powersystem stability 01

7/31/2019 PowerSystem Stability 01

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Power System Stability

The Swing EquationThe relation between the mechanical angular velocity, m(t) (in rad/s), and the mechanical rotor

angular position, m(t) (in rad), with respect to a stationary axis is given by:

rad/s)(

)(dt

tdt mm

=

The equation governing rotor motion of a synchronous machine is based on the elementary

principle in dynamics which states that accelerating torque is the product of the moment of inertia

of the rotor times its angular acceleration. In the MKS (meter-kilogram-second) system of units this

equation can be written for the synchronous generator in the form

mN)()(

)(2

2

==== aemmm

m TTTdt

tdJ

dt

tdJtJ

Where the symbols have the following meaning

J: The total moment of inertia of the rotor mass, in (Kg-m2).

m: The mechanical rotor angular acceleration (in rad/s2).

Tm: The mechanical or shaft torque supplied by the prime mover less retarding torque due to

rotational loss, in (N-m).Te: The electrical torque that accounts for the total three-phase electrical power output of the

generator, plus electrical losses, in (N-m).

Ta: The net accelerating torque, in (N-m)

t: Time in seconds (s)

Tm and Te are positive for generator operation. In steady-state, Tm equals Te, the accelerating torque,

Ta, is zero as well as the rotor acceleration, m, is zero.

When Tm is greater than Te, Ta is positive and m is therefore positive resulting in increasing rotor

speed.

When Tm is less than Te, Ta is negative and m is therefore negative resulting in decreasing rotor

speed.

It is convenient to measure the rotor angular position with respect to a synchronously rotating

reference axis instead of a stationary reference axis. Accordingly we define:

)()( , tt msynmm +=

dt

td

dt

td

dt

tdt m

msynmmm

)()]([)()(

,=

+

==

2

2

2

2 )()(

dt

td

dt

td mm=

m,syn: synchronous angular velocity of the rotor (in rad/s)

m: The angular position with respect to a synchronously rotating reference axis, (in rad)

Thus the previous equation becomes

mN)(

2

2

== aemm TTT

dt

tdJ

It is also convenient to work with power rather than torque, and to work in per-unit rather than in

actual units. Accordingly, the previous equation is multiplied by m(t) and divided by Srated, the

three-phase voltampere rating of the generator:


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rated

am

rated

em

rated

mmm

rated

m

S

Tt

S

Tt

S

Tt

dt

td

S

tJ )()()()()(2

2

==

Defining per-unit rotor angular velocity, mechanical power, electrical power and accelerating power

synm

mpu

tt

,

)()(

= ,

rated

mmpum

S

TtP

)(,

= ,

rated

empue

S

TtP

)(,

= ,

rated

ampua

S

TtP

)(,

=

puapuepumm

pu

rated

synmPPP

dt

tdt

S

J,,,2

2, )(

)( ==

Where,

Pm,pu: The mechanical power supplied by the prime mover minus mechanical losses, per-unit

Pe,pu: The electrical power output plus electrical losses, per-unit

Pa,pu: The accelerating power, per-unit

Finally, it is convenient to work with a normalized inertia constant, called theH constant, which is

defined as

ratedS

KEH ==

ratingvoltampereGenerator

speedssynchronouatenergykineticStored

secondunitperorJ/VA)2/1(

2

,=

rated

synm

S

JH

The Hconstant has the advantage that is falls within a fairly narrow range, normally between 1 and

10 pu s, whereas Jvaries widely depending on the generator unit size and type.

Using H constant, the previous equation becomes

puapuepumm

pu

synm

PPPdt

tdt

H,,,2

2

,

)()(

2==

For a synchronous generator with P poles, the electrical angular acceleration , electrical radianfrequency , power angle , and the synchronous electrical radian frequency syn are

)(2

)( tP

t m = , )(2

)( tP

t m = , )(2

)( tP

t m = and )(2

)( , tP

t synmsyn =

The per-unit electrical frequency is

synsynm

mpu

ttt

)()()(

,

==

Thus, the previous equation becomes

puapuepumpu

syn

PPP

dt

tdt

H,,,2

2)(

)(2

==

Frequently the previous equation is modified to also including a term that represents a damping

torque anytime the generator deviates from its synchronous speed, with its value proportional to the

speed deviation

pua

syn

puepumpu

syn

Pdt

tdDPP

dt

tdt

H,,,2

2)()(

)(2

==

Where D is either zero or a relatively small positive number with typical values between 0 and 2.

The units ofD are per-unit power divided by per-unit speed deviation.


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The previous equation is called the per-unit swing equation, is the fundamental equation that

determines rotor dynamics in transient stability studies.

In practice the rotor speed does not vary significantly from synchronous speed during transients i.e.

pu(t) = 1.0.

The previous equation can be written as

syntdt

td

= )(

)(

pua

syn

puepumpu

syn

Pdt

tdDPP

dt

tdt

H,,,

)()()(

2==

The previous two equations are two first-order differential equations.

When the swing equation is solved, the expression ofis as a function of time. Thus, the curve of

as function of time (t) is calledswing curve.

powersystem stability 01

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