20

CHAPTER 2

DYNAMIC STABILITY MODEL OF THE POWER SYSTEM

2.1 GENERAL

Dynamic stability of a power system is concerned with the dynamic

behavior of the system under small perturbations around an operating

condition and more specifically it is a phenomena of slow and poorly damped

or sustained or even diverging power oscillations which are essentially due to

varying system loads and ill controlled controllers of the system. Computer

analysis of this problem requires mathematical models which simulate as

accurately as the behavior of physical system but at the same time not very

complex to handle.

This chapter presents the development of the mathematical model

for the dynamic stability analysis of Single Machine Infinite Bus System

(SMIB) and Multi- machine power system.

2.2 SIMPLIFIED LINEAR MODEL FOR SMIB SYSTEM

In stability analysis, the mathematical model is used for dynamic

analysis of power systems.

2.2.1 Assumptions

The following assumptions are made for the development of

simplified linear model of SMIB system:

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1. Damper windings both in the d and q axes are neglected.

2. Armature resistance of the machine is neglected.

3. Excitation system is represented by a single time constant.

4. Balanced conditions are assumed and saturation effects are

neglected.

2.2 2 Classical Machine Model

In the classical methods of analysis, the simplified model or

classical model of the generator is used (Kundur 1994). Here, the machine is

modeled by an equivalent voltage source behind impedance connected to an

infinite-bus as shown in Figure 2.1.

Infinite Bus

An infinite bus is a source of invariable frequency and voltage

(both in magnitude and angle). A major bus of a power system of very large

capacity compared to the rating of the machine under consideration is

approximately an infinite bus.

Figure 2.1 One-line diagram of SMIB

The state space classical machine model is shown in Figure 2.2.

xe

22

Figure 2.2 Classical Machine Model

The state equations of the classical model are given in equation (2.1):

1

p( ) T K Dm S2H ∆ω = ∆ − ∆δ − ∆ω

0p( )∆δ = ω ∆ω (2.1)

State vector Tx ( )= ∆ω ∆δ (2.2)

And when the effect of flux linkage is included, three states are used to model

the generator: ∆ω, ∆δ and ∆Eq'. The state equations are given in equation (2.3)

as follows:

'm 1 2q

j j j j

T K K Dp ω= - δ- E - ω

τ τ τ τ

∆∆ ∆ ∆ ∆

0p( )∆δ = ω ∆ω

' ' 4q q FD' ' '

3 do do do

-1 1 Kp E = E + E - δ

K τ τ τ∆ ∆ ∆ ∆ (2.3)

where τj = 2H, H is the inertia constant.

Ks

23

State Vector of the SMIB system including the effect of flux

linkage is given by equation (2.4)

t

qx [ E ' ]= ∆ω ∆δ ∆ (2.4)

where the variables are

Eq'

- quadrature axis component of voltage behind transient

reactance

ω - angular velocity of rotor

δ - rotor angle in radians

K1 to K6 is the Heffron Philips constants (Padiyar 2002).

2.3 EXCITATION SYSTEM REPRESENTATION

The excitation system model considered is the simplified form of

ST1A model shown in Figure 2.3. A high exciter gain, without derivative

feedback, is used. By inspection of Figure 2.3, the state space equations can

be written as,

Figure 2.3 Excitation System Representation

p∆EFD = -1/TA (KA ∆Vt +∆ EFD) (2.5)

with TR is neglected, Vref =constant.

And ∆Vt = K5 ∆δ + K6 ∆Eq'

∆V1

∆Vt A

A

K

1 sT+

∆EFD

24

where EFD - Equivalent stator emf proportional to field voltage

KA - Gain of the Exciter

TA - Time constant of the exciter

TR - Terminal Voltage Transuder Time Constant

Vt - Terminal voltage of the Synchronous machine

Vref - Reference voltage of the Synchronous machine

Combining the Equations (2.3) with the exciter equation (2.5), the

complete state space description of SMIB system including exciter is given in

equation (2.6).

'm 1 2q

j j j j

T K K Dp ω= - δ- E - ω

τ τ τ τ

∆∆ ∆ ∆ ∆

0p∆δ = ω ∆ω

' ' 4q q FD' ' '

3 do do do

-1 1 Kp E = E + E - δ

K τ τ τ∆ ∆ ∆ ∆

pEFD = -1/TA (KA K5 ∆δ + KA K6 ∆Eq' + EFD) (2.6)

The state vector is thus defined by Equation (2.7):

t

q FDx [ E ' E ]= ∆ω ∆δ ∆ ∆ (2.7)

2.4 SMIB SYSTEM REPRESENTATION WITH CPSS

The block diagram of the CPSS is shown in Figure 2.4. The state

equations for the same can be written as follows.

p∆V2 = KPSS p∆ω – (1/Tw) ∆V2 (2.8)

p∆Vs = (T1/T2) p∆V2 + (1/T2) ∆V2 – (1/T2) ∆ Vs (2.9)

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where KPSS - CPSS gain

T1, T2 - Phase compensator time constants

Tw - Wash out time constant

Figure 2.4 CPSS Representation

State vector of the synchronous machine model including PSS is

given by equation (2.10):

T

q FD 2 sx = [ E ' E V V ]∆ω ∆δ ∆ ∆ ∆ ∆ (2.10)

The block diagram of simplified linear model of a synchronous

machine connected to an infinite bus with exciter and PSS is shown in

Figure 2.5.

Figure 2.5 State Space Model of SMIB system representation with CPSS

∆Vt

CPSS

∆Vs

∆Vt

Conventional Power System Stabilizer

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2.5 DYNAMIC STABILITY MODEL OF MULTI MACHINE

POWER SYSTEM

In stability analysis of a multi-machine system, modelling of all the

machines in a more detailed manner is exceedingly complex in view of the

large number of synchronous machines to be simulated. Therefore simplifying

assumptions and approximations are usually made in modelling the system.

In this thesis two axis model is used for all machines in the sample system

taken for investigation.

2.5.1 Assumptions Made

In this work the synchronous machine is modeled using the two-

axis model (Anderson and Fouad 2003). In the two-axis model the transient

effects are accounted for, while the sub transient effects are neglected. The

transient effects are dominated by the rotor circuits, which are the field circuit

in d-axis and an equivalent circuit in the q-axis formed by the solid rotor. The

amortisseur winding effects are neglected. An additional assumption made in

this model is that in stator voltage equations the terms pλd

and pλq are

negligible compared to the speed voltage terms and that ω ω =1p.u.R

≅ The

block diagram representation of the synchronous machine in two-axis model

is shown in Figure 2.6.

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Figure 2.6 Block diagram representation of two axis model for synchronous machine

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2.5.2 Synchronous Machine Representation

Using the block diagram reduction technique and with the

simplifying assumptions the state equations for the two-axis model in p.u.

form

pEd'

= {-Ed' - (xq-xq') Iq} / τqo'

pEq'

= {EFD - Eq' - ( xd - xd' ) Id } / τdo'

pω = {Tm - Dω - Te } / τj

pδ = ω - 1 (2.11)

where the state variables are

Ed' - direct axis component of voltage behind transient reactance

Eq' - quadrature axis component of voltage behind transient

reactance

ω - angular velocity of rotor

δ - rotor angle in radians

and

Te = Ed'Id + Eq'Iq – (xq' – xd' ) Id Iq

τj = 4πfH

xd - direct axis synchronous reactance

xq - quadrature axis synchronous reactance

xd' - direct axis transient reactance

xq' - quadrature axis transient reactance

τdo' - direct axis open circuit time constant

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τqo' - quadrature axis open circuit time constant

Te - electrical torque of synchronous machine

Tm - mechanical torque of synchronous machine

D - damping coefficient of synchronous machine

EFD - Equivalent stator emf corresponding to field voltage

Iq - quadrature axis armature current

Id - direct axis armature current

H - inertia constant of synchronous machine in sec

f - frequency in Hz

A multi-machine power system is shown in Figure 2.7 and the

network has n machines and r loads. The active source nodal voltages in

Figure 2.7 are taken as the terminal voltages i

V , i = 1.2….n instead of the

internal EMF`s. The loads are represented by constant impedances and the

network has n active sources representing the synchronous machines.

Figure 2.7 Multi-machine with constant impedance loads

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This network is reduced to a n-node network shown in

Figure 2.8 in which the current and voltage phases of each node are expressed

in terms of the respective machine reference frame.

Figure 2.8 Reduced n-port network

The objective here is to derive relations between vdi and

vqi, i=1,2,….n, and the state variables. This will be obtained in the form of a

relation between these voltages, the machine currents iqi and idi , and the

angles δi , i=1,2,….n. For convenience we will use a complex notation as

follows.

For a machine i we define the phasors i iV and I as

V V jV ; I I jIi qi di i qi di

= + = + (2.12)

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where

V v / 3 : V v / 3qi qi di di

= =

I i / 3 : I i / 3qi qi di di

= =

and where the axis qi is taken as the phasor reference in each case. Then we

define the complex vectors V and I by

q1 d1

q 2 d 22

qn dnn

V V jV

V jVVV

... .............

V jVV

+

+ = = +

(2.13)

1 q1 d1

q 2 d 22

qn dnn

I I jI

I jIII

... .............

I jII

+

+ = = +

(2.14)

The voltage iiV and thecurrent I are referred to the q and d axes of

machine i. In the other words the different voltages and currents are expressed

in terms of different reference. To obtain general network relationships, it is

desirable to express the various branch quantities to the same reference which

is given by equation (2.15):

The node voltages and currents are expressed as i iˆ ˆV and I ,

i = 1,2,…..n, and

ˆ ˆI = YV (2.15)

where Y is the short circuit admittance of the network.

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2.5.3 Converting to Common Reference Frame

Let us assume that we want to convert the phasor i qi diV V jV= + to

the common reference frame (moving at synchronous speed). Let the same

voltage, expressed in new notation, be i Qi DiV V jV= + as shown in Figure 2.9.

where, i qi diV V jV= + and i Qi Di

V V jV= + (2.16)

Figure 2.9 Two frames of reference for phasor quantities

From the Figure 2.9

Qi qi i di iV [V cos V cos ]= δ − δ (2.17)

Di di i qi iV [V cos V cos ]= δ − δ (2.18)

Qi Di qi i di i qi i di iV V (V cos V cos ) (V sin V cos )+ = δ − δ + δ + δ (2.19)

j i

i iV V e

δ= (2.20)

iδ i

δ

Qref

Dref

iV = ˆi

V VDi

VQi

Vqi

Vdi

qi di

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The equation (2.20) can be written in generalized matrix form as below

j 1e 0 0V jV V jV

Q1 D1 q1 d1j 2

e 0V jV V jVQ2 D2 q2 d2

0............. .............

0V jV V jVqnDn j nQn dn0 0 0 0 e

δ ⋅ ⋅+ + δ

⋅ ⋅ ⋅ + + = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

+ + δ

(2.21)

j 1e 0 0

j 2e 0

T 0

0

j n0 0 0 0 e

δ ⋅ ⋅

δ ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ δ

(2.22)

The equation (2.20) can be written as

V = TV (2.23)

Thus T is a transformation that transforms the d and q quantities of all

machines to the system frame, which a common frame is moving at

synchronous speed. The transformation matrix T contains elements only at the

leading diagonal and hence we can show that T is orthogonal, i.e. T-1

= T*.

Now the equation (2.23) can be rewritten as

* ˆV=T V (2.24)

Similarly for node current

I = TI (2.25)

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* ˆI = T I (2.26)

Substituting equation (2.25) and equation (2.23) in equation (2.15),

we get

I = MV (2.27)

where M = T-1

YT (2.28)

Linearizing equation (2.27) and making necessary substitutions

(Anderson and Fouad 2003), the following equations are obtained.

∆Iqi = Gii ∆Vqi – Bii ∆Vdi + n

[j 1

i

∑=

≠

Yii cos(θij - δij0 ) ∆Vqj]

- n

[j 1

i

∑=

≠

Yij sin (θij - δij0 ) ∆Vdj] + n

[j 1

i

∑=

≠

Yij {sin(θij - δij0) Vqj0 + cos(θij - δij0)Vdj0]∆δij

; i = 1, ...n (2.29)

∆Idi = Bii ∆Vqi + Gii ∆Vdi + n

[j 1

i

∑=

≠

Yij cos (θij - δij0 ) ∆Vdj]

+n

[j 1

i

∑=

≠

Yij sin (θij - δij0 ) ∆Vqj] + n

[j 1

i

∑=

≠

Yij {sin (θij - δij0 )Vdj0 - cos(θij - δij0)Vqj0]∆δij

; i = 1, ...n (2.30)

The state space model for linearized system is obtained by

linearizing the differential and algebraic equations at an operating point.

