lang li, shi, minghui zheng gg ne y ne l ne v vpe.csu.edu.cn/lunwen/a decentralized control for...

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2019.2945266, IEEETransactions on Industrial Electronics

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

Abstract—In this letter, a decentralized control for cascaded inverters is introduced, in which one inverter is controlled as a current source and the others are controlled as voltage sources. The power sharing and synchronization of the inverters are realized without high-bandwidth communication. Meanwhile, it is robust against the voltage sag\swell and frequency deviation of grid. Finally, the feasibility of the proposed method is verified by simulation and experimental results.

Index Terms—Cascaded inverters, decentralized control, power sharing control.

I. INTRODUCTION

HE cascaded H-bridge inverters have been widely studied

and applied in middle- and high-voltage level power

network [1]. They are early used for high-voltage motor drivers,

STATCOM, and flexible AC transmission systems (FACTS)

[2]. Recently, they have been extended to distributed

generation, such as AC-stacked PV generation, energy storage

system and micro-grids [3-4].

In the past, the centralized control methods [5-6] were

widely used for grid-connected cascaded inverters. However,

these methods depend on real-time communication networks

and powerful centralized controller, which will lead to the

reduced reliability due to communication failure, and higher

capital costs. Moreover, it becomes more difficult for the

centralized methods to be implemented when the number of

modules is large.

Recently, there has been some interests on the distributed

control strategies [7-8]. A novel power regulation controller for

the grid-connected cascaded inverters is designed in [7], but the

communication is always needed to maintain the

Manuscript received January 15, 2019; revised May 5, 2019 and July

29, 2019; accepted September 22, 2019. This work was supported in part by the National Natural Science Foundation of China under Grant 61622311, the Joint Research Fund of Chinese Ministry of Education under Grant 6141A02033514, the Major Project of Changzhutan Self-dependent Innovation Demonstration Area under Grant 2018XK2002, the Project of Innovation-driven Plan in Central South University under Grant 2019CX003, and the Fundamental Research Funds for the Central Universities of Central South University under Grant 2019zzts276. (Corresponding author: Yao Sun.)

L. Li, Y. Sun, H. Han, G. Shi, and M. Su are with the School of Automation, Central South University and the Hunan Provincial Key Laboratory of Power Electronics Equipment and Gird, Changsha 410083, China (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected]).

M. Zheng is with Department of Mechanical and Aerospace Engineering, University at Buffalo, The State University of New York, New York, USA (e-mail: [email protected]).

synchronization of system. In addition, the decentralized

control methods have also been presented for cascaded

H-bridge inverters [9-12]. The control schemes could be

classified into two categories according to operation modes. In

the islanded mode, [9-10] proposed the inverse power factor

droop control for power sharing and autonomous

synchronization of system. In grid-connected mode, [11-12]

have been presented without any communication, in which all

modules are controlled as voltage sources. However, the

methods are sensitive to the variations of grid voltage and may

lead to instability. Thus, a more robust control approach against

the grid voltage variations should be developed.

To address the above concerns, a hybrid current-voltage

control scheme is proposed for grid-connected cascaded

inverters in this letter. Compared to centralized methods, the

proposed scheme have advantages of no need of high

bandwidth communications. Therefore, it is more attractive in

the applications where the number of inverter modules is large,

and the distance between the modules is long. The main

features of the proposed scheme are summarized as follows:1)

the phase synchronization of all modules is realized by using

the common current information, thus the decentralized manner

is obtained; 2) It is robust against the grid voltage sag,

distortion and frequency deviations.

II. ANALYSIS OF THE PROPOSED CONTROL SCHEME

A. Equivalent models of grid-connected cascaded inverters

H-bridge #1

PW

M

PR

Current control mode

*

lineilinei

H-bridge #i

PW

M

Inn

er l

oo

ps

Voltage control mode

linei

The grid power

requirements

,g gP Q

lineI

line

*P

g gV

*P

fL

fC

2 2V

n nV

g gV

line lineI

Voltage

control mode

Current

control mode

line lineZ Inverter #1

Inverter #2

Inverter #n

Utility

Grid

Inverter #1

Inverter #i

Fig. 1. Structure of grid-connected cascaded inverters.

