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Complete set of electrical equations in per unit

In the following equations, two q-axis amortisseur circuits are considered, and the subscript

1

and 2 are used to identify them. Only one d-axis amortisseur circuit is considered, and it isindentified by the subscript 1.Per unit stator voltage equations:

= 8.1 = + 8.2 = 8.3

Per unit rotor voltage equations:

= + 8.40 = + 8.50 = + 8.60 = + 8.7

Per unit stator flux linkage equations:

= + + + 8.8 = + + + 8.9 = 8.10

Per unit rotor flux linkage equations:

= + 8.11

= + 8.12 = + 8.13 = + 8.14

Per unit air-gap torque:

= 8.15


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In the above equations, we have assumed that the per unit mutual inductance = . Thisimplies that the stator and rotor circuits in the q-axis all link a single mutual flux represented by

. This is acceptable because the rotor circuits represent the overall rotor body effects, and theactual winding with physically measurable voltages and current do not exist.For power system stability analysis, the machine equations are normally solved with all

quantities expressed in per unit, with the exception of time. Usually time is expressed inseconds, in which case the per unit p in above equations is replaced by 1 !" .Synchronous Machine Representation in Stability Studies

Simplifications essentials for large-scale studies

For stability analysis of large system, it is necessary to neglect the following from Equations(8.1) and (8.2) for stator voltage:

The transformer voltage terms, #$ . The effect of speed variations.

The reasons for and the effects of these simplifications are discussed below.

Neglect of Stator &' termTransformer voltages are lesser than speed voltages, and only due to transformer voltage terms

the voltage equation (8.1) and (8.2) are differential equations. If we neglect transformer voltage

terms, then stator voltage equation become algebraic equation.

Stator term represents stator circuit transients. Stator is usually connected to transmissionline and transmission line has very low time constants i.e. associated transients quickly dies out,

so in stability studies, so we generally ignore transmission line transients. On the other hand, if

we consider stator transients and ignore transmission line transients than it is inconsistent,

therefore we also neglect the stator transient #$ .Neglecting the Effect of Speed Variations on Stator Voltages

Another simplifying assumption normally made is that the per unit value of is equal to 1.0 inthe stator voltage equations. This is not the same as saying that speed is constant; it assumes thatspeed changes are small and do not have a significant effect on the voltage.

The assumption of per unit = 1.0 (i.e., = ) *# ) in the stator voltage equations doesnot contribute to computational simplicity in itself. The primary reason for making this

assumption is that it counterbalances the effect of neglecting , terms so far as the lowfrequency rotor oscillations are concerned.


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With per unit = 1.0, the stator voltage equations reduced to

= 8.16

= 8.17Relationship between per unit / and /The terminal electric power in per unit is given by

= + Substituting for and from Equations (8.16) and (8.17) gives

= + = + 8.18=

The air-gap power, measured behind , is given by = + 8.19= 8.20The per unit air gap power so computed is in fact the power at synchronous speed and is equal

to the per unit air-gap torque . Normally, = , however, the assumption of = 1.0 puin the stator voltage equation is also reflected in the torque equation, making = . This fact isoften overlooked.

Simplified Model with Amortisseurs Neglected

The first order of simplification to the synchronous machine model is to neglect the

ammortisseur effects. This minimizes data requirements since the machine parameters related to

the amortisseurs are often not readily available. In addition, it may contribute to reduction in

computational effort by reducing the order of the model and allow large integration steps in time

domain simulations.

With the amortisseurs neglected, the stator voltage Equations (8.16) and (8.17) are unchanged.

The remaining equations (8.4) to (8.15) simplify as follow.

Flux linkages:

= + 8.21


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= 8.22

= + 8.23

Rotor voltage:

= + = 8.24

Equation (8.24) is now the only differential equation associated with the electrical characteristics

of the machine. In the above equation all quantities, including time, are in per unit.

