Short Notes: Power System Stability | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download

 Page 1


 
 
 
 
 
                      
                                          
Two types of stability are studied: 
1) Steady State Stability 
2) Transient Stability 
Steady State Stability 
For Steady State Stability 
            
dP
0
d
?
?
 
And for this condition to be true. 
            
e max
PP ? 
If power demand is greater than maximum demand than machine goes out of synchronous. 
For a loss less machine, 
max
S
EV
P
X
? 
Transient Stability 
Swing Equation 
              
? ?
2
me 2
Md
PP
dt
?
?? 
              M = inertia constant   ( MJ-S / elect - rad) 
              P m= mechanical input (MW) 
              
e
P = electrical output (MW) 
              ? = rotor angle 
Another Form 
       
? ?
2
me 2
Hd
PP
f dt
?
??
?
 
        H = inertia constant ( MJ / MVA) 
        
m
P & 
e
P both are in pu 
 
Page 2


 
 
 
 
 
                      
                                          
Two types of stability are studied: 
1) Steady State Stability 
2) Transient Stability 
Steady State Stability 
For Steady State Stability 
            
dP
0
d
?
?
 
And for this condition to be true. 
            
e max
PP ? 
If power demand is greater than maximum demand than machine goes out of synchronous. 
For a loss less machine, 
max
S
EV
P
X
? 
Transient Stability 
Swing Equation 
              
? ?
2
me 2
Md
PP
dt
?
?? 
              M = inertia constant   ( MJ-S / elect - rad) 
              P m= mechanical input (MW) 
              
e
P = electrical output (MW) 
              ? = rotor angle 
Another Form 
       
? ?
2
me 2
Hd
PP
f dt
?
??
?
 
        H = inertia constant ( MJ / MVA) 
        
m
P & 
e
P both are in pu 
 
 
 
 
 
 
 
        
GH
M
180f
?   (MJ – S / elect - deg) 
        
GH
M
f
?
?
      (MJ – S / elect - rad) 
        G = machine rating (MVA) 
? If two alternators are swinging coherently. Then they can be replaced by a single 
alternator having  
  
eq 1 2
M M M ?? 
But “ H “ cannot be added directly, they must first be on same base. 
? If machines are not swinging coherently, then  
  
12
eq
12
MM
M
MM
?
?
 
 
? Accelerating Power,  
? ?
a m e
P P P ?? 
In steady state 
me
PP ? 
In transient, 
me
PP ? so rotor accelerate or decelerate. 
Equal area criterion  
For system to possess transient stability 
a
P d 0 ??
?
 
There are basically 3 stages in stability analysis 
? Before Fault 
We say maximum power transferrable is 
max,1
P 
 & 
e max,1
P P sin ?? 
 
? During fault 
We say maximum power transferrable is 
max,2
P 
 
e max,2
P P sin ?? 
 
? After Fault 
We say maximum power transferrable is 
max,3
P 
 
e max,3
P P sin ?? 
 
 
Page 3


 
 
 
 
 
                      
                                          
Two types of stability are studied: 
1) Steady State Stability 
2) Transient Stability 
Steady State Stability 
For Steady State Stability 
            
dP
0
d
?
?
 
And for this condition to be true. 
            
e max
PP ? 
If power demand is greater than maximum demand than machine goes out of synchronous. 
For a loss less machine, 
max
S
EV
P
X
? 
Transient Stability 
Swing Equation 
              
? ?
2
me 2
Md
PP
dt
?
?? 
              M = inertia constant   ( MJ-S / elect - rad) 
              P m= mechanical input (MW) 
              
e
P = electrical output (MW) 
              ? = rotor angle 
Another Form 
       
? ?
2
me 2
Hd
PP
f dt
?
??
?
 
        H = inertia constant ( MJ / MVA) 
        
m
P & 
e
P both are in pu 
 
 
 
 
 
 
 
        
GH
M
180f
?   (MJ – S / elect - deg) 
        
GH
M
f
?
?
      (MJ – S / elect - rad) 
        G = machine rating (MVA) 
? If two alternators are swinging coherently. Then they can be replaced by a single 
alternator having  
  
eq 1 2
M M M ?? 
But “ H “ cannot be added directly, they must first be on same base. 
? If machines are not swinging coherently, then  
  
12
eq
12
MM
M
MM
?
?
 
 
? Accelerating Power,  
? ?
a m e
P P P ?? 
In steady state 
me
PP ? 
In transient, 
me
PP ? so rotor accelerate or decelerate. 
Equal area criterion  
For system to possess transient stability 
a
P d 0 ??
?
 
There are basically 3 stages in stability analysis 
? Before Fault 
We say maximum power transferrable is 
max,1
P 
 & 
e max,1
P P sin ?? 
 
? During fault 
We say maximum power transferrable is 
max,2
P 
 
e max,2
P P sin ?? 
 
