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1949-3053 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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
Authorized licensed use limited to: Central South University. Downloaded on February 12,2020 at 12:55:32 UTC from IEEE Xplore. Restrictions apply.
1949-3053 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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
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
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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
Authorized licensed use limited to: Central South University. Downloaded on February 12,2020 at 12:55:32 UTC from IEEE Xplore. Restrictions apply.
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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)
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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
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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.
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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.
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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.
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Case 2: 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 = 2.58 kV.
Case 3: The maximal CPL vector P is the same with q2, and
the voltage reference is min 1= V v q , where 1 10 101 ;1.1 1q =
and vmin = 1.12 kV.
Case 4: The maximal CPL vector P is the same with q2, and
the voltage reference is min 1= V v q , where 1 10 101 ;1.1 1q =
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
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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.
VI. REFERENCES
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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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