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Electrical Power and Energy Systems

journal homepage: www.elsevier.com/locate/ijepes

Locally-distributed and globally-decentralized control for hybrid series-parallel microgridsXiaohai Ge, Hua Han, Wenjing Xiong⁎, Mei Su, Zhangjie Liu, Yao SunSchool of Automation, Central South University, Changsha, People’s Republic of China

A R T I C L E I N F O

Keywords:Hybrid series-parallel microgridDistributed cooperative controlDroop controlPower sharingLow-bandwidth communication

A B S T R A C T

Hybrid series-parallel microgrid is a new promising structure for integrating low voltage (LV) and remotesources efficiently. DG units are connected in series as a string locally, and the strings supply power in parallel.This paper proposes a locally-distributed and globally-decentralized (LDGD) control for this microgrid. Local DGunits are controlled as a consistent string by distributed cooperative control, and global power demand amongstrings is shared in a decentralized manner. The frequency synchronization and proper power sharing areachieved under both the resistive-inductive and resistive-capacitive loads. Moreover, only local sparse low-bandwidth communication (LBC) networks are employed, which improves the reliability and reduces thecommunication costs. The overall system stability is analyzed and reasonable ranges of control parameters aregiven. Finally, the feasibility and effectiveness of the proposed solution are verified by simulation tests.

1. Introduction

Microgrid has been developed for decades since it was proposed,and now it has become an effective approach [1] to integrate andcontrol various distributed generation (DG) units [2]. Because of itsvital role in maintaining the stability of electrical networks, microgridwill gradually become a powerful support for the main power grid [3].With the growing penetration of DG units, some structural changes willinevitably occur in microgrid [4]. Microgrids are typically classifiedinto parallel-type and cascaded-type [5] according to the connectiontype of interface converters. Recently, a hybrid series-parallel structurehas been proposed to form a medium voltage (MV) or high voltage (HV)level microgrid with better flexibility and less costs [6]. For microgridsof any structures, power sharing is always a significant control target toavoid overstressing and aging of DGs.

The power control strategies for parallel-type microgrid have beenextensively and intensively studied. In grid-connected mode, DG unitsare usually controlled as current sources to follow the grid. In islandedmode, decentralized control methods, mainly including droop controland its variants [7–9], are widely used to achieve frequency synchro-nization and power sharing. However, conventional droop control hastwo drawbacks: imprecise reactive power sharing and frequency/vol-tage deviations which are difficult to be overcome without commu-nication [10]. The centralized control relying on high-bandwidthcommunication links can achieve excellent frequency/voltage

regulation and proper power sharing. However, all local controllersneed to communicate with a central controller, which weakens thereliability and scalability of the system. The distributed control [11–13]is considered as a promising method to address the above concerns. Asparse low-bandwidth communication (LBC) network is used for thelimited information passing among adjacent inverters [14]. Globalvoltage regulation, frequency synchronization, and proportional loadsharing are achieved by cooperation among local controllers. In addi-tion, the distributed control has no central controller and every moduleis symmetric so that the single point-of-failure can be avoided and thescalability, modularity and plug-and-play capability can be featured[15].

Cascaded-type microgrid, also called series-type [16] microgrid insome papers, is developed from the cascaded H-bridge inverters appliedto multilevel topology [17]. It has attracted increasing attentions owingto its HV level application and high efficiency [18,19]. Compared toparallel-type microgrid, it can be used to integrate low-voltage (LV)DGs into MV/HV power system with only a single-stage conversion andLV devices [20]. For cascaded microgrid in grid-connected mode, adecentralized droop control is proposed in [5] to achieve power bal-ance. For grid-connected cascaded PV inverters system, [21] proposes acommunication-free decentralized control to achieve maximum powerpoint tracking (MPPT) and frequency self-synchronization. In islandedmode, in addition to centralized control, a series of decentralized powercontrol methods have been proposed recently [22–24]. [22] first

https://doi.org/10.1016/j.ijepes.2019.105537Received 16 April 2019; Received in revised form 2 September 2019; Accepted 8 September 2019

⁎ Corresponding author.E-mail address: csu.xiong@163.com (W. Xiong).

Electrical Power and Energy Systems 116 (2020) 105537

0142-0615/ © 2019 Elsevier Ltd. All rights reserved.

T

proposes an inverse power factor droop control for power sharingwithout communication. However, this method is only applicable to thecases of the resistive-inductive (RL) loads. To overcome this limitation,an f-P/Q droop control scheme which can realize frequency synchro-nization and power sharing under both RL and resistive-capacitive (RC)loads is proposed in [23]. On this basis, [24] presents a power factorangle consistency control method to improve load voltage quality. Fromthe above, power sharing in cascaded microgrid can be well controlledwithout using communication links.