? After Fault 
We say maximum power transferrable is 
max,3
P 
 
e max,3
P P sin ?? 
 
 
 
 
 
 
 
Critical clearing angle 
It is the maximum value of ? beyond which if the fault is cleared system will be unstable. The 
time instant corresponding to this angle is called as critical clearing time assuming fault occurs 
at t = 0. 
 
Case-1 : Fault occurs on TL near to bus  
max,2
P0 ? 
max,3 max,1
PP ? 
Cr
clearing angle ?? 
By equal area criteria  
                      
? ?
2
0
m max,1
P P sin d 0
?
?
? ? ? ?
?
 
                      
1 m
0
max,1
P
sin
P
?
??
??
??
??
??
 
For critical clearing 
2 max
? ? ? 
                      
0 max
? ? ? ? ? 
                      
? ?
Cr 0
Cr
m
2H
t
fP
? ? ?
?
?
 = Critical Clearing Time 
 
Case-2 : Fault occurs on one of parallel lines close to bus 
Before Fault 
? ?
max,1
g 1 2
EV
P
X X X
?
?
 
During Fault 
max,2
eq
EV
P0
X
?? 
After Fault 
max,3
g1
EV
P
XX
?
?
 
Page 4


 
 
 
 
 
                      
                                          
Two types of stability are studied: 
1) Steady State Stability 
2) Transient Stability 
Steady State Stability 
For Steady State Stability 
            
dP
0
d
?
?
 
And for this condition to be true. 
            
e max
PP ? 
If power demand is greater than maximum demand than machine goes out of synchronous. 
For a loss less machine, 
max
S
EV
P
X
? 
Transient Stability 
Swing Equation 
              
? ?
2
me 2
Md
PP
dt
?
?? 
              M = inertia constant   ( MJ-S / elect - rad) 
              P m= mechanical input (MW) 
              
e
P = electrical output (MW) 
              ? = rotor angle 
Another Form 
       
? ?
2
me 2
Hd
PP
f dt
?
??
?
 
        H = inertia constant ( MJ / MVA) 
        
m
P & 
e
P both are in pu 
 
 
 
 
 
 
 
        
GH
M
180f
?   (MJ – S / elect - deg) 
        
GH
M
f
?
?
      (MJ – S / elect - rad) 
        G = machine rating (MVA) 
? If two alternators are swinging coherently. Then they can be replaced by a single 
alternator having  
  
eq 1 2
M M M ?? 
But “ H “ cannot be added directly, they must first be on same base. 
? If machines are not swinging coherently, then  
  
12
eq
12
MM
M
MM
?
?
 
 
? Accelerating Power,  
? ?
a m e
P P P ?? 
In steady state 
me
PP ? 
In transient, 
me
PP ? so rotor accelerate or decelerate. 
Equal area criterion  
For system to possess transient stability 
a
P d 0 ??
?
 
There are basically 3 stages in stability analysis 
? Before Fault 
We say maximum power transferrable is 
max,1
P 
 & 
e max,1
P P sin ?? 
 
? During fault 
We say maximum power transferrable is 
max,2
P 
 
e max,2
P P sin ?? 
 
? After Fault 
We say maximum power transferrable is 
max,3
P 
 
e max,3
P P sin ?? 
 
 
 
 
 
 
 
Critical clearing angle 
It is the maximum value of ? beyond which if the fault is cleared system will be unstable. The 
time instant corresponding to this angle is called as critical clearing time assuming fault occurs 
at t = 0. 
 
Case-1 : Fault occurs on TL near to bus  
max,2
P0 ? 
max,3 max,1
PP ? 
Cr
clearing angle ?? 
By equal area criteria  
                      
? ?
2
0
m max,1
P P sin d 0
?
?
? ? ? ?
?
 
                      
1 m
0
max,1
P
sin
P
?
??
??
??
??
??
 
For critical clearing 
2 max
? ? ? 
                      
0 max
? ? ? ? ? 
                      
? ?
Cr 0
Cr
m
2H
t
fP
? ? ?
?
?
 = Critical Clearing Time 
 
Case-2 : Fault occurs on one of parallel lines close to bus 
Before Fault 
? ?
max,1
g 1 2
EV
P
X X X
?
?
 
During Fault 
max,2
eq
EV
P0
X
?? 
After Fault 
max,3
g1
EV
P
XX
?
?
 
 
 
 
 
 
1 m
0
max,1
P
sin
P
??
??
??
??
?? 
1 m
max
max,3
P
sin
P
?
??
??
??
??
? ? ? ? 
For transient stability 
              
2
0
a
P d 0
?
?
??
?
 
             
? ? ? ?
c 2
c 0
mm max,3
P 0 d P P sin d 0
? ?
??
? ? ? ? ? ? ?
??
 
For critical Clearing 
             
2 max
? ? ? 
             
