Research ArticleFirst-Order and High-Order Repetitive Control for Single-PhaseGrid-Connected Inverter

Cheng Guo 1 Linzhen Zhong 2 Jun Zhao 2 Guanbin Gao 2 and Yingbo Huang2

1Yunnan Electrical Power Experiment Institute Co Kunming Yunnan 650217 China2Faculty of Mechanical and Electrical Engineering Kunming University of Science and Technology Kunming 650500 China

Correspondence should be addressed to Jun Zhao junzhao1993163com

Received 19 June 2020 Revised 11 July 2020 Accepted 16 July 2020 Published 12 August 2020

Guest Editor Shubo Wang

Copyright copy 2020 Cheng Guo 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

With the increasing demand of users for power sources and quality how to provide high-quality renewable clean energy hasbecome a key issue of power electronics e main idea of this paper is to develop a composite control including a PI control andrepetitive control for a single-phase grid-connected inverter to eliminate the effects of harmonics which can obtain better steady-state and dynamic responses of the single-phase inverter system and reduce the net current harmonics e modelling of a single-phase inverter is first introduced then a first-order repetitive control is developed for the proposed grid-connected inverterMoreover a high-order repetitive controller is adopted to further improve the robustness against the uncertainties in the period ofsignals e stability and performance analysis are given for the first-order repetitive control and high-order repetitive controlFinally comparative simulations are conducted in a circuit-level inverter model which show the effectiveness of theproposed method

1 Introduction

e development of traditional fuel-based energy is be-coming stringent due to its serious pollution problem andother shortcomings while with the rapid development ofhuman economy and society the investigation of clean andhigh-efficiency energy has been gradually expanded [1 2] Infact the exploration of renewable energy and alternativeenergy has been gradually becoming a key research area invarious countries and industries In the fields of powergeneration thermal power is now being gradually replacedby renewable energy such as hydropower photovoltaic andwind power nuclear energy and solar energy [3 4]

However there are some critical issues to be addressed inthe renewable energy powered grid-connected systems Forinstance photovoltaic power generation is greatly affectedby many factors such as weather and region and thus thepower supply is not stable [3 5] In addition due to thecharacteristics of large investment and wide land occupa-tion the centralized photovoltaic grid-connected system haslimited application though the distributed photovoltaic

grid-connected system has more potential applications thatis small-scale photovoltaic grid-connected system How-ever it remains as an open problem to ensure that a largenumber of distributed systems can be successfully connectedto the grid through power electronic methods that is grid-connected inverter [6ndash9] which can retain the stability ofpublic grid while ensuring the efficiency of the grid con-nection and avoiding the interference of large-scale gridconnection to public grid is is of great significance andnecessity not only for the photovoltaic power generationsystem but also for other renewable energy powered systems[5] Among these grid-connected systems the grid-con-nected inverter is the core part [10ndash12] As an interfacedevice the grid-connected inverter used to transform theunstable DC power received by photovoltaic cells into amore stable AC power supply and feed it into the public gridplays an important role which has also attracted significantattentions in the power electronics industry [13ndash15]

e grid-connected inverter is to convert the DC poweroutput extracted from the photovoltaic array into the ACpower with the same frequency and phase as the grid voltage

HindawiComplexityVolume 2020 Article ID 1094386 10 pageshttpsdoiorg10115520201094386

while the output current harmonic should be small and thesinusoidal AC is incorporated into the grid Hence propercontrol strategies are essential for this purposeere are twobasic control strategies of the grid-connected current that isindirect current control and direct current control [16ndash19]Indirect current control uses the vector relationship of theoutput voltage so that the grid-connected current canachieve the expected amplitude and phase without theintroduction of AC current feedback Direct current controlapplies the AC input current command and AC currentfeedback so as to make the input current tracking commandchange through the regulatorrsquos tracking control it has gooddynamic performance In this line there are many controlmethods developed for grid-connected inverter such as PIcontrol hysteresis control deadbeat control robust controlrepetitive control and adaptive control [20ndash26] PI control isone of the commonly used control methods since this al-gorithm is simple and reliable but it cannot realize thecurrent adjustment without static difference It may containoscillations so that the current quality of the network side islimited Hysteresis control is designed to control the error ofthe inverter output current to track a sinusoidal referencecurrent within the hysteresis width which has the advan-tages of real-time control and fast response However thismethod suffers the problems of switching loss and trackingaccuracy Deadbeat control is a method based on the precisemathematical model of the controlled plant e basicprinciple is to calculate the duty cycle of the next switchingcycle of the inverter power device according to the stateequation and the output feedback signal and the requiredreference signal at the next time Deadbeat control has theadvantages of fast response and low harmonic contentHowever it has specific requirements for the calculationaccuracy and speed of the controller

Based on the above discussions we will develop a re-petitive control for a grid-connected inverter Repetitivecontrol is a control method developed based on the internalmodel principle where the basic principle is that the har-monic distortion of the previous cycle will be eliminated inthe next cyclee controller is used to add control signals inthe next cycle for correction and compensation so as toeliminate the periodic interference To this end we firstintroduce the modelling of a single-phase inverter en afirst-order repetitive control is developed for the proposedgrid-connected inverter Moreover a high-order repetitivecontroller is also adopted to further improve the robustnessagainst the uncertainties in the period of signalse stabilityand performance analysis are all given for the proposed twocontrol strategies Finally comparative simulations areconducted in a circuit-level inverter model to show that thehigh-order repetitive control can obtain better steady-stateand dynamic features of the single-phase inverter system andreduce the net current harmonics

e major contributions of this paper include thefollowing

(1) e repetitive control is used to eliminate the totalharmonics in the current of the grid-connectedinverter

(2) A high-order repetitive control is proposed to ad-dress the variation in the period of grid which canguarantee the control system robustness of the grid-connected inverter

(3) e stability and performance analysis for the pro-posed first-order repetitive control and high-orderrepetitive control are all given

is paper is organized as follows In Section 2 weintroduce the single-phase inverter type and modelling InSection 3 a first-order repetitive control and high-orderrepetitive control are introduced based on the proposedgrid-connected inverter to suppress the total harmonics inthe current e stability and performance analysis are alsogiven Comparative simulations are conducted in Section 4and some conclusions are stated in Section 5

2 Preliminaries and Problem Formulation

21 Single-Phase Inverter Type Grid-connected inverter isthe poststage structure of photovoltaic grid which is a deviceto convert DC into AC Inverters are widely used in theelectric traffic military and other fields ere are threemain topologies of single-phase inverter push-pull inverterhalf-bridge inverter and full-bridge inverter [27ndash31]

211 Push-Pull Inverter e push-pull inverter has a simplestructure which consists of two common negative powerswitches and a step-up transformer with a central tap on itsoriginal side e main disadvantage is that the outputtransformer is easy to saturate e utilization ratio of trans-former is relatively low and it is difficult to drive the inductiveload which is suitable for the occasion of low DC bus voltagee topology of push-pull inverter is given in Figure 1

212 Half-Bridge Inverter e half-bridge inverter circuit iscomposed of two capacitors in series a pair of controllabledevices and a bridge arm for antiparallel diodes When thecapacity of the two voltage dividing capacitors is large enoughthe capacitor voltage of the power switch device keeps U d2when being in switching on and off state which has a strongability to resist voltage output imbalance However its disad-vantage is that the AC output voltage amplitude of the maincircuit is half of the input voltage and theDC side still needs thevoltage balance of two capacitors theDC side voltage utilizationis low and the harmonic of the grid current is large Half-bridgeinverter has been widely used in the low power level invertere topology of half-bridge inverter is shown in Figure 2

213 Full-Bridge Inverter e full-bridge inverter circuithas two bridge arms which can be composed of two half-bridge circuitse topology is given in Figure 3 One pair offull-control devices isV1 andV4 and the other pair is V2 andV3 e same pair of full-control devices is on at the sametime and the two pairs of devices are complementary

Under the same DC input voltage the maximum outputvoltage of the full-bridge inverter is twice of that of the half-bridge inverter When the power is the same the output

2 Complexity

current and the current through the switch element are halfof the half-bridge inverter circuit Full-bridge inverter circuitis simple in terms of structure and easy to control thereforeit has been widely used in the high-power occasions

At the same time the inverter is divided into the activeinverter and passive inverter according to whether there is apower supply or not Active and passive refer to whether theinverter is connected to the power supply In this project thephotovoltaic grid-connected inverter needs to be connectedto the grid which belongs to the grid-connected inverter sothat the active inverter circuit is selected

From Figure 3 for the inverter with load its filter circuitis usually composed of an LC filter circuit with inductanceand capacitance For the grid-connected inverter the outputterminal needs to be connected to the grid e filter ca-pacitor C will be clamped by the power grid and cannot playthe role of filtering and the parallel capacitor can be re-moved equivalently which can be seen in Figure 4

is paper mainly focusses on the later stage of the grid-connected inverter system which consists of a DC powersupply provided by the former DC-DC part four powerswitch devices (IGBT) inverter bridge and filter inductore inverter outputs sinusoidal current with the same fre-quency and phase the same as the grid voltage and enters thegrid

22 Modelling of Single-Phase Inverter e main circuit ofthe single-phase grid-connected inverter is shown in Fig-ure 5 which is a voltage type full-bridge circuit composed offour IGBTs and continuous current diodes in reverse par-allel T1 minus T4 is the IGBT VD1 minus VD4 is the continuouscurrent diode Ls is the filter inductance on one side of thepower grid and Rs is the parasitic resistance of the filterinductance Us and Udc are the corresponding AC powersupply and DC power supply respectively

According to Figure 5 we can obtain the flowingequation

Ls

disdt

UAB minus Us minus Rsis (1)

en we have

Is(s)

UAB(s) minus Us(s)

1Rs + Lss

(2)

For the DC side we get

Udc(s)

Idc(s) minus IL(s)

1sCdc

(3)

en we know that the single-phase grid-connectedinverter adopts the double loop control strategy of voltageand current loop and the outer loop is the DC side voltageloop whose function is to keep the DC side voltage stablee inner loop is a current loop the output current of theinverter is in the same phase as the grid voltage and thecurrent amplitude is determined by the output of the voltageloop regulator is paper only studies the inverter functionof the later stage of the grid-connected system therefore thefront end of the inverter bridge can be regarded as a constantvoltage DC power supply while the voltage loop is notconsidered

3 Repetitive Control Design of Single-PhaseGrid-Connected Inverter

Repetitive control [32] is derived from the internal modelprinciple (IMP) e merit of IMP is that in a stable closed-loop control system the internal control includes the

~ACUd

Figure 2 Half-bridge inverter circuit

DC

V1

V2

V3

V4

Figure 3 Full-bridge inverter circuit without AC

DC

+ndash

Figure 1 Push-pull inverter circuit

Complexity 3

generator of external controlled signals so as to realize theadjustment without steady-state error and suppress theperiodic interference When the steady-state error of asystem is zero thus the input of the controller is zero At thistime the reference signal and the feedback signal still existthe periodic disturbance still changes according to theoriginal dynamic characteristics and the controller stilloutputs the corresponding control signal to keep the systemadjusted without static error erefore the essence of re-alizing no steady-state error is that the controller is astructural model which can reflect and process externalsignals

e proposed controller should be designed to suppressthe current harmonics to reduce the distortion of sine waveof current feed into the network However the jammingsignal in the power electronic system is periodic then aparticular control structure shown in Figure 6 will be used

Remark 1 is control system in Figure 6 consists of an RCcontroller and PI controller e PI controller is designed tostabilize the closed-loop system whose parameters can betuned as [33] is paper will focus on the design of

repetitive control (eg FORC and HORC) and theircomparisons

31 First-Order Repetitive Control (FORC)

311 First-Order Repetitive Control Structure As shown inFigure 7 repetitive control can be designed based on aninternal model which can learn a signal of the period T andduplicate it even if the input of this model is set to zeroConsider the frequency-domain response this internalmodel introduces infinite gains (without filter H(s)) at thespecific frequencies ω 2nπT rads foralln isin N+ enaccording to the well-known IMP zero-error tracking orrejection of period signals at these frequencies can beguaranteed if the closed-loop system is stable

A low-pass filter H(s) is usually used to improve therobustness of controlled systems although it may reduce thegains in these specific frequencies H(s) is set as a second-order Butterworth filter in this paper by designing its cut-offfrequency e compensator K(s) is utilized to retain thesystem stability when the internal model is added to theclosed-loop system us the transfer function of RC isdescribed by

RC(s) Z(s)

E(s)

eminus sT

1 minus H(s)eminussT middot K(s) (4)

It is found that the use of low-pass filter H(s) will lead toan unavoidable phase lag which will shift the frequencies atwhich the maximum gains are obtained Specifically thefrequency shift is mainly related to its phase However aproper compensation can be further incorporated to addressthis issue as [34]

312 Stability and Performance Analysis In this part wewill give a proposition regarding the system stability efollowing conditions should be fulfilled to guarantee thestability of the closed-loop system

DC

V1 V3

V2 V4

~AC

Figure 4 Full-bridge inverter circuit with AC

~Udc UAB US

iL

iS

RS

LS

T1 T3

T2 T4

VD1

VD2VD4

VD3

Figure 5 e main circuit of single phase

R (s) +

ndash

E ZRC (s)

PI (s)

Controller D (s)

P (s)Y (s)

Figure 6 Schematic of the proposed control

E

Internal model

+H (s)

K (s)Z

endashsT

Figure 7 Block diagram of FORC

4 Complexity

Proposition 1 6e closed-loop system in Figure 6 withFORC in Figure 7 is stable if the following conditions arefulfilled [35 36]

(1) 6e closed-loop system without RC is stable that isT0(s) PI(s)P(s)(1 + PI(s)P(s)) is stable

(2) H(s)infin le 1 that is the filter should have a gain closeto 1 within the suitable bandwidth

(3) (K(s)T0(s) minus H(s))eminus sTinfin K(s)T0(s)

minusH(s)infinle 1

Proof e proof is similar to that given in [36] and thus isnot presented here

According to these three conditions we have differentdesign methods to satisfy the conditions e PI control isdesigned to fulfill condition (1) a low-pass filter H(s) is usedto fulfill condition (2) and K(s) can be designed to satisfycondition (3) For any minimum-phase plant P(s) a con-structive selection of K(s) is given as

K(s) krH(s)T0(s) (5)

where kr is a constantBy substituting (5) into condition (3) of Proposition 1

we have the following

Corollary 1 6e closed-loop control system shown in Fig-ure 6 with a stable minimum-phase P(s) and compensator (5)is stable if the following conditions are fulfilled

(1) H(s) is a stable system H(s)infin lt 1(2) kr is a con-stant fulfilling |kr minus 1|lt 1H(s)infin

Remark 1 e compensator K(s) of (5) is valid for stableminimum-phase plants because there are no zero-polecancellations in the right-half plane in this case For non-minimum-phase plants an alternative approach is to cancelthe minimum-phase zeroes and poles and to compensate forthe phase of the nonminimum-phase ones as [37]

32 High-Order Repetitive Control (HORC)

321 High-Order Repetitive Control Structure AlthoughFORC in Figure 7 is easy to implement its performance maydegrade when the period T of the reference or disturbancehas uncertainties that is it is sensitive to the variation in theperiod of signals In order to tackle these disadvantageshigh-order repetitive control (HORC) was proposed in[38ndash40] because of its strong robustness which is given inFigure 8

In Figure 8 W(s) is a weighted sum function of re-petitive loops us the transfer function of HORC can bewritten as

HORC(s) W(s)K(s)

1 minus W(s)H(s) (6)

It is shown that HORC uses a multiple loop internalmodel which replaces the delay eminus sT by a weighted sumfunction of several delays given as follows

W(s) 1113944M

m1Wme

minusmsT (7)

where M is the number of delays that is the order of RCloop

Similar to FORC the gains of HORC are infinite at themultiples of the input signal frequency In particular thegain of (6) will tend to infinity if W(s)H(s) 1 As shown in[38] we substitute s jk2πT into (7) and have

W(jk2πT) 1113944M

m1Wme

minus jmk2π 1113944

M

m1Wm 1 (8)

It is noted that for M 1 and W1 1 (8) is reduced toFORC For the case Mgt 1 we need to determine the weightparameters Wm Inspired by the idea of [38] we can makethe first-order derivative of W(s) with respect to T zero atthe specific frequency is imposes the condition

zW(s jk2πT)

zT 1113944

M

m1

minusWmmjk2πT

0 (9)

According to (9) the following equation is true

1113944

M

m1Wmm 0 (10)

As shown in [38] to further decrease the sensitivity forperiod-time variations by using the weight parameters ofHORC we can make (M minus 1)-th derivatives equal zero sothat

1113944

M

m1Wmm

(Mminus 1) 0 (11)

e weight parameters can be calculated based on(7)ndash(11) For example when M 2 and 3 we can set W1 2and W2 minus1 W1 3 W2 minus3 and W3 1 respectively

322 Stability and Performance Analysis e stabilityanalysis of HORC system is similar to that given in Prop-osition 1 and thus we have the following

E

H (s)

K (s) Z

W (s)

endashsT endashsT endashsT

W1

W2

WM

Figure 8 Block diagram of HORC

Complexity 5

Proposition 2 6e closed-loop system in Figure 6 withHORC in Figure 8 is stable if the following conditions hold

(1) 6e closed-loop system without the HORC is stablethat isT0(s) PI(s)P(s)(1 + PI(s)P(s)) is stable

(2) H(s)infin le 1(3) (K(s)T0(s) minus H(s))W(s)infinle 1

Proof e proof is similar to that of Proposition 1 byreplacing eminus sT with W(s)

For any minimum-phase P(s) we design K(s) as

K(s) krH(s)

T0(s) (12)

By substituting (12) into the third condition in Propo-sition 2 we can get (krH(s) minus H(s))W(s)infinle 1 Fur-thermore condition (2) in Corollary 2 can be derived as|kr minus 1|lt 1H(s)W(s)infin Hence we have thefollowing

Corollary 2 For stable minimum-phase P(s) the closed-loop control system shown in Figure 8 with HORC is stable ifthe following conditions are true

