Output Voltage Control of MMC-Based Microgrid Based on...

Research ArticleOutput Voltage Control of MMC-Based Microgrid Based onVoltage Fluctuation Compensation Sliding Mode Control

Sheng Xue Xinggui Wang and Xiaoying Li

School of Electrical and Information Engineering Lanzhou University of Technology Lanzhou China

Correspondence should be addressed to Sheng Xue xueshenglut126com and Xinggui Wang wangxg689126com

Received 17 July 2020 Revised 31 July 2020 Accepted 1 August 2020 Published 20 August 2020

Academic Editor Yang Li

Copyright copy 2020 Sheng Xue et al -is 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

As a novel topology of microgrid the output voltage control of MMC half bridge series microgrid (MMC-MG) is rarely studied Inthis paper on the basis of fully analyzing the mechanism of output voltage fluctuation of MMC-MG under the condition ofislanded mode a control strategy of a hybrid energy storage system is proposed to reduce the generating module (GM) DC-linkvoltage fluctuation caused by the randomness of renewable energy microsource output power Moreover in order to furtherimprove the stabilization of the MMC-MG output voltage and meet the requirements of fast voltage recovery and antijamming asliding mode controller is designed -en a voltage fluctuation compensation controller is designed to suppress the DCcomponent and fundamental frequency deviation of system output voltage caused by GM DC-link voltage fluctuation -eproposed control approach is validated against simulations usingMMC-MGmodels with 4-GM per arm-e results show that theproposed hybrid energy storage control strategy can suppress the GMDC-link voltage fluctuation the slidingmode controller canstabilize the system output voltage when the load drastic changes and the fluctuation compensation strategy can suppress the DCcomponent and the fundamental frequency deviation of system output voltage

1 Introduction

Microgrid (MG) is an autonomous system that can achieveself-control protection and management which includesdistributed generation (DG) load energy storage devicesand monitoring and protection devices [1ndash3] For the iso-lated ac MG the system output voltage loses the power gridsupport and effective control methods are needed tomaintain the frequency and amplitude of the system outputvoltage within the standard range [4ndash6] -e P-Q controlalgorithm is adopted for renewable energy source (RES)inverters to improve the output power utilization and theV-f control algorithm is adopted for nonrenewable energysource inverters to ensure the stabilization of system voltageand frequency [7 8]

A novel MG topology based on modular multilevelconverter half-bridge series structure (MMC-MG) is pro-posed in [9] and the islanded system output characteristicsis analyzed Different from themulti-inverter parallel acMGthe output voltage of MMC-MG is superposed by the output

voltage of each generating module (GM) Affected by therandomness of the microsource (MS) output power the DC-link voltage of GM fluctuates which leads to the amplitudefluctuation of MMC-MG output voltage and the increase ofharmonic -erefore in order to realize the output voltagestable control of MMC-MG it is necessary to realize the GMDC-link voltage stable control and then realize the outputvoltage closed-loop control of the system inverter

-e hybrid energy storage system (HESS) composed ofthe supercapacitor (SC) and battery can be used in MG Itplays an important role in stability control power qualityimprovement and uninterrupted power supply of MG [10]A new chance constrained programming-based schedulingmodel is proposed in [11] which makes full use of ESS toprovide spinning reserve services for isolated MG A mul-tiagent sliding mode control strategy for state of chargebalancing between distributed DC-microgrid battery energystorage systems is proposed in [12] A supervisory powermanagement system for a hybrid MG is proposed in [13]when MG in the islanded mode the HESS can effectively

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9404259 11 pageshttpsdoiorg10115520209404259

suppress the frequency and voltage fluctuation of thesystem For the photovoltaic (PV) MG a control strategyof virtual synchronous generator (VSG) which combinesenergy storage and photovoltaic technology is proposedin [14] to stabilize the fluctuation of PV power andimprove the frequency stability Fang et al [15] proposeda batterySC HESS to implement VSGs which is used toachieve frequency regulation for RES in islanded MG Aconverter topology to interface a RES with a HESS in aMG is investigated in [16] and the proposed controlstrategy can improve the power quality and stabilityHowever the research works mentioned above are mainlyconcentrated on HESS control for the AC MG withmultiple MS inverters connected in parallel while theHESS for MMC-MG is seldom studied because of itsseries structure

For the problem of output voltage closed-loop control ofthe MS inverter a robust virtual inertia control for islandedMG is proposed in [17] which enhancing the robust per-formance and stability of the MG during contingencies Anovel framework of coordinated voltage and frequencycontrol strategy for islanded MG is studied in [18] theproposed control strategy improves the voltage and fre-quency regulation transient response and MG stability Acontrol scheme for distributed energy resource based onsliding mode control (SMC) is proposed in [19] whichprovides fast and stable control on the voltage and frequencyof islanded MG A higher order SMC strategy is studied forislanded MG to chattering alleviation and improve thesystem stability in [20] However the method studied in theabove literature is only applicable to ordinary MG withparallel inverters not to MMC-MG with microsource in-verters in series

A control scheme of output and circulating current ofMMC using sliding mode control (SMC) is proposed in [21]while suppressing second harmonics contents in circulatingcurrent For the MMC system the voltage of each sub-module is supported by the system common DC busHowever for MMC-MG there is no common DC-busvoltage supported and the voltage of each GM is generatedby MS Due to its series structure the voltage deviation ofeach GM will cause the system output voltage contain DCcomponent and fundamental deviation which need to besuppressed

However the previous research on SMC for MMC wasaimed at the control of output current and circulatingcurrent rather than the output voltage control of the islandsystem And its control algorithm cannot eliminate theinfluence of GM voltage deviation on the output voltage ofMMC-MG

In this paper in order to solve the problem of outputvoltage control of MMC-MG in the islanded mode a HESScontrol strategy suitable for GM DC-link voltage control isproposed To further stabilize the system output voltage thefeedback linearization is used to decouple the system pre-cisely and the SMC strategy based on voltage fluctuationcompensation(VFC) is designed for the decoupled systemSimulation results of an MMC-MG system with 4-GM perarm is obtained to verify the effectiveness of the proposed

strategy -e main contributions of this paper are listed asfollows

(1) -e MMC-MG studied in this paper is a new type ofMG which has the advantages of higher outputvoltage levels and less harmonics than the ordinaryMG with multiple inverters in parallel

(2) A SMC strategy for MMC-MG output voltagecontrol is proposed whichmeets the requirements ofislanding system for fast voltage recovery and anti-interference antijamming

(3) -e controller of GM voltage fluctuation compen-sation is designed to reduce the influence of GMvoltage fluctuation on MMC-MG output voltage

-e rest of this paper is organized as follows Section 2analyses the voltage fluctuation factors of MMC-MG outputac side and GM DC link Section 3 focuses on the controlstrategy of HESS Section 4 presents the modeling anddecoupling of the system inverter including topologymodels and mathematic models and state feedback line-arization decoupled Section 5 presents design of the systemoutput controller based on SMC Simulation of the proposedarchitecture and control strategy is shown in Section 6Conclusions are given in Section 7

2 MMC-MG Output Voltage Analysis

-eMMC-MG topology is shown in Figure 1 where the GMis a power generation unit composed of MS ACDC or DCDC converter HESS and half-bridge converter (HC)

-e MS in GM can be categorized into renewable energysources such as wind turbine and PV and nonrenewableenergy sources such as diesel generator andmicrogas turbinegenerator And the HESS is only equipped in GM with RESWith GM as the basic module the three-phase inverter isconstructed according to MMC structure connected to theac bus through the filter then connected to the public powergrid with a static transfer switch (STS) and the local load isconnected to the ac bus-e system is composed of two armsz (z isin p n) in each phase X (X isin A B C) Each arm isconstituted by N series-connected GMs with a series in-ductor L

Compared with the parallel inverters ac MG the MMC-MG series structure inverter makes the stable of frequencyhigher-emultilevel output voltage of the system is formedby the superposition of several GM output voltages whichreduces the pollution of harmonics to the power grid At thesame output voltage level the operating voltage of the MSinverter is reduced then save the investments of powerelectronic equipment Increasing GM can improve thesystem output voltage level and provide a new grid structurefor low and medium voltage MG

21 Inverter Output Voltage Analysis Ignoring the highharmonics the output voltage uAB of the inverters in theislanded mode under the phase-shifted PWM (PSC-PWM)can be given by

2 Mathematical Problems in Engineering

uAB uZAB + uSAB minus

3

radicNulowastdc

2M sin ω0t +

π6

1113874 1113875 (1)

where uZAB is the output DC component which satisfies

uZAB 14ΔupA minus ΔunA + ΔunB minus ΔupB1113960 1113961 (2)

where uSAB is the fundamental frequency deviation which isformed by superimposing two sine waves with the samefrequency and different amplitude of the line voltage fun-damental component which satisfies

uSAB M

4ΔunB + ΔupB1113872 1113873sin ω0t minus

2π3

1113874 11138751113876

minus ΔunA + ΔupA1113872 1113873sinω0t1113961

(3)

whereM (0leMle 1) denotes the modulation depth ω0 is theangular frequency of the output voltage ulowastdc is the voltagereference value of GM DC-link ΔupXi

ΔunXi(i 1 2 N)

are the deviation of GMDC-link voltage and ulowastdc in each armwhen MMC-MG operation and ΔupX 1113936

