Stability analysis of wide area power system under the influence of interval time-varying delay

Abstract

In view of the problem that time delay always existing in wide area power system can cause severe effects on the operation performance of the whole system, this paper studies the stability of the wide area power system with interval time-varying delays. Firstly, the model of wide area power system with interval time-varying delay is established, based on that, a new augmented vector and new Lyapunov-Krasoskii functional (LKF) are constructed. Then, the delay-partitioning approach, Wirtinger integral inequality, free-matrix-based inequality and convex combination approach are used to estimate the derivative of the functional, and as a result, a less conservative stability criterion for the delayed power system is obtained. Finally, numerical simulations of the typical second-order system, the single machine system and two-area four-generator power system are given to illustrate that the proposed method in this paper expands the stability margin of the system effectively.

Full text

SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL
2018, VOL. 6, NO. 3, 1–9
https://doi.org/10.1080/21642583.2018.1491905
Stability analysis of wide area power system under the influence of interval
time-varying delay
Chen-chen Wang, Wei Qian, Bing-feng Li and Yunji Zhao
Department of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo, People’s Republic of China
ABSTRACT
In view of the problem that time delay always existing in wide area power system can cause severe
effects on the operation performance of the whole system, this paper studies the stability of the
wide area power system with interval time-varying delays. Firstly, the model of wide area power sys-
tem with interval time-varying delay is established, based on that, a new augmented vector and
new Lyapunov-Krasoskii functional (LKF) are constructed. Then, the delay-partitioning approach,
Wirtinger integral inequality, free-matrix-based inequality and convex combination approach are
used to estimate the derivative of the functional, and as a result, a less conservative stability criterion
for the delayed power system is obtained. Finally, numerical simulations of the typical second-order
system, the single machine system and two-area four-generator power system are given to illustrate
that the proposed method in this paper expands the stability margin of the system effectively.
ARTICLE HISTORY
Received 4 June 2018
Accepted 19 June 2018
KEYWORDS
Power system; interval
time-varying delay;
Lyapunov-Krasovskii
functional (LKF);
delay-partitioning; stability
margin
1. Introduction
With the expansion of the scale of modern power sys-
tem and the interconnection of power grids, the dynamic
process of the power system becomes more and more
complex. The traditional local control method was unable
to meet the requirements of security and stability in the
current wide area power system. In recent years, the wide
area measurement system (WAMS) based on phasor mea-
surement unit (PMU) has rapidly developed and widely
used in power system, which promoted the develop-
ment of the wide area control in power system (Hadidi &
Jeyasurya, 2013; Manousakis, Korres, & Georgilakis, 2012;
Yan, Govindarasu, Liu, Ming, & Vaidya, 2015). In the wide
area environment, time delay exists in the process of sig-
nal transmission and processing, especially in long dis-
tance transmission. It has been shown that even small
time delay can cause serious negative effects to the stable
operation of the power system (Zhang, Zhan, Wei, Shi, &
Xie, 2016). So, it is of great practical significance to study
the stability of power system under the influence of time
delay (Hailati & Wang, 2014;Yang&Sun,2014).
There are two main methods for analyzing the stabil-
ity of power systems with time delay: frequency domain
method and time domain method. The frequency domain
method is mainly based on the transformation of the
characteristic equation and the distribution of eigenval-
ues to determine the stability of the system (Hua, Jian,
CONTACT Wei Qian qwei@hpu.edu.cn
& Liu, 2013;Li,2015), the necessary and sufficient condi-
tions for the stability of the system can be obtained by
this method, but the calculation process is so complicated
that it is difficult to be applied when the operation state
of power system jumps or contains time-varying param-
eters. Compared with the frequency domain method, the
time domain method has obvious advantages (Ma, Li, Li,
Zhu, & Wang, 2015; Liu, Ding, Wang, & Zhou, 2011), and
it is the main method for the stability analysis of time
delay power systems. The LKF method based on the Lya-
punov stability theory is used most widely in time domain
method. This method gives the sufficient conditions for
the stability of the system, which leads to a certain con-
servatism. Therefore, how to reduce the conservatism to
expand the stable operation area of the system becomes
a hot issue in recent years, and different research meth-
ods have been proposed by many researchers. In aspect
of LKF construction, by constructing one-integral LKF
and double integral LKF (Chen & Cai, 2009;Sunetal.,
2015), augmented LKF (Li, Sun, & Wei, 2017), the sta-
bility and controller design of power systems with con-
stant time delay and time-varying delay are studied. In
aspect of estmating functional derivatives, many new
methods are proposed such as free matrices method
(Jia, An, & Yu, 2010), the Jesen integral inequality (Dong,
Jia, & Jiang, 2015), the free-weighting matrix approach
(Huang, Guo, & Sun, 2014), the generalized eigenvalue
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2C.-C. WANG ET AL.