(1) H(s)infin lt 1(2) kr is a constant fulfilling |kr minus 1|lt 1H(s)W(s)infin

4 Simulation Results and Analysis

In this section we validate the effectiveness of the proposedcontrol algorithms by using Simpower systems Toolbox inMatlabSimulink e simulation model of a single-phase

grid-connected inverter is built to compare the currentharmonic suppression of FORC and HORC e FFTanalysis tool in Simulink is used to get the total harmonicdistortion (THD) rough the block diagram of currentloop control system in Figure 6 and the topology structure inFigure 5 of the single-phase grid-connected inverter thesimulation diagram of the single-phase inverter can beobtained as Figure 9 In addition the main circuit param-eters are shown in Table 1

emain circuit simulationmodule is composed of a DCpower supply an IGBT inverter bridge a filter inductor anAC source of public power grid and an output currentdetection module e control part consists of a controlcircuit a PWM generator module and a PLL module ecurrent loop is used in the control loop e control moduleincludes a PI control a first-order repetitive control or ahigh-order repetitive control respectively In this paper PIcontrol PI + FORC and PI +HORC are all simulated Inorder to fully analyze the characteristics of these threecontrol methods the steady-state waveform harmonicsuppression and dynamic tracking are all given

41 6e Steady-State Waveform Figures 10ndash12 show thevoltage and current of AC side when the PI controlPI + FORC and PI +HORC reach the steady state respec-tively Figures 13ndash15 are the corresponding harmonic analysisresults It can be seen from Figures 10ndash12 that the threemethods can achieve the same phase of voltage and current inthe steady state However the current harmonic is relativelylarge and the THD reaches 210 in Figure 12 In Figure 11the harmonics are correspondingly less and the THD is

Ud+ndash

gE

c gE

c

gE

cgE

c

P Uref

+ indash

+ vndash

PI (s)

+

+

+

+

ndash

Frist-order repetitive control

ird-order repetitive control

Out1In1

Out1In1

In Freqwt Sin

643

times

THD

Discrete2e-05 s

Powergui

Simout

Simout1

Figure 9 e simulation diagram of single-phase inverter

6 Complexity

reduced to 198 when PI +FORC is adopted FinallyPI +HORC is used to suppress the current harmonics morerestrictively where the THD is reduced to 184 in Figure 15

42 Harmonic Suppression In order to further verify theharmonic suppression of the three controlmethods high-orderharmonics are added to the systemmanually e frequency of

Table 1 e simulation parameters

Parameters ValuesGrid phase voltage Us(V) 220DC side voltage UDC(V) 800Switching frequency fs(kHz) 10Filter inductance Ls(mH) 30Inductance equivalent resistance Rs(Ω) 01Input voltage frequency f(Hz) 50

VoltageCurrent

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 10 Simulation results of PI control

2

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20

Harmonic order

Fundamental (50Hz) = 6285 THD = 210

Figure 11 e THD of PI + FORC control

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA

VoltageCurrent

012 013 014 015 016 017 018 019 0201101

Time (s)

Figure 12 e THD of PI control

2

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20Harmonic order

Fundamental (50Hz) = 6293 THD = 198

Figure 13 Simulation results of PI +HORC control

VoltageCurrent

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA

011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 14 Simulation results of PI + FORC control

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20

Harmonic order

Fundamental (50Hz) = 6303 THD = 184

Figure 15 e THD of PI +HORC control

VoltageCurrent

ndash400

ndash200

0

200

400

UA

ndash200

ndash100

0

100

200

IA

011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 16 Simulation results of PI control with high-orderharmonics

Complexity 7

the grid is set as 50Hz then the 7th and 9th harmonics areinjected It is shown in Figure 16 that PI control has poorperformance to suppress harmonics and it leads to greatdistortions in the current It can be seen from Figure 17 thatafter the introduction of FORC the high-order harmonics aresuppressed and the output current retains a better sinusoidalwaveform After using HORC we can get a smoother sinecurve as shown in Figure 18

43 Dynamic Tracking rough the above analysis we canfind that the HORC has a better harmonic suppression

performance In order to further verify the tracking per-formance of HORC the reference current is suddenly re-duced by 30 in 026s to simulate the sudden drop ofcurrent It is shown in Figure 19 that the current can berecovered in 028s erefore when the reference currentchanges suddenly the current can respond quickly and thereference current can be tracked after a short transient thatis the system has fast dynamic response

5 Conclusion

In this paper a novel control method combining PI controland repetitive control is proposed for a single-phase grid-connected inverter After introducing the single-phase in-verter type and modelling a first-order repetitive control anda high-order repetitive control are developed for the grid-connected inverter respectively e stability and perfor-mance analysis are all given for the proposed two repetitivecontrols Finally comparative simulations are conductedbased on the proposed single-phase grid-connected inverterto show that the high-order repetitive control can obtainbetter steady-state and dynamic features and reduce the netcurrent harmonics Future work will focus on the reactivepower compensation control and its practical application

Data Availability

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

Conflicts of Interest

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

Acknowledgments

is work was supported by China Southern Power GridCorporation (no YNKJXM20180366) and the ScientificResearch Fund of Yunnan Education Department underGrant 2020J0067

References

[1] C Zhang Y Ye and H Chen ldquoPhotovoltaic grid connectedinverter based on output current controlrdquo Journal of ElectricalTechnology vol 22 no 8 pp 41ndash45 2007

[2] F Wang S Yu and J Su ldquoResearch on solar photovoltaic gridconnected power generation systemrdquo Journal of ElectricalTechnology vol 20 no 5 pp 72ndash74 2005

[3] Z Yao F Yu and Q Zhao ldquoSimulation of large photovoltaicgrid connected system based on modular multilevel con-verterrdquo Chinese Journal of Electrical Engineering vol 33no 36 pp 27ndash33 2013

[4] T Cheng M Chen Y Wang et al ldquoAdaptive robust methodfor dynamic economic emission dispatch incorporating re-newable energy and energy storagerdquo Complexity vol 2018Article ID 2517987 13 pages 2018

[5] X Guo X Zhang Z Lu B Wang H Qi and X Sun ldquoPowercurrent quality coordinated control strategy of photovoltaicgrid connected inverter under unbalanced grid voltagerdquo

VoltageCurrent

ndash400

ndash200

0

200

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 17 Simulation diagram of PI + FORC control with high-order harmonics

VoltageCurrent

ndash400

ndash200

0

200

400

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 18 Simulation results of PI +HORC control with high-order harmonics

Abrupt changeof current

VoltageCurrent

ndash400ndash300ndash200ndash100

0100200300400

UV

ndash200ndash150ndash100ndash50050100150200

IA

02 022 024 026 028 03 032 034 036018

Time (s)

Figure 19 e simulation diagram when current breaks inPI +HORC control

8 Complexity

Chinese Journal of Electrical Engineering vol 34 no 3pp 346ndash353 2014

[6] Z Yao and L Xiao ldquoControl of single-phase grid-connectedinverters with nonlinear loadsrdquo IEEE Transactions on In-dustrial Electronics vol 60 no 4 pp 1384ndash1389 2013

[7] Q Yan X Wu X Yuan and Y Geng ldquoAn improved grid-voltage feedforward strategy for high-power three-phase grid-connected inverters based on the simplified repetitive pre-dictorrdquo IEEE Transactions on Power Electronics vol 31 no 5pp 3880ndash3897 2016

[8] Y He H S-H Chung C-T Lai X Zhang and W WuldquoActive cancelation of equivalent grid impedance for im-proving stability and injected power quality of grid-connectedinverter under variable grid conditionrdquo IEEE Transactions onPower Electronics vol 33 no 11 pp 9387ndash9398 2018

[9] C Zhao J Liu Z Xie and F Zhou ldquoCoordinate control ofpowercurrent for grid-connected inverter based on PCIcontroller under unbalanced grid conditionsrdquo Complexityvol 2019 Article ID 5968984 14 pages 2019

[10] Q L Zhao X Q Guo and W Y Wu ldquoResearch on gridconnected control technology of single-phase inverterrdquo ChineseJournal of Electrical Engineering vol 27 no 16 pp 60ndash64 2007

[11] M Dong and A Luo ldquoDesign and control method of inverterin photovoltaic grid connected power generation systemrdquoPower System Automation vol 30 no 20 pp 97ndash102 2006

[12] X B Ruan and Y G Yan ldquoControl strategy of four leg three-phase inverterrdquo Transactions of China Electrotechnical Societyvol 15 no 1 pp 61ndash64 2000

[13] B Sahan A N Vergara N Henze A Engler and P ZachariasldquoA single-stage PVmodule integrated converter based on a low-power current-source inverterrdquo IEEE Transactions on Indus-trial Electronics vol 55 no 7 pp 2602ndash2609 2008

[14] J Selvaraj and N A Rahim ldquoMultilevel inverter for grid-con-nected PV system employing digital PI controllerrdquo IEEE Trans-actions on Industrial Electronics vol 56 no 1 pp 149ndash158 2009

[15] S Wang L Tao Q Chen J Na and X Ren ldquoUSDE-basedsliding mode control for servo mechanisms with unknownsystem dynamicsrdquo IEEEASME Transactions onMechatronicsvol 25 no 2 pp 1056ndash1066 2020

[16] J Fei T Li F Wang and W Juan ldquoA novel sliding modecontrol technique for indirect current controlled active powerfilterrdquo Mathematical Problems in Engineering vol 2012Article ID 549782 18 pages 2012

[17] Y-P Liu K-Z Liu and X Yang ldquoNonlinear current controlfor reluctance actuator with hysteresis compensationrdquo Jour-nal of Control Science and Engineering vol 2014 Article ID150345 7 pages 2014

[18] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[19] M Ruiz-Ruigomez J Badiola S M Schmidt-Malan et alldquoDirect electrical current reduces bacterial and yeast biofilmformationrdquo International Journal of Bacteriology vol 2016Article ID 9727810 6 pages 2016

[20] R Grino R Cardoner R Costa-Castello and E FossasldquoDigital repetitive control of a three-phase four-wire shuntactive filterrdquo IEEE Transactions on Industrial Electronicsvol 54 no 3 pp 1495ndash1503 2007

[21] R Costa-Castello R Grino and E Fossas ldquoOdd-harmonicdigital repetitive control of a single-phase current active fil-terrdquo IEEE Transactions on Power Electronics vol 19 no 4pp 1060ndash1068 2004

[22] G Weiss Q-C Zhong T C Green and J Liang ldquoRepetitivecontrol of DC-AC converters in microgridsrdquo IEEE Transac-tions on Power Electronics vol 19 no 1 pp 219ndash230 2004

[23] K Zhou and D Wang ldquoDigital repetitive learning controllerfor three-phase CVCF PWM inverterrdquo IEEE Transactions onIndustrial Electronics vol 48 pp 820ndash830 2001

[24] S Wang J Na and Y Xing ldquoAdaptive optimal parameterestimation and control of servo mechanisms theory andexperimentsrdquo IEEE Transactions on Industrial Electronicsvol 99 p 1 2020

[25] S Wang and J Na ldquoParameter estimation and adaptivecontrol for servo mechanisms with friction compensationrdquoIEEE Transactions on Industrial Informatics vol 99 p 12020

[26] J Zhao J Na and G Gao ldquoAdaptive dynamic programmingbased robust control of nonlinear systems with unmatcheduncertaintiesrdquo Neurocomputing vol 395 pp 56ndash65 2020

[27] P Giri A Das and A Bhattacharya ldquoPush-pull inverter basedwireless power transfer systemrdquo in Proceedings of the 7thInternational Conference on Signal Processing and IntegratedNetworks (SPIN) pp 489ndash493 Noida India February 2020

[28] Z Kaczmarczyk andW Jurczak ldquoA push-pull class-E inverterwith improved efficiencyrdquo IEEE Transactions on IndustrialElectronics vol 55 no 4 pp 1871ndash1874 2008

[29] E Babaei E Shokati Asl and M Hasan Babayi ldquoSteady-stateand small-signal analysis of high-voltage gain half-bridgeswitched boost inverterrdquo IEEE Transactions on IndustrialElectronics vol 63 no 6 pp 3546ndash3553 2016

[30] S-H Ryu D-G Woo M-K Kim and B-K Lee ldquoAnalysisand design of modified half-bridge series-resonant inverterwith DC-link neutral-point-clamped cellrdquo IEEE Transactionson Power Electronics vol 31 no 3 pp 2282ndash2295 2016

[31] L Zhang K Sun Y Xing and J Zhao ldquoA family of five-leveldual-buck full-bridge inverters for grid-tied applicationsrdquoIEEE Transactions on Power Electronics vol 31 pp 7029ndash7042 2016

[32] X Liu G Ma P R Pagilla and S S Ge ldquoState-estimator-based asynchronous repetitive control of discrete-time mar-kovian switching systemsrdquo Complexity vol 2020 Article ID6195162 13 pages 2020

[33] R Yi and C Boshi Electric Drive Automatic Control System-Motion Control System pp 70-71 China Machine PressBeijing China 2009

[34] J Na X Ren R Costa-Castello and Y Guo ldquoRepetitivecontrol of servo systems with time delaysrdquo Robotics andAutonomous Systems vol 62 no 3 pp 319ndash329 2014

[35] R Grintildeo and R Costa-Castello ldquoDigital repetitive plug-incontroller for odd-harmonic periodic references and distur-bancesrdquo Automatica vol 41 no 1 pp 153ndash157 2005

[36] R Costa-Castello J Nebot and R Grino ldquoDemonstration ofthe internal model principle by digital repetitive control of aneducational laboratory plantrdquo IEEE Transactions on Educa-tion vol 48 no 1 pp 73ndash80 2005

[37] W S Chang I H Suh and T W Kim ldquoAnalysis and designof two types of digital repetitive control systemsrdquoAutomaticavol 31 no 5 pp 741ndash746 1995

[38] M Steinbuch ldquoRepetitive control for systems with uncertainperiod-timerdquo Automatica vol 38 no 12 pp 2103ndash21092002

[39] T Inoue ldquoPractical repetitive control system designrdquo inProceedings of the 29th IEEE Conference on Decision andControl pp 1673ndash1678 Honolulu HI USA December 1990

[40] R Costa-Castello G A Ramos J M Olm and M SteinbuchldquoSecond-order odd-harmonic repetitive control and its

Complexity 9

application to active filter controlrdquo in Proceedings of the IEEEConference on Decision amp Control Atlanta GA USA De-cember 2011

10 Complexity

current and the current through the switch element are halfof the half-bridge inverter circuit Full-bridge inverter circuitis simple in terms of structure and easy to control thereforeit has been widely used in the high-power occasions

At the same time the inverter is divided into the activeinverter and passive inverter according to whether there is apower supply or not Active and passive refer to whether theinverter is connected to the power supply In this project thephotovoltaic grid-connected inverter needs to be connectedto the grid which belongs to the grid-connected inverter sothat the active inverter circuit is selected

From Figure 3 for the inverter with load its filter circuitis usually composed of an LC filter circuit with inductanceand capacitance For the grid-connected inverter the outputterminal needs to be connected to the grid e filter ca-pacitor C will be clamped by the power grid and cannot playthe role of filtering and the parallel capacitor can be re-moved equivalently which can be seen in Figure 4

is paper mainly focusses on the later stage of the grid-connected inverter system which consists of a DC powersupply provided by the former DC-DC part four powerswitch devices (IGBT) inverter bridge and filter inductore inverter outputs sinusoidal current with the same fre-quency and phase the same as the grid voltage and enters thegrid

22 Modelling of Single-Phase Inverter e main circuit ofthe single-phase grid-connected inverter is shown in Fig-ure 5 which is a voltage type full-bridge circuit composed offour IGBTs and continuous current diodes in reverse par-allel T1 minus T4 is the IGBT VD1 minus VD4 is the continuouscurrent diode Ls is the filter inductance on one side of thepower grid and Rs is the parasitic resistance of the filterinductance Us and Udc are the corresponding AC powersupply and DC power supply respectively

According to Figure 5 we can obtain the flowingequation

Ls

disdt

UAB minus Us minus Rsis (1)

en we have

Is(s)

UAB(s) minus Us(s)

1Rs + Lss

(2)

For the DC side we get

Udc(s)

Idc(s) minus IL(s)

1sCdc

(3)

en we know that the single-phase grid-connectedinverter adopts the double loop control strategy of voltageand current loop and the outer loop is the DC side voltageloop whose function is to keep the DC side voltage stablee inner loop is a current loop the output current of theinverter is in the same phase as the grid voltage and thecurrent amplitude is determined by the output of the voltageloop regulator is paper only studies the inverter functionof the later stage of the grid-connected system therefore thefront end of the inverter bridge can be regarded as a constantvoltage DC power supply while the voltage loop is notconsidered

3 Repetitive Control Design of Single-PhaseGrid-Connected Inverter

Repetitive control [32] is derived from the internal modelprinciple (IMP) e merit of IMP is that in a stable closed-loop control system the internal control includes the

~ACUd

Figure 2 Half-bridge inverter circuit

DC

V1

V2

V3

V4

Figure 3 Full-bridge inverter circuit without AC

DC

+ndash

Figure 1 Push-pull inverter circuit

Complexity 3

generator of external controlled signals so as to realize theadjustment without steady-state error and suppress theperiodic interference When the steady-state error of asystem is zero thus the input of the controller is zero At thistime the reference signal and the feedback signal still existthe periodic disturbance still changes according to theoriginal dynamic characteristics and the controller stilloutputs the corresponding control signal to keep the systemadjusted without static error erefore the essence of re-alizing no steady-state error is that the controller is astructural model which can reflect and process externalsignals

e proposed controller should be designed to suppressthe current harmonics to reduce the distortion of sine waveof current feed into the network However the jammingsignal in the power electronic system is periodic then aparticular control structure shown in Figure 6 will be used

