Introduction to Modal Analysis in Power Systems

Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 1/63
Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor.
Copyright©2009.http:www.fglongatt.org.ve
Francisco M. Gonzalez-Longatt, Dr.Sc
Manchester, UK, November, 2009
Tutorial:
Introduction to Modal
Analysis

This tutorial is a simple theoretical introduction to modal
analysis
Tutorial:
Introduction to Modal Analysis
Francisco M. Gonzalez-Longatt, Dr.Sc
fglogatt@fglongatt.org.ve
Manchester, 28th October 2009

1. Introduction
Introduction to Modal Analysis in Power
System

1. Introduction: Small Signal Stability
• Small signal stability is the ability of the power system to
maintain synchronism when subjected to small
disturbances [1].
• A disturbance is considered to be small if the equations
that describe the resulting response of power system
may be linearized for the purpose of analysis.
• Instability that result can be of two forms [1]:
– Steady increase in generator rotor angle due to lack
synchronizing torque,
– Rotor oscillation of increasing amplitude due to lack of sufficient
damping torque.
[1] P. Kundur, Power System Stability and Control. New York: McGraw- Hill, 1994.

Instable
• TS Negative
• TDPositive
δ∆ ST∆
DT∆eT∆
ω∆
δ∆
P
0 t
δ∆
P
0 t
Stable
• TS Positive
• TDPositive
δ∆
ST∆
DT∆ eT∆
ω∆
(a) With constant field voltage
Non-oscillatory

δ∆
P
0 t
Stable
• TS Positive
• TDPositive
δ∆
ST∆
DT∆ eT∆
ω∆
δ∆
P
0 t
Instable
• TS Positive
• TD Negative δ∆ST∆
DT∆
eT∆
ω∆
(b) With excitation control

• ln today's practical power systems, the small-signal
stability problem is usually one of insufficient damping of
system oscillations.
• Small signal inherent analysis using linear techniques
provides valuable information about the dynamic
characteristics of the power system and assists its
design.

2. Fundamental
Concepts of Small
Signal Stability

2. State-Space Representation
• The behaviour of a dynamic system, such as a power
system, may be described by a set of n first order non-
linear ordinary differential equations.
• This can be written using vector-matrix notation [1]:
where:
( )t,,uxfx =












=
nx
x
x

2
1
x












=
ru
u
u

2
1
u












=
nf
f
f

2
1
f

2. State Space Representation
• The column vector x is deferred to as the state vector.
• xi are referred as state variables.
The column vector u is the vector of inputs to the
system.
• u are the external signal that influence the performance
of the system.
• t denote time
• is the derivate of a state variable respect to time.












=
nx
x
x

2
1
x












=
ru
u
u

2
1
u












=
nf
f
f

2
1
f
x

• The system is called autonomous if the derivatives of
the stated variables are not explicit functions of time.
• The output variables can be observed in the system.
• These may be expressed in terms of the state variables
and the inputs variables [1]:
• where:
( )uxfx ,=
( )uxgy ,=












=
my
y
y

2
1
y












=
rg
g
g

2
1
g

• The column vector y is the vector of outputs
• g is a vector of non-linear factions relating state and
input variables to output variables.












=
my
y
y

2
1
y












=
rg
g
g

2
1
g

2.1. State Concept
• The concept of state is fundamental to the state-space
approach.
• The state of a system represents the minimum amount
of information about the system at any instant in time t0
that is necessary so that its future behaviour can be
determined without the input before t0.
• Any set of n linearly independent system variables may
be used to describe the state of the system –state
variables [1].
• State variables form a minimal set of dynamic variables
that, along with the inputs to the system, provide a
complete description of the system behaviour

2.1. State-Space Concept
• The system state may be represented in a n-dimensional
Euclidean space, called state space.

2.2. Equilibrium (or singular) points
• Those points where all derivatives of state variables are
simultaneously zero [1].
• They define the points on the trajectory with zero
velocity.
• This system is at rest since all the variables are constant
and unvarying with time.












=
=
=
=
0
0
0
2
1
nx
x
x




x

2.2. Equilibrium (or singular) points
• The equilibrium or singular point must therefore satisfy
the equation:
where x0 is the state vecto x at the equilibrium point.
( ) 00 0 === xfx

2.3. Stability of a Dynamic System
• Linear system: stability is entirely independent of the
input.
• State of a stable system with zero input will always
return to the origin of the state space, independent of the
finite initial state.
• Non-linear system: Stability depends on the type and
magnitude of input and the initial state.

2.4. Classification of Stability
• Classification of stability of non-linear system, depending
on the region of state space in which the state vector
ranges:
– Local stability or stability in the small.
– Finite stability.
– Global stability or stability large.

