Masters of Engineering
Small Signal Stability
Aaron CowanElectrical 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
delta _wr
delta _deltadelta _Tedelta _Psi_fd
v_s
v_1
delta _E_t
V_refw_0/s
Voltage Transducer
1
T_R.s+1
1
2*Hs+K_D
PSS
delta_wrv _s
K_6
K_5
K_4
K_2
K_1
1s
Field Circuit
K_3
T_3.s+1Exciter
K_AV_ref
delta _Tm
Problem details in section 12.3 of Power System Stability and Control, Kundur
Results
Matlabωd = 1.21Hzξ = 0.1447KS = 1.1062KD = 15.6306
Kundurωd = 1.05Hzξ = 0.15KS = 0.829KD = 14.08
State Matrix and eigenvalues agree
[0 −0.109 −0.123 0 0 0
376.99 0 0 0 0 00 −0.193 −0.4229 −27.317 0 27.3170 −7.312 20.839 −50 0 00 −1.037 −1.173 0 −0.714 00 −4.840 −5.477 0 26.969 −30.303
][Δ𝜔𝑟Δ𝛿Δ𝜓 𝑓𝑑Δ𝜈 1Δ𝜈 2Δ𝜈𝑠
]A =
Power World Transient Stability
slack
Bus1
72 MW 27 Mvar
Bus 4
Bus 5
125 MW 50 Mvar
Bus 2
163 MW 7 Mvar
Bus 7 Bus 8 Bus 9 Bus 3
85 MW -11 Mvar
100 MW 35 Mvar
Bus 6
90 MW 30 Mvar
1.026 pu1.025 pu
0.996 pu
1.016 pu1.032 pu 1.025 pu
1.013 pu
1.026 pu
1.040 pu
WECC equivalent in Power World
Exciter Models
Exciter Models
Exciter Models
PSS Model
IEEE 421.2
SMIB – Power World
• Equivalent SMIB• State Matrix• Eigenvalues{
Power World Transient Stability
slack
Bus1
72 MW 27 Mvar
Bus 4
Bus 5
125 MW 50 Mvar
Bus 2
163 MW 7 Mvar
Bus 7 Bus 8 Bus 9 Bus 3
85 MW -11 Mvar
100 MW 35 Mvar
Bus 6
90 MW 30 Mvar
1.026 pu1.025 pu
0.996 pu
1.016 pu1.032 pu 1.025 pu
1.013 pu
1.026 pu
1.040 pu
WECC equivalent in Power World
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 Terminal_Gen '1' '1'gfedcb
2520151050-5-10-15-20-25-30-35-40
220
200
180
160
140
120
100
80
60
40
20
0
-20
-40
MW vs. Rotor Angle Generator 1
MW Terminal_Gen '1' '1'gfedcb
20151050-5-10-15-20-25-30-35-40
240
220
200
180
160
140
120
100
80
60
40
20
0
-20
𝑀𝑊 0=71.6 𝛿0=3.5° 𝑀𝑊 𝑐𝑙𝑒𝑎𝑟=70.6 𝛿𝑐𝑙𝑒𝑎𝑟=−5.3 °
Generator 2
Generator 2: ESDC1A
MW vs. Rotor Angle Generator 2
MW Terminal_Gen '2' '1'gfedcb
100959085807570656055
2102001901801701601501401301201101009080706050403020100
MW vs. Rotor Angle Generator 2
MW Terminal_Gen '2' '1'gfedcb
100959085807570656055
220210200190180170160150140130120110100
9080706050403020100
𝑀𝑊 0=163 𝛿0=61.1° 𝑀𝑊 𝑐𝑙𝑒𝑎𝑟=163 𝛿𝑐𝑙𝑒𝑎𝑟=70.7 °
Generator 3
Generator 3: SEXS_GE
MW vs. Rotor Angle Generator 3
MW Terminal_Gen '3' '1'gfedcb
6059585756555453525150494847
100
95
90
85
80
75
70
65
60
55
50
45
MW vs. Rotor Angle Generator 3
MW Terminal_Gen '3' '1'gfedcb
5958575655545352515049
95
90
85
80
75
70
65
60
55
50
45
𝑀𝑊 0=85 𝛿0=54.1 ° 𝑀𝑊 𝑐𝑙𝑒𝑎𝑟=85 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 Terminal_Gen '1' '1'gfedcb
151050-5-10-15-20-25-30-35-40-45
1701601501401301201101009080706050403020100
-10-20-30
MW vs. Rotor Angle Generator 1
MW Terminal_Gen '1' '1'gfedcb
151050-5-10-15-20-25-30-35-40-45
200
180
160
140
120
100
80
60
40
20
0
-20
Generator 1: SEXS_GE
MW vs. Rotor Angle Generator 1
MW Terminal_Gen '1' '1'gfedcb
20151050-5-10-15-20-25-30
180
160
140
120
100
80
60
40
20
0
-20
MW vs. Rotor Angle Generator 1
MW Terminal_Gen '1' '1'gfedcb
1614121086420-2-4-6-8-10-12-14-16-18-20-22-24-26-28-30
1701601501401301201101009080706050403020100
-10-20
Generator 2: ESAC1A
MW vs. Rotor Angle Generator 2
MW Terminal_Gen '2' '1'gfedcb
989694929088868482807876747270686664626058565452504846
2102001901801701601501401301201101009080706050403020100
MW vs. Rotor Angle Generator 2
MW Terminal_Gen '2' '1'gfedcb
95908580757065605550
2202102001901801701601501401301201101009080706050403020100
Generator 2: SEXS_GE
MW vs. Rotor Angle Generator 2
MW Terminal_Gen '2' '1'gfedcb
908886848280787674727068666462605856545250
2202102001901801701601501401301201101009080706050403020100
MW vs. Rotor Angle Generator 2
MW Terminal_Gen '2' '1'gfedcb
90888684828078767472706866646260585654
2001901801701601501401301201101009080706050403020100
Generator 3: ESDC1A
MW vs. Rotor Angle Generator 3
MW Terminal_Gen '3' '1'gfedcb
64636261605958575655545352515049
100
95
90
85
80
75
70
65
60
55
50
45
MW vs. Rotor Angle Generator 3
MW Terminal_Gen '3' '1'gfedcb
646362616059585756555453525150494847
105
100
95
90
85
80
75
70
65
60
55
50
45
Generator 3: ESAC1A
MW vs. Rotor Angle Generator 3
MW Terminal_Gen '3' '1'gfedcb
6160595857565554535251504948474645
100
95
90
85
80
75
70
65
60
55
50
45
MW vs. Rotor Angle Generator 3
MW Terminal_Gen '3' '1'gfedcb
61605958575655545352515049484746
105
100
95
90
85
80
75
70
65
60
55
50
45
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