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Abstract— Constant power loads (CPLs) are often the cause of

instability and no equilibrium of DC microgrids. In this study, we

analyze the existence and stability of equilibrium of DC

microgirds with CPLs and the sufficient conditions for them are

provided. To derive the existence of system equilibrium, we

transform the problem of quadratic equation solvability into the

existence of a fixed point for an increasing fractional mapping.

Then, the sufficient condition based on the Tarski fixed-point

theorem is derived. It is less conservative compared with the

existing results. Moreover, we adopt the small-signal model to

predict the system qualitative behavior around equilibrium. The

analytic conditions of robust stability are determined by

analyzing quadratic eigenvalue. Overall, the obtained conditions

provide the references for building reliable DC microgrids. The

simulation results verify the correctness of the proposed

conditions.

Index Terms-- DC microgrid, solvability, nonlinear equations,

fixed-point theorem, quadratic eigenvalue problem, stability,

constant power load. I. INTRODUCTION

Microgrid has been identified as a key component of

modern electrical system to integrate renewable distributed

generation units [1]-[3]. Microgrids can be divided into two

types: alternating-current (AC) and direct-current (DC)

microgrid [4]-[5]. DC microgrids have attracted much

attention given three main advantages over their AC

counterparts, namely high efficiency, simple control and

robustness. They are increasingly used in applications such as

aircraft, spacecraft and electric vehicle [6]–[8]. In DC

microgrid, most loads are connected to the DC-bus through

power electronic interface circuits which make the loads

behave constant power loads (CPLs) [9]-[10].

Nevertheless, the negative impedance of CPLs often result

in instability. Thus, stability has been widely investigated in

DC microgrids with CPLs, and several stabilization methods

have been proposed [11]–[26]. The topologies of DC

microgrids in these studies can be divided into two groups

regarding the number of converters and loads: 1) n converters

and one CPL; 2) n converters and m CPLs. Manuscript received December 07, 2017; revised **, **; accepted June 10,

2018. This work was supported by the National Natural Science Foundation

of China under Grants 51677195 and 61573384, the Natural Science

Foundation of Hunan Province of China under Grant no. 2016JJ1019, and the Joint Research Fund of Chinese Ministry of Education under Grant

6141A02033514. (Corresponding author: Yao Sun.)

Zhangjie Liu, Mei Su, Yao Sun, Wenbin Yuan, and Hua Han are with the School of Information Science and Engineering, Central South University,

Changsha 410083, China (e-mail: [email protected]).

Jianghua Feng is with the CRRC Zhuzhou Institute Co. Ltd., Zhuzhou

412001, China (e-mail:[email protected]).

Various studies focused on stability criteria and stabilization

methods of DC microgrids composed of n converters and one

CPL [11]-[25]. In this topology, the distributed generations

(DGs) are connected to the CPLs through a DC bus, and when

the DC-bus resistance is neglected, the loads can be modeled

as a single CPL. Therefore, this system is equivalent to

star-connection topology consisting of n DGs and one CPL.

The existence of equilibrium in this system is determined by a

quadratic equation with one unknown, and hence the sufficient

conditions can be easily obtained [18], [25]. Therefore, related

studies mainly focus on overcoming the instability due to CPL.

Several linear techniques are used to stabilize the system

[11]–[14]. Based on the idea that increasing damping can

mitigate oscillations, several stabilization methods such as

passivity based control [11], active damping [12], and virtual

impedance [13]–[14] have been proposed. Furthermore,

several nonlinear methods such as phase-plane analysis [15],

feedback linearization [16] and sliding-mode control [17] have

been applied to overcome CPL instability. To estimate the

region of attraction around known equilibria, Lyapunov-like

functions have been proposed, including Lure Lyapunov

function [18], Brayton-Moser’s mixed potential [19]–[20],

block diagonalized quadratic Lyapunov function [21] and

Popov criterion [22]. In addition, the stability of DC

microgrids under droop control was analyzed in [23]–[25]. A

stability condition is derived based on the reduced-order

model of DC microgrids [23]. Another stability condition is

derived under the assumption that all the DG filter inductances

have the same ratio, R/L [24]. To obtain more accurate

stability conditions, a high dimensional model is used in

stability analysis [25].

Due to the transmission loss, the increase of CPL may result

in the loss of system equilibrium (i.e., voltage collapse) [26]

-[28]. Hence, finding the condition for the existence of the

equilibrium is a prerequisite. The condition for equilibrium

existence can be easily obtained under the assumption that the

DC-bus resistance can be neglected. However, in most cases,

this assumption is not reasonable. In the practical

DC-microgrid consisting of n converters and m CPLs, the

existence of equilibrium is determined by m-dimensional

simultaneous quadratic equations with m unknowns, which is

a difficult problem. By using quadratic mapping, a necessary

condition for the existence of equilibrium based on a linear

matrix inequality (LMI) is obtained in [26], where the

condition is sufficient if and only if the quadratic mapping is

convex. Although a test to verify the convexity of quadratic

mappings is given in [27], the mappings are usually

nonconvex for the case m > 2. To determine the sufficient

condition for the solvability of the m-dimensional quadratic

Existence and Stability of Equilibrium of DC

Microgrid with Constant Power Loads Zhangjie Liu, Mei Su, Yao Sun, Member IEEE, Wenbin Yuan, Hua Han and Jianghua Feng

mailto:[email protected]



equations, a contraction mapping is constructed ingeniously in

[28]. However, the obtained condition is a little conservative

because contraction mapping requires that the norm of the

mapping Jacobian matrix is less than 1. Under the assumption

that system has a constant steady-state, the stability of the

equilibrium is analyzed in [26] and [9]. In [26], the Jacobian

matrix is derived for the given equilibrium, thus, the system is

stable if the Jacobian matrix is Hurwitz. Considering that the

equilibrium depends on the uncertain loads, the locally roust

stability conditions based on LMI are obtained [9]. However,

in these studies, the existence of equilibrium and the analytical

relation among the cable resistances, capacitances, voltage

reference, the maximal loads and the control parameters have

not been addressed.