While doing this linearization process, additional terms involving terminal

voltage components (which are not state variables) remain in the differential

35

equations. To express the voltage components in terms of state variables, the

machine currents are also linearized and expressed in terms of state variables

and voltage components. Finally the current components are eliminated using

the interconnecting network algebraic equations. From the initial conditions,

Ed'i0, Eq'i0, Iqi0, Idi0, EFDi0 and δi0 are determined.

Linearizing equation (2.11) we get

p∆Ed'i = {- ∆Ed'i - (xqi - xq'i) ∆Iqi } / τqo'i ; i = 1,...n

p∆Eq' i = {∆EFDi - Eq'i + ( xdi - xd'i ) ∆Idi } / τdo'i ; i = 1,...n

p∆ωi = {∆Tmi –(Idi0 ∆Ed'i + Iqi0 ∆Eq'i + Ed'i0 ∆Idi +Eq'i0 ∆Iqi)-

Diωi } / τj ; i = 1,...n

p∆δi = ∆ωi ; i = 1, …n (2.31)

Substituting equations (2.29) and (2.30) in equation (2.31).

(replacing V by E'):

p∆Ed’i =

1

'qo iτ

{[(xqi - xq'i) Bii –1] ∆Ed'i

+ (xqi - xq'i) n

[k 1

i

∑=

≠

Yik {sin (θik - δik0 ) ∆Ed'k -(xqi - xq'i) Gii ∆Eq'i

- (xqi - xq'i) n

[k 1

i

∑=

≠

Yik cos (θik - δik0 ) ] ∆Eq'k

- (xqi - xq'i)n

[k 1

i

∑=

≠

Yik cos(θik-δik0)] Ed'k0 +Yik sin ((θik - δik0) Ed'k0] ∆δik}

i =1,2 ….n (2.32)

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p∆Eq' i = 1

'do i

τ {[(xdi – xd'i) Bii –1] ∆Eq'i

+ (xdi – xd’i) n

[k 1

i

∑=

≠

Yik {sin (θik - δik0 )] ∆Ed'k + (xdi – xd'i) Gii ∆Ed'i

+ (xdi – xd'i) n

[k 1

i

∑=

≠

Yik sin (θik - δik0 ) ] ∆Eq'k

- (xdi–xd'i)n

[k 1

i

∑=

≠

Yik cos(θik-δik0) Eq'k0-Yik sin((θik-δik0)Ed'k0]∆δik+ ∆EFDi)}

i =1,2 ….n (2.33)

p∆ωi = ji

1

τ {[∆Tmi - Di∆ωi -[Idi0 + GiiEd'i0 - Bii Eq'i0] ∆Ed'i

- [ Iqi0 + Bii Ed'i0 + Gii Eq'i0 ] ∆Eq'i

- n

[k 1

i

∑=

≠

Yik cos (θik - δik0 ) Ed'i0 - Yik sin (θik - δik0 ) Eq'i0 ] ∆Ed'k

- n

[k 1

i

∑=

≠

Yik sin (θik - δik0 ) Ed'i0 + Yik cos (θik - δik0 ) Eq'i0 ] ∆Eq'k

- n

[k 1

i

∑=

≠

Yik cos (θik - δik0 ) (-Eq'k0Ed'i0 +Ed'k0 Eq'i0) + Yik sin ((θik - δik0)

(-Ed'k0Ed'i0 +Eq'k0 Eq'i0)∆δik} i =1,2 ….n (2.34)

p∆δ1i = ω1 - ωi (2.35)

i = 2,3 ….n taking machine 1 as reference.

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The above set of equations (2.32 to 2.35) gives the state space

model of n-machine system.

2.6 EXCITER REPRESENTATION

The state space equation of the exciter can be derived from the

block diagram of the exciter shown in the Figure 2.3.

