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Power Swings and Distance Relaying
In this lecture, we will introduce the concept of power swings. It will be shown that the
post fault power swings may encroach the relay characteristics. This can lead to nuisance
tripping of distance relays which can sacrifice the system security.
Analysis of Two Area System
Power swings refer to oscillation in active and reactive power flows on a transmission
line consequent to a large disturbance like a fault. The oscillation in the apparent power
and bus voltages is seen by the relay as an impedance swing on the R – X plane. If the
impedance trajectory enters a relay zone and stays there for sufficiently long time, then
the relay will issue a trip decision on power swing. Tripping on power swings is not
desirable. We now investigate this phenomenon and then discuss remedial measures.
Let us consider a simple two machines system connected by a transmission line of
impedance ZL as shown in Figure 23.1(a) ES and ER are the generator voltages at two
ends and we assume that the system is purely reactive. The voltage ES leads ER by an
angle  so that power flows from A to B during steady state. The relay under
consideration is located at bus A end. The power angle curve is shown in Figure 23.1(b).
The system is operating at initial steady operating point A with P mo as output power and
 0 as initial rotor angle.
From the power angle curve, initial rotor angle,
P 
 0  sin 1  m0  --------------------------------------(1)
 Pmax 
A
ES
ZS
Relay
Irelay
B
ZL
(a) Single line diagram
(b) Power Angle curve
Figure 2.1 Simple two machine system
ZR
ER0
Now, suppose, a self clearing transient three phase short circuit fault occurs on the line.
During the fault, the electrical output power Pe drops to zero. The resulting rotor
acceleration advances rotor angle to 1 . After a time interval tcr , corresponding to
angle 1 , the fault is cleared and the operating point jumps back to the sinusoidal curve.
Rotor angle correspond to this instant is 1. As per equal area criteria, the rotor will
swings up to maximum rotor angle  max , satisfying the following condition,
Accelerating Area (A1) = Decelerating Area (A2)
Rotor angle 1corresponding to fault clearing time tcr can be computed by swing
equation,
2 H d 2
 Pmo  Pe  Pa ----------------------------(2)
s dt 2
where H is the equivalent rotor angle inertia.
During fault, Pe = 0, hence,
2 H d 2
 Pmo ---------------------------------------(3)
s dt 2
On integrating both the sides with respect to variable t,
d s Pmo

(t  t0 ) -------------------------------------(4)
dt
2H
Prior to fault  0 is a stationary point.
d
The initial condition of
is specified as follows
dt
d
0
dt t t
0
Integrating equation (4) and substituting   1 at time t = t1, with t1  t 2  t cr ,
P
1  s mo (tcr2 )   0 -------------------------------------(5)
4H
Thus, accelerating area A1 is given by,
1
1   Pmo d  Pmo (1 - 0 ) --------------------------------(6)
0
 Pmo (1   0 )
Substituting in equation (6),
 P 2 (t 2 )
1  s mo cr ------------------------------------------(7)
4H
Similarly, decelerating area, A2, can be calculated as follows.
2 
 max

1
Pmax Sin .d  Pmax (Cos1  Cos max )  Pmo ( max  1 ) -------(8)
Since for a stable swing, 1  2
Pmo  (1   0 )  Pmax (Cos1  Cos max )  Pmo ( max  1 ) --------------(9)
P
i.e. cos  max  cos  1  m0 ( max   0 ) -----------------------------(10)
Pmax
Since δ0 is function of Pmo from equation (1) and δ1 is function of Pmo as well as
t cr from equation (5), it follows from equation (10) that
δmax depends on Pmo and tcr.
i.e.  max  f ( Pm0 , t cr ) ------------------------------------------------(11)
The variation of max verses Pmo for different values of tcr is shown in Figure 23.2.
Figure 23.2 Plots of max verses Pmo, for different values of tcr
B) Determination of power swing locus
A distance relay may classify power swing as a phase fault if impedance trajectory enters
the operating characteristic of the relay. We will now derive the apparent impedance seen
by the relay R on R-X plane. Again consider simple two machine system connected by a
transmission line of impedance ZL as shown in Figure 23.1(a), where machine B is treated
as reference.
E   E R
----------------------(12)
I relay  S
ZT
Where, Z T  Z S  Z L  Z R ------------(13)
Now, impedance seen by relay is given by the following equation,
Z seen (relay ) 
Vrelay
I relay

E S   I relay Z S
I relay
 E S  
Z T -----------------------(14)
  Z S  
 E S   E R 