Remark 1 is control system in Figure 6 consists of an RCcontroller and PI controller e PI controller is designed tostabilize the closed-loop system whose parameters can betuned as [33] is paper will focus on the design of

repetitive control (eg FORC and HORC) and theircomparisons

31 First-Order Repetitive Control (FORC)

311 First-Order Repetitive Control Structure As shown inFigure 7 repetitive control can be designed based on aninternal model which can learn a signal of the period T andduplicate it even if the input of this model is set to zeroConsider the frequency-domain response this internalmodel introduces infinite gains (without filter H(s)) at thespecific frequencies ω 2nπT rads foralln isin N+ enaccording to the well-known IMP zero-error tracking orrejection of period signals at these frequencies can beguaranteed if the closed-loop system is stable

A low-pass filter H(s) is usually used to improve therobustness of controlled systems although it may reduce thegains in these specific frequencies H(s) is set as a second-order Butterworth filter in this paper by designing its cut-offfrequency e compensator K(s) is utilized to retain thesystem stability when the internal model is added to theclosed-loop system us the transfer function of RC isdescribed by

RC(s) Z(s)

E(s)

eminus sT

1 minus H(s)eminussT middot K(s) (4)

It is found that the use of low-pass filter H(s) will lead toan unavoidable phase lag which will shift the frequencies atwhich the maximum gains are obtained Specifically thefrequency shift is mainly related to its phase However aproper compensation can be further incorporated to addressthis issue as [34]

312 Stability and Performance Analysis In this part wewill give a proposition regarding the system stability efollowing conditions should be fulfilled to guarantee thestability of the closed-loop system

DC

V1 V3

V2 V4

~AC

Figure 4 Full-bridge inverter circuit with AC

~Udc UAB US

iL

iS

RS

LS

T1 T3

T2 T4

VD1

VD2VD4

VD3

Figure 5 e main circuit of single phase

R (s) +

ndash

E ZRC (s)

PI (s)

Controller D (s)

P (s)Y (s)

Figure 6 Schematic of the proposed control

E

Internal model

+H (s)

K (s)Z

endashsT

Figure 7 Block diagram of FORC

4 Complexity

Proposition 1 6e closed-loop system in Figure 6 withFORC in Figure 7 is stable if the following conditions arefulfilled [35 36]

(1) 6e closed-loop system without RC is stable that isT0(s) PI(s)P(s)(1 + PI(s)P(s)) is stable

(2) H(s)infin le 1 that is the filter should have a gain closeto 1 within the suitable bandwidth

(3) (K(s)T0(s) minus H(s))eminus sTinfin K(s)T0(s)

minusH(s)infinle 1

Proof e proof is similar to that given in [36] and thus isnot presented here

According to these three conditions we have differentdesign methods to satisfy the conditions e PI control isdesigned to fulfill condition (1) a low-pass filter H(s) is usedto fulfill condition (2) and K(s) can be designed to satisfycondition (3) For any minimum-phase plant P(s) a con-structive selection of K(s) is given as

K(s) krH(s)T0(s) (5)

where kr is a constantBy substituting (5) into condition (3) of Proposition 1

we have the following

Corollary 1 6e closed-loop control system shown in Fig-ure 6 with a stable minimum-phase P(s) and compensator (5)is stable if the following conditions are fulfilled

(1) H(s) is a stable system H(s)infin lt 1(2) kr is a con-stant fulfilling |kr minus 1|lt 1H(s)infin

Remark 1 e compensator K(s) of (5) is valid for stableminimum-phase plants because there are no zero-polecancellations in the right-half plane in this case For non-minimum-phase plants an alternative approach is to cancelthe minimum-phase zeroes and poles and to compensate forthe phase of the nonminimum-phase ones as [37]

32 High-Order Repetitive Control (HORC)

321 High-Order Repetitive Control Structure AlthoughFORC in Figure 7 is easy to implement its performance maydegrade when the period T of the reference or disturbancehas uncertainties that is it is sensitive to the variation in theperiod of signals In order to tackle these disadvantageshigh-order repetitive control (HORC) was proposed in[38ndash40] because of its strong robustness which is given inFigure 8

In Figure 8 W(s) is a weighted sum function of re-petitive loops us the transfer function of HORC can bewritten as

HORC(s) W(s)K(s)

1 minus W(s)H(s) (6)

It is shown that HORC uses a multiple loop internalmodel which replaces the delay eminus sT by a weighted sumfunction of several delays given as follows

W(s) 1113944M

m1Wme

minusmsT (7)

where M is the number of delays that is the order of RCloop

Similar to FORC the gains of HORC are infinite at themultiples of the input signal frequency In particular thegain of (6) will tend to infinity if W(s)H(s) 1 As shown in[38] we substitute s jk2πT into (7) and have

W(jk2πT) 1113944M

m1Wme

minus jmk2π 1113944

M

m1Wm 1 (8)

It is noted that for M 1 and W1 1 (8) is reduced toFORC For the case Mgt 1 we need to determine the weightparameters Wm Inspired by the idea of [38] we can makethe first-order derivative of W(s) with respect to T zero atthe specific frequency is imposes the condition

zW(s jk2πT)

zT 1113944

M

m1

minusWmmjk2πT

0 (9)

According to (9) the following equation is true

1113944

M

m1Wmm 0 (10)

As shown in [38] to further decrease the sensitivity forperiod-time variations by using the weight parameters ofHORC we can make (M minus 1)-th derivatives equal zero sothat

1113944

M

m1Wmm

(Mminus 1) 0 (11)

e weight parameters can be calculated based on(7)ndash(11) For example when M 2 and 3 we can set W1 2and W2 minus1 W1 3 W2 minus3 and W3 1 respectively

322 Stability and Performance Analysis e stabilityanalysis of HORC system is similar to that given in Prop-osition 1 and thus we have the following

E

H (s)

K (s) Z

W (s)

endashsT endashsT endashsT

W1

W2

WM

Figure 8 Block diagram of HORC

Complexity 5

Proposition 2 6e closed-loop system in Figure 6 withHORC in Figure 8 is stable if the following conditions hold

(1) 6e closed-loop system without the HORC is stablethat isT0(s) PI(s)P(s)(1 + PI(s)P(s)) is stable

(2) H(s)infin le 1(3) (K(s)T0(s) minus H(s))W(s)infinle 1

Proof e proof is similar to that of Proposition 1 byreplacing eminus sT with W(s)

For any minimum-phase P(s) we design K(s) as

K(s) krH(s)

T0(s) (12)

By substituting (12) into the third condition in Propo-sition 2 we can get (krH(s) minus H(s))W(s)infinle 1 Fur-thermore condition (2) in Corollary 2 can be derived as|kr minus 1|lt 1H(s)W(s)infin Hence we have thefollowing

Corollary 2 For stable minimum-phase P(s) the closed-loop control system shown in Figure 8 with HORC is stable ifthe following conditions are true

(1) H(s)infin lt 1(2) kr is a constant fulfilling |kr minus 1|lt 1H(s)W(s)infin

4 Simulation Results and Analysis

In this section we validate the effectiveness of the proposedcontrol algorithms by using Simpower systems Toolbox inMatlabSimulink e simulation model of a single-phase

grid-connected inverter is built to compare the currentharmonic suppression of FORC and HORC e FFTanalysis tool in Simulink is used to get the total harmonicdistortion (THD) rough the block diagram of currentloop control system in Figure 6 and the topology structure inFigure 5 of the single-phase grid-connected inverter thesimulation diagram of the single-phase inverter can beobtained as Figure 9 In addition the main circuit param-eters are shown in Table 1

emain circuit simulationmodule is composed of a DCpower supply an IGBT inverter bridge a filter inductor anAC source of public power grid and an output currentdetection module e control part consists of a controlcircuit a PWM generator module and a PLL module ecurrent loop is used in the control loop e control moduleincludes a PI control a first-order repetitive control or ahigh-order repetitive control respectively In this paper PIcontrol PI + FORC and PI +HORC are all simulated Inorder to fully analyze the characteristics of these threecontrol methods the steady-state waveform harmonicsuppression and dynamic tracking are all given

41 6e Steady-State Waveform Figures 10ndash12 show thevoltage and current of AC side when the PI controlPI + FORC and PI +HORC reach the steady state respec-tively Figures 13ndash15 are the corresponding harmonic analysisresults It can be seen from Figures 10ndash12 that the threemethods can achieve the same phase of voltage and current inthe steady state However the current harmonic is relativelylarge and the THD reaches 210 in Figure 12 In Figure 11the harmonics are correspondingly less and the THD is

Ud+ndash

gE

c gE

c

gE

cgE

c

P Uref

+ indash

+ vndash

PI (s)

+

+

+

+

ndash

Frist-order repetitive control

ird-order repetitive control

Out1In1

Out1In1

In Freqwt Sin

643

times

THD

Discrete2e-05 s

Powergui

Simout

Simout1

Figure 9 e simulation diagram of single-phase inverter

6 Complexity

reduced to 198 when PI +FORC is adopted FinallyPI +HORC is used to suppress the current harmonics morerestrictively where the THD is reduced to 184 in Figure 15

42 Harmonic Suppression In order to further verify theharmonic suppression of the three controlmethods high-orderharmonics are added to the systemmanually e frequency of

Table 1 e simulation parameters

Parameters ValuesGrid phase voltage Us(V) 220DC side voltage UDC(V) 800Switching frequency fs(kHz) 10Filter inductance Ls(mH) 30Inductance equivalent resistance Rs(Ω) 01Input voltage frequency f(Hz) 50

VoltageCurrent

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 10 Simulation results of PI control

2

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20

Harmonic order

Fundamental (50Hz) = 6285 THD = 210

Figure 11 e THD of PI + FORC control

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA

VoltageCurrent

012 013 014 015 016 017 018 019 0201101

Time (s)

Figure 12 e THD of PI control

2

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20Harmonic order

Fundamental (50Hz) = 6293 THD = 198

Figure 13 Simulation results of PI +HORC control

VoltageCurrent

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA

011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 14 Simulation results of PI + FORC control

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20

Harmonic order

Fundamental (50Hz) = 6303 THD = 184

Figure 15 e THD of PI +HORC control

VoltageCurrent

ndash400

ndash200

0

200

400

UA

ndash200

ndash100

0

100

200

IA

011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 16 Simulation results of PI control with high-orderharmonics

Complexity 7

the grid is set as 50Hz then the 7th and 9th harmonics areinjected It is shown in Figure 16 that PI control has poorperformance to suppress harmonics and it leads to greatdistortions in the current It can be seen from Figure 17 thatafter the introduction of FORC the high-order harmonics aresuppressed and the output current retains a better sinusoidalwaveform After using HORC we can get a smoother sinecurve as shown in Figure 18

43 Dynamic Tracking rough the above analysis we canfind that the HORC has a better harmonic suppression

performance In order to further verify the tracking per-formance of HORC the reference current is suddenly re-duced by 30 in 026s to simulate the sudden drop ofcurrent It is shown in Figure 19 that the current can berecovered in 028s erefore when the reference currentchanges suddenly the current can respond quickly and thereference current can be tracked after a short transient thatis the system has fast dynamic response

5 Conclusion

In this paper a novel control method combining PI controland repetitive control is proposed for a single-phase grid-connected inverter After introducing the single-phase in-verter type and modelling a first-order repetitive control anda high-order repetitive control are developed for the grid-connected inverter respectively e stability and perfor-mance analysis are all given for the proposed two repetitivecontrols Finally comparative simulations are conductedbased on the proposed single-phase grid-connected inverterto show that the high-order repetitive control can obtainbetter steady-state and dynamic features and reduce the netcurrent harmonics Future work will focus on the reactivepower compensation control and its practical application

Data Availability

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

Conflicts of Interest

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

Acknowledgments

is work was supported by China Southern Power GridCorporation (no YNKJXM20180366) and the ScientificResearch Fund of Yunnan Education Department underGrant 2020J0067

References

[1] C Zhang Y Ye and H Chen ldquoPhotovoltaic grid connectedinverter based on output current controlrdquo Journal of ElectricalTechnology vol 22 no 8 pp 41ndash45 2007

[2] F Wang S Yu and J Su ldquoResearch on solar photovoltaic gridconnected power generation systemrdquo Journal of ElectricalTechnology vol 20 no 5 pp 72ndash74 2005

[3] Z Yao F Yu and Q Zhao ldquoSimulation of large photovoltaicgrid connected system based on modular multilevel con-verterrdquo Chinese Journal of Electrical Engineering vol 33no 36 pp 27ndash33 2013

[4] T Cheng M Chen Y Wang et al ldquoAdaptive robust methodfor dynamic economic emission dispatch incorporating re-newable energy and energy storagerdquo Complexity vol 2018Article ID 2517987 13 pages 2018

[5] X Guo X Zhang Z Lu B Wang H Qi and X Sun ldquoPowercurrent quality coordinated control strategy of photovoltaicgrid connected inverter under unbalanced grid voltagerdquo

VoltageCurrent

ndash400

ndash200

0

200

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 17 Simulation diagram of PI + FORC control with high-order harmonics

VoltageCurrent

ndash400

ndash200

0

200

400

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 18 Simulation results of PI +HORC control with high-order harmonics

Abrupt changeof current

VoltageCurrent

ndash400ndash300ndash200ndash100

0100200300400

UV

ndash200ndash150ndash100ndash50050100150200

IA

02 022 024 026 028 03 032 034 036018

Time (s)

Figure 19 e simulation diagram when current breaks inPI +HORC control

8 Complexity

Chinese Journal of Electrical Engineering vol 34 no 3pp 346ndash353 2014

[6] Z Yao and L Xiao ldquoControl of single-phase grid-connectedinverters with nonlinear loadsrdquo IEEE Transactions on In-dustrial Electronics vol 60 no 4 pp 1384ndash1389 2013

[7] Q Yan X Wu X Yuan and Y Geng ldquoAn improved grid-voltage feedforward strategy for high-power three-phase grid-connected inverters based on the simplified repetitive pre-dictorrdquo IEEE Transactions on Power Electronics vol 31 no 5pp 3880ndash3897 2016

[8] Y He H S-H Chung C-T Lai X Zhang and W WuldquoActive cancelation of equivalent grid impedance for im-proving stability and injected power quality of grid-connectedinverter under variable grid conditionrdquo IEEE Transactions onPower Electronics vol 33 no 11 pp 9387ndash9398 2018

[9] C Zhao J Liu Z Xie and F Zhou ldquoCoordinate control ofpowercurrent for grid-connected inverter based on PCIcontroller under unbalanced grid conditionsrdquo Complexityvol 2019 Article ID 5968984 14 pages 2019

[10] Q L Zhao X Q Guo and W Y Wu ldquoResearch on gridconnected control technology of single-phase inverterrdquo ChineseJournal of Electrical Engineering vol 27 no 16 pp 60ndash64 2007

[11] M Dong and A Luo ldquoDesign and control method of inverterin photovoltaic grid connected power generation systemrdquoPower System Automation vol 30 no 20 pp 97ndash102 2006

[12] X B Ruan and Y G Yan ldquoControl strategy of four leg three-phase inverterrdquo Transactions of China Electrotechnical Societyvol 15 no 1 pp 61ndash64 2000

[13] B Sahan A N Vergara N Henze A Engler and P ZachariasldquoA single-stage PVmodule integrated converter based on a low-power current-source inverterrdquo IEEE Transactions on Indus-trial Electronics vol 55 no 7 pp 2602ndash2609 2008

[14] J Selvaraj and N A Rahim ldquoMultilevel inverter for grid-con-nected PV system employing digital PI controllerrdquo IEEE Trans-actions on Industrial Electronics vol 56 no 1 pp 149ndash158 2009

[15] S Wang L Tao Q Chen J Na and X Ren ldquoUSDE-basedsliding mode control for servo mechanisms with unknownsystem dynamicsrdquo IEEEASME Transactions onMechatronicsvol 25 no 2 pp 1056ndash1066 2020

[16] J Fei T Li F Wang and W Juan ldquoA novel sliding modecontrol technique for indirect current controlled active powerfilterrdquo Mathematical Problems in Engineering vol 2012Article ID 549782 18 pages 2012

[17] Y-P Liu K-Z Liu and X Yang ldquoNonlinear current controlfor reluctance actuator with hysteresis compensationrdquo Jour-nal of Control Science and Engineering vol 2014 Article ID150345 7 pages 2014

[18] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[19] M Ruiz-Ruigomez J Badiola S M Schmidt-Malan et alldquoDirect electrical current reduces bacterial and yeast biofilmformationrdquo International Journal of Bacteriology vol 2016Article ID 9727810 6 pages 2016

[20] R Grino R Cardoner R Costa-Castello and E FossasldquoDigital repetitive control of a three-phase four-wire shuntactive filterrdquo IEEE Transactions on Industrial Electronicsvol 54 no 3 pp 1495ndash1503 2007

[21] R Costa-Castello R Grino and E Fossas ldquoOdd-harmonicdigital repetitive control of a single-phase current active fil-terrdquo IEEE Transactions on Power Electronics vol 19 no 4pp 1060ndash1068 2004

[22] G Weiss Q-C Zhong T C Green and J Liang ldquoRepetitivecontrol of DC-AC converters in microgridsrdquo IEEE Transac-tions on Power Electronics vol 19 no 1 pp 219ndash230 2004

[23] K Zhou and D Wang ldquoDigital repetitive learning controllerfor three-phase CVCF PWM inverterrdquo IEEE Transactions onIndustrial Electronics vol 48 pp 820ndash830 2001

[24] S Wang J Na and Y Xing ldquoAdaptive optimal parameterestimation and control of servo mechanisms theory andexperimentsrdquo IEEE Transactions on Industrial Electronicsvol 99 p 1 2020

[25] S Wang and J Na ldquoParameter estimation and adaptivecontrol for servo mechanisms with friction compensationrdquoIEEE Transactions on Industrial Informatics vol 99 p 12020

[26] J Zhao J Na and G Gao ldquoAdaptive dynamic programmingbased robust control of nonlinear systems with unmatcheduncertaintiesrdquo Neurocomputing vol 395 pp 56ndash65 2020

[27] P Giri A Das and A Bhattacharya ldquoPush-pull inverter basedwireless power transfer systemrdquo in Proceedings of the 7thInternational Conference on Signal Processing and IntegratedNetworks (SPIN) pp 489ndash493 Noida India February 2020