Fig.1. Diagram of the DC microgrid.

In this study, we investigate the following two questions:

1) Under what conditions should DC microgrid admit a

constant steady state and how can we reduce the

conservativeness?

2) How to design a stabilization method to guarantee

system stability and how to obtain the analytical robust

stability conditions?

The main contributions of this paper can be summarized as

A control method based on virtual resistance and virtual

inductance concept to overcome instability is proposed.

The sufficient condition for the system to admit an

equilibrium is obtained, and it is less conservative

compared with [28].

The equilibrium stability is analyzed and the analytical

stability conditions are determined using eigenvalue

analysis.

The rest of this paper is organized as follows: Section II

introduces some preliminaries and notations. Section III

describes the basic models and the proposed control scheme.

The existence of equilibrium for DC microgrids is presented

in Section IV. The stability analysis and the sufficient

conditions are detailed in Section V. Simulation results are

presented in section VI. Finally, we draw our conclusions in

Section VII.

II. PRELIMINARIES AND NOTATION

Definition 1. We denote A−B > 0 if matrix A−B is positive

definite. Matrix A (or a vector) is called nonnegative

(respectively positive, negative and nonpositive) if its entries

are nonnegative (respectively positive, negative and

nonpositive) and we denote , , ,A B A B A B A B if the

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

nonpositive, respectively. In addition, 1m (0m) is the vector of

ones (zeros). For Hermitian matrix A∈ m mR , we denote its

eigenvalue as 1 mA A .

Definition 2. Square matrix A is a Z-matrix if all the

off-diagonal elements are zero or negative, and it is also an

M-matrix [29] if and only if one of these statements is true:

1) the eigenvalues of A are in the right half-plane;

2) there exists a positive vector, x, such that 0nAx .

Definition 3. Matrix A∈ n nR is irreducible if there exists no

permutation matrix P such that PTAP can be represented as

11 12

22

TA A

P APO A

where A11, and A22 are square matrices, and O is the zero

matrix of proper dimension [30].

Lemma 1. If A is an M-matrix, then

1) A + D is an M-matrix for every nonnegative diagonal

matrix D;

2) if A is irreducible, A-1 is positive [31].

Lemma 2. Tarski fixed-point theorem [32]. Given D n nR

convex, let :f D D be a continuous function such that

1) f(x) is strictly increasing, i.e., 1 2, Dx x , if 1 2x x ,

1 2f x f x ;

2) 1 2,x x D such that 11 22x xf x f x ,

there is a unique vector 2*

1x xx such that f (x*) = x*.

Lemma 3. Let A be a real positive matrix. Perron root χ and

Perron vector η satisfy Aη = χη, where η 0 and ηTη = 1.

Moreover, χ is also the spectral radius of A, denoted ρ(A).

If A B O , then ρ(A)> ρ(B) [30].

Lemma 4. Let2( )Q M K C . Let M, K and C be

positive definite Hermitian matrices. For a quadratic



eigenvalue problem2 0M K C , Re (λ) <0 [33].

III. DC-MICROGRID MODEL AND CONTROL SCHEME

A general DC microgrid with n converters (DGs) and m

CPLs is illustrated in Fig. 1 and consists of three main

components: sources, loads and cables. In a low-voltage DC

microgrid, the cable can be regarded as purely resistive and we

assume the loads as CPLs. In addition, we consider the

DC-microgrid topology as a graph (such as Fig.1 (b)) with the

sources and loads representing nodes, and the cables

representing edges, respectively. Furthermore, we assume the

graph as being strongly connected, i.e., every source has

access to every load.

The dynamics of the ith converter can be described by

i

i

L

i i i i

ii L i

diL V d u

dt

duC i i

dt

(1)

where Vi, di, iLi , ui and ii are the input voltage, duty cycle,

inductance current, output voltage and current of the converter,

respectively.

The system is usually unstable when there is no

stabilization measures as Fig.2 (a). To prevent the instability

caused by CPLs in a DC microgrid, we propose a controller

based on virtual resistance and virtual inductance whose duty

cycle is given by

ref 1,2, ,i

i ii i L i

i i i

L ud u k i u i n

V X V , (2)

where Xi, ki, and uref are the virtual inductance, virtual

resistance, and reference voltage of the ith converter,

respectively. It needs to be emphasized that Xi, ki and uref are

all the control parameters to guarantee a stable equilibrium.

iLiC

iV1/Vi

refu

ii

iLiC

iV1/ViLi/Xi

refu

ki

ui

PWM

(a)

(b)

PWM

Fig.2. (a) The control diagram without stabilization method. (b) The control

diagram of the proposed method.

The control diagram of the proposed stabilization method is

presented in Fig.2, and it does not need communication. Then, substituting (2) into (1) and writting the result in

matrix form, we obtain

LL S

SL S

diX V Ki u

dt

duC i i

dt

(3)

where 1 2 n

T

L L L Li i i i , uS= [u1 u2 … un]T, iS = [i1 i2 …

in]T, V*= uref1n, C = diag{Ci}, and X = diag{Xi}, i ∈{1, 2, …,

n}. In addition, the load voltages are denoted by uL = [un+1

un+2 … un+m]T.