2.4.a. Local Stability
• The system is locally stable about equilibrium point if,
when subjected to small perturbation, it remain within a
small region surrounding the equilibrium point.
• If, a t increase, the system return to the original state, it
is said to be asymptotically stable in the small.

2.4.b. Finite Stability
• If the state of a system remains within a finite region R, it
is said to be stable within R.
• If, further, the state of the system returns to the original
equilibrium point form any point within R, it is
asymptotically stable within the finite region R.

2.4.c. Global Stability
• The system is said to be globally stable if R include the
entire finite space.

3. Linealization

3. Linearization
• Let x0 be the initial state vector and u0 the input vector
corresponding to the equilibrium point [2] ,[3].
• Let include a perturbation from the above state
where the prefix Δ denote a small deviation.
• As the perturbations are assumed to be small, the
nonlinear functions f(x,u) can be expressed in terms of
Taylor’s series expansion.
( ) 0, 000 == uxfx
xxx ∆+= 0 uuu ∆+= 0

3. Linearization
• The original equations are [1]-[3]:
• The lienarized forms are:
– Δx is the state vector of dimension n
– Δy Is the output vector of dimension m
– Δu is the input vector of dimension r
– A is state of plant matrix of size nxn
– B is the control or input matrix of size nxr
– C is the output matrix of size mxn
– D is the (feedforward) matrix which defines the proportion of
inputs which appears directly in the output, size nxr
( )t,,uxfx =
( )uxgy ,=
uDxCy
uBxAx
ΔΔΔ
ΔΔΔ
+=
+=

3. Linearization
• Taking the Laplace transform the state equations in the
frequency domain are obtained:
uDxCy
uBxAx
ΔΔΔ
ΔΔΔ
+=
+=
( ) ( ) ( )
( ) ( ) ( )sΔsΔsΔ
sΔsΔΔsΔ
uDxCy
uBxAxx
+=
+=− 0

3. Linearization
• Block diagram of the state-space representation [1]-[3]
B
D
I
s
1
A
ΣΣ C
u∆
y∆
x∆x∆ +
+
+
+

3. Linearization
• The initial conditions are Δx(0) assumed zero.
• The solution of the state equations can be obtained [1]:
• The Laplace transform of Δx and Δy are seen to have
two components:
(i) Dependent on the initial conditions and
(ii) Dependent on the inputs.
• These are the Laplace transforms of the free and zero-
state components of the state and output vectors
( ) ( )
( )
( ) ( )[ ] ( )sΔsΔΔ
s
sadj
sΔ uDuBx
AI
AI
Cy ++
−
−
= 0
det

3. Linearization
• The poles of Δx and Δy are the roots of the equation:
• The values of s which satisfy this conditions are known
as eigenvalues of matrix A [1]-[3].
• The equation [1],[2]:
is referred as he characteristic equation of matrix A.
( ) 0det =− AIs
( ) 0det =− AIs

4. Eigenvalues and
Eigenvectors

4. Eigenvalues
• The eigenvalues of a matrix are given by the values of
the scalar parameter which there exist non-trivial
solutions to the equation [1]:
• For a non-trivial solution [2]:
• Expansion of the determinant give the characteristic
equation.
• The n solution of λ = λ1, λ2, …λn are the eigenvalues of A
[2].
( ) 0det =− AIs
( ) 0=− φλIA

4. Eigenvectors
• For every eigenvalue λi, there is an eigenvector ϕi which
satisfies:
• ϕi is called the right eigenvector of the state matrix A
associated with the eigenvalue λi.
• Each right eigenvector is a column vector with the length
equal to the number of the states.
• The right eigenvector is called mode shape.
iii φλφ =A

4. Eigenvectors
• Left eigenvector associated with the eigenvalue λi is the
n-row vector which satisfies:
• The right eigenvector describes how each mode of
oscillation is distributed among the system states.
• It indicates on which system variables the mode is
more observable.
• The left eigenvector, together with the system’s initial
state, determines the amplitude of the mode.
• A left eigenvector carries mode controllability
information.
iii ψλψ =A

4. Eigenvectors
The left eigenvector indicates on which system
variables the mode is more observable.
The right eigenvector is called mode shape.

5. Eigenvalues and
stability

5. Eigenvalues and Stability
• The stability of the system is determined by the
eigenvalues λi.
• Real eigenvalues: Non oscillatory mode.
– Negative real eigenvalue represent a decaying mode.
– Magnitude define the decay.
– Positive real eigenvalues represent aperiodic instability.
• Complex eigenvalues: Occurs in conjugate pair, and
each pairs correspond a oscillatory mode.

