Transcript MEngr Presentation - Electrical and Computer Engineering

Masters of Engineering
Small Signal Stability
Aaron Cowan
Electrical Engineering
Power
Small Signal Stability
• Exciter
– Field current
– Terminal voltage
• Power System Stabilizer
– Enhance stability
– Rotor angle
• Equal Area Criterion (Fig 13.5, Kundur)
– Aa < Ad
– Aa > Ad
SMIB Example
v _s
delta_wr
PSS
K_4
delta _Tm
v_s
delta _Psi_fd
V _ref
K_3
K _A
delta _Te
1
K_2
T_3.s+1
Exciter
V_ref
2*Hs+K _D
delta _wr
1
s
delta _delta
w_0/s
Field Circuit
v_1
K_6
K _1
1
delta _E_t
K_5
T_R.s+1
Voltage Transducer
Problem details in section 12.3 of Power System Stability and Control, Kundur
Results
Kundur
ωd = 1.05Hz
ξ = 0.15
KS = 0.829
KD = 14.08
A=
0
376.99
0
0
0
0
Matlab
ωd = 1.21Hz
ξ = 0.1447
KS = 1.1062
KD = 15.6306
−0.109
0
−0.193
−7.312
−1.037
−4.840
−0.123
0
−0.4229
20.839
−1.173
−5.477
0
0
−27.317
−50
0
0
State Matrix and
eigenvalues agree
0
0
0
0
−0.714
26.969
0
0
27.317
0
0
−30.303
𝜆2 = 0.504 + 𝑗7.23 ← 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑒𝑑 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑟𝑜𝑡𝑜𝑟 𝑎𝑛𝑔𝑙𝑒
Δ𝜔𝑟
Δ𝛿
Δ𝜓𝑓𝑑
Δ𝜈1
Δ𝜈2
Δ𝜈𝑠
Power World Transient Stability
Bus 2
163 MW
7 Mvar
Bus 7
Bus 8
Bus 9
Bus 3
1.016 pu
1.025 pu
1.026 pu
Bus 5
1.032 pu
0.996 pu
100 MW
Bus 6
1.025 pu
1.013 pu
35 Mvar
125 MW
50 Mvar
Bus 4
1.026 pu
90 MW
30 Mvar
Bus1
1.040 pu
slack
72 MW
27 Mvar
WECC equivalent in Power World
85 MW
-11 Mvar
Exciter Models
Exciter Models
Exciter Models
PSS Model
IEEE 421.2
SMIB – Power World
{
• Equivalent SMIB
• State Matrix
• Eigenvalues
Power World Transient Stability
Bus 2
163 MW
7 Mvar
Bus 7
Bus 8
Bus 9
Bus 3
1.016 pu
1.025 pu
1.026 pu
Bus 5
1.032 pu
0.996 pu
100 MW
Bus 6
1.025 pu
1.013 pu
35 Mvar
125 MW
50 Mvar
Bus 4
1.026 pu
90 MW
30 Mvar
Bus1
1.040 pu
slack
72 MW
27 Mvar
WECC equivalent in Power World
85 MW
-11 Mvar
Stability Simulation
• Default values used
– Did change TR to 0.02 in all cases
• SEXS_GE and STAB1 ↔ Fig 17.5, Kundur
• Set all generator stability models equal
– Innumerable permutations
Stability Simulation
• Fault on line 7-5
– Both breakers open
– Cleared in 0.07 sec
• Three cases for each Exciter
– Each generator
• Three cases for each Exciter+PSS
– Each generator
Generator 1
Generator 1: ESAC1A
MW vs. Rotor Angle Generator 1
MW vs. Rotor Angle Generator 1
240
220
220
200
200
180
180
160
160
140
140
120
120
100
100
80
80
60
60
40
40
20
20
0
0
-20
-20
-40
-40
-35
-30
-25
-20
-15
-10
b
c
d
e
f
g
𝑀𝑊0 = 71.6
-5
0
5
10
15
20
MW Terminal_Gen '1' '1'
𝛿0 = 3.5°
25
-40
-35
-30
-25
-20
-15
b
c
d
e
f
g
-10
-5
0
5
10
15
20
MW Terminal_Gen '1' '1'
𝑀𝑊𝑐𝑙𝑒𝑎𝑟 = 70.6
𝛿𝑐𝑙𝑒𝑎𝑟 = −5.3°
Generator 2
Generator 2: ESDC1A
MW vs. Rotor Angle Generator 2
MW vs. Rotor Angle Generator 2
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
220
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
55
60
65
70
75
b
c
d
e
f
g
𝑀𝑊0 = 163
80
85
90
95
100
MW Terminal_Gen '2' '1'
𝛿0 = 61.1°
55
60
65
70
75
b
c
d
e
f
g
𝑀𝑊𝑐𝑙𝑒𝑎𝑟 = 163
80
85
90
95
100
MW Terminal_Gen '2' '1'
𝛿𝑐𝑙𝑒𝑎𝑟 = 70.7°
Generator 3
Generator 3: SEXS_GE
MW vs. Rotor Angle Generator 3
MW vs. Rotor Angle Generator 3
100
95
95
90
90
85
85
80
80
75
75
70
70
65
65
60
60
55
55
50
50
45
45
47
48
49
50
51
52
53
b
c
d
e
f
g
𝑀𝑊0 = 85
54
55
56
57
58
59
MW Terminal_Gen '3' '1'
𝛿0 = 54.1°
60
49
50
51
52
53
b
c
d
e
f
g
𝑀𝑊𝑐𝑙𝑒𝑎𝑟 = 85
54
55
56
57
58
59
MW Terminal_Gen '3' '1'
𝛿𝑐𝑙𝑒𝑎𝑟 = 51.9°
Summary
• Power World Transient Stability
– Block Diagrams
– SMIB Eigenvalues
• ESDC1A without PSS
• SEXS_GE with PSS
• PSS stability enhancement
Small Signal Stability
Questions?
Generator 1: ESDC1A
MW vs. Rotor Angle Generator 1
MW vs. Rotor Angle Generator 1
200
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
-10
-20
-30
180
160
140
120
100
80
60
40
20
0
-20
-45
-40
-35
-30
-25
-20
b
c
d
e
f
g
-15
-10
-5
MW Terminal_Gen '1' '1'
0
5
10
15
-45
-40
-35
-30
-25
-20
b
c
d
e
f
g
-15
-10
-5
MW Terminal_Gen '1' '1'
0
5
10
15
Generator 1: SEXS_GE
MW vs. Rotor Angle Generator 1
MW vs. Rotor Angle Generator 1
170
180
160
150
160
140
140
130
120
120
110
100
100
90
80
80
70
60
60
50
40
40
30
20
20
10
0
0
-10
-20
-20
-30
-25
-20
-15
-10
b
c
d
e
f
g
-5
0
5
MW Terminal_Gen '1' '1'
10
15
20
-30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8
b
c
d
e
f
g
-6
-4
-2
0
2
MW Terminal_Gen '1' '1'
4
6
8
10 12 14 16
Generator 2: ESAC1A
MW vs. Rotor Angle Generator 2
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
MW vs. Rotor Angle Generator 2
220
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98
b
c
d
e
f
g
MW Terminal_Gen '2' '1'
50
55
60
65
70
b
c
d
e
f
g
75
80
MW Terminal_Gen '2' '1'
85
90
95
Generator 2: SEXS_GE
MW vs. Rotor Angle Generator 2
MW vs. Rotor Angle Generator 2
220
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
50
52
54
56
58
60
62
64
66
b
c
d
e
f
g
68
70
72
74
76
MW Terminal_Gen '2' '1'
78
80
82
84
86
88
90
54
56
58
60
62
64
66
68
70
b
c
d
e
f
g
72
74
76
78
80
MW Terminal_Gen '2' '1'
82
84
86
88
90
Generator 3: ESDC1A
MW vs. Rotor Angle Generator 3
MW vs. Rotor Angle Generator 3
105
100
100
95
95
90
90
85
85
80
80
75
75
70
70
65
65
60
60
55
55
50
50
45
45
49
50
51
52
53
54
55
b
c
d
e
f
g
56
57
58
59
60
MW Terminal_Gen '3' '1'
61
62
63
64
47
48
49
50
51
52
53
54
b
c
d
e
f
g
55
56
57
58
59
MW Terminal_Gen '3' '1'
60
61
62
63
64
Generator 3: ESAC1A
MW vs. Rotor Angle Generator 3
MW vs. Rotor Angle Generator 3
100
105
95
100
90
95
85
90
80
85
80
75
75
70
70
65
65
60
60
55
55
50
50
45
45
45
46
47
48
49
50
51
52
b
c
d
e
f
g
53
54
55
56
MW Terminal_Gen '3' '1'
57
58
59
60
61
46
47
48
49
50
51
52
b
c
d
e
f
g
53
54
55
56
57
MW Terminal_Gen '3' '1'
58
59
60
61