Applying the Kirchoff’s and Ohm’s laws, we have

1 2

1 2

,

T

S nSS SLS S S

TLS LLL L L L n n n m

i i i iY Yi u uY

Y Yi u u i i i i

(4)

where Y=[yij] is the symmetric admittance matrix of the graph,

where yij is the cable conductance, yij=0 if there is no cable

connecting nodes i and j, and rij =1/yij represents the resistance.

iS and iL are the current vectors of the sources and loads,

respectively. For a CPL, it yields

, 1, 2, ,i i iu i P i n n n m (5)

where Pi is the power of load at node i, and the right-hand side

of the equation is negative because the actual current direction

and reference are opposite.

According to (3), (4) and (5), the dynamics of the whole

system can be expressed as the following differential-algebraic

equations

0

LL

L

S

SL S

S SS

LL L L LS S

S SL L

m

diX V Ki u

dt

duC i i

dt

i Y u

U Y u U Y u

Y u

P

(6)

where UL = diag{uL}, P = [Pm+1 Pm+2 …Pm+n]T.

Remark 1: According to (6), the load voltages are determined

by m-dimensional simultaneous quadratic equations, the

increase of CPL may result in the loss of system equilibrium.

Thus, to maintain the system stable, finding the condition for

the existence of the equilibrium is a prerequisite.

IV. EXISTENCE OF EQUILIBRIUM OF DC MICROGRID

Besides instability, CPLs can cause inexistence of

equilibrium in DC microgrids [25]-[26]. For simplicity, all the

loads can be equivalent to a common CPL under the

assumption that the DC-bus resistance can be neglected [24]-

[25]. Consequently, the existence of equilibrium is formulated

by the solvability of a quadratic equation with one unknown,

whose conditions are easily obtained. However, in most cases,

the DC bus resistance cannot be neglected, then, the

equilibrium should be determined from quadratic equations

with multiple unknowns, and the topology of the solution set

is complicated [26].

A. Problem Formulation

According to (3), when the system achieves steady-state,

the output voltage is given by uS = V*– KiS, and substituting it

into (4) yields

S SS SS S SL L

L LS LS S LL L

i Y V Y Ki Y u

i Y V Y Ki Y u

, (7)

whose simplification yields to



1 11 1 1

1 11 1 1

S SS SS SS SL L

L LS LS SS LL LS SS SS SL L

i Y K V Y K Y Y u

i Y Y K Y K V Y Y K Y K Y Y u

. (8)

Combining this result with (5), the system admits a constant

steady state if and only if

1 0L L mLU Y u U P (9)

is solvable where

1 1

1 1 1 11= 1 ,ref LS SS SS n LL LS SS SS SLu Y Y Y K Y Y Y K Y K Y Y

(10)

Clearly, the system admits an equilibrium if and only if, for

given values of uref, Y, and P, the multidimensional quadratic

equations in (9) admit a real solution. For this issue, this paper

mainly aims at answering the following questions:

Q1: Given the maximum CPL power, how should the voltage

reference uref be regulated to keep (9) solvable?

Q2: Given the fixed uref, how to obtained the maximum CPL

power to keep (9) solvable?

B. Related Results

To analyze the solvability of m-dimensional quadratic

equations such as (9), there are mainly two methods:

“completing the square” [26] and “contraction mapping” [28].

In the next, we will start the story-line of m-dimensional

quadratic equations by one-dimensional quadratic equation.

Some existed studies neglect the resistance of the common

DC bus [24], [25]. Under this assumption, all CPLs in a DC

microgrid are equivalent to a common CPL, as shown in Fig.

2, and (9) becomes an one-dimensional quadratic equation as

the following

eq LS LL eq equ Y V Y u P (11)

where 1Teq mP P is the equivalent load, ueq∈R is its voltage,

1

11 1TLL n SS nY Y K

and

11=1T

LS n SSY Y K

are the equivalent

admittance matrices. Obviously, (11) can be solved by

completing the square as the following

22

20

2 4

LS eq LLLSeq

LL LL

Y V P YY Vu

Y Y

(12)

Thus, (11) admits an equilibrium if and only if

2

4 0LS LL eqY V Y P (13)

Then, the following question naturally arise: can the

m-dimensional quadratic equations such as (9) be solved by

completing the square? Although (9) cannot be completely

solved by this method, some valuable conclusions can still be

obtained. Equation (9) must have no solution if the following

one-dimensional quadratic equation is unsolvable

1 1 1 01 T Tm

Tm L L L mU Y u U P (14)

By completing the square, (14) can be expressed as

1 1

1 1 1 1 1 1

1

1 1

1

2

TT T T

i

T

L L

mT

n i

u Y Y Y Y u Y Y

P Y Y

(15)

Obviously, (15) is unsolvable if

1 1

1

1 11

0

2

T

m T Tn ii

Y Y

P HY Y H

(16)

So, (9) is unsolvable if (16) holds. Thus, a necessary

condition of solvability of (9) is that (16) does not hold. In fact,

(14) is sum of all the sub-equations of (9). Likewise, if the

“weighed sum”,

1 1 01 1TTm L L L

Tm mHU Y u HPUH (17)

where H = diag{hi}, has no solution, (9) is unsolvable [26].

Reshaping (17) into the square form, an extension necessary

condition of (16) can be obtained as the following

Proposition 1: Assume there exists a diagonal matrix H such

that

1 1

11 11

0

2n m T

i ii n

HY Y H

h P H HY Y H H

(18)

Then, (9) has no real solution. (The proof, that consists of a

simple “completion of squares” procedure, is given in [26].)

The necessary condition in Proposition 1 implies that if

there exists a constant steady-state, LMI (18) has no solutions.

However, is it true that the lack of solutions of (18) implies

solvability of (9)? Under what conditions (18) is necessary

and sufficient? The following results can be found in [26].

Define the multidimensional quadratic mapping, f : Rm→Rm , of the form

2 11 L L m L L Lf f u f u u U β Y uf .