method (Ma, Li, Gao, & Wang, 2014), the convex combina-
tion approach (Qian & Gao, 2015), the Wirtinger integral
inequality method (Qian, Jiang, & Che, 2016), to study the
stability analysis and control of the wide area time delay
power system. Although the above literatures reduce
the conservatism of stability criterion for the time delay
power system, they still have some shortcomings, such as
the simple LKF, the limitations of the analytic method in
reducing conservatism and so on, all of which cause the
conservatism of the stability criteria.
Motivated by the discussion mentioned above, the
main purpose of this paper is to study the stability of
the wide area power system with interval time-varying
delays. By establishing the model of wide area power
system with interval time-varing delay, construting new
augmented vector and a new LKF with triple integral
terms, dividing the delay interval into two parts, using
wirtinger integral inequality, free-matrix-based inequal-
ity and convex combination approach to estimate the
derivative of the functional, the less conservative stabil-
ity conditions are proposed. The numerical examples are
also given to show that the proposed method expands
the stable operating area of the system effectively.
2. Model of power system with time delay
In this section, based on the traditional power system
model, by introducing time-varying delay to describe,
the model of of power system with time-varying delay is
established.
To the power system, the dynamic model of generator
is described as:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
dδ
dt=ω−ω0
Tjdω
dt=(pm−pe)ω0−D(ω −ω0)
TddEq
dt=Ef−Eq−(Xd−Xd)Id
Ud=XqIq,Uq=Eq−XdId
(1)
where
Id=Eq−Vcos δ
Xd+Xe
=EQ−Vcos δ
Xd+Xe
,pe=EQV
Xq+Xe
sin δ,
Vg=XqVsin δ
Xq+Xe2
+XeEq+XdVcos δ
Xd+Xe2
The meanings of parameters in the differential equations
are given in (Li, 2015). In order to ensure the reliability
of the power system, AVR excitation control method is
used. Considering the time delay existing in the system,
the dynamic equations of the excitation system can be
expressed as follows:
TdE
dt=−K[V(t−τ(t)) −V]−(E−E)(2)
According to (1) and (2), the model of time delay power
system can be expressed as follows:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
dδ
dt=ω−ω0
dω
dt=(pm−pe)ω0
Tj−D
Tj(ω −ω0)
dEq
dt=Ef−Eq
Td−(Xd−Xd)Id
Td
dEf
dt=−
Ka
Ta[Vg(t−τ(t)) −Vg0]−1
Ta(Ef−Ef0)
(3)
Linearizing the equation (3) at the equilibrium point
can obtain:
˙
x(t)=Ax(t)+A1x(t−h(t)) t≥0
x(t)=φ(t)t∈[−h,0] (4)
where x(t)∈Rnis the state vector of power system,
the initial condition φ(t)is a continuously differen-
tiable vector-valued function in [−h0], A,A1∈Rn×nare
known constant matries, h(t)is the time-varying delay
and satisfying 0 ≤h(t)≤h,μ1≤˙
h(t)≤μ2<1, where
h,μ1,μ2are constants.