Ni1 ΔupXi

ΔunX 1113936Ni1 ΔunXi

According to (1)ndash(3) the voltage deviation of each GM

DC link is superimposed on each other resulting in theoutput voltage of the MMC-MG including DC componentand fundamental frequency deviation

22 GM DC-Link Voltage Fluctuation Analysis As shownin Figure 2 the ith GM internal currents of phase A positivearm are used for analysis ig is the output current of MS ic isthe charge current of capacitor C im is the input current ofHC ipA is the arm current and upAi

is the output voltage of

HC When GM is switched on im ipA and upAi udc When

GM is switched off im 0 and upAi 0

MS output power PM and GM output power PG can bemodeled as follows

PM udcig

PG upAiipA udcim

(4)

-e system is cascaded structure GM is not directlyconnected to the load so PG is called equivalent load power-en the GM internal power satisfies

PM minus PG udcCdudc

dt PC (5)

where PC is capacitiveC absorption powerWithin t to t+Δtthe variation of GM DC-link voltage satisfies

Δudc

2C

1113946t+Δt

tPM minus PG( 1113857dt

1113971

(6)

To keep the GM DC-link voltage stable Δudc must to be0 that means MS output power should be follow the

Power grid PCC

STSAC BUS

Filter Load

C

HC

HESS

C

HC

HESS

DC

DC

DC

AC

C

HC

HESS HESS

DC

AC

GMPA1 GMNA1

C

HC

DC

AC

GMNANGMPAN

AB

C

L L

Figure 1 MMC-MG topology structure

Con

vert

er

MS

HC

igic

ipA

upAi

im

udc C

+

+

ndash ndash

Figure 2 Schematic diagram of GM

Mathematical Problems in Engineering 3

equivalent load power variation While the output power ofMS such as wind turbine and PV are random it is difficult tomaintain the stable of the GM DC-link voltage by means ofcontrol So it is necessary to add HESS in GM to dynam-ically adjust the output power and keep voltage stable

3 GM DC-Link Voltage Control

Different from the full-bridge MS inverter used in ordinaryAC MG the GM uses a half-bridge converter During theswitching process im is affected by the arm current whichcontains high harmonic content and AC component

31 GM Internal Power Analysis -e HESS is composed ofbatteries and SC and connected to GMDC link in parallel bybidirectional DCDC converters -is topology can give fullplay to their advantages in energy storage Battery has theadvantages of high energy density which is suitable forregulating low-frequency power fluctuation SC has theadvantages of high power density and fast charge-dischargeresponse which is suitable for regulating high-frequencypower fluctuation [22] -e structure of HESS is shown inFigure 3 where usc and isc are the SC voltage and outputcurrent ib is the battery output current udc is GM DC-linkvoltage and BC1 and BC2 are the bidirectional DCDCconverters

When GM is switched on the internal power is satisfied

PM minus PG minus PF PC (7)

where PF is the absorption energy of HESS which is used tosuppress the GM DC-link voltage fluctuation caused byfluctuation of MS output power and arm current When theDC-link voltage is stable PC 0 and (7) should be rewrittenas follows

PM minus PG PF (8)

PF is caused by the MS output power fluctuation and theequivalent load change When the HESS is used to smoothpower fluctuation PF is divided into a low-frequency fluc-tuation PL and a high-frequency fluctuation PH -e SCmainly absorb or release the PH that can reducing the re-sponse frequency of the battery to the PH then reducing thenumber of charges and discharges of battery and improvingHESS service life PL and PH can be expressed as follows

PL PF1

τ1s + 1

PH PF minus PL

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)

where s is the differential operator 1(τ1s + 1) is a low-passfilter (LPF) and τ1 is a time constant

32Hybrid Energy Storage SystemControl -e SC operatingvoltage is divided into the following five intervals [23]

(1) High voltage over-limit zone (HOZ) uscgt uscminusmax

(2) High voltage limit zone (HLZ) uscminushlt usclt uscminusmax

(3) Normal working zone (NZ) uscminusllt usclt uscminush(4) Low voltage limit zone (LLZ) uscminusminlt usclt uscminusl(5) Low voltage over-limit zone (LOZ) usclt uscminusmin

uscminusmax and uscminusmin are the maximum voltage andminimum voltage of SC and uscminush and uscminusl are the upper-limit value and lower-limit value of the SC voltage controllerIn order to prevent overcharge and discharge of the SC it isnecessary to correct the charge and discharge poweraccording to the voltage -e power correction value ΔPsc inthe five working intervals is shown in Table 1

-e battery bidirectional DCDC converter operates in asingle-ended regulated mode to ensure that the GMDC-linkvoltage is stable and the power correction value of battery isminusΔPsc -e control diagram of the HESS is shown in Fig-ure 4 and PC is the power correction controller based onTable 1

4 Modeling and Decoupling of Inverter

41 Mathematical Model of System Inverter As showN inFigure 5 the equivalent circuit diagram of the MMC-MGinverter in the islanded mode is used for analysis

In Figure 5 Lf is the filter inductor Cf is the filter ca-pacitor Lam is the arm equivalent inductor and Lam L2 iXis the phase current uX is the output voltage of phase X uoXis the filter capacitor voltage Rload is the load and R is thearm equivalent resistance

According to KCL (Kirchhoffrsquos current law) and KVL(Kirchhoffrsquos voltage law) the voltage-to-current equation ofthe inverter in dq0 coordinate can be expressed as follows

ud R

2id + Lm

did

dtminus ω0Lmiq + uod

id Cf

duod

dtminus ω0Cfuq + iod

uq R

2iq + Lm

diq

dt+ ω0Lmid + uoq

iq Cf

duoq

dt+ ω0Cfud + ioq

(10)

where ud and id are the active components of the phasevoltage and current and uq and iq are the reactive compo-nents of the phase voltage and current Similarly uod iod anduoq ioq are the active and reactive components of the

BC1 BC2Battery

ib

L1

V1

V2

C1 C2

V3

V4

L2

isc

uscudc

SCndash

+

ndash

+

Figure 3 Structure diagram of HESS


capacitor voltage and load current Lm is the sum of the Lfand Lam and ω0 is the voltage angular frequency

42 State Feedback LinearizationDecoupled -e voltage-to-current equations under dq0 coordinate of the inverter canrewritten as follows

d

dt

id

iq

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

minusR

2Lm

ω0

minusω0minusR

2Lm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

id

iq

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ +

ud minus uod

Lm

uq minus uoq

Lm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)

In order to improve the decoupling accuracy the statefeedback linearization (SFL) method is used for systemdecoupling Take state variables as x [id iq]T input variablesas u [ud uq]T and output variables as y [h1(x) h2(x)]Twhere h1(x) id h2(x) iq -e state equation of affine 2-input 2-output nonlinear system is obtained as follows

_x f(x) + g1(x)u1 + g2(x)u2

y1 h1(x)

y2 h2(x)

⎧⎪⎪⎨

⎪⎪⎩(12)

where g1(x) 1Lm

01113890 1113891 g2(x) 0

1Lm

1113890 1113891 and

f(x) minusR2Lmx1 + ω0x2 minus uodLm

minusω0x1 minus R2Lmx2 minus uoqLm1113890 1113891

-e solvable condition of the system full-state feedbacklinearization problem is that the system has relative degreeρ1 ρ2 and satisfies ρ1 + ρ2 n where n is the dimension ofthe system variables According to the relative degreedefinition there is any point x0 of the state variable xsatisfied

A x0

1113872 1113873 Lg1

L0fh1 x

01113872 1113873 Lg2

L0fh1 x

01113872 1113873

Lg1L0fh2 x

01113872 1113873 Lg2

L0fh2 x

01113872 1113873

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (13)

where L0fh(x) h(x) Lgj

hi(x) is the Lie derivative of h(x) forg(x) and Lgj

hi(x) zhi(x)zxgj(x) (i j 1 2)-en matrix A is

A x0

1113872 1113873 1Lm 0

0 1Lm

1113890 1113891 (14)

Based on (14) the relative degree of system ρ1 ρ2 1and ρ1 + ρ2 n 2 -e system satisfies the conditions ofexact linearization Exact linearization decoupling of thesystem constructing a new system input variable v there arematrix A(x) b(x) and the input variable v satisfies

v b(x) + A(x)u (15)

where b(x) Lfh1(x)

Lfh2(x)1113890 1113891

minusR2Lmx1 + ω0x2 minus uoqLm


and A(x) Lg1

h1(x) Lg2h1(x)

Lg1h2(x) Lg2

h2(x)1113890 1113891

1Lm 00 1Lm

1113890 1113891

-en the relationship between new input variables andoutput variables is satisfied

v1

v21113890 1113891 b(x) + A(x)u

_y1

_y21113890 1113891 (16)

It can be seen from (16) that the system is decoupled intoa first-order linear system -en the system becomes atypical tracking system and the control target is to track thegiven values ilowastd and ilowastq of the system output variables id andiq

Table 1 Power correction value ΔPsc

Intervals ΔPsc (PHgt 0) ΔPsc (PHlt 0)HOZ 0 minusPHHLZ usc minus uscminushuscminusmax minus uscminushPH uscminush minus uscuscminusmax minus uscminushPHNZ 0 0LLZ usc minus uscminusluscminusl minus uscminusminPH uscminusl minus uscuscminusl minus uscminusminPHLOZ minusPH 0