The hybrid series-parallel microgrid as shown in Fig. 1 is a newemerging structure, which promotes the integration of low voltage (LV)and remote DG units [20]. A number of local LV sources are cascaded toform a higher voltage DG string, then these strings are connected inparallel to supply the common load for high power. Thus the hybridseries-parallel microgrid features high flexibility, high efficiency andeasy HV level application. However, power sharing strategies for hybridseries-parallel microgrid are rarely investigated. In [6], a hybrid mi-crogrid with series- and parallel-connected microconverters is firstelaborated and a hierarchical power regulation method with LBC isproposed. The droop control is used to synchronize parallel strings andcentral controllers are implemented to allocate the power among cas-caded inverters. Proper power sharing performance and voltage qualitycan be achieved with this method. Nevertheless, the reliability of thesystem is impaired by introducing one central controller in each DGstring. In [25], a unified distributed control strategy is proposed. Thereis a DG in each string selected as the leader to exchange informationwith other leaders. Though accurate power sharing can be guaranteed,the communication network throughout the whole microgrid limits theintegration of remote DG units. Therefore, for better reliability andflexibility as well as less communication costs, a locally-distributed andglobally-decentralized (LDGD) control is proposed in this paper. Themain contributions in this work are explained as follows.

(1) A LDGD control. In local, DGs in series are synchronized by dis-tributed cooperative control, then the consistent string runs as anintegrated DG unit. Thus, global load sharing among parallel stringscan be ensured by the droop control, and even some improveddecentralized methods such as virtual impedance [26].

(2) Sparse local LBC networks. Compared with existing methods, sparselocal LBC networks are adopted without central controllers orleader sources. Each DG only needs to exchange data with its localneighbors, which strengthens the reliability and redundancy. Theplug-and-play capability of DG strings is also obtained due to no

communication among strings.(3) Modeling and stability analysis. In this work, small signal modeling

and eigenvalue analysis are carried out to study the system stabilityand dynamic performance for the first time. It is of great sig-nificance for system modeling and further research of complexmicrogrids.

The remainder of this paper is outlined as follows. In Section 2, theproposed LDGD control is elaborated. In Section 3, the small signalmodel is established and the system dynamic performance is analyzed.In Section 4, the feasibility of the proposed method is verified by si-mulation results. Finally, this paper is concluded in Section 5.

2. The proposed LDGD control

In this section, the hybrid series-parallel microgrid is described andthe proposed LDGD control is presented.

2.1. Hybrid series-parallel microgrid

The configuration of hybrid series-parallel microgrid is shown inFig. 1. A number of local DG units are cascaded as a DG string and thensupply power to the point of common coupling (PCC) in parallel. N andMi represent the total number of DG strings and DG units in string#i,respectively. Here the interface inverters adopt the structure of con-ventional H-bridge with an output LC filter for easy independent con-trol. The DG string supplies a controllable high voltage which is equalto the vector sum of each in-string output voltage. Therefore, with thistopology, multiple LV sources can be integrated into a higher voltagenetwork without back stage converters, unlike in parallel microgridwhere two stage dc/ac power conversion is commonly needed.

The PCC of microgrid is connected to the grid with a static transferswitch (STS). In the grid-connected operation mode, the PCC voltage isdetermined by the grid, hence each DG string in hybrid series-parallelmicrogrid can be controlled independently as a separate cascadedsystem. Decentralized power control methods proposed for grid-con-nected cascaded microgrid in [5] and [21] are valid to this hybridsystem as well. When the grid is not available and the STS turns off,namely the microgrid runs in islanded mode, PCC load is powered byparallel DG strings. Actually, the series-parallel microgrid can be re-garded as a more comprehensive system that includes both cascadedand parallel systems. The well-known droop control for parallel systemis expressed as (1),

= m Pi i i (1)

where Pi and Qi are output active power reactive power; ωi is the an-gular frequency; ω* represents the rated angle frequency; mi is a coef-ficient of P−ω;

On the other hand, from the power transmission characteristics ofthe i-th DG in islanded cascaded microgrid [23], the P -i i re-lationship is derived as,

+P sin( )i load ij i_0 (2)

where θij_0 is the power angle in the steady-state; θload is the load angleof. Compared with θload, θij_0 is usually small enough to be ignored.With RL loads, sin(θload) > 0, the P -i i correlation is negative. Thusthe P-ω inverse droop control is adopted [25]. On the contrary, droopcontrol should be employed under RC loads. Therefore, the inversedroop control (RL loads) in [22] and [23] for cascaded microgrid can beessentially expressed as

= + m Q Psgn( )i i i i (3)

where sgn (·) is the signum function. Due to the incompatibility be-tween (1) and (3), it is difficult to find a fully decentralized method forpower sharing in hybrid series-parallel microgrid [25]. Based on theabove analysis, a power control method with low communication costs

Fig. 1. Configuration of hybrid series-parallel microgrid.

