltage of SRC Zr ( =radic(LrCr) ) is characteristicimpedance defined by parameter of resonant tanks 120572k (=120574푘-120573푘) is the transistor and diode conduction angle during theswitching interval (event k) and 120579s (=120596푠 119905) is represented bythe switching frequency of converter

The steady-state solution of discrete state equation foroutput variable io is obtained by substituting steady-statecondition into (A21) as

119868표

= 119894표1003816100381610038161003816(훽119896=훽훼119896=훼훾119896=훾퐼1199030(k)=퐼119903푉1198621199030(k)=푉119888119903 푉1199000(k)=푉119900 푉1198920(k)=푉119892)

(A23)

Step 5 Define State Variable Since the discrete large-signalstate equations in (A15) (A16) and (A21) have a highnonlinearity control design technique based on the linearcontrol theory cannot directly be applied To obtain a linearstate-space model therefore the linearization of large-signalequation is necessary Equation (A24) gives the definitionsof interesting state variables in both the kth switching event(t0(k) le t le t2(k)) and the (119896 + 1)푡ℎ switching event (t2(k) le t let2(k+1)) Finally the equations of approximation of derivativein (A24) and (A25) are used to convert the discrete state-equation (large-signal model) into continuous time [10]

1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)

119868푟2(푘) = minus1199091(푘+1)119881퐶푟2(푘) = minus1199092(푘+1)

(A24)

sdot119909푖 (t푘) = 119909푖(k+1) minus 119909푖(푘)1199050(k+1) minus 1199050(푘)

= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where

120574푘 = 120596푠 (1199052(k) minus 1199050(k)) = 120596푠 (1199050(k+1) minus 1199050(푘)) (A26)

By replacing the state variables in (A15) and (A16) with thedefined state variables in (A24) and applying the approxima-tion of (A25) for derivative the nonlinear state-space modelis given by

sdot1199091(k) = 120596푠120574푘

sdot [sin (120596푟푠120573퐾) sdot sin (120596푟푠120572퐾) minus 1] sdot 1199091(푘)

+ 120596푠120574푘

sdot [ 1119885푟

sdot cos (120596푟푠120573퐾) sdot sin (120596푟푠120572퐾)] sdot 1199092(푘)

+ 120596푠120574푘

sdot [minus2119885푟

sdot sin (120596푟푠120572퐾) + 1119885푟

sdot cos (120596푟푠120573퐾)sdot sin (120596푟푠120572퐾)] sdot 119881표0(푘) + 120596푠120574푘

sdot [ 1119885푟

sdot sin (120596푟푠120572퐾)+ minus1119885푟


= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘

sdot [minus119885푟 sdot sin (120596푟푠120573퐾) sdot cos (120596푟푠120572퐾)]sdot 1199091(푘) + 120596푠120574푘

sdot [minuscos (120596푟푠120573퐾) sdot cos (120596푟푠120572퐾) minus 1]sdot 1199092(푘) + 120596푠120574푘

sdot [2 sdot cos (120596푟푠120572퐾) minus cos (120596푟푠120573퐾)sdot cos (120596푟푠120572퐾) minus 1] sdot 119881표0(푘) + 120596푠120574푘

sdot [minuscos (120596푟푠120572퐾)+ cos (120596푟푠120573퐾) sdot cos (120596푟푠120572퐾)] sdot 119881푔0(푘)

= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)

where the output equation is defined as

119894표 = 119891표푢푡 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 119891표푢푡 1199091 1199092 V표 V푔 120572= 1120574푘

sdot 119891lowast표푢푡 1199091 1199092 V표 V푔 120572

(A29)

Step 6 Linearization and Small-Signal Model Consider thatall the interesting state variables in pervious steps are in thesteady-state (near the certain operating point OP) with asmall perturbation therefore the nonlinear state equationscan be formalized with Taylor Series Expansion in terms ofthe operating point (OP) and the perturbations

(i) Resonant inductor currentsdot[1199091 + 1]= 1198911 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)

where the subscript OP indicates the steady-state pointwhere the derivatives are evaluated at that point

119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)

(ii) Resonant capacitor voltagesdot[1199092 + 2]= 1198912 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)

(iii) Output current equation

119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot = 1120574 sdot 1

minus (120574) + ( 120574)2 minus + sdot 119891lowast

표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where

119868표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572표 = 1120574 sdot [ 120597119891lowast

표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표] sdot

(A41)

Neglect the higher-order terms of perturbation signals andretain only the linear terms in Taylor Series Expansion toobtain the linearized equations for (A32) (A36) and (A40)

Step 7 State-Space Model Equation (A42) gives a linearizedstate-space model of SRC in subresonant mode from (A32)(A36) and (A40) and the transfer functions between inputstate variables and the defined states are summarized in(A50) and (A52)

[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표] 1120574 sdot [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)

For derivative of equations 1198911 and 1198912

120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)

With the steady-state operating conditions

120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)

The same approach as the derivative of 1198911 and 1198912 with respectto input states 1199091 and 1199092 can be used to evaluate the derivativeof 119891lowast

표푢푡 and the derivative of 1198911 and 1198912 with respect to inputstates 120572 V푔 and vo

According to the derivation of large signal model in (A8)the angle 120573 and its steady-state solution can be expressed by

tan (120596푟푠120573) = minus1199091119885푟(V푔 minus V표 minus 1199092) = tan (120596푟푠120573 minus 120587) (A47)

1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠

sdot tanminus1 [ minus119868푟119885푟(119881푔 minus 119881표 minus 119881퐶푟)](A48)

The derivatives of 120573with respect to input states 1199091 1199092 V푔 andV표 at the given operating points are

1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot minus119885푟 sdot (V푔 minus V표 minus 1199092)(V푔 minus V표 minus 1199092)2 + (1199091119885푟)2

]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠

sdot minus119885푟 sdot (119881푔 minus 119881표 minus 119881퐶푟)(119881푔 minus 119881표 minus 119881퐶푟)2 + (119868푟119885푟)2

1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot minus1199091119885푟

(V푔 minus V표 minus 1199092)2 + (1199091119885푟)2]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠

sdot minus119868푟119885푟

(119881푔 minus 119881표 minus 119881퐶푟)2 + (119868푟119885푟)2

120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)

where120596푟푠 (=120596푟120596푠) is defined as the ratio between the naturalfrequency (120596푟) of resonant tank and the switching frequencyof converter (120596푠)

Step 8 Transfer FunctionThe transfer functions between theconverter output current (output rectifier current) and inputstate variables can be obtained with (A42)

표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)

and transfer functions between defined internal state vari-ables and input state are

= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)


The derivation of the linearized state-space model and theexpression of elements in [A] [B] [C] and [D] matrix aregiven in (A42)The transfer functions 1198921(119904)1198922(119904) and 1198923(119904)in (A50) can be obtained by the formula of 119862(119904119868-119860)minus1119861 + 119863

Nomenclature

119871푟 Inductor in resonant tank119862푟 Capacitor in resonant tank119894푟 Resonant inductor currentV퐶푟 Resonant capacitor voltageV푔 Input voltage of resonant tank referred to

as secondary side of medium-frequencytransformer

V표 Output voltage of resonant tank119881퐿푉퐷퐶 Low voltage DC119881푀푉퐷퐶 Medium voltage DC119894표푢푡Rec Output current of diode rectifier119894turb Output current of DC wind turbineconverter119871푓 Inductor in output filter119862푓 Capacitor in output filter119891푠 Switching frequency of series resonantconverter defined by 119891푠 = 120596푠2120587120596푟 Natural resonant frequency of tankdefined by 120596푟 = 1radic119871푟119862푟120572푘 Transistor and diode conduction angleduring event 119896120573푘 Transistor conduction angle during event119896120574푘 Total duration of event (120574푘 = 120572푘 + 120573푘)

Data Availability

The authors of the manuscript declare that the data used tosupport the findings of this study are included within thearticle

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] C Yu-Hsing et al ldquoStudies for Characterisation of ElectricalProperties of DC Collection System in Offshore Wind Farmsrdquoin Proceedings of the of Cigre General Session 2016 2016 articleno B4-301

