DYNAMIC STABILITY ANALYSIS Small Signal Stability

DYNAMIC
STABILITY
ANALYSIS
(Small Signal
Stability)
By
Prof. C. Radhakrishna

210/17/2010 10:25 AM
CONTENTS
DYNAMIC STABILITY ANALYSIS (Small Signal Stability) – 1
Small-Signal Stability of Multi-machine Systems
Special techniques for analysis of very large systems
Characteristics of Small-Signal Stability Problems
Local problems
Global problems
DYNAMIC STABILITY ANALYSIS – 2
Introduction
Overview of the Proposed Method
Generating Unit
Synchronous Machine
Calculation of Equilibrium State Conditions
Excitation and Governor Control Systems
Excitation System
Turbine-Governor System
Combined Model of Generating Unit

310/17/2010 10:25 AM
CONTENTS cont………
Load Representation
Multi-Component Models
Network Representation
State Space Model of the Overall System
Conclusions

410/17/2010 10:25 AM
DYNAMIC STABILITY ANALYSIS (Small Signal Stability)-1
• Small-signal stability, is the ability of the power system to
maintain synchronism when subjected to small
disturbances.
• A disturbance is considered to be small if the equations
that describe the resulting response of the system may be
linearized for the purpose of analysis.
• The small-signal stability problem is usually one of
insufficient damping of system oscillations.
Small-Signal Stability of Multi-machine Systems
Analysis of practical power systems involves the simultaneous
solution of equations representing the following:
• Synchronous machines, and the associated excitation systems
and prime movers.
• Interconnecting transmission network.
• Static and dynamic (motor) loads
• Other devices such as HVDC converters, static var compensators

510/17/2010 10:25 AM
• For system stability studies it is appropriate to neglect
the transmission network and machine stator transients.
• The dynamics of machine rotor circuits, excitation
systems, prime mover and other devices are represented by
differential equations.
• The result is that the complete system model consists of a
large number of ordinary differential and algebraic equations.

610/17/2010 10:25 AM
* Algebraic equations
** Differential equations
Figure.1: Structure of the complete power system model

2010/17/2010 10:25 AM
• Analysis of inter-area oscillations in a large interconnected
power system requires a detailed modelling of the entire system.
• Special techniques have been developed that focus on
evaluating a selected subset of eigenvalues associated with the
complete system response.
• AESOPS algorithm. It uses a novel frequency response
approach to calculate the eigenvalues associated with the rotor
angle modes.
• The selective modal analysis (SMA) approach computes
eigenvalues associated with selected modes of interest by using
special techniques to identify variables that are relevant to the
selected modes, and then constructing a reduced-order model that
involves only the relevant variables.
• The PEALS (Program for Eigenvalue Analysis of Large Systems)
uses two of these techniques.
• The AESOPS algorithm and the modified Arnoldi method.
• These two methods have been found to be efficient and reliable,
and they complement each other in meeting the requirements of
small-signal stability analysis of large complex power systems.
Special techniques for analysis of very large systems

2110/17/2010 10:25 AM
Characteristics of Small-Signal Stability Problems
Local problems
• Associated with rotor angle oscillations of a single generator
or a single plant against the rest of the power system. Such
oscillations are called local plant mode oscillations.
• Most commonly encountered small-signal stability problems
are of this category.
• Local problems may also be associated with oscillations
between the rotors of a few generators close to each other.
• Such oscillations are called inter-machine or inter-plant
mode oscillations.
• The local plant mode and interplant mode oscillations have
frequencies in the range of 0.7 to 2.0 Hz.
• Analysis of local small-signal stability problems requires a
detailed representation of a small portion of the complete
interconnected power system.
• The rest of the system representation may be appropriately
simplified by use of simple models and system equivalents.

2210/17/2010 10:25 AM
Global problems
• Global small-signal stability problems are caused by
interactions among large groups of generators and have
widespread effects.
• They involve oscillations of a group of generators in one
area swinging against a group of generators in another
area. Such oscillations are called inter-area mode oscillations.
Large interconnected systems usually have two distinct forms of
interarea oscillations:
(a)A very low frequency mode involving all the generators in the
system. The frequency of this mode of oscillation is on the
order of 0.1 to 0.3 Hz.
(b)Higher frequency modes involving subgroups of generators
swinging against each other. The frequency of these
oscillations is typically in the range of 0.4 to 0.7 Hz.

