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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/TSG.2020.2967353, IEEETransactions on Smart Grid

Abstract— DC Microgrids have been widely used due to their

high efficiency, high reliability and flexibility. A sine qua non

condition for the correct operation of systems is the existence of a

feasible power-flow solution. This paper analyzes the existence of

the feasible power-flow solution of the DC microgrid under droop

control. Firstly, the power-flow mathematical model of DC

microgrid is established. Then, based on the nested interval

theorem, we obtain the sufficient conditions of the existence of

the feasible power-flow solution, and the uniqueness of the

feasible power-flow solution is proved. Moreover, the iterative

algorithm of the feasible power-flow solution is proposed, which

is proved to be monotonically exponentially convergent. The

proposed algorithm’s domain of attraction is derived, thus, the

initial iterative value of which can easily be chosen to guarantee

its convergence. Finally, case studies are given in this paper to

verify the correctness and effectiveness of the proposed theorems.

Index Terms-- DC microgrids, power-flow solution, nested

interval theorem, solvability, convergence analysis.

I. INTRODUCTION

Microgrid, which mainly consists of renewable generations

such as photovoltaic (PV) and wind power, has been identified

as an effective complement to the traditional power systems.

In general, it can be divided into DC microgrid and AC

microgrid [1]-[2]. At present, the domestic research on

microgrid mainly focuses on AC microgrid [3]-[6]. Compared

with AC microgrid, DC microgrid has the following

advantages: high transmission efficiency and high reliability;

no frequency synchronization problem; easier integration of

renewable energy; easy to stabilize [7]-[13]. Therefore, DC

microgrids are increasingly being used in applications such as

aircrafts, space crafts and electric vehicles [14]-[15].

Manuscript received XX, 2019; revised XX, 2019; accepted December 27,

2019. Date of publication June 25, 2018; date of current version October 18,

2018. This work was supported in part by Singapore ACRF Tier 1 Grant: RG

85/18, the NTU Start-up Grant for Prof Zhang Xin, BCA 94.23.1.3, in part by the National Natural Science Foundation of China under Grants 61933011 &

61903383, in part by the Major Project of Changzhutan Self-dependent

Innovation Demonstration Area under Grant 2018XK2002, in part by the key R & D program of Hunan Province of China under Project 2019GK2211 and .

Paper no. TSG-00950-2019. (Corresponding author: Xin Zhang. e-mail: [email protected])

Z. Liu, R. Su, M. Su, Y. Sun, and H. Han are with the School of

Automation, Central South University and with Hunan Provincial Key Laboratory of Power Electronics Equipment and Grid, Changsha 410083,

China. Z. Liu is also with Energy Research Institute, Nanyang Technological

University, Singapore 639798. X. Zhang and P. Wang are with the School of Electrical and Electronic

Engineering, Nanyang Technological University, Singapore 639798.

In DC microgrid, the load is typically connected to the DC

bus through a DC/DC or DC/AC converter. When the

response of the load-end converter is fast, the load exhibits a

negative impedance characteristic, which is equivalent to a

constant power load (CPL) [15]-[16]. CPLs can easily lead to

system instability and even loss of equilibrium [16]-[25]. The

existing studies mainly focus on the small signal stability of

DC microgrid, that is, the stability near the known equilibrium

is analyzed, and corresponding stabilization control strategies

is proposed [7][8][14]-[24]. All these studies are based on the

assumption that the system has an equilibrium. However, with

the increasing of the CPLs, DC microgrid may lose

equilibrium due to the transmission loss, thus leading to

voltage collapse [25].

When the line resistance of DC bus can be neglected, all

the loads can be equivalent as a common CPL [15] [24]. Thus,

the system equilibrium are determined by a quadratic equation

with one unknown, which is easy to solve. When the line

resistance of DC bus can not be ignored, the system will

become a meshed DC microgrid with multiple CPLs, whose

equilibrium is determined by a multi-dimensional quadratic

equation (MDQE) with multiple unknowns [25]-[27]. Thus,

the problem of existence of system equilibrium will become

complex.

To analyze the solvability of the multi-dimensional

quadratic equation with multiple unknowns, there are mainly

two methods: “completing the quadratic form” [25] and

“contraction mapping” [27]-[31]. The first method is based on

the fact that if the weighted sum of all the sub-equations of the

MDQE has no solution, then, the MDQE must have no

solution. Then, the MDQE can be transformed into a

one-dimensional quadratic equation, and we can analyze the

solvability by completing the quadratic form. Thus, a

necessary condition based on linear matrix inequality (LMI)

aimed for the existence of the equilibrium is obtained in [25].

When the necessary condition based on LMI is also sufficient,

which is discussed in [32], and it is necessary and sufficient

only when the system has at most two CPLs. To obtain the

sufficient conditions for solvability of MDQE, several

methods based on contraction mapping theory are proposed.

Firstly, they transform the problem of the MDQE solvability

into the existence of the fixed point of the constructed

mapping. Then, the sufficient solvability condition for MDQE

can be obtained by using the Banach’s fixed-point theorem

[27]-[28], Brouwer’s fixed-point theorem [29], Kantorovitch’s

theorem [30] and Tarski’s fixed-point theorem [31]. For DC

microgrid, the sufficient conditions in [27] and [29] are

equivalent and more conservative than the solvability

condition in [31]. However, how to design iterative algorithm

Feasible Power-Flow Solution Analysis of DC

Microgrids under Droop Control Zhangjie Liu, Ruisong Liu, Xin Zhang, Mei Su, Yao Sun, Hua Han and Peng Wang

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to obtain the power-flow solution of DC microgrid has not

been discussed in [31], and the solvability conditions in [31]

are obtained based on the assumption that all the voltage

references are equal, which may be harsh in the practical

engineering.

This paper aims to analyze the existence of the equilibrium

of DC microgrid and propose the effective iterative algorithm

to solve the MDQE. The main contributions of this paper can

be summarized as the following:

1) This paper analyzes the solvability of the MDQE and

obtains the sufficient conditions for the existence of

power-flow solution of DC microgrid. Moreover, the proposed

solvability condition (presented in (50)) is stronger than the

result in [27] and [29] (presented in (13)).