+ +

+ndash

ndash

ndash

+

+

PWM PI

SC control f ( )ndash1

ΔPscPlowast

scilowastsc

isc

uscudc

11 + τ1sig

im

PF PL

PHPC

(a)

ndash+

++ndash

f ( )ndash1

ilowastbulowast

dc ib

udc

PI

PI

ndashΔPsc

PWM

Battery control

(b)

Figure 4 Block diagram of energy storage converter control


5 Design of System Output Controller

In order to ensure the stable operation of MMC-MG in theislanded mode the double-closed-loop control of voltageand current is adopted on the inverter to provide voltage andfrequency support -e outer voltage loop determines thereference value of the instruction current and stabilizes theAC side voltage of the inverter -e inner current loopcontrols the current according to the instruction current torealize the fast tracking

51 SlidingMode Controller Design Firstly an outer voltageloop controller is designed to determine ilowastd and ilowastq -enthe SMC is adopted in the current inner loop controller toresist the influence of parameter perturbation and externaldisturbance on the feedback linearization model and im-prove the robustness of the system Define the systemtracking error as follows

e1

e21113890 1113891

ylowast1 minus y1

ylowast2 minus y2

⎡⎣ ⎤⎦ ilowastd minus id

ilowastq minus iq

⎡⎣ ⎤⎦ (17)

-e control effect of the sliding mode controller is re-lated to the selection of the sliding surface -e traditionalnonlinear integral sliding surface is

si ci1ei minus ci2 1113946t

0eidτ (18)

In order to improve the robustness of the system thefunction f(t) is introduced to form the global integral slidingsurface so that the initial state of the system is on the slidingsurface eliminating the arrival process -e system globalintegral sliding surface is


0eidτ minus fi(t) (19)

where fi(t) ei(0)eiminuspt pgt 0 and ci1 and ci2 are the coefficientof SMC

According to the reaching law of the sliding mode inorder to reduce the system chattering the reaching law is

_si minuskisi minus εisat si( 1113857 (20)

where sat(s) is a saturation function ki and εi are thereaching law coefficients and kigt 0 and εigt 0 -e existenceof the boundary layer Δ makes sat(s) satisfy

sat(s)

1 sgtΔ

ks |s|leΔ

minus1 sgtΔ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

k 1Δ

(21)

Derivation of (19) is

_si ci1 _ei minus ci2ei + ei(0)peminuspt

i (22)

Combining (17) (20) and (22) we obtain

vi ci2ei minus kisi minus εisat si( 1113857 minus ei(0)pe

minuspti1113960 1113961

ci1 (23)

In this system the input variables can be written asfollows

v1 k1s1 + ε1sat s1( 1113857 + e1(0)pe

minuspt1 minus c12e11113960 1113961

c11

v2 k2s2 + ε2sat s2( 1113857 + e2(0)pe


c21

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)

Combining (24) and (16) the output control amountafter system feedback linearization can be written as follows

upA upB

RRloadLf

Cf

O

Lam

uoA uoB uoC

R

uC

uB

uA iB

iC

iA

+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

R

R R R

+

ndashupC

unA unB unC

Figure 5 System equivalent circuit in the islanded mode


ud uod + Lmv1 + Rid2 minus ω0Lmiq

uq uoq + Lmv2 + Riq2 + ω0Lmid

⎧⎨

⎩ (25)

52 Voltage Fluctuation Compensation Controller DesignAffected by the arm current there is still a small deviation inthe GM DC-link voltage under the control of HESS -esuperposition of the deviations makes the output voltage ofthe system include DC and fundamental frequency deviationcomponents Firstly the influence of the deviation under theclosed-loop on the system control is analyzed Let ΔupA aand the rest be 0 Based on (2) and (3) the output voltagedeviation caused by a is superimposed on the uod and uoqafter dq0 transformation can be given by

uzd 1

23

radic a sin α +π3

1113874 1113875

uzq 1

23

radic a sin α minusπ6

1113874 1113875

usd minus3 +

3

radic

24aM sin 2α

usq

3

radic

8aM sin 2α +

π3

1113874 1113875 minus316

aM

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

It means that the DC and fundamental frequency de-viation components are converted to fundamental frequencycomponent and double-frequency component by dq0transformation

-erefore it is necessary to modify the SMC controllerand suppress ΔupX and ΔunX influences on the outputvoltage by voltage fluctuation compensation (VFC) con-troller -e VFC controller is shown in Figure 6Δud and Δuq are the corrections then the actual output

of the controller is

udminusref ud + Δud

uqminusref uq + Δuq

⎧⎨

⎩ (27)

Combining (17) (25) and (27) we can obtain the systemoutput voltage control block diagram as shown in Figure 7

6 Results and Discussion

In order to validate the effectiveness of the HESS controlstrategy and MMC-MG output voltage control strategy asimulation mode of MMC-MG was built in MatlabSimu-link -e MMC-MG configuration is given in Figure 1 -eHESS configuration is given in Figure 3 -e NO of GMs ineach arm is N 4 the arm inductor L 1mH and the HCcapacitor C 4400 μF GM DC-link voltage ulowastdc 160V

61 Simulation Verification of HESS A PV battery is used asthe RES of GM and a SC and battery which are used to forma HESS For comparative analysis the simulation models of

PV DC power generation PV inverter nine-level cascadedH-bridge PV inverter and MMC-MG were established inturn -e PV simulation parameters are shown in Table 2

-e simulation results of PV DC-link voltages withdifferent topologies under different solar incident irradi-ances are shown in Figure 8 a is the output voltage of PVDCpower generation b is the DC-link voltage of PV inverter cis the DC-link voltage of cascaded H-bridge PV inverter andd is the DC-link voltage of GM

It can be seen from the graph that when the solar ra-diation intensity changes at 1 s 2 s 3 s and 4 s the voltage ab and c have the same trend and the amplitude fluctuationare small while voltage amplitude of d fluctuates greatly-is shows that under the same solar radiation intensityconditions due to the series structure of MMC-MG thevoltage fluctuation of GM is large

-e HESS is connected to the GM DC link to suppressthe DC voltage fluctuation caused by the change of solarradiation intensity -en design the energy storage con-verter controller according to Figure 4 Under the HESScontrol the GM DC-link voltages udc is shown in Figure 9-e amplitude fluctuation is small and the output voltage isstable when the solar radiation intensity changes -econtrol strategy can achieve good voltage stability controleffect of GM DC-link

-e battery output current and the power PG-PPV areshown in Figure 10(a) -e SC output current is shown inFigure 10(b) It can be seen that the trend of ib is similar tothat of PG-PPV and the trend of isc is similar to that of high-frequency component of PG-PPV which further reflects thedifference and complementarities of battery and SC inenergy balance Besides the SC is in frequent charge dis-charge state switching while the battery state is relativelystable

62 Simulation Verification of SMC -e HESS is used toensure that the GM DC-link voltage is stable at 160V undervarying solar incident irradiances Design the system outputvoltage controller according to Figure 7 design SMCaccording to (24) and design SFL according to (25) WhenMMC-MG is in the islanded mode the output voltage andcurrent of the system inverter is shown in Figures 11ndash15

Figures 11(a) and 11(b) show the root mean square(RMS) of the line voltage and phase current -e load in-creases at 1 s 2 s and 5 s and decreases at 3 s and 4 s As canbe seen from the graph the line voltage is adjusted rapidlyafter a small fluctuation to maintain the voltage stability andthe output phase current changes with the change of theload

abc

uox dq

abc

dquoq

uod 11 + τ2s

11 + τ2s PIΔuq

PIΔud

ud

ud

uXref

+ndash

ndash

++

++ +

Figure 6 Block diagram of voltage suppression control


Figures 12(a) and 12(b) show the voltage and current ofsystem when the load changes from 14Ω to 10Ω at 2 sWhen the load increases the line voltage amplitude recoversafter a slight fluctuation and the phase current increases

Figure 13 shows the output voltage frequency In case ofsudden load change the output frequency deviation is al-ways within plusmn02Hz which meets the frequency require-ment of the power grid

Design the VFC controller according to Figure 6 -e DCcomponent of line voltage is shown in Figure 14 When theVFC strategy is applied to the system at 05 s the DC com-ponent is suppressed At the moment of load switching from1 s to 5 s the DC component changes abruptly but decreasesrapidly under the control effect-is strategy can achieve goodDC component suppression effect of output line voltage

Fundamental frequency deviation uSAB is the same as thefrequency of fundamental frequency component of uABwith different phase and smaller amplitude In order toverify the suppression effect of VFC strategy on uSAB uSBCand uSCA more intuitively the FFT analysis of uod after dq0transformation of the line voltage before and after the VFCcontrol is carried out

As shown in Figure 15(a) without VFC control thefundamental component uzd caused by the DC componentof the line voltage is 098V and the second harmoniccomponent usd caused by the fundamental frequency de-viation component of the line voltage is 026V As shown inFigure 15(b) with VFC control the amplitude of uzd and usddecreased to 02V and 011V It shows that the VFC strategycan achieve good fundamental frequency deviation sup-pression effect of line voltage

Figure 16 compares the response curves of the currentinner loop using traditional PI control and SMC It can beseen from the graph that under the condition of sudden loadchanges both control strategies can maintain the systemoutput voltage stability but the overshoot and adjustmenttime of the SMC are better than the PI control It can be seenfrom Figures 11 to 15 that under the condition of GM DC-link voltage fluctuation and load mutation the designedSMC controller has good anti-interference effect and canrealize stable control of system output voltage