[28] Z Kaczmarczyk andW Jurczak ldquoA push-pull class-E inverterwith improved efficiencyrdquo IEEE Transactions on IndustrialElectronics vol 55 no 4 pp 1871ndash1874 2008

[29] E Babaei E Shokati Asl and M Hasan Babayi ldquoSteady-stateand small-signal analysis of high-voltage gain half-bridgeswitched boost inverterrdquo IEEE Transactions on IndustrialElectronics vol 63 no 6 pp 3546ndash3553 2016

[30] S-H Ryu D-G Woo M-K Kim and B-K Lee ldquoAnalysisand design of modified half-bridge series-resonant inverterwith DC-link neutral-point-clamped cellrdquo IEEE Transactionson Power Electronics vol 31 no 3 pp 2282ndash2295 2016

[31] L Zhang K Sun Y Xing and J Zhao ldquoA family of five-leveldual-buck full-bridge inverters for grid-tied applicationsrdquoIEEE Transactions on Power Electronics vol 31 pp 7029ndash7042 2016

[32] X Liu G Ma P R Pagilla and S S Ge ldquoState-estimator-based asynchronous repetitive control of discrete-time mar-kovian switching systemsrdquo Complexity vol 2020 Article ID6195162 13 pages 2020

[33] R Yi and C Boshi Electric Drive Automatic Control System-Motion Control System pp 70-71 China Machine PressBeijing China 2009

[34] J Na X Ren R Costa-Castello and Y Guo ldquoRepetitivecontrol of servo systems with time delaysrdquo Robotics andAutonomous Systems vol 62 no 3 pp 319ndash329 2014

[35] R Grintildeo and R Costa-Castello ldquoDigital repetitive plug-incontroller for odd-harmonic periodic references and distur-bancesrdquo Automatica vol 41 no 1 pp 153ndash157 2005

[36] R Costa-Castello J Nebot and R Grino ldquoDemonstration ofthe internal model principle by digital repetitive control of aneducational laboratory plantrdquo IEEE Transactions on Educa-tion vol 48 no 1 pp 73ndash80 2005

[37] W S Chang I H Suh and T W Kim ldquoAnalysis and designof two types of digital repetitive control systemsrdquoAutomaticavol 31 no 5 pp 741ndash746 1995

[38] M Steinbuch ldquoRepetitive control for systems with uncertainperiod-timerdquo Automatica vol 38 no 12 pp 2103ndash21092002

[39] T Inoue ldquoPractical repetitive control system designrdquo inProceedings of the 29th IEEE Conference on Decision andControl pp 1673ndash1678 Honolulu HI USA December 1990

[40] R Costa-Castello G A Ramos J M Olm and M SteinbuchldquoSecond-order odd-harmonic repetitive control and its

Complexity 9

application to active filter controlrdquo in Proceedings of the IEEEConference on Decision amp Control Atlanta GA USA De-cember 2011

10 Complexity

generator of external controlled signals so as to realize theadjustment without steady-state error and suppress theperiodic interference When the steady-state error of asystem is zero thus the input of the controller is zero At thistime the reference signal and the feedback signal still existthe periodic disturbance still changes according to theoriginal dynamic characteristics and the controller stilloutputs the corresponding control signal to keep the systemadjusted without static error erefore the essence of re-alizing no steady-state error is that the controller is astructural model which can reflect and process externalsignals

e proposed controller should be designed to suppressthe current harmonics to reduce the distortion of sine waveof current feed into the network However the jammingsignal in the power electronic system is periodic then aparticular control structure shown in Figure 6 will be used

Remark 1 is control system in Figure 6 consists of an RCcontroller and PI controller e PI controller is designed tostabilize the closed-loop system whose parameters can betuned as [33] is paper will focus on the design of

repetitive control (eg FORC and HORC) and theircomparisons

31 First-Order Repetitive Control (FORC)

311 First-Order Repetitive Control Structure As shown inFigure 7 repetitive control can be designed based on aninternal model which can learn a signal of the period T andduplicate it even if the input of this model is set to zeroConsider the frequency-domain response this internalmodel introduces infinite gains (without filter H(s)) at thespecific frequencies ω 2nπT rads foralln isin N+ enaccording to the well-known IMP zero-error tracking orrejection of period signals at these frequencies can beguaranteed if the closed-loop system is stable

A low-pass filter H(s) is usually used to improve therobustness of controlled systems although it may reduce thegains in these specific frequencies H(s) is set as a second-order Butterworth filter in this paper by designing its cut-offfrequency e compensator K(s) is utilized to retain thesystem stability when the internal model is added to theclosed-loop system us the transfer function of RC isdescribed by

RC(s) Z(s)

E(s)

eminus sT

1 minus H(s)eminussT middot K(s) (4)

It is found that the use of low-pass filter H(s) will lead toan unavoidable phase lag which will shift the frequencies atwhich the maximum gains are obtained Specifically thefrequency shift is mainly related to its phase However aproper compensation can be further incorporated to addressthis issue as [34]

312 Stability and Performance Analysis In this part wewill give a proposition regarding the system stability efollowing conditions should be fulfilled to guarantee thestability of the closed-loop system

DC

V1 V3

V2 V4

~AC

Figure 4 Full-bridge inverter circuit with AC

~Udc UAB US

iL

iS

RS

LS

T1 T3

T2 T4

VD1

VD2VD4

VD3

Figure 5 e main circuit of single phase

R (s) +

ndash

E ZRC (s)

PI (s)

Controller D (s)

P (s)Y (s)

Figure 6 Schematic of the proposed control

E

Internal model

+H (s)

K (s)Z

endashsT

Figure 7 Block diagram of FORC

4 Complexity

Proposition 1 6e closed-loop system in Figure 6 withFORC in Figure 7 is stable if the following conditions arefulfilled [35 36]

(1) 6e closed-loop system without RC is stable that isT0(s) PI(s)P(s)(1 + PI(s)P(s)) is stable

(2) H(s)infin le 1 that is the filter should have a gain closeto 1 within the suitable bandwidth

(3) (K(s)T0(s) minus H(s))eminus sTinfin K(s)T0(s)

minusH(s)infinle 1

Proof e proof is similar to that given in [36] and thus isnot presented here

According to these three conditions we have differentdesign methods to satisfy the conditions e PI control isdesigned to fulfill condition (1) a low-pass filter H(s) is usedto fulfill condition (2) and K(s) can be designed to satisfycondition (3) For any minimum-phase plant P(s) a con-structive selection of K(s) is given as

K(s) krH(s)T0(s) (5)

where kr is a constantBy substituting (5) into condition (3) of Proposition 1

we have the following

Corollary 1 6e closed-loop control system shown in Fig-ure 6 with a stable minimum-phase P(s) and compensator (5)is stable if the following conditions are fulfilled

(1) H(s) is a stable system H(s)infin lt 1(2) kr is a con-stant fulfilling |kr minus 1|lt 1H(s)infin

Remark 1 e compensator K(s) of (5) is valid for stableminimum-phase plants because there are no zero-polecancellations in the right-half plane in this case For non-minimum-phase plants an alternative approach is to cancelthe minimum-phase zeroes and poles and to compensate forthe phase of the nonminimum-phase ones as [37]

32 High-Order Repetitive Control (HORC)

321 High-Order Repetitive Control Structure AlthoughFORC in Figure 7 is easy to implement its performance maydegrade when the period T of the reference or disturbancehas uncertainties that is it is sensitive to the variation in theperiod of signals In order to tackle these disadvantageshigh-order repetitive control (HORC) was proposed in[38ndash40] because of its strong robustness which is given inFigure 8

In Figure 8 W(s) is a weighted sum function of re-petitive loops us the transfer function of HORC can bewritten as

HORC(s) W(s)K(s)

1 minus W(s)H(s) (6)

It is shown that HORC uses a multiple loop internalmodel which replaces the delay eminus sT by a weighted sumfunction of several delays given as follows

W(s) 1113944M

m1Wme

minusmsT (7)

where M is the number of delays that is the order of RCloop

Similar to FORC the gains of HORC are infinite at themultiples of the input signal frequency In particular thegain of (6) will tend to infinity if W(s)H(s) 1 As shown in[38] we substitute s jk2πT into (7) and have

W(jk2πT) 1113944M

m1Wme

minus jmk2π 1113944

M

m1Wm 1 (8)

It is noted that for M 1 and W1 1 (8) is reduced toFORC For the case Mgt 1 we need to determine the weightparameters Wm Inspired by the idea of [38] we can makethe first-order derivative of W(s) with respect to T zero atthe specific frequency is imposes the condition

zW(s jk2πT)

zT 1113944

M

m1

minusWmmjk2πT

0 (9)

According to (9) the following equation is true

1113944

M

m1Wmm 0 (10)

As shown in [38] to further decrease the sensitivity forperiod-time variations by using the weight parameters ofHORC we can make (M minus 1)-th derivatives equal zero sothat

1113944

M

m1Wmm

(Mminus 1) 0 (11)

e weight parameters can be calculated based on(7)ndash(11) For example when M 2 and 3 we can set W1 2and W2 minus1 W1 3 W2 minus3 and W3 1 respectively

322 Stability and Performance Analysis e stabilityanalysis of HORC system is similar to that given in Prop-osition 1 and thus we have the following

E

H (s)

K (s) Z

W (s)

endashsT endashsT endashsT

W1

W2

WM

Figure 8 Block diagram of HORC

Complexity 5

Proposition 2 6e closed-loop system in Figure 6 withHORC in Figure 8 is stable if the following conditions hold

(1) 6e closed-loop system without the HORC is stablethat isT0(s) PI(s)P(s)(1 + PI(s)P(s)) is stable

(2) H(s)infin le 1(3) (K(s)T0(s) minus H(s))W(s)infinle 1

Proof e proof is similar to that of Proposition 1 byreplacing eminus sT with W(s)

For any minimum-phase P(s) we design K(s) as

K(s) krH(s)

T0(s) (12)

By substituting (12) into the third condition in Propo-sition 2 we can get (krH(s) minus H(s))W(s)infinle 1 Fur-thermore condition (2) in Corollary 2 can be derived as|kr minus 1|lt 1H(s)W(s)infin Hence we have thefollowing

Corollary 2 For stable minimum-phase P(s) the closed-loop control system shown in Figure 8 with HORC is stable ifthe following conditions are true

(1) H(s)infin lt 1(2) kr is a constant fulfilling |kr minus 1|lt 1H(s)W(s)infin

4 Simulation Results and Analysis

In this section we validate the effectiveness of the proposedcontrol algorithms by using Simpower systems Toolbox inMatlabSimulink e simulation model of a single-phase

grid-connected inverter is built to compare the currentharmonic suppression of FORC and HORC e FFTanalysis tool in Simulink is used to get the total harmonicdistortion (THD) rough the block diagram of currentloop control system in Figure 6 and the topology structure inFigure 5 of the single-phase grid-connected inverter thesimulation diagram of the single-phase inverter can beobtained as Figure 9 In addition the main circuit param-eters are shown in Table 1

emain circuit simulationmodule is composed of a DCpower supply an IGBT inverter bridge a filter inductor anAC source of public power grid and an output currentdetection module e control part consists of a controlcircuit a PWM generator module and a PLL module ecurrent loop is used in the control loop e control moduleincludes a PI control a first-order repetitive control or ahigh-order repetitive control respectively In this paper PIcontrol PI + FORC and PI +HORC are all simulated Inorder to fully analyze the characteristics of these threecontrol methods the steady-state waveform harmonicsuppression and dynamic tracking are all given

41 6e Steady-State Waveform Figures 10ndash12 show thevoltage and current of AC side when the PI controlPI + FORC and PI +HORC reach the steady state respec-tively Figures 13ndash15 are the corresponding harmonic analysisresults It can be seen from Figures 10ndash12 that the threemethods can achieve the same phase of voltage and current inthe steady state However the current harmonic is relativelylarge and the THD reaches 210 in Figure 12 In Figure 11the harmonics are correspondingly less and the THD is

Ud+ndash

gE

c gE

c

gE

cgE

c

P Uref

+ indash

+ vndash

PI (s)

+

+

+

+

ndash

Frist-order repetitive control

ird-order repetitive control

Out1In1

Out1In1

In Freqwt Sin

643

times

THD

Discrete2e-05 s

Powergui

Simout

Simout1

Figure 9 e simulation diagram of single-phase inverter

6 Complexity

reduced to 198 when PI +FORC is adopted FinallyPI +HORC is used to suppress the current harmonics morerestrictively where the THD is reduced to 184 in Figure 15

42 Harmonic Suppression In order to further verify theharmonic suppression of the three controlmethods high-orderharmonics are added to the systemmanually e frequency of

Table 1 e simulation parameters

Parameters ValuesGrid phase voltage Us(V) 220DC side voltage UDC(V) 800Switching frequency fs(kHz) 10Filter inductance Ls(mH) 30Inductance equivalent resistance Rs(Ω) 01Input voltage frequency f(Hz) 50

VoltageCurrent

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 10 Simulation results of PI control

2

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20

Harmonic order

Fundamental (50Hz) = 6285 THD = 210

Figure 11 e THD of PI + FORC control

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA

VoltageCurrent

012 013 014 015 016 017 018 019 0201101

Time (s)

Figure 12 e THD of PI control

2

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20Harmonic order

Fundamental (50Hz) = 6293 THD = 198

Figure 13 Simulation results of PI +HORC control

VoltageCurrent

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA

011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 14 Simulation results of PI + FORC control

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20

Harmonic order

Fundamental (50Hz) = 6303 THD = 184

Figure 15 e THD of PI +HORC control

VoltageCurrent

ndash400

ndash200

0

200

400

UA

ndash200

ndash100

0

100

200

IA

011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 16 Simulation results of PI control with high-orderharmonics

Complexity 7

the grid is set as 50Hz then the 7th and 9th harmonics areinjected It is shown in Figure 16 that PI control has poorperformance to suppress harmonics and it leads to greatdistortions in the current It can be seen from Figure 17 thatafter the introduction of FORC the high-order harmonics aresuppressed and the output current retains a better sinusoidalwaveform After using HORC we can get a smoother sinecurve as shown in Figure 18

43 Dynamic Tracking rough the above analysis we canfind that the HORC has a better harmonic suppression

performance In order to further verify the tracking per-formance of HORC the reference current is suddenly re-duced by 30 in 026s to simulate the sudden drop ofcurrent It is shown in Figure 19 that the current can berecovered in 028s erefore when the reference currentchanges suddenly the current can respond quickly and thereference current can be tracked after a short transient thatis the system has fast dynamic response

5 Conclusion

In this paper a novel control method combining PI controland repetitive control is proposed for a single-phase grid-connected inverter After introducing the single-phase in-verter type and modelling a first-order repetitive control anda high-order repetitive control are developed for the grid-connected inverter respectively e stability and perfor-mance analysis are all given for the proposed two repetitivecontrols Finally comparative simulations are conductedbased on the proposed single-phase grid-connected inverterto show that the high-order repetitive control can obtainbetter steady-state and dynamic features and reduce the netcurrent harmonics Future work will focus on the reactivepower compensation control and its practical application

Data Availability

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

Conflicts of Interest

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

Acknowledgments

is work was supported by China Southern Power GridCorporation (no YNKJXM20180366) and the ScientificResearch Fund of Yunnan Education Department underGrant 2020J0067

References

[1] C Zhang Y Ye and H Chen ldquoPhotovoltaic grid connectedinverter based on output current controlrdquo Journal of ElectricalTechnology vol 22 no 8 pp 41ndash45 2007

[2] F Wang S Yu and J Su ldquoResearch on solar photovoltaic gridconnected power generation systemrdquo Journal of ElectricalTechnology vol 20 no 5 pp 72ndash74 2005

[3] Z Yao F Yu and Q Zhao ldquoSimulation of large photovoltaicgrid connected system based on modular multilevel con-verterrdquo Chinese Journal of Electrical Engineering vol 33no 36 pp 27ndash33 2013

[4] T Cheng M Chen Y Wang et al ldquoAdaptive robust methodfor dynamic economic emission dispatch incorporating re-newable energy and energy storagerdquo Complexity vol 2018Article ID 2517987 13 pages 2018

[5] X Guo X Zhang Z Lu B Wang H Qi and X Sun ldquoPowercurrent quality coordinated control strategy of photovoltaicgrid connected inverter under unbalanced grid voltagerdquo

VoltageCurrent

ndash400

ndash200

0

200

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 17 Simulation diagram of PI + FORC control with high-order harmonics

VoltageCurrent

ndash400

ndash200

0

200

400

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 18 Simulation results of PI +HORC control with high-order harmonics

Abrupt changeof current

VoltageCurrent

ndash400ndash300ndash200ndash100

0100200300400

UV

ndash200ndash150ndash100ndash50050100150200

IA

02 022 024 026 028 03 032 034 036018

Time (s)

Figure 19 e simulation diagram when current breaks inPI +HORC control

8 Complexity

Chinese Journal of Electrical Engineering vol 34 no 3pp 346ndash353 2014

[6] Z Yao and L Xiao ldquoControl of single-phase grid-connectedinverters with nonlinear loadsrdquo IEEE Transactions on In-dustrial Electronics vol 60 no 4 pp 1384ndash1389 2013

[7] Q Yan X Wu X Yuan and Y Geng ldquoAn improved grid-voltage feedforward strategy for high-power three-phase grid-connected inverters based on the simplified repetitive pre-dictorrdquo IEEE Transactions on Power Electronics vol 31 no 5pp 3880ndash3897 2016

[8] Y He H S-H Chung C-T Lai X Zhang and W WuldquoActive cancelation of equivalent grid impedance for im-proving stability and injected power quality of grid-connectedinverter under variable grid conditionrdquo IEEE Transactions onPower Electronics vol 33 no 11 pp 9387ndash9398 2018

[9] C Zhao J Liu Z Xie and F Zhou ldquoCoordinate control ofpowercurrent for grid-connected inverter based on PCIcontroller under unbalanced grid conditionsrdquo Complexityvol 2019 Article ID 5968984 14 pages 2019

[10] Q L Zhao X Q Guo and W Y Wu ldquoResearch on gridconnected control technology of single-phase inverterrdquo ChineseJournal of Electrical Engineering vol 27 no 16 pp 60ndash64 2007

