example: modal analysis using digsilent powerfactory

Dr. Francisco M. Gonzalez-Longatt, [email protected] .Copyright © 2009 1/63

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Francisco M. Gonzalez-Longatt, Dr.ScManchester, UK, November, 2009

Tutorial:Introduction to Modal Analysis


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This tutorial is a simple theoretical introduction to modalanalysis

Tutorial:Introduction to Modal Analysis

Francisco M. Gonzalez-Longatt, [email protected]

Manchester, 28th October 2009


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1. IntroductionIntroduction to Modal Analysis in Power

System


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1. Introduction: Small Signal Stability• Small signal stability is the ability of the power system to

maintain synchronism when subjected to smalldisturbances [1].

• A disturbance is considered to be small if the equationsthat describe the resulting response of power systemmay 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.


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1. Introduction: Small Signal Stability

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


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1. Introduction: Small Signal Stabilityδ∆

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


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1. Introduction: Small Signal Stability• ln today's practical power systems, the small-signal

stability problem is usually one of insufficient damping ofsystem oscillations.

• Small signal inherent analysis using linear techniquesprovides valuable information about the dynamiccharacteristics of the power system and assists itsdesign.


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2. Fundamental Concepts of Small

Signal Stability


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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

xx

2

1

x

=

ru

uu

2

1

u

=

nf

ff

2

1

f


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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 thesystem.

• u are the external signal that influence the performanceof the system.

• t denote time• is the derivate of a state variable respect to time.

=

nx

xx

2

1

x

=

ru

uu

2

1

u

=

nf

ff

2

1

f

x


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2. State Space Representation• 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

yy

2

1

y

=

rg

gg

2

1

g


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2. State Space Representation

• 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

yy

2

1

y

=

rg

gg

2

1

g


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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 t0that is necessary so that its future behaviour can bedetermined without the input before t0.

• Any set of n linearly independent system variables maybe used to describe the state of the system –statevariables [1].

• State variables form a minimal set of dynamic variablesthat, along with the inputs to the system, provide acomplete description of the system behaviour


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

Euclidean space, called state space.


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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 zerovelocity.

• This system is at rest since all the variables are constantand unvarying with time.

=

==

=

0

00

2

1

nx

xx

x


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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


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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 thefinite initial state.

• Non-linear system: Stability depends on the type andmagnitude of input and the initial state.


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2.4. Classification of Stability• Classification of stability of non-linear system, depending

on the region of state space in which the state vectorranges:– Local stability or stability in the small.– Finite stability.– Global stability or stability large.


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2.4.a. Local Stability

• The system is locally stable about equilibrium point if,when subjected to small perturbation, it remain within asmall region surrounding the equilibrium point.

• If, a t increase, the system return to the original state, itis said to be asymptotically stable in the small.


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2.4.b. Finite Stability

• If the state of a system remains within a finite region R, itis said to be stable within R.

• If, further, the state of the system returns to the originalequilibrium point form any point within R, it isasymptotically stable within the finite region R.


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2.4.c. Global Stability

• The system is said to be globally stable if R include theentire finite space.


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3. Linealization


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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 ofTaylor’s series expansion.

( ) 0, 000 == uxfx

xxx ∆+= 0 uuu ∆+= 0


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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 ,=

uDxCyuBxAx

ΔΔΔΔΔΔ

+=+=


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3. Linearization

• Taking the Laplace transform the state equations in thefrequency domain are obtained:

uDxCyuBxAx

ΔΔΔΔΔΔ

+=+=

( ) ( ) ( )( ) ( ) ( )sΔsΔsΔ

sΔsΔΔsΔuDxCy

uBxAxx+=

+=− 0


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3. Linearization• Block diagram of the state-space representation [1]-[3]

B

D

Is1

A

ΣΣ Cu∆ y∆x∆x∆ +

+

++


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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 havetwo 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

( ) ( )( ) ( ) ( )[ ] ( )sΔsΔΔssadjsΔ uDuBx

AIAICy ++

−−

= 0det


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3. Linearization• The poles of Δx and Δy are the roots of the equation:

• The values of s which satisfy this conditions are knownas eigenvalues of matrix A [1]-[3].

• The equation [1],[2]:

is referred as he characteristic equation of matrix A.