? ?
Cr 0
Cr
m
2H
t
fP
? ? ?
?
?
 
Case-3 : Fault occurs in middle of one of parallel lines 
The equivalent reactance during the fault is highest and thus 
max,2
P is lowest 
                  
max,1 max,3 max,2
P P P ?? 
                 
2
0
a
P d 0
?
?
??
?
 
                  
? ? ? ?
c 2
c
0
mm
max,2 max,3
P P sin d P sin P d
?
?
??
? ? ? ? ? ? ?
??
 
For critical clearing,  
                     
2 max
? ? ?
1 m
max,3
P
sin
P
?
??
??
??
??
? ? ? 
Page 5


 
 
 
 
 
                      
                                          
Two types of stability are studied: 
1) Steady State Stability 
2) Transient Stability 
Steady State Stability 
For Steady State Stability 
            
dP
0
d
?
?
 
And for this condition to be true. 
            
e max
PP ? 
If power demand is greater than maximum demand than machine goes out of synchronous. 
For a loss less machine, 
max
S
EV
P
X
? 
Transient Stability 
Swing Equation 
              
? ?
2
me 2
Md
PP
dt
?
?? 
              M = inertia constant   ( MJ-S / elect - rad) 
              P m= mechanical input (MW) 
              
e
P = electrical output (MW) 
              ? = rotor angle 
Another Form 
       
? ?
2
me 2
Hd
PP
f dt
?
??
?
 
        H = inertia constant ( MJ / MVA) 
        
m
P & 
e
P both are in pu 
 
 
 
 
 
 
 
        
GH
M
180f
?   (MJ – S / elect - deg) 
        
GH
M
f
?
?
      (MJ – S / elect - rad) 
        G = machine rating (MVA) 
? If two alternators are swinging coherently. Then they can be replaced by a single 
alternator having  
  
eq 1 2
M M M ?? 
But “ H “ cannot be added directly, they must first be on same base. 
? If machines are not swinging coherently, then  
  
12
eq
12
MM
M
MM
?
?
 
 
? Accelerating Power,  
? ?
a m e
P P P ?? 
In steady state 
me
PP ? 
In transient, 
me
PP ? so rotor accelerate or decelerate. 
Equal area criterion  
For system to possess transient stability 
a
P d 0 ??
?
 
There are basically 3 stages in stability analysis 
? Before Fault 
We say maximum power transferrable is 
max,1
P 
 & 
e max,1
P P sin ?? 
 
? During fault 
We say maximum power transferrable is 
max,2
P 
 
e max,2
P P sin ?? 
 
? After Fault 
We say maximum power transferrable is 
max,3
P 
 
e max,3
P P sin ?? 
 
 
 
 
 
 
 
Critical clearing angle 
It is the maximum value of ? beyond which if the fault is cleared system will be unstable. The 
time instant corresponding to this angle is called as critical clearing time assuming fault occurs 
at t = 0. 
 
Case-1 : Fault occurs on TL near to bus  
max,2
P0 ? 
max,3 max,1
PP ? 
Cr
clearing angle ?? 
By equal area criteria  
                      
? ?
2
0
m max,1
P P sin d 0
?
?
? ? ? ?
?
 
                      
1 m
0
max,1
P
sin
P
?
??
??
??
??
??
 
For critical clearing 
2 max
? ? ? 
                      
0 max
? ? ? ? ? 
                      
? ?
Cr 0
Cr
m
2H
t
fP
? ? ?
?
?
 = Critical Clearing Time 
 
Case-2 : Fault occurs on one of parallel lines close to bus 
Before Fault 
? ?
max,1
g 1 2
EV
P
X X X
?
?
 
During Fault 
max,2
eq
EV
P0
X
?? 
After Fault 
max,3
g1
EV
P
XX
?
?
 
 
 
 
 
 
1 m
0
max,1
P
sin
P
??
??
??
??
?? 
1 m
max
max,3
P
sin
P
?
??
??
??
??
? ? ? ? 
For transient stability 
              
2
0
a
P d 0
?
?
??
?
 
             
? ? ? ?
c 2
c 0
mm max,3
P 0 d P P sin d 0
? ?
??
? ? ? ? ? ? ?
??
 
For critical Clearing 
             
2 max
? ? ? 
             
? ?
Cr 0
Cr
m
2H
t
fP
? ? ?
?
?
 
Case-3 : Fault occurs in middle of one of parallel lines 
The equivalent reactance during the fault is highest and thus 
max,2
P is lowest 
                  
max,1 max,3 max,2
P P P ?? 
                 
2
0
a
P d 0
?
?
??
?
 
                  
? ? ? ?
c 2
c
0
mm
max,2 max,3
P P sin d P sin P d
?
?
??
? ? ? ? ? ? ?
??
 
For critical clearing,  
                     
2 max
? ? ?
1 m
max,3
P
sin
P
?
??
??
??
??
? ? ?