VFC

VFC

PI

PI

PI

PI

SMC

SFLdq

abc

+

+

+

+

+

++ +

+

+

+

++

θ

ndash

ndash ndash

ndash

ndash

ndash

ndash

ulowast

od

uoq

uodid

iX

iq

ilowastq

ilowastd

uox

Δud

Δuq

θ θ

ω0Cf

ω0Cf

e2

e1

dq

abc

dq

abc

v2

v1

11 + τ2s

11 + τ2s

uXref

ulowast

oq

Figure 7 Block diagram of the system output voltage control

Table 2 PV parameters

Parameter ValuePV maximum power point voltage (V) 48PV maximum power point voltage (A) 102PV open circuit voltage (V) 56PV short circuit current (A) 108Temperature (degC) 25

a

b cd

0

40

80

120

160

200

Am

plitu

de (V

)

454 525 3 35050 151 2Time (s)

Figure 8 PV DC-link voltages

0

40

80

120

160

Am

plitu

de (V

)

21 15 4525 3 35 4 50 05Time (s)

Figure 9 udc under HESS compensates


PGndashPPV

ib

ndash300

0

300

Pow

er (W

)

ndash15

0

15Cu

rren

t (A

)

1 205 25 3 35 4 45 515Time (s)

(a)

ndash8

0

8

Curr

ent (

A)

21 15 25 3 35 4 45 505Time (s)

(b)

Figure 10 Battery output current power PG-PPV and SC output current

U

250

300

350

400

Am

plitu

de (V

)

2 3 4 5 61Time (s)

(a)

2 3 4 5 61Time (s)

0

12

24

36

Am

plitu

de (V

)

(b)

Figure 11 Line voltage and phase current RMS under time-varying load

498

40

502

Freq

uenc

y (H

z)

1 2 3 4 5 60Time (s)

Figure 13 Voltage frequency under time-varying load

ndash30

0

30

Am

plitu

de (V

)

1 2 3 4 5 60Time (s)

Figure 14 DC component of line voltage under time-varying load

ndash600

0

600

Am

plitu

de (V

)

199 201 203 205197Time (s)

(a)

ndash40

0

40

Am

plitu

de (A

)

199 201 203 205197Time (s)

(b)

Figure 12 Line voltage and phase current at the moment of load change


7 Conclusions

In this paper the voltage fluctuation mechanism of GM DC-link caused by the randomness of RES output power is analyzedand the GM DC-link voltage stability control is realizedby HESS control -e sliding mode controller based on GMvoltage fluctuation compensation is designed to realize thestable control of the systemoutput voltage in the islandedmode

-e simulation results show that the HESS controlstrategy can effectively suppress the GM DC-link voltagefluctuation caused by the RES output power variation -esliding mode controller has fast response speed and strongantidisturbance capability than the PI controller Under thecondition of sudden load change the system voltage reg-ulation speed is fast the amplitude change is small and thefrequency fluctuation is plusmn02Hz -e VFC controller caneffectively suppress DC components and fundamental de-viation components of the system output voltage and im-prove the power quality Hence the proposed controlstrategy has a good adaptability to MMC-MG

Nomenclature

MG MicrogridDG Distributed generationRES Renewable energy sourcesGM Generating moduleMS MicrosourcesHESS Hybrid energy storage systemSC SupercapacitorSMC Sliding mode control

VFC Voltage fluctuation compensationHC Half-bridge converterSTS Static transfer switchSFL State feedback linearizationuZAB DC component of uAB (V)uSAB Fundamental deviation of uAB (V)ulowastdc Voltage reference value of GM DC link (V)upXi

unXi ith GM output voltage of positive and negativearm in X phase (V)

ΔupXi Deviation of ith GM DC-link voltage in X phase

positive arm (V)ΔunXi

Deviation of ith GM DC-link voltage in X phasenegative arm (V)

ig Output current of microsource (A)ic Charge current of C (A)im Input current of half-bridge converter (A)ipX inX Positive arm and negative arm currents of X

phase (A)PM PG Microsource output power and GM output

power (A)usc isc SC voltage and output current (V A)ub ib Battery voltage and output current (V A)PF Absorption energy of HESS (W)usc-max Maximum voltage of SC (V)usc-min Minimum voltage of SC (V)usc-h usc-l Upper limit value and lower limit value of SC

voltage controller (V)uX iX Output current and output voltage of phase X (V

A)uoX Filter capacitor voltage of phase X (V)R Arm equivalent resistance (Ω)ud id Active components of the phase voltage and

current (V A)uq iq Reactive components of the phase voltage and

current (V A)uod iod Active components of the capacitor voltage and

load current (V A)uoq ioq Reactive components of the capacitor voltage and

load current (V A)

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Fundamental (50Hz) = 098 THD = 4731

0 200 400 600 800 1000Frequency (Hz)

020406080

100

Am

plitu

de

(a)


0 200 400 600 800 1000Frequency (Hz)

04080

120160

Am

plitu

de

(b)

Figure 15 uod frequency analysis

101 103 1041 105102099Time (s)

370

385

Am

plitu

de (V

)

Inner loop PIInner loop SMC

Figure 16 Control system response curve


Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is research was funded by the National Natural ScienceFoundation of China Grant no 51967011

References

[1] R H Lasseter and P Paigi ldquoMicrogrid a conceptual solu-tionrdquo in Proceedings of the 2004 IEEE 35th Annual PowerElectronics Specialists Conference (IEEE Cat No04CH37551)pp 4285ndash4290 Aachen Germany June 2004

[2] N Hatziargyriou H Asano R Iravani and C MarnayldquoMicrogridsrdquo IEEE Power and Energy Magazine vol 5 no 4pp 78ndash94 2007

[3] E J Ng and R A El-Shatshat ldquoMulti-microgrid controlsystems (MMCS)rdquo in Proceedings of the IEEE PES GeneralMeeting pp 1ndash6 Providence RI USA July 2010

[4] D Wu F Tang T Dragicevic J C Vasquez andJ M Guerrero ldquoA control architecture to coordinate re-newable energy sources and energy storage systems inislanded microgridsrdquo IEEE Transactions on Smart Grid vol 6no 3 pp 1156ndash1166 2015

[5] J W Simpson-Porco Q Shafiee F Dorfler J C VasquezJ M Guerrero and F Bullo ldquoSecondary frequency andvoltage control of islanded microgrids via distributed aver-agingrdquo IEEE Transactions on Industrial Electronics vol 62no 11 pp 7025ndash7038 2015

[6] Y Xu H Sun W Gu Y Xu and Z Li ldquoOptimal distributedcontrol for secondary frequency and voltage regulation in anislanded microgridrdquo IEEE Transactions on Industrial Infor-matics vol 15 no 1 pp 225ndash235 2019

[7] X Tang X Hu N Li W Deng and G Zhang ldquoA novelfrequency and voltage control method for islanded microgridbased on multienergy storagesrdquo IEEE Transactions on SmartGrid vol 7 no 1 pp 410ndash419 2016

[8] S Adhikari and F Li ldquoCoordinated V-f and P-Q control ofsolar photovoltaic generators with MPPT and battery storagein microgridsrdquo IEEE Transactions on Smart Grid vol 5 no 3pp 1270ndash1281 2014

[9] X G Wang S Xue and X Y Li ldquoAnalysis of outputcharacteristics of a microgrid based on modular multilevelconverter half-bridge series structurerdquo Transactions of ChinaElectrotechnical Society vol 34 no 10 pp 2130ndash2140 2019

[10] B V Solanki K Bhattacharya and C A Cantildeizares ldquoAsustainable energy management system for isolated micro-gridsrdquo IEEE Transactions on Sustainable Energy vol 8 no 4pp 1507ndash1517 2017

[11] Y Li Z Yang G Li D Zhao and W Tian ldquoOptimalscheduling of an isolated microgrid with battery storageconsidering load and renewable generation uncertaintiesrdquoIEEE Transactions on Industrial Electronics vol 66 no 2pp 1565ndash1575 2019

[12] T Morstyn A V Savkin B Hredzak and V G AgelidisldquoMulti-agent sliding mode control for state of charge bal-ancing between battery energy storage systems distributed in aDCmicrogridrdquo IEEE Transactions on Smart Grid vol 9 no 5pp 4735ndash4743 2018

[13] S Kotra and M K Mishra ldquoA supervisory power manage-ment system for a hybrid microgrid with HESSrdquo IEEE

Transactions on Industrial Electronics vol 64 no 5pp 3640ndash3649 2017

[14] A Anzalchi M M Pour and A Sarwat ldquoA combinatorialapproach for addressing intermittency and providing inertialresponse in a grid-connected photovoltaic systemrdquo in Pro-ceedings of the 2016 IEEE Power and Energy Society GeneralMeeting (PESGM) pp 1ndash5 Boston MA USA July 2016

[15] J Fang Y Tang H Li and X Li ldquoA batteryultracapacitorhybrid energy storage system for implementing the powermanagement of virtual synchronous generatorsrdquo IEEETransactions on Power Electronics vol 33 no 4 pp 2820ndash2824 2018

[16] Q Tabart I Vechiu A Etxeberria and S Bacha ldquoHybridenergy storage system microgrids integration for powerquality improvement using four-leg three-level NPC inverterand second-order sliding mode controlrdquo IEEE Transactionson Industrial Electronics vol 65 no 1 pp 424ndash435 2018

[17] T Kerdphol F S Rahman Y Mitani M Watanabe andS Kufeoglu ldquoRobust virtual inertia control of an islandedmicrogrid considering high penetration of renewable energyrdquoIEEE Access vol 6 pp 625ndash636 2018