2310/17/2010 10:25 AM
DYNAMIC STABILITY ANALYSIS
The analysis of dynamic stability can be performed by deriving a
linearized state space model of the system in the following
form
p X = A X + B u
Where the matrices A and B depend on the system parameters
and the operating conditions.
• The Eigen values of the system matrix A determine the
stability of the operating point.
• The Eigen value analysis can be used not only for the
determination of the stability regions, but also for the design
of the controllers in the system.
The novel features of the proposed method :
• It is not necessary to reduce the power system network to
eliminate non-generator buses. The same network used for load flow
studies can also be used for the dynamic stability calculations.
• The development of system model proceeds systematically by the
development of the individual models of various components and
subsystems and their interconnection through the network model.

2410/17/2010 10:25 AM
Overview of the Proposed Method
At any bus k of an N-bus network the following equations apply
















kIj
j
j
k
j
j
k
k V
V
PP
P 
 















kIj
j
j
k
j
j
k
k V
V
QQ
Q 

where Ik is the set of buses that are connected to bus k. Also it
would be shown that for each bus, (P, Q) or (, V)
Fig. 1 Block diagram for power system network
can be eliminated depending on the type of bus. The A matrix
formulation is based on identifying the interconnections among the
various subsystems of the power system as shown in Figure 2.

2510/17/2010 10:25 AM
• Development of the system model is based on the formulation of
the individual component models and identifying the various
interconnections between the subsystems.
• The linearized network algebraic equations are solved in terms of
the system state variables resulting in the final system model.
Fig. 2 Block diagram showing the interconnections among the
various subsystems of the power system

2610/17/2010 10:25 AM
The rotor circuit differential equations, including its motion, are
given by
p Xm = [Am] Xm + bme vfd + b mg  Pm + [Bp] Sg
Ym = [Cm] Xm
where, Xm = [Id Iq f k  ]t
Ym = [Id Iq  ]t
Sg = [Pg Qg]t
Also, the generator terminal bus voltage magnitude and phase angle are
expressed in the form
Zg = [Dm] Ym + [Dp] Sg
where Zg = [g Vg]t
Id and Iq are state variables derived from the rotor flux linkages.
GENERATING UNIT
Synchronous Machine
Calculation of Equilibrium State Conditions
The values at the operating point (equilibrium state) of the
power system are calculated from the load flow results of the
system.

2710/17/2010 10:25 AM
• The excitation and governor control systems used in modern
generators fall into standard categories compiled in IEEE
Committee reports.
Excitation and Governor Control Systems
While it was initially thought that high gain voltage regulator loop with
a fast acting static exciter would improve transient stability, the
practical experience was that it led to dynamic instability.
Power system stabilizer (PSS) which introduces supplementary
stabilizing signal to suppress rotor oscillations has become a
desirable part of any excitation system.
The change in the magnitude of the terminal voltage, Vg, is one of the
inputs for the excitation system and this has to be expressed in terms
of the state variables and is given in the equation (5).
The state space model of excitation system is represented in the form
p Xe = [Ae] Xe + [Bem] Ym + [Bep] Sg + be ue
ye = [Ce] Xe
ye = Vfd; ue = Vref
where Xe, ue and ye are respectively the state, input and output
quantities; and the structures of the associated matrices are
obtained for the IEEE Type 1 excitation system.
Excitation System

2810/17/2010 10:25 AM
The state space model of governor control system can be
represented in the form
p Xg = [Ag] Xg + [Bgm] Ym + bg ug
yg = [Cg] Xg
yg = Pm; ug = Pmo
where Xg, ug and yg are respectively the state, input and output
quantities; and the structures of the associated matrices are
obtained for an IEEE system model.
Turbine-Governor System
The following state space model is obtained, where all the
component elements are matrices.
Combined Model of Generating Unit
m mm me mg p
e
e eem e ep
g
g ggm g
p X A B B X B
u
p X B A X B
u
p X B O A X O
g e
g
O O
O S b O
O b
        
                      
                

2910/17/2010 10:25 AM
[Bme] = bme Tt
el ; [Bmg] = bmg Tt
gl ;
[B’em] = [Bem] [Cm] ; [B’gm] = [Bgm] [Cm] ;
and Tel and Tgl are vectors containing only one non zero element
each equal to one and defined by the following equations
Vfd = Tt
el Xe; Pm = Tt
gl Xg































g
e
m
g
e
m
g
e
m
X
X
X
C
C
C
y
y
Y
The usual constant power, constant current and constant
impedance type loads and any other voltage dependent
nonlinear loads can be represented in the general form
Load Representation
where consent coefficients kp, kq and the exponents np and nq
depend upon the type of load under consideration. Linearizing,
we get
nq
LqL
np
LpL VkQVkP  ;

3010/17/2010 10:25 AM
The nonlinear loads dependent on the bus frequency, if present
in the system, can also be handled without any difficulty, if
desired.
where,
= [Al] Zl
= [PL QL ]t
= [L VL ]t

S


S

Z










 

1
1
][ nq
Lqq
np
Lpp
l
VknO
VknO
A
Multi-Component Models
The various subsystems described earlier can be assembled
together for the analysis of large-scale power systems
including large number of machines and loads.

3110/17/2010 10:25 AM
[AM] = Block diag. [[Aml], [Am2],…………… [Amn]]
and XM = [Xt
ml Xt
m2…………… Xt
mn]t



























































g
e
EP
P
G
E
M
GGM
EEM
MGMEM
G
E
M
u
u
O
B
B
X
X
X
AOB
AB
BBA
Xp
Xp
Xp
G
EG
B
O
O
O
B
O
SO































G
E
M
G
E
M
G
E
M
X
X
X
C
C
C
Y
Y
Y
where all the components are matrices.
Also,
ZG = [DM] YM + [Dp] SG
SL = [AL] ZL

3210/17/2010 10:25 AM
The network equations are
[SG
t SL
t]t = [J] [ZG
t ZL
t]t
where [J] is the Jacobian matrix of the network and is given by
where all the components are matrices. Substituting the
equations (17) in (18) and simplifying, we get,
where [J’LL] = [JLL] + [AL]
Network Representation
• The network is represented by its Jacobian matrix in the
polar form.
• For a N-bus power system network the Jacobian is of (2N x
2N) dimension and the identity of all the buses is preserved.







LLLG
GLGG
JJ
JJ
J][




















L
G
LLLG
GLGGG
Z
Z
JJ
JJ
O
S

3310/17/2010 10:25 AM
ZG = [DM] [CM] XM + [Dp] SG
M
M
M
MLG
MGG
M
L
G
X
C
C
DJ
DJ
J
Z
S















 1
][





 

LLPLG
GLPGG
M
JDJ
JDJU
J
'
)(
][where
State Space Model of the Overall System
State space model of the overall system
p X = [A] X + [B] U
Y = [C] X
where X = [XM
t XE
t XG
t] t
U = [ue
t ug
t ] t
Y = [YM
t YE
t YG
t] t











GGM
EEM
MGMEM
AOB
AB'
BBA'
][ OA











G
E
B
O
O
O
B
O
B][











G
E
M
C
C
C
][C

3410/17/2010 10:25 AM
• This state space model is amenable to the application of linear
control theory and eigenvalue analysis.
• This allows one to study the overall dynamic performance of
power systems, including the interaction between machine
controls.
CONCLUSIONS
REFERENCES :
[1] Prabha Kundur: “Power System Stability and control”, The EPRI Power
System Engineering Series, McGraw-Hill, Inc., 1994.
[2] C. Radhakrishna : “Stability Studies of AC/DC Power Systems” , Ph. D.
Thesis , submitted to Indian Institute of Technology Kanpur, India, 1980.

3510/17/2010 10:25 AM
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YOU

DYNAMIC STABILITY ANALYSIS Small Signal Stability

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