Alternative form of machine equations

In the literature on synchronous machines, Equations (8.21) to (8.24) are often written in terms

of the following variables:

56 = = Voltage proportional to 57 = :;;: = Voltage proportional to 5 = :;:= :;;:

Then


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7 =

Substituting in Equation (8.26) gives

57 = 56 7 8.27Multiplying Equation (8.24) by

:;;: throughout, we have ? :;;: @ = :


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=

= J + J

= J + 56 Therefore, 56 = + J + Multiplying by j, we have

K56 = K + KJ + KIn terms of phasor notation,

56 = + KJ + 8.29From equation (8.27) withJ = 7 ,

57 = + J + J + J= + J +

Multiplying by j gives

K57

= K +KJ + K

In terms of phasor notation,

57 = +KJ + 8.30We see that phasorsPQand PR7 both lie along the q-axis and PR7 also lies along q-axis.w.k.t. the following Equation and substituting 56 forJ, we get

56 = 5 +KJ J 8.31Figure (8.1) shows the phasor diagram representingPR7 , 56 and 5 given by equations (8.29),(8.30) and (8.31). 5 = + K = +

5 7 = 5 + + KJ7 SPR7 =q-axis component of5 7

= + + KJ7


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PQ = 5 +KJ J

Figure (8.1) Synchronous machine phasor diagram in terms ofUV7 , UV and UWConstant flux linkage model (Classical model):

For studies in which the period of analysis is small as compared to the machine model isoften simplified by assuming

57(or

) constant throughout the study period. The assumption

eliminates the only differential equation associated with the electrical characteristics of the

machine.

A further approximation to simplify the machine model is to ignore transient saliency by

assumingJ7 = J7 , and to assume that the flux linkage also remains constant. With theseassumptions, the voltage behind the transient impedance + KJ7 has a constant magnitude.The per unit flux linkages identified in the d-axis are given by

= + 8.32

= 8.33 = + 8.34From Equation (8.34)

= ;:Y:;: 8.35


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Figure (8.2) Thed- and q-axis equivalent circuits with one rotor circuit in each axis

Substitute in Eq. (8.32) gives, = + :;: = 7 ? + ;:;: @

Where, 7 = :" > ;:" = Similarly, for the q-axis

= 7 ? + Z[Z[ @Where, 7 = From Equation (8.16), the d-axis stator voltage is given by

= = +

Substituting for from equation (8.35) gives = + 7 ? + Z[Z[ @= + + 7 7 ?Z[Z[ @ 8.36= + J + 57


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Where, 57 = ?Z[Z[ @ 8.37Similarly, the q-axis stator voltage is given by

= + J + 57 8.38Where, 57 = ?;:;: @ 8.39With transient saliency neglected J7 = J7 the stator terminal voltage is

+ K = 5 +K5 + K + J K= 5 +K5 + K KJ + KUsing phasor notation, we have

5] = 5P +KJSP 8.40Where, 5P = 5 +K5

= ? Z[Z[ + K ;:;: @The corresponding equivalent is shown in figure (8.3)

Figure (8.3) Simplified transient model

With rotor flux linkages and constant, 57 and 57 are constant. Therefore the magnitudeof5 is constant. As the rotor speed changes, the d- and q-axes move with respect to any generalreference coordinate system whose R-I axes rotate at synchronous speed, as shown in figure ( ).

Hence, the components 5


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Figure (8.4) TheR-Iandd-q coordinate system

This model offers considerable computational simplicity; it allows the transient electrical

performance of the machine to be represented by a simple voltage source of fixed magnitude

behind an effective reactance. It is commonly referred to as the classical model, since it was used

extensively in early stability studies.

8. SMALL SIGNAL STABILITY OF A SINGLE MACHINE INFINITE BUS SYSTEM:Small signal stability is the ability of the power system to maintain synchronism when subjected

to small disturbance. The small perturbation continuously occurs in any power system n due to

small changes in load and generation. For analyzing the small signal stability of any system the

system model can be linearized around an operating point i.e the disturbances are considered tobe so small or incremental in nature so that we can develop a linear model of the system around

an operating point. Once we develop the linear model of the system we can understand behavior

of the system under small perturbation, various parameter of the system which affects the

stability of the system. Further the moment we have linear model of the system we can apply the

linear control system theory for designing the controllers. The controllers particularly are

excitation control system (i.e voltage regulator) and power system stabilizer. But as we can see

that in any power system the actual system is somewhat complex and it is not as simple as a

machine connected to an infinite bus. In multi machine system there are different modes of

oscillations such as control mode of oscillation, local modes of oscillations, inter-area modes of

oscillations. Primary requirement of the system to have stability is that system should have thepositive synchronizing torque, positive damping torque. The system stability gets affected if any

of these two torques becomes negative.