Considering the disturbance in the system, the system
(4) should be expressed as:
⎧
⎪
⎨
⎪
⎩
˙
x(t)=(A+A)x(t)
+(A1+A1)x(t−h(t))
x(t)=φ(t)t∈[−h,0]
(5)
where A,A1are disturbance terms satisfying [A,
A1]=HF0[Ea,Eb], and H,Ea,Ebare known constant
matrices, F0is free matrix satisfying FT
0F0≤I.
In order to obtained the main result, the following
lemmas are needed.
Lemma 1: (Seuret & Gouaisbaut, 2013) For any constant
symmetric matrices M =MT>0, real scalars a,bwhichsat-
isfying a >b, and vector-valued function ϕ:[ba]→Rn,
then the following inequality holds
a
b
˙ϕ(s)TM˙ϕ(s)ds ≥1
a−ba
b
˙ϕ(s)dsT
Ma
b
˙ϕ(s)ds
+3
a−bTM,
where =ϕ(a)+ϕ(b)−2
a−ba
bϕ(s)ds.
Lemma 2: (Park, Kwon, Park, Lee, & Cha, 2015)Foragiven
symmetric matrix M =MT>0, scalars a,b which satisfying

SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 3
a>b, and vector-valued function ϕ:[ba]→Rn, then
the following inequality holds
(a−b)2
2a
ba
s
ϕT(u)Mϕ(u)duds
≥a
ba
s
ϕ(u)dudsT
Ma
ba
s
ϕT(u)duds
+2T
dMd,
where d=−
a
ba
sϕ(u)duds +3
a−ba
ba
sa
uϕ(v)
dvduds.
Lemma 3: (Park, Ko, & Jeong, 2011). For given positive inte-
gers m,n, variable α∈(0, 1), for given matrices Z ∈Rn×n>
0, K1∈Rn×m,K2∈Rm×n, the function
ϒ=1
αζTKT
1ZK1ζ+1
1−αζTKT
2ZK2ζ.
if there exists a matirx Y ∈Rn×nandsatisfying ZY
∗Z>0,
then the following inequality holds
min
α∈(0,1)ϒ≥K1ζ
K2ζTZY
∗ZK1ζ
K2ζ.
Lemma 4: (Zeng, He, & Wu, 2015). Let x(s)be a differen-
tiable function {x(s)|s∈[a,b]}, for symmetric matrices F ∈
Rn×n,and B,L∈R3n×3n, any matrices G ∈R3n×3nand C,N∈
R3n×nsatisfying BGN
∗LC
∗∗F≥0, then the following inequality
holds:
−b
a
˙
xT(s)F˙
x(s)ds
≤θT[(b−a)B+b−a
3L+He(N1+C2)]θ
where 1=[I,−I,0],2=[−I, −I,2 I],θ=xT(b),xT(a).
3. Stability analysis of time varying delay power
system
In this section, based on the established model of time
varying delay power system, by constructing new aug-
mented terms and new Lyapunov-Krasoskii functional,
applying less conservative methods to dealing with the
derivatives of the functional, the developed stability cri-
teria of the wide area power system with time varying
delays is obtained.
Firstly, let:
˜
xT(t)=xT(t),t
t−h(t)
xT(s)ds,t−h(t)
t−h
xT(s)ds,
×1
h(t)t
t−h(t)t
s
xT(u)duds,
×1
h−h(t)t−h(t)
t−ht−h(t)
s
xT(u)duds
ηT
1(t)=xT(t),1
h(t)t
t−h(t)
xT(s)ds
ηT
2(t)=xT(t),1
h−h(t)t−h(t)
t−h
xT(s)ds
The following LKF is constructed for system (4):
V(t)=
5

i=1
Vi(t)
where
V1(t)=˜
xT(t)P˜
x(t)
V2(t)=h(t)ηT
1(t)Wη1(t)+(h−h(t))h(t)ηT
2(t)Mη2(t)
V3(t)=t
t−h(t)
xT(s)Qx(s)ds +t−h(t)
t−h
xT(s)Rx(s)ds
V4(t)=h0
−ht
t+s
˙
xT(u)Z˙
x(u)duds
V5(t)=h2
2t
t−ht
st
u
˙
xT(v)F˙
x(v)dvduds
and P=PT>0, W=WT>0, M=MT>0, Q=QT>0,
R=RT>0, Z=ZT>0, F=FT>0.