1


 Z S  Z T


ER
  
1
ES


Let us define k 
ES
. Assuming for simplicity, both the voltages as equal to 1pu, i.e.
ER
k=1,

1
Z seen (relay )   Z S  Z T 
 1  cos   j sin 
 Z S 
ZT
2 sin

(sin





ZT 
1


   Z S 


2  2

 j sin cos 
 sin
2
2
2


 j cos )
2
2
2
ZT

 Z S 
(1  j cot ) -------------------------(15)
2
2
Z
Z

  Z S  T  j T cot
2
2 
2


a cons tan t offset
perpendicular line segment
ZT
2
There is a geometrical interpretation of above equation. The vector component
Z
Z

 Z S  T in equation (15) is a constant in R – X plane. The component  j T cot is
2
2
2
Z
a straight line, perpendicular to line segment T . Thus the trajectory of the impedance
2
measured by relay during the power swing is a straight line as shown in fig 23.3. The
angle subtended by a point in the locus on S and R end points is angle  . For simplicity,
angle of Z S , Z R and Z L are considered identical. It intersect the line AB at mid point,
when  180 . The corresponding point of intersection of swing impedance trajectory to
the impedance line is known as electrical center of the swing. (Fig 23.4(a)). The angle, 
between two sources can be mapped graphically as the angle subtended by source points
E S and E R on the swing trajectory. At the electrical center, angle between two sources
is 180 . The existence of electrical center is an indication of system instability, the two
generators now being out of step.
From equation (15) at  180 , cot   0, Z seen  Z S 
If the power swing is stable, i.e. if the post fault system is stable, then  max will be less
than 180   . In such event, the power swing retraces its path at  max .
ES
 k  1 , then the power swing locus on the R – X is an arc of the circle. (See fig
ER
22.4(b))
It can be easily shown that
ES
k (cos   sin  )
k[(k  cos  )  j sin  ]
---------------(16)


E S  E R k (cos   j sin  )  1 (k  cos  ) 2  sin 2 
Then,
k (k  cos  )  j sin  
Z seen   Z S 
ZT
(k  cos  ) 2  sin 2 
If
Figure 23.4 Impedance Trajectories at relay point during power swing
It is also clear from Fig 23.4 (b), that the location of the electrical center is dependent
E
upon the S ratio. Appearance o f electrical center on a transmission line is a transient
ER
phenomenon. The voltage profile across the transmission at the point of occurrence of
electrical center is shown in fig 23.5.
At the electrical center, the voltage is exactly zero. This means that relays at both the
ends of line perceive it as a bolted three phase fault and immediately trip the line. Thus,
we can conclude that existence of electrical center indicates (1) system instability (2)
possibility of nuisance tripping of distance relay.
Now consider a double-end-feed transmission line with three stepped distance protection
scheme having Z1, Z2 and Z3 protection zones as shown in Figure 23.6. The mho relays
are used and characteristics are plotted on R-X plane as shown in Figure 23.7. Swing
impedance trajectory is also overlapped on relay characteristics for a simple case of equal
end voltages (i.e. k = 1) and it is perpendicular to line AB.
T3
Z3
T2
Z2
EA
O
T1
Z1
EB0
M
N
ZL1
ZL2
2
1
Reverse
fault
3
Forward
Fig 22.5 Single line diagram showing three protection zones
X
Zone-3
Zone-2
Zone-1
R
Impedance Trajectories entering three protection zones
Figure22.6 Three stepped distance protection to double-end-feed line
From Figure 22.6,  Z 1 ,  Z 2 and  Z 3 are rotor angles when swing just enters the zone Z1,
Z2 and Z3 respectively and it can be obtained at the intersection of swing trajectory to the
relay characteristics. Recalling δmax is the maximum rotor angle for stable power swing,
following inferences can be drawn.
If  max   Z 3 , then swing will not enter the relay characteristics.
If  Z 3   max   Z 2 , swing will enter in zone Z 3 . If it stays in zone - Z 3 for larger interval
than its TDS, then the relay will trip the line.
If  Z 2   max   Z 1 , swing will enter in both the zones Z 2 and Z 3 . If it stays in zone 2, for
larger interval than its TDS, then the relay will trip on Z 2
If  max   Z 1 , swing will enter in the zones Z1 , Z 2 and Z 3 and operate zone 1 protection
will operate without instantaneous delay.
Evaluation of power swings on a multimachine system requires usage of transient
stability program. Using transient stability program, post fault the relay end node voltage
and line currents can be monitored and then the swing trajectory traced on impedance.
Fig23.-- shows one such trajectory observed on a node system when a fault on a
line for internal
Es     0 1  (rad)  max