2) This paper proposes the effective iterative algorithm for

the MDQE, which is monotonic exponential convergent.

Moreover, the proposed algorithm’s domain of attraction is

derived. Thus, the initial iteration value can easily be chosen

to guarantee the monotonic exponential convergence of the

proposed algorithm.

This paper is organized as follows: Section II introduces the

topologies and control strategies of DC Microgrid. Section III

presents the existence condition and iterative algorithm for the

feasible power-flow solution. The case studies are in the

Section IV. Finally, the conclusion and future work are drawn

in Section V.

II. TOPOLOGIES AND CONTROL STRATEGIES OF DC MICROGRID

The basic topological structure of DC microgrids can be

divided into two types: single bus and multiple buses. For the

DC microgrid with single bus structure, under the condition

that the resistance of the DC bus can be neglected, all loads

connected on the bus can be equivalent to a common load.

Thus, the topology of DC microgrid can be equivalent to a star.

For a DC microgrid with multiple buses, its topology is

equivalent to the meshed topology with n distributed

generations (DGs) and m CPLs.

A typical meshed DC microgrid with n DGs and m CPLs is

shown in Fig 1, which consists of three main components:

sources, loads and cables. The sources (i.e. DGs) are under

droop control mode, and the cable impedances are pure

resistances. All the loads connect on the DC bus through the

power electronic interfaces, and they usually show the

instantaneous CPL behaviors. Thus, all the loads are modeled

as CPLs. Meanwhile, according to the graph theory, the

topology of the DC microgrid can be equivalent to a graph:

where the DGs and loads are the nodes of the graph, the

transmission cables are the edges of the graph, and its

conductance are the connection weights of the edges.

Generally, the graph of the transmission network of the DC

microgrid is strongly connected, that is, the power of any DG

can be transmitted to any load through the transmission

network.

12 1314

15

2 31

45

16

17

6

78910

11

iu

ivii

1

siu

PWM

DC/DC

Converter

ik

(a) The topology of a meshed DC micrrogrid.

(b) The equivalent graph of the

dc micrrogrid.

(c) The control diagram of the

converter under droop control.

DGCPL

Fig.1. The topology of a meshed DC microgrid under droop control. In (a), the

black and blue block are the cable resistance and CPL, respectively. The

equivalent graph of the DC microgrid is presented in (b), the red and blue

point represent the DG and load node, respectively. The control diagram is

presented in (c).

III. EXISTENCE CONDITIONS OF POWER-FLOW SOLUTION FOR

DC MICROGRID

3.1. Notations and Preliminaries

Notations. Denote , , ,m n m+

as the set of the real

numbers, positive real numbers, real m-dimensional vector,

and real n m matrices, respectively. Denote O as the zero

matrix.

The next parts will employ three definitions and four

lemmas:

Definition 1. Denoting A−B > 0, if matrix A−B is positive

definite, i.e., the quadratic ( ) 0Tx A B x− is set up for

any 0Tx x . Deonte , , ,A B A B A B A B if all the

entries of A−B are positive, nonnegative, negative and

nonpositive, respectively. Matrix A (or a vector) is called

positive if its entries are all positive (i.e., A O ).

Definition 2. For a positive vector 1 2

T

mx x x x= ，we

define 1x− as 1 1 1 12

T

mx x x x− − − − = . Define ( )1 0m m as the

m-dimensional vector which all entries are 1 (0).

Definition 3. m mA is a Z-matrix if all the off-diagonal

elements are zero or negative. A is also an M-matrix if and

only if the real parts of all eigenvalues of A are positive. [33]

Lemma 1. Let m mA be a positive matrix. The Perron

root χ and Perron vector η of A satisfy Aη = χη, where 0m

and 1T = . Moreover, χ is the spectral radius of A (denoted

as ρ(A)). [34]

Lemma 2. Letm mA be a Z-matrix, then A is an M-matrix

if and only if there exists a positive vector x ,which makes

0mAx true. [33]

Lemma 3. Let m mA be an M-matrix, then 1A O− .

Particularly, if A is irreducible, 1A O− . [35]

Lemma 4. Let m mA be a Laplacian matrix of a strongly

connected graph, then all the leading pricipal submatrices are

positive definite M-matrices. [35]

3.2. Power-Flow Model of DC Microgrid




According to the Ohm's law, the current injected into the

transmission network by each node can be described as

follows:

1 2 1 2

1 2 1 2

,

,

GG GLG G G

LG LLL L L

T T

G n G n

T T

L n n n m L n n n m

B Bi u uB

B Bi u u

i i i i u u u u

i i i i u u u u+ + + + + +

= =

= =

= =

, (1)

where iG, uG, iL and uL are the vector of DG’s currents, DG’s

voltages, load currents and voltages, respectively. Matrix B is

the Laplacian matrix of the graph of the transmission network.

For the DG under droop control, its output voltage uG can

be given by

G Gu V Ki= − , (2)

where 1 2 ,T

n iV v v v K diag k= = . +iv and

+ik are the voltage reference and droop gain of the i-th

DG, respectively. Clearly, K is a positive definite matrix.

For a CPL, the voltage-current characteristic can be

expressed as

, 1, 2, ,i i iu i P i n n n m= − + + + . (3)

Since the current reference direction is opposite to the

voltage reference direction, the right side of (3) is negative.

Substituting (2) into (1), the following can be obtained

G GG GG G GL L

L LG LG G LL L

i B V B Ki B u

i B V B Ki B u

= − +

= − + . (4)

Clearly, BGG is a leading principal submatrix of B. Since the

DC microgrid is strongly connected, according to Lemma 4,

BGG is a positive definite M-matrix. Since K is positive

definite, 1+ GGK B− and 1+GGB K − are all positive definite (i.e.,

invertible).