[11] M Dong and A Luo ldquoDesign and control method of inverterin photovoltaic grid connected power generation systemrdquoPower System Automation vol 30 no 20 pp 97ndash102 2006

[12] X B Ruan and Y G Yan ldquoControl strategy of four leg three-phase inverterrdquo Transactions of China Electrotechnical Societyvol 15 no 1 pp 61ndash64 2000

[13] B Sahan A N Vergara N Henze A Engler and P ZachariasldquoA single-stage PVmodule integrated converter based on a low-power current-source inverterrdquo IEEE Transactions on Indus-trial Electronics vol 55 no 7 pp 2602ndash2609 2008

[14] J Selvaraj and N A Rahim ldquoMultilevel inverter for grid-con-nected PV system employing digital PI controllerrdquo IEEE Trans-actions on Industrial Electronics vol 56 no 1 pp 149ndash158 2009

[15] S Wang L Tao Q Chen J Na and X Ren ldquoUSDE-basedsliding mode control for servo mechanisms with unknownsystem dynamicsrdquo IEEEASME Transactions onMechatronicsvol 25 no 2 pp 1056ndash1066 2020

[16] J Fei T Li F Wang and W Juan ldquoA novel sliding modecontrol technique for indirect current controlled active powerfilterrdquo Mathematical Problems in Engineering vol 2012Article ID 549782 18 pages 2012

[17] Y-P Liu K-Z Liu and X Yang ldquoNonlinear current controlfor reluctance actuator with hysteresis compensationrdquo Jour-nal of Control Science and Engineering vol 2014 Article ID150345 7 pages 2014

[18] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[19] M Ruiz-Ruigomez J Badiola S M Schmidt-Malan et alldquoDirect electrical current reduces bacterial and yeast biofilmformationrdquo International Journal of Bacteriology vol 2016Article ID 9727810 6 pages 2016

[20] R Grino R Cardoner R Costa-Castello and E FossasldquoDigital repetitive control of a three-phase four-wire shuntactive filterrdquo IEEE Transactions on Industrial Electronicsvol 54 no 3 pp 1495ndash1503 2007

[21] R Costa-Castello R Grino and E Fossas ldquoOdd-harmonicdigital repetitive control of a single-phase current active fil-terrdquo IEEE Transactions on Power Electronics vol 19 no 4pp 1060ndash1068 2004

[22] G Weiss Q-C Zhong T C Green and J Liang ldquoRepetitivecontrol of DC-AC converters in microgridsrdquo IEEE Transac-tions on Power Electronics vol 19 no 1 pp 219ndash230 2004

[23] K Zhou and D Wang ldquoDigital repetitive learning controllerfor three-phase CVCF PWM inverterrdquo IEEE Transactions onIndustrial Electronics vol 48 pp 820ndash830 2001

[24] S Wang J Na and Y Xing ldquoAdaptive optimal parameterestimation and control of servo mechanisms theory andexperimentsrdquo IEEE Transactions on Industrial Electronicsvol 99 p 1 2020

[25] S Wang and J Na ldquoParameter estimation and adaptivecontrol for servo mechanisms with friction compensationrdquoIEEE Transactions on Industrial Informatics vol 99 p 12020

[26] J Zhao J Na and G Gao ldquoAdaptive dynamic programmingbased robust control of nonlinear systems with unmatcheduncertaintiesrdquo Neurocomputing vol 395 pp 56ndash65 2020

[27] P Giri A Das and A Bhattacharya ldquoPush-pull inverter basedwireless power transfer systemrdquo in Proceedings of the 7thInternational Conference on Signal Processing and IntegratedNetworks (SPIN) pp 489ndash493 Noida India February 2020

[28] Z Kaczmarczyk andW Jurczak ldquoA push-pull class-E inverterwith improved efficiencyrdquo IEEE Transactions on IndustrialElectronics vol 55 no 4 pp 1871ndash1874 2008

[29] E Babaei E Shokati Asl and M Hasan Babayi ldquoSteady-stateand small-signal analysis of high-voltage gain half-bridgeswitched boost inverterrdquo IEEE Transactions on IndustrialElectronics vol 63 no 6 pp 3546ndash3553 2016

[30] S-H Ryu D-G Woo M-K Kim and B-K Lee ldquoAnalysisand design of modified half-bridge series-resonant inverterwith DC-link neutral-point-clamped cellrdquo IEEE Transactionson Power Electronics vol 31 no 3 pp 2282ndash2295 2016

[31] L Zhang K Sun Y Xing and J Zhao ldquoA family of five-leveldual-buck full-bridge inverters for grid-tied applicationsrdquoIEEE Transactions on Power Electronics vol 31 pp 7029ndash7042 2016

[32] X Liu G Ma P R Pagilla and S S Ge ldquoState-estimator-based asynchronous repetitive control of discrete-time mar-kovian switching systemsrdquo Complexity vol 2020 Article ID6195162 13 pages 2020

[33] R Yi and C Boshi Electric Drive Automatic Control System-Motion Control System pp 70-71 China Machine PressBeijing China 2009

[34] J Na X Ren R Costa-Castello and Y Guo ldquoRepetitivecontrol of servo systems with time delaysrdquo Robotics andAutonomous Systems vol 62 no 3 pp 319ndash329 2014

[35] R Grintildeo and R Costa-Castello ldquoDigital repetitive plug-incontroller for odd-harmonic periodic references and distur-bancesrdquo Automatica vol 41 no 1 pp 153ndash157 2005

[36] R Costa-Castello J Nebot and R Grino ldquoDemonstration ofthe internal model principle by digital repetitive control of aneducational laboratory plantrdquo IEEE Transactions on Educa-tion vol 48 no 1 pp 73ndash80 2005

[37] W S Chang I H Suh and T W Kim ldquoAnalysis and designof two types of digital repetitive control systemsrdquoAutomaticavol 31 no 5 pp 741ndash746 1995

[38] M Steinbuch ldquoRepetitive control for systems with uncertainperiod-timerdquo Automatica vol 38 no 12 pp 2103ndash21092002

[39] T Inoue ldquoPractical repetitive control system designrdquo inProceedings of the 29th IEEE Conference on Decision andControl pp 1673ndash1678 Honolulu HI USA December 1990

[40] R Costa-Castello G A Ramos J M Olm and M SteinbuchldquoSecond-order odd-harmonic repetitive control and its

Complexity 9

application to active filter controlrdquo in Proceedings of the IEEEConference on Decision amp Control Atlanta GA USA De-cember 2011

10 Complexity

Proposition 1 6e closed-loop system in Figure 6 withFORC in Figure 7 is stable if the following conditions arefulfilled [35 36]

(1) 6e closed-loop system without RC is stable that isT0(s) PI(s)P(s)(1 + PI(s)P(s)) is stable

(2) H(s)infin le 1 that is the filter should have a gain closeto 1 within the suitable bandwidth

(3) (K(s)T0(s) minus H(s))eminus sTinfin K(s)T0(s)

minusH(s)infinle 1

Proof e proof is similar to that given in [36] and thus isnot presented here

According to these three conditions we have differentdesign methods to satisfy the conditions e PI control isdesigned to fulfill condition (1) a low-pass filter H(s) is usedto fulfill condition (2) and K(s) can be designed to satisfycondition (3) For any minimum-phase plant P(s) a con-structive selection of K(s) is given as

K(s) krH(s)T0(s) (5)

where kr is a constantBy substituting (5) into condition (3) of Proposition 1

we have the following

Corollary 1 6e closed-loop control system shown in Fig-ure 6 with a stable minimum-phase P(s) and compensator (5)is stable if the following conditions are fulfilled

(1) H(s) is a stable system H(s)infin lt 1(2) kr is a con-stant fulfilling |kr minus 1|lt 1H(s)infin

Remark 1 e compensator K(s) of (5) is valid for stableminimum-phase plants because there are no zero-polecancellations in the right-half plane in this case For non-minimum-phase plants an alternative approach is to cancelthe minimum-phase zeroes and poles and to compensate forthe phase of the nonminimum-phase ones as [37]

32 High-Order Repetitive Control (HORC)

321 High-Order Repetitive Control Structure AlthoughFORC in Figure 7 is easy to implement its performance maydegrade when the period T of the reference or disturbancehas uncertainties that is it is sensitive to the variation in theperiod of signals In order to tackle these disadvantageshigh-order repetitive control (HORC) was proposed in[38ndash40] because of its strong robustness which is given inFigure 8

In Figure 8 W(s) is a weighted sum function of re-petitive loops us the transfer function of HORC can bewritten as

HORC(s) W(s)K(s)

1 minus W(s)H(s) (6)

It is shown that HORC uses a multiple loop internalmodel which replaces the delay eminus sT by a weighted sumfunction of several delays given as follows

W(s) 1113944M

m1Wme

minusmsT (7)

where M is the number of delays that is the order of RCloop

Similar to FORC the gains of HORC are infinite at themultiples of the input signal frequency In particular thegain of (6) will tend to infinity if W(s)H(s) 1 As shown in[38] we substitute s jk2πT into (7) and have

W(jk2πT) 1113944M

m1Wme

minus jmk2π 1113944

M

m1Wm 1 (8)

It is noted that for M 1 and W1 1 (8) is reduced toFORC For the case Mgt 1 we need to determine the weightparameters Wm Inspired by the idea of [38] we can makethe first-order derivative of W(s) with respect to T zero atthe specific frequency is imposes the condition

zW(s jk2πT)

zT 1113944

M

m1

minusWmmjk2πT

0 (9)

According to (9) the following equation is true

1113944

M

m1Wmm 0 (10)

As shown in [38] to further decrease the sensitivity forperiod-time variations by using the weight parameters ofHORC we can make (M minus 1)-th derivatives equal zero sothat

1113944

M

m1Wmm

(Mminus 1) 0 (11)

e weight parameters can be calculated based on(7)ndash(11) For example when M 2 and 3 we can set W1 2and W2 minus1 W1 3 W2 minus3 and W3 1 respectively

322 Stability and Performance Analysis e stabilityanalysis of HORC system is similar to that given in Prop-osition 1 and thus we have the following

E

H (s)

K (s) Z

W (s)

endashsT endashsT endashsT

W1

W2

WM

Figure 8 Block diagram of HORC

Complexity 5

Proposition 2 6e closed-loop system in Figure 6 withHORC in Figure 8 is stable if the following conditions hold

(1) 6e closed-loop system without the HORC is stablethat isT0(s) PI(s)P(s)(1 + PI(s)P(s)) is stable

(2) H(s)infin le 1(3) (K(s)T0(s) minus H(s))W(s)infinle 1

Proof e proof is similar to that of Proposition 1 byreplacing eminus sT with W(s)

For any minimum-phase P(s) we design K(s) as

K(s) krH(s)

T0(s) (12)

By substituting (12) into the third condition in Propo-sition 2 we can get (krH(s) minus H(s))W(s)infinle 1 Fur-thermore condition (2) in Corollary 2 can be derived as|kr minus 1|lt 1H(s)W(s)infin Hence we have thefollowing

Corollary 2 For stable minimum-phase P(s) the closed-loop control system shown in Figure 8 with HORC is stable ifthe following conditions are true

(1) H(s)infin lt 1(2) kr is a constant fulfilling |kr minus 1|lt 1H(s)W(s)infin

4 Simulation Results and Analysis

In this section we validate the effectiveness of the proposedcontrol algorithms by using Simpower systems Toolbox inMatlabSimulink e simulation model of a single-phase

grid-connected inverter is built to compare the currentharmonic suppression of FORC and HORC e FFTanalysis tool in Simulink is used to get the total harmonicdistortion (THD) rough the block diagram of currentloop control system in Figure 6 and the topology structure inFigure 5 of the single-phase grid-connected inverter thesimulation diagram of the single-phase inverter can beobtained as Figure 9 In addition the main circuit param-eters are shown in Table 1

emain circuit simulationmodule is composed of a DCpower supply an IGBT inverter bridge a filter inductor anAC source of public power grid and an output currentdetection module e control part consists of a controlcircuit a PWM generator module and a PLL module ecurrent loop is used in the control loop e control moduleincludes a PI control a first-order repetitive control or ahigh-order repetitive control respectively In this paper PIcontrol PI + FORC and PI +HORC are all simulated Inorder to fully analyze the characteristics of these threecontrol methods the steady-state waveform harmonicsuppression and dynamic tracking are all given

41 6e Steady-State Waveform Figures 10ndash12 show thevoltage and current of AC side when the PI controlPI + FORC and PI +HORC reach the steady state respec-tively Figures 13ndash15 are the corresponding harmonic analysisresults It can be seen from Figures 10ndash12 that the threemethods can achieve the same phase of voltage and current inthe steady state However the current harmonic is relativelylarge and the THD reaches 210 in Figure 12 In Figure 11the harmonics are correspondingly less and the THD is

Ud+ndash

gE

c gE

c

gE

cgE

c

P Uref

+ indash

+ vndash

PI (s)

+

+

+

+

ndash

Frist-order repetitive control

ird-order repetitive control

Out1In1

Out1In1

In Freqwt Sin

643

times

THD

Discrete2e-05 s

Powergui

Simout

Simout1

Figure 9 e simulation diagram of single-phase inverter

6 Complexity

reduced to 198 when PI +FORC is adopted FinallyPI +HORC is used to suppress the current harmonics morerestrictively where the THD is reduced to 184 in Figure 15

42 Harmonic Suppression In order to further verify theharmonic suppression of the three controlmethods high-orderharmonics are added to the systemmanually e frequency of

Table 1 e simulation parameters

Parameters ValuesGrid phase voltage Us(V) 220DC side voltage UDC(V) 800Switching frequency fs(kHz) 10Filter inductance Ls(mH) 30Inductance equivalent resistance Rs(Ω) 01Input voltage frequency f(Hz) 50

VoltageCurrent

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 10 Simulation results of PI control

2

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20

Harmonic order

Fundamental (50Hz) = 6285 THD = 210

Figure 11 e THD of PI + FORC control

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA

VoltageCurrent

012 013 014 015 016 017 018 019 0201101

Time (s)

Figure 12 e THD of PI control

2

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20Harmonic order

Fundamental (50Hz) = 6293 THD = 198

Figure 13 Simulation results of PI +HORC control

VoltageCurrent

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA

011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 14 Simulation results of PI + FORC control

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20

Harmonic order

Fundamental (50Hz) = 6303 THD = 184

Figure 15 e THD of PI +HORC control

VoltageCurrent

ndash400

ndash200

0

200

400

UA

ndash200

ndash100

0

100

200

IA

011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 16 Simulation results of PI control with high-orderharmonics

Complexity 7

the grid is set as 50Hz then the 7th and 9th harmonics areinjected It is shown in Figure 16 that PI control has poorperformance to suppress harmonics and it leads to greatdistortions in the current It can be seen from Figure 17 thatafter the introduction of FORC the high-order harmonics aresuppressed and the output current retains a better sinusoidalwaveform After using HORC we can get a smoother sinecurve as shown in Figure 18

43 Dynamic Tracking rough the above analysis we canfind that the HORC has a better harmonic suppression

performance In order to further verify the tracking per-formance of HORC the reference current is suddenly re-duced by 30 in 026s to simulate the sudden drop ofcurrent It is shown in Figure 19 that the current can berecovered in 028s erefore when the reference currentchanges suddenly the current can respond quickly and thereference current can be tracked after a short transient thatis the system has fast dynamic response

5 Conclusion

In this paper a novel control method combining PI controland repetitive control is proposed for a single-phase grid-connected inverter After introducing the single-phase in-verter type and modelling a first-order repetitive control anda high-order repetitive control are developed for the grid-connected inverter respectively e stability and perfor-mance analysis are all given for the proposed two repetitivecontrols Finally comparative simulations are conductedbased on the proposed single-phase grid-connected inverterto show that the high-order repetitive control can obtainbetter steady-state and dynamic features and reduce the netcurrent harmonics Future work will focus on the reactivepower compensation control and its practical application

Data Availability

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

Conflicts of Interest

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

Acknowledgments

is work was supported by China Southern Power GridCorporation (no YNKJXM20180366) and the ScientificResearch Fund of Yunnan Education Department underGrant 2020J0067

References

[1] C Zhang Y Ye and H Chen ldquoPhotovoltaic grid connectedinverter based on output current controlrdquo Journal of ElectricalTechnology vol 22 no 8 pp 41ndash45 2007

[2] F Wang S Yu and J Su ldquoResearch on solar photovoltaic gridconnected power generation systemrdquo Journal of ElectricalTechnology vol 20 no 5 pp 72ndash74 2005

[3] Z Yao F Yu and Q Zhao ldquoSimulation of large photovoltaicgrid connected system based on modular multilevel con-verterrdquo Chinese Journal of Electrical Engineering vol 33no 36 pp 27ndash33 2013

[4] T Cheng M Chen Y Wang et al ldquoAdaptive robust methodfor dynamic economic emission dispatch incorporating re-newable energy and energy storagerdquo Complexity vol 2018Article ID 2517987 13 pages 2018

[5] X Guo X Zhang Z Lu B Wang H Qi and X Sun ldquoPowercurrent quality coordinated control strategy of photovoltaicgrid connected inverter under unbalanced grid voltagerdquo

VoltageCurrent

ndash400

ndash200

0

200

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 17 Simulation diagram of PI + FORC control with high-order harmonics

VoltageCurrent

ndash400

ndash200

0

200

400

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 18 Simulation results of PI +HORC control with high-order harmonics

Abrupt changeof current

VoltageCurrent

ndash400ndash300ndash200ndash100

0100200300400

UV

ndash200ndash150ndash100ndash50050100150200

IA

02 022 024 026 028 03 032 034 036018

Time (s)

Figure 19 e simulation diagram when current breaks inPI +HORC control

8 Complexity

Chinese Journal of Electrical Engineering vol 34 no 3pp 346ndash353 2014

[6] Z Yao and L Xiao ldquoControl of single-phase grid-connectedinverters with nonlinear loadsrdquo IEEE Transactions on In-dustrial Electronics vol 60 no 4 pp 1384ndash1389 2013