Remark 1: Different from the existing references, in this
paper, a new augmented term is constructed as
˜
xT(t)=xT(t),t
t−h(t)
xT(s)ds,t−h(t)
t−h
xT(s)ds,
×1
h(t)t
t−h(t)t
s
xT(u)duds,
×1
h−h(t)t−h(t)
t−ht−h(t)
s
xT(u)duds
and the proposed LKF contains triple integral term V5(t),
so, more information of the time delay is employed, which
play a key role in the further reduction of conservation.

4C.-C. WANG ET AL.
Taking the time derivatives of V(t)along the trajectory
of system (4) yield:
˙
V1(t)=2˜
xT(t)P˙
˜
x(t)(11)
˙
V2(t)=˙
h(t)ηT
1(t)Wη1(t)+2h(t)ηT
1(t)W˙η1(t)
−˙
h(t)ηT
2(t)Mη2(t)+2(h−h(t))ηT
2(t)M˙η2(t)
(12)
˙
V3(t)=xT(t)Qx(t)−(1−˙
h(t))xT(t−h(t))Qx(t−h(t))
+(1−˙
h(t))xT(t−h(t))Rx(t−h(t))
−xT(t−h)Rx(t−h)(13)
˙
V4(t)=h2˙
xT(t)Z˙
x(t)−ht
t−h
˙
xT(s)Z˙
x(s)ds (14)
Deviding the integral interval −ht
t−h˙
xT(s)Z˙
x(s)ds in ˙
V4,
and using lemma 1 and lemma 3, the following can be
obtained:
−ht
t−h
˙
xT(s)Z˙
x(s)ds
=−ht
t−h(t)
˙
xT(s)Z˙
x(s)ds −ht−h(t)
t−h
˙
xT(s)Z˙
x(s)ds
≤− h
h(t)ξTT
1Z1ξ−3h
h(t)ξTT
2Z2ξ
−h
h−h(t)ξTT
3Z3ξ−3h
h−h(t)ξTT
4Z4ξ
=−1
αξT1
2T
˜
Z1
2ξ−1
1−αξT3
4T
˜
Z3
4ξ
(15)
where ˜
Z=diag(Z,3Z),α=h(t)/h,1=[1−100000
],
2=[110−2000
],3=[01−10000
],4=
[0110−200
].
Remark 2: The integral interval [t−h,t]ofthe−ht
t−h˙
xT
(s)Z˙
x(s)ds in ˙
V4is decomposed into [t−h,t−h(t)]and
[t−h(t),t], compared with the existing works (Ma et al.,
2014; Qian et al., 2016;Sunetal.,2015), more infor-
mation of the time delay is employed, which efficiently
reduces the conservatism of the proposed approach.
Then, Wirtinger integral inequality is applied to tackling
with the integral terms on two subintervals, compared
with the methods in (Park, Kwon, Park, & Lee, 2011;Qian&
Gao, 2015; Seuret & Gouaisbaut, 2013) which use Jensen
integral inequalities, this method is beneficial to expand
the stable operating area of the system.