Simplifying (4), it yields

( ) ( )

( )( )( )( )

1 11 1 1

11

11 1

G GG GG GG GL L

L LG LG GG

LL LG GG GG GL L

i B K V B K B B u

i B B K B K V

B B K B K B B u

− −− − −

−−

−− −

= + + +

= − +

+ − +

. (5)

Combining (3) with (5), the power-flow solution are

determined by the following equations

( )

( )

1

1

11

1

1, 2, ,

L L

i i i

LG GG

LL LG GG GL

i B u

u i P i n n n m

B I KB V

B B B B K B

−

−−

= − +

= − + + + = − +

= − +

. (6)

Denote [ ]L Lu diag u= and 1 2

T

n mn nP P P P+ + += .

Writing (6) into the compact form, we can obtain the

power-flow equation as

1[ ] [ ] 0L L L mu B u u P + =− . (7)

Clearly, equation (7) is a MDQE. The system admits a

constant steady-state if and only if nonlinear equation (7) has

at least one positive solution. Therefore, the core problem of

the feasible power-flow solution for DC microgrid can be

described as the following: Under what conditions among

reference voltage V, transmission network admittance matrix

B and load P, equation (7) is solvable?

In this paper, we will investigate the following two

questions:

Q1. Given the maximum CPL power vector P and the

transmission network admittance matrix B, how should the

voltage reference V be regulated to keep (7) solvable?

Q2. Given the fixed voltage reference V and the transmission

network admittance matrix B, how to obtain the maximum

power of CPL vector P to keep (7) solvable?

3.3. Recent Related Results

To analyze the solvability of the MDQE, there are mainly

two methods: “completing the quadratic form” [12] and

“contraction mapping” [31][27]-[29]. The main idea of the

former can be described as the following: if an m-dimensional

equation is solvable, then, the sum of all sub-equations must

have solution. In other words, if the weight sum of all

sub-equations of (7) has no solution, equation (7) must have

no solution. We denote 1 2= m . Multiplying by τ,

equation (7) becomes

1 1 0T T TL L L mu TB u u T TP + =+ . (8)

where T = diag{τ}. By completing the quadratic form, (8)

becomes

( )( ) ( ) ( )( )

( )

1 1

1 1 1 1 1 1

1

1 1

1

2

T

L L

m

nT

i

i

u TB B T T TB B T u TB B T T

TB B T TT P

− −

−

+

=

+ + + + +

= + −

.(9)

Obviously, if (9) is unsolvable, then, (7) is unsolvable.

Denote

( ) 1 1=

2 1

T

T Tm

TB B T TH T

T P T

+

.

According to Schur’s complement theorem, H(T) is positive

definite if and only if ( )1

1 112 0Tm

n iiP TB B T

−

+=− + and

TB1 + B1T is positive definite. Clearly, if there is a diagonal

matrix T such that H(T) is positive definite, then (9) must have

no solution because the left side of (9) is a positive definite

quadratic form and the right side is a negative constant. The

necessary condition for the solvability of (7) is that (9) is

solvable. Therefore, (7) is solvable only if the following LMI

problem has no solution

1 10

2 1

T

T Tm

TB B T T

T P T

+

. (10 )

Furthermore, reference [19] shows that (10) is a necessary

and sufficient condition if and only if the system has at most

two CPLs, i.e., 2m . Therefore, condition (10) can not ensure

that the meshed DC microgrid admits a feasible power-flow

solution.

To obtain the sufficient conditions, several methods based

on contraction mapping have been proposed. The main

process of the contraction mapping method can be




summarized as the following.

Firstly, they transform the problem of solvability of MDQE

into the existence of a constructed mapping. According to

Proposition 1 (in Appendix), B1 is invertible. Multiplying by 1 1

1 [ ]LB u− − , (7) becomes

1 11L L Lu V B u − −= − . (11)

where ( )11

2 2 1,L LG GGV B V B B B I KB− −= = − + and diag P= .

Obviously, LV is the open-circuit voltage of load nodes, and

11B− are the equivalent impedance matrix of the node network.

Define ( ) 1 11LV B xx − −= − , then, (11) is equivalent to the

following equation

( )x x= . (12)

Thus, the solvability of (7) has been equivalent to the

existence of the fixed-point of function ( )x . Based on

Banach’s fixed-point theorem, reference [27] obtains the

sufficient condition as the following

4 1A

. (13)

where ( )1

1L LA diag V B diag V−

= . In addition, based on

Brouwer’s fixed-point theorem, [29] obtains the sufficient

condition as the following

2 2 1AP AP A + . (14)

According to proposition 1, 0L mV and 11B O− , then,

we obtain A O and 1mAP A A = = . Therefore,

condition (13) and (14) are equivalent.

The proof of condition (13) and (14) are detailed in [27]

and [29], respectively. In this paper, we will provide an

alternative proof along with a stronger condition.

Under the assumption that 1 2 n refv v v v= = = = , reference

[31] obtains the following sufficient condition

( )1 1min 2 ,refv B B

+

. (15)

where ( )1B and are the spectral radius and Perron

eigenvector of 1B , respectively, max i = and

min i = . However, the assumption that all the voltage

references are equal may be too harsh. Next, we will

investigate more general sufficient conditions than the

conditions in [31].

3.4. The Proposed Sufficient Conditions for Existence of

Equilibrium

Multiplying by 1

Ldiag V−

, (11) becomes

1

1 11

1

1L m LL Ldiag V diu Bag V u− −

− −= − . (16)

Denote 1

L Lx diag V u−

= , substituting it into (16), it yields

11mx A x−− = . (17)

Define ( ) 11mx A x −= − . Considering that A O , the

following two important properties of ( )x can be easily

obtained:

(i) ( ) 1mx , for any 0mx ;

(ii) ( ) ( )1 2x x , for any 2 1 0mx x .

Next, this paper will derive the sufficient conditions to

guarantee that (7) is solvable according to the above

properties.

The main results of this paper are as the following.

Theorem 1. If there is a positive vector y such that

( )y y . (18)

then, there exists a unique vector x in the domain

1mx y x such that ( )x x = .