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and high reliability needs to be studied.

2.2. Communication network design

The equivalent circuit of the autonomous series-parallel microgrid isshown in Fig. 2 where the adopted LBC links are indicated by red lines.From the control perspective, microgrid can be deemed as a cyber-physical system with a communication network to facilitate data ex-change among DGs for different targets [27]. The graphical re-presentation of communication network is illustrated in Fig. 3(a). In thegraph, nodes represent DGs and edges represent communication linksthat can be bidirectional to form an undigraph or not (digraph). Eachnode broadcasts its information, such as active power measurementsand power estimations, to its neighbors with different communicationweights. Note ij denotes the j-th DG unit in string#i and l Nij is definedas the set of all local in-string neighbors of Node ij, then the commu-nication weight >c 0ij l_ if Node ij receives data from Node l, otherwise

=c 0ij l_ . All communication weights in the graph are carried by anassociated adjacency matrix = ×C c[ ]ij l

M M_ where M is the number

of nodes.The communication network is designed to improve the reliability

and minimize the complexity and costs. In the proposed control, onlyin-string communication is sufficient for data exchange among localadjacent inverters. To simplify the analysis, the local communicationnetwork of each string is assumed to be identical. Taking string#i as anexample, a bidirectional sparse LBC network with the minimum re-dundancy is designed as shown in Fig. 3(b). The circular communica-tion graph is turned out to be an effective structure [28] that containsenough redundant links to enable system running in the case of anysingle link failure [29].

2.3. Proposed LDGD control

Based on the designed communication network, this paper proposesa locally-distributed and globally-decentralized control for islandedhybrid series-parallel microgrid. Objectives of the controllers are (1)

frequency synchronization, (2) accurate active power sharing, (3)proper reactive power sharing, (4) unified control under RL loads andRC loads. Typically, the output active and reactive power can bemanaged by adjusting frequency and voltage amplitude of each DG,respectively.

= mP Q k c P P¯ sgn( ) ( )ij ij ij il N

ij l l ij_ij (4)

=V V n Q̄ij i i ij (5)

where ωij and ω* are reference angular frequency and nominal angularfrequency, respectively. m, ki and ni are control parameters. P̄ij and Q̄ijrepresent the estimations of local averaged active power and reactivepower at Node ij. The rated voltage amplitude Vij, the filtered outputactive power Pij and reactive power Qij are expressed as,

=V V M/i p i (6)

=+

=+

Ps

p Qs

q,ijc

cij ij

c

cij (7)

where VP* is the rated voltage amplitude of PCC, Mi represents the totalnumber of DGs units in string#i. pij and qij are the measured power, c isthe cutoff frequency of the low-pass filter. Eventually, the in-stringactive power will be shared accurately when all local averaged activepower estimations converge to a value.

Each node has active and reactive power estimators which generatethe estimations of averaged power and exchange them with its localneighbors. The average value estimators implement the following dy-namic consensus protocol [30].

= +P t P t c P P d¯ ( ) ( ) ( ¯ ( ) ¯ ( ))ij ijt

l Nij l l ij0 _

ij (8)

The outputs of estimators at each node are updated by processingestimations of adjacent nodes and its output power [31]. The overallestimator dynamic of string#i can be formulated in matrix form as,

= + =P I L P H Ps s¯ ( )i i i i i i1 (9)

where ×IiM Mi i is the identity matrix, Hi is the active power esti-

mator transfer function matrix of string#i. =L D Ci iin

i is a Laplacianmatrix carrying the information of communication graph instring#i. =D diag d{ }i

inijin is the in-degree matrix of

string#i. = ×C c[ ]i ij lM M

_ i i is an associated adjacency matrix ofstring#i.

It is presented in [15] that all elements of P̄i converge to the overallaverage output active power of string#i in steady state if Li is balanced,namely

= = = =P P P P¯ ¯ ¯ ¯iss

iss

iMss

iss

1 2 i (10)

where X ss represents the steady-state value of X. P̄iss is the average value

of all output active power in string#i in steady state. Similarly, theaveraged reactive power estimations of string#i converge to the trueaverage value.