˙
V5(t)=h4
4˙
xT(t)F˙
x(t)−h2
2t
t−ht
s
˙
xT(u)F˙
x(u)duds
(16)
Deviding the integral into two parts in ˙
V5,wecanget:
−h2
2t
t−ht
s
˙
xT(u)F˙
x(u)duds
=−
h2
2t
t−h(t)t
s
˙
xT(u)F˙
x(u)duds
+t−h(t)
t−ht−h(t)
s
˙
xT(u)F˙
x(u)duds
+(h−h(t)) t
t−h(t)
˙
xT(s)F˙
x(s)ds(17)
Using the lemma 2 to deal with t
t−h(t)t
s˙
xT(u)F˙
x(u)duds
and t−h(t)
t−ht−h(t)
s˙
xT(u)F˙
x(u)duds, we can obtain:
t
t−h(t)t
s
˙
xT(u)F˙
x(u)duds
≥2
h2(t)h(t)x(t)−t
t−h(t)
x(s)dsT
Fh(t)x(t)
−t
t−h(t)
x(s)ds+4
h2(t)1
2h(t)x(t)+t
t−h(t)
x(s)ds
−3
h(t)t
t−h(t)t
s
x(u)dudsT
F1
2h(t)x(t)
+t
t−h(t)
x(s)ds −3
h(t)t
t−h(t)t
s
x(u)duds(18)
t−h(t)
t−ht−h(t)
s
˙
xT(u)F˙
x(u)duds
≥2
(h−h(t))2(h−h(t))x(t−h(t)) −t−h(t)
t−h
x(s)ds
T
×F(h−h(t))x(t−h(t)) −t−h(t)
t−h
x(s)ds
+4
(h−h(t))2h−h(t)
2x(t−h(t)) +t−h(t)
t−h
x(s)ds
−3
h−h(t)t−h(t)
t−ht−h(t)
s
x(u)dudsT
×Fh−h(t)
2x(t−h(t))
+t−h(t)
t−h
x(s)ds −3
h−h(t)t−h(t)
t−ht−h(t)
s
x(u)duds
(19)

SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 5
Using the lemma 4 to deal with t
t−h(t)˙
xT(s)F˙
x(s)ds can be
obtained:
t
t−h(t)
˙
xT(s)F˙
x(s)ds ≤θTh(t)B+L
3+He NI−I0
+C−I−I2Iθ(20)
where θ=xT(t)xT(t−h(t)) 1
h(t)t
t−h(t)xT(s)dsT.
Remark 3: Similar to the method dealing with ˙
V4,the
delay interval of the functional ˙
V5is also decomposed into
two subintervals. To the double integral terms, double
Wirtinger integral inequality and Free-matrix-based inte-
gral equality are used to reduce the estimating error, so
that the obtained result is much less conservative than
those in (Chaibi & Tissir, 2012; Seuret & Gouaisbaut, 2013;
Zeng et al., 2015).
Combining (11)–(20), it can be obtained:
˙
V(t)=
5

i=1
˙
Vi(t)=ξTξ (21)
where
=[Eij](i,j=1, 2, ...,7)
ξT=xT(t),xT(t−h(t)),xT(t−h),1
h(t)t
t−h(t)
xT(s)ds,
×1
h−h(t)t−h(t)
t−h
xT(s)ds 1
h2(t)t
t−h
xT(s)ds,
×1
(h−h(t))2t
t−ht
s
xT(u)duds
By applying Lyapunov Stability Theorem, when ˙
V=
ξTξ < 0, system (4) is asymptotically stable.
