Power System Stability

1. BEF43303
POWER SYSTEM ANALYSIS AND PROTECTION
POWER SYSTEM STABILITY
WEEK 7

2. CONTENT
7.1 Introduction
7.2 Swing Equation
7.3 Synchronous Machine Models for Stability Studies
7.4 Transient Stability – Equal Area Criterion

3. 7.1 INTRODUCTION
• Stability is the tendency of a power system to develop restoring forces
equal to or greater than the disturbing forces to maintain the state of
equilibrium.
• The stability problem is concerned with the behavior of the
synchronous machines after a disturbance.
• Generally divided into two major categories - steady-state stability
and transient stability.
• Steady-state stability refers to the ability of the power system to
regain synchronism after small and slow disturbances such as gradual
power changes.

4. 7.1 INTRODUCTION
• Transient stability studies deal with the effects of large, sudden
disturbances (fault, the sudden outage of a line, sudden increment or
decrement of loads).
• Needed to ensure that the system can withstand the transient
condition following a major disturbance.
• Conducted when new generating and transmitting facilities are
planned.
• Helpful in determining such things as the nature of the relaying
system needed, critical clearing time of circuit breakers, voltage level
of, and transfer capability between systems.

5. 7.2 SWING EQUATION
• Under normal operating conditions the relative position of the rotor
axis and the resultant magnetic field axis is fixed.
• The angle between the two is known as the power angle.
• During any disturbance, rotor will decelerate or accelerate with
respect to the synchronously rotating air gap mmf and a relative
motion begins.
• The equation describing this relative motion is known as the swing
equation.

6. 7.2 SWING EQUATION
• Consider a synchronous generator developing an electromagnetic
torque 𝑇𝑇𝑒𝑒 and running at the synchronous speed 𝜔𝜔𝑠𝑠𝑠𝑠.
• If 𝑇𝑇𝑚𝑚 is the driving mechanical torque then under steady-state
operation with losses neglected we have:
𝑇𝑇𝑚𝑚 = 𝑇𝑇𝑒𝑒
• A departure from steady slate due to a disturbance results in an
accelerating or decelerating torque on the rotor.
𝑇𝑇𝑎𝑎 = 𝑇𝑇𝑚𝑚 − 𝑇𝑇𝑒𝑒

7. 7.2 SWING EQUATION
• If 𝐽𝐽 is the combined moment of inertia of the prime mover and
generator, neglecting frictional and damping torques, from laws of
rotation we have:
𝐽𝐽
𝑑𝑑2𝜃𝜃𝑚𝑚
𝑑𝑑𝑡𝑡2
= 𝑇𝑇𝑎𝑎 = 𝑇𝑇𝑚𝑚 − 𝑇𝑇𝑒𝑒
• where 𝜃𝜃𝑚𝑚 is the angular displacement of the rotor with respect to the
stationary reference axis on the stator.
• The swing equation is rewritten as:
𝐻𝐻
180𝑓𝑓0
𝑑𝑑2
𝛿𝛿
𝑑𝑑𝑡𝑡2
= 𝑃𝑃𝑚𝑚 − 𝑃𝑃𝑒𝑒

8. 7.3 SYNCHRONOUS MACHINE MODELS FOR
STABILITY STUDIES
• The cylindrical rotor machine was modeled with a constant voltage
source behind proper reactances, which may be 𝑋𝑋𝑋𝑑𝑑, 𝑋𝑋𝑋𝑑𝑑 or 𝑋𝑋𝑑𝑑.
• The simplest model for stability analysis is the classical model where
saliency is ignored, and the machine is represented by a constant
voltage 𝐸𝐸𝐸 behind the direct axis transient reactance 𝑋𝑋𝑋𝑑𝑑.
• Consider a generator connected to a major substation of a very large
system through a transmission line as shown in Figure 1.

9. 7.3 SYNCHRONOUS MACHINE MODELS FOR
STABILITY STUDIES
Figure 1

10. 7.3 SYNCHRONOUS MACHINE MODELS FOR
STABILITY STUDIES
• The bus voltage and frequency is assumed to remain constant.
• This is commonly referred to as an infinite bus.
• It does not change regardless of the power supplied or consumed by
any device connected to it.
• The generator is represented by a constant voltage behind the direct
axis transient reactance 𝑋𝑋𝑋𝑑𝑑.

11. 7.3 SYNCHRONOUS MACHINE MODELS FOR
STABILITY STUDIES
• The node representing the generator terminal voltage 𝑉𝑉
𝑔𝑔 can be
eliminated by converting the Y-connected impedances to an
equivalent ∆ with admittances as shown in Figure 2.
Figure 2

12. 7.3 SYNCHRONOUS MACHINE MODELS FOR
STABILITY STUDIES
• Writing the nodal equations, we have:
𝐼𝐼1 = 𝑦𝑦10 + 𝑦𝑦12 𝐸𝐸′
− 𝑦𝑦12𝑉𝑉
𝐼𝐼2 = −𝑦𝑦12𝐸𝐸′
+ 𝑦𝑦20 + 𝑦𝑦12 𝑉𝑉
• Rewritten the previous equation in matrix form:
𝐼𝐼1
𝐼𝐼2
=
𝑌𝑌11 𝑌𝑌12
𝑌𝑌21 𝑌𝑌22
𝐸𝐸𝐸
𝑉𝑉
• The real power at node 1 is given by:
𝑃𝑃𝑒𝑒 = 𝐸𝐸𝐸 2 𝑌𝑌11 cos 𝜃𝜃11 + 𝐸𝐸𝐸 𝑉𝑉 𝑌𝑌12 cos 𝛿𝛿 − 𝜃𝜃12