= = = =Q Q Q Q¯ ¯ ¯ ¯iss

iss

iMss

iss

1 2 i (11)

2.4. Controllers performance analysis

For concise analysis, it is assumed that the rated capacities of allsources are equal in steady state. The block diagram of proposed LDGDcontrol at Node ij is shown in Fig. 4. In active power regulator, ij is acorrection term carrying the active power mismatch between DG#ij andits neighbors, and it is expressed as,

= Q k c P Psgn( ) ( )ij ij il N

ij l l ij_ij (12)

Here the sign function sgn(Qij) is introduced to unify the control

Fig. 2. Equivalent circuit of hybrid series-parallel microgrid in island mode.

i1

i2

ijCommunication in String#i

iMi

Node

i1

i2

ij

(Inverter)

iMi

Fig.3. (a) Graphical representation of communication network, (b) DesignedLBC network in string#i.

X. Ge, et al. Electrical Power and Energy Systems 116 (2020) 105537

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mode under both RL and RC loads. In the steady-state, the frequency ofDG#ij is derived from (4) and (12),

= mP̄ij ijss

ijss (13)

P̄ijss converges to the true averaged active power of string#i, all

frequencies synchronize to the steady-state value, hence

= = =iss

iss

iMss

1 2 i (14)

It can be deduced that ijss decays to zero, which means in-string

accurate active power sharing is guaranteed.

= = =P P Piss

iss

iMss

1 2 i (15)

Combining (5), (6) and (11), the voltage amplitude is same for allDGs in string#i in steady state and they share the same current, so theapparent power of all modules in string#i is equal. Furthermore, (16)can be obtained due to the identical active power.

= = =Q Q Qiss

iss

iMss

1 2 i (16)

From aforementioned analysis, accurate in-string power sharing isachieved and the steady-state phase angles of all DGs in string#i areequal. So the voltage of PCC can be regulated easily by setting VP* in(6), which can be expressed by the vectors diagram (see Fig. 5).

On the other hand, combing (10), (11) (15), and (16), the Eqs. (4)and (5) in the steady-state can be rewritten as follows.

= mP̄ij iss (17)

=VM

V n M Q1 ( ¯ )iji

p i i iss

(18)

The consistent string could be considered as an integrated DG unit,then load sharing among parallel strings can be ensured by the droopcontrol, which is similar to the conventional droop control. The outputpower among strings is shared properly by designing the parameters as,

= = = =m m m mN1 2 (19)

= = =n M n M n MN N1 1 2 2 (20)

The proposed LDGD control achieves frequency synchronizationand load sharing under both RL and RC loads. It strengthens the re-liability and flexibility of the system and provides a referable applica-tion approach to microgrid with complex structure. It is worth notingthat proportional load sharing can also be realized if the rated capa-cities of DGs are unequal. In this case, the output power ought to benormalized and the control parameters should be redesigned as follows.

=

=

P P P

Q Q Q

/

/ijpu

ij ijrated

ijpu

ij ijrated

(21)

= == =

VQ

QV n

QQ

n;ijijrated

xM

ixrated P ij

ijrated

xM

ixrated i

*

1 1i i (22)

where Pijrated and Qij

rated are the rated active and reactive power of DG#ij.

3. Stability analysis

In this part, to study the system stability and dynamic performanceswith the proposed control strategy, small signal modeling and eigen-value analysis are carried out [32], and reasonable ranges of para-meters are given.

3.1. Small signal modeling

3.1.1. Distribution network modelingIn the presented hybrid series-parallel microgrid in Fig. 1, the

output active power pij and reactive power qij of DG#ij can be expressedas

+ ==

p jqV e

V e V e Y e2

| |ij ijij

j

b

M

ibj

Pj

ij

1

ij iib P i

(23)

where VP and δP represent the voltage amplitude and phase angle ofPCC. |Yi| and φi are amplitude and angle of the equivalent line ad-mittance in string#i. Usually, the line impedance is highly inductive

Fig. 4. Block diagram of the proposed LDGD control at Node ij.

Vp Vp

Vi1Vi2

ViMiVi,line

Fig. 5. The voltage vectors diagram in string#i.

X. Ge, et al. Electrical Power and Energy Systems 116 (2020) 105537

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(φi ≈ −π/2) in medium/high voltage power system.Then V eP

j P can be derived as follows according to Kirchhoff’s laws.