Then, E={f(uL): uL∈Rm} is the image of the space of variables

uL under this map.

Proposition 2: If E is convex, (9) have a real solution if and

only if LMI (13) is unfeasible.

Reference [27] provides a test to check convexity of E.

However, for m > 2, set E is usually nonconvex. So, it is

difficult to obtain the sufficient conditions of solvability of (9)

by completing the square.

eqP

DG4 DG3

DG2DG1 Fig.3. Equivalent topology of DC microgrid with n DGs and one CPL.

To obtain the sufficient conditions, a method based on

contraction mapping is proposed in [28]. Equation (9) are

solvable if

1

14 1diag Y diag

(19)

where 11Y and idiag P . The proof, that consists

of an ingenious “constructing contraction mapping” procedure,

is detailed in [28]. Next, we provide an alternative proof along

with two stronger conditions.

C. Sufficient Conditions for Existence of Equilibrium

According to Lemma 2, if we can transform (9) into the

form x = f(x), where f(x) satisfies 1) and 2), (9) must be

solvable. Next, the main goal of this paper is to construct a

mapping that satisfies the above two conditions.

Multiplying by 1 11 LY U , (9) becomes

1 1 1 11 1 2Θ

T

L L L L n n n mu F u Y g u g u u u u

， (20)

Then, the solvability of (9) is equivalent to the existence of



fixed point in the fractional functions from (20). If 1) and 2) of

Lemma 2 are satisfied, the system admits a constant steady

state.

Then, we start by proving F(uL) is strictly increasing. Firstly,

we prove the following proposition.

Proposition 3: The following three statements are true

1) Y is irreducible;

2) 11Y is positive ( 1

1Y O );

3) ζ = uref 1m.

Proof. First, we assume that Y is reducible, and as Y is

symmetric, there exists a permutation matrix E such that '

11

'22

TY O

E AEO Y

. (21)

Hence, the DC microgrid can be divided into two separate

microgrids, which contradicts the assumption that all sources

and loads are strongly connected, thus proving 1).

Next, Y1 can be simplified as 1

11 LL LS SS SLY Y Y Y K Y

and we define 1

1ΓSS SL

LS LL

K Y Y

Y Y

. (22)

Clearly, Y1 is a Schur complement of Г1. According to 1) of

Lemma 1, Г1 is a positive-definite M-matrix. Clearly, Г1 is

irreducible as Y, and according to 2) of Lemma 1, 11 is

strictly 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

ΓSS SL LL LS SS SL

LL LS SS SL LL LS

K Y Y Y Y K Y Y Y

Y Y K Y Y Y Y Y

.

Because 11 is strictly positive, 1

1Y is positive and

irreducible, thus proving 2).

Finally, given that Y is a Laplacian matrix, Y1n+m=0n+m, i.e., 1

1

1 1 0

1 1 0

n SS SL m n

LL LS n m n

Y Y

Y Y

. (23)

Then, we have

1 11 1 1

11 1

1 1

1 1 1 1 0

LS LS SS n LL LS SS SS SL m

LS n LL m LS SS n SS SL m m

Y Y K Y K Y Y K Y K Y Y

Y Y Y K Y K Y Y

(24)

According to (24), we obtain1 1

1 1 0ref m mu Y , i.e., ζ =

uref 1m, thus proving 3).

According to 2) in Proposition 3, 11Y is strictly positive,

and hence 11 ΘY is also strictly positive. Thus, we have

11 2 1 2 1 1 20 , 0m mF x F x Y g x g x x x (25)

Thus, F(uL) is strictly increasing, satisfying 1) of Lemma 2. If

F(uL) also satisfies the other condition, (9) must be solvable.

In addition, since 11Y is positive, the following necessary

condition for that (9) has a positive real solution can be

obtained by completing the square.

Theorem 1. A necessary condition for that (20) has a positive

real solution is

2refu , (26)

where χ is the Perron root of 11 ΘY .

Proof. First, let 11 1 2Θ ; ; ; nY a a a where ai is the ith row

vector of 11 ΘY . We assume 0 is a real solution of (20) of

the form

1

1, 1,2, ,

m

i ref ij

j j

u a i m

, (27)

where aij represents the vector entry. Thus, (27) can be

expressed as

2 2

1

1, 1,2, ,

2 4

mref ref

i i ij

j j

u ua i m

. (28)

Then, the following can be obtained

2

1

1 10, 1,2, ,

4

mref

ij

ji j

ua i m

, (29)

whose matrix form is given by

2 1 1 1 11 1 24 0 ,

T

ref m nu I Y (30)

According to 2) of Proposition 3, 11Y is positive, i.e.,

2 114refu I Y is a Z-matrix. In addition, (30) satisfies 2) of

Definition 2, and hence 2 114refu I Y is an M-matrix.

Therefore, according to 1) in Definition 2, (26) must be

satisfied, completing the proof.

Usually, for a practical DC power system, the voltage is

positive. Therefore, Theorem 1 provides a necessary condition

for that the system admits a constant steady-state.

Because F(uL) is strictly increasing, according to Lemma 2,

if we can find positive vector x1, x2 such that x1 f(x1) f(x2)

x2, the system admits a constant steady-state. Then, the main

results are stated as follows.

