A Generating Unit
U ref
Exciter AVR
U
Valve/gate Ef
U
Water
Turbine ~ I
Power
system
or Shaft
steam f
Governor
Pmset
1
Synchronous machine
• Almost all generators are synchronous
generators.
• They are the most important components
in the analysis of electromechanical
dynamic.
• It is of fundamental importance to under-
stand the behavior and dynamics of these
machines in power system stability study.
2
Generator
I
Tm ω m Te Stator
+ Pe
Pm
Rotor U
Turbine Stator
Shaft I
3
a-axis
d-axis
βm
q-axis
b′ c
a a′
if
c′ c-axis
b
b-axis
4
a-axis
d-axis
βm
q-axis
b′ c
a a′
if
c′ c-axis
b
b-axis
• a, b and c denote the stator three windings.
• They are 120 electrical degrees apart.
5
a-axis
d-axis
βm
q-axis
b′ c
a a′
if
c′ c-axis
b
b-axis
• The field winding.
• It carries a direct current to produce a
magnetomotive force (mmf) which drives
the field flux around the magnetic circuit.
6
a-axis
d-axis ia q-axis
+
ua
βm
u + if
f
n
b-axis
ib ub uc + c-axis
+
ic
• The stator is represented by three magnetic
axes a, b and c each corresponding to one of
the phase windings. 7
a-axis
d-axis ia q-axis
+
ua
βm
u + if
f
n
b-axis
ib ub uc + c-axis
+
ic
• The rotor is represented by two axes. The
direct axis (d-axis), which is the magnetic
axis of the field winding. 8
a-axis
d-axis ia q-axis
+
ua
βm
u + if
f
n
b-axis
ib ub uc + c-axis
+
ic
• The quadrature axis (q-axis) which is located
90 electrical degrees behind the d-axis.
9
d-axis a-axis
r βm ia
,L q-axis
f
+
ra , Laa
Q
u + f
i
ua ,L Q
ff
r rQ
,L
f
D
iD
0
u
=
iQ
D
D
=
Q
u
0 n
, L bb rc ,
rb Lc
c
b-axis
ib ub uc + c-axis
+
ic
• The rotor can be equipped with additional
short-circuited damper windings to reduce
the mechanical oscillations of the rotor. 10
a-axis
d-axis ia q-axis
+
ua
βm
u + if
f
n
b-axis
ib ub uc + c-axis
+
ic
• The mechanical rotor angle.
• It defines the instantaneous position of the
rotor d-axis with respect to a stationary
reference. 11
a-axis
d-axis ia q-axis
+
ua
βm
u + if
f
b-axis
ib ub uc + c-axis
+
ic
12
a-axis
d-axis
βm
q-axis
b′ c
a a′
if
c′ c-axis
b
b-axis 13
14
a-axis
d-axis
βm
q-axis
b′ c
a a′
if
c′ c-axis
b
b-axis
15
a-axis
d-axis
βm
q-axis
b′ c
a a′
if
c′ c-axis
b
b-axis
16
a-axis
d-axis
βm
q-axis
b′ c
a a′
if
c′ c-axis
b
b-axis
and it doesn’t give any useful information
17
• In power system, one is interested in how
much the rotor speed deviates from the
synchronous speed.
• Therefore, the position of the rotor angle is
measured with respect to a reference axis
which rotates at synchronous speed.
• Furthermore, due to practical issues, the rotor
angle position is measured from the q-axis.
18
19
20
Generator
I
Tm ω m Te Stator
+ Pe
Pm
Rotor U
Turbine Stator
Shaft I
21
Generator
I
Tm ω m Te Stator
+ Pe
Pm
Rotor U
Turbine Stator
Shaft I
• The turbine with a torque (Tm) rotates the
rotor shaft and thereby the field winding on
the rotor (which carries a direct current)
produces a rotating flux in the air gap.
22
Generator
I
Tm ω m Te Stator
+ Pe
Pm
Rotor U
Turbine Stator
Shaft I
• When the generator is loaded, currents
flowing in the stator windings also produce
rotating flux in the air gap.
23
Generator
I
Tm ω m Te Stator
+ Pe
Pm
Rotor U
Turbine Stator
Shaft I
• Then, the resultant (or combined) flux
across the air gap provides an electro-
magnetic torque (Te) which opposes the
torque of the turbine (Tm).
24
Swing equation
• J total moment of inertia (kgm2)
• βm rotor mechanical angle
• ωm rotor mechanical speed
• Tm mechanical torque (Nm)
• Te electrical torque (Nm)
25
26
• WKs is the total kinetic energy stored in the
generator in the steady-state.
• Sng is the generator rated three-phase VA.