=+= = =

V e Y V eY Y

Pj

a

N

b

Ma ab

j

load cN

c1 1 1

Pa ab

(24)

Assume

=+

==

Y YY Y

Y e| |aa

cN

ca

j

load 1

a

(25)

Hence the voltage of PCC can be simplified to

== =

+V e Y V e| |Pj

a

N

b

M

a abj

1 1

( )Pa

ab a

(26)

Assume that ωs is the system synchronous angle frequency of steadystate. Defining = =ij ij s ij s, and combining (23)–(26), thepower transmission characteristic is given as,

=

=

+

= =

=

= =

=

p Y V Y V sin

Y V V sin

q Y V Y V cos

Y V V cos

| | | | ( )

| | ( )

| | | | ( )

| | ( )

ij i ija

N

b

M

a ab ij ab a

i ijb

M

ib ij ib

ij i ija

N

b

M

a ab ij ab a

i ijb

M

ib ij ib

12

1 1

12

1

12

1 1

12

1

a

i

a

i

(27)

By differentiating (7),

= +

= +

P P p

Q Q q

( )

( )ij c ij ij

ij c ij ij (28)

Combining (27), (28) and linearizing them around the steady statepoint yield,

= +

+

= +

+

= =

= =

= =

= =

P P k ~

k V

Q Q k ~

k V

ij ijij

ab

ij ijij

ab

c ca 1

N

b 1

M

p_~

ca 1

N

b 1

M

p_Vij

ab

c ca 1

N

b 1

M

q_~

ca 1

N

b 1

M

q_Vij

ab

ab

ab

a

a

ab

a

a

ab(29)

where Δ denotes small perturbations around the equilibrium point and

= =

= =

= == =

k k

k k

i N j Ma N b M

;

;

1, , ; 1, ,1, , ; 1, ,

pij p

p Vij p

V

qij q

q Vij q

V

i

a

_ _

_ _

ab

ij

ab abijab

ab

ij

ab abij

ab (30)

Then (29) is rewritten in matrix form,

= + +

= + +

P P K K V

Q Q K K V

~

~c c p c p V

c c q c q V

_ _

_ _ (31)

where ΔP, ΔQ, ~, ΔV, P, Q are variable vectors and the partialdifferential coefficients matrixes are given as follows (Kp_V, Kq_ andKq_V are omitted due to its similar form).

=K

k k k

k k k

k k k

p

p p p

p p p

pNM

pNM

pNM

_

_11

_11

_11

_12

_12

_12

_ _ _

NMN

NMN

N NNMNN

11 12

11 12

11 12 (32)

3.1.2. Controllers modelingWith the proposed control, the controllers of adjacent DGs in one

string process and exchange information to update the frequency andvoltage set points, which is shown in Fig. 4. In the frequency domain,there is

=

=

mP Q k c P P

V n Q

¯ sgn( ) ( )

¯ij ref s ij ij i l N ij l l ij

ij i ij

_ij

(33)

Linearizing them around the steady state point yield,

=

=

m P Q k c P P

V n Q

¯ sgn( ) ( )

¯ij ij ij i l N ij l l ij

ij i ij

_ij

(34)

Combining (9) and (34), writing them in matrix form, the smallsignal models of controllers are obtained.

==

DkL H PV nH Q

m~ ( )(35)

where D= diag{sgn(Qij)} is a diagonal matrix. L is a Laplacian matrixcontaining the information of system communication graph topology. His the entire active/reactive power estimator transfer function matrix. kand n are matrixes of controller parameters. They can be expressed as,

=

=

= =

= = +

=

n

k

L

LL

L

L D C

H

HH

H

H I L

diag n

diag n

diag k

diag k

s s

i N

{ }

{ }

{ }

{ }

;

; ( )

( 1, , ; )

n

n

n

i iin

i

n

i i i

1

1

1

2

1

2 1

(36)

3.1.3. Dynamic model of the entire microgridCombining the small signal models of the distribution network (31)

and controllers (35), the small signal dynamic model of the entireseries-parallel microgrid is constructed as follows.

x Ax= (37)

where =x P Q V~[ ]T ,

(38)

and = =M MiN

i1 denotes the total number of DGs in the microgrid.

3.2. Eigenvalue analysis

To evaluate the stability and dynamic performance of the system,

X. Ge, et al. Electrical Power and Energy Systems 116 (2020) 105537

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the eigenvalue analysis of matrix A in (38) is adopted. Based on thesimulation model described in Section 4 which consists of 3 × 3 DGswith only Zl1, the system eigenvalues are analyzed while varying thecontrol parameters: m, ki and ni. And the corresponding eigenvaluetrajectory diagrams are illustrated. To simplify the analysis, it is as-sumed that the control parameters of all DGs are equal. The dynamicresponse under RC loads is omitted due to space constraints.

