Appendix A Data on IEEE Benchmark Models

A.1 IEEE FIRST BENCHMARK MODEL ( FBