Energy and Power Engineering, 2011, 3, 24-28
doi:10.4236/epe.2011.31004 Published Online February 2011 (http://www.SciRP.org/journal/epe)
Some Design and Simulation of Sliding Mode Variable
Structure Control for Hopf Bifurcation in Power Systems*
Qiwei Du1, Zhonghua Su2, Sheng Li2
1Zhejiang Electric Power Corporation, Hangzhou, China
2School of Electric Power Engineering, Nanjing Institute of Technology, Nanjing, China
Received September 11, 2010; revised November 12, 2010; accepted November 15, 2010
Abstract
In power systems, the Hopf bifurcation (HB) can occur before the saddle-node bifurcation (SNB) and be-
comes one of main reasons of voltage instability and collapse, so the bifurcation control for HB has impor-
tant significance in improving power system voltage stability. In this paper, the numerical bifurcation analy-
sis software MATCONT was used to study bifurcation behavior of a single-machine dynamic-load (SMDL)
system with SVC, and the simulation analysis results show that a unstable Hopf bifurcation (UHB) point oc-
curring before SNB point and engendering potential harm to voltage stability. To delay or eliminate the UHB
phenomenon and enhance voltage stability of the SMDL system with SVC, we designed a sliding mode
variable structure controller. The switching function and control variables of the controller are clearly de-
scribed and the derivations are directly provided in detail in this paper. The MATLAB simulation results
prove that the designed controller can eliminate the UHB point effectively and ensure safe and stable opera-
tion of the system.
Keywords: Voltage Stability, Hopf Bifurcation, SMVSC, Simulation
1. Introduction
With many voltage instability and collapse accidents
occurred in the last 30 years, voltage stability problem
has been drawn extensive attention and become a hot
spot in power system research. Power systems are com-
plex nonlinear dynamic systems in essence, therefore, the
bifurcation theory of nonlinear dynamics has been
widely used for studying power system voltage stability
problem. Now, saddle-node bifurcation (SNB), limit in-
duced bifurcation (LIB), Hopf bifurcation (HB) and sin-
gularity induced bifurcation (SIB) are generally consid-
ered to be the main bifurcation types that cause voltage
instability and collapse in power systems. HB is one of
the typical dynamic bifurcations, when it occurs, the
system will produce periodic oscillations which lead to
voltage collapse finally. The past researches show that
the HB can occur before the SNB and becomes one of
main reasons that lead to voltage instability and collapse
in power systems [1-5]. Obviously, the bifurcation con-
trol aiming at the HB has important significance in im-
proving voltage stability.
Sliding mode variable structure control (SMVSC) is a
kind of nonlinear variable structure control strategy, it
can act purposefully in the dynamic process according to
the current state of the system (such as state variable
deviation and its derivative, etc.), which forces the sys-
tem moving according to the state trajectory of the ex-
pected ‘sliding mode’. There is no need to design an ac-
curacy object model and online identification system for
SMVSC, and with the advantages of fast response and
being physically implemented easily, when the system
comes into sliding mode motion, it is almost unacted on
parameters change and external disturbances. Therefore,
SMVSC has a good adaptability and robustness [6,7].
In power system bifurcation analysis, when the HB
occurs, the power system has to take a short time to os-
cillate before the voltage collapse occurs, and at the same
time, SMVSC can make the trajectory of the system
reach the designed manifold quickly, and slid the stable
equilibrium point. Based on the above analysis, we can
realize the control aiming at the HB by using SMVSC
method in power systems.
*Jiangsu Province University Natural Science Research Project o
f
China (08KJD470008)

Q. W. DU ET AL.25
2. System Model and Bifurcation Analysis
2.1. Power System Model
A classical single-machine dynamic-load (SMDL) sys-
tem model was proposed to study voltage stability prob-
lem in [8,9]. To delay saddle-node bifurcation and im-
prove voltage stability, Reference [10,11] added a static
var compensator (SVC) to the load bus in the SMDL
system, as shown in Figure 1.
This system in Figure 1 can be described by the fol-
lowing differential state equations [10]:

2
1sin
11
cos
1
ref
EV
PD
MX
EV
VkPVB
XX
BVV
T






 





 
 










(1)
Where δ is the generator power angle; ω is the gen-
erator rotor angle speed; X is the line reactance; E is the
generator potential; D is the generator damping coeffi-
cient; M is the generator moment of inertia; τ is the time
constant of PQ dynamic load; P is the system active
power demand; V is the load bus voltage; k is a scalar
representing constant power factor of the PQ dynamic
load; B is the compensation susceptance of SVC; T is the
control time constant of SVC; Vref is the reference volt-
age value of load bus. The SVC controller is modeled as
a first order pure integrator.
The research results of [10] show that the SVC control
can delay SNB point effectively, but also induces a new
bifurcation phenomenon — HB in the system.
2.2. Bi furc at ion An alys is
We can use MATCONT, a MATLAB package for nu-
merical bifurcation analysis of ordinary differential
equations (ODEs), to analyze bifurcation phenomena of
the simple system shown in Figure 1.
The system parameters appearing in (1) are:
D = 0.1 p.u., M = 0.1 s, X = 0.5 p.u., E = 1 p.u., k = 0.5,
τ = 0.001 s, T = 0.01 s, Vref = 1 p.u.
The MATCONT bifurcation analysis result for the
SMDL system with SVC is depicted in Figure 2. Ac-
cording to Figure 2, a HB point (H) occurs before SNB
point (LP) in the system.
Bifurcation parameter of the HB point (H) is:
(δ, ω, V, B, P) = (0.785469, 0, 1, 1.293043, 1.414314).
First Lyapunov coefficient is 6.406133e + 002 (> 0),
Figure 1. The SMDL system with SVC.
Figure 2. P-δ curve (based on MATCONT).
so the HB point is a subcritical Hopf bifurcation (namely
unstable Hopf bifurcation, UHB), it is harmful to the
system voltage stability.
Figure 3 shows the time domain simulation result for
a change in P from 1.414314 p.u. to 1.42 p.u. (a small
disturbance) at the UHB point. From figure 3 we can
know when the UHB occurs, the load bus voltage will
lose its stability with the increasing oscillation phe-
nomenon and lead to collapse finally, which can be a
mechanism to explain voltage oscillation and instability.
Hence, SNB point couldn’t be the reference standard of
voltage instability, and UHB point can be seen the start-
ing point of voltage instability in power systems.
3. Design of SMVSC
It’s undoubtedly of great significance to delay or elimi-
nate UHB phenomena in power systems which can im-
prove load limits and enhance voltage stability. Refer-
ences [12,13] designed some new types of SVC control-
lers based on SMVSC so that the load bus voltage turns
more stable. In this section, we will use the SMVSC
method to design a SVC controller in the SMDL system,
with the purpose of regulating reactive power output of
SVC and eliminating or delaying UHB phenomenon.

Q. W. DU ET AL. 26
Figure 3. t-V curve (based on MATCONT).
The differential equations describing the system with
SMVSC are:

2
1sin
11
cos
1
ref
EV
PD
MX
EV
VkPVB
XX
BVVu
T






 





 
 










(2)
Where u is the control input to the SVC.
Set




1234
,,, ,,,
x
xx xxVB  

x
, take
yh

 as the output. Change the nonlinear sys-
tem to the linear system directly:


 
 
0
11
10
2
21
3
32
4
f
fff
fff
fff
zLhhx
zLhLLhf
zLhLLhf
zLhLLhf








 


 

 




 

(3)
Namely
1
2
3
4
22
22
22
1sin
1cos sin
sin sin
sinsin cos
z
z
EV
zP D
MX
EVDP DEV
zMX MMX
DkPE EVB
MX X
EVE V
XX















 
 



 



 



 




 (4)
Suppose




4
zxx  
u, and make a nonlinear
transformation as


vx  





2
1
ux 

. After deriva-
tion calculus to (4), the original system can be translated
into the following standard system:
11
2
33
44
0100 0
0010 0
0001 0
0000 1
zz
zz
v
zz
zz
 


 


 



 



 



  




(5)
In the new coordinates, suppose the switching function
of the new system:


11223 34
SZCzCzCzz

 (6)
where


1234
,,,
Z
zzzz.
When the system goes into the sliding mode, S = 0, i.e.,


411223
zCzCzCz 3 . Therefore, the sliding mode
equations can be given as follows:
11
22
3123
010
001
zz
zz
zCCCz
3

 

 


 

 


 



(7)
The eigenvalue equation of (7) is:
32
321
0CCC 

 (8)
Supply the poles


123
,,kkk, which enable the sliding
mode Equations (7) to be stable in advance, then


123
,,CC C are derived:

1123
212231
3123
Ckkk
Ckkkkkk
Ckkk





3
lS
(9)
Substitute the nonlinear transfer equations Z = T(x)
(namely (4)) for the switching function S(Z), then the
switching function of the system in the original coordi-
nates can be derived.
Let equal the exponential reaching law:
S

sgn SS 


 (10)
Where ε > 0 and l > 0, then
12 23 34
sgn SlS CzCzCzv


   (11)
So


sgnvCZ SlS   ,


123
0, ,,CCCC, and
now u can be derived as follows:




 
   
  
1
1
1
sgn
sgn
uxvx
x
CZS lSx
xCTx STx
lS Txx







 







 




(12)
Now we have finished designing the switching function
and the law of the SMVSC. In the next section, the simu-

Q. W. DU ET AL.
27
lation analyses about control effect of the designed con-
troller will be introduced.
SMVSC is very complex and it keeps regulating the load
bus voltage. However, the swing amplitudes are very
small, and the voltage can remain in the qualified area.
4. Time Domain Simulation 5. Conclusions
To observe and study control effect of the designed
variable structure controller, a MATLAB-procedure was
written and executed for the time domain simulation
analyses. The flow chart of extracting the control vari-
able u is depicted in Figure 4.
In this paper, a simple but meaningful and representative
power system model, the single-machine dynamic-load
system with SVC, was used to analyze Hopf bifurcation
and voltage stability. We also used the SMVSC method
to design a controller to stabilize the load bus voltage
and control UHB phenomenon of the simple system by
controlling the reactive power output of the SVC. Simu-
According to the previous analyses based on MAT-
CONT, we can know that in the SMDL system with SVC,
the UHB occurs when bifurcation parameter is: (δ, ω, V,
B, P) = (0.785469, 0, 1, 1.293043, 1.414314).
Make the UHB point be the initial state of the SMDL
system with SVC, when the system is subjected to a
small disturbance (P is from 1.414314p.u. to 1.42p.u.),
the time domain simulation results without SMVSC and
with SMVSC are depicted in Figure 5 and Figure 6 re-
spectively. According to Figures 5-6, conclusions below
can be got:
Linearization for
system model
Determining the
switching function
Choosing the
exponential
reaching law
Determining the
sliding mode
equations
Extracting the
variable v
Extracting the
variable u
1) When the system without SMVSC operates at the
HB point, after the small disturbance occurs, the load bus
voltage will have a sudden drop at about 0.57s after the
increasing oscillation, and at the same time, the generator
power angle also loses its stability.
2) When the system with SMVSC operates at the HB
point, after the small disturbance occurs, the load bus
voltage won’t have an oscillation and keeps stable, and at
the same time, the generator power angle also keeps its
stability.
3) The load bus voltage after variable structure control
has a little erratic swing at all time, this is because the Figure 4. Flow chart of extracting the variable u.
Figure 5. Time domain simulation results without SMVSC.
Figure 6. Time domain simulation results with SMVSC.

Q. W. DU ET AL. 28
lation results show that the UHB of the original system
can be eliminated effectively and the voltage stability
margin is improved.
The SMVSC method can eliminate the UHB, but also
bring some new problems. For example, we can derive
the voltage stability index and the voltage stability mar-
gin of the original system by the methods that have been
found in [14,15], but there is no method for the system
with SMVSC at present. In addition, the structure of the
SMVSC is very complex, and it’s very difficult to
change the controller into a practical product now. These
problems will take a long time to be studied. Therefore,
the SMVSC method in many areas including power sys-
tems is still in the stage of theoretical research. There is
an urgent need to solve above problems depending on a
variety of advanced automatic control technologies.
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