13. 7.3 SYNCHRONOUS MACHINE MODELS FOR
STABILITY STUDIES
• If all resistances are neglected,
the previous equation becomes:
𝑃𝑃𝑒𝑒 =
𝐸𝐸𝐸 𝑉𝑉
𝑋𝑋12
sin 𝛿𝛿 = 𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 sin 𝛿𝛿
• The curve 𝑃𝑃𝑒𝑒 versus 𝛿𝛿 is known
as power angle curve shown in
the following figure:
Figure 3

14. 7.3 SYNCHRONOUS MACHINE MODELS FOR
STABILITY STUDIES
Example 1
Consider a synchronous machine characterized by the following
parameters:
𝑋𝑋𝑑𝑑 = 1.0, 𝑋𝑋𝑞𝑞 = 0.6, 𝑋𝑋𝑋𝑑𝑑 = 0.3 per unit
And negligible armature resistance. The machine is connected directly to
an infinite bus of voltage 1.0 per unit. The generator is delivering a real
power of 0.5 per unit at 0.8 power factor lagging. Determine the voltage
behind transient reactance and the transient power-angle equation if the
saliency effect is neglected.

15. 7.4 TRANSIENT STABILITY – EQUAL AREA
CRITERION
• The determination whether or not synchronism is maintained after
the machine has been subjected to severe disturbance.
• A method known as the equal-area criterion is used for a prediction of
stability.
• It is based on the graphical interpretation of the energy stored in the
rotating mass to determine if the machine maintains its stability after
disturbance.
• Only applicable to a single machine infinite bus system (SMIB).

16. 7.4 TRANSIENT STABILITY – EQUAL AREA
CRITERION
• Consider a synchronous machine connected to an infinite bus. The
swing equation with the damping neglected is given by:
𝐻𝐻
𝜋𝜋𝑓𝑓0
𝑑𝑑2𝛿𝛿
𝑑𝑑𝑡𝑡2
= 𝑃𝑃𝑚𝑚 − 𝑃𝑃𝑒𝑒 = 𝑃𝑃𝑎𝑎
• The previous equation can be rewritten as:
𝑑𝑑𝛿𝛿
𝑑𝑑𝑑𝑑
=
2𝜋𝜋𝑓𝑓0
𝐻𝐻
�
𝛿𝛿0
𝛿𝛿
𝑃𝑃𝑚𝑚 − 𝑃𝑃𝑒𝑒 𝑑𝑑𝛿𝛿

17. 7.4 TRANSIENT STABILITY – EQUAL AREA
CRITERION
• The equation gives the relative speed of the machine with respect to
the synchronously revolving reference frame.
• For stability, this speed must become zero after the disturbance.
• Thus, for the stability criterion:
�
𝛿𝛿0
𝛿𝛿
𝑃𝑃𝑚𝑚 − 𝑃𝑃𝑒𝑒 𝑑𝑑𝛿𝛿 = 0
• From the equation, the machine operates at the equilibrium point 𝛿𝛿0,
corresponding to the mechanical power input 𝑃𝑃𝑚𝑚𝑚 = 𝑃𝑃𝑒𝑒𝑒.
• This condition is illustrated in Figure 4.

18. 7.4 TRANSIENT STABILITY – EQUAL AREA
CRITERION
• Consider a sudden increase in
input power represented by the
horizontal line 𝑃𝑃𝑚𝑚𝑚.
• Since 𝑃𝑃𝑚𝑚𝑚 > 𝑃𝑃𝑒𝑒𝑒, the
accelerating power on the rotor
is positive and the 𝛿𝛿 increases.
• The excess energy stored is:
�
𝛿𝛿0
𝛿𝛿1
𝑃𝑃𝑚𝑚1 − 𝑃𝑃𝑒𝑒 𝑑𝑑𝛿𝛿 = 𝑎𝑎𝑎𝑎𝑎𝑎 = 𝐴𝐴1 Figure 4

19. 7.4 TRANSIENT STABILITY – EQUAL AREA
CRITERION
• With increase in 𝛿𝛿, the electrical power increases.
• The electrical power matches the new input power 𝑃𝑃𝑚𝑚𝑚 when 𝛿𝛿 = 𝛿𝛿1.
• Even though the accelerating power is zero at this point, the rotor is
running above synchronous speed.
• Hence, 𝛿𝛿 and 𝑃𝑃𝑒𝑒 will continue to increase.
• Now 𝑃𝑃𝑚𝑚 < 𝑃𝑃𝑒𝑒, causing the rotor to decelerate toward synchronous
speed until 𝛿𝛿 = 𝛿𝛿max.
• The rotor must swing past point 𝑏𝑏 until an equal amount of energy is
given up by the rotating masses.

20. 7.4 TRANSIENT STABILITY – EQUAL AREA
CRITERION
• The energy given up by the rotor as it decelerates back to
synchronous speed is:
�
𝛿𝛿1
𝛿𝛿𝑚𝑚𝑚𝑚𝑚𝑚
𝑃𝑃𝑚𝑚𝑚 − 𝑃𝑃𝑒𝑒 𝑑𝑑𝛿𝛿 = 𝑏𝑏𝑏𝑏𝑏𝑏 = 𝐴𝐴2
• The result is that the rotor swings to point b and the angle 𝛿𝛿max, at
which point:
𝐴𝐴1 = 𝐴𝐴2
• This is known as the equal-area criterion.