[2] V Vorperian and S Cuk ldquoA complete DC analysis of the seriesresonant converterrdquo in Proceedings of the 13th Annual IEEEPower Electronics Specialists Conference (PESC rsquo82) pp 85ndash1001982

[3] A F Witulski and R W Erickson ldquoSteady-State Analysis ofthe Series Resonant Converterrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 21 no 6 pp 791ndash799 1985

[4] R U Lenke J Hu and R W De Doncker ldquoUnified steady-state description of phase-shift-controlledZVS-operated series-resonant and non-resonant single-active-bridge convertersrdquo in

Proceedings of the 2009 IEEE Energy Conversion Congress andExposition ECCE 2009 pp 796ndash803 IEEE 2009

[5] G Ortiz H Uemura D Bortis J W Kolar and O ApeldoornldquoModeling of soft-switching losses of IGBTs in high-powerhigh-efficiency dual-active-bridge DCDC convertersrdquo IEEETransactions on Electron Devices vol 60 no 2 pp 587ndash5972013

[6] C Dincan P Kjaer Y Chen S Munk-Nielsen and C L BakldquoAnalysis of aHigh-Power Resonant DCndashDCConverter for DCWindTurbinesrdquo IEEETransactions on Power Electronics vol 33no 9 pp 7438ndash7454 2017

[7] R J King and T A Stuart ldquoInherent Overload Protection forthe Series Resonant Converterrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 19 no 6 pp 820ndash830 1983

[8] H Wang T Saha and R Zane ldquoControl of series connectedresonant convertermodules in constant current dc distributionpower systemsrdquo in Proceedings of the 17th IEEE Workshop onControl and Modeling for Power Electronics (COMPEL rsquo6) pp1ndash7 2016

[9] C Dincan P Kjaer Y Chen S Munk-Nielsen and C L Bak ldquoAHigh-Power Medium-Voltage Series-Resonant Converter forDC Wind Turbinesrdquo IEEE Transactions on Power Electronics2017

[10] R J King and T A Stuart ldquoSmall-Signal Model for theSeries Resonant Converterrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 21 no 3 pp 301ndash319 1985

[11] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Science amp Business Media 2007

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The Scientific World Journal

Volume 2018

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Research ArticleModel-Based Control Design of Series ResonantConverter Based on the Discrete Time Domain ModellingApproach for DC Wind Turbine

Yu-Hsing Chen Catalin Gabriel Dincan Philip Kjaeligr Claus Leth Bak Xiongfei WangCarlos Enrique Imbaquingo Eduard Sarragrave Nicola Isernia and Alberto Tonellotto

Department of Energy Technology Aalborg University Aalborg Denmark

Correspondence should be addressed to Catalin Gabriel Dincan cgdetaaudk

Received 24 August 2018 Accepted 31 October 2018 Published 2 December 2018

Academic Editor Shuhui Li

Copyright copy 2018 Yu-Hsing Chen et alThis 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

This paper focuses on the modelling of the series resonant converter proposed as a DCDC converter for DC wind turbines Theclosed-loop control design based on the discrete time domain modelling technique for the converter (named SRC) operated incontinuous-conduction mode (CCM) is investigated To facilitate dynamic analysis and design of control structure the designprocess includes derivation of linearized state-space equations design of closed-loop control structure and design of gainscheduling controller The analytical results of system are verified in z-domain by comparison of circuit simulator response (inPLECS) to changes in pulse frequency and disturbances in input and output voltages and show a good agreement Furthermorethe test results also give enough supporting arguments to proposed control design

1 Introduction

MEDIUM-voltage DC (MVDC) collection of wind power isan attractive candidate to reduce overall losses and installa-tion cost especially within offshore HVDC-connected windgeneration as illustrated in Figure 1 [1] To connect DC windturbine with MVDC network (plusmn50kVDC) the series resonantconverter (SRC) serves as a step-up solid-state transformer asshown in Figure 2With the series resonant converter the DCturbine converter can take advantages of high efficiency highvoltage transformation ratio and galvanic fault isolation fordifferent ratings of turbine generator [2ndash6]

Traditional closed-loop control of SRC for the DC distri-bution system is easily implemented by detecting the zero-crossing of the resonant inductor current 119894푟 and controllingthe length of transistor and diode conduction angle 120572withoutconsidering circuit parameters of SRC [7] Additionally theoutput power flow control of SRC for DC network is achievedby controlling the phase-shift angle and frequency betweenthe two arms of H-bridge inverter [6 8 9]

Based on the discrete time domain modelling approachthe small-signal model of an improved SRC (named SRC)

is proposed [9 10] This paper continues with the small-signal plant model addressed in Section 3 and the Appendixand mainly focuses on the closed-loop control design for thesystem In the following sections the mode of operation ofSRC and small-signal plant model based on the discretetime domain modelling approach will be briefly introducedfirst The structure of closed-loop control based on theproposed small-signal plant model and the improvement inthe disturbance rejection capabilitywill be revealed To satisfythe power flow control with variable switching frequencythe gain scheduling technique will be given Finally theanalytical solution of overall system is revealed and verifiedby comparing with time-domain trace in circuit simulationmodel implemented in PLECS under different operatingpoints Furthermore the proposed control deign will bedemonstrated by a scaled-down laboratory test bench

2 Mode of Operation of SeriesResonant Converter

The mode of operation of series resonant converter (SRC)in Figure 2 is decided by the ratio between natural frequency

HindawiJournal of Renewable EnergyVolume 2018 Article ID 7898679 18 pageshttpsdoiorg10115520187898679


MVDC


~

HVDC

Feeder 1

Onshore substation

AC GridFeeder 2


Energy dump


~

~ ~Onshore DCAC

converter





Array network

GGG

LVDC


VLVDC

Vturb

+

-frac12Lf

iturb

2Cf

2Cf

VoutRec

+

-

frac12LfioutRec

VMVDC

Lr CrN1 N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

-Vg+

-Vo+

-Vrsquog

+ＣＬ


Output Power

vg(t)N2N1x VLVDC

-N2N1x VLVDC

t

t

t

0

0

0

10 pu














Draw the equivalent


model





converter

Create interesting

transfer functions








[ 11990911199092

] = [119860] [12

] + [119861] [[[[

푔

표

]]]]

표 = [119862] [12

] + [119863][[[[

푔

표

]]]]

(1)

where

1 = 푟2 = V푐푟

(2)


0

0

0

Kthevent

(K+1)thevent

= =

ＣＬ(ＭＮ)


ＭＮ

ＭＮ

ＭＮ

ＥＥ

Ｅ

ＥＥ

Ｅ

６Ｌ(ＭＮ)



ＭＮ1(Ｅ+1)

ＭＮ2(Ｅ+1)ＭＮ0(Ｅ+2)

|


-

+

-

+


ir(t)

vg(t)=Vg0(k)=

N1 N2

+ vTank(t) -

Lr Crir(t)

rsquog(t)

= N2N1 times VLVDC


-

+

-

+


vg(t)=Vg1(k)=0

ir(t)N1 N2

+ vTank(t) -

Lr Crir(t)

rsquog(t)


-

+

-

+


vg(t)=Vg0(k+1)

+ vTank(t) -

N1 N2Lr Crir(t)ir

(t)

rsquog(t)

=-

(c) t0(k+1) le t le t1(k+1) (T2 T3 ON)


vg(t)=Vg1(k+1)=0

N1 N2

-

++ vTank(t) -

Lr Crir(t)

-

+

ir(t)

rsquog(t)

(d) t1(k+1) le t le t2(k+1) (D2 T4 ON)




PREF1

６－６＄

IoREFＡ＝(Ｍ)

fs

Vg

Ａ1(Ｍ)

Ａ2(Ｍ)

Ａ3(Ｍ)

Io

Vo

-

Ｌ



= [ 푟

V푐푟

] = [12

] = [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(3)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(4)


표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(5)


[1198921 (s) 1198922 (s) 1198923 (s)] = 119862 (119878119868 minus 119860)minus1 119861 + 119863 (6)

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(7)