• The inertia constant states how many
seconds it would take to bring the generator
from synchronous speed to standstill if
rated power is extracted from it while no
mechanical power is supplied by the
turbine. 27
⇒
28
Swing equation (2)
29
Swing equation (3)
30
31
Swing equation with damping
32
Swing equation (4)
33
Electrical d-axis a-axis
rf βm ia
,L q-axis
equations +
ra , Laa
ff Q
u + f
i
rD ua ,L Q
,L rQ
f
D 0
iD
Linear flux-current u D
iQ =
D
= uQ
0 n
relationships, rb , L bb rc ,
Lc
c
b-axis
ib ub uc + c-axis
+
ic
34
i r d-axis a-axis
rf βm ia
,L q-axis
e = − dψ
+
u
ra , Laa
ff Q
u + f
i
r ua ,L Q
dt ,L rQ
D
iD
0
u
=
iQ
D
D Q
=
e = − dψ = u + ri
u
0 n
, L bb rc ,
Lc
dt b-axis
rb
ib ub uc +
c
c-axis
+
ic
35
Electrical d-axis a-axis
f
βm ia q-axis
r
equations
,L
+
ra , Laa
ff
u + f
i
,L
Q
ua
Q
r r
,L
Q
f
D
iD
0
u
=
iQ
D
D Q
=
u
0 n
, L bb rc ,
rb Lc
c
b-axis
ib ub uc + c-axis
+
ic
36
Park’s transformation
37
xis
d-a
id
+
ud
, Ld
rd
io n
iD tat
Ro )
=0
(ω g q-a
xis
uD Lq
D
,
D
rq
L
iq
,
rD
if
+
uf Q
, LQ
+
rQ iQ uq
, L ff
rf
=0
uQ
38
rf , L ff rD , LDD rd , Ld
if
id
d-axis
iD
+ +
uf
uD = 0 ud
39
rQ , LQQ rq , Lq q-axis
iQ iq
+
uQ = 0 uq
40
SG modelling
Im-axis q-axis
U ∠θ Uq
I U Im U
SG Power System
δ
d-axis
θ Re-axis
U Re
( −) U d
SG
U ∠θ
Eq′ ∠ δ
jxd′ I
Power System
41
Classical model
42
One-axis model
43
Transient stability
44
45
46
47
48
xo x (to )
γ
ε
xo x (t )
o
ε γ
49
50
51
52
xo x (to )
γ
ε
xo x (t )
o
ε γ
53
54
55
56
Transient stability of SMIB system
• The SMIB system is transiently stable if
the generator remains in synchronism
with the infinite bus after being subjected
to a large disturbance.
• Instability that may result occurs in the
form of increasing rotor angle (or speed)
of the generator leading to its loss of
synchronism.
57
Transient stability of SMIB system
• The SMIB system is transiently stable if
the initial point of the post-disturbance
system lies within the stability region of
the stable e.p of the post-disturbance
system.
58
59
60
61
BUS T BUS 1
Horred
GR3 I R3
G jxL1 BUS N
TR3 Infinite
GR4 jxL 2 Bus
I R4
G T2
TR4
62
63
Pre-fault (1)
Eq′ U1 UN
jxL
jxd′ jxT 1 jxT 2 Infinite
Bus
I jxL
64
Pre-fault (2)
Eq′ U1 UN
jxL
jxd′ jxT 1 jxT 2 Infinite
Bus
I jxL
65
During-fault (1)
Eq′
jxL
jxd′ jxT 1
If jxL
66
During-fault (2)
Eq′
jxL
jxd′ jxT 1
If jxL
67
Post-fault (1)
Eq′ UN
jxd′ jxT 1 jxL jxT 2 Infinite
Bus
I post
68
Post-fault (2)
Eq′ UN
jxd′ jxT 1 jxL jxT 2 Infinite
Bus
I post
69
Post-fault (3)
Eq′ UN
jxd′ jxT 1 jxL jxT 2 Infinite
Bus
I post
70
71
72
73
74
75
76
77
78
79
80
81
82
83
180
160
140
120
100
80
60
40
20
84
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Equal Area Criterion (EAC)
• A necessary condition for transient
stability (or first-swing stability) of the
SMIB system is that at some time tm
85
86
87
88
89
90
91
92
93
94
95
• Aa represents the kinetic energy injected
into the system during the fault. It is also
called accelerating area.
96
• Ad represents the ability of the post-fault
system to absorb energy, i.e. potential
energy. It is also called decelerating area.