• For a complex pairs of eigenvalues [1]:
• The frequency of oscillation (f) in Hz is given by [1]:
• This represents the actual or damped frequency (f).
• T he damping ratio (ζ) is given by:
• The damping ratio ζ determines the rate of decay of the
amplitude of the oscillation [1]-[2].
ωσλ j±=
π
ω
2
=f
22
ωσ
σ
ς
+
−
=

σ
ωj
X
X
z1
z2
(a) Stable focus
Eigenvalues
ωσλ j±=
Trajectory Type of singularity
(b) Unstable focus
σ
ωj
X
X
z1
z2

(f) Vortex
Eigenvalues
ωσλ j±=
(g) Saddle
σ
ωj
X
X
z1
z2
σ
ωj
XX
z1
z2

(c) Stable focus
Eigenvalues
ωσλ j±=
(d) Unstable focus
σ
ωj
XX
z1
z2
σ
ωj
XX
z1
z2

6. Indexes

6. Indexes
• Numerous indices, can be calculated from eigenvectors
such as [4]:
– Participation factors,
– Transfer function residues and
– Mode sensitivities.
• Those are very useful in system analysis and controller
design.

6.1. Participation factors
• The Participation matrix (P), combine the right and left
eigenvectors [1]:
where
where ψki is the kth element in the ith row of the the left
eigenvector ψi, and φki is the kth element in the ith column
of the right eigenvector φi [4].
[ ]npppP 21=












=












=
nini
ii
ii
ni
i
i
p
p
p
ψφ
ψφ
ψφ

22
11
2
1
1p

6.1. Participation factors
• The participation factor pki is a measure of the relative
participation of the kth state variable in the ith mode, and
vice versa.
• The sensitivity of a particular eigenvalue λi to the
changes in the diagonal elements of the state matrix A.
kikikip ψφ=
kk
i
ki
a
p
∂
∂
=
λ
kikikip ψφ=

6.2. Controllability and observability
• The system response in presence of input is given as:
• Expressing in terms of the transform ed variables Z:
where Φ is the modal matrix of A.
• Then yield to:
uDxCy
uBxAx
ΔΔΔ
ΔΔΔ
+=
+=
zx Φ=Δ
uDzCy
uBxAz
ΔΔ
ΔΔ
+Φ=
+=Φ

• The state equations in the normal form (decoupled) may
therefore [1], [4]:
where:
• They are the modal controllability (B’) and modal
observability matrices (C’).
uDzCy
uBΛzzΦ
ΔΔ
Δ
+=
+=
'
'
CΦC
BΦB
=
= −
'
' 1

• If the ith row of matrix B’ is zero, the inputs have not
effect on the ith mode.
• ith mode is said to be uncontrollable [1]
• If the ith row of matrix C’ determines whether or not
variable zi contribute to the formation of outputs.
• If the ith Coolum of matrix C’ is zero, then the
corresponding mode is unobservable [1].
BΦB 1
' −
= Mode controllability matrix
CΦC =' Mode observability matrix

6.3. Residues
• For small-signal stability analysis of power systems, this
primarily is related on the eigenvalue of the state matrix.
• For control design the open-loop transfer function
between specific variables is useful [4].
• Consider transfer function between the variables y and
u:
• Let asume y is not direct function of u (D = 0)
xy
uxAx
cΔΔ
bΔΔΔ
=
+=
( ) ( )
( )su
sy
sG
∆
∆
=
( ) ( ) bAIc
1−
−= ssG

6.3. Residues
• G(s) can be factorized [1], [2], [4]:
• Using partial fractions:
where and Ri is known as the residue of G(s) function at
pole pi.
( ) ( )( ) ( )
( )( ) ( )n
l
pspsps
zszszs
KsG
−−−
−−−
=


21
21
( )
n
n
ps
R
ps
R
ps
R
sG
−
++
−
+
−
= 
2
2
1
1
( ) ( )
( )su
sy
sG
∆
∆
= ( ) [ ] ΨbΛIcΦG
1−
−= ss

6.3. Residues
• Since Λ is a diagonal matrix [4]:
• This equation gives the residues in terms of eigenvalues.
( ) ∑= −
=
n
j j
j
s
R
sG
1 λ
bΨcΦ iiiR =

7. References

7. References
[1] P. Kundur, Power System Stability and Control. New
York: McGraw- Hill, 1994.
[2] J. Arrillaga and C.P. Arnold, Computer Modelling of
Electrical Power Systems, John Wiley & Sons, 1983.
[3] P.M. Anderson and A.A. Fouad, Power System Control
and Stability, The Iowa State University Press, 1977.
[4] R.Sadikovi’c, Use of FACTS Devices for Power Flow
Control and Damping of Oscillations in Power Systems.
PhD Thesis in Swiss federal Institute of Technology,
Zurich, 2006.

Please visit:
http://www.fglongatt.org.ve
Comments and suggestion are welcome:
fglongatt@fglongatt.org.ve

Introduction to Modal Analysis in Power Systems

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Introduction to Modal Analysis in Power Systems