[18] H M Ibrahim M S El Moursi and P-H Huang ldquoAdaptiveroles of islanded microgrid components for voltage andfrequency transient responses enhancementrdquo IEEE Transac-tions on Industrial Informatics vol 11 no 6 pp 1298ndash13122015

[19] M B Delghavi and A Yazdani ldquoSliding-mode control of ACvoltages and currents of dispatchable distributed energy re-sources in master-slave-organized inverter-based micro-gridsrdquo IEEE Transactions on Smart Grid vol 10 no 1pp 980ndash991 2019

[20] M Cucuzzella G P Incremona and A Ferrara ldquoDesign ofrobust higher order sliding mode control for microgridsrdquoIEEE Journal on Emerging and Selected Topics in Circuits andSystems vol 5 no 3 pp 393ndash401 2015

[21] W Uddin K Zeb M A Adil Khan et al ldquoControl of outputand circulating current of modular multilevel converter usinga sliding mode approachrdquo Energies vol 12 no 21 p 40842019

[22] A Khaligh and Z Zhihao Li ldquoBattery ultracapacitor fuel celland hybrid energy storage systems for electric hybrid electricfuel cell and plug-in hybrid electric vehicles state of the artrdquoIEEE Transactions on Vehicular Technology vol 59 no 6pp 2806ndash2814 2010

[23] Y Zhang L Guo H J Jia and C S Wang ldquoAn energymanagement method of hybrid energy storage system basedon smoothing controlrdquo Automation of Electric Power Systemsvol 36 no 16 pp 36ndash41 2012


suppress the frequency and voltage fluctuation of thesystem For the photovoltaic (PV) MG a control strategyof virtual synchronous generator (VSG) which combinesenergy storage and photovoltaic technology is proposedin [14] to stabilize the fluctuation of PV power andimprove the frequency stability Fang et al [15] proposeda batterySC HESS to implement VSGs which is used toachieve frequency regulation for RES in islanded MG Aconverter topology to interface a RES with a HESS in aMG is investigated in [16] and the proposed controlstrategy can improve the power quality and stabilityHowever the research works mentioned above are mainlyconcentrated on HESS control for the AC MG withmultiple MS inverters connected in parallel while theHESS for MMC-MG is seldom studied because of itsseries structure

For the problem of output voltage closed-loop control ofthe MS inverter a robust virtual inertia control for islandedMG is proposed in [17] which enhancing the robust per-formance and stability of the MG during contingencies Anovel framework of coordinated voltage and frequencycontrol strategy for islanded MG is studied in [18] theproposed control strategy improves the voltage and fre-quency regulation transient response and MG stability Acontrol scheme for distributed energy resource based onsliding mode control (SMC) is proposed in [19] whichprovides fast and stable control on the voltage and frequencyof islanded MG A higher order SMC strategy is studied forislanded MG to chattering alleviation and improve thesystem stability in [20] However the method studied in theabove literature is only applicable to ordinary MG withparallel inverters not to MMC-MG with microsource in-verters in series

A control scheme of output and circulating current ofMMC using sliding mode control (SMC) is proposed in [21]while suppressing second harmonics contents in circulatingcurrent For the MMC system the voltage of each sub-module is supported by the system common DC busHowever for MMC-MG there is no common DC-busvoltage supported and the voltage of each GM is generatedby MS Due to its series structure the voltage deviation ofeach GM will cause the system output voltage contain DCcomponent and fundamental deviation which need to besuppressed

However the previous research on SMC for MMC wasaimed at the control of output current and circulatingcurrent rather than the output voltage control of the islandsystem And its control algorithm cannot eliminate theinfluence of GM voltage deviation on the output voltage ofMMC-MG

In this paper in order to solve the problem of outputvoltage control of MMC-MG in the islanded mode a HESScontrol strategy suitable for GM DC-link voltage control isproposed To further stabilize the system output voltage thefeedback linearization is used to decouple the system pre-cisely and the SMC strategy based on voltage fluctuationcompensation(VFC) is designed for the decoupled systemSimulation results of an MMC-MG system with 4-GM perarm is obtained to verify the effectiveness of the proposed

strategy -e main contributions of this paper are listed asfollows

(1) -e MMC-MG studied in this paper is a new type ofMG which has the advantages of higher outputvoltage levels and less harmonics than the ordinaryMG with multiple inverters in parallel

(2) A SMC strategy for MMC-MG output voltagecontrol is proposed whichmeets the requirements ofislanding system for fast voltage recovery and anti-interference antijamming

(3) -e controller of GM voltage fluctuation compen-sation is designed to reduce the influence of GMvoltage fluctuation on MMC-MG output voltage

-e rest of this paper is organized as follows Section 2analyses the voltage fluctuation factors of MMC-MG outputac side and GM DC link Section 3 focuses on the controlstrategy of HESS Section 4 presents the modeling anddecoupling of the system inverter including topologymodels and mathematic models and state feedback line-arization decoupled Section 5 presents design of the systemoutput controller based on SMC Simulation of the proposedarchitecture and control strategy is shown in Section 6Conclusions are given in Section 7

2 MMC-MG Output Voltage Analysis

-eMMC-MG topology is shown in Figure 1 where the GMis a power generation unit composed of MS ACDC or DCDC converter HESS and half-bridge converter (HC)

-e MS in GM can be categorized into renewable energysources such as wind turbine and PV and nonrenewableenergy sources such as diesel generator andmicrogas turbinegenerator And the HESS is only equipped in GM with RESWith GM as the basic module the three-phase inverter isconstructed according to MMC structure connected to theac bus through the filter then connected to the public powergrid with a static transfer switch (STS) and the local load isconnected to the ac bus-e system is composed of two armsz (z isin p n) in each phase X (X isin A B C) Each arm isconstituted by N series-connected GMs with a series in-ductor L

Compared with the parallel inverters ac MG the MMC-MG series structure inverter makes the stable of frequencyhigher-emultilevel output voltage of the system is formedby the superposition of several GM output voltages whichreduces the pollution of harmonics to the power grid At thesame output voltage level the operating voltage of the MSinverter is reduced then save the investments of powerelectronic equipment Increasing GM can improve thesystem output voltage level and provide a new grid structurefor low and medium voltage MG

21 Inverter Output Voltage Analysis Ignoring the highharmonics the output voltage uAB of the inverters in theislanded mode under the phase-shifted PWM (PSC-PWM)can be given by



3

radicNulowastdc

2M sin ω0t +

π6

1113874 1113875 (1)




uSAB M


2π3

1113874 11138751113876


(3)


ΔunXi(i 1 2 N)


Ni1 ΔupXi









PM udcig

PG upAiipA udcim

(4)



dt PC (5)


Δudc

2C

1113946t+Δt


1113971

(6)


Power grid PCC

STSAC BUS

Filter Load

C

HC

HESS

C

HC

HESS

DC

DC

DC

AC

C

HC

HESS HESS

DC

AC

GMPA1 GMNA1

C

HC

DC

AC

GMNANGMPAN

AB

C

L L


Con

vert

er

MS

HC

igic

ipA

upAi

im

udc C

+

+

ndash ndash










PM minus PG PF (8)


PL PF1

τ1s + 1

PH PF minus PL

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)












ud R

2id + Lm

did


id Cf

duod


uq R

2iq + Lm

diq

dt+ ω0Lmid + uoq

iq Cf

duoq

dt+ ω0Cfud + ioq

(10)


BC1 BC2Battery

ib

L1

V1

V2

C1 C2

V3

V4

L2

isc

uscudc

SCndash

+

ndash

+





d

dt

id

iq

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

minusR

2Lm

ω0

minusω0minusR

2Lm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

id

iq

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ +

ud minus uod

Lm

uq minus uoq

Lm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)


_x f(x) + g1(x)u1 + g2(x)u2

y1 h1(x)

y2 h2(x)

⎧⎪⎪⎨

⎪⎪⎩(12)

where g1(x) 1Lm

01113890 1113891 g2(x) 0

1Lm

1113890 1113891 and




A x0

1113872 1113873 Lg1

L0fh1 x

01113872 1113873 Lg2

L0fh1 x

01113872 1113873

Lg1L0fh2 x

01113872 1113873 Lg2

L0fh2 x

01113872 1113873

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (13)




A x0

1113872 1113873 1Lm 0

0 1Lm

1113890 1113891 (14)


v b(x) + A(x)u (15)

where b(x) Lfh1(x)

Lfh2(x)1113890 1113891



and A(x) Lg1

h1(x) Lg2h1(x)

Lg1h2(x) Lg2

h2(x)1113890 1113891

1Lm 00 1Lm

1113890 1113891


v1

v21113890 1113891 b(x) + A(x)u

_y1

_y21113890 1113891 (16)




+ +

+ndash

ndash

ndash

+

+

PWM PI


ΔPscPlowast

scilowastsc

isc

uscudc

11 + τ1sig

im

PF PL

PHPC

(a)

ndash+

++ndash

f ( )ndash1

ilowastbulowast

dc ib

udc

PI

PI

ndashΔPsc

PWM

Battery control

(b)






e1

e21113890 1113891

ylowast1 minus y1

ylowast2 minus y2


ilowastq minus iq

⎡⎣ ⎤⎦ (17)



0eidτ (18)








sat(s)