Based on the above proof, the main result for the
asymptotic stability of the system (4) is given as follows:
Theorem 1: For the given scalar h,μ1,μ2,system(4)is
asymptotically stable if there exist definite symmertric matri-
ces P ∈R5n×5n,W∈R2n×2n,M∈R2n×2n,Q∈Rn×n,R∈
Rn×n,Z∈Rn×n,F∈Rn×n,symmertric matrices B ∈R3n×3n,
L∈R3n×3n,and matrices G ∈R3n×n,N∈R3n×n,C ∈R3n×n
satisfying that:
⎡
⎣
BGN
∗LC
∗∗F⎤
⎦≥0, =[Eij]<0, (i,j=1, 2, ...,7)(22)
where
E11 =P11A+ATP11 +P12 +PT
12 +P14 +PT
14 +˙
h(t)W11
+h(t)W11A+h(t)ATW11 +W12 +WT
12 −˙
h(t)M11
+(h−h(t))M11A+(h−h(t))ATM11 +Q
+h2ATZA +h2
4ATFA −4Z−3
2h2(h−h(t))
×h(t)B11 +1
3L11+N11 +NT
11 −C11 −CT
11

E12 =P11A1−(1−˙
h(t))P12 +(1−˙
h(t))P13
+(1−˙
h(t))P15 +h(t)W11A1−(1−˙
h(t))W12
+(h−h(t))M11A1+(1−˙
h(t))M12 +h2ATZA1
+h4
4ATFA1−2Z−Y11 −Y21 −Y12 −Y22
+h2
2(h−h(t)) h(t)B12 +1
3L12
−N11 −C11 +NT
21 −CT
21
E13 =−P13 −M12 +Y11 +Y21 −Y12 −Y22
E14 =(˙
h(t)−1)P14 +h(t)P21A+h(t)P22 +h(t)P24
+h(t)W21A+W22 +6Z+h2
2(h−h(t))
×h(t)B13 +1
3L13−2C11 +NT
31 −CT
31
E15 =−P15 +(h−h(t))(P31A+P32 +P34 +M21 A)
+2Y12 +2Y22
E16 =−
˙
h(t)P14 +h(t)(P41A+P42 +P44 )+3h2F,
E17 =˙
h(t)P15 +(h−h(t))(P51A+P52 +P54 )
E22 =(˙
h(t)−1)Q+(1−˙
h(t))R+h2AT
1ZA1
+h4
4AT
1FA1−8Z+2Y11 +2Y12 −2Y21
−2Y22 −3
2h2F+h2
2(h−h(t)) h(t)B22 +1
3L22
−N21 −NT
21 −C21 −CT
21
E23 =−Y11 +Y21 +Y12 −Y22 −2Z
E24 =h(t)P21A1−h(t)(1−˙
h(t))(P22 +P23 +P25)
+h(t)W21A1−(1−˙
h(t))W22 +6Z+2Y21

6C.-C. WANG ET AL.
+2Y22 +h2
2(h−h(t)) h(t)B23 +1
3L23
+2C21 −NT
31 −CT
31
E25 =(h−h(t))(P31A1+M21 A1)
−(h−h(t))(1−˙
h(t))(P32 +P33 +P35)
+(1−˙
h(t))M22 −2Y12 +2Y22 +6Z
E26 =h(t)P41A1−h(t)(1−˙
h(t))(P42 +P43 +P45)
E27 =(h−h(t))P51A1+(h−h(t))((1−˙
h(t)))
×(P52 +P53 +P55)+3h2F
E33 =−R−4Z,E34 =−h(t)P23 −2Y21 +2Y22,
E35 =(h(t)−h)P33 −M22 +6Z
E36 =−h(t)P43,E37 =(h(t)−h)P53
E44 =−h(t)(1−˙
h(t))P24 −˙
h(t)W22 −12Z−3h2F
+h2
2(h−h(t)) h(t)B33 +1
3L33+2C31 +2CT
31
E45 =−h(t)P25 −(h−h(t))((1−˙
h(t)))P34 −4Y22,
E46 =−h(t)˙
h(t)P24 −h(t)(1−˙
h(t)P44 +6h2F
E47 =h(t)˙
h(t)P25 −(h−h(t))((1−˙
h(t)))P54,
E55 =(h(t)−h)P35 −12Z−3h2F
E56 =(h(t)−h)˙
h(t)P34 −h(t)P45,
E57 =(h−h(t))˙
h(t)P35 −(h−h(t))P55 +6h2F
E66 =−h(t)˙
h(t)P44 −18h2F,
E67 =h(t)˙
h(t)P45 −(h−h(t))˙
h(t)P54,
E77 =(h−h(t))˙
h(t)P55 −18h2F
4. Numerical simulations
In this section, four typical numerical examples are given
to show the less conservatism and the effectiveness of the
proposed method in this paper.
Example 1: Consider system (4) with time-varying delay
and the parameters as follows:
A=−20
0−0.9,A1=−10
−1−1
The purpose of this example is to compare the con-
Tab le 1 . Maximal time delay for different μ.
