 Open Access
 Total Downloads : 1064
 Authors : M.Chandra Prasad, M.Ramu, I.E.S.Naidu
 Paper ID : IJERTV2IS3067
 Volume & Issue : Volume 02, Issue 03 (March 2013)
 Published (First Online): 08032013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Effect Of Nonlinearitiy On Dynamic Stability Of A Single Machine Infinite Bus System
M.Chandra Prasad, P.G Scholar, M.Ramu, Asst. Professor, I.E.S.Naidu,Assoc. Professor, GITAM Institute of Technology, GITAM University, Visakhapatnam, India,
Abstract Electrical power system is a huge complex system consists of the components needed for generation, transmission and large scale distribution of electrical energy. Generating station and distribution system are connected through transmission lines, which also connect one power system to another. A distribution system connects all the loads in a particular area to the transmission lines. The system must be able to meet the continually changing load demand for active and reactive power.
Power system stability may be defined as that property of a power system that enables it to remain in a state of operating equilibrium under normal operating conditions. In power system stability, synchronous machine has great importance and receives increasing attention. By using the modern voltage regulators and excitation systems with fast speed of response and high ceiling voltage can be used to improve the transient stability by increasing the synchronizing torque of the machine, their effects on the damping torque are rather small.
Generating system equipped with modern fast acting static excitation systems with larger percentage of generating capacity, excitation system had an impact on dynamic stability (small signal response) of power system.
To analyze the smallsignal stability of the single system connected to a large system through transmission lines system which is a single machine infinite bus system is extended by considering single Kaplan turbinegenerator with exciter and governor in a low head hydro power plant connected to local load and an infinite bus for the study. Nonlinearity has been considered in this work as it is inherently present in the system, and the nonlinearity considered is governor dead band. Turbinegovernor dead bands are found due to backlash in the linkage connecting the piston to the shaft. Backlash is the nonlinearity which causes governor dead band tending to produce continuous sinusoidal oscillations. The dual regulation of hydroturbine
I.INTRODUCTION
Electric power system varies in size and structural components. Synchronous machine is used to power generation. Industrial loads are threephase; single phase residential and commercial loads are distributed equally form a three phase system. Power system stability can be defined as if the oscillatory response of a power system during the transient period following a disturbance is damped and the system settles in a finite time to a new steady operating condition, we say the system is stable. If the system is not stable, it is considered unstable.The power system is a highly non linear system whose dynamic response is influenced by a wide array of devices with different response rates and characteristics. The power system instability of interconnecting system depends on rotor angle instability and voltage instability.

SINGLE MACHINE INFINITE BUS SYSTEM For smallsignal stability analysis, dynamic modeling is required for the major components of the power system. Consider a single machine system as shown in fig1, neglecting damper windings both in the d and q axis. Also the armature resistance of the machine is neglected and the excitation system represented by a single time constant of the system is shown in fig1.
Fig 1 : A Single Machine Power System Model.

MODIFIED HEFFRONPHILLIPS MODEL The standard Heffron Phillips model can be obtained by linearizing the system equations around an operating
condition. From fig1 the following equations are obtained. The development of the model is explained in [].
is incorporated through the operation of both wicket gate
E' X ' i V
( 1)
and runner blade. The results of extended single machine infinite bus system are plotted for speed deviation and successfully analysed using Matlab simulink program.
KeywordsPower system stability, SMIB, Kaplan turbine, Nonlinearity.
q d d q
xqiq = Vd ( 2)
Here q and d refers to the q and d axis respectively. The complex voltage can be expressed as
V jV (V jV )e j (i ji )(R jX )e j E 0
IV: EXTENDED SMIB
q d q d q d e e b
V jV
(i ji )(R jX ) E e j
(3)
Single Machine Infinite Bus System is extended to
q d q d e e b
Separating Real and Imaginary parts of eq(3) can be expressed as
dual regulation of lowhead power plant with wicket gate opening and runner blade position for controlling the water pressure. The generation of
Vq Reiq Xeid Eb cos
Vd Reid Xeiq Eb sin
Substituting(4) and (5) in (1) and (2) gives
(4)
(5)
hydroelectric power is accomplished by means of hydraulic turbines that are directly connected to synchronous generators. Kaplan turbine shown in
( X ' X )
R id
E cos E'
figure 3 was an evaluation of the Francis turbine.
d e e
b q
(6)
Re
( Xq Xe ) iq
Eb sin
The system mechanical equations, electrical equations and eqn.(6) are linearized as in [] to obtain the following modified Kconstants.
K Eb Eqo coso ( Xq Xd ') E i
sin
1 ( X
X ) ( X X ')
b qo o
e q e d
K ( Xe Xq ) i ;
e d
e d
2 ( X X ' ) qo
( X X ' )
K e d ;
3 ( X X )
e d
( X X ' )
K4 d d Eb sin o ;
e d
e d
( X X ' )
Fig 3: Kaplan Propeller Turbine
This design has the advantage of fairly high efficiency over
X V E cos X 'V E sin
K q do b o d qo b 0 ;
a wide range of head and wicket gate settings. Adjustments
5 ( X

X )V
( X X
')V
e q to e d to
K Vqo Xe
of wicket gate setting and blade angle can both be made
with the unit running. This permits optimization of turbine
6 V ( X X ')
to e d
efficiency over a wide range of head and load conditions.
V.MODELING OF EXTENDED SMIB
The hydraulic flow in the penstock is modelled with the assumption of inelastic water column effect. The stiff water hammer equation can be expressed as
dh T dw
(7)
dt w dt
Fig 2: Modified block diagram of SMIB with Heffron Phillips constants.
The turbine flow q and torque m in case of Kaplan turbine are nonlinear functions of head h, wicket gate opening z, machine speed wand runner blade position .
The corresponding servomotor equations are
described as
dz (U dt
gov Z ) / Tgv
(8)
d (U dt
gov ) / Tr ;
Where Tgv and Trb are wicket gate and runner blade servomotor constants respectively.
The dynamic characteristics of the extended SMIB system are expressed in terms of constants K1 to K9. The constants K7, K8, K9 are due to head h, wicket gate opening z, machine speed w and runner blade position and their equations are given by;
K7 T7 (T3T5 ) / T1 ; (9)
Fig 5: Model of governor for hydraulic turbines.
VII: SIMULATION RESULTS
The results are obtained using MATLABSIMULINK
K8 T6 (T2T5 ) / T1;
K9 T8 (T4T5 ) / T1;
(10)
(11)
software for the block diagrams shwn below at different operating conditions.
VI: NONLINEAR TURBINE MODEL
The primary function of a governor is to control speed and /or load. Here we discuss the special requirements of governing hydraulic turbines, their physical structure and modelling in system studies. For a stable parallel operation of multiple units, the speed governor is provided with the droop characteristic. The purpose of droop is to ensure equitable load sharing between generating units. For a stable control performance, a large transient droop with a long setting time is required. This is accomplished by the provision of a rate feed back or transient gain reduction compensation as shown in fig (4).