120572 = ( 12119891푠

minus 12119891푟

) 2120587119891푠 = 120587 minus 120587119891푟

119891푠 (8)

where

119891푟 = 12120587radic119871푟119862푟

(9)


120572 + = 120587 minus 120587119891푟

(119891푠 + 푠) (10)


01Hz 10Hz 10Hz 100Hz 1000Hz 10000Hz

0 dB

minus20 dB

minus40 dB

20 dB

40 dB

60 dBfp1

fz

fp2

fc

Crossover frequency

-20 dBdecade

-40 dBdecade

f (Hz)

MAG (dB)

minus60 dB

T0dB

|T(1+T)|

|T|



= minus 120587119891푟

푠 (11)



푅퐸퐹

+ 11989221 + 119892푐 (minus120587119891푟) 1198921

푔

+ 11989231 + 119892푐 (minus120587119891푟) 1198921

표

(12)


표 (s) = 1119881푀푉퐷퐶

1198791 + 119879푅퐸퐹 + 11989221 + 119879푔 + 11989231 + 119879표 (13)

with a loop gain

119879 (s) = 119892푐

minus120587119891푟

1198921 (14)




표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉0=0

= 11989221 + 119879 (15)



표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

= 11989231 + 119879 (16)


표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0

= 1119881푀푉퐷퐶

1198791 + 119879 (17)


119868표119875푅퐸퐹


(18)


119879 (s) = 119892푐

minus120587119891푟

1198921

= 1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

(19)




표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0


(20)




표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0푉0=0

asymp 1198922119879 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

asymp 1198923119879

(21)



1003816100381610038161003816푂푃

= 119891푟minus120587sdot 11198921

1003816100381610038161003816푂푃

1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃target

(22)


119891푐 = 01119891푠 (23)

119891푝1 = 145119891푐 (24)





119879표 = ( 119891푐119891푝1


(27)








119892푐 (z) = 11988651199115 + 11988641199114 + 11988631199113 + 11988621199112 + 1198861119911 + 119886011988751199115 + 11988741199114 + 11988731199113 + 11988721199112 + 1198871119911 + 1198870

(28)






sf

Switching model



Feedforward control

PREFIoREF

gC(z)

Vg Vo

1

６－６＄

1

６－６＄


fs

PREF

Io = Io + Io

Io











10MW (full load)


1M (Hz)

Duty cycle 50







216 (VDC)


400 (VDC)


1 2





10 (mF)



(ie 119891푖푛푡 = 2119891푠)16k (Hz)


1M (Hz)

Duty cycle 50







(A)

96

955

95

945

94

97

96

95

94

93

96

95

94

93

92

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1



755

75

745

74


76

755

75

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1


775

77

765

755

75

76

78

(A)

t (Sec)093 094 095 096 097 098 099 1



6 7 8 9 10x 106

0

minus01

minus02


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

05

04

03

02

minus02

minus06

0

minus04

06

02

minus02

04

0

08

minus08

minus1minus09

minus11

1

06

08

04

a0(PREF)

a0(PREF)|PolyFit

a2(PREF)

a2(PREF)|PolyFit

a4(PREF)

a4(PREF)|PolyFit

a1(PREF)

a1(PREF)|PolyFit

a3(PREF)

a3(PREF)|PolyFit

a5(PREF)

a5(PREF)|PolyFit

0

1

2 3

4

5


6 7 8 9 10x 106

0

minus005

minus01


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

minus3

minus2

minus1

minus35

minus3

minus25

b0(PREF)

b0(PREF)|PolyFit

b2(PREF)

b2(PREF)|PolyFit

b4(PREF)

b4(PREF)|PolyFit

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

1

05

0

5

3

4

2

1

05

0

b1(PREF)

b1(PREF)|PolyFit

b3(PREF)

b3(PREF)|PolyFit

b5(PREF)

b5(PREF)|PolyFit

b0

b1

b2

b3

b4

b5



Io

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(A)


Io(A)

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25



SRC (Fig 2)

+

VMVDC

iturbVMVDCh

VMVDC0

Perturbation


DAux

+Vturb

minus

minus




9 Conclusion



Time [s]


098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]


Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]


1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]



Appendix






V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



MVDC


~

HVDC

Feeder 1

Onshore substation

AC GridFeeder 2


Energy dump


~

~ ~Onshore DCAC

converter





Array network

GGG

LVDC


VLVDC

Vturb

+

-frac12Lf

iturb

2Cf

2Cf

VoutRec

+

-

frac12LfioutRec

VMVDC

Lr CrN1 N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

-Vg+

-Vo+

-Vrsquog

+ＣＬ


Output Power

vg(t)N2N1x VLVDC

-N2N1x VLVDC

t

t

t

0

0

0

10 pu














Draw the equivalent


model





converter

Create interesting

transfer functions








[ 11990911199092

] = [119860] [12

] + [119861] [[[[

푔

표

]]]]

표 = [119862] [12

] + [119863][[[[

푔

표

]]]]

(1)

where

1 = 푟2 = V푐푟

(2)


0

0

0

Kthevent

(K+1)thevent

= =

ＣＬ(ＭＮ)


ＭＮ

ＭＮ

ＭＮ

ＥＥ

Ｅ

ＥＥ

Ｅ

６Ｌ(ＭＮ)



ＭＮ1(Ｅ+1)

ＭＮ2(Ｅ+1)ＭＮ0(Ｅ+2)

|


-

+

-

+


ir(t)

vg(t)=Vg0(k)=

N1 N2

+ vTank(t) -

Lr Crir(t)

rsquog(t)

= N2N1 times VLVDC


-

+

-

+


vg(t)=Vg1(k)=0

ir(t)N1 N2

+ vTank(t) -

Lr Crir(t)

rsquog(t)


-

+

-

+


vg(t)=Vg0(k+1)

+ vTank(t) -

N1 N2Lr Crir(t)ir

(t)

rsquog(t)

=-

(c) t0(k+1) le t le t1(k+1) (T2 T3 ON)


vg(t)=Vg1(k+1)=0

N1 N2

-

++ vTank(t) -

Lr Crir(t)

-

+

ir(t)

rsquog(t)

(d) t1(k+1) le t le t2(k+1) (D2 T4 ON)




PREF1

６－６＄

IoREFＡ＝(Ｍ)

fs

Vg

Ａ1(Ｍ)

Ａ2(Ｍ)

Ａ3(Ｍ)

Io

Vo

-

Ｌ



= [ 푟

V푐푟

] = [12

] = [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(3)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(4)


표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(5)


[1198921 (s) 1198922 (s) 1198923 (s)] = 119862 (119878119868 minus 119860)minus1 119861 + 119863 (6)

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(7)




120572 = ( 12119891푠

minus 12119891푟

) 2120587119891푠 = 120587 minus 120587119891푟

119891푠 (8)

where

119891푟 = 12120587radic119871푟119862푟

(9)


120572 + = 120587 minus 120587119891푟

(119891푠 + 푠) (10)


01Hz 10Hz 10Hz 100Hz 1000Hz 10000Hz

0 dB

minus20 dB

minus40 dB

20 dB

40 dB

60 dBfp1

fz

fp2

fc

Crossover frequency

-20 dBdecade

-40 dBdecade

f (Hz)

MAG (dB)

minus60 dB

T0dB

|T(1+T)|

|T|



= minus 120587119891푟

푠 (11)



푅퐸퐹

+ 11989221 + 119892푐 (minus120587119891푟) 1198921

푔

+ 11989231 + 119892푐 (minus120587119891푟) 1198921

표

(12)


표 (s) = 1119881푀푉퐷퐶

1198791 + 119879푅퐸퐹 + 11989221 + 119879푔 + 11989231 + 119879표 (13)

with a loop gain

119879 (s) = 119892푐

minus120587119891푟

1198921 (14)




표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉0=0

= 11989221 + 119879 (15)



표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

= 11989231 + 119879 (16)


표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0

= 1119881푀푉퐷퐶

1198791 + 119879 (17)


119868표119875푅퐸퐹


(18)


119879 (s) = 119892푐

minus120587119891푟

1198921

= 1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

(19)




표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0


(20)




표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0푉0=0

asymp 1198922119879 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

asymp 1198923119879

(21)



1003816100381610038161003816푂푃

= 119891푟minus120587sdot 11198921

1003816100381610038161003816푂푃

1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃target

(22)


119891푐 = 01119891푠 (23)

119891푝1 = 145119891푐 (24)





119879표 = ( 119891푐119891푝1


(27)








119892푐 (z) = 11988651199115 + 11988641199114 + 11988631199113 + 11988621199112 + 1198861119911 + 119886011988751199115 + 11988741199114 + 11988731199113 + 11988721199112 + 1198871119911 + 1198870

(28)






sf

Switching model



Feedforward control

PREFIoREF

gC(z)

Vg Vo

1

６－６＄

1

６－６＄


fs

PREF

Io = Io + Io

Io











10MW (full load)


1M (Hz)

Duty cycle 50







216 (VDC)


400 (VDC)


1 2





10 (mF)



(ie 119891푖푛푡 = 2119891푠)16k (Hz)


1M (Hz)

Duty cycle 50







(A)

96

955

95

945

94

97

96

95

94

93

96

95

94

93

92

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1



755

75

745

74


76

755

75

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1


775

77

765

755

75

76

78

(A)

t (Sec)093 094 095 096 097 098 099 1



6 7 8 9 10x 106

0

minus01

minus02


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

05

04

03

02

minus02

minus06

0

minus04

06

02

minus02

04

0

08

minus08

minus1minus09

minus11

1

06

08

04

a0(PREF)

a0(PREF)|PolyFit

a2(PREF)

a2(PREF)|PolyFit

a4(PREF)

a4(PREF)|PolyFit

a1(PREF)

a1(PREF)|PolyFit

a3(PREF)

a3(PREF)|PolyFit

a5(PREF)

a5(PREF)|PolyFit

0

1

2 3

4

5


6 7 8 9 10x 106

0

minus005

minus01


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

minus3

minus2

minus1

minus35

minus3

minus25

b0(PREF)

b0(PREF)|PolyFit

b2(PREF)

b2(PREF)|PolyFit

b4(PREF)

b4(PREF)|PolyFit

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

1

05

0

5

3

4

2

1

05

0

b1(PREF)

b1(PREF)|PolyFit

b3(PREF)

b3(PREF)|PolyFit

b5(PREF)

b5(PREF)|PolyFit

b0

b1

b2

b3

b4

b5



Io

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(A)


Io(A)

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25



SRC (Fig 2)

+

VMVDC

iturbVMVDCh

VMVDC0

Perturbation


DAux

+Vturb

minus

minus




9 Conclusion



Time [s]


098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]


Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]


1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]



Appendix






V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























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[ 11990911199092

] = [119860] [12

] + [119861] [[[[

푔

표

]]]]

표 = [119862] [12

] + [119863][[[[

푔

표

]]]]

(1)

where

1 = 푟2 = V푐푟

(2)


0

0

0

Kthevent

(K+1)thevent

= =

ＣＬ(ＭＮ)


ＭＮ

ＭＮ

ＭＮ

ＥＥ

Ｅ

ＥＥ

Ｅ

６Ｌ(ＭＮ)



ＭＮ1(Ｅ+1)

ＭＮ2(Ｅ+1)ＭＮ0(Ｅ+2)

|


-

+

-

+


ir(t)

vg(t)=Vg0(k)=

N1 N2

+ vTank(t) -

Lr Crir(t)

rsquog(t)

= N2N1 times VLVDC


-

+

-

+


vg(t)=Vg1(k)=0

ir(t)N1 N2

+ vTank(t) -

Lr Crir(t)

rsquog(t)


-

+

-

+


vg(t)=Vg0(k+1)

+ vTank(t) -

N1 N2Lr Crir(t)ir

(t)

rsquog(t)

=-

(c) t0(k+1) le t le t1(k+1) (T2 T3 ON)


vg(t)=Vg1(k+1)=0

N1 N2

-

++ vTank(t) -

Lr Crir(t)

-

+

ir(t)

rsquog(t)

(d) t1(k+1) le t le t2(k+1) (D2 T4 ON)




PREF1

６－６＄

IoREFＡ＝(Ｍ)

fs

Vg

Ａ1(Ｍ)

Ａ2(Ｍ)

Ａ3(Ｍ)

Io

Vo

-

Ｌ



= [ 푟

V푐푟

] = [12

] = [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(3)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(4)


표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(5)


[1198921 (s) 1198922 (s) 1198923 (s)] = 119862 (119878119868 minus 119860)minus1 119861 + 119863 (6)

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(7)




120572 = ( 12119891푠

minus 12119891푟

) 2120587119891푠 = 120587 minus 120587119891푟

119891푠 (8)

where

119891푟 = 12120587radic119871푟119862푟

(9)


120572 + = 120587 minus 120587119891푟

(119891푠 + 푠) (10)


01Hz 10Hz 10Hz 100Hz 1000Hz 10000Hz

0 dB

minus20 dB

minus40 dB

20 dB

40 dB

60 dBfp1

fz

fp2

fc

Crossover frequency

-20 dBdecade

-40 dBdecade

f (Hz)

MAG (dB)

minus60 dB

T0dB

|T(1+T)|

|T|



= minus 120587119891푟

푠 (11)



푅퐸퐹

+ 11989221 + 119892푐 (minus120587119891푟) 1198921

푔

+ 11989231 + 119892푐 (minus120587119891푟) 1198921

표

(12)


표 (s) = 1119881푀푉퐷퐶

1198791 + 119879푅퐸퐹 + 11989221 + 119879푔 + 11989231 + 119879표 (13)

with a loop gain

119879 (s) = 119892푐

minus120587119891푟

1198921 (14)




표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉0=0

= 11989221 + 119879 (15)



표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

= 11989231 + 119879 (16)


표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0

= 1119881푀푉퐷퐶

1198791 + 119879 (17)


119868표119875푅퐸퐹


(18)


119879 (s) = 119892푐

minus120587119891푟

1198921

= 1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

(19)




표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0


(20)




표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0푉0=0

asymp 1198922119879 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

asymp 1198923119879

(21)



1003816100381610038161003816푂푃

= 119891푟minus120587sdot 11198921

1003816100381610038161003816푂푃

1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃target

(22)


119891푐 = 01119891푠 (23)

119891푝1 = 145119891푐 (24)





119879표 = ( 119891푐119891푝1


(27)








119892푐 (z) = 11988651199115 + 11988641199114 + 11988631199113 + 11988621199112 + 1198861119911 + 119886011988751199115 + 11988741199114 + 11988731199113 + 11988721199112 + 1198871119911 + 1198870

(28)






sf

Switching model



Feedforward control

PREFIoREF

gC(z)

Vg Vo

1

６－６＄

1

６－６＄


fs

PREF

Io = Io + Io

Io











10MW (full load)


1M (Hz)

Duty cycle 50







216 (VDC)


400 (VDC)


1 2





10 (mF)



(ie 119891푖푛푡 = 2119891푠)16k (Hz)


1M (Hz)

Duty cycle 50







(A)

96

955

95

945

94

97

96

95

94

93

96

95

94

93

92

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1



755

75

745

74


76

755

75

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1


775

77

765

755

75

76

78

(A)

t (Sec)093 094 095 096 097 098 099 1



6 7 8 9 10x 106

0

minus01

minus02


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

05

04

03

02

minus02

minus06

0

minus04

06

02

minus02

04

0

08

minus08

minus1minus09

minus11

1

06

08

04

a0(PREF)

a0(PREF)|PolyFit

a2(PREF)

a2(PREF)|PolyFit

a4(PREF)

a4(PREF)|PolyFit

a1(PREF)

a1(PREF)|PolyFit

a3(PREF)

a3(PREF)|PolyFit

a5(PREF)

a5(PREF)|PolyFit

0

1

2 3

4

5


6 7 8 9 10x 106

0

minus005

minus01


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

minus3

minus2

minus1

minus35

minus3

minus25

b0(PREF)

b0(PREF)|PolyFit

b2(PREF)

b2(PREF)|PolyFit

b4(PREF)

b4(PREF)|PolyFit

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

1

05

0

5

3

4

2

1

05

0

b1(PREF)

b1(PREF)|PolyFit

b3(PREF)

b3(PREF)|PolyFit

b5(PREF)

b5(PREF)|PolyFit

b0

b1

b2

b3

b4

b5



Io

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(A)