Figure 1 shows the configuration of the grid-connected

cascaded inverters, which consists of n H-bridge modules. iV

and i represent the voltage amplitudes and phase angle of the

ith module. lineI and line are grid-connected current amplitudes

A Decentralized Control for Cascaded Inverters in Grid-connected Applications

Lang Li, Student Member, IEEE, Yao Sun, Member, IEEE, Hua Han, Member, IEEE, Guangze Shi, Student Member, IEEE, Mei Su, Member, IEEE, Minghui Zheng

T




and phase-angle. gV and g are the voltage amplitudes and

phase angle of the utility grid. lineZ and line are the grid

impedance amplitudes and phase angle. Only the inverter

(inverter #1) near to the grid is controlled as a current source

line lineI . All other inverters 2,3, ,i n are controlled as

voltage sources i iV .

B. Proposed control strategy

Assume that the active and reactive power requirement of the

grid are gP and gQ , which are expressed as

1

cos2

g g line g lineP V I (1)

1

sin2

g g line g lineQ V I (2)

The proposed decentralized control is a hybrid

current-voltage control scheme, which is depicted in Fig. 1.

For the inverter #1, its current reference is line lineI . From

(1) and (2), lineI and line are

2

cos

g

line

g

PI

V (3)

line g (4)

where arctan g gQ P . Then, the line current is regulated

to track line lineI by a proportional-resonance (PR) controller

[13].

The other inverters #i 2,3, ,i n are controlled as

voltage sources, and the corresponding reference voltage is

i iV . Assume that the output active power and reactive power

reference of ith module are *P and *Q , respectively. Then we

have

,i line i (5)

*

,

2

cosi

line i

PV

I (6)

where ,line i and ,line iI are the real-time phase angle and

amplitudes of line current. Then, the output voltage of the ith

module is controlled by the double-loop voltage-current

controller [13-14]. Since they could be obtained locally by each

inverter, these inverters are controlled in decentralized

manners.

In the proposed control frame, the inverter #1 is responsible

for regulating the grid current. Meanwhile, the others are

controlled as voltage sources to maintain the system

synchronization and power regulations according to the

common current.

C. Steady-state analysis

Due to the series structure, the output currents of all inverters

are completely the same, (7) is obtained in the steady-state.

,2 ,3 ,line line line n line (7)

,2 ,3 ,line line line n lineI I I I (8)

From (5) and (6), we have

2 3 n g (9)

2 3 nV V V (10)

That is to say, these inverters could keep pace with the grid. In

this study we let *gP P n and *

gQ Q n . Combining

(9)-(10), , cosi i i lineP V I and , sini i i lineQ V I , the following

equations are derived *

2 3 nP P P P (11)

*

2 3 nQ Q Q Q (12)

According to the power balance principle, the profit and loss

of power of the system is provided by the inverter #1. Due to

the inductive grid impedance, the output active and reactive

power of inverter #1 are

*

1

2

n

g i

i

P P P P

(13)

*

1

2

n

g line i

i

Q Q Q Q Q

(14)

where lineQ is the reactive power loss of the grid impedance.

In practice, grid voltage may change within contain limits.

Then gP and gQ will change accordingly. Thus, from (13)-(14),

the power rating of inverter #1 will be determined by the

fluctuation ranges of grid and safety margin.

D. Stability analysis

From Fig. 1, the model of system is expressed as

,1

2

,

,

,

n

out g i

i

L i

f out i i

i

f L i

diL u u u

dt

diL u u

dt

duC i i

dt

(15)

where L is the grid inductor, fL and fC are the inductor and

capacitance of LC filter, the subscript 2,3, ,i n . gu is the

grid voltage, i is the grid current, ,out iu is the output voltage of

the inverter #i, ,L ii is the filter inductor current of inverter #i.

The PR controller for inverter #1 is expressed as

*

,1 ,1

,1 ,2

2

,2 ,1 ,22 2

out P I

I I

I c r I c I

u k i i x

x x

x k i x x

(16)

where Pk and rk are the proportional and resonance coefficient

of inverter #1. c is the cut-off frequency, is the

synchronous angle frequency of grid, *i is the reference grid

current.

The voltage-current dual closed loop controllers for the

inverters #i 2,3, ,i n are expressed as

*

, ,1 ,

,1 ,2

2

,2 ,1 ,22 2

out i Pi Pv i i Pi vi Pi L i

vi vi

vi c rv i vi c vi

u k k u u k x k i

x x

x k u x x

(17)




where Pik , Pvk , rvk are the control coefficients of PR controller.

*

iu and *

,L ii are the reference capacitor voltage and inductor

current.

Because this system is a periodic system, the sinusoidal

analysis method is applied [15] to prove the stability of the

system. Let Re j t

g gu V e , Re j ti Ie , , ,Re j t

out i out iu V e ,

, ,Re j t

L i L ii I e , ,1 ,1Re j t

I Ix X e , ,2 ,2Re j t

I Ix X e ,

,1 ,1Re j t

vi vix X e , ,2 ,2Re j t

vi vix X e , where Re returns

the real part of the complex argument.

Combining (15)-(17) yields (18).

,1

2

,1 ,1 ,2

2

,2 ,2 ,1 ,2

,

, ,

,

,1 ,1 ,2

2

,2 ,2 ,1 ,2

2 X 2

2 2

n

out g i

i

I I I

I I c r I c I

L i

f L i f out i i

i

f i f L i

vi vi I

vi vi c rv i i vi c vi

dIj LI L V V V

dt

j X X X

j X X k j I I X

dIj L I L V V

dt

dVj C V C I I

dt

j X X X

j X X k j V V X X

(18)

Further, the small signal model of system is expressed as

,1

2

,1 ,2 ,1

2

,2 ,1 ,2

2

, , ,1

1 1

2 2 22

1

1

nP

I i

i

I I I

nc r P c r c r

I I c I i

i

Pi Pv Pi Pi

L i i L i vi

f f f f

i

f

kI j I X V

L L L

X X j X

k k k kX I X j X V

L L L

k k k kI V j I X

L L L L

V IC

,

,1 ,2 ,1

2

,2 , ,1 ,2

1

2 22

L i i

f

vi vi vi

c rv c rv

vi L i vi c vi

f f

I j VC

X X j X

k kX I I X j X

C C

(19)

For brevity, (19) is rewritten as matrix form

X AX (20)

where the state variable vector X and the system matrix A are

-3000 -2000 -1000 0-2

-1

0

1

2x 10

4

1

2

3

4

56

7

-200 -150 -100 -50-1500

-1000

-500

0

500

1000

8 9

10

11

0.01 8pk

100rk

3pik

0.5pvk

50rvk

Real

Imag

inar

y

(a)

-3000 -2000 -1000 0-2

-1

0

1

2x 10

4

1

2

3

4

5 6

7

-100 -50 0

-1000

-500

0

500

11

8

9

10

0.2pk

3pik

0.5pvk

50rvk

50 200rk

Real

Imag

inar

y

(b)

35L e35L e

-4000 -3000 -2000 -1000 0-2

-1

0

1

2 x 104

1

2

3

4

-1000 -500 0

-600

-400

-200

0

200

400 56

7

89

10

11

0.2pk

100rk

0.5pvk

50rvk

0.01 6pik

Real(c)

Imag

inar

y

-3000 -2000 -1000 0-4

-2

0

2

4 x 104

1

2

3

4

-300 -200 -100

-500

0

5005

6

7

8

9

10

11

0.2pk 100rk

3pik 50rvk

0.01 2.5pvk

Real(d)

Imag

inar

y 35L e

35L e

-3000 -2000 -1000 0-2

-1

0

1

2 x 104

1

2

3

4

5

-300 -200 -100

-500

0

5006

7

89

10

11

0.2pk 100rk

3pik 0.5pvk

30 120pvk

Real(e)

Imag

inar

y

-6000 -4000 -2000 0-4

-2

0

2

4 x 104

1

3

2

5

4

-300 -200 -100 0

-500

0

500

6

7

8

9

10

11

0.2pk 100rk

3pik 0.5pvk

50pvk

Real(f)

Imag

inar

y

3 30.1 8L e e

35L e

Fig. 2. Root locus diagram as parameter changes.

shown in Appendix. The root locus of the closed loop system is

depicted based on the experimental tests, the effect of Pk , rk

Pik , Pvk , rvk and L are studied. As seen the root locus diagrams

in Fig. 2, all eigenvalues are always in the left half-plane. That

is to say, the system is stable when the appropriate control and

physics parameters are selected.

III. SIMULATION RESULTS

H-brideg #1

H-brideg #2

H-brideg #6

Utility

Grid

g gV

2 2V

6 6V

line lineI

1.2mH

600V

50Hz

Module #1

Module #2

Module #6

Fig. 3. Simulation model consisting of six modules.

TABLE I PARAMETERS FOR SIMULATIONS

Method Parameters Values Parameters Values

Proposed

scheme

*gV (V)

600 *P (W) 1000

lineL (H) 1.2e-3 *Q (Var) 500

Method in

[11]

*gV (V)

600 *P (W) 1000

*V (V) 98 k 1.2e-3

lineL (H) 1.2e-3 * (rad/s) 100




The circuit level simulation model consisting of six modules

are carried out on the platform of MATLAB/SIMULINK using

SimPowerSystems (see Fig.3). This simulation is implemented

based on the method in [11] and the proposed scheme. The

associated parameters are listed in Table I.

0 2 4 6 8 10-1000

0

1000

2000

3000

Time(sec)(a1)

Act

ive

Po

wer

(W)

0 2 4 6 8 10-4000

-2000

0

2000

4000

6000

8000

Time(sec)(b1)

Rea

ctiv

e P

ow

er(V

ar)

0 2 4 6 8 1049.649.749.849.9

5050.150.250.350.4

Time(sec)(c1)

Fre

qu

ency

(Hz)

0

200

400

600

800

1000

1200

0 2 4 6 8 10Time(sec)

(a2)

Act

ive

Pow

er(W

)

0

100200300

400500600700

0 2 4 6 81

0Time(sec)

Rea

ctiv

e P

ow

er(V

ar)

(b2)

49.8

49.9

50

50.1

50.2

0 2 4 6 8 10Time(sec)

(c2)

Fre

qu

ency

(Hz)

1 6f f

1 6Q Q

1 6P P

2 6Q Q

1 6P P

1Q

1 6f f

Fig. 4. Simulation results.

A. Case1: the method in [11]

This simulation with 10% grid voltage sag at t=5s is

performed based on the method in [11]. The active power and

reactive power sharing results are depicted in Fig. 4(a1) and

(b1). The reactive power is negative to maintain the system

stable operation [11] before t=5s. The frequencies are shown in

Fig. 4(c1). Therefore, it is concluded that the system is unstable

under the grid voltage sag condition based on the method in

[11].

B. Case2: the proposed scheme

Under the same setup as case 1, this simulation is

implemented with the proposed scheme under the 10% grid

voltage sag condition at t=5s. The waveforms of active power,

reactive power and frequency are shown in Fig. 4(a2), (b2) and

(c2), respectively. The reactive power is always positive, which

indicates that the proposed scheme holds the capability of

reactive power compensation to the grid. When the grid voltage

sag occurs after t=5s, the system could always maintain the

system stable and inject the desired powers into the grid. The

simulation results show that the proposed scheme is more

robust compared to the method in [11] under the same grid

voltage sag condition.

C. Case3: fault-tolerant operation with the proposed scheme

This simulation with module #6 suddenly lost at t=5s is

carried out based on the proposed scheme. When a failure

occurs in module #6, the bypass method [16] is applied. The

waveforms of active and reactive power are shown in Fig. 5(a)

and (b), in which the power shortage of module #6 is supplied

automatically by module #1 because it is a free variable.

Therefore, the proposed scheme can realize the fault-tolerant

operation within a certain extent.