Theorem 2. There must exist a unique vector Lu such that

,L L Lu F u h u provided that

2

1 ,maxref ij

i j mu f q

(31)

where q=[q1 q2 … qm]T is an arbitrary positive vector and

2 1 1 11 2min 4 ,

2

Ti i

ref ref m

i

q a qh u u q q q

q

,

2

4 max , if 2 max ,

if 2 max ,

j j ji i i

i j j i i j

jiij

j i j ji i

j ji i j i i j

j i i j

a q a q a qa q a q a q

q q q q q q

a qa qf q

q q a q a qa q a q

a q a qa q a q q q q q

q q q q

The proof is detailed in Appendix. Remark 2: We transform the quadratic equation solvability

into the existence of a fixed point in a nonlinear mapping. The

key is to construct an appropriate mapping that satisfies the

conditions of Lemma 2. In this process, the positivity of 11Y

play a crucial role. However, for a passive-transmission

network, its admittance matrix Y is a symmetric positive

semidefinite Z-matrix. Thus, Γ1 and Y1 must be irreducible



M-matrices. Therefore, 11Y is positive, and function F(x) is

strictly increasing. This way, we can obtain the sufficient

conditions for the system to admit an equilibrium by using the

Tarski fixed-point theorem. It is important to underscore that 1

1Y is positive whether the microgrid topology is radial or

meshed, hence, (31) can always ensure the existence of

equilibrium.

Furthermore, positive vector q in (31) is arbitrary.

Consequently, we can obtain the optimal q that minimizes the

right side of (31). Thus, the result is less conservative, and the

optimal sufficient conditions can be formulated as

2

1 ,min maxref ij

q i j mu f q

, (32)

Then, to obtain an explicit analytic condition, we take q as

several special vectors into (31).

Corollary 1. For given Y, K and P, the system must admit a

constant steady-state if the following holds

11m 2 Θin ,ref Yu

, (33)

where η is the Perron vector of 11 ΘY , max i and

min i .

Proof. Consider q = 1m in (31), and then

2

1 , 1 ,

11

1

max max 4max 1 , 1

4max 1 4 Θ

ref ij i m j mi j m i j m

i mi m

u f q a a

a Y

. (34)

Substituting ζ = uref 1m into (13), it turns equivalent to (34).

Thus, (19) is proved.

Let q = η, we have

, 2max , 2j j ji i i

j i j i i j

a aa a

. (35)

According to (35), fij(η) becomes

2

2

2

4 max , 4 if

= if

jii j

i j

jiij

j i jii j

j iji

j i

f

, (36)

and hence the following is straightforward

2

max ijf

. (37)

Thus, (33) is obtained, completing the proof.

Remark 3: Corollary 1 shows that for fixed load P, the system

must admit a constant steady-state as long as uref is large

enough, as can be expected. Meanwhile, (32) and (33) provide

numerical and analytical condition to guarantee the existence

of the equilibrium, respectively. Moreover, both (32) and (33)

are less conservative than the results from the recent works

[28].

In [28], the solvability condition is obtained by constructing

a contraction mapping, which requires that the norm of the

mapping Jacobian matrix is less than 1 ( / 1F x ).

Compared with [28], we transform (9) into an increasing

mapping, which does not need / 1F x , thus decreasing the

conservativeness.

For the given maximum CPL vector Pmax and K, the voltage

reference uref can be determined by (32), thus addressing Q1.

For the given uref and K, according to (34), the maximum CPL

vector Pmax yields 2

11 max 1

4

ref

m

uY P . (38)

Denote 11 ijY t Then, the following can be easily obtained

2

max,

1

max,

4

0

mref

ij j

j

j

ut P

P

, (39)

thus addressing Q2.

The relation among line resistances, virtual resistances,

loads and DG output voltages that make (9) solvable is

described by (32) and (33). This allows to determine design

guideline to build reliable DC microgrid.

V. STABILITY ANALYSIS OF DC MICROGRID

A. Small-signal Model around Equilibrium

According to Theorem 2, if condition (33) holds, the system

will admit a constant steady state, we denote it by

, , , ,L S S L Li u i i u . Linearizing (6) around the equilibrium, the

equivalent CPL resistances can be obtained from

2

11, 1, 2, ,i i i i i

i

i u r P u i n n n mr

(40)

where Δ represents the small-signal variation around the

equilibrium and ri is the negative CPL resistance. Substituting

(42) in linearized (5), we have

1

1

L SS SL LL L LS Si Y Y Y R Y u

, (41)

where RL = diag{ri}.

Obviously, Lu increases when uref increases.

1 1 11 1

1 11

1 Θ 0

1

L L ref ref m L L L m

ref m L L

u du u du Y g u Y R du

du I Y R du

(42)

where duref and duL are the increments of uref and uL,

respectively. When duref is positive, duL is a positive vector.

According to Definition 2, 1 11 LI Y R is an M-matrix, i.e.,

1 11 1, 1,2, ,i LI Y R i m (43)

It needs to be emphasized that (43) holds as long as the system

admits a constant steady-state.

Combining the linearized (5) with (43), the equivalent

linearized model of the system is given by

ΔΔ Δ

ΔΔ Δ

LL S

SL eq S

d iX K i u

dt

d uC i Y u

dt

, (44)



where 1

1eq SS SL LL L LSY Y Y Y R Y

. The system Jacobian

matrix is given by 1 1

2 1 1=

eq

X K XJ

C C Y

. (45)

B. Robust Stability Conditions

According to the Hartman–Grobman theorem, the

equilibrium is stable if and only if J2 is Hurwitz. The

characteristic polynomial of J2 is obtained as

1 1 1 1

1 1 1 1

11 1 1 1 1

1 1 1 1= +

eq eq

eq

eq

X K X λI X K XλI

C C Y C λI C Y

λI X K λI C Y C λI X K X

λI X K λI C Y X C

,(46)

which results in

2 1 1 1 1 1 1 0eq eqλ I λ X K C Y X KC Y X C . (47)

According to Lemma 4, if the coefficient matrix of equation

(47) are all symmetric and positive definite, J2 is Hurwitz.