[7] Q Yan X Wu X Yuan and Y Geng ldquoAn improved grid-voltage feedforward strategy for high-power three-phase grid-connected inverters based on the simplified repetitive pre-dictorrdquo IEEE Transactions on Power Electronics vol 31 no 5pp 3880ndash3897 2016

[8] Y He H S-H Chung C-T Lai X Zhang and W WuldquoActive cancelation of equivalent grid impedance for im-proving stability and injected power quality of grid-connectedinverter under variable grid conditionrdquo IEEE Transactions onPower Electronics vol 33 no 11 pp 9387ndash9398 2018

[9] C Zhao J Liu Z Xie and F Zhou ldquoCoordinate control ofpowercurrent for grid-connected inverter based on PCIcontroller under unbalanced grid conditionsrdquo Complexityvol 2019 Article ID 5968984 14 pages 2019

[10] Q L Zhao X Q Guo and W Y Wu ldquoResearch on gridconnected control technology of single-phase inverterrdquo ChineseJournal of Electrical Engineering vol 27 no 16 pp 60ndash64 2007

[11] M Dong and A Luo ldquoDesign and control method of inverterin photovoltaic grid connected power generation systemrdquoPower System Automation vol 30 no 20 pp 97ndash102 2006

[12] X B Ruan and Y G Yan ldquoControl strategy of four leg three-phase inverterrdquo Transactions of China Electrotechnical Societyvol 15 no 1 pp 61ndash64 2000

[13] B Sahan A N Vergara N Henze A Engler and P ZachariasldquoA single-stage PVmodule integrated converter based on a low-power current-source inverterrdquo IEEE Transactions on Indus-trial Electronics vol 55 no 7 pp 2602ndash2609 2008

[14] J Selvaraj and N A Rahim ldquoMultilevel inverter for grid-con-nected PV system employing digital PI controllerrdquo IEEE Trans-actions on Industrial Electronics vol 56 no 1 pp 149ndash158 2009

[15] S Wang L Tao Q Chen J Na and X Ren ldquoUSDE-basedsliding mode control for servo mechanisms with unknownsystem dynamicsrdquo IEEEASME Transactions onMechatronicsvol 25 no 2 pp 1056ndash1066 2020

[16] J Fei T Li F Wang and W Juan ldquoA novel sliding modecontrol technique for indirect current controlled active powerfilterrdquo Mathematical Problems in Engineering vol 2012Article ID 549782 18 pages 2012

[17] Y-P Liu K-Z Liu and X Yang ldquoNonlinear current controlfor reluctance actuator with hysteresis compensationrdquo Jour-nal of Control Science and Engineering vol 2014 Article ID150345 7 pages 2014

[18] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[19] M Ruiz-Ruigomez J Badiola S M Schmidt-Malan et alldquoDirect electrical current reduces bacterial and yeast biofilmformationrdquo International Journal of Bacteriology vol 2016Article ID 9727810 6 pages 2016

[20] R Grino R Cardoner R Costa-Castello and E FossasldquoDigital repetitive control of a three-phase four-wire shuntactive filterrdquo IEEE Transactions on Industrial Electronicsvol 54 no 3 pp 1495ndash1503 2007

[21] R Costa-Castello R Grino and E Fossas ldquoOdd-harmonicdigital repetitive control of a single-phase current active fil-terrdquo IEEE Transactions on Power Electronics vol 19 no 4pp 1060ndash1068 2004

[22] G Weiss Q-C Zhong T C Green and J Liang ldquoRepetitivecontrol of DC-AC converters in microgridsrdquo IEEE Transac-tions on Power Electronics vol 19 no 1 pp 219ndash230 2004

[23] K Zhou and D Wang ldquoDigital repetitive learning controllerfor three-phase CVCF PWM inverterrdquo IEEE Transactions onIndustrial Electronics vol 48 pp 820ndash830 2001

[24] S Wang J Na and Y Xing ldquoAdaptive optimal parameterestimation and control of servo mechanisms theory andexperimentsrdquo IEEE Transactions on Industrial Electronicsvol 99 p 1 2020

[25] S Wang and J Na ldquoParameter estimation and adaptivecontrol for servo mechanisms with friction compensationrdquoIEEE Transactions on Industrial Informatics vol 99 p 12020

[26] J Zhao J Na and G Gao ldquoAdaptive dynamic programmingbased robust control of nonlinear systems with unmatcheduncertaintiesrdquo Neurocomputing vol 395 pp 56ndash65 2020

[27] P Giri A Das and A Bhattacharya ldquoPush-pull inverter basedwireless power transfer systemrdquo in Proceedings of the 7thInternational Conference on Signal Processing and IntegratedNetworks (SPIN) pp 489ndash493 Noida India February 2020

[28] Z Kaczmarczyk andW Jurczak ldquoA push-pull class-E inverterwith improved efficiencyrdquo IEEE Transactions on IndustrialElectronics vol 55 no 4 pp 1871ndash1874 2008

[29] E Babaei E Shokati Asl and M Hasan Babayi ldquoSteady-stateand small-signal analysis of high-voltage gain half-bridgeswitched boost inverterrdquo IEEE Transactions on IndustrialElectronics vol 63 no 6 pp 3546ndash3553 2016

[30] S-H Ryu D-G Woo M-K Kim and B-K Lee ldquoAnalysisand design of modified half-bridge series-resonant inverterwith DC-link neutral-point-clamped cellrdquo IEEE Transactionson Power Electronics vol 31 no 3 pp 2282ndash2295 2016

[31] L Zhang K Sun Y Xing and J Zhao ldquoA family of five-leveldual-buck full-bridge inverters for grid-tied applicationsrdquoIEEE Transactions on Power Electronics vol 31 pp 7029ndash7042 2016

[32] X Liu G Ma P R Pagilla and S S Ge ldquoState-estimator-based asynchronous repetitive control of discrete-time mar-kovian switching systemsrdquo Complexity vol 2020 Article ID6195162 13 pages 2020

[33] R Yi and C Boshi Electric Drive Automatic Control System-Motion Control System pp 70-71 China Machine PressBeijing China 2009

[34] J Na X Ren R Costa-Castello and Y Guo ldquoRepetitivecontrol of servo systems with time delaysrdquo Robotics andAutonomous Systems vol 62 no 3 pp 319ndash329 2014

[35] R Grintildeo and R Costa-Castello ldquoDigital repetitive plug-incontroller for odd-harmonic periodic references and distur-bancesrdquo Automatica vol 41 no 1 pp 153ndash157 2005

[36] R Costa-Castello J Nebot and R Grino ldquoDemonstration ofthe internal model principle by digital repetitive control of aneducational laboratory plantrdquo IEEE Transactions on Educa-tion vol 48 no 1 pp 73ndash80 2005

[37] W S Chang I H Suh and T W Kim ldquoAnalysis and designof two types of digital repetitive control systemsrdquoAutomaticavol 31 no 5 pp 741ndash746 1995

[38] M Steinbuch ldquoRepetitive control for systems with uncertainperiod-timerdquo Automatica vol 38 no 12 pp 2103ndash21092002

[39] T Inoue ldquoPractical repetitive control system designrdquo inProceedings of the 29th IEEE Conference on Decision andControl pp 1673ndash1678 Honolulu HI USA December 1990

[40] R Costa-Castello G A Ramos J M Olm and M SteinbuchldquoSecond-order odd-harmonic repetitive control and its

Complexity 9

application to active filter controlrdquo in Proceedings of the IEEEConference on Decision amp Control Atlanta GA USA De-cember 2011

10 Complexity

Proposition 2 6e closed-loop system in Figure 6 withHORC in Figure 8 is stable if the following conditions hold

(1) 6e closed-loop system without the HORC is stablethat isT0(s) PI(s)P(s)(1 + PI(s)P(s)) is stable

(2) H(s)infin le 1(3) (K(s)T0(s) minus H(s))W(s)infinle 1

Proof e proof is similar to that of Proposition 1 byreplacing eminus sT with W(s)

For any minimum-phase P(s) we design K(s) as

K(s) krH(s)

T0(s) (12)

By substituting (12) into the third condition in Propo-sition 2 we can get (krH(s) minus H(s))W(s)infinle 1 Fur-thermore condition (2) in Corollary 2 can be derived as|kr minus 1|lt 1H(s)W(s)infin Hence we have thefollowing

Corollary 2 For stable minimum-phase P(s) the closed-loop control system shown in Figure 8 with HORC is stable ifthe following conditions are true

(1) H(s)infin lt 1(2) kr is a constant fulfilling |kr minus 1|lt 1H(s)W(s)infin

4 Simulation Results and Analysis

In this section we validate the effectiveness of the proposedcontrol algorithms by using Simpower systems Toolbox inMatlabSimulink e simulation model of a single-phase

grid-connected inverter is built to compare the currentharmonic suppression of FORC and HORC e FFTanalysis tool in Simulink is used to get the total harmonicdistortion (THD) rough the block diagram of currentloop control system in Figure 6 and the topology structure inFigure 5 of the single-phase grid-connected inverter thesimulation diagram of the single-phase inverter can beobtained as Figure 9 In addition the main circuit param-eters are shown in Table 1

emain circuit simulationmodule is composed of a DCpower supply an IGBT inverter bridge a filter inductor anAC source of public power grid and an output currentdetection module e control part consists of a controlcircuit a PWM generator module and a PLL module ecurrent loop is used in the control loop e control moduleincludes a PI control a first-order repetitive control or ahigh-order repetitive control respectively In this paper PIcontrol PI + FORC and PI +HORC are all simulated Inorder to fully analyze the characteristics of these threecontrol methods the steady-state waveform harmonicsuppression and dynamic tracking are all given

41 6e Steady-State Waveform Figures 10ndash12 show thevoltage and current of AC side when the PI controlPI + FORC and PI +HORC reach the steady state respec-tively Figures 13ndash15 are the corresponding harmonic analysisresults It can be seen from Figures 10ndash12 that the threemethods can achieve the same phase of voltage and current inthe steady state However the current harmonic is relativelylarge and the THD reaches 210 in Figure 12 In Figure 11the harmonics are correspondingly less and the THD is

Ud+ndash

gE

c gE

c

gE

cgE

c

P Uref

+ indash

+ vndash

PI (s)

+

+

+

+

ndash

Frist-order repetitive control

ird-order repetitive control

Out1In1

Out1In1

In Freqwt Sin

643

times

THD

Discrete2e-05 s

Powergui

Simout

Simout1

Figure 9 e simulation diagram of single-phase inverter

6 Complexity

reduced to 198 when PI +FORC is adopted FinallyPI +HORC is used to suppress the current harmonics morerestrictively where the THD is reduced to 184 in Figure 15

42 Harmonic Suppression In order to further verify theharmonic suppression of the three controlmethods high-orderharmonics are added to the systemmanually e frequency of

Table 1 e simulation parameters

Parameters ValuesGrid phase voltage Us(V) 220DC side voltage UDC(V) 800Switching frequency fs(kHz) 10Filter inductance Ls(mH) 30Inductance equivalent resistance Rs(Ω) 01Input voltage frequency f(Hz) 50

VoltageCurrent

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 10 Simulation results of PI control

2

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20

Harmonic order

Fundamental (50Hz) = 6285 THD = 210

Figure 11 e THD of PI + FORC control

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA

VoltageCurrent

012 013 014 015 016 017 018 019 0201101

Time (s)

Figure 12 e THD of PI control

2

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20Harmonic order

Fundamental (50Hz) = 6293 THD = 198

Figure 13 Simulation results of PI +HORC control

VoltageCurrent

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA

011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 14 Simulation results of PI + FORC control

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20

Harmonic order

Fundamental (50Hz) = 6303 THD = 184

Figure 15 e THD of PI +HORC control

VoltageCurrent

ndash400

ndash200

0

200

400

UA

ndash200

ndash100

0

100

200

IA

011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 16 Simulation results of PI control with high-orderharmonics

Complexity 7

the grid is set as 50Hz then the 7th and 9th harmonics areinjected It is shown in Figure 16 that PI control has poorperformance to suppress harmonics and it leads to greatdistortions in the current It can be seen from Figure 17 thatafter the introduction of FORC the high-order harmonics aresuppressed and the output current retains a better sinusoidalwaveform After using HORC we can get a smoother sinecurve as shown in Figure 18

43 Dynamic Tracking rough the above analysis we canfind that the HORC has a better harmonic suppression

performance In order to further verify the tracking per-formance of HORC the reference current is suddenly re-duced by 30 in 026s to simulate the sudden drop ofcurrent It is shown in Figure 19 that the current can berecovered in 028s erefore when the reference currentchanges suddenly the current can respond quickly and thereference current can be tracked after a short transient thatis the system has fast dynamic response

5 Conclusion

In this paper a novel control method combining PI controland repetitive control is proposed for a single-phase grid-connected inverter After introducing the single-phase in-verter type and modelling a first-order repetitive control anda high-order repetitive control are developed for the grid-connected inverter respectively e stability and perfor-mance analysis are all given for the proposed two repetitivecontrols Finally comparative simulations are conductedbased on the proposed single-phase grid-connected inverterto show that the high-order repetitive control can obtainbetter steady-state and dynamic features and reduce the netcurrent harmonics Future work will focus on the reactivepower compensation control and its practical application

Data Availability

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

Conflicts of Interest

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

Acknowledgments

is work was supported by China Southern Power GridCorporation (no YNKJXM20180366) and the ScientificResearch Fund of Yunnan Education Department underGrant 2020J0067

References

[1] C Zhang Y Ye and H Chen ldquoPhotovoltaic grid connectedinverter based on output current controlrdquo Journal of ElectricalTechnology vol 22 no 8 pp 41ndash45 2007

[2] F Wang S Yu and J Su ldquoResearch on solar photovoltaic gridconnected power generation systemrdquo Journal of ElectricalTechnology vol 20 no 5 pp 72ndash74 2005

[3] Z Yao F Yu and Q Zhao ldquoSimulation of large photovoltaicgrid connected system based on modular multilevel con-verterrdquo Chinese Journal of Electrical Engineering vol 33no 36 pp 27ndash33 2013

[4] T Cheng M Chen Y Wang et al ldquoAdaptive robust methodfor dynamic economic emission dispatch incorporating re-newable energy and energy storagerdquo Complexity vol 2018Article ID 2517987 13 pages 2018

[5] X Guo X Zhang Z Lu B Wang H Qi and X Sun ldquoPowercurrent quality coordinated control strategy of photovoltaicgrid connected inverter under unbalanced grid voltagerdquo

VoltageCurrent

ndash400

ndash200

0

200

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 17 Simulation diagram of PI + FORC control with high-order harmonics

VoltageCurrent

ndash400

ndash200

0

200

400

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 18 Simulation results of PI +HORC control with high-order harmonics

Abrupt changeof current

VoltageCurrent

ndash400ndash300ndash200ndash100

0100200300400

UV

ndash200ndash150ndash100ndash50050100150200

IA

02 022 024 026 028 03 032 034 036018

Time (s)

Figure 19 e simulation diagram when current breaks inPI +HORC control

8 Complexity

Chinese Journal of Electrical Engineering vol 34 no 3pp 346ndash353 2014

[6] Z Yao and L Xiao ldquoControl of single-phase grid-connectedinverters with nonlinear loadsrdquo IEEE Transactions on In-dustrial Electronics vol 60 no 4 pp 1384ndash1389 2013

[7] Q Yan X Wu X Yuan and Y Geng ldquoAn improved grid-voltage feedforward strategy for high-power three-phase grid-connected inverters based on the simplified repetitive pre-dictorrdquo IEEE Transactions on Power Electronics vol 31 no 5pp 3880ndash3897 2016

[8] Y He H S-H Chung C-T Lai X Zhang and W WuldquoActive cancelation of equivalent grid impedance for im-proving stability and injected power quality of grid-connectedinverter under variable grid conditionrdquo IEEE Transactions onPower Electronics vol 33 no 11 pp 9387ndash9398 2018

[9] C Zhao J Liu Z Xie and F Zhou ldquoCoordinate control ofpowercurrent for grid-connected inverter based on PCIcontroller under unbalanced grid conditionsrdquo Complexityvol 2019 Article ID 5968984 14 pages 2019

[10] Q L Zhao X Q Guo and W Y Wu ldquoResearch on gridconnected control technology of single-phase inverterrdquo ChineseJournal of Electrical Engineering vol 27 no 16 pp 60ndash64 2007

[11] M Dong and A Luo ldquoDesign and control method of inverterin photovoltaic grid connected power generation systemrdquoPower System Automation vol 30 no 20 pp 97ndash102 2006

[12] X B Ruan and Y G Yan ldquoControl strategy of four leg three-phase inverterrdquo Transactions of China Electrotechnical Societyvol 15 no 1 pp 61ndash64 2000

[13] B Sahan A N Vergara N Henze A Engler and P ZachariasldquoA single-stage PVmodule integrated converter based on a low-power current-source inverterrdquo IEEE Transactions on Indus-trial Electronics vol 55 no 7 pp 2602ndash2609 2008

[14] J Selvaraj and N A Rahim ldquoMultilevel inverter for grid-con-nected PV system employing digital PI controllerrdquo IEEE Trans-actions on Industrial Electronics vol 56 no 1 pp 149ndash158 2009

[15] S Wang L Tao Q Chen J Na and X Ren ldquoUSDE-basedsliding mode control for servo mechanisms with unknownsystem dynamicsrdquo IEEEASME Transactions onMechatronicsvol 25 no 2 pp 1056ndash1066 2020

[16] J Fei T Li F Wang and W Juan ldquoA novel sliding modecontrol technique for indirect current controlled active powerfilterrdquo Mathematical Problems in Engineering vol 2012Article ID 549782 18 pages 2012

[17] Y-P Liu K-Z Liu and X Yang ldquoNonlinear current controlfor reluctance actuator with hysteresis compensationrdquo Jour-nal of Control Science and Engineering vol 2014 Article ID150345 7 pages 2014

[18] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[19] M Ruiz-Ruigomez J Badiola S M Schmidt-Malan et alldquoDirect electrical current reduces bacterial and yeast biofilmformationrdquo International Journal of Bacteriology vol 2016Article ID 9727810 6 pages 2016