1 sgtΔ

ks |s|leΔ

minus1 sgtΔ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

k 1Δ

(21)



i (22)



minuspti1113960 1113961

ci1 (23)


v1 k1s1 + ε1sat s1( 1113857 + e1(0)pe


c11

v2 k2s2 + ε2sat s2( 1113857 + e2(0)pe


c21

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


upA upB

RRloadLf

Cf

O

Lam

uoA uoB uoC

R

uC

uB

uA iB

iC

iA

+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

R

R R R

+

ndashupC

unA unB unC





⎧⎨

⎩ (25)


uzd 1

23

radic a sin α +π3

1113874 1113875

uzq 1

23


1113874 1113875

usd minus3 +

3

radic

24aM sin 2α

usq

3

radic

8aM sin 2α +

π3

1113874 1113875 minus316

aM

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)






⎧⎨

⎩ (27)












abc

uox dq

abc

dquoq

uod 11 + τ2s

11 + τ2s PIΔuq

PIΔud

ud

ud

uXref

+ndash

ndash

++

++ +









VFC

VFC

PI

PI

PI

PI

SMC

SFLdq

abc

+

+

+

+

+

++ +

+

+

+

++

θ

ndash

ndash ndash

ndash

ndash

ndash

ndash

ulowast

od

uoq

uodid

iX

iq

ilowastq

ilowastd

uox

Δud

Δuq

θ θ

ω0Cf

ω0Cf

e2

e1

dq

abc

dq

abc

v2

v1

11 + τ2s

11 + τ2s

uXref

ulowast

oq




a

b cd

0

40

80

120

160

200

Am

plitu

de (V

)

454 525 3 35050 151 2Time (s)


0

40

80

120

160

Am

plitu

de (V

)

21 15 4525 3 35 4 50 05Time (s)



PGndashPPV

ib

ndash300

0

300

Pow

er (W

)

ndash15

0

15Cu

rren

t (A

)

1 205 25 3 35 4 45 515Time (s)

(a)

ndash8

0

8

Curr

ent (

A)

21 15 25 3 35 4 45 505Time (s)

(b)


U

250

300

350

400

Am

plitu

de (V

)

2 3 4 5 61Time (s)

(a)

2 3 4 5 61Time (s)

0

12

24

36

Am

plitu

de (V

)

(b)


498

40

502

Freq

uenc

y (H

z)

1 2 3 4 5 60Time (s)


ndash30

0

30

Am

plitu

de (V

)

1 2 3 4 5 60Time (s)


ndash600

0

600

Am

plitu

de (V

)

199 201 203 205197Time (s)

(a)

ndash40

0

40

Am

plitu

de (A

)

199 201 203 205197Time (s)

(b)



7 Conclusions



Nomenclature















load current (V A)

Data Availability



0 200 400 600 800 1000Frequency (Hz)

020406080

100

Am

plitu

de

(a)


0 200 400 600 800 1000Frequency (Hz)

04080

120160

Am

plitu

de

(b)


101 103 1041 105102099Time (s)

370

385

Am

plitu

de (V

)






Acknowledgments


References



























3

radicNulowastdc

2M sin ω0t +

π6

1113874 1113875 (1)




uSAB M


2π3

1113874 11138751113876


(3)


ΔunXi(i 1 2 N)


Ni1 ΔupXi









PM udcig

PG upAiipA udcim

(4)



dt PC (5)


Δudc

2C

1113946t+Δt


1113971

(6)


Power grid PCC

STSAC BUS

Filter Load

C

HC

HESS

C

HC

HESS

DC

DC

DC

AC

C

HC

HESS HESS

DC

AC

GMPA1 GMNA1

C

HC

DC

AC

GMNANGMPAN

AB

C

L L


Con

vert

er

MS

HC

igic

ipA

upAi

im

udc C

+

+

ndash ndash










PM minus PG PF (8)


PL PF1

τ1s + 1

PH PF minus PL

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)












ud R

2id + Lm

did


id Cf

duod


uq R

2iq + Lm

diq

dt+ ω0Lmid + uoq

iq Cf

duoq

dt+ ω0Cfud + ioq

(10)


BC1 BC2Battery

ib

L1

V1

V2

C1 C2

V3

V4

L2

isc

uscudc

SCndash

+

ndash

+





d

dt

id

iq

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

minusR

2Lm

ω0

minusω0minusR

2Lm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

id

iq

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ +

ud minus uod

Lm

uq minus uoq

Lm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)


_x f(x) + g1(x)u1 + g2(x)u2

y1 h1(x)

y2 h2(x)

⎧⎪⎪⎨

⎪⎪⎩(12)

where g1(x) 1Lm

01113890 1113891 g2(x) 0

1Lm

1113890 1113891 and




A x0

1113872 1113873 Lg1

L0fh1 x

01113872 1113873 Lg2

L0fh1 x

01113872 1113873

Lg1L0fh2 x

01113872 1113873 Lg2

L0fh2 x

01113872 1113873

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (13)




A x0

1113872 1113873 1Lm 0

0 1Lm

1113890 1113891 (14)


v b(x) + A(x)u (15)

where b(x) Lfh1(x)

Lfh2(x)1113890 1113891



and A(x) Lg1

h1(x) Lg2h1(x)

Lg1h2(x) Lg2

h2(x)1113890 1113891

1Lm 00 1Lm

1113890 1113891


v1

v21113890 1113891 b(x) + A(x)u

_y1

_y21113890 1113891 (16)




+ +

+ndash

ndash

ndash

+

+

PWM PI


ΔPscPlowast

scilowastsc

isc

uscudc

11 + τ1sig

im

PF PL

PHPC

(a)

ndash+

++ndash

f ( )ndash1

ilowastbulowast

dc ib

udc

PI

PI

ndashΔPsc

PWM

Battery control

(b)






e1

e21113890 1113891

ylowast1 minus y1

ylowast2 minus y2


ilowastq minus iq

⎡⎣ ⎤⎦ (17)



0eidτ (18)








sat(s)

1 sgtΔ

ks |s|leΔ

minus1 sgtΔ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

k 1Δ

(21)



i (22)



minuspti1113960 1113961

ci1 (23)


v1 k1s1 + ε1sat s1( 1113857 + e1(0)pe


c11

v2 k2s2 + ε2sat s2( 1113857 + e2(0)pe


c21

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


upA upB

RRloadLf

Cf

O

Lam

uoA uoB uoC

R

uC

uB

uA iB

iC

iA

+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

R

R R R

+

ndashupC

unA unB unC





⎧⎨

⎩ (25)


uzd 1

23

radic a sin α +π3

1113874 1113875

uzq 1

23


1113874 1113875

usd minus3 +

3

radic

24aM sin 2α

usq

3

radic

8aM sin 2α +

π3

1113874 1113875 minus316

aM

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)






⎧⎨

⎩ (27)












abc

uox dq

abc

dquoq

uod 11 + τ2s

11 + τ2s PIΔuq

PIΔud

ud

ud

uXref

+ndash

ndash

++

++ +









VFC

VFC

PI

PI

PI

PI

SMC

SFLdq

abc

+

+

+

+

+

++ +

+

+

+

++

θ

ndash

ndash ndash

ndash

ndash

ndash

ndash

ulowast

od

uoq

uodid

iX

iq

ilowastq

ilowastd

uox

Δud

Δuq

θ θ

ω0Cf

ω0Cf

e2

e1

dq

abc

dq

abc

v2

v1

11 + τ2s

11 + τ2s

uXref

ulowast

oq




a

b cd

0

40

80

120

160

200

Am

plitu

de (V

)

454 525 3 35050 151 2Time (s)


0

40

80

120

160

Am

plitu

de (V

)

21 15 4525 3 35 4 50 05Time (s)



PGndashPPV

ib

ndash300

0

300

Pow

er (W

)

ndash15

0

15Cu

rren

t (A

)

1 205 25 3 35 4 45 515Time (s)

(a)

ndash8

0

8

Curr

ent (

A)

21 15 25 3 35 4 45 505Time (s)

(b)


U

250

300

350

400

Am

plitu

de (V

)

2 3 4 5 61Time (s)

(a)

2 3 4 5 61Time (s)

0

12

24

36

Am

plitu

de (V

)

(b)


498

40

502

Freq

uenc

y (H

z)

1 2 3 4 5 60Time (s)


ndash30

0

30

Am

plitu

de (V

)

1 2 3 4 5 60Time (s)


ndash600

0

600

Am

plitu

de (V

)

199 201 203 205197Time (s)

(a)

ndash40

0

40

Am

plitu

de (A

)

199 201 203 205197Time (s)

(b)



7 Conclusions



Nomenclature















load current (V A)

Data Availability



0 200 400 600 800 1000Frequency (Hz)

020406080

100

Am

plitu

de

(a)


0 200 400 600 800 1000Frequency (Hz)

04080

120160

Am

plitu

de

(b)


101 103 1041 105102099Time (s)

370

385

Am

plitu

de (V

)






Acknowledgments


References

































PM minus PG PF (8)


PL PF1

τ1s + 1

PH PF minus PL

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)












ud R

2id + Lm

did


id Cf

duod


uq R

2iq + Lm

diq

dt+ ω0Lmid + uoq

iq Cf

duoq

dt+ ω0Cfud + ioq

(10)


BC1 BC2Battery

ib

L1

V1

V2

C1 C2

V3

V4

L2

isc

uscudc

SCndash

+

ndash

+





d

dt

id

iq

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

minusR

2Lm

ω0

minusω0minusR

2Lm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

id

iq

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ +

ud minus uod

Lm

uq minus uoq

Lm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)