Hence, to maintain symmetry in (47), we take X = bK, where b

is a proportionality coefficient. Then, the duty cycle is

designed as

1,2, ,i

i ii ref i L i

i i i

L ud u k i u i n

bV k V , (48)

and the system Jacobian matrix is

1

21 1

1 1

=

eq

I Kb bJ

C C Y

. (49)

Then, simplifying (47), we have

2 1 1 1 11 1 10eq eqλ I λ I C Y C Y K C

b b b

(50)

Thus, we can determine the stability condition according to

Lemma 2.The results are obtained as follows.

Theorem 3. Matrix J2 is Hurwitz if the following holds

1

0

0

eq

eq

C bY

K Y

(51)

Proof. Multiplying (50) by C, we have

2 11 10eq eqλ C λ C Y Y K

b b

(52)

Then, according to Lemma 3, (51) can be easily obtained.

Combining with the existence condition of equilibrium point,

the system admit a stable operation point if (32) and (51) hold.

Proposition 4: For any K satisfying (33), the following two

statements are true.

1

1 0

0q

q

e

e

K

Y

Y

(53)

Proof. Clearly, Yeq is the Schur complement of Г2 which is

defined as

2 1

SS SL

LS LL L

Y Y

Y Y R

(54)

Given that 12 1

1 1 0n mT

n m n m ii nr

, Γ2 must have at

least one negative eigenvalue. Let 1 11 1 LR I Y R , whose

eigenvalues clearly have a positive real part according to (43).

Thus, we have 1/2 1/2 1/2 1 1/2

2 1 1 1 1 1 0LR Y R Y I Y R Y . (55)

In fact, 1 1/2 1/21 1 2 1LY R Y R Y , and according to (55),

11 0LY R . Since

11

LS SS SLY K Y Y

is positive definite,

according to eigenvalue perturbations theorem, we have

1

1 1 11 0LL L L LS SS SLY R Y R Y K Y Y

, (56)

i.e., 1LLLY R is positive definite. Next, as Γ2 has at least one

negative eigenvalue and 1LLLY R is positive definite, according

to Schur’s theorem [29], Yeq must have at least one negative

eigenvalue, thus proving λ1(Yeq) < 0.

Because 1

1 1 11 0L LL L LS SS SLY R Y R Y K Y Y

and

1 0SSK Y , according to lemma on Schur complement, we

have 1

3 10

SS SL

LS LL L

Y K Y

Y Y R

. (57)

Since Γ3 is positive definite, according to lemma on Schur

complement, we have

1

1 1 1 0eq SS LS L LL SLK Y K Y Y R Y Y

, (58)

thus proving (53).

Remark 4: According to Proposition 4, the results in (53) are

properties that is inherited by the equilibrium point, i.e., (53)

always holds as long as the system admits an equilibrium.

According to (51) and (53), the equilibrium is stable if

min

1

0eq

Cb

Y (59)

However, different CPL vectors may lead to different

equilibrium. We assume all the CPLs are bounded in the set Ψ

defined as max0mP P P . Then, the following question

naturally arises:

Q3. How to design the control parameter b to keep system

stable for any P ?

The results are obtained as follows.

Corollary 2. For given K and uref that satisfy (32) and (33), the

equilibrium is locally robust stable if

min0C

b

, (60)

where Cmin = min{Ci}, 21

,max ,max ,maxL i iR diag u P , ,maxiu is the

steady voltage of the ith load when the CPL vector is Pmax , σ is

the minimal eigenvalue of matrix 1

1,maxSS SL LL L LSY Y Y R Y

.

Proof. When CPL power vector increases, the output currents

of DGs would have a corresponded increase, and thus the CPL

voltage will decrease due to the voltage drop of line resistance.

Consequently, for an arbitrary P , the corresponding



steady voltages Lu of loads yield ,maxL Lu u .Then, we obtain

1 1,maxL LR R and 1 1

,max 0LL L LL LY R Y R .Thus, we have

1 11 1

,max

1 11 1,max

0 LL L LL L

SS SL LL L LS SS SL LL L LS

Y R Y R

Y Y Y R Y Y Y Y R Y

(61)

According to eigenvalue perturbation theory, we obtain

1 0eqY (62)

Substituting (62) into (59), (60) is obtained, thus completing

the proof.

Remark 5: Corollary 2 implies that if (60) holds, the system is

locally robust stable for any P , thus Q3 is answered. In

fact, system topology just affects the eigenvalue of Yeq, and

Corollary 2 always holds subject to arbitrary topology of DC

microgrid as long as b is enough small according to (60).

Overall, the system admits a stable equilibrium when

following these two steps:

Step 1. For given maximal CPL vector Pmax and Y, selecting

the appropriate uref and K to ensure a constant steady state

according to Theorem 2 and Corollary 1; for given uref and Y,

selecting the appropriate K and determining the maximal CPL

vector Pmax to ensure a constant steady state according to

Theorem 2 and Corollary 1.

Step 2. Selecting the appropriate b to ensure the equilibrium is

stable according to Corollary 2.

Compared with [9], this paper has investigated the existence

condition of equilibrium. The sufficient existence condition of

equilibrium in this paper is less conservative compared with

[28]. Moreover, Corollary 2 provides an analytical locally

robust stability conditions as a function of the system

parameters, and it does not depend on Lyapunov equation.

VI. CASE STUDY

To verify the presented analyses, we simulate a DC

microgrid with the structure in Fig.4 using MATLAB/

Simulink. The ideal CPL is modeled as a controlled current

source. The converter inductances are L1 = L2 =…= L10 = 2mH,

C1 = C2 =…= C5 = 2mF, C6 = C7 =…= C10 = 2.5mF. The input

voltage of converter V1 = V2 =…= V10 =3000V.