Io(A)

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25



SRC (Fig 2)

+

VMVDC

iturbVMVDCh

VMVDC0

Perturbation


DAux

+Vturb

minus

minus




9 Conclusion



Time [s]


098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]


Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]


1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]



Appendix






V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



0

0

0

Kthevent

(K+1)thevent

= =

ＣＬ(ＭＮ)


ＭＮ

ＭＮ

ＭＮ

ＥＥ

Ｅ

ＥＥ

Ｅ

６Ｌ(ＭＮ)



ＭＮ1(Ｅ+1)

ＭＮ2(Ｅ+1)ＭＮ0(Ｅ+2)

|


-

+

-

+


ir(t)

vg(t)=Vg0(k)=

N1 N2

+ vTank(t) -

Lr Crir(t)

rsquog(t)

= N2N1 times VLVDC


-

+

-

+


vg(t)=Vg1(k)=0

ir(t)N1 N2

+ vTank(t) -

Lr Crir(t)

rsquog(t)


-

+

-

+


vg(t)=Vg0(k+1)

+ vTank(t) -

N1 N2Lr Crir(t)ir

(t)

rsquog(t)

=-

(c) t0(k+1) le t le t1(k+1) (T2 T3 ON)


vg(t)=Vg1(k+1)=0

N1 N2

-

++ vTank(t) -

Lr Crir(t)

-

+

ir(t)

rsquog(t)

(d) t1(k+1) le t le t2(k+1) (D2 T4 ON)




PREF1

６－６＄

IoREFＡ＝(Ｍ)

fs

Vg

Ａ1(Ｍ)

Ａ2(Ｍ)

Ａ3(Ｍ)

Io

Vo

-

Ｌ



= [ 푟

V푐푟

] = [12

] = [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(3)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(4)


표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(5)


[1198921 (s) 1198922 (s) 1198923 (s)] = 119862 (119878119868 minus 119860)minus1 119861 + 119863 (6)

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(7)




120572 = ( 12119891푠

minus 12119891푟

) 2120587119891푠 = 120587 minus 120587119891푟

119891푠 (8)

where

119891푟 = 12120587radic119871푟119862푟

(9)


120572 + = 120587 minus 120587119891푟

(119891푠 + 푠) (10)


01Hz 10Hz 10Hz 100Hz 1000Hz 10000Hz

0 dB

minus20 dB

minus40 dB

20 dB

40 dB

60 dBfp1

fz

fp2

fc

Crossover frequency

-20 dBdecade

-40 dBdecade

f (Hz)

MAG (dB)

minus60 dB

T0dB

|T(1+T)|

|T|



= minus 120587119891푟

푠 (11)



푅퐸퐹

+ 11989221 + 119892푐 (minus120587119891푟) 1198921

푔

+ 11989231 + 119892푐 (minus120587119891푟) 1198921

표

(12)


표 (s) = 1119881푀푉퐷퐶

1198791 + 119879푅퐸퐹 + 11989221 + 119879푔 + 11989231 + 119879표 (13)

with a loop gain

119879 (s) = 119892푐

minus120587119891푟

1198921 (14)




표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉0=0

= 11989221 + 119879 (15)



표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

= 11989231 + 119879 (16)


표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0

= 1119881푀푉퐷퐶

1198791 + 119879 (17)


119868표119875푅퐸퐹


(18)


119879 (s) = 119892푐

minus120587119891푟

1198921

= 1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

(19)




표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0


(20)




표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0푉0=0

asymp 1198922119879 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

asymp 1198923119879

(21)



1003816100381610038161003816푂푃

= 119891푟minus120587sdot 11198921

1003816100381610038161003816푂푃

1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃target

(22)


119891푐 = 01119891푠 (23)

119891푝1 = 145119891푐 (24)





119879표 = ( 119891푐119891푝1


(27)








119892푐 (z) = 11988651199115 + 11988641199114 + 11988631199113 + 11988621199112 + 1198861119911 + 119886011988751199115 + 11988741199114 + 11988731199113 + 11988721199112 + 1198871119911 + 1198870

(28)






sf

Switching model



Feedforward control

PREFIoREF

gC(z)

Vg Vo

1

６－６＄

1

６－６＄


fs

PREF

Io = Io + Io

Io











10MW (full load)


1M (Hz)

Duty cycle 50







216 (VDC)


400 (VDC)


1 2





10 (mF)



(ie 119891푖푛푡 = 2119891푠)16k (Hz)


1M (Hz)

Duty cycle 50







(A)

96

955

95

945

94

97

96

95

94

93

96

95

94

93

92

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1



755

75

745

74


76

755

75

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1


775

77

765

755

75

76

78

(A)

t (Sec)093 094 095 096 097 098 099 1



6 7 8 9 10x 106

0

minus01

minus02


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

05

04

03

02

minus02

minus06

0

minus04

06

02

minus02

04

0

08

minus08

minus1minus09

minus11

1

06

08

04

a0(PREF)

a0(PREF)|PolyFit

a2(PREF)

a2(PREF)|PolyFit

a4(PREF)

a4(PREF)|PolyFit

a1(PREF)

a1(PREF)|PolyFit

a3(PREF)

a3(PREF)|PolyFit

a5(PREF)

a5(PREF)|PolyFit

0

1

2 3

4

5


6 7 8 9 10x 106

0

minus005

minus01


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

minus3

minus2

minus1

minus35

minus3

minus25

b0(PREF)

b0(PREF)|PolyFit

b2(PREF)

b2(PREF)|PolyFit

b4(PREF)

b4(PREF)|PolyFit

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

1

05

0

5

3

4

2

1

05

0

b1(PREF)

b1(PREF)|PolyFit

b3(PREF)

b3(PREF)|PolyFit

b5(PREF)

b5(PREF)|PolyFit

b0

b1

b2

b3

b4

b5



Io

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(A)


Io(A)

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25



SRC (Fig 2)

+

VMVDC

iturbVMVDCh

VMVDC0

Perturbation


DAux

+Vturb

minus

minus




9 Conclusion



Time [s]


098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]


Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]


1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]



Appendix






V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018




PREF1

６－６＄

IoREFＡ＝(Ｍ)

fs

Vg

Ａ1(Ｍ)

Ａ2(Ｍ)

Ａ3(Ｍ)

Io

Vo

-

Ｌ



= [ 푟

V푐푟

] = [12

] = [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(3)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(4)


표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(5)


[1198921 (s) 1198922 (s) 1198923 (s)] = 119862 (119878119868 minus 119860)minus1 119861 + 119863 (6)

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(7)




120572 = ( 12119891푠

minus 12119891푟

) 2120587119891푠 = 120587 minus 120587119891푟

119891푠 (8)

where

119891푟 = 12120587radic119871푟119862푟

(9)


120572 + = 120587 minus 120587119891푟

(119891푠 + 푠) (10)


01Hz 10Hz 10Hz 100Hz 1000Hz 10000Hz

0 dB

minus20 dB

minus40 dB

20 dB

40 dB

60 dBfp1

fz

fp2

fc

Crossover frequency

-20 dBdecade

-40 dBdecade

f (Hz)

MAG (dB)

minus60 dB

T0dB

|T(1+T)|

|T|



= minus 120587119891푟

푠 (11)



푅퐸퐹

+ 11989221 + 119892푐 (minus120587119891푟) 1198921

푔

+ 11989231 + 119892푐 (minus120587119891푟) 1198921

표

(12)


표 (s) = 1119881푀푉퐷퐶

1198791 + 119879푅퐸퐹 + 11989221 + 119879푔 + 11989231 + 119879표 (13)

with a loop gain

119879 (s) = 119892푐

minus120587119891푟

1198921 (14)




표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉0=0

= 11989221 + 119879 (15)



표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

= 11989231 + 119879 (16)


표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0

= 1119881푀푉퐷퐶

1198791 + 119879 (17)


119868표119875푅퐸퐹


(18)


119879 (s) = 119892푐

minus120587119891푟

1198921

= 1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

(19)




표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0


(20)




표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0푉0=0

asymp 1198922119879 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

asymp 1198923119879

(21)



1003816100381610038161003816푂푃

= 119891푟minus120587sdot 11198921

1003816100381610038161003816푂푃

1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃target

(22)


119891푐 = 01119891푠 (23)

119891푝1 = 145119891푐 (24)





119879표 = ( 119891푐119891푝1


(27)








119892푐 (z) = 11988651199115 + 11988641199114 + 11988631199113 + 11988621199112 + 1198861119911 + 119886011988751199115 + 11988741199114 + 11988731199113 + 11988721199112 + 1198871119911 + 1198870

(28)






sf

Switching model



Feedforward control

PREFIoREF

gC(z)

Vg Vo

1

６－６＄

1

６－６＄


fs

PREF

Io = Io + Io

Io











10MW (full load)


1M (Hz)

Duty cycle 50







216 (VDC)


400 (VDC)


1 2





10 (mF)



(ie 119891푖푛푡 = 2119891푠)16k (Hz)


1M (Hz)

Duty cycle 50







(A)

96

955

95

945

94

97

96

95

94

93

96

95

94

93

92

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1



755

75

745

74


76

755

75

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1


775

77

765

755

75

76

78

(A)

t (Sec)093 094 095 096 097 098 099 1



6 7 8 9 10x 106

0

minus01

minus02


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

05

04

03

02

minus02

minus06

0

minus04

06

02

minus02

04

0

08

minus08

minus1minus09

minus11

1

06

08

04

a0(PREF)

a0(PREF)|PolyFit

a2(PREF)

a2(PREF)|PolyFit

a4(PREF)

a4(PREF)|PolyFit

a1(PREF)

a1(PREF)|PolyFit

a3(PREF)

a3(PREF)|PolyFit

a5(PREF)

a5(PREF)|PolyFit

0

1

2 3

4

5


6 7 8 9 10x 106

0

minus005

minus01


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

minus3

minus2

minus1

minus35

minus3

minus25

b0(PREF)

b0(PREF)|PolyFit

b2(PREF)

b2(PREF)|PolyFit

b4(PREF)

b4(PREF)|PolyFit

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

1

05

0

5

3

4

2

1

05

0

b1(PREF)

b1(PREF)|PolyFit

b3(PREF)

b3(PREF)|PolyFit

b5(PREF)

b5(PREF)|PolyFit

b0

b1

b2

b3

b4

b5



Io

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(A)


Io(A)

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25



SRC (Fig 2)

+

VMVDC

iturbVMVDCh

VMVDC0

Perturbation


DAux

+Vturb

minus

minus




9 Conclusion



Time [s]


098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]


Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]


1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]



Appendix






V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



01Hz 10Hz 10Hz 100Hz 1000Hz 10000Hz

0 dB

minus20 dB

minus40 dB

20 dB

40 dB

60 dBfp1

fz

fp2

fc

Crossover frequency

-20 dBdecade

-40 dBdecade

f (Hz)

MAG (dB)

minus60 dB

T0dB

|T(1+T)|

|T|



= minus 120587119891푟

푠 (11)



푅퐸퐹

+ 11989221 + 119892푐 (minus120587119891푟) 1198921

푔

+ 11989231 + 119892푐 (minus120587119891푟) 1198921

표

(12)


표 (s) = 1119881푀푉퐷퐶

1198791 + 119879푅퐸퐹 + 11989221 + 119879푔 + 11989231 + 119879표 (13)

with a loop gain

119879 (s) = 119892푐

minus120587119891푟

1198921 (14)




표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉0=0

= 11989221 + 119879 (15)



표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

= 11989231 + 119879 (16)


표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0

= 1119881푀푉퐷퐶

1198791 + 119879 (17)


119868표119875푅퐸퐹


(18)


119879 (s) = 119892푐

minus120587119891푟

1198921

= 1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

(19)




표 (s)푅퐸퐹

100381610038161003816100381610038161003816100381610038161003816푉119892=0푉119900=0


(20)




표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0푉0=0

asymp 1198922119879 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

asymp 1198923119879

(21)



1003816100381610038161003816푂푃

= 119891푟minus120587sdot 11198921

1003816100381610038161003816푂푃

1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃target

(22)


119891푐 = 01119891푠 (23)

119891푝1 = 145119891푐 (24)





119879표 = ( 119891푐119891푝1


(27)








119892푐 (z) = 11988651199115 + 11988641199114 + 11988631199113 + 11988621199112 + 1198861119911 + 119886011988751199115 + 11988741199114 + 11988731199113 + 11988721199112 + 1198871119911 + 1198870

(28)






sf

Switching model



Feedforward control

PREFIoREF

gC(z)

Vg Vo

1

６－６＄

1

６－６＄


fs

PREF

Io = Io + Io

Io











10MW (full load)


1M (Hz)

Duty cycle 50







216 (VDC)


400 (VDC)


1 2





10 (mF)



(ie 119891푖푛푡 = 2119891푠)16k (Hz)


1M (Hz)

Duty cycle 50







(A)

96

955

95

945

94

97

96

95

94

93

96

95

94

93

92

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1



755

75

745

74


76

755

75

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1


775

77

765

755

75

76

78

(A)

t (Sec)093 094 095 096 097 098 099 1



6 7 8 9 10x 106

0

minus01

minus02


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

05

04

03

02

minus02

minus06

0

minus04

06

02

minus02

04

0

08

minus08

minus1minus09

minus11

1

06

08

04

a0(PREF)

a0(PREF)|PolyFit

a2(PREF)

a2(PREF)|PolyFit

a4(PREF)

a4(PREF)|PolyFit

a1(PREF)

a1(PREF)|PolyFit

a3(PREF)

a3(PREF)|PolyFit

a5(PREF)

a5(PREF)|PolyFit

0

1

2 3

4

5


6 7 8 9 10x 106

0

minus005

minus01


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

minus3

minus2

minus1

minus35

minus3

minus25

b0(PREF)

b0(PREF)|PolyFit

b2(PREF)

b2(PREF)|PolyFit

b4(PREF)

b4(PREF)|PolyFit

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

1

05

0

5

3

4

2

1

05

0

b1(PREF)

b1(PREF)|PolyFit

b3(PREF)

b3(PREF)|PolyFit

b5(PREF)

b5(PREF)|PolyFit

b0

b1

b2

b3

b4

b5



Io

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(A)


Io(A)

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25



SRC (Fig 2)

+

VMVDC

iturbVMVDCh

VMVDC0

Perturbation


DAux

+Vturb

minus

minus




9 Conclusion



Time [s]


098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]


Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]


1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]



Appendix






V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018





표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0푉0=0

asymp 1198922119879 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816푃119877119864119865=0

푉119892=0

asymp 1198923119879

(21)



1003816100381610038161003816푂푃

= 119891푟minus120587sdot 11198921

1003816100381610038161003816푂푃

1198790 times (1 + 119904120596푧)(1 + 119904119876120596푝1 + (119904119876120596푝1)2) (1 + 119904120596푝2)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃target

(22)


119891푐 = 01119891푠 (23)

119891푝1 = 145119891푐 (24)





119879표 = ( 119891푐119891푝1


(27)








119892푐 (z) = 11988651199115 + 11988641199114 + 11988631199113 + 11988621199112 + 1198861119911 + 119886011988751199115 + 11988741199114 + 11988731199113 + 11988721199112 + 1198871119911 + 1198870

(28)






sf

Switching model



Feedforward control

PREFIoREF

gC(z)

Vg Vo

1

６－６＄

1

６－６＄


fs

PREF

Io = Io + Io

Io











10MW (full load)


1M (Hz)

Duty cycle 50







216 (VDC)


400 (VDC)


1 2





10 (mF)