So to start with we should first study a simple model considering the constant flux linkages in

the field winding. Next we will include field dynamics i.e changes of flux linkage in the field

winding. It is here first considered that the field winding is manually controlled. Then we will

include the automatic field voltage regulator (AVR) and we will study the parameter of the

excitation system and gain setting of the automatic voltage regulator on stability of the system.

Then next step will be we include the auxiliary control of the controller that is the power system


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stabilizer (PSS). Then we will extend the model with amortisseurs. So that the step by step study

will give the complete insight into the small signal stability of the system.

8.1. Generator represented by the classical model:With the generator represented by the classical model and all resistances neglected, the

system representation is as shown in fig. 8.1.

Fig.8.1

Here 57is the voltage behind J7 . Its magnitude is remains constant at the pre-disturbance value.Let the be the angle by which 57 leads the infinite bus voltage 5^.As the rotor oscillates duringa disturbance, changes.

With

57 as reference phasor,

S_ = `Ca)bY`caYdefg = `CY`chHidYeijkdefg 8.41Where 57] = 5] + KJ7 S

J = J7 + J`The complex power behindJ7 is given by

l7 = + Km7 = 5S= `C`c ijkdfg + K `C`CY`c hHidfg 8.42

With stator resistance neglected, the air gap power () is equal the terminal power. In perunit, the air gap torque is equal to the air gap power. Hence,

= = ``cfg sn 8.43Linearizing about an initial operating condition represented by = ) yields


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= uAvud = ``cfg os) 8.44The equations of motion in per unit are

= y z {| 8.45 = 8.46

Where is the per unit speed deviation, is rotor angle in electrical radians, }) is the baserotor electrical speed in radians per second, and p is the differential operator " with time t inseconds.

Linearizing Equation (8.45) and substituting for given by Equation (8.44) we obtain = y ~z {! {| 8.47Where {!is given by

{! = ``cfg os) 8.48Linearizing Equation (8.46) we have

= )8.49

Writing Equations 8.47&8.49 in the vector matrix form, we obtain

= y y) 0

+

y0 z 8.50

This is of the form = + . The elements of the state matrix A are seen to bedependent on the system parameters {|, , JA and the initial operating condition represented bythe values of

5and

). The block diagram representation shown in fig.(8.5) can be used to

describe the small signal performance.


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Fig.8.6. Block diagram of a single machine infinite bus system

with classical generator model

Where

{! =Synchronizing torque coefficient in pu torque/rad{| =Damping torque coefficient in pu torque /pu speed deviation =Inertia constant in MW.s/MVA =Speed deviation in pu = ( ))/) =Rotor angle deviation in elec. Rad

=Laplace operator

) =Rated speed in elec. Rad/s = 2)From the block diagram of fig.8.6, we have

= b! y! {! {| + z= b! y! {! {| db + z


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Rearranging, we get

+ y + y ) = by z

Therefore, the characteristic equation is given by

+ y + by = 0 8.51This is of the general form

+2 + = 0Thus the undamped natural frequency is

= {! by *# And the damped ratio is

= y= yb

As the synchronizing torque coefficient

{!increases, the natural frequency increases and the

damping ratio decreases. An increase in damping torque coefficient {| increases the dampingratio, whereas an increase in inertia constant decreases both and .Effect of Synchronous Machine Field Circuit Dynamics

In this case we consider the system performance including the effect of field flux variations. The

amortisseur effects will be neglected and the field voltage will be assumed constant (manual

excitation control).

Here we will develop the state-space model of the system by first reducing the synchronous

machine equations to an appropriate form and then combining them with the network equations.

Synchronous machine equations:

As in the case of classical generator model, the acceleration equations are

p = T T 8.52p = ) 8.53

The field circuit dynamic equation is given by


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pE = )E EE

= b E )EE8.54

In the above three equations, we can say the following statements:

(i) The left hand side terms ie, , E are the state variables(ii) On the right hand side the terms T E represent input to the system(iii) On the right hand side the terms T E are neither state variables nor input variables.Hence in order to develop the complete system equations in the state-space form, we need to

express

T Ein terms of the state variables as determined by the machine flux linkage

equations and network equations.