97
98
99
100
101
102
103
104
105
106
Gen3
T3
BUS 3
S L3
BUS 1 BUS 4 BUS 2
Gen1 Gen2
T1 S L1 SL4 SL2 T2
BUS 1
Eq′ ∠ δ U Th ∠θTh
jxd′ ZTh
ITh
Gen1
107
BUS 1
Eq′ ∠ δ U Th ∠θTh
jxd′ ZTh
ITh
Gen1
BUS 1 BUS 2
Eq′ ∠ δ 0.3 ZTh U Th ∠θTh
jxd′ 0.85 ZTh
ITh 0.3 ZTh
Gen1
108
U1 , U 2 ,U 3 ,U 4
0 0 0 0
Run LFC: Pg1 , Pg 2 , Pg 3 , Pg 4
Qg1 , Qg 2 , Qg 3 , Qg 4
*
S gi0
I gi ⇒ Eqi′ =Eqi′ ∠δ ispre =U i + jxdi
′ I gi
0
U i*0 0 0 0 0
109
Convert all given loads to impedance loads
and disconnect the selected generator from
its terminal bus. 2
U4
Z L4 = *
0
BUS 6
Eq′ 3 SL4
jxd′ 3
I6
BUS 3
jx34
BUS 1 BUS 4 BUS 2 Eq′ 2
Eq′1
jxd′ 1 jx14 jx24 jxd′ 2
I5 BUS 5
ZL4
110
I = YU ⇒ U = ZI U Th
= U=
1 Z15 I 5 + Z16 I 6
U1 Z11 Z12 Z13 Z14 Z15 Z16 0
U 2 Z 21 Z 22 Z 23 Z 24 Z 25 Z 26 0
U3 Z Z32 Z33 Z 34 Z35 Z36 0
= 31
U 4 Z 41 Z 42 Z 43 Z 44 Z 45 Z 46 0
E′ Z Z52 Z53 Z 54 Z55 Z56 I 5
q 20 51
Eq′ 3 Z 61
0 Z 62 Z 63 Z 64 Z 65 Z 66 I 6
−1
Eq′ 20 Z 55 Z56 I 5 I 5 Z55 Z56 Eq′ 20
= ⇒
′ Z 65 Z 66 I 6 I 6 Z 65 Z 66 ′
Eq 30 Eq 30
111
U Th
= U=
1 Z15 I 5 + Z16 I 6
Eq′ 3
BUS 6
jxd′ 3
I6
BUS 3
jx34
BUS 1 BUS 4 BUS 2 Eq′ 2
Eq′1
jxd′ 1 jx14 jx24 jxd′ 2
I5 BUS 5
ZL4
112
BUS 1
Eq′ ∠ δ U Th ∠θTh
jxd′ ZTh
ITh
Gen1
Eq′ 1 − UTh
ITh = ⇒ U1 = UTh + ZTh ITh = U1
jxd′ + ZTh 0
*
S g1 =Pg1 +
0
jQg1 =U1ITh
113
BUS 1
Eq′ ∠ δ U Th ∠θTh
jxd′ ZTh
ITh
Gen1
Z =jxd′ + ZTh =R + jX
Eq′ R
Pe = 2 Eq′ R + U Th Z sin δ − arctan − θTh
Z X
Z= Z
114
BUS 1
Eq′ ∠ δ U Th ∠θTh
jxd′ ZTh
ITh
Gen1
Eq′ R
Pe = 2 Eq′ R + U Th Z sin δ − arctan − θTh
Z X
K1 + K 2 sin (δ − K 3 )
Pe =
115
BUS 1
Eq′ ∠ δ U Th ∠θTh
jxd′ ZTh
ITh
Gen1
BUS 1 BUS 2
Eq′ ∠ δ 0.3 ZTh U Th ∠θTh
jxd′ 0.85 ZTh
ITh 0.3 ZTh
Gen1
Pe pre
K +K
= 1
pre
2
pre
sin (δ s
pre
−K 3
pre
)
116
BUS 1 BUS 2
Eq′ ∠ δ 0.3 ZTh U Th ∠θTh
jxd′ 0.85 ZTh
Gen1
Pe post
K
=+ K 1
post
2
post
sin (δ − K 3
post
)
=δ ???
= , δ max ???
s
post
117
δ max = 180 − δ s + 2 K 3
118
Case 1-3
BUS 1 BUS 2
Eq′ ∠ δ 0.3 ZTh U Th ∠θTh
jxd′ 0.85 ZTh
ITh 0.3 ZTh
Gen1
K + K sin (δ − K
Pe = f
1
f
2
f
3
f
)
f f f
K
1 ???
= , K ???
= , K ??? 2 3
119
( )
δ cc
∫δ ( P − Pm ) dδ
δ max
∫δ δf
A1 pre
Pm − Pe d= A2 e
post
s cc
A
=1 A2 ⇒ δ cc g (δ cc ) = A1 − A2 = 0
( Pm K
A1 =− 1)(δ
f
cc −δ s
pre
)
+K f
cos (δ cc − K
f
) − cos (δ pre
−K f
)
2 3 s 3
(K
A2 = 1
post
− Pm ) (δ max − δ cc )
−K post
cos (δ max − K 3post ) − cos (δ cc − K 3post )
2
120
121
122
123
Lyapunov Function
124
Transient Energy Function (TEF)
for an SMIB system
125
126
127
128
129
130
131
132
133
134
• Thermal limit
• Transient stability limit
• POD limit
• Voltage stability limit
N-1 criterion
135
P = P0 +∆P
SI S II
136
Transient stability enhancement
137
138
U1U N
Pm sin (θ1 ) ⇒θ1
xL1 + xL 2 + xt 2
U1 − U N
I= ⇒ Eq′ ∠δ spre= j ( xd′ + xt1 ) I + U1
j ( xL1 + xL 2 + xt 2 )
A1 = A2 ⇒ δ cc ⇒ tcc
139
140
Braking resistors (1)
141
Braking resistors (2)
142
•143
Series capacitor(1)
144
Series capacitor(2)
145
146
Shunt capacitor(1)
147
Shunt capacitor(2)
148