1

23

4

5 6

78

9

1011

1213

14

15

16 17 18 19

20

21

2223

24

25

26

27

2829

30

1

2

5

54

4

2

5

10

5

4

2

1

2 4

5

5

2

2

5

1 2

5

2

4

1

2 4

4

2

Fig.4. A general DC microgrid with 10 DGs and CPLs. 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.

A. Stabilization Design

According to the proposed stabilization controller, the

converter duty cycles are designed according to (48). Take k1 =

k2 =… = k10 =2, then, the duty cycles take the form

6

21,2, ,10

30001 10 i

ii ref L i

ud u i u i

b

where b and uref are the parameters that should be selected to

guarantee that the system admits a stable equilibrium.

B. Voltage Reference to Guarantee the Existence of

Equilibrium

System equilibrium Lu is determined by 1L LU Y u P ,

which can be expressed as 11 ΘL L Lu F u Y g u , where ζ

= uref [1 1 1 1 1 1 1 1 1 1]T.

According to Theorem 2 and Corollary 1, provided that

either (32) or (33) holds, there must exist a unique vector

Lu D such that L Lu F u , i.e., the system admits a

constant steady state. Let

11 2 3 4 1

1 ,2 , min max , 2, Θij

q i j mf q Y

We use q* to denote the optimal vector that minimizes

1 ,

max iji j m

f q

. If uref > τ2, the system admits an equilibrium.

Then, to test the correctness of the existence condition of

euilibirium, we evaluated five cases:

Case 1: Assume the given maximal CPL vector is Pmax =

104[15 11 10 11 11 7 8 10 8 7 11 14 9 10 8 7 6 10 14 11]T.

According to (26), (32) and (33), τi and q* can be obtained as

τ1=2006.3, τ2=2095.7, τ3=2266.5, τ4 = 2365.9, q* = [0.3221

0.3664 0.3805 0.3636 0.3542 0.4415 0.4062 0.3652 0.3898

0.4306 0.3413 0.3243 0.3732 0.3455 0.4176 0.4665 0.6993

0.3475 0.3233 0.3638]T, respectively. Take uref = 2096V,

according to Theorem 1, the system would admit an

equilibrium;

Case 2: The loads are same with Case 1.Take uref =2093V;

Case 3: Assume the given maximal CPL vector is Pmax=

104[10 10 10 10 10 8 8 8 8 8 10 10 10 10 10 10 12 12 12 12]T.

Then, τi and q* can be obtained as τ1 = 2234.1, τ2 =2462.6, τ3 =

3651.4, τ4=2761.3, q*=[0.2649 0.2822 0.2879 0.2768 0.2799

0.3196 0.3210 0.2892 0.2976 0.3144 0.2769 0.2626 0.2928

0.2801 0.3099 0.3674 0.5359 0.2937 0.2717 0.2874]T. Take

uref =2463V, according to Theorem 1, the system would admit

an equilibrium;

Case 4: The loads are same with Case 3.Take uref =2459V.

The corresponding results, obtained from MATLAB, are listed

in Table II. TABLE II.

Equilibrium of the Evaluated Cases

Cases The corresponding hξ, ζ such

that h F h F The solution of equation

uL=F(uL)

Case 1

hξ = 103[1.5061 1.3240 1.2749

1.3342 1.3696 1.0988 1.1943 1.3283 1.2445 1.1266 1.4213

1.4959 1.2999 1.4041 1.1617

1.0399]T, ζ=2096×120.

uL=103[1.5210 1.3403 1.2988 1.3868 1.3922 1.1411 1.2523

1.3641 1.2824 1.1619 1.4445

1.5529 1.3248 1.4228 1.1904 1.1054 0.8085 1.4257 1.5156

1.3530]T

Case 2 Inexistence Unsolvable

Case 3

hξ = 103[2.0712 1.9438 1.9054 1.9817 1.9602 1.7165 1.7092

1.8971 1.8435 1.7448 1.9811

2.0895 1.8735 1.9585 1.7702

1.4931 1.0237 1.8680 2.0196

1.9091]T, ζ=2263×120.

uL=103[2.0782 1.9587 1.9151 1.9862 1.9666 1.7291 1.7167

1.9007 1.8482 1.7571 1.9906

2.1014 1.8873 1.9617 1.7746

1.5022 1.0412 1.8751 2.0374

1.9119]T

Case 4 Inexistence Unsolvable



The results shows that equation uL=F(uL) has a solution if

uref > τ2, and it may have no solution otherwise. In the

evaluated cases, τ2 < τ4, which shows that solvability condition

(32) and (33) are stronger than the result in [28]. Moreover,

the results shows that there is a solution when uref >2095.7 and

no solution when uref =2093. Hence, condition (32) is less

conservative.

C. Performances of the proposed stabilization method

According to Corollary 2, the equilibrium is locally robust

stable if (60) holds. Let us define b0 as

min0

Cb

Hence, the system equilibrium is robust stable if b < b0. To

verify the correctness of the existence and stability condition

of euilibirium, we evaluated three cases:

Case 5. The load is same as case 1. Take uref =2500, then σ and

b0 can be obtained as σ = −0.059, b0 = 0.034. Take b = 3×10−3.

Case 6. Assume the maximal CPL vector Pmax is same as case

3, P =6×104×120 for t < 0.05s, P = 8×105×120 for 0.05 ≤ t <

0.1s, and P = Pmax for t ≥ 0.1s.Take uref = 2463, then σ and b0

can be obtained as σ = −0.1872, b0 = 0.011. Take b =1×10−3.

Case 7: The load is same as case 6. Take uref = 2459, b =

1×10-3;

Moreover, CPLs were activated at t=0.001s in all these

cases. In Case 5, ki = 0 for t < 0.2 and ki =1 onwards, i.e., the

proposed stabilization acted for t ≥ 0.2. In Case 6 and 7, ki=1

throughout the simulation. The simulation results are depicted

in Fig.4−6.