Fig4: Governor with transient droop compensation
The governor model as shown in fig 5 has provision for representing the effect of dead bands.
fig 6 : Simulink block diagram of extended SMIB without nonlinearity for operating point p=1.0 and q=1.0
Fig 6.1 : Simulation results for extended SMIB without nonlinearity for operating point p=1.0 and q=0
Fig 6.2 : Simulation block diagram for extended SMIB with nonlinearity for operating point p=1.0 and q=0
Fig 6.3 : Simulation results for extended SMIB with nonlinearity for operating point p=1 and q=0
Fig 6.4 : simulation block diagram for extended SMIB with and without nonlinearity fro operating point p=1.0 and q=0
fig 6.5 : Simulation block diagram results for an extended SMIB with and without nonlinearity for operating point p=1.0 and q=0
VIII: CONCLUSIONS
Dual regulation of extended low hydro power plant is achieved by controlling both wicket opening and runner blade position which is connected to SMIB. Turbine governor dead band is considered which is inherently present in the system which is found due to backlash in the linkage connecting the piston to the shaft. From the step responses it has been observed that the speed deviation settling time is increased for an extended SMIB with the inherent backlash nonlinearity is considered. From the step responses of two operating points of extended SMIB it can be observed that the dynamic stability is affected.
Table 1: Data considered for Extended SMIB for an operating point p=1.0 and q=1.0
3
x 10
1
0
without nonlinearity with nonlinearity
1
Speed deviation
Speed deviation
2
3
4
5
6
7
0 1 2 3 4 5 6 7 8 9 10
Time in seconds
Ta=0.05
T
' =7.76
G+jB=
Rt =0.4
0.248+j0.26
2
Tgv=0. 5
KA=130
Vd0=1.34
Tr = 5.0
Vt
=1.05
D=0
Kf = 1.0
Tw=2.2 3
0=65.1
M=9.6
Id0=0.40
T1= 0.30
Xq=057
0=377.17
Tp=0.05
T2
=0.82
Xd= 1.6
Ta,Te=0.05,0.0 5
Ks= 5.0
T3
=0.038
X
'
K1 =0.55,
Tg= 0.2
T4
=0.32
K2 =1.16,
=0.42
K3 =0.66,
K4 =0.67,
K 5 = 0.99,
K 6=0.82
Ke=1.0
R+jX=
0.34+j0.926
Rp=0.04
T5 1.276
=
Ta=0.05
T
' =7.76
G+jB=
Rt =0.4
0.248+j0.26
2
Tgv=0. 5
KA=130
Vd0=1.34
Tr = 5.0
Vt
=1.05
D=0
Kf = 1.0
Tw=2.2 3
0=65.1
M=9.6
Id0=0.40
T1= 0.30
Xq=057
0=377.17
Tp=0.05
T2
=0.82
Xd= 1.6
Ta,Te=0.05,0.0 5
Ks= 5.0
T3
=0.038
X
'
K1 =0.55,
Tg= 0.2
T4
=0.32
K2 =1.16,
=0.42
K3 =0.66,
K4 =0.67,
K 5 = 0.99,
K 6=0.82
Ke=1.0
R+jX=
0.34+j0.926
Rp=0.04
T5 1.276
=
Table 2 : Settling time with and without nonlinearity for an opearting point.
Operating point
Settling Time without nonlinearity
Settling Time with nonlinearity
P=1 and q=0
8.99sec
15 sec
IX: REFERENCES

F.P.Demello and C.Concordia, Concepts of synchronous machine stability as affected by excitation control, IEE Trans. Power Apparatus and Systems, vol. PAS88, No.4, pp. 316329, 1969.

E. Larsen and D. Swan, Applying power system stabilizers, parts I,II and III, IEEE Trans.Power Apparatus and Systems, vol. PAS100, pp. 3017 3046, June 1981.

K.R.Padiyar, POWER SYSTEM DYNAMICS Stability and Control. John Wiley; Interline Publishing, 1996.

KISHOR, N.SAINI, R. P.SINGH, S. P. :
Coordinated control for the exciter and governor in a
small hydropower plant, Proc. of National Power . System Conference, Indian Institute of Technology,
Madras, Dec. 2730, 2004, 431436.

Power System Control and Stability Second Edition ,

P. M. Anderson, A. A. Fouad, A John Wlley & Sons, Inc., Publication.
6 Power System Stability and Control , P.Kundur, McGrawHill Inc.
7. Mlijenko Brezovec, Igor Kuzle, Nonlinear Digital Simulation Model of Hydroelectric Power Unit With Kaplan Turbine, IEEE Trans. Energy conversion, vol.21, No. 1, MAR 2006.