Proof of Theorem 1. Assume there is a positive vector y such

that ( )y y , conbining with the property (i) of ( )x , one

obtains

( ) ( )1 1m my y . (19)

According to the property (ii) of ( )x , the following can be

easily obtained

( ) ( )( ) ( )( ) ( )1 1 1m m my y y . (20)

Define infinite sequences ( ) ( )1 1,n n n na a b b + += = ,

where 1a y= and 1 1mb = . Then, the following can be

obtained

1 2 3 3 2 1n na a a a b b b b . (21)

According to the monotone convergence theorem, infinite

sequences an and bn will converge to their limit values.

Denote lim , limn nn n

b b a a→ →

= = , and the following can be

obtained

( ) ( )1 1lim lim , lim limn n n nn n n n

b b b a a a + +→ → → →

= = = = . (22)

According to (22), we have

( ) ( ),a a b b = = . (23)

Equation (23) shows that equation (7) has at least one solution.

Next, this paper will prove that a = b, i.e., the feasible

solution of (7) is unique. According to (21), the following can

easily be obtained

1 2 3

3 2 1

n

n

a a a a b

b b b b b

. (24)

Then, the following can be derived

( ) ( )

( )

1 1

1 1

n n n n

n n n n

b a b a

A diag b diag a b a

+ +

− −

− = −

= −. (25)

Since 0n n mb a− and nb b for any positive integer n,

there,we have the equation as

( )

( )

11

1 1

1 1

1

0m n n n n n

n n

b a A diag b diag a b a

A diag b diag a b a

−−

+ +

− −

− −

−. (26)




On the other hand, substituting 11m A b b−= + into (19), the

following can be obtained 1 1

1 1A b b A a a− − + − . (27)

From (27), we obtain

( )( )1 1

1 1 0mI A diag b diag a b a− −

− − . (28)

Define 1 1

1H I A diag b diag a− −

= − . According to

Definition 3, H is a Z-matrix. Since ( )1 0mb a− , according

to Lemma 2, H is a M-matrix, i.e., ( ) 1H . According to

(26), the following can be obtained

( ) ( )

( )

21 1 1 1

1 1

0

m n n n n n n

n

b a H b a H b a

H b a

+ + − −− − −

−. (29)

According to (29), we can easily obtain equation as

( ) ( ) ( )1 1 1 1lim lim 0n

n n mn n

b a H b a+ +→ →

− = − = . (30)

Combining (30) with (21), according to the nested interval

theorem, we obtain

1 1lim limn nn n

b a+ +→ →

= . (31)

Assume there is another vector 1mb x y x such

that ( )b b = . Then, since 1my b , we can get the

following

( ) ( ) ( )1 1m my y b b = . (32)

Then, the following can be obtained

1 2 2 1n na a a b b b b . (33)

According to (30), there, we have the equation as

1 1lim limn nn n

b a b+ +→ →

= = . (34)

Because the limit of sequence is uniqueness when it exists, we

obtain b b= . Therefore, there is no other solution in the

interval 1mx y x . The proof is accomplished.

Remark 1. In this part, we tansform the solvability problem of

the MDQE into the convergence problem of two monotone

infinite sequences. Then, the sufficient existence condition is

derived by using monotone convergence theorem. Moreover,

the uniqueness of the feasible power-flow solution is proved

by using the nested interval theorem. In this process, the key is

to construct a sequence of nested, unbounded and nonempty

intervals according to the monotonicity of ( )x .

Remark 2. Since the DGs are under the droop control,

according to Proposition 1, 1

1Y − is posotive, which plays a

crucial role in the monotonicity of ( )x . For a DC microgrid

under master-slave control (i.e., only a few of dominant

converters are under droop control and the rest are set based

on MPPT), the DG’s volt-ampere characteristics are highly

nonlinear and the equivalent output impedances are negative.

As such, the existence and stability of equilibrium of DC

microgrid under master-slave control is still a challenge

problem.

Next, this paper will derive the explicit analytic solvable

condition. Denote 1 2; ; mA a a a = , where ai is the

i-th row vector of A . The main results are as the following.

Theorem 2. Equation (7) is solvable if the system parameters

satisfy

4 1A

. (35)

Proof of Theorem 2. According to Theorem 1, if there is a

positive vector y such that ( )y y , equation (7) is solvable.

Take 1my = , and ε is a positive undetermined scalar. Then,

the MDQE (7) is solvable if

( )1 1m m . (36)

Cleary, (36) can be decomposed into m quadratic

inequalities of ε as the following

11 1 , 1,2, ,i ma i m

− = . (37)

Solve the quadratic inequalities in (37), we obtain

( ) ( )

( ) ( )

1 1

1 11 1 4 1 1 1 4 1

2 2

1 11 1 4 1 1 1 4 1

2 2

m m

m m m m

a a

a a

− − + −

− − + −

. (38)

Denote

( ) ( )1

1 11 1 4 1 1 1 4 1

2 2

m

i i i m i m

i

a a=

= = − − + −

，， . (39)

Then， if is true, for any , ( )1 1m m

always holds, i.e., (7) is solvable. Obviously, is nonempty

if and only if

1,2,...,

4 max 1 1i mi m

a=

. (40)

Since A is a positive matrix, the following can be obtained

1,2,...,

4 max 1 4 1i mi m

a A=

= . (41)

The proof is accomplished.

Theorem 3. Equation (7) is solvable if the system parameters

satisfy

1

+ . (42)

where and are the Perron eigenvalue and eigenvector

of A , and the maximum and minimum values of are

min , max = = , respectively.

Proof of Theorem 3. Since A is positive, according to

Lemma 1, A has a Perron eigenvalue and Perron

eigenvector ξ. Take y = εξ−1, where ε is a positive

undetermined scalar. Then, the MDQE (7) is solvable if ε

satisfies ( )1 1 − −. Likewise, it can be decomposed

into m quadratic inequalities of ε as the following

11 , 1,2, ,i

i

a i m

− = . (43)

Since A = , we obtain i ia = . Substituting it into

(43), it yields




2 2 0, 1,2, ,i i i m − + = . (44)

Solving the quadratic inequalities in (44), we obtain

( ) ( )

( ) ( )

1 11 1 4 1 1 42 2

1 1 4 1 1 42 2

m m

− − + −

− − + −

. (45)

Denote

( ) ( )1

1 1 4 1 1 42 2

mi i

i i

i

=

= = − − + −

，， . (46)

Similarly, equation (7) is solvable if is nonempty.