[20] R Grino R Cardoner R Costa-Castello and E FossasldquoDigital repetitive control of a three-phase four-wire shuntactive filterrdquo IEEE Transactions on Industrial Electronicsvol 54 no 3 pp 1495ndash1503 2007

[21] R Costa-Castello R Grino and E Fossas ldquoOdd-harmonicdigital repetitive control of a single-phase current active fil-terrdquo IEEE Transactions on Power Electronics vol 19 no 4pp 1060ndash1068 2004

[22] G Weiss Q-C Zhong T C Green and J Liang ldquoRepetitivecontrol of DC-AC converters in microgridsrdquo IEEE Transac-tions on Power Electronics vol 19 no 1 pp 219ndash230 2004

[23] K Zhou and D Wang ldquoDigital repetitive learning controllerfor three-phase CVCF PWM inverterrdquo IEEE Transactions onIndustrial Electronics vol 48 pp 820ndash830 2001

[24] S Wang J Na and Y Xing ldquoAdaptive optimal parameterestimation and control of servo mechanisms theory andexperimentsrdquo IEEE Transactions on Industrial Electronicsvol 99 p 1 2020

[25] S Wang and J Na ldquoParameter estimation and adaptivecontrol for servo mechanisms with friction compensationrdquoIEEE Transactions on Industrial Informatics vol 99 p 12020

[26] J Zhao J Na and G Gao ldquoAdaptive dynamic programmingbased robust control of nonlinear systems with unmatcheduncertaintiesrdquo Neurocomputing vol 395 pp 56ndash65 2020

[27] P Giri A Das and A Bhattacharya ldquoPush-pull inverter basedwireless power transfer systemrdquo in Proceedings of the 7thInternational Conference on Signal Processing and IntegratedNetworks (SPIN) pp 489ndash493 Noida India February 2020

[28] Z Kaczmarczyk andW Jurczak ldquoA push-pull class-E inverterwith improved efficiencyrdquo IEEE Transactions on IndustrialElectronics vol 55 no 4 pp 1871ndash1874 2008

[29] E Babaei E Shokati Asl and M Hasan Babayi ldquoSteady-stateand small-signal analysis of high-voltage gain half-bridgeswitched boost inverterrdquo IEEE Transactions on IndustrialElectronics vol 63 no 6 pp 3546ndash3553 2016

[30] S-H Ryu D-G Woo M-K Kim and B-K Lee ldquoAnalysisand design of modified half-bridge series-resonant inverterwith DC-link neutral-point-clamped cellrdquo IEEE Transactionson Power Electronics vol 31 no 3 pp 2282ndash2295 2016

[31] L Zhang K Sun Y Xing and J Zhao ldquoA family of five-leveldual-buck full-bridge inverters for grid-tied applicationsrdquoIEEE Transactions on Power Electronics vol 31 pp 7029ndash7042 2016

[32] X Liu G Ma P R Pagilla and S S Ge ldquoState-estimator-based asynchronous repetitive control of discrete-time mar-kovian switching systemsrdquo Complexity vol 2020 Article ID6195162 13 pages 2020

[33] R Yi and C Boshi Electric Drive Automatic Control System-Motion Control System pp 70-71 China Machine PressBeijing China 2009

[34] J Na X Ren R Costa-Castello and Y Guo ldquoRepetitivecontrol of servo systems with time delaysrdquo Robotics andAutonomous Systems vol 62 no 3 pp 319ndash329 2014

[35] R Grintildeo and R Costa-Castello ldquoDigital repetitive plug-incontroller for odd-harmonic periodic references and distur-bancesrdquo Automatica vol 41 no 1 pp 153ndash157 2005

[36] R Costa-Castello J Nebot and R Grino ldquoDemonstration ofthe internal model principle by digital repetitive control of aneducational laboratory plantrdquo IEEE Transactions on Educa-tion vol 48 no 1 pp 73ndash80 2005

[37] W S Chang I H Suh and T W Kim ldquoAnalysis and designof two types of digital repetitive control systemsrdquoAutomaticavol 31 no 5 pp 741ndash746 1995

[38] M Steinbuch ldquoRepetitive control for systems with uncertainperiod-timerdquo Automatica vol 38 no 12 pp 2103ndash21092002

[39] T Inoue ldquoPractical repetitive control system designrdquo inProceedings of the 29th IEEE Conference on Decision andControl pp 1673ndash1678 Honolulu HI USA December 1990

[40] R Costa-Castello G A Ramos J M Olm and M SteinbuchldquoSecond-order odd-harmonic repetitive control and its

Complexity 9

application to active filter controlrdquo in Proceedings of the IEEEConference on Decision amp Control Atlanta GA USA De-cember 2011

10 Complexity

reduced to 198 when PI +FORC is adopted FinallyPI +HORC is used to suppress the current harmonics morerestrictively where the THD is reduced to 184 in Figure 15

42 Harmonic Suppression In order to further verify theharmonic suppression of the three controlmethods high-orderharmonics are added to the systemmanually e frequency of

Table 1 e simulation parameters

Parameters ValuesGrid phase voltage Us(V) 220DC side voltage UDC(V) 800Switching frequency fs(kHz) 10Filter inductance Ls(mH) 30Inductance equivalent resistance Rs(Ω) 01Input voltage frequency f(Hz) 50

VoltageCurrent

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 10 Simulation results of PI control

2

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20

Harmonic order

Fundamental (50Hz) = 6285 THD = 210

Figure 11 e THD of PI + FORC control

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA

VoltageCurrent

012 013 014 015 016 017 018 019 0201101

Time (s)

Figure 12 e THD of PI control

2

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20Harmonic order

Fundamental (50Hz) = 6293 THD = 198

Figure 13 Simulation results of PI +HORC control

VoltageCurrent

ndash300ndash200ndash100

0100200300

UV

ndash200

ndash100

0

100

200

IA

011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 14 Simulation results of PI + FORC control

15

1

05

0Mag

( o

f fun

dam

enta

l)

0 2 4 6 8 10 12 14 16 18 20

Harmonic order

Fundamental (50Hz) = 6303 THD = 184

Figure 15 e THD of PI +HORC control

VoltageCurrent

ndash400

ndash200

0

200

400

UA

ndash200

ndash100

0

100

200

IA

011 012 013 014 015 016 017 018 019 0201

Time (s)

Figure 16 Simulation results of PI control with high-orderharmonics

Complexity 7

the grid is set as 50Hz then the 7th and 9th harmonics areinjected It is shown in Figure 16 that PI control has poorperformance to suppress harmonics and it leads to greatdistortions in the current It can be seen from Figure 17 thatafter the introduction of FORC the high-order harmonics aresuppressed and the output current retains a better sinusoidalwaveform After using HORC we can get a smoother sinecurve as shown in Figure 18

43 Dynamic Tracking rough the above analysis we canfind that the HORC has a better harmonic suppression

performance In order to further verify the tracking per-formance of HORC the reference current is suddenly re-duced by 30 in 026s to simulate the sudden drop ofcurrent It is shown in Figure 19 that the current can berecovered in 028s erefore when the reference currentchanges suddenly the current can respond quickly and thereference current can be tracked after a short transient thatis the system has fast dynamic response

5 Conclusion

In this paper a novel control method combining PI controland repetitive control is proposed for a single-phase grid-connected inverter After introducing the single-phase in-verter type and modelling a first-order repetitive control anda high-order repetitive control are developed for the grid-connected inverter respectively e stability and perfor-mance analysis are all given for the proposed two repetitivecontrols Finally comparative simulations are conductedbased on the proposed single-phase grid-connected inverterto show that the high-order repetitive control can obtainbetter steady-state and dynamic features and reduce the netcurrent harmonics Future work will focus on the reactivepower compensation control and its practical application

Data Availability

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

Conflicts of Interest

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

Acknowledgments

is work was supported by China Southern Power GridCorporation (no YNKJXM20180366) and the ScientificResearch Fund of Yunnan Education Department underGrant 2020J0067

References

[1] C Zhang Y Ye and H Chen ldquoPhotovoltaic grid connectedinverter based on output current controlrdquo Journal of ElectricalTechnology vol 22 no 8 pp 41ndash45 2007

[2] F Wang S Yu and J Su ldquoResearch on solar photovoltaic gridconnected power generation systemrdquo Journal of ElectricalTechnology vol 20 no 5 pp 72ndash74 2005

[3] Z Yao F Yu and Q Zhao ldquoSimulation of large photovoltaicgrid connected system based on modular multilevel con-verterrdquo Chinese Journal of Electrical Engineering vol 33no 36 pp 27ndash33 2013

[4] T Cheng M Chen Y Wang et al ldquoAdaptive robust methodfor dynamic economic emission dispatch incorporating re-newable energy and energy storagerdquo Complexity vol 2018Article ID 2517987 13 pages 2018

[5] X Guo X Zhang Z Lu B Wang H Qi and X Sun ldquoPowercurrent quality coordinated control strategy of photovoltaicgrid connected inverter under unbalanced grid voltagerdquo

VoltageCurrent

ndash400

ndash200

0

200

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 17 Simulation diagram of PI + FORC control with high-order harmonics

VoltageCurrent

ndash400

ndash200

0

200

400

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 18 Simulation results of PI +HORC control with high-order harmonics

Abrupt changeof current

VoltageCurrent

ndash400ndash300ndash200ndash100

0100200300400

UV

ndash200ndash150ndash100ndash50050100150200

IA

02 022 024 026 028 03 032 034 036018

Time (s)

Figure 19 e simulation diagram when current breaks inPI +HORC control

8 Complexity

Chinese Journal of Electrical Engineering vol 34 no 3pp 346ndash353 2014

[6] Z Yao and L Xiao ldquoControl of single-phase grid-connectedinverters with nonlinear loadsrdquo IEEE Transactions on In-dustrial Electronics vol 60 no 4 pp 1384ndash1389 2013

[7] Q Yan X Wu X Yuan and Y Geng ldquoAn improved grid-voltage feedforward strategy for high-power three-phase grid-connected inverters based on the simplified repetitive pre-dictorrdquo IEEE Transactions on Power Electronics vol 31 no 5pp 3880ndash3897 2016

[8] Y He H S-H Chung C-T Lai X Zhang and W WuldquoActive cancelation of equivalent grid impedance for im-proving stability and injected power quality of grid-connectedinverter under variable grid conditionrdquo IEEE Transactions onPower Electronics vol 33 no 11 pp 9387ndash9398 2018

[9] C Zhao J Liu Z Xie and F Zhou ldquoCoordinate control ofpowercurrent for grid-connected inverter based on PCIcontroller under unbalanced grid conditionsrdquo Complexityvol 2019 Article ID 5968984 14 pages 2019

[10] Q L Zhao X Q Guo and W Y Wu ldquoResearch on gridconnected control technology of single-phase inverterrdquo ChineseJournal of Electrical Engineering vol 27 no 16 pp 60ndash64 2007

[11] M Dong and A Luo ldquoDesign and control method of inverterin photovoltaic grid connected power generation systemrdquoPower System Automation vol 30 no 20 pp 97ndash102 2006

[12] X B Ruan and Y G Yan ldquoControl strategy of four leg three-phase inverterrdquo Transactions of China Electrotechnical Societyvol 15 no 1 pp 61ndash64 2000

[13] B Sahan A N Vergara N Henze A Engler and P ZachariasldquoA single-stage PVmodule integrated converter based on a low-power current-source inverterrdquo IEEE Transactions on Indus-trial Electronics vol 55 no 7 pp 2602ndash2609 2008

[14] J Selvaraj and N A Rahim ldquoMultilevel inverter for grid-con-nected PV system employing digital PI controllerrdquo IEEE Trans-actions on Industrial Electronics vol 56 no 1 pp 149ndash158 2009

[15] S Wang L Tao Q Chen J Na and X Ren ldquoUSDE-basedsliding mode control for servo mechanisms with unknownsystem dynamicsrdquo IEEEASME Transactions onMechatronicsvol 25 no 2 pp 1056ndash1066 2020

[16] J Fei T Li F Wang and W Juan ldquoA novel sliding modecontrol technique for indirect current controlled active powerfilterrdquo Mathematical Problems in Engineering vol 2012Article ID 549782 18 pages 2012

[17] Y-P Liu K-Z Liu and X Yang ldquoNonlinear current controlfor reluctance actuator with hysteresis compensationrdquo Jour-nal of Control Science and Engineering vol 2014 Article ID150345 7 pages 2014

[18] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[19] M Ruiz-Ruigomez J Badiola S M Schmidt-Malan et alldquoDirect electrical current reduces bacterial and yeast biofilmformationrdquo International Journal of Bacteriology vol 2016Article ID 9727810 6 pages 2016

[20] R Grino R Cardoner R Costa-Castello and E FossasldquoDigital repetitive control of a three-phase four-wire shuntactive filterrdquo IEEE Transactions on Industrial Electronicsvol 54 no 3 pp 1495ndash1503 2007

[21] R Costa-Castello R Grino and E Fossas ldquoOdd-harmonicdigital repetitive control of a single-phase current active fil-terrdquo IEEE Transactions on Power Electronics vol 19 no 4pp 1060ndash1068 2004

[22] G Weiss Q-C Zhong T C Green and J Liang ldquoRepetitivecontrol of DC-AC converters in microgridsrdquo IEEE Transac-tions on Power Electronics vol 19 no 1 pp 219ndash230 2004

[23] K Zhou and D Wang ldquoDigital repetitive learning controllerfor three-phase CVCF PWM inverterrdquo IEEE Transactions onIndustrial Electronics vol 48 pp 820ndash830 2001

[24] S Wang J Na and Y Xing ldquoAdaptive optimal parameterestimation and control of servo mechanisms theory andexperimentsrdquo IEEE Transactions on Industrial Electronicsvol 99 p 1 2020

[25] S Wang and J Na ldquoParameter estimation and adaptivecontrol for servo mechanisms with friction compensationrdquoIEEE Transactions on Industrial Informatics vol 99 p 12020

[26] J Zhao J Na and G Gao ldquoAdaptive dynamic programmingbased robust control of nonlinear systems with unmatcheduncertaintiesrdquo Neurocomputing vol 395 pp 56ndash65 2020

[27] P Giri A Das and A Bhattacharya ldquoPush-pull inverter basedwireless power transfer systemrdquo in Proceedings of the 7thInternational Conference on Signal Processing and IntegratedNetworks (SPIN) pp 489ndash493 Noida India February 2020

[28] Z Kaczmarczyk andW Jurczak ldquoA push-pull class-E inverterwith improved efficiencyrdquo IEEE Transactions on IndustrialElectronics vol 55 no 4 pp 1871ndash1874 2008

[29] E Babaei E Shokati Asl and M Hasan Babayi ldquoSteady-stateand small-signal analysis of high-voltage gain half-bridgeswitched boost inverterrdquo IEEE Transactions on IndustrialElectronics vol 63 no 6 pp 3546ndash3553 2016

[30] S-H Ryu D-G Woo M-K Kim and B-K Lee ldquoAnalysisand design of modified half-bridge series-resonant inverterwith DC-link neutral-point-clamped cellrdquo IEEE Transactionson Power Electronics vol 31 no 3 pp 2282ndash2295 2016

[31] L Zhang K Sun Y Xing and J Zhao ldquoA family of five-leveldual-buck full-bridge inverters for grid-tied applicationsrdquoIEEE Transactions on Power Electronics vol 31 pp 7029ndash7042 2016

[32] X Liu G Ma P R Pagilla and S S Ge ldquoState-estimator-based asynchronous repetitive control of discrete-time mar-kovian switching systemsrdquo Complexity vol 2020 Article ID6195162 13 pages 2020

[33] R Yi and C Boshi Electric Drive Automatic Control System-Motion Control System pp 70-71 China Machine PressBeijing China 2009

[34] J Na X Ren R Costa-Castello and Y Guo ldquoRepetitivecontrol of servo systems with time delaysrdquo Robotics andAutonomous Systems vol 62 no 3 pp 319ndash329 2014

[35] R Grintildeo and R Costa-Castello ldquoDigital repetitive plug-incontroller for odd-harmonic periodic references and distur-bancesrdquo Automatica vol 41 no 1 pp 153ndash157 2005

[36] R Costa-Castello J Nebot and R Grino ldquoDemonstration ofthe internal model principle by digital repetitive control of aneducational laboratory plantrdquo IEEE Transactions on Educa-tion vol 48 no 1 pp 73ndash80 2005

[37] W S Chang I H Suh and T W Kim ldquoAnalysis and designof two types of digital repetitive control systemsrdquoAutomaticavol 31 no 5 pp 741ndash746 1995

[38] M Steinbuch ldquoRepetitive control for systems with uncertainperiod-timerdquo Automatica vol 38 no 12 pp 2103ndash21092002

[39] T Inoue ldquoPractical repetitive control system designrdquo inProceedings of the 29th IEEE Conference on Decision andControl pp 1673ndash1678 Honolulu HI USA December 1990

[40] R Costa-Castello G A Ramos J M Olm and M SteinbuchldquoSecond-order odd-harmonic repetitive control and its

Complexity 9

application to active filter controlrdquo in Proceedings of the IEEEConference on Decision amp Control Atlanta GA USA De-cember 2011

10 Complexity

the grid is set as 50Hz then the 7th and 9th harmonics areinjected It is shown in Figure 16 that PI control has poorperformance to suppress harmonics and it leads to greatdistortions in the current It can be seen from Figure 17 thatafter the introduction of FORC the high-order harmonics aresuppressed and the output current retains a better sinusoidalwaveform After using HORC we can get a smoother sinecurve as shown in Figure 18

43 Dynamic Tracking rough the above analysis we canfind that the HORC has a better harmonic suppression

performance In order to further verify the tracking per-formance of HORC the reference current is suddenly re-duced by 30 in 026s to simulate the sudden drop ofcurrent It is shown in Figure 19 that the current can berecovered in 028s erefore when the reference currentchanges suddenly the current can respond quickly and thereference current can be tracked after a short transient thatis the system has fast dynamic response