μ0 0.1 0.5 0.8
[18] 4.606 3.703 2.337 1.524
[24] 4.664 3.768 2.529 2.209
[25] 5.120 4.081 2.528 2.152
Theorem 1 6.0594 4.7261 2.6317 2.2539
Figure 1. Stability margin of typical second-order system with
time delay.
servatism of the stability conditions by applying differ-
ent existing methods. When h1=0, μ=0.5, the upper
bound of time delay given in (Chen & Cai, 2009) is 2.42,
using the method of this paper, the upper bound of time
delay is 2.63. When μ=0, 0.1, 0.5, 0.8, by Theorem 1 of
this paper, the obtained upper delay bounds are 6.0594,
4.7261, 2.6317, 2.2539 respectively.
Table 1lists the results of the maximum allowable
delay bounds when μ(μ =−μ1=μ2)takes different
values. From Table 1, it can be clearly seen that the results
obtained by Theorem 1 are less conservative than the
methods presented in (Ariba & Gouaisbaut, 2009;Park,
Kwon, et al., 2011; Qian & Gao, 2015).
Figure 1gives stability margin of typical second-order
system with time delay by different methods, it can be
seen that the larger stability margin of typical of the sys-
tem is obtained in this paper compared with the results in
(Ariba & Gouaisbaut, 2009; Park, Kwon, et al., 2011;Qian&
Gao, 2015).
Example 2: Consider system (4) with time-varying delay
and the parameters as follows:
A=01
−1−2,A1=00
−11

For different μ, Table 2lists the results of the maximum
allowable delay bounds obtained by Theorem 1 in this
paper and results in (Ariba & Gouaisbaut, 2009;Park,
Kwon, et al., 2011; Seuret & Gouaisbaut, 2013). Figure 2

SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 7
Tab le 2 . Maximal time delay for different μ
μ0.1 0.2 0.5 0.8
[24] 5.901 3.839 2.003 1.404
[25] 5.823 3.824 2.008 1.357
[20] 6.059 3.672 1.411 1.275
Theorem 1 6.3715 4.1502 2.1344 1.6251
Figure 2. Stability margin of typical second-order system with
time delay.
Figure 3. Single-machine infinite-bus system.
gives stability margin of typical second-order system with
time-varying delay. It is clear that the proposed method in
this paper has less conservatism and larger stability mar-
gin than those in (Ariba & Gouaisbaut, 2009; Park, Kwon,
et al., 2011; Seuret & Gouaisbaut, 2013).
Example 3: The single-machine infinite-bus system is
choosed to verify the effectiveness of Theorem 1, and the
parameters of the system resource from (Qian et al., 2016)
(Figure 3).
where
A=⎡
⎢
⎢
⎣
0 376.9911 0 0
−0.0963 −0.7000 −0.0801 0
−0.0480 0 −0.1667 0.1000
000−0.1000
⎤
⎥
⎥
⎦
,
A1=⎡
⎢
⎢
⎣
0000
0000
0000
38.0187 0 −95.2560 0
⎤
⎥
⎥
⎦
Table 3lists the maximal allowable time delays of the
single-machine infinite-bus system by using different
Tab le 3 . Maximal time delay for different methods
Method [14] [19] [11] [12] Theorem 1
h/ms 65.4 65.4 65.29 61.3 71. 90
methods when there is no disturbance in the system.
The upper time-delay bound obtained by Theorem 1 is
71.9 ms, which is lager than the results in (Chen & Cai,
2009; Jia et al., 2010; Qian et al., 2016;Sunetal.,2015).
It is clear that the proposed method in this paper has
less conservatism and larger stability margin than those
in (Chen & Cai, 2009; Jia et al., 2010; Qian et al., 2016;Sun
et al., 2015).
Further, if there is random disturbance in the system,
the excitation amplification factor should be:
K
A=KA+r
where, KA: Excitation amplification factor setting value;
KA: Excitation amplification factor with disturbance; r:A
scalar that reflects the disturbance to KA.