Clearly, is non-empty if and only if

( ) ( )

( ) ( )

1 1 4 1 1 42 2

1 1 4 1 1 42 2

ji

j i

− − + −

− − + −

, (47)

holds for every i, j∈{1,2,…, m}. Simplifying (47), it is

equivalent to the following

( )2

1i j

i j

+ . (48)

Since (48) holds for every i, j∈{1,2,…, m}, the following

can be obtained

( )2 2

1 ,max 1

i j

i j mi j

+ = +

. (49)

According to (49), (42) is derived. The proof is accomplished.

Corollary 1. The system admits a feasible power-flow

solution if the system parameters satisify

min 2 , 1A

+

. (50)

Proof of the Corollary 1. Clearly, equation (7) is solvable as

long as one of conditions (35) and (42) holds. Therefore,

conditions (50) can be easily derived. The proof is

accomplished.

Remark 3. According to Theorem 1, equation (7) is solvable

if there exists a positive vector y ,which satisfies (18).

Substitute 1my = and 1y −= into (18) respectively, then

the explicit condition (50) is derived. Condition (35) is the

main result of [27]. Condition (35) is firstly derived in [27] by

using Banach’s fixed-point theorem, and this paper provides

an alternative proof by using the nested interval theorem.

Clearly, condition (35) is completely covered by (50).

Therefore, the proposed existence condition is stronger than

the results in [27].

3.5 Solutions for Q1 and Q2

Next, this paper will compare the proposed condition with

the main results in [31] by answering Q1 and Q2 discussed in

Section 3.2.

For Q1, according to Corollary 1, the system admits a

feasible power-flow solution if the system parameters satisfy

(50). When the voltage references are different, we write V as

min 1V v q= , where 11 minq v V−= and min minv V= . Clearly,

1q is the proportion among the voltage references. Then,

according to (50), the system admits a feasible power-flow

solution if

1 1

1 1 min

1 1

min 2 ,A v

+

, (51)

holds,where 1 11

1 1 1 1 1 2 1,A diag B diag B q − −−= = , 1 and

1 are the Perron eigenvalue and eigenvector of 1A ,

respectively, and 1 1 1 1min , max = = .

For Q2, we write P as min 2P P q= , where min minP P=

and 12 minq P P−= is the proportion among the loads. According

to (50), the system admits a feasible power-flow solution if

( )min 2

2 2

2 2

2 2

1

min 4 ,

P

Aq

+

, (52)

holds, where 2 and 2 are the Perron eigenvalue and

eigenvector of 2Adiag q , and 2 2 1 2min , max = = .

The flowchart of solutions for Q1 and Q2 is depicted in

Fig.2.

Laplacian matrix

Droop gains

B

K

Calculate

( )( )

11

1

112 1

LL LG GG GL

LG GG

B B B B K B

B B B I KB

−−

−−

= − +

= − +

Q1

1

Maximal CPL power

Proportion

P

q

Calculate

1 2 1

1 111 1 1 1

,B q diag P

A diag B diag

− −−

= =

=

min 1

The voltage reference is

designed as V v q=

1 1

1 1

1 1

min 2 ,A

+

1 1

min 1 1

1 1

The system admits a solution if

min 2 ,v A

+

Q2

min 2

The maximal CPL is

designed as P P q=

2

Voltage reference

Proportion

V

q

( )2

1

1

L

L L

V B V

A diag V B diag V

−

=

=

2 2

2 2

2 2

min 2 ,Aq

+

( )min 2

2 2

2 2

2 2

The system admits a solution if

1

min 4 ,

P

Aq

+

Calculate

1 2

1 2

1 2

1

2

1 1 1 1

2 2 2 2

χ and χ are the Perron

roots of and ,

and are the Perron

eigenvectors of and

, respectively.

= max , = min

= max , = min

A Adiag q

A

Adiag q

Calculate

End

Calculate

Fig.2. The flowchart of the solutions for Q1 and Q2. Remark 4. In DC microgrid, the existence of a stable

steady-state behavior is critical for the correct operation of DC

distribution, which is usually difficult to analyze. This paper

provides the analytical existence condition as a function of the

system parameters (i.e., V, B, K and P), which leads to a

design guideline to plan a reliable DC microgrid.

3.6 Comparing with main result of [31]

Next, this paper will compare the proposed condition with

the main results in [31] by answering Q1 discussed in Section

3.2.




When 1 1mq = , according to Proposition 1, min1L mV v = .

Then, equation (11) becomes 1 1

min 11L m Lu v B u− −= − . (53)

According to the main result of [31, Corollary 1], equation (53)

is solvable if the following holds

( )1 1 minmin 2 ,B B v

+

. (54)

On the other hand, substituting 1 1mq = into (51), (54) can

also be obtained. Hence, the result of [31] can be seen as a

special case of the proposed condition. When 1 1mq ,

equation (11) does not take the form of (53), because the

entries of LV are different. Thus, the result of [31] can not

directly apply to equation (11) when 1 1mq . However, we

can obtain the solvability condition of (11) by the following

ways. If (54) holds, according to the result of [31], there exists

a positive vector 1y such that 1

1 1 121

11my B y

u

−− . Since

min min1mv q v , we obtain

( ) ( )1 11 1

1 1 1 min 1 1LG GG LG GG mu B B I KB q u B B I KB− −− −= − + − +

. (55)

From (55), 1 11mu is ture, i.e., 1 121

1B A

u. Then, the

following is obtained

1 11 1 1 1 12

1

11 1m my B y A y

u

− −− − . (56)

Therefore, according to Theorem 1, the system admits a

feasible solution.