(1) Eigenvalue trajectory with respect to m. Fig. 6 shows the eigenvaluetrajectory diagram as m increases from 2e to 5 to 1e-3, with k= 8e-3 and n = 5e-4. It can be seen that the conjugate poles λ1 and λ2 lieon the right half-plane when m is small, which means that thesystem is unstable. λ1 and λ2 move to the left half-plane of when mis greater than 1.1e-4 and gradually move away from the imaginaryaxis as m increases. Meanwhile, the dominant poles move awayfrom the imaginary axis to increase the damping ratio of system. Inother words, the system becomes stable in this case.

(2) Eigenvalue trajectory with respect to k. The active power regulatorcoefficient k has a significant influence on the system stability.Fig. 7 shows the eigenvalue trajectory diagram as k changes from 2eto 3 to 2 with m= 4e-4 and n= 5e-4. In Fig. 7, the conjugate polesλ1 and λ2 move from the left half-plane to the right half-plane whenk= 0.35, making the system unstable. Thus, a too large k is notconductive to the stability of the system.

(3) Eigenvalue trajectory with respect to n. The droop coefficient nshould be designed properly to ensure the system stable. Fig. 8shows the eigenvalue trajectory diagram as n varies from 1e to 5 to1e-3, with m= 4e-4 and k= 8e-3. In the beginning, λ1 lies in theright half-plane, indicating that the system is unstable. When nincreases to 6e-5, all eigenvalues stay in the left half-plane so thesystem becomes stable.

In conclusion, the control parameters mi, ki and n have crucial ef-fects on the system stability and dynamic performance. These controlparameters should be designed in the stable regions. In addition, the

LDGD control (4) and (5) can be essentially expressed as (17) and (18)in the steady-state. The voltage and frequency deviations in steady stateshould be as small as possible while ensuring fast transients response ofthe system. The steady-state frequency deviation should not exceed 1%of the rated frequency and the steady-state voltage deviation should bekept within 5% of the rated voltage [33]. Therefore, the design ofcontrol parameters should satisfy the following equation,

= =mP

nV

Q¯ , ¯ssp

ssmaxmax

maxmax

(39)

where P̄ ss and Q̄ss is average active power of DGs and average re-active power of strings in steady state. Considering the overshootduring transients as well, a reasonable design region of those para-meters under RL loads and RC loads is given as shown in Table 1.

4. Simulation results

In this section, to verify the feasibility of the proposed method inthis paper, a series-parallel single-phase AC microgrid model consistingof 3 × 3 DGs is built in Matlab/Simulink. The equivalent circuit of thesimulation model is presented in Fig. 9. The physical and controlparameters are listed in Table 2, and three simulation cases are carriedout.

4.1. Case I: Performance of the proposed LDGD control

Compared with conventional droop control [3] and inverse droopcontrol [23], the performance of the proposed LDGD control is tested inthis case. The rated capacity of each DG is equal and the system oper-ates with RL loads Zl1 + Zl2. Droop control is enabled at first and in-verse droop control is activated during 4–6 s, the LDGD control is em-ployed during 2–4 s and 6–8 s. The simulation results of frequency andactive power are illustrated in Fig. 10. It can be seen that the systemtends to be unstable when droop or inverse droop control is adopted. Inparticular, frequencies in different strings diverge gradually during4–6 s, which is consistent with our previous theoretical analysis inSection 2.1. By contrast, the system stability can be ensured when theproposed LDGD control is enabled. All frequencies synchronize quicklywith a slight deviation to the rated frequency and proper power sharingis guaranteed.

Fig. 6. Eigenvalue trajectory as m increases from 2e to 5 to 1e-3.

Fig. 7. Eigenvalue trajectory as k increases from 2e to 3 to 2.

Fig. 8. Eigenvalue trajectory as n increases from 1e to 5 to 1e-3.

Table 1Design regions of control parameters.

Parameters RL loads RC loads

m [1.1e−4, 2e−3] [1.3e−4, 2e−3]k [2e−3, 5e−2] [2e−3, 5e−2]n [6e−5, 5e−3] [9e−5, 5e−3]

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4.2. Case II: Performance with different rated capacities

In this case, the rated active and reactive power of DGs are set asshown in Table 2 and power sharing performance is tested under RLloads and RC loads. Zl1 is connected to the PCC at first and Zl2 is addedat 2 s, then the PCC load changes to Zl3 at 4 s and Zl4 is plugged at 6 s.The corresponding simulation results are illustrated in Fig. 11. It showsthat the system stabilizes rapidly after running. In the steady-state, theactive power among all inverters is shared exactly in proportional to therated active power. The output active power is equal for DGs with thesame rated value, such as P33 and P21. Similarly, the in-string reactivepower is shared proportionally and the output reactive power of DGs indifferent strings with the same rated value has a small difference due tothe line impedance mismatch, such as Q33 and Q21. Besides, when theload suddenly changes, the system has a fast response and the activepower and reactive power can be quickly adjusted. As a result, pro-portional load sharing can also be realized under both RL and RC loads.

4.3. Case III: Simulation with communication latency

Communication links between adjacent DGs are indispensable fordistributed control methods [34]. Generally, communication protocolswith high bandwidth and low latency are expected to be used to ensuresatisfactory performance [35]. However, undesirable communicationconditions may occur in practical applications. Thus in this case, theeffects of communication bandwidth and time delay on the controllerperformance are studied. max and min represent the maximum timedelay and the minimum bandwidth, respectively. The system operatesinitially with Zl1, Zl2 is attached at 2 s. Ideally, time delay is so muchsmaller than the controller dynamics that its impact on performancecan be negligent, such as simulation in case I and II with = µ2 smax and

min = 1GHZ. With the decrease of communication bandwidth and theincrease of time delay, the system performance deteriorates gradually.Fig. 12 shows the frequency waveforms with max = 50 ms and

min = 10KHZ. It can be seen that only slight transient fluctuations arecaused and good controller performance is still guaranteed. The active

and reactive power waveforms are omitted because they are similarwith those in case II.

However, from the simulation results with max = 120 ms andmin = 1KHZ in Fig. 13, it can be observed that the system frequency

and active power waveforms have large fluctuations and even thesystem oscillation is triggered. Furthermore, it can be deduced that thesystem will be unstable if the delay time is higher than the threshold ofstability region. The proposed controller is immune to time delaywithin 50 ms and bandwidth as low as 10 Khz, which ensures that thecommonly used communication protocols are applicable [15]. Sparselocal low-bandwidth communication network is sufficient for data ex-change in the proposed method.

4.4. Case IV: Simulation with communication links failure

In this case, to study the system resiliency to the communicationlinks failure, the controller performance with sudden communicationlinks failure is presented in Fig. 14 (results under RC loads are omitted).Links 11–12, 22–23 and 33–31 fail at 1 s, 2 s and 3 s respectively, whichis illustrated in Fig. 14(a). Load Zl5 = 10 + j5 Ω is attached at 2 s anddetached at 3 s to test the response to load change. The rated active andreactive power of DGs are set as shown in Table 2. From Fig. 15, pro-portional power sharing can be achieved in the steady-state at first.Then it can be seen that the steady-state performance is not affectedwhen the link 11–12 is disconnected at 1 s. The steady-state powersharing performance in the interval [3 s, 4 s] gradually returns to theinceptive state when Zl5 is detached. However, the frequency syn-chronization and power balance are achieved more slowly than case II

DG#13 DG#12 DG#11

DG#23 DG#22 DG#21

DG#33 DG#32 DG#31

Z1

Z2

0.2+j0.75

0.3+j1.0

Z3

0.1+j0.5

Zl1=5+j5 Zl2=5+j5

Zl3=5-j5 Zl4=5-j5

Fig. 9. Equivalent circuit of the simulation model.

Table 2Simulation parameters.

Items Symbol & value

Voltage reference of PCC VP* = 311 V; fP* = 50 HzRated angle frequency ω* = 100π rad/sFilter inductance Lf = 1.6 mHFilter capacitance Cf = 20 μFCommunication weights cij l l Nij_ , = 1Case I Rated active power =P kW2 (Equal for all.)

Rated reactive power =Q kVar2 (Equal for all.)Control coefficients m= 4e−4; k= 5e−3; n= 5e−4

Case II,III,IV,V

Rated active power = = =P kW P kW P kW3 ; 3. 5 ; 4 ;11 12 13 = = =P kW P kW P kW2 ; 2.5 ; 3 ;12 22 23 = = =P kW P kW P kW1 ; 1.5 ; 2 ;31 32 33Rated reactive power = = =Q kVar Q kVar Q kVar3 ; 3.5 ; 4 ;11 12 13 = = =Q kVar Q kVar Q kVar2 ; 2.5 ; 3 ;21 22 23 = = =Q kVar Q kVar Q kVar1 ; 1.5 ; 2 ;31 32 33Control coefficients m= 1; k= 5; n= 2

Fig. 10. Simulation results of case I. (a) Frequency, (b) Active power.

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after load changes. In a word, single link failure does not affect thesteady-state performance while it slows down the system dynamics tosome extent.