_x f(x) + g1(x)u1 + g2(x)u2

y1 h1(x)

y2 h2(x)

⎧⎪⎪⎨

⎪⎪⎩(12)

where g1(x) 1Lm

01113890 1113891 g2(x) 0

1Lm

1113890 1113891 and




A x0

1113872 1113873 Lg1

L0fh1 x

01113872 1113873 Lg2

L0fh1 x

01113872 1113873

Lg1L0fh2 x

01113872 1113873 Lg2

L0fh2 x

01113872 1113873

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (13)




A x0

1113872 1113873 1Lm 0

0 1Lm

1113890 1113891 (14)


v b(x) + A(x)u (15)

where b(x) Lfh1(x)

Lfh2(x)1113890 1113891



and A(x) Lg1

h1(x) Lg2h1(x)

Lg1h2(x) Lg2

h2(x)1113890 1113891

1Lm 00 1Lm

1113890 1113891


v1

v21113890 1113891 b(x) + A(x)u

_y1

_y21113890 1113891 (16)




+ +

+ndash

ndash

ndash

+

+

PWM PI


ΔPscPlowast

scilowastsc

isc

uscudc

11 + τ1sig

im

PF PL

PHPC

(a)

ndash+

++ndash

f ( )ndash1

ilowastbulowast

dc ib

udc

PI

PI

ndashΔPsc

PWM

Battery control

(b)






e1

e21113890 1113891

ylowast1 minus y1

ylowast2 minus y2


ilowastq minus iq

⎡⎣ ⎤⎦ (17)



0eidτ (18)








sat(s)

1 sgtΔ

ks |s|leΔ

minus1 sgtΔ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

k 1Δ

(21)



i (22)



minuspti1113960 1113961

ci1 (23)


v1 k1s1 + ε1sat s1( 1113857 + e1(0)pe


c11

v2 k2s2 + ε2sat s2( 1113857 + e2(0)pe


c21

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


upA upB

RRloadLf

Cf

O

Lam

uoA uoB uoC

R

uC

uB

uA iB

iC

iA

+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

R

R R R

+

ndashupC

unA unB unC





⎧⎨

⎩ (25)


uzd 1

23

radic a sin α +π3

1113874 1113875

uzq 1

23


1113874 1113875

usd minus3 +

3

radic

24aM sin 2α

usq

3

radic

8aM sin 2α +

π3

1113874 1113875 minus316

aM

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)






⎧⎨

⎩ (27)












abc

uox dq

abc

dquoq

uod 11 + τ2s

11 + τ2s PIΔuq

PIΔud

ud

ud

uXref

+ndash

ndash

++

++ +









VFC

VFC

PI

PI

PI

PI

SMC

SFLdq

abc

+

+

+

+

+

++ +

+

+

+

++

θ

ndash

ndash ndash

ndash

ndash

ndash

ndash

ulowast

od

uoq

uodid

iX

iq

ilowastq

ilowastd

uox

Δud

Δuq

θ θ

ω0Cf

ω0Cf

e2

e1

dq

abc

dq

abc

v2

v1

11 + τ2s

11 + τ2s

uXref

ulowast

oq




a

b cd

0

40

80

120

160

200

Am

plitu

de (V

)

454 525 3 35050 151 2Time (s)


0

40

80

120

160

Am

plitu

de (V

)

21 15 4525 3 35 4 50 05Time (s)



PGndashPPV

ib

ndash300

0

300

Pow

er (W

)

ndash15

0

15Cu

rren

t (A

)

1 205 25 3 35 4 45 515Time (s)

(a)

ndash8

0

8

Curr

ent (

A)

21 15 25 3 35 4 45 505Time (s)

(b)


U

250

300

350

400

Am

plitu

de (V

)

2 3 4 5 61Time (s)

(a)

2 3 4 5 61Time (s)

0

12

24

36

Am

plitu

de (V

)

(b)


498

40

502

Freq

uenc

y (H

z)

1 2 3 4 5 60Time (s)


ndash30

0

30

Am

plitu

de (V

)

1 2 3 4 5 60Time (s)


ndash600

0

600

Am

plitu

de (V

)

199 201 203 205197Time (s)

(a)

ndash40

0

40

Am

plitu

de (A

)

199 201 203 205197Time (s)

(b)



7 Conclusions



Nomenclature















load current (V A)

Data Availability



0 200 400 600 800 1000Frequency (Hz)

020406080

100

Am

plitu

de

(a)


0 200 400 600 800 1000Frequency (Hz)

04080

120160

Am

plitu

de

(b)


101 103 1041 105102099Time (s)

370

385

Am

plitu

de (V

)






Acknowledgments


References




























d

dt

id

iq

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

minusR

2Lm

ω0

minusω0minusR

2Lm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

id

iq

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ +

ud minus uod

Lm

uq minus uoq

Lm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)


_x f(x) + g1(x)u1 + g2(x)u2

y1 h1(x)

y2 h2(x)

⎧⎪⎪⎨

⎪⎪⎩(12)

where g1(x) 1Lm

01113890 1113891 g2(x) 0

1Lm

1113890 1113891 and




A x0

1113872 1113873 Lg1

L0fh1 x

01113872 1113873 Lg2

L0fh1 x

01113872 1113873

Lg1L0fh2 x

01113872 1113873 Lg2

L0fh2 x

01113872 1113873

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (13)




A x0

1113872 1113873 1Lm 0

0 1Lm

1113890 1113891 (14)


v b(x) + A(x)u (15)

where b(x) Lfh1(x)

Lfh2(x)1113890 1113891



and A(x) Lg1

h1(x) Lg2h1(x)

Lg1h2(x) Lg2

h2(x)1113890 1113891

1Lm 00 1Lm

1113890 1113891


v1

v21113890 1113891 b(x) + A(x)u

_y1

_y21113890 1113891 (16)




+ +

+ndash

ndash

ndash

+

+

PWM PI


ΔPscPlowast

scilowastsc

isc

uscudc

11 + τ1sig

im

PF PL

PHPC

(a)

ndash+

++ndash

f ( )ndash1

ilowastbulowast

dc ib

udc

PI

PI

ndashΔPsc

PWM

Battery control

(b)






e1

e21113890 1113891

ylowast1 minus y1

ylowast2 minus y2


ilowastq minus iq

⎡⎣ ⎤⎦ (17)



0eidτ (18)








sat(s)

1 sgtΔ

ks |s|leΔ

minus1 sgtΔ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

k 1Δ

(21)



i (22)



minuspti1113960 1113961

ci1 (23)


v1 k1s1 + ε1sat s1( 1113857 + e1(0)pe


c11

v2 k2s2 + ε2sat s2( 1113857 + e2(0)pe


c21

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


upA upB

RRloadLf

Cf

O

Lam

uoA uoB uoC

R

uC

uB

uA iB

iC

iA

+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

R

R R R

+

ndashupC

unA unB unC





⎧⎨

⎩ (25)


uzd 1

23

radic a sin α +π3

1113874 1113875

uzq 1

23


1113874 1113875

usd minus3 +

3

radic

24aM sin 2α

usq

3

radic

8aM sin 2α +

π3

1113874 1113875 minus316

aM

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)






⎧⎨

⎩ (27)












abc

uox dq

abc

dquoq

uod 11 + τ2s

11 + τ2s PIΔuq

PIΔud

ud

ud

uXref

+ndash

ndash

++

++ +









VFC

VFC

PI

PI

PI

PI

SMC

SFLdq

abc

+

+

+

+

+

++ +

+

+

+

++

θ

ndash

ndash ndash

ndash

ndash

ndash

ndash

ulowast

od

uoq

uodid

iX

iq

ilowastq

ilowastd

uox

Δud

Δuq

θ θ

ω0Cf

ω0Cf

e2

e1

dq

abc

dq

abc

v2

v1

11 + τ2s

11 + τ2s

uXref

ulowast

oq




a

b cd

0

40

80

120

160

200

Am

plitu

de (V

)

454 525 3 35050 151 2Time (s)


0

40

80

120

160

Am

plitu

de (V

)

21 15 4525 3 35 4 50 05Time (s)



PGndashPPV

ib

ndash300

0

300

Pow

er (W

)

ndash15

0

15Cu

rren

t (A

)

1 205 25 3 35 4 45 515Time (s)

(a)

ndash8

0

8

Curr

ent (

A)

21 15 25 3 35 4 45 505Time (s)

(b)


U

250

300

350

400

Am

plitu

de (V

)

2 3 4 5 61Time (s)

(a)

2 3 4 5 61Time (s)

0

12

24

36

Am

plitu

de (V

)

(b)


498

40

502

Freq

uenc

y (H

z)

1 2 3 4 5 60Time (s)


ndash30

0

30

Am

plitu

de (V

)

1 2 3 4 5 60Time (s)


ndash600

0

600

Am

plitu

de (V

)

199 201 203 205197Time (s)

(a)

ndash40

0

40

Am

plitu

de (A

)

199 201 203 205197Time (s)

(b)



7 Conclusions



Nomenclature















load current (V A)

Data Availability



0 200 400 600 800 1000Frequency (Hz)

020406080

100

Am

plitu

de

(a)


0 200 400 600 800 1000Frequency (Hz)

04080

120160

Am

plitu

de

(b)


101 103 1041 105102099Time (s)

370

385

Am

plitu

de (V

)