To sum up, when 1 1mq = , the solvability condition of (11)

derived from this paper and [31] are equivalent; when 1 1mq ,

the result of [31] can not directly apply to equation (11). Based

on the fact that equation (11) is solvable if (53) is solvable, a

solvability condition for (11) is obtained as (54). Comparing

with [31], condition (50) is a generalization of the result of

[31].

3.7. The Proposed Iterative Algorithm and Its Domain of

Attraction

According to corollary 1, the system admits a feasible

power-flow solution if (50) holds. Next, this paper will design

the effective iterative algorithm to obtain the feasible power-

flow solution. The main results are as the following.

Theorem 4. If (18) holds, for any x x y , the

following iterative algorithm will monotonically converge to

the solution of (7): ( )1 1,n nx x x + = = . Moreover,

(i) if x y x x , the convergence rate of the

proposed iterative algorithm can be expressed as

( ) ( )1 1

1 1

n

nx x A diag x diag x x− −

+− − . (57)

(ii) if x x x , the convergence rate of the proposed

iterative algorithm can be expressed as

( ) ( )2

1 1

n

nx x A diag x x x−

+ − − . (58)

Proof of the Theorem 4. If y x , accoding to (21), we

have

1 2 3 nx x x x x . (59)

According to (59), the following can be obtained

( ) ( )

( )

1

11

n n

n n n

x x x x

A diag x diag x x x x x

+

−−

− = −

= − −. (60)

From (60), 11

nI A diag x diag x−− − is an M-matrix.

According to Lemma 2, ( )11

1nA diag x diag x−−

holds for any positive integer n, i.e.,

( )11

1A diag diag x −− . From (59) and (60), the

following is derived

( )

( )

( ) ( )

1 1

1

11

1 1

1

n n n

n

n

x x A diag x diag x x x

A diag diag x x x

A diag x diag x x

− − +

−−

− −

− = −

−

−

. (61)

Since ( )11

1A diag diag x −− , we obtain

1lim nn

x x+→

= , which proves the statement (i) of Theorem 4.

If x , then, the following can be obtained

3 2 1nx x x x x . (62)

Likewise, there, we have the equation as

( )

( )

( )

( ) ( )

11

11

1

2

2

1

n n n

n n n

n

n

A diag x diag x x x x x

x x A diag x diag x x x

A diag x x x

A diag x x x

−−

−− +

−

−

− −

− = −

−

−

. (63)

Similarly, we have ( )2

1A diag x−

and

1lim nn

x x+→

= , thus proving the statement (i) of Theorem 4.

Remark 5: In this section, this paper proposes an effective

iterative algorithm to obtain the feasible power-flow solution.

Moreover, the proposed iterative algorithm has been proved,

which has a wide domain of attraction by theoretical analysis.

Therefore, one can easily choose the initial iteration value to

guarantee monotonic convergence according to Theorem 4.




23

24

28

29

68 3

1

1

45

4 2

5

10

2

2

2

1

41

5

2

1

4

539

37

36

353127

21

22

44

47

51

52

8

1

1011

4 2

5

10

2

2

2

1

41

5 4

1

4

560

59

58

57535046

45

32

69 66

26

2

4 2

9

2

225

5 5

412 2

33

34

1

48

49

1

2

22

2 22

1

2

54

55

56

2

67 70

2

76

4

12

230 38

2

2

402 2

61

62

63

65

64

2

2

2

2

242

43

13

14

15

2

2

2

2

2

2

2 2 2 2

16

17

18

2

2

2

19

4

20

4

？？

？？

?

DGCPL

Fig. 3. The structure of the simulation DC microgrid

IV. CASE STUDY

To verify the presented analyses, we simulate a meshed DC

microgrid with 20 DGs and 50 CPLs which is shown in Fig 3.

The red and blue points represent DGs and CPLs, respectively.

The black line represents the cables. The green numbers are

the resistances of cables, and the black numbers are the

identifiers of nodes. The droop gain coefficients are set as

1 2 20 2k k k= = = = .

4.1 The Answers to Q1 and Q2

In this part, this paper will answer the question Q1 and Q2

discussed in section 3.2 by the following specific cases.

q1. Assume the maximal CPL vector is P = 100 501 kW, how

to design the voltage reference V to ensure the system admits a

feasible power-flow solution?

q2. Assume the maximal CPL vector is P = [133 64 133 6 121T

44 6 6 40 6 111T 36 6 91T 104 65 105 60 71T ]T kW, how to

design the voltage reference V to ensure the system admits a

feasible power-flow solution?

q3. Assume the voltage reference is V = 10 101.1 1 ;1 1 kV,

how to design the maximal CPL vector P to ensure the system

admits a feasible power-flow solution?

For q1, we assume that V is designed as min 1= V v q and

1 10 101 ;1.2 1q = . According to the flowchart presented in

Fig. 2, the system admits a feasible power-flow solution if vmin

satisfies (51). By calculating, the following solvability

condition is obtained

1 1

min 1 1

1 1

min 2 , 2.84kVv A

+ =

. (64)

For q2, we assume that V is designed as min 1= V v q and

1 10 101 ;1.1 1q = . Likewise, the solvability condition for q2 is

obtained as

1 1

min 1 1

1 1

min 2 , 1.12kVv A

+ =

. (65)

Then, the question Q1 (at the end of Section 3.2) is

answered.

For q3, we assume P is designed as min 2P P q= where

2 40 101 ;3 1q = . According to the flowchart presented in Fig.

2, the system admits a feasible power-flow solution if Pmin

satisfies (51). Then, the solvability condition for q3 is obtained

as

( )min 2

2 2

2 2

2 2

1=10.23kW

min 4 ,

P

Aq

+

. (66)

Then, the question Q2 (at the end of Section 3.2) is

answered.

4.2 Compared with the existing results

For q1 and q2, according to the results of [27], the system

admits a feasible power-flow solution if min 12v A

.