(ie 119891푖푛푡 = 2119891푠)16k (Hz)


1M (Hz)

Duty cycle 50







(A)

96

955

95

945

94

97

96

95

94

93

96

95

94

93

92

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1



755

75

745

74


76

755

75

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1


775

77

765

755

75

76

78

(A)

t (Sec)093 094 095 096 097 098 099 1



6 7 8 9 10x 106

0

minus01

minus02


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

05

04

03

02

minus02

minus06

0

minus04

06

02

minus02

04

0

08

minus08

minus1minus09

minus11

1

06

08

04

a0(PREF)

a0(PREF)|PolyFit

a2(PREF)

a2(PREF)|PolyFit

a4(PREF)

a4(PREF)|PolyFit

a1(PREF)

a1(PREF)|PolyFit

a3(PREF)

a3(PREF)|PolyFit

a5(PREF)

a5(PREF)|PolyFit

0

1

2 3

4

5


6 7 8 9 10x 106

0

minus005

minus01


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

minus3

minus2

minus1

minus35

minus3

minus25

b0(PREF)

b0(PREF)|PolyFit

b2(PREF)

b2(PREF)|PolyFit

b4(PREF)

b4(PREF)|PolyFit

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

1

05

0

5

3

4

2

1

05

0

b1(PREF)

b1(PREF)|PolyFit

b3(PREF)

b3(PREF)|PolyFit

b5(PREF)

b5(PREF)|PolyFit

b0

b1

b2

b3

b4

b5



Io

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(A)


Io(A)

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25



SRC (Fig 2)

+

VMVDC

iturbVMVDCh

VMVDC0

Perturbation


DAux

+Vturb

minus

minus




9 Conclusion



Time [s]


098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]


Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]


1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]



Appendix






V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



sf

Switching model



Feedforward control

PREFIoREF

gC(z)

Vg Vo

1

６－６＄

1

６－６＄


fs

PREF

Io = Io + Io

Io











10MW (full load)


1M (Hz)

Duty cycle 50







216 (VDC)


400 (VDC)


1 2





10 (mF)



(ie 119891푖푛푡 = 2119891푠)16k (Hz)


1M (Hz)

Duty cycle 50







(A)

96

955

95

945

94

97

96

95

94

93

96

95

94

93

92

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1



755

75

745

74


76

755

75

(A)

t (Sec)093 094 095 096 097 098 099 1

(A)

t (Sec)093 094 095 096 097 098 099 1


775

77

765

755

75

76

78

(A)

t (Sec)093 094 095 096 097 098 099 1



6 7 8 9 10x 106

0

minus01

minus02


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

05

04

03

02

minus02

minus06

0

minus04

06

02

minus02

04

0

08

minus08

minus1minus09

minus11

1

06

08

04

a0(PREF)

a0(PREF)|PolyFit

a2(PREF)

a2(PREF)|PolyFit

a4(PREF)

a4(PREF)|PolyFit

a1(PREF)

a1(PREF)|PolyFit

a3(PREF)

a3(PREF)|PolyFit

a5(PREF)

a5(PREF)|PolyFit

0

1

2 3

4

5


6 7 8 9 10x 106

0

minus005

minus01


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

minus3

minus2

minus1

minus35

minus3

minus25

b0(PREF)

b0(PREF)|PolyFit

b2(PREF)

b2(PREF)|PolyFit

b4(PREF)

b4(PREF)|PolyFit

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

1

05

0

5

3

4

2

1

05

0

b1(PREF)

b1(PREF)|PolyFit

b3(PREF)

b3(PREF)|PolyFit

b5(PREF)

b5(PREF)|PolyFit

b0

b1

b2

b3

b4

b5



Io

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(A)


Io(A)

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25



SRC (Fig 2)

+

VMVDC

iturbVMVDCh

VMVDC0

Perturbation


DAux

+Vturb

minus

minus




9 Conclusion



Time [s]


098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]


Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]


1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]



Appendix






V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018





(A)

96

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t (Sec)093 094 095 096 097 098 099 1

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t (Sec)093 094 095 096 097 098 099 1

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t (Sec)093 094 095 096 097 098 099 1



6 7 8 9 10x 106

0

minus01

minus02


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

05

04

03

02

minus02

minus06

0

minus04

06

02

minus02

04

0

08

minus08

minus1minus09

minus11

1

06

08

04

a0(PREF)

a0(PREF)|PolyFit

a2(PREF)

a2(PREF)|PolyFit

a4(PREF)

a4(PREF)|PolyFit

a1(PREF)

a1(PREF)|PolyFit

a3(PREF)

a3(PREF)|PolyFit

a5(PREF)

a5(PREF)|PolyFit

0

1

2 3

4

5


6 7 8 9 10x 106

0

minus005

minus01


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

minus3

minus2

minus1

minus35

minus3

minus25

b0(PREF)

b0(PREF)|PolyFit

b2(PREF)

b2(PREF)|PolyFit

b4(PREF)

b4(PREF)|PolyFit

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

1

05

0

5

3

4

2

1

05

0

b1(PREF)

b1(PREF)|PolyFit

b3(PREF)

b3(PREF)|PolyFit

b5(PREF)

b5(PREF)|PolyFit

b0

b1

b2

b3

b4

b5



Io

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(A)


Io(A)

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25



SRC (Fig 2)

+

VMVDC

iturbVMVDCh

VMVDC0

Perturbation


DAux

+Vturb

minus

minus




9 Conclusion



Time [s]


098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]


Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]


1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]



Appendix






V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



6 7 8 9 10x 106

0

minus01

minus02


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

05

04

03

02

minus02

minus06

0

minus04

06

02

minus02

04

0

08

minus08

minus1minus09

minus11

1

06

08

04

a0(PREF)

a0(PREF)|PolyFit

a2(PREF)

a2(PREF)|PolyFit

a4(PREF)

a4(PREF)|PolyFit

a1(PREF)

a1(PREF)|PolyFit

a3(PREF)

a3(PREF)|PolyFit

a5(PREF)

a5(PREF)|PolyFit

0

1

2 3

4

5


6 7 8 9 10x 106

0

minus005

minus01


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

minus3

minus2

minus1

minus35

minus3

minus25

b0(PREF)

b0(PREF)|PolyFit

b2(PREF)

b2(PREF)|PolyFit

b4(PREF)

b4(PREF)|PolyFit

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

6 7 8 9 10x 106


PREF (W)

1

05

0

5

3

4

2

1

05

0

b1(PREF)

b1(PREF)|PolyFit

b3(PREF)

b3(PREF)|PolyFit

b5(PREF)

b5(PREF)|PolyFit

b0

b1

b2

b3

b4

b5



Io

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(A)


Io(A)

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25



SRC (Fig 2)

+

VMVDC

iturbVMVDCh

VMVDC0

Perturbation


DAux

+Vturb

minus

minus




9 Conclusion



Time [s]


098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]


Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]


1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]



Appendix






V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



Io

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25

(A)


Io(A)

(sec)0

10

20

30

40

50

60

70

80

90

100

110

05 10 15 20 25



SRC (Fig 2)

+

VMVDC

iturbVMVDCh

VMVDC0

Perturbation


DAux

+Vturb

minus

minus




9 Conclusion



Time [s]


098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]


Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]


1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]



Appendix






V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



Time [s]


098 1 102 104 106 108

0

05

1

15

2ＣＮＯＬ＜

[A]


Time [s]09 094 098 104 106 108


1

15

2

25

3

092 096 102105

ＣＮＯＬ＜

[A]


1 105 11 115 12 125


13

15

19

2

18

17

16

14

Time [s]

ＣＮＯＬ＜

[A]



Appendix






V푔 = 119871푟

119889119894푟119889119905 + V퐶푟 + V표


119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



119894푟 = 119862푟

119889V퐶푟119889119905(A1)

where

V푔 = 119881푔0(푘)V표 = 119881표0(푘)

(A2)


119894푟 = 1119885푟


(A3)


(A4)

where

119885푟 = radic119871푟119862푟

120596푟 = 1radic119871푟119862푟

(A5)