Acknowledgments


References





























e1

e21113890 1113891

ylowast1 minus y1

ylowast2 minus y2


ilowastq minus iq

⎡⎣ ⎤⎦ (17)



0eidτ (18)








sat(s)

1 sgtΔ

ks |s|leΔ

minus1 sgtΔ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

k 1Δ

(21)



i (22)



minuspti1113960 1113961

ci1 (23)


v1 k1s1 + ε1sat s1( 1113857 + e1(0)pe


c11

v2 k2s2 + ε2sat s2( 1113857 + e2(0)pe


c21

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


upA upB

RRloadLf

Cf

O

Lam

uoA uoB uoC

R

uC

uB

uA iB

iC

iA

+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

R

R R R

+

ndashupC

unA unB unC





⎧⎨

⎩ (25)


uzd 1

23

radic a sin α +π3

1113874 1113875

uzq 1

23


1113874 1113875

usd minus3 +

3

radic

24aM sin 2α

usq

3

radic

8aM sin 2α +

π3

1113874 1113875 minus316

aM

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)






⎧⎨

⎩ (27)












abc

uox dq

abc

dquoq

uod 11 + τ2s

11 + τ2s PIΔuq

PIΔud

ud

ud

uXref

+ndash

ndash

++

++ +









VFC

VFC

PI

PI

PI

PI

SMC

SFLdq

abc

+

+

+

+

+

++ +

+

+

+

++

θ

ndash

ndash ndash

ndash

ndash

ndash

ndash

ulowast

od

uoq

uodid

iX

iq

ilowastq

ilowastd

uox

Δud

Δuq

θ θ

ω0Cf

ω0Cf

e2

e1

dq

abc

dq

abc

v2

v1

11 + τ2s

11 + τ2s

uXref

ulowast

oq




a

b cd

0

40

80

120

160

200

Am

plitu

de (V

)

454 525 3 35050 151 2Time (s)


0

40

80

120

160

Am

plitu

de (V

)

21 15 4525 3 35 4 50 05Time (s)



PGndashPPV

ib

ndash300

0

300

Pow

er (W

)

ndash15

0

15Cu

rren

t (A

)

1 205 25 3 35 4 45 515Time (s)

(a)

ndash8

0

8

Curr

ent (

A)

21 15 25 3 35 4 45 505Time (s)

(b)


U

250

300

350

400

Am

plitu

de (V

)

2 3 4 5 61Time (s)

(a)

2 3 4 5 61Time (s)

0

12

24

36

Am

plitu

de (V

)

(b)


498

40

502

Freq

uenc

y (H

z)

1 2 3 4 5 60Time (s)


ndash30

0

30

Am

plitu

de (V

)

1 2 3 4 5 60Time (s)


ndash600

0

600

Am

plitu

de (V

)

199 201 203 205197Time (s)

(a)

ndash40

0

40

Am

plitu

de (A

)

199 201 203 205197Time (s)

(b)



7 Conclusions



Nomenclature















load current (V A)

Data Availability



0 200 400 600 800 1000Frequency (Hz)

020406080

100

Am

plitu

de

(a)


0 200 400 600 800 1000Frequency (Hz)

04080

120160

Am

plitu

de

(b)


101 103 1041 105102099Time (s)

370

385

Am

plitu

de (V

)






Acknowledgments


References




























⎧⎨

⎩ (25)


uzd 1

23

radic a sin α +π3

1113874 1113875

uzq 1

23


1113874 1113875

usd minus3 +

3

radic

24aM sin 2α

usq

3

radic

8aM sin 2α +

π3

1113874 1113875 minus316

aM

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)






⎧⎨

⎩ (27)












abc

uox dq

abc

dquoq

uod 11 + τ2s

11 + τ2s PIΔuq

PIΔud

ud

ud

uXref

+ndash

ndash

++

++ +









VFC

VFC

PI

PI

PI

PI

SMC

SFLdq

abc

+

+

+

+

+

++ +

+

+

+

++

θ

ndash

ndash ndash

ndash

ndash

ndash

ndash

ulowast

od

uoq

uodid

iX

iq

ilowastq

ilowastd

uox

Δud

Δuq

θ θ

ω0Cf

ω0Cf

e2

e1

dq

abc

dq

abc

v2

v1

11 + τ2s

11 + τ2s

uXref

ulowast

oq




a

b cd

0

40

80

120

160

200

Am

plitu

de (V

)

454 525 3 35050 151 2Time (s)


0

40

80

120

160

Am

plitu

de (V

)

21 15 4525 3 35 4 50 05Time (s)



PGndashPPV

ib

ndash300

0

300

Pow

er (W

)

ndash15

0

15Cu

rren

t (A

)

1 205 25 3 35 4 45 515Time (s)

(a)

ndash8

0

8

Curr

ent (

A)

21 15 25 3 35 4 45 505Time (s)

(b)


U

250

300

350

400

Am

plitu

de (V

)

2 3 4 5 61Time (s)

(a)

2 3 4 5 61Time (s)

0

12

24

36

Am

plitu

de (V

)

(b)


498

40

502

Freq

uenc

y (H

z)

1 2 3 4 5 60Time (s)


ndash30

0

30

Am

plitu

de (V

)

1 2 3 4 5 60Time (s)


ndash600

0

600

Am

plitu

de (V

)

199 201 203 205197Time (s)

(a)

ndash40

0

40

Am

plitu

de (A

)

199 201 203 205197Time (s)

(b)



7 Conclusions



Nomenclature















load current (V A)

Data Availability



0 200 400 600 800 1000Frequency (Hz)

020406080

100

Am

plitu

de

(a)


0 200 400 600 800 1000Frequency (Hz)

04080

120160

Am

plitu

de

(b)


101 103 1041 105102099Time (s)

370

385

Am

plitu

de (V

)






Acknowledgments


References
































VFC

VFC

PI

PI

PI

PI

SMC

SFLdq

abc

+

+

+

+

+

++ +

+

+

+

++

θ

ndash

ndash ndash

ndash

ndash

ndash

ndash

ulowast

od

uoq

uodid

iX

iq

ilowastq

ilowastd

uox

Δud

Δuq

θ θ

ω0Cf

ω0Cf

e2

e1

dq

abc

dq

abc

v2

v1

11 + τ2s

11 + τ2s

uXref

ulowast

oq




a

b cd

0

40

80

120

160

200

Am

plitu

de (V

)

454 525 3 35050 151 2Time (s)


0

40

80

120

160

Am

plitu

de (V

)

21 15 4525 3 35 4 50 05Time (s)



PGndashPPV

ib

ndash300

0

300

Pow

er (W

)

ndash15

0

15Cu

rren

t (A

)

1 205 25 3 35 4 45 515Time (s)

(a)

ndash8

0

8

Curr

ent (

A)

21 15 25 3 35 4 45 505Time (s)

(b)


U

250

300

350

400

Am

plitu

de (V

)

2 3 4 5 61Time (s)

(a)

2 3 4 5 61Time (s)

0

12

24

36

Am

plitu

de (V

)

(b)


498

40

502

Freq

uenc

y (H

z)

1 2 3 4 5 60Time (s)


ndash30

0

30

Am

plitu

de (V

)

1 2 3 4 5 60Time (s)


ndash600

0

600

Am

plitu

de (V

)

199 201 203 205197Time (s)

(a)

ndash40

0

40

Am

plitu

de (A

)

199 201 203 205197Time (s)

(b)



7 Conclusions



Nomenclature















load current (V A)

Data Availability



0 200 400 600 800 1000Frequency (Hz)

020406080

100

Am

plitu

de

(a)


0 200 400 600 800 1000Frequency (Hz)

04080

120160

Am

plitu

de

(b)


101 103 1041 105102099Time (s)

370

385

Am

plitu

de (V

)






Acknowledgments


References


























PGndashPPV

ib

ndash300

0

300

Pow

er (W

)

ndash15

0

15Cu

rren

t (A

)

1 205 25 3 35 4 45 515Time (s)

(a)

ndash8

0

8

Curr

ent (

A)

21 15 25 3 35 4 45 505Time (s)

(b)


U

250

300

350

400

Am

plitu

de (V

)

2 3 4 5 61Time (s)

(a)

2 3 4 5 61Time (s)

0

12

24

36

Am

plitu

de (V

)

(b)


498

40

502

Freq

uenc

y (H

z)

1 2 3 4 5 60Time (s)


ndash30

0

30

Am

plitu

de (V

)

1 2 3 4 5 60Time (s)


ndash600

0

600

Am

plitu

de (V

)

199 201 203 205197Time (s)

(a)

ndash40

0

40

Am

plitu

de (A

)

199 201 203 205197Time (s)

(b)



7 Conclusions



Nomenclature















load current (V A)

Data Availability



0 200 400 600 800 1000Frequency (Hz)

020406080

100

Am

plitu

de

(a)


0 200 400 600 800 1000Frequency (Hz)

04080

120160

Am

plitu

de

(b)


101 103 1041 105102099Time (s)

370

385

Am

plitu

de (V

)






Acknowledgments


References


























7 Conclusions



Nomenclature















load current (V A)

Data Availability



0 200 400 600 800 1000Frequency (Hz)

020406080

100

Am

plitu

de

(a)


0 200 400 600 800 1000Frequency (Hz)

04080

120160

Am

plitu

de

(b)


101 103 1041 105102099Time (s)

370

385

Am

plitu

de (V

)






Acknowledgments


References




























Acknowledgments


References


























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