0 2 4 6 8 10-500

0

500

1000

1500

2000

2500

Time(sec)

Act

ive

Po

wer

(W)

1P

6P

2 5P P

0

200

400

600

800

1000

1200

0 2 4 6 8 10Time(sec)

Rea

ctiv

e P

ow

er(V

ar)

1Q

6Q

2 5Q Q

(a) (b)

Fig. 5. Simulation results when module #6 is by-passed.

IV. EXPERIMENTAL RESULTS

Personal

Computer

Scope

Inverter-1

Inverter-2

Inverter-3

Driver-1

DSP+FPGA

Control Platform

DC Power

Supply-1

DC Power

Supply-2

DC Power

Supply-3

Driver-3

Driver-2

LC filterLC filter

Line

Inductor Switch

Programmable

Power Supply

Grid

Fig. 6. Prototype setup of the grid-connected cascaded inverters.

The prototype setup of the grid-connected cascaded inverters

shown in Fig. 6 is built in the lab, which is consisting of three

modules. The control diagram of the proposed scheme is shown

in Fig. 1. The experimental parameters are listed in Table II. TABLE II

PARAMETERS FOR EXPERIMENTS

Parameters Values Parameters Values

*gV (V)

60 *P (W) 20

lineL (H) 5e-3 *Q (Var) 4

fL (H) 0.6e-3 fC F 20

1dcV (V) 40 2 3,dc dcV V (V) 30

A. Case1: normal grid condition

Vg

V2

V3

Iline

Fig. 7. Experimental waveforms under the normal-grid condition.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

Time(sec)

Act

ive

Po

wer

(W)

Module #1Module #2Module #3

(a)

1P2 3P P




0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

Time(sec)

Rea

ctiv

e P

ow

er(V

ar)


(b)

1Q

2 3Q Q

Fig. 8. Experimental results of case 1 (a) active power (b) reactive power.

This experiment is performed under the normal grid

condition. The experimental waveforms of voltage and current

are presented in Fig. 7, in which the output voltages of module

#2 and #3 are always in-phase with grid. The active power

sharing results are shown in Fig. 8(a), in which the active

powers of module #1 are higher than others due to the active

power losses in experiment. Figure 8(b) shows that the reactive

powers of module #1 are more than others to compensate the

absorbed reactive power of line. From the experimental results,

the proposed scheme can inject the desired powers into grid and

maintain the self-synchronized operation.

B. Case2: grid voltage sag condition

Vg

V2

V3

Iline

Fig. 9. Experimental waveforms under the grid voltage sag condition.

This experiment with the 10% grid voltage sag is carried out.

The measured waveforms are depicted in Fig. 9, in which the

output voltage of module #2 and #3 reduces while the line

current increasing after the grid voltage sag occurring. The test

results are shown in Fig. 10(a) and (b), in which the powers of

module #1 are increased as the line current increases. The

experimental results indicate that the proposed scheme is

adaptive to the grid voltage sag condition.

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

Time(sec)(a)

Act

ive

Po

wer

(W)


Voltage sag

1P2 3~P P

0 1 2 3 4 5 6 7 8 9 100123456789

Time(sec)(b)

Rea

ctiv

e P

ow

er(V

ar)

Voltage sag


1Q

2 3Q Q


C. Case3: grid frequency deviations

Vg

V2

V3

Iline

Fig. 11. Experimental waveforms under the grid frequency deviations.

0 1 2 3 4 5 6 7 8 9 1048.849

49.249.449.649.8

5050.2

Time(sec)(a)

Fre

quen

cy(H

z) Grid

Module #1

Module #2

Module #3

Frequency deviation

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

Time(sec)(b)

Act

ive

Pow

er(W

)Frequency deviation Module #1

Module #2Module #3

1P2 3P P

0 1 2 3 4 5 6 7 8 9 10-30-20-10

0102030

Time(sec)(c)

Rea

ctiv

e P

ow

er(V

ar)

Frequency deviation


1Q

2 3Q Q

Fig. 12. Experimental results of case 3 (a) frequency (b) active power (c) reactive power.