From the Figure 2.3, we get

AFD t

A

KE V

1 ST

−∆ = ∆

+ (2.36)

For n, number of exciters, the state equations is as follows:

( )i

Aifdi Ref i fdi

Ai Ai

-K 1p∆E = -∆V +∆V - ∆E ;

T T i=1,…n (2.37)

Now the state vector of the n machine state model including exciter

equation is as follows.

XT

i = [∆Ed'i ∆Eq' i ∆ω i ∆δ i ∆EFD i] ; i=1,…n (2.38)

2.7 CONVENTIONAL POWER SYSTEM STABILIZER

REPRESENTATION

The Conventional Power System Stabilizer (PSS) adds damping to

the generator rotor oscillations by controlling its excitation using auxiliary

stabilizing signals. To provide damping, the stabilizer must produce a

component of electrical torque in phase with the rotor speed deviations.

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The important blocks in a power system stabilizer are:

• Washout circuit.

• Phase compensator.

• Stabilizer gain.

The state space equation for the power system stabilizer (PSS) can be

obtained from the block diagram shown in Figure 2.10.

Figure 2.10 Conventional Power System Stabilizer Structure (CPSS)

From the wash out block, we get

w2 PSS

w

sTV (K )

1 sT∆ = ∆ω

+ (2.39)

p∆V2i = KPSSi

p∆ωi – (1/Twi) ∆V2i

; i = 1,...n (2.40)

From the phase compensator block we get

1

s 2

2

1 sTV V

1 sT

+∆ = ∆

+ (2.41)

From equation (2.41) we get

p∆Vsi = (T1i

/T2i) p∆V2i

+(1/T2i) ∆V2i

–(1/T2i) ∆Vsi

; i = 1,...n (2.42)

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The state vector of the complete system after the inclusion of power

system stabilizer is as follows:

xT

i = [∆E'di ∆E'qi ∆ωi ∆δi ∆EFDi ∆V2i ∆Vsi] ; i=1,…n (2.43)

2.8 FUZZY LOGIC BASED POWER SYSTEM STABILIZER

(FPSS)

Figure 2.11 shows the schematic block diagram of the system with

FPSS.

Figure 2.11 Structure of the Power system with FPSS

Speed Deviation of the synchronous machine ( )∆ω and its deviation

( )•

∆ω are chosen as inputs to the FPSS. Simulation of the sample SMIB

system without PSS is carried out for several operating conditions and

different disturbances and the inputs are normalized using their estimated

peak values. Seven labels are taken for both the inputs and output. The labels

are LP (large positive), MP (medium positive), SP (small positive), VS (very

small), SN (small negative), MN (medium negative) and LN (Large negative).

Linear triangular membership function is used in the design of FPSS. In our

design of FPSS, the fuzzy sets with triangular membership function for ∆ω

are shown in Figure 2.12. The membership function for •

∆ω and Vs are

similar to the above Figure 2.12.

FPSS Power System

Generator and

Exciter

dt

d

∆ω ∆ω

∆ω +∆Vref

∆Vt

∆Vs

+

-

40

Figure 2.12 Triangular membership function of ∆ω

Table 2.1 shows the rules of fuzzy logic based PSS (Lakshmi and

Khan 2000).

Table 2.1 Rule Table of fuzzy logic PSS

∆ω

ω•

∆

LP MP SP VS SN MN LN

LP LP LP LP LP MP SP VS

MP LP LP MP MP SP VS SN

SP LP MP SP SP VS SN MN

VS MP MP SP VS SN MN N

SN MP SP VS SN SN MN LN

MN SP VS SN MN MN LN LN

LN VS SN MN LN LN LN LN

LN

-1 -0.66 -0.33 0 0.33 0.66 1

SN VS SP MP LP MN

41

2.9 CONCLUSION

Mathematical model of SMIB system for dynamic stability analysis

is presented in this chapter. Various state variables with PSS, system matrix

including static exciter and CPSS are included in this chapter. Block diagram

of simplified linear model of SMIB including exciter and CPSS is also neatly

presented in this chapter. Non –linear mathematical model representing the

dynamics of the multi machine power system combining the synchronous

machine model, excitation system (IEEE Type ST 1A), with conventional

power system stabilizers are described in this chapter. The fuzzy logic based

PSS model is also described.