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Masters of Engineering
Small Signal Stability
Aaron CowanElectrical Engineering
Power
Small Signal Stability
• ExciterField current– Field current
– Terminal voltageP S t St bili• Power System Stabilizer– Enhance stability– Rotor angle
• Equal Area Criterion (Fig 13.5, Kundur)g– Aa < Ad
– Aa > Ada d
SMIB Example
PSS
delta_wrv _s
delta _deltadelta _Tedelta _Psi_fd
v_s
1
K_4
K 21K_3
K AV ref
delta _Tm
delta _wr
v_1V_ref
w_0/s2*Hs+K_D
K_6
K_2
K_1
s
Field Circuit
T_3.s+1Exciter
K_AV_ref
delta _E_t
Voltage Transducer
1
T_R.s+1K_5
Problem details in section 12.3 of Power System Stability and Control, Kundur
Results
Matlabωd = 1.21Hz
Kundurωd = 1.05Hz dωd 1.21Hz
ξ = 0.1447KS = 1.1062K 15 6306
ωd 1.05Hzξ = 0.15KS = 0.829K 14 08
State Matrix and eigenvalues agree
KD = 15.6306KD = 14.08
A
Power World Transient Stability
Bus 2
163 MW
Bus 7 Bus 8 Bus 9 Bus 3
85 MW1.016 pu
Bus 5
163 MW 7 Mvar
85 MW -11 Mvar
100 MW 35 Mvar
Bus 6
1.026 pu1.025 pu
0.996 pu
1.032 pu 1.025 pu
1.013 pu
Bus 4
125 MW 50 Mvar
90 MW 30 Mvar
1.026 pu
slack
Bus1
72 MW 27 Mvar
1.040 pu
WECC equivalent in Power World
Exciter Models
Exciter Models
Exciter Models
PSS Model
IEEE 421.2
SMIB – Power World
l{• Equivalent SMIB• State Matrix• Eigenvalues
{• Eigenvalues
{
Power World Transient Stability
Bus 2
163 MW
Bus 7 Bus 8 Bus 9 Bus 3
85 MW1.016 pu
Bus 5
163 MW 7 Mvar
85 MW -11 Mvar
100 MW 35 Mvar
Bus 6
1.026 pu1.025 pu
0.996 pu
1.032 pu 1.025 pu
1.013 pu
Bus 4
125 MW 50 Mvar
90 MW 30 Mvar
1.026 pu
slack
Bus1
72 MW 27 Mvar
1.040 pu
WECC equivalent in Power World
Stability Simulation
• Default values usedDid change T to 0 02 in all cases– 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 7-5Both breakers open– Both breakers open
– Cleared in 0.07 secTh f h E it• 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 1220
200
MW vs. Rotor Angle Generator 1240
220200
180
160
140
120
100
200
180
160
140
120
10080
60
40
20
0
-20
80
60
40
20
0
-20
MW Terminal_Gen '1' '1'gfedcb
2520151050-5-10-15-20-25-30-35-40-40
MW Terminal_Gen '1' '1'gfedcb
20151050-5-10-15-20-25-30-35-40
20
Generator 2
Generator 2: ESDC1A
MW vs. Rotor Angle Generator 2210200190
MW vs. Rotor Angle Generator 2220210200190
180170160150140130120110100
200190180170160150140130120110100
9080706050403020
1101009080706050403020
MW Terminal_Gen '2' '1'gfedcb
100959085807570656055
100
MW Terminal_Gen '2' '1'gfedcb
100959085807570656055
100
Generator 3
Generator 3: SEXS_GE
MW vs. Rotor Angle Generator 3
100
MW vs. Rotor Angle Generator 3
95
95
90
85
80
75
70
90
85
80
75
7070
65
60
55
50
45
65
60
55
50
45
MW Terminal_Gen '3' '1'gfedcb
6059585756555453525150494847
45
MW Terminal_Gen '3' '1'gfedcb
5958575655545352515049
Summary
• Power World Transient StabilityBlock Diagrams– Block Diagrams
– SMIB EigenvaluesESDC1A ith t PSS• 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 1MW vs. Rotor Angle Generator 1170160150140130120110100
MW vs. Rotor Angle Generator 1200
180
160
140
120
9080706050403020100
100
80
60
40
20
MW Terminal_Gen '1' '1'gfedcb
151050-5-10-15-20-25-30-35-40-45
0-10-20-30
MW Terminal_Gen '1' '1'gfedcb
151050-5-10-15-20-25-30-35-40-45
0
-20
Generator 1: SEXS_GE
MW vs Rotor Angle Generator 1 MW vs Rotor Angle Generator 1MW vs. Rotor Angle Generator 1
180
160
140
120
MW vs. Rotor Angle Generator 1170160150140130120110100
100
80
60
40
20
1009080706050403020
MW Terminal_Gen '1' '1'gfedcb
20151050-5-10-15-20-25-30
0
-20
MW Terminal_Gen '1' '1'gfedcb
1614121086420-2-4-6-8-10-12-14-16-18-20-22-24-26-28-30
100
-10-20
Generator 2: ESAC1A
MW vs Rotor Angle Generator 2 MW vs Rotor Angle Generator 2MW vs. Rotor Angle Generator 2210200190180170160150140130
MW vs. Rotor Angle Generator 2220210200190180170160150140130
120110100908070605040
140130120110100908070605040
MW Terminal_Gen '2' '1'gfedcb
989694929088868482807876747270686664626058565452504846
403020100
MW Terminal_Gen '2' '1'gfedcb
95908580757065605550
403020100
Generator 2: SEXS_GE
MW vs. Rotor Angle Generator 2220210200190180170160150
MW vs. Rotor Angle Generator 2200190180170160150140
1401301201101009080706050
1301201101009080706050
MW Terminal_Gen '2' '1'gfedcb
908886848280787674727068666462605856545250
50403020100
MW Terminal_Gen '2' '1'gfedcb
90888684828078767472706866646260585654
403020100
Generator 3: ESDC1A
MW vs Rotor Angle Generator 3 MW vs Rotor Angle Generator 3MW vs. Rotor Angle Generator 3
100
95
90
85
80
MW vs. Rotor Angle Generator 3
105
100
95
90
8580
75
70
65
60
55
80
75
70
65
60
55
MW Terminal_Gen '3' '1'gfedcb
64636261605958575655545352515049
50
45
MW Terminal_Gen '3' '1'gfedcb
646362616059585756555453525150494847
50
45
Generator 3: ESAC1A
MW vs. Rotor Angle Generator 3 MW vs. Rotor Angle Generator 3100
95
90
85
80
105
100
95
90
85
8075
70
65
60
55
50
80
75
70
65
60
55
MW Terminal_Gen '3' '1'gfedcb
6160595857565554535251504948474645
50
45
MW Terminal_Gen '3' '1'gfedcb
61605958575655545352515049484746
50
45