For comparison, resiliency to single link failures of the controlmethod in [6] is studied in Fig. 16. The droop control is used to syn-chronize parallel strings and central controllers are implemented toallocate the power among cascaded inverters. The cyber configurationis shown in Fig. 14(b), where “CC” stands for the central controller anda single link failure occurs at 1 s, 2 s, and 3 s, respectively. From Fig. 16,it can be seen that the system has good steady-state performance in thebeginning. Frequency synchronization, accurate active power sharingand approximate reactive power sharing can be guaranteed. However,when a single link in string#1 fails at 1 s, the steady-state performanceis affected. DGs in string#1 supply more active power, which meansthat active power in different strings cannot be shared exactly in

proportional to their rated capacities. When more single link failuresoccur, power sharing performance of the system is worse and thetransient response is slower. In addition, all in-string DGs will be out ofservice if the central controller fails. In conclusion, the proposed LDGDcontrol has better resiliency to the communication link failure becausesparse local LBC networks are adopted without central controllers orleader sources.

4.5. Case V: Simulation with plug-and-play

In this case, the system is tested via unplugging the string#3 at 1 sand plugging it back at 3 s, whose physical configuration is depicted inFig. 17. The simulation results are shown in Fig. 18. When string#3 isunplugged, the system quickly reaches a new steady state. Because thecontrollers are still in action, the frequency of DGs in string#3 changes

Fig. 11. Simulation results of case II. (a) Frequency, (b) Active power, (c)Reactive power.

Fig. 12. Simulation results with = 50 msmax and = 10 kHZmin .

Fig. 13. Simulation results with = 120 msmax and = 1 kHZmin . (a)Frequency, (b) Active power, (c) Reactive power.

13 12 11

23 22 21

33 32 31

1s

2s

3s

Cyber Configurations in Case IV

13 12 11

23 22 21

33 32 31

1s

2s

CC1

CC2

3sCC3

(a) The Proposed LDGD Control Method

(b) The Control Method in [6]

Fig. 14. Cyber configurations in case IV, (a) the proposed LDGD controlmethod, (b) the control method in [6].

X. Ge, et al. Electrical Power and Energy Systems 116 (2020) 105537

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to 50HZ while the frequency of remaining DGs drops to a new value.Accordingly, the output power of DGs in string#3 reduces to zero whileproper power sharing is still maintained among the remaining DGs.After string#3 is plugged back, the controllers have a fast response andthe steady-state performance is consistent with the interval [0 s, 1 s].The load demand is shared among all DGs again. So the plug-and-playcapability of DG strings can be obtained with the proposed controlmethod.

5. Conclusions

In this paper, a locally-distributed and globally-decentralized con-trol is proposed for the hybrid series-parallel microgrid. The frequencysynchronization and proper power sharing are achieved in islandedmode with only local sparse LBC networks. The reliability and re-dundancy of the system are improved and the communication costs arereduced. Small signal modeling and eigenvalue analysis are carried outto study the system stability and dynamic performances, and reasonabledesign ranges of parameters are given. Finally, the power sharing

Fig. 15. Simulation results of case IV with the LDGD control. (a) Frequency, (b)Active power, (c) Reactive power.

Fig. 16. Simulation results of case IV with the control method in (6). (a) Fre-quency, (b) Active power, (c) Reactive power.

(a) State I

Physical Configurations in Case V

String#1

String#2

String#2

(b) State II

String#1

String#2

String#3

Fig. 17. Physical configurations in case V.

Fig. 18. Simulation results of case V. (a) Frequency, (b) Active power, (c)Reactive power.

X. Ge, et al. Electrical Power and Energy Systems 116 (2020) 105537

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performance is verified by simulation results compared with the ex-isting methods. The resiliency to communication latency, communica-tion links failure and plug-and-play capability are tested as well. Inconclusion, the proposed control method has good performance, whichis of great significance for further research of complex microgrids.

Declaration of Competing Interest

We declare that we have no financial and personal relationshipswith other people or organizations that can inappropriately influenceour work, there is no professional or other personal interest of anynature or kind in any product, service and/or company that could beconstrued as influencing the position presented in, or the review of, themanuscript entitled.

Acknowledgements

This work was supported by the National Natural ScienceFoundation of China under Grants 61573384, the Joint Research Fundof Chinese Ministry of Education under Grant 6141A02033514, theMajor Project of Changzhutang Self-dependent InnovationDemonstration Area under Grant 2018XK2002, the National NaturalScience Foundation of China under Grant 61622311, and theFundamental Research Funds for the Central Universities of CentralSouth University under Grant 2018zzts580.

Appendix A. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijepes.2019.105537.

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