( ) 0det =− AIs

( ) 0det =− AIs


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4. Eigenvalues and Eigenvectors


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4. Eigenvalues• The eigenvalues of a matrix are given by the values of

the scalar parameter which there exist non-trivialsolutions to the equation [1]:

• For a non-trivial solution [2]:

• Expansion of the determinant give the characteristicequation.

• The n solution of λ = λ1, λ2, …λn are the eigenvalues of A[2].

( ) 0det =− AIs

( ) 0=− φλIA


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4. Eigenvectors• For every eigenvalue λi, there is an eigenvector ϕi which

satisfies:

• ϕi is called the right eigenvector of the state matrix Aassociated with the eigenvalue λi.

• Each right eigenvector is a column vector with the lengthequal to the number of the states.

• The right eigenvector is called mode shape.

iii φλφ =A


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4. Eigenvectors• Left eigenvector associated with the eigenvalue λi is the

n-row vector which satisfies:

• The right eigenvector describes how each mode ofoscillation is distributed among the system states.

• It indicates on which system variables the mode ismore observable.

• The left eigenvector, together with the system’s initialstate, determines the amplitude of the mode.

• A left eigenvector carries mode controllabilityinformation.

iii ψλψ =A


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4. Eigenvectors

The left eigenvector indicates on which systemvariables the mode is more observable.

The right eigenvector is called mode shape.


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5. Eigenvalues and stability


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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, andeach pairs correspond a oscillatory mode.


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5. Eigenvalues and Stability• 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 theamplitude of the oscillation [1]-[2].

ωσλ j±=

πω2

=f

22 ωσσς+

−=


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5. Eigenvalues and Stability

σ

ωjX

X

z1

z2(a) Stable focus

Eigenvaluesωσλ j±=

Trajectory Type of singularity

(b) Unstable focusσ

ωjX

X

z1

z2


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(f) Vortex



(g) Saddle

σ

ωj

X

X

z1

z2

σ

ωj

XX

z1

z2


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(c) Stable focus



(d) Unstable focusσ

ωj

XX

z1

z2

σ

ωj

XX

z1

z2


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6. Indexes


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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 controllerdesign.


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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 lefteigenvector ψi, and φki is the kth element in the ith columnof the right eigenvector φi [4].

[ ]npppP 21=

=

=

nini

ii

ii

ni

i

i

p

pp

ψφ

ψφψφ

22

11

2

1

1p


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6.1. Participation factors• The participation factor pki is a measure of the relative

participation of the kth state variable in the ith mode, andvice versa.

• The sensitivity of a particular eigenvalue λi to thechanges in the diagonal elements of the state matrix A.

kikikip ψφ=

kk

iki a

p∂∂

=λ

kikikip ψφ=


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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:

uDxCyuBxAx

ΔΔΔΔΔΔ

+=+=

zx Φ=Δ

uDzCyuBxAz

ΔΔΔΔ

+Φ=+=Φ


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6.2. Controllability and observability• The state equations in the normal form (decoupled) may

therefore [1], [4]:

where:

• They are the modal controllability (B’) and modalobservability matrices (C’).

uDzCyuBΛzzΦ

ΔΔΔ

+=+=

''

CΦCBΦB

== −

'' 1


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6.2. Controllability and observability• 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 notvariable zi contribute to the formation of outputs.

• If the ith Coolum of matrix C’ is zero, then thecorresponding mode is unobservable [1].

BΦB 1' −= Mode controllability matrix

CΦC =' Mode observability matrix


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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)

xyuxAx

cΔΔbΔΔΔ

=+=

( ) ( )( )susysG

∆∆

=

( ) ( ) bAIc 1−−= ssG


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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 atpole pi.

( ) ( )( ) ( )( )( ) ( )n

l

pspspszszszsKsG

−−−−−−

=

21

21

( )n

n

psR

psR

psRsG

−++

−+

−=

2

2

1

1

( ) ( )( )susysG

∆∆

= ( ) [ ] ΨbΛIcΦG 1−−= ss


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6.3. Residues• Since Λ is a diagonal matrix [4]:

• This equation gives the residues in terms of eigenvalues.

( ) ∑= −

=n

j j

j

sR

sG1 λ

bΨcΦ iiiR =


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7. References


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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.


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