Fig.4. Load voltages for Case 5

Fig.5. Load voltages for Case 6, uref > τ2

Fig.6. Load voltages for Case 7, uref <τ2

In Fig. 3, the system is unstable for t < 0.2 and stabilize

after activating the proposed stabilization method, verifying its

effectiveness. In Case 6, uref > τ2 and the system admits a

stable equilibrium when the loads increase to maximal values

as shown in Fig.5, which also verified that the system is robust

stable under the uncertain CPLs. In contrast, in Case 7, uref <

τ2 and the load voltages collapse as shown in Fig.5. These

results confirm the correctness of the sufficient conditions to

the existence and robust stability of equilibrium presented in

this paper.

VII. CONCLUSIONS

We investigate the existence and stability of equilibrium of

in general DC microgrids with multiple CPLs. A stabilization

method is proposed and the sufficient conditions for the

system admitting a stable equilibrium are derived. We

transform the problem of nonlinear equation solvability into

the existence of fixed-point of an increasing mapping and

obtain the sufficient condition based on Tarski fixed-point

theorem. The sufficient condition is less conservative

comparing with the existing results. We adopt the linearized

equivalent model around the equilibrium and obtained the

locally robust stability conditions by analyzing the eigenvalue

of the Jacobian matrix. These conditions provide a design

guideline to build reliable DC microgrids. Finally, the

simulation results verify the correctness of the proposed

conditions.

Appendix. Proof of Theorem 2.

Proof. According to 2) in Proposition 3, 11Y is strictly

positive, and hence 11 ΘY is also strictly positive. Consequently,

11 2 1 2 1 0mF x F x Y g x g x for every 1 2 0mx x ,

satisfying 1) of Lemma 2. Likewise, the system admits an

equilibrium if 2) of Lemma 2 is also satisfied. Let x2 =ζ and

x1=hξ, where 1 1 11 2

T

mq q q

and h is an undetermined

positive scalar. Given that 11Y is positive, the following

can be obtained:

F . (63)

Then, the quadratic equation in (14) is solvable if

h F h . (64)

Clearly, (64) can be expressed as

1, 1,2, ,ref i

i

hu a k i m

q h , (65)

and (65) is equivalent to

2 21 1 1 1

1 1

2 2

4 42 2

4 42 2

ref ref ref ref

m m m mref ref ref ref

m m

q a q q a qu u h u u

q q

q a q q a qu u h u u

q q

. (66)

Next, let

2 2

1

4 42 2

mi i i i

i i ref ref ref refi ii

q a q q a qu u u u

q q

，， .

If Ω ≠ (i.e., F and F h h ), according to



Lemma 2, there must exist a unique vector, Lh u ,

such that L Lu F u .

Clearly, Ω is non-empty if and only if

2 2

2 2

4 42 2

4 42 2

j ji iref ref ref ref

i j

j j i iref ref ref ref

j i

q a qq a qu u u u

q q

q a q q a qu u u u

q q

(67)

holds for every i, j∈{1,2,…, m}. For specific i and j, if qi=qj,

(67) is solvable as

2 4max ,ji

refi j

a qa qu

q q

(68)

If qi ≠ qj, (67) can be expressed as

2 2 22 4 4j ji i

ref ref refj i i j

a q a qa q a qu u u

q q q q

(69)

By solving (69), the solution of (67) is given by

2

2

2

4max , 2max ,

2max ,

j j ji i iref

i j j i i j

ji

j i j ji iref

j i i jj ji i

j i i j

a q a q a qa q a q a qu

q q q q q q

a qa q

q q a q a qa q a qu

q q q qa q a qa q a q

q q q q

, (70)

thus, obtaining (31) and completing the proof.

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Zhangjie Liu received the B.S. degree in Detection Guidance and Control Techniques

from the Central South University,

Changsha, China, in 2013, where he is currently working toward Ph.D. degree in

control engineering.

His research interests include Renewable Energy Systems, Distributed generation and

DC micro-grid.

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.

Wenbin Yuan received the B.S. degree in

Electrical Intelligent Building from the

Xiangtan University, Xiangtan, China, in 2015, and he is currently working toward Master’s

degree in Electrical Engineering in Central South University, Changsha, China.

His research interests include Renewable

Energy Systems, Distributed generation and AC microgrid.

Hua Han was was born in Hunan, China, in

1970. She received the M.S. and Ph.D. degrees from the School of Information

Science and Engineering, Central South

University, Changsha, China, in 1998 and 2008, respectively. She was a visiting scholar

of University of Central. Florida, Orlando, FL,

USA, from April 2011 to April 2012. She is currently an associate professor with the

School of Information Science and Engineering, Central South University, China.

Her research interests include microgrid, renewable energy power

generation system and power electronic equipment.

Jianghua Feng received his B.S. degree and

M.S. degree in Electric Machine and Control from Zhejiang University, Hangzhou, China

respectively in 1986 and 1989 and his Ph.D.

degree in Control Theory and Control Engineering from Central South University,

Changsha, China in 2008. He joined CSR

Zhuzhou Institute Co., Ltd., Zhuzhou, China in 1989.

His research interest is electrical system and

its control in rail transportation field. He is

now a professorate senior engineer and has several journal papers

published in Proceedings of China Internat, IEEE International Symposium on Industrial Electronics, International Power Electronics and

Motion Control Conference, IEEE Conference on Industrial Electronics

and Applications, IPEC, IECON, ICEMS.

Documents

Existence and Stability of Equilibrium of DC Microgrid ...pe.csu.edu.cn/lunwen/Existence and Stability of Equilibrium of DC Microgrid with...in the loss of system equilibrium (i.e.,