This experiment with the grid frequency deviation (-1Hz) is

implemented. The voltage and current waveforms are shown in

Fig. 11. The frequencies are shown in Fig. 12(a), in which all

modules always maintain synchronization with the grid even

under such large disturbances. The active and reactive power

allocations are shown in Fig. 12(b) and (c). Clearly, the

proposed scheme is robust against the grid frequency deviation.

D. Case4: DC input voltage sag condition

Vg

V2

VDC2

Iline

Fig. 13. Experimental waveforms under DC input voltage sag condition.




0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

Time(sec)(a)

Act

ive

Pow

er(W

)


Input voltage sag

1P2 3~P P

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

Time(sec)(b)

Rea

ctiv

e P

ow

er(V

ar)

Input voltage sag

1Q

2 3~Q QModule #1Module #2Module #3


This experiment with the DC input voltage sag is carried out.

The test waveforms are shown in Fig. 13. The active power and

reactive power sharing results are depicted in Fig. 14(a) and (b),

in which the powers remain the same even as DC input voltage

sag occurs. Therefore, the proposed scheme is feasible to the

input voltage sag condition to some extent.

E. Case5: grid impedance variation

Vg

V2

V3

Iline

Fig. 15. Experimental waveforms under grid impedance variation.

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

Time(sec)(a)

Act

ive

Po

wer

(W)


1P2 3~P P

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

Time(sec)(b)


1Q

2 3~Q Q

Rea

ctiv

e P

ow

er(V

ar)


This test with grid inductance variation (from 5mH to 6.2mH)

is performed. The related waveforms are shown in Fig. 15. The

active and reactive powers are depicted in Fig. 16(a) and (b),

respectively. Compared to the results in case 1, the reactive

power of module #1 is increased to compensate more reactive

powers when the grid inductor increases. As seen, the proposed

scheme is suitable for the grid impedance variations.

F. Case6: the method in [11]

Vg

Iline

Fig. 17. Experimental waveforms with the method in [11].

This test is carried out with the control method in [11], and

the corresponding experiment waveforms are shown in Fig. 17.

When the 10% grid voltage sag occurs, the grid current begins

to divergent until the overcurrent protection is triggered.

Compared to case 2, it is concluded that the proposed scheme is

insensitive to the grid voltage condition. That is to say that the

proposed scheme is more robust than the method [11].

G. Case7: grid harmonics condition

The test under the distorted grid voltage (THD=5.26%) is

implemented. The experimental waveforms are shown in Fig.

18. As seen, the proposed scheme is still feasible under the

condition of the grid distortion.

Vg

V2

V3

Iline

Fig. 18. Experimental waveforms under the harmonic grid condition.

V. CONCLUSION

This letter proposes a decentralized control scheme of the

grid-connected cascaded inverters, where the synchronization

operation is realized via the common line current information.

Due to the decentralized control, the number of cascaded

modules is unlimited in theory. Experimental results show that

the cascaded system could operate normally under the

abnormal-grid conditions (grid voltage sag, distortion and

frequency deviation). That is to say the method is robust against

the grid voltage variations to a certain extent. Based on these

features, it has the potential to be extended to the medium/high




voltage level photovoltaic, storage and STATCOM cascaded

systems, where the number of series modules is very large.

APPENDIX

T

1 2 n

X X X X (21)

where ,1 ,2I II X X 1

X ,, ,1 ,2i L i i vi viI V X X X .

1 2 2

3 4

3 4

A A A

A A 0 0A

0 0

A 0 0 A

(22)

where

2

10

0 1

2 22

P

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REFERENCES

[1] M. Malinowski, K. Gopakumar, J. Rodriguez and M. A. Perez, "A Survey on Cascaded Multilevel Inverters," in IEEE Transactions on Industrial Electronics, vol. 57, no. 7, pp. 2197-2206, July 2010.

[2] J. Rodriguez, Jih-Sheng Lai and Fang Zheng Peng, "Multilevel inverters: a survey of topologies, controls, and applications," in IEEE Transactions on Industrial Electronics, vol. 49, no. 4, pp. 724-738, Aug. 2002.