119868푟1(푘) = 119894푟 (t1(푘))= 1119885푟


(A6)

where

120596푟t1(푘) = 120596푟120596푠

120573퐾 = 120596푟푠 sdot 120573퐾 for t0(푘) = 0 (A7)


0 lt (120596푟푠120573퐾) le 120587t1(푘) = 120573퐾120596푠

(A9)


sdot cos (120596푟푠120573퐾) + 119868푟0(푘)119885푟


(A10)


119894푟 (t耠) = minus1119885푟

(119881표1(푘) + 119881퐶푟1(k)) sin (120596푟t耠) (A11)

where



where


(A14)



+ [minus 1119885푟


+ [ 2119885푟



sdot sin (120596푟푠120572퐾) + 1119885푟


(A15)



(A16)

where





(A18)



119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018




119868푟 = 119868푟0(푘) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A19)

119881퐶푟 = 119881퐶푟0(k) = 119891 (119881표0(푘) 119881푔0(푘) 120573퐾 120572푘) (A20)



119894표 = 1120574푘

int훽119896

0119894표푢푡Rec (120579푠) 119889120579푠 + 1120574푘

int훾119896

훽119896

119894표푢푡Rec (120579푠) 119889120579푠

= 1120574푘

sdot 1120596푟푠

sin (120596푟푠120573푘) + 1120596푟푠



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟



1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


sdot 1120596푟푠

1119885푟

[1 minus cos (120596푟푠120573푘)]+ 1120596푟푠

1119885푟


(A21)

where

120579푠 = 120596푠119905120572푘 = 120574푘 minus 120573푘

(A22)



119868표


(A23)


1199091(푘) = 119868푟0(푘)1199092(푘) = 119881퐶푟0(k)


(A24)


= 120596푠120574푘

(119909푖(k+1) minus 119909푖(푘)) (A25)

where



sdot1199091(k) = 120596푠120574푘


+ 120596푠120574푘

sdot [ 1119885푟


+ 120596푠120574푘


sdot sin (120596푟푠120572퐾) + 1119885푟


sdot [ 1119885푟



= 1198911 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198911 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast1 1199091 1199092 V표 V푔 120572

(A27)


sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



sdot1199092(k) = 120596푠120574푘





= 1198912 1199091(푘) 1199092(푘) 119881표0(푘) 119881푔0(푘) 120572푘= 1198912 1199091 1199092 V표 V푔 120572 = 120596푠120574푘

sdot 119891lowast2 1199091 1199092 V표 V푔 120572

(A28)




(A29)



+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast1 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A30)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A31)

and thensdot[1199091 + 1] = 1198911 1199091 1199092 119881표 119881푔 120572 + 12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989111205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A32)


119874119875 = 119868푟 119881퐶푟 119881표 119881푔 1205721198911 1199091 1199092 119881표 119881푔 120572 = 119868푟 = 0 (A33)


+ 푔 120572 + = 120596푠120574푘

sdot 119891lowast2 1199091 + 1 1199092 + 2 119881표

+ 표 119881푔 + 푔 120572 + (A34)

where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 +

(A35)

and thensdot[1199092 + 2] = 1198912 1199091 1199092 119881표 119881푔 120572 + 12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔

+ 1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표 + 120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 120597211989121205971199091

2

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A36)

where

1198912 1199091 1199092 119881표 119881푔 120572 = sdot119881퐶푟= 0 (A37)


119868표 + 표 = 119891표푢푡 1199091 + 1 1199092 + 2 119881표 + 표 119881푔

+ 푔 120572 + = 1(120574 + ) sdot 119891lowast표푢푡 1199091 + 1 1199092

+ 2 119881표 + 표 119881푔 + 푔 120572 + (A38)


where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



where

1199091 = 1199091 + 11199092 = 1199092 + 2V푔 = 119881푔 + 푔V표 = 119881표 + 표120572 = 120572 + = +

(A39)

and then

119868표 + 표 = 119891표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891표푢푡1205971199091

10038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891표푢푡1205971199092

10038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891표푢푡12059712057210038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃



표푢푡 1199091 1199092 119881표 119881푔 120572 + 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

1

+ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

2 + 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

푔 + 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

표

+ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃

+ 12 1205972119891lowast표푢푡12059711990912

100381610038161003816100381610038161003816100381610038161003816푂푃

12 + sdot sdot sdot

(A40)

where


표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 1

+ [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 2

+ [ 120597119891lowast표푢푡120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] sdot 푔

+ [ 120597119891lowast표푢푡120597V표

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

] sdot 표

+ [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


(A41)



[ 11990911199092

] = [[[[

12059711989111205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989111205971199092

10038161003816100381610038161003816100381610038161003816푂푃12059711989121205971199091

10038161003816100381610038161003816100381610038161003816푂푃

12059711989121205971199092

10038161003816100381610038161003816100381610038161003816푂푃

]]]]

[12

] + [[[[[

120597119891112059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198911120597V표

10038161003816100381610038161003816100381610038161003816푂푃120597119891212059712057210038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

1205971198912120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]]]]]

[[[[

푔

표

]]]]

표 = [1120574 sdot [ 120597119891lowast표푢푡1205971199091

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891lowast표푢푡1205971199092

100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

] ][12

]

+ [1120574 sdot [ 120597119891lowast표푢푡120597120572

100381610038161003816100381610038161003816100381610038161003816푂푃


1003816100381610038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

] 1120574 sdot [ 120597119891표푢푡120597V표

10038161003816100381610038161003816100381610038161003816푂푃

minus 119868표 sdot 120597120573120597V표

10038161003816100381610038161003816100381610038161003816푂푃

]][[[[

푔

표

]]]]

(A42)


120597119891푖1205971199096

= ( 1205971205971199096

120596푠120574 ) sdot 119891lowast푖

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

+ 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A43)

where

120574 = 120572 + 120573 for 119894 = 1 2 119895 = 1 2 (A44)


120596푠12057410038161003816100381610038161003816100381610038161003816푂푃

= 0119891lowast

푖1003816100381610038161003816푂푃 = 0

(A45)


Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018



Therefore

120597119891푖1205971199096

= 120596푠120574 120597119891lowast푖1205971199096

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

(A46)





1205731003816100381610038161003816푂푃 = 120587120596푟푠

+ 1120596푟푠



1205971205731205971199091

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠


]]100381610038161003816100381610038161003816100381610038161003816100381610038161003816푂푃

= 1120596푟푠


1205971205731205971199092

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V0

10038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠



= 1120596푟푠



120597120573120597V푔

1003816100381610038161003816100381610038161003816100381610038161003816푂푃

= [[

1120596푟푠

sdot 1199091119885푟


= 1120596푟푠

sdot 119868푟119885푟


(A49)



표 (s) = [1198921 (s) 1198922 (s) 1198923 (s)] [[[[

푔

표

]]]]

(A50)

where

1198921 (s) = 표 (s) (s)100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

1198922 (s) = 표 (s)푔 (s)

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

1198923 (s) = 표 (s)0 (s)

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119891푠 = (s) sdot 119891푟minus120587

(A51)


= [ 푟

V푐푟

] = [12

]

= [119892푥푢11 119892푥푢12 119892푥푢13119892푥푢21 119892푥푢22 119892푥푢23

][[[[

푔

표

]]]]

(A52)

where

119892푥푢11 = 푟100381610038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢12 = 푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢13 = 푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

119892푥푢21 = V푐푟10038161003816100381610038161003816100381610038161003816푉119892(s)=0

푉0(s)=0

119892푥푢22 = V푐푟푔

1003816100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉0(s)=0

119892푥푢23 = V푐푟표

100381610038161003816100381610038161003816100381610038161003816훼(s)=0푉119892(s)=0

(A53)



Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018




Nomenclature




Data Availability




References

























Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018














Advances in



Journal of


Renewable Energy



EnergyJournal of









Shock and Vibration




RotatingMachinery








Volume 2018


Documents

Aalborg Universitet Model-based control design of Series ...Xiongfei; Imbaquingo, Carlos Enrique; Sarrá, Eduard; Isernia, Nicola; Tonellotto, Alberto Published in: Journal of Renewable