The stator and rotor flux linkages are given by

E = LE + LEiE + E= LE + E 8.55R = LR + LRiR

= LR + R8.56

E = LEiE + E + LEE= E + LEE 8.57In the above equations ER are the air-gap (mutual) flux linkages, and LEi &LRi aresaturated values of mutual inductances.

From equation 8.57, the field current may be expressed as

E =Y

8.58

The d-axis mutual flux linkage can be written in terms ofE&E as follows:E = LEiE + LEiE = LEiE + E E= L7Ei E + 8.59


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Where

L7Ei = Z> ZSince there are no rotor circuits considered in the q-axis, the mutual flux linkage is given by

R = LRiR 8.60The air-gap toque is given by

T = ER RE

= ER RE 8.61With p terms and speed variations neglected, the stator voltage equations are

E = E R= E + LR R 8.62R = R + E= R LE E 8.63As a first step, we have expressed ET in terms ofE, E, R, E n R.

Network equations:


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Figure 8.7

Referring to fig.8.5, the machine terminal and infinite bus voltages in terms of the d and q

components are

= E + R 8.64 = E + R 8.65

The network constraint equation for the system of fig.8.6 is

Figure (8.8) SMIB system

= + + XIE + R = E + R + + XE + R 8.66

Resolving into d and q components gives

E = E XR + E 8.67R = R + XE + R 8.68

Where

E = sn 8.69

R = os8.70

Using equations 8.58 & 8.59 to eliminate E R in equations 8.55 &8.56 and using theexpressions for ER given by equations 8.61 & 8.62, we obtain the following expressionsfor En R in terms of state variablesE n :

E = ? @YhHiYijk 8.71R = ? @YhHi>ijk 8.72


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Where

= +

XR = X + LRi + L = X + XRiXE = X + L7Ei + L = X + X7EiD = + XRXE

Linearized system equation

For small disturbance or perturbed values

Since #$ is a function of state variable #$ so after incremental change = + 8.73 = $ + $ 8.74

Where

, , $#$$ are function of synchronous machine operating condition and operatingparameters.

= `cfg[ ijkdbY;: 8.75$ = ;:

Now linearizing

#$then we get

= ! ? + ;:;: @= ? ;: @ ! ! 8.76 = != $! $!


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Now incremental change in field current

=

;:Y

:;: = ;: ?1 :;: + !@ + ;: ! 8.77For torque

= = ! !

Now linearizing this equation then we get

= ) ) ) )Substituting for , ,#$ from above equations then we get

= { + { 8.78Where

{ = $) + !) ) + !) 8.79{ = $) + !) ) + !) + :;: ) 8.80{=Synchronizing torque component.

{ =Torque component due to variation in field flux componentBy linearizing equations (8.52) to (8.54) and substitute the expression for and given byequation (8.77) and (8.78) we obtain the system equations in the desired final form:

= # # ## 0 00 # #

+ 00 00

z5 8.81

Where

# = y # = Zy

#

=

y #

= )

= 2)


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# = b


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This is the model for synchronous machine filed winding dynamics.

Effect of field flux linkage variation on system stability

From above block-diagram; with constant field voltage (5 = 0), the field flux variation arecaused only by feedback of through coefficient {.This represents the demagnetizing effect of the armature reaction.

The change in air-gap torque due field flux variation caused by rotor angle changes is given by

Av

d = >!A

{,{,#${ usually positive.Now we examine frequency of oscillation of rotor.

In steady state and at very low oscillation frequency.

= K 0 = {{{

From above equation we conclude that (air gap torque); the field flux variation due to feedback (due to armature reaction) introduction a negative synchronizing torque component.The system is unstable when

{{{ {Steady state stability reached when

= { {{{!

When field flux linkages are constant i.e. = 0 then synchronizing torque coefficient isonly{.Let rotor is oscillate with some frequency

Avd = >!A

Put = K and rationalize it then we get


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;: = d>A!z

+ eAd>Azz

We put in above equation

= b! Change in electrical torque or air gap torque due to change in flux linkages only

;: = >A + A>A ) At oscillating frequency much higher than 1 " :

eA KThus, component of air gap torque due to to 90) ahead ofOr in phase with .Hence, results in a positive damping torque

r

Ts

Te Te

K2fd

Figure (8.10)

result in positive damping torque and a negative synchronizing torque component .The neteffect is to reduce slightly the synchronizing torque component and increases the damping torque

component.

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