[3] S. Kouro et al., "Recent Advances and Industrial Applications of Multilevel Converters," in IEEE Transactions on Industrial Electronics, vol. 57, no. 8, pp. 2553-2580, Aug. 2010.

[4] L. Wang, D. Zhang, Y. Wang, B. Wu and H. S. Athab, "Power and Voltage Balance Control of a Novel Three-Phase Solid-State Transformer Using Multilevel Cascaded H-Bridge Inverters for Microgrid Applications," in IEEE Transactions on Power Electronics, vol. 31, no. 4, pp. 3289-3301, April 2016.

[5] A. Mortezaei, M. G. Simões, A. S. Bubshait, T. D. C. Busarello, F. P. Marafão and A. Al-Durra, "Multifunctional Control Strategy for Asymmetrical Cascaded H-Bridge Inverter in Microgrid Applications," in IEEE Transactions on Industry Applications, vol. 53, no. 2, pp. 1538-1551, March-April 2017.

[6] B. Xiao, L. Hang, J. Mei, C. Riley, L. M. Tolbert and B. Ozpineci, "Modular Cascaded H-Bridge Multilevel PV Inverter With Distributed

MPPT for Grid-Connected Applications," in IEEE Transactions on Industry Applications, vol. 51, no. 2, pp. 1722-1731, March-April 2015.

[7] L. Zhang, K. Sun, Y. W. Li, X. Lu and J. Zhao, "A Distributed Power Control of Series-Connected Module-Integrated Inverters for PV Grid-Tied Applications," in IEEE Transactions on Power Electronics, vol. 33, no. 9, pp. 7698-7707, Sept. 2018.

[8] J. He, Y. Li, C. Wang, Y. Pan, C. Zhang and X. Xing, "Hybrid Microgrid With Parallel- and Series-Connected Microconverters," in IEEE Transactions on Power Electronics, vol. 33, no. 6, pp. 4817-4831, June 2018.

[9] L. Du and J. He, "A Simple Autonomous Phase-Shifting PWM Approach for Series-Connected Multi-Converter Harmonic Mitigation," in IEEE Transactions on Power Electronics, vol. 34, no. 12, pp. 11516-11520, Dec. 2019. doi: 10.1109/TPEL.2019.2919386

[10] J. He, X. Liu, C. Mu and C. Wang, "Hierarchical Control of Series-Connected String Converter Based Islanded Electrical Power System," in IEEE Transactions on Power Electronics, early access, 2019. doi: 10.1109/TPEL.2019.2913212

[11] X. Hou, Y. Sun, H. Han, Z. Liu, W. Yuan and M. Su, "A Fully Decentralized Control of Grid-Connected Cascaded Inverters," in IEEE Transactions on Sustainable Energy, vol.10, no.1, pp. 315-317, 2019.

[12] X. Hou et al., "A General Decentralized Control Scheme for Medium-/High-Voltage Cascaded STATCOM," in IEEE Transactions on Power Systems, vol. 33, no. 6, pp. 7296-7300, Nov. 2018.

[13] J. M. Guerrero, M. Chandorkar, T. Lee and P. C. Loh, "Advanced Control Architectures for Intelligent Microgrids—Part I: Decentralized and Hierarchical Control," in IEEE Transactions on Industrial Electronics, vol. 60, no. 4, pp. 1254-1262, April 2013.

[14] J. He, Y. Pan, B. Liang and C. Wang, "A Simple Decentralized Islanding Microgrid Power Sharing Method Without Using Droop Control," in IEEE Transactions on Smart Grid, vol. 9, no. 6, pp. 6128-6139, Nov. 2018.

[15] E. C. Johnson, "Sinusoidal analysis of feedback-control systems containing nonlinear elements," in Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry, vol. 71, no. 3, pp. 169-181, July 1952.

[16] L. Maharjan, T. Yamagishi, H. Akagi and J. Asakura, "Fault-Tolerant Operation of a Battery-Energy-Storage System Based on a Multilevel Cascade PWM Converter With Star Configuration," in IEEE Transactions on Power Electronics, vol. 25, no. 9, pp. 2386-2396, Sept. 2010.

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