Let the matrix H,Ea,Ebas follows:
H=⎡
⎢
⎢
⎣
0000
0000
0000
000r
⎤
⎥
⎥
⎦
,Ea=0, Eb=⎡
⎢
⎢
⎣
0000
0000
0000
1010
⎤
⎥
⎥
⎦
Let r∈[0, 10], Table 4lists the maximal allowable
time delays obtained by different methods when r=
0.5, 1, 1.5, 2 ..., 10 (takeing 0.5 as an interval), and Figure
4gives stability margin of the single-machine infinite-bus
system in this paper and (Jia et al., 2010; Qian et al., 2016;
Sun et al., 2015).
It can be seen, with the increase of disturbance term r,
the stability margin of the system with time-varying delay
becomes smaller. It is clear that our results has less conser-
vatism and larger stability margin than those in (Jia et al.,
2010; Qian et al., 2016;Sunetal.,2015).
Example 4: In this example, the two-area-four-machine
power system shown in Figure 5is used to verify the effec-
tiveness of the main result in this paper. The detailed
parameters of the two-area-four-machine power system
are given in (Chen & Cai, 2009). Where
A=⎡
⎢
⎢
⎢
⎢
⎣
0 0 0 376.9 0 0
0 0 0 0 376.9 0
0 0 0 0 0 376.9
−0.073 0.065 0.004 −0.730 0.272 0.076
0.058 −0.087 0.009 1.160 −0.343 −0.134
0.008 0.011 −0.082 −0.020 0.047 −0.554
⎤
⎥
⎥
⎥
⎥
⎦

8C.-C. WANG ET AL.
Tab le 4 . Maximal time delay for different r.
r[14] [12] [19] Theorem 1 r[14] [12] [19] Theorem 1
0.5 0.0570 0.0587 0.0650 0.0713 5 0.0397 0.0457 0.0617 0.0642
1 0.0552 0.0576 0.0647 0.0702 6 0.0353 0.0444 0.0609 0.0631
1.5 0.0534 0.0557 0.0643 0.0694 7 0.0307 0.0417 0.0601 0.0619
2 0.0516 0.0545 0.0639 0.0685 8 0.0263 0.0392 0.0593 0.0608
3 0.0478 0.0515 0.0632 0.0669 9 0.0220 0.0363 0.0586 0.0594
4 0.0439 0.0479 0.0624 0.0654 10 0.0180 0.0343 0.0578 0.0582
Figure 4. Comparison of stability margins obtained by different
methods.
Figure 5. Two-area-four-machine power system.
Tab le 5 . Maximal time delay for different methods.
Method [5] [17] [26] [19] Theorem 1
h/s0.195 0.288 0.328 0.440 0.519
A1=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
000000
000000
000000
00 0−0.234 −0.839 0.010
0−0.0011 0.001 −0.348 −1.362 −0.138
0 0.001 0 0.049 −0.290 −0.638
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Table 5shows the upper bounds of time-delay for the
two-area-four-machine power system by different meth-
ods, it can be seen that Theorem 1 in this paper has less
conservatism than those in (Chaibi & Tissir, 2012;Maetal.,
2014;Yang&Sun,2014).
5. Conclusion
This paper studies the stability of the wide area power
system with interval time-varying delays. By establishing
the model of the wide area power system with interval
time-varying delay, a novel LKF with augmented vec-
tor is constructed. Then, the delay-partitioning approach,
wirtinger-based integral inequality, free-matrix-based
inequality and convex combination approach are used
to estimate the derivative of the functional, and as a
results, the new stability criterion with less conservatism
is obtained. Finally, the numerical examples are given to
show the effectiveness of the proposed method in this
paper.
Acknowledgement
This work is supported by the National Natural Science Founda-
tion of China under Grant (61573130); the Innovation Scientists
and Technicians Troop Construction Projects of Henan Poly-
technic University and Henan Province under Grant (T2017-1)
and (CXTD2016054).
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work is supported by the National Natural Science Foun-
dation of China [grant number 61573130]; the Innovation Sci-
entists and Technicians Troop Construction Projects of Henan
Polytechnic University and Henan Province [grant number
T2017-1 and CXTD2016054].
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