5 Conclusion

In this paper a novel control method combining PI controland repetitive control is proposed for a single-phase grid-connected inverter After introducing the single-phase in-verter type and modelling a first-order repetitive control anda high-order repetitive control are developed for the grid-connected inverter respectively e stability and perfor-mance analysis are all given for the proposed two repetitivecontrols Finally comparative simulations are conductedbased on the proposed single-phase grid-connected inverterto show that the high-order repetitive control can obtainbetter steady-state and dynamic features and reduce the netcurrent harmonics Future work will focus on the reactivepower compensation control and its practical application

Data Availability

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

Conflicts of Interest

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

Acknowledgments

is work was supported by China Southern Power GridCorporation (no YNKJXM20180366) and the ScientificResearch Fund of Yunnan Education Department underGrant 2020J0067

References

[1] C Zhang Y Ye and H Chen ldquoPhotovoltaic grid connectedinverter based on output current controlrdquo Journal of ElectricalTechnology vol 22 no 8 pp 41ndash45 2007

[2] F Wang S Yu and J Su ldquoResearch on solar photovoltaic gridconnected power generation systemrdquo Journal of ElectricalTechnology vol 20 no 5 pp 72ndash74 2005

[3] Z Yao F Yu and Q Zhao ldquoSimulation of large photovoltaicgrid connected system based on modular multilevel con-verterrdquo Chinese Journal of Electrical Engineering vol 33no 36 pp 27ndash33 2013

[4] T Cheng M Chen Y Wang et al ldquoAdaptive robust methodfor dynamic economic emission dispatch incorporating re-newable energy and energy storagerdquo Complexity vol 2018Article ID 2517987 13 pages 2018

[5] X Guo X Zhang Z Lu B Wang H Qi and X Sun ldquoPowercurrent quality coordinated control strategy of photovoltaicgrid connected inverter under unbalanced grid voltagerdquo

VoltageCurrent

ndash400

ndash200

0

200

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 17 Simulation diagram of PI + FORC control with high-order harmonics

VoltageCurrent

ndash400

ndash200

0

200

400

UV

ndash200

ndash100

0

100

200

IA

004 006 008 01 012 014002

Time (s)

Figure 18 Simulation results of PI +HORC control with high-order harmonics

Abrupt changeof current

VoltageCurrent

ndash400ndash300ndash200ndash100

0100200300400

UV

ndash200ndash150ndash100ndash50050100150200

IA

02 022 024 026 028 03 032 034 036018

Time (s)

Figure 19 e simulation diagram when current breaks inPI +HORC control

8 Complexity

Chinese Journal of Electrical Engineering vol 34 no 3pp 346ndash353 2014

[6] Z Yao and L Xiao ldquoControl of single-phase grid-connectedinverters with nonlinear loadsrdquo IEEE Transactions on In-dustrial Electronics vol 60 no 4 pp 1384ndash1389 2013

[7] Q Yan X Wu X Yuan and Y Geng ldquoAn improved grid-voltage feedforward strategy for high-power three-phase grid-connected inverters based on the simplified repetitive pre-dictorrdquo IEEE Transactions on Power Electronics vol 31 no 5pp 3880ndash3897 2016

[8] Y He H S-H Chung C-T Lai X Zhang and W WuldquoActive cancelation of equivalent grid impedance for im-proving stability and injected power quality of grid-connectedinverter under variable grid conditionrdquo IEEE Transactions onPower Electronics vol 33 no 11 pp 9387ndash9398 2018

[9] C Zhao J Liu Z Xie and F Zhou ldquoCoordinate control ofpowercurrent for grid-connected inverter based on PCIcontroller under unbalanced grid conditionsrdquo Complexityvol 2019 Article ID 5968984 14 pages 2019

[10] Q L Zhao X Q Guo and W Y Wu ldquoResearch on gridconnected control technology of single-phase inverterrdquo ChineseJournal of Electrical Engineering vol 27 no 16 pp 60ndash64 2007

[11] M Dong and A Luo ldquoDesign and control method of inverterin photovoltaic grid connected power generation systemrdquoPower System Automation vol 30 no 20 pp 97ndash102 2006

[12] X B Ruan and Y G Yan ldquoControl strategy of four leg three-phase inverterrdquo Transactions of China Electrotechnical Societyvol 15 no 1 pp 61ndash64 2000

[13] B Sahan A N Vergara N Henze A Engler and P ZachariasldquoA single-stage PVmodule integrated converter based on a low-power current-source inverterrdquo IEEE Transactions on Indus-trial Electronics vol 55 no 7 pp 2602ndash2609 2008

[14] J Selvaraj and N A Rahim ldquoMultilevel inverter for grid-con-nected PV system employing digital PI controllerrdquo IEEE Trans-actions on Industrial Electronics vol 56 no 1 pp 149ndash158 2009

[15] S Wang L Tao Q Chen J Na and X Ren ldquoUSDE-basedsliding mode control for servo mechanisms with unknownsystem dynamicsrdquo IEEEASME Transactions onMechatronicsvol 25 no 2 pp 1056ndash1066 2020

[16] J Fei T Li F Wang and W Juan ldquoA novel sliding modecontrol technique for indirect current controlled active powerfilterrdquo Mathematical Problems in Engineering vol 2012Article ID 549782 18 pages 2012

[17] Y-P Liu K-Z Liu and X Yang ldquoNonlinear current controlfor reluctance actuator with hysteresis compensationrdquo Jour-nal of Control Science and Engineering vol 2014 Article ID150345 7 pages 2014

[18] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[19] M Ruiz-Ruigomez J Badiola S M Schmidt-Malan et alldquoDirect electrical current reduces bacterial and yeast biofilmformationrdquo International Journal of Bacteriology vol 2016Article ID 9727810 6 pages 2016

[20] R Grino R Cardoner R Costa-Castello and E FossasldquoDigital repetitive control of a three-phase four-wire shuntactive filterrdquo IEEE Transactions on Industrial Electronicsvol 54 no 3 pp 1495ndash1503 2007

[21] R Costa-Castello R Grino and E Fossas ldquoOdd-harmonicdigital repetitive control of a single-phase current active fil-terrdquo IEEE Transactions on Power Electronics vol 19 no 4pp 1060ndash1068 2004

[22] G Weiss Q-C Zhong T C Green and J Liang ldquoRepetitivecontrol of DC-AC converters in microgridsrdquo IEEE Transac-tions on Power Electronics vol 19 no 1 pp 219ndash230 2004

[23] K Zhou and D Wang ldquoDigital repetitive learning controllerfor three-phase CVCF PWM inverterrdquo IEEE Transactions onIndustrial Electronics vol 48 pp 820ndash830 2001

[24] S Wang J Na and Y Xing ldquoAdaptive optimal parameterestimation and control of servo mechanisms theory andexperimentsrdquo IEEE Transactions on Industrial Electronicsvol 99 p 1 2020

[25] S Wang and J Na ldquoParameter estimation and adaptivecontrol for servo mechanisms with friction compensationrdquoIEEE Transactions on Industrial Informatics vol 99 p 12020

[26] J Zhao J Na and G Gao ldquoAdaptive dynamic programmingbased robust control of nonlinear systems with unmatcheduncertaintiesrdquo Neurocomputing vol 395 pp 56ndash65 2020

[27] P Giri A Das and A Bhattacharya ldquoPush-pull inverter basedwireless power transfer systemrdquo in Proceedings of the 7thInternational Conference on Signal Processing and IntegratedNetworks (SPIN) pp 489ndash493 Noida India February 2020

[28] Z Kaczmarczyk andW Jurczak ldquoA push-pull class-E inverterwith improved efficiencyrdquo IEEE Transactions on IndustrialElectronics vol 55 no 4 pp 1871ndash1874 2008

[29] E Babaei E Shokati Asl and M Hasan Babayi ldquoSteady-stateand small-signal analysis of high-voltage gain half-bridgeswitched boost inverterrdquo IEEE Transactions on IndustrialElectronics vol 63 no 6 pp 3546ndash3553 2016

[30] S-H Ryu D-G Woo M-K Kim and B-K Lee ldquoAnalysisand design of modified half-bridge series-resonant inverterwith DC-link neutral-point-clamped cellrdquo IEEE Transactionson Power Electronics vol 31 no 3 pp 2282ndash2295 2016

[31] L Zhang K Sun Y Xing and J Zhao ldquoA family of five-leveldual-buck full-bridge inverters for grid-tied applicationsrdquoIEEE Transactions on Power Electronics vol 31 pp 7029ndash7042 2016

[32] X Liu G Ma P R Pagilla and S S Ge ldquoState-estimator-based asynchronous repetitive control of discrete-time mar-kovian switching systemsrdquo Complexity vol 2020 Article ID6195162 13 pages 2020

[33] R Yi and C Boshi Electric Drive Automatic Control System-Motion Control System pp 70-71 China Machine PressBeijing China 2009

[34] J Na X Ren R Costa-Castello and Y Guo ldquoRepetitivecontrol of servo systems with time delaysrdquo Robotics andAutonomous Systems vol 62 no 3 pp 319ndash329 2014

[35] R Grintildeo and R Costa-Castello ldquoDigital repetitive plug-incontroller for odd-harmonic periodic references and distur-bancesrdquo Automatica vol 41 no 1 pp 153ndash157 2005

[36] R Costa-Castello J Nebot and R Grino ldquoDemonstration ofthe internal model principle by digital repetitive control of aneducational laboratory plantrdquo IEEE Transactions on Educa-tion vol 48 no 1 pp 73ndash80 2005

[37] W S Chang I H Suh and T W Kim ldquoAnalysis and designof two types of digital repetitive control systemsrdquoAutomaticavol 31 no 5 pp 741ndash746 1995

[38] M Steinbuch ldquoRepetitive control for systems with uncertainperiod-timerdquo Automatica vol 38 no 12 pp 2103ndash21092002

[39] T Inoue ldquoPractical repetitive control system designrdquo inProceedings of the 29th IEEE Conference on Decision andControl pp 1673ndash1678 Honolulu HI USA December 1990

[40] R Costa-Castello G A Ramos J M Olm and M SteinbuchldquoSecond-order odd-harmonic repetitive control and its

Complexity 9

application to active filter controlrdquo in Proceedings of the IEEEConference on Decision amp Control Atlanta GA USA De-cember 2011

10 Complexity

Chinese Journal of Electrical Engineering vol 34 no 3pp 346ndash353 2014

[6] Z Yao and L Xiao ldquoControl of single-phase grid-connectedinverters with nonlinear loadsrdquo IEEE Transactions on In-dustrial Electronics vol 60 no 4 pp 1384ndash1389 2013

[7] Q Yan X Wu X Yuan and Y Geng ldquoAn improved grid-voltage feedforward strategy for high-power three-phase grid-connected inverters based on the simplified repetitive pre-dictorrdquo IEEE Transactions on Power Electronics vol 31 no 5pp 3880ndash3897 2016

[8] Y He H S-H Chung C-T Lai X Zhang and W WuldquoActive cancelation of equivalent grid impedance for im-proving stability and injected power quality of grid-connectedinverter under variable grid conditionrdquo IEEE Transactions onPower Electronics vol 33 no 11 pp 9387ndash9398 2018

[9] C Zhao J Liu Z Xie and F Zhou ldquoCoordinate control ofpowercurrent for grid-connected inverter based on PCIcontroller under unbalanced grid conditionsrdquo Complexityvol 2019 Article ID 5968984 14 pages 2019

[10] Q L Zhao X Q Guo and W Y Wu ldquoResearch on gridconnected control technology of single-phase inverterrdquo ChineseJournal of Electrical Engineering vol 27 no 16 pp 60ndash64 2007

[11] M Dong and A Luo ldquoDesign and control method of inverterin photovoltaic grid connected power generation systemrdquoPower System Automation vol 30 no 20 pp 97ndash102 2006

[12] X B Ruan and Y G Yan ldquoControl strategy of four leg three-phase inverterrdquo Transactions of China Electrotechnical Societyvol 15 no 1 pp 61ndash64 2000

[13] B Sahan A N Vergara N Henze A Engler and P ZachariasldquoA single-stage PVmodule integrated converter based on a low-power current-source inverterrdquo IEEE Transactions on Indus-trial Electronics vol 55 no 7 pp 2602ndash2609 2008

[14] J Selvaraj and N A Rahim ldquoMultilevel inverter for grid-con-nected PV system employing digital PI controllerrdquo IEEE Trans-actions on Industrial Electronics vol 56 no 1 pp 149ndash158 2009

[15] S Wang L Tao Q Chen J Na and X Ren ldquoUSDE-basedsliding mode control for servo mechanisms with unknownsystem dynamicsrdquo IEEEASME Transactions onMechatronicsvol 25 no 2 pp 1056ndash1066 2020

[16] J Fei T Li F Wang and W Juan ldquoA novel sliding modecontrol technique for indirect current controlled active powerfilterrdquo Mathematical Problems in Engineering vol 2012Article ID 549782 18 pages 2012

[17] Y-P Liu K-Z Liu and X Yang ldquoNonlinear current controlfor reluctance actuator with hysteresis compensationrdquo Jour-nal of Control Science and Engineering vol 2014 Article ID150345 7 pages 2014

[18] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[19] M Ruiz-Ruigomez J Badiola S M Schmidt-Malan et alldquoDirect electrical current reduces bacterial and yeast biofilmformationrdquo International Journal of Bacteriology vol 2016Article ID 9727810 6 pages 2016

[20] R Grino R Cardoner R Costa-Castello and E FossasldquoDigital repetitive control of a three-phase four-wire shuntactive filterrdquo IEEE Transactions on Industrial Electronicsvol 54 no 3 pp 1495ndash1503 2007

[21] R Costa-Castello R Grino and E Fossas ldquoOdd-harmonicdigital repetitive control of a single-phase current active fil-terrdquo IEEE Transactions on Power Electronics vol 19 no 4pp 1060ndash1068 2004

[22] G Weiss Q-C Zhong T C Green and J Liang ldquoRepetitivecontrol of DC-AC converters in microgridsrdquo IEEE Transac-tions on Power Electronics vol 19 no 1 pp 219ndash230 2004

[23] K Zhou and D Wang ldquoDigital repetitive learning controllerfor three-phase CVCF PWM inverterrdquo IEEE Transactions onIndustrial Electronics vol 48 pp 820ndash830 2001

[24] S Wang J Na and Y Xing ldquoAdaptive optimal parameterestimation and control of servo mechanisms theory andexperimentsrdquo IEEE Transactions on Industrial Electronicsvol 99 p 1 2020

[25] S Wang and J Na ldquoParameter estimation and adaptivecontrol for servo mechanisms with friction compensationrdquoIEEE Transactions on Industrial Informatics vol 99 p 12020

[26] J Zhao J Na and G Gao ldquoAdaptive dynamic programmingbased robust control of nonlinear systems with unmatcheduncertaintiesrdquo Neurocomputing vol 395 pp 56ndash65 2020

[27] P Giri A Das and A Bhattacharya ldquoPush-pull inverter basedwireless power transfer systemrdquo in Proceedings of the 7thInternational Conference on Signal Processing and IntegratedNetworks (SPIN) pp 489ndash493 Noida India February 2020

[28] Z Kaczmarczyk andW Jurczak ldquoA push-pull class-E inverterwith improved efficiencyrdquo IEEE Transactions on IndustrialElectronics vol 55 no 4 pp 1871ndash1874 2008

[29] E Babaei E Shokati Asl and M Hasan Babayi ldquoSteady-stateand small-signal analysis of high-voltage gain half-bridgeswitched boost inverterrdquo IEEE Transactions on IndustrialElectronics vol 63 no 6 pp 3546ndash3553 2016

[30] S-H Ryu D-G Woo M-K Kim and B-K Lee ldquoAnalysisand design of modified half-bridge series-resonant inverterwith DC-link neutral-point-clamped cellrdquo IEEE Transactionson Power Electronics vol 31 no 3 pp 2282ndash2295 2016

[31] L Zhang K Sun Y Xing and J Zhao ldquoA family of five-leveldual-buck full-bridge inverters for grid-tied applicationsrdquoIEEE Transactions on Power Electronics vol 31 pp 7029ndash7042 2016

[32] X Liu G Ma P R Pagilla and S S Ge ldquoState-estimator-based asynchronous repetitive control of discrete-time mar-kovian switching systemsrdquo Complexity vol 2020 Article ID6195162 13 pages 2020

[33] R Yi and C Boshi Electric Drive Automatic Control System-Motion Control System pp 70-71 China Machine PressBeijing China 2009

[34] J Na X Ren R Costa-Castello and Y Guo ldquoRepetitivecontrol of servo systems with time delaysrdquo Robotics andAutonomous Systems vol 62 no 3 pp 319ndash329 2014

[35] R Grintildeo and R Costa-Castello ldquoDigital repetitive plug-incontroller for odd-harmonic periodic references and distur-bancesrdquo Automatica vol 41 no 1 pp 153ndash157 2005

[36] R Costa-Castello J Nebot and R Grino ldquoDemonstration ofthe internal model principle by digital repetitive control of aneducational laboratory plantrdquo IEEE Transactions on Educa-tion vol 48 no 1 pp 73ndash80 2005

[37] W S Chang I H Suh and T W Kim ldquoAnalysis and designof two types of digital repetitive control systemsrdquoAutomaticavol 31 no 5 pp 741ndash746 1995

[38] M Steinbuch ldquoRepetitive control for systems with uncertainperiod-timerdquo Automatica vol 38 no 12 pp 2103ndash21092002

[39] T Inoue ldquoPractical repetitive control system designrdquo inProceedings of the 29th IEEE Conference on Decision andControl pp 1673ndash1678 Honolulu HI USA December 1990

[40] R Costa-Castello G A Ramos J M Olm and M SteinbuchldquoSecond-order odd-harmonic repetitive control and its

Complexity 9

application to active filter controlrdquo in Proceedings of the IEEEConference on Decision amp Control Atlanta GA USA De-cember 2011

10 Complexity

application to active filter controlrdquo in Proceedings of the IEEEConference on Decision amp Control Atlanta GA USA De-cember 2011

10 Complexity