Then, the solvability conditions for q1 and q2 are obtained as

(67) and (68), respectively.

min 12 =2.84kVv A

, (67)

min 12 =1.15kVv A

. (68)

According to the main results of [31], the system admits a

feasible power-flow solution if vmin satisfies (54). Then, the

solvability conditions for q1 and q2 are obtained as (69) and

(70), respectively.

( )min 1 1min 2 , =3.08kVv B B

+

, (69)

( )min 1 1min 2 , =1.17kVv B B

+

. (70)

The theoretical results in (65) and (68) show that the

proposed solvability condition (50) is stronger than [27]. The

theoretical results in (65) and (68) show that the proposed

solvability condition (50) is stronger than [31] when the

voltage references are different. Next, we will verify the

correctness of the above theoretical analysis results by

Matlab/Simulink.

Define 1 2, and as the follows:

( )

1 1

1 min 1 1 2 min 1

1 1

3 min 1 1

min 2 , 2

min 2 , .

v A v A

v B B

+ = − = −

+ = −

，

TABLE I

Existence of feasible power-flow solutions of Case 1-4 Cases Case 1 Case 2 Case 3 Case 4

τ1 > 0 ? Yes No Yes No

τ2 > 0 ? Yes No No No

τ3 > 0 ? No No No No

Is (7) solvable? Yes No Yes No

Clearly, the solvability conditions derived in this paper, [27]

and [31] are 1 20, 0 and 3 0 , respectively. Then, we

evaluate the following cases.

Case 1: The maximal CPL vector P is the same with q1, and

the voltage reference is min 1= V v q , where 1 10 101 ;1.2 1q =

and vmin = 2.85 kV.






and vmin = 2.58 kV.



and vmin = 1.12 kV.



and vmin = 1.1 kV.

Case 5: The voltage reference V is same with q3, and the CPL

vector is min 2P P q= , where 2 40 101 ;3 1q = and Pmin =10.2 kW.

Case 6: The voltage reference V is same with q3, and the CPL

vector is min 2P P q= , where 2 40 101 ;3 1q = and Pmin =11 kW.

0 0.01 0.02 0.03 0.04

800

1600

2400

3200

Times(s)

Vo

ltag

e (V

)

0 0.02 0.04 0.06 0.08

1600

2000

2400

2800

3200

Times(s)

Vo

ltag

e (V

)

(a) Case 1 (b) Case 2

0 0.02 0.04 0.06 0.080

500

1000

1500

Vo

ltag

e (V

)

0 0.04 0.08 0.12

600

700

800

900

1000

1100

Times(s)

(c) Case 3

Vo

ltag

e (V

)

(d) Case 4

Times(s)

0 0.01 0.02 0.03 0.04 0.05500

600

700

800

900

1000

Times(s)

(e) Case 5

Vo

ltag

e (V

)

0.02 0.04 0.06 0.080

400

800

1200

0

Vo

ltag

e (V

)

Times(s)

(f) Case 6

Fig. 4. The load voltages of case 1-6. The subfigure (a), (b) and (c) are load

voltages of case 1-6, respectively.

Case 1, 3 and 5 are designed to validate the correctness of the

proposed solvability condition, while case 2, 4 and 6 are

designed as the comparisons.

The simulation results of case 1-6 are in Table I and Fig. 4.

The results in Fig.4 shows that case 1, 3 and 5 are stable,

which verifies the correctness of the proposed solvability

conditions. Moreover, case 3 is stable when 1 20, 0

and 3 0 , which shows that solvability condition (51) is

stronger than the results in [31] and [27]. Case 2, 4 and 6

shows that the system will lose equilibrium resulting in

voltage collapse when the load is too heavy or the reference

voltage is too low.

4.3 Test of the Proposed Iteration Algorithm

According to Theorem 4, the proposed algorithm is

convergent if 1x y is true, where x1 is the initial iteration

value and y is defined by (18).

To verify the presented results, we evaluate the following

four cases:

Case 7: The CPL and voltage reference are same with case 1.

By calculating, (18) holds when y = 0.5 501 . The initial

iteration value x1 is set as 1 500.6 1x = .

Case 8: The CPL and voltage reference are the same as case 1.

The initial iteration value x1 is set as 1 500.4 1x = .

Case 9: The CPL and voltage reference are the same as case 2.

The initial iteration value x1 is set as 1 501x = .

In case 7 and 8, P and V are same with case 1 which has

been proved to be stable. According to Theorem 4, the

proposed iterative algorithm is monotonically convergent if

1 500.5 1x . Thus, we take 1 500.6 1x = in case 7 and take

1 500.4 1x = in case 8 as a comparison. In case 9, P and V are

same with case 2 which has been proved that the system has

no equilibrium. Then, the proposed iterative algorithm will be

divergent.

ϕ (

x)

0 4 8 12 16 20

0.6

0.7

0.8

0.9

Iteration numbers

(a) Case 7

0 10 20 30 40

0

1

2

3

ϕ (

x)

Iteration numbers

(b) Case 8

0 40 80 120 160 200 4

0

4

8

12

16

Iteration numbers

(c) Case 9

ϕ (

x)

Fig. 5. The iteration processes of the proposed algorithm: ( )1n nx x+ = . The

subfigure (a), (b) and (c) are the iteration processes of case 7-9, respectively.

The iterative process of the algorithm is shown in Figure 5.

Fig.5 (a) shows that the proposed iterative algorithm

monotonically converges to the solution if the initial value is

in the proposed domain of attraction, which verifies the

correctness of Theorem 4. Subfigure (b) shows that the

proposed algorithm may not be monotonically convergent if

the initial value is not in domain of attraction. Case 9 shows

that the proposed iterative algorithm is not divergent if the

system has no equilibrium.

In summary, the simulation results are consistent with the

theoretical analysis, thus verifying the correctness and

effectiveness of the proposed existence conditions and

iterative algorithm of the feasible power-flow solution.

V. CONCLUSIONS

The existence of power flow solution for DC microgrid is

investigated in this paper. A stronger condition for the

existence of the feasible power-flow solution is obtained based

on the nested interval theorem. Meanwhile, this paper proves




the uniqueness of feasible power-flow solution according to the

properties of limitations, and presents an iterative algorithm for

feasible power-flow solution. Although the convergence rate of

the proposed iterative algorithm is linear, it has well

convergence in terms of the initial iterative value. In the future

research work, we will focus on the following issues: 1) the

sufficient and necessary conditions for the existence of feasible

power-flow solution and the iterative algorithm with faster

convergence speed; 2) the existence and stability of equilibrium

of DC microgrid under master-slave control.

Appendix

Proposition 1. For a strong connected DC microgrid, the

following three statements hold true.

1) 11B− is positive ( 1

1B O− );

2) 21 1m mB = ;

3) 0L mV .

Proof. Define 1

1ΓGG GL

LG LL

K B B

B B

− +

. (71)

Clearly, B1 is a Schur complement of Г1. Since the DC

microgrid is strong connected, B is irreducible. Г1 is an

irreducible positive definite M-matrix because K is positive

definite diagonal matrix and B is a Laplacian matrix.

According to Lemma 3, 1

1− is positive. Applying the

formula for the inverse of a block matrix, we obtain

( ) ( )

( )

1 11 1 1 1

11

1 11 1 1 1

1

ΓGG GL LL LG GG GL

LL LG GG GL LL LG

K B B B B K B B B

B B K B B B B B

− −− − − −

−

−− − − −

+ − − +

= − + −

.

Because 1

1− is positive,

11B−

is positive, thus proving

statement 1).

Given that B is a Laplacian matrix, Y1n+m =0n+m, i.e., 1

1

1 1 0

1 1 0

n GG GL m n

LL LG n m n

B B

B B

−

−

+ =

+ =

. (72)

According to (72), 21 1m mB− can be calculated as

( )( )( )( )

( )( )( )( )

( )

( ) ( )

112 1 1

11 1 1

1 1

11 1

1

11 1

1

1 11

11 1 1

1

1 1 1 1

= 1 1

1

1

1 1

1 1

0

m n m LG GG n

m LG LG GG m

LL LG GG GL m

LG LG GG n

LL LL LG n m

LG GG n GG GL m

m

B B B B I KB

B B B B K B K

B B B B K B

B B Y K Y K

B B B B

B B K B K B B

−−

−− − −

−− −

−− −

− −

−− − −

− = + +

+ − +

= − + +

− +

= + −

+ +

=

.(73)

According to (73), we obtain 21 1 0m n mB− = , thus proving

2).

Since B is Laplacian matrix of a strong connected DC

microgrid, BGG is a positive definite M-matrix. Clearly, K−1 +

BGG is also an M-matrix. Then, we have ( )1

1 1K B K O−

− −+ .

Since 11B O− and LGB O , we obtain 2B O . Given

that min1nV v , we obtain 2 2 min min1 1L n nV B V B v v = = .

Then, the statement 3) of Proposition 1 is proved.

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Zhangjie Liu received the B.S. degree, in 2013, in

Detection Guidance and Control Techniques and the Ph. D. degree in Control Science and Engineering from the

Central South University, Changsha, China. He is

currently an Associate Professor at School of Automation of Central South University. He is also a Postdoctoral

Research Fellow at the Nanyang Technological

University.

He is interested in DC microgrid, power system stability, complex network

and nonlinear circuit.

Ruisong Liu received the B.S degree in electrical engineering and automation from the Guizhou University

Guiyang, China, in 2019, and he is currently working

toward master's degree in electrical engineering in Central South University, Changsha, China.

His research interests include Renewable system and

DC microgrid.

Xin Zhang (M'15) received the Ph.D. degree in

Automatic Control and Systems Engineering from the

University of Sheffield, U.K., in 2016 and the Ph.D.

degree in Electronic and Electrical Engineering from

Nanjing University of Aeronautics & Astronautics,

China, in 2014. Currently, he is an Assistant Professor of

Power Engineering at the School of Electrical and

Electronic Engineering of Nanyang Technological

University. He is also the Cluster Director of Energy

Research Institute @NTU. Dr Xin Zhang has received the highly-prestigious

Chinese National Award for Outstanding Students Abroad in 2016. He is the

Associated Editor of IEEE TIE/JESTPE/Access/OJPE, IET Power Electronics.

He is generally interested in power electronics, power system, and advanced

control theory, together with their applications in various sectors.

Mei Su was born in Hunan, China, in 1967. She received

the B.S., M.S., and Ph.D. degrees from the School of

Information Science and Engineering, Central South

University, Changsha, China, in 1989, 1992, and 2005,

respectively. Since 2006, she has been a Professor with

the School of Information Science and Engineering,

Central South University. Her research interests include

matrix converter, adjustable speed drives, and wind

energy conversion system.

Yao Sun (M’13) was born in Hunan, China, in 1981.He

received the B.S., M.S., and Ph.D. degrees from the

School of Information Science and Engineering, Central

South University, Changsha, China, in 2004, 2007, and

2010, respectively. He is currently with the School of

Information Science and Engineering, Central South

University, China, as an Associate Professor.

His research interests include matrix converter,

micro-grid, and wind energy conversion system.

Hua Han was born in Hunan, China, in 1970. She

received the M.S. and Ph.D. degrees from the School of

Automation, Central South University, Changsha, China,

in 1998 and 2008, respectively. She was a Visiting

Scholar with the University of Central Florida, Orlando,

FL, USA, from 2011 to 2012. She is currently a

Professor with the School of Automation, Central South

University.

Her research interests include microgrids, renewable

energy power generation systems, and power electronic

equipment.

Peng Wang (F'18) received the B.Sc. degree in

electronic engineering from Xi'an Jiaotong University,

Xi'an, China, in 1978, the M.Sc. degree from Taiyuan

University of Technology, Taiyuan, China, in 1987, and

the M.Sc. and Ph.D. degrees in electrical engineering

from the University of Saskatchewan, Saskatoon, SK,

Canada, in 1995 and 1998, respectively. Currently, he is

a Professor with the School of Electrical and Electronic

Engineering at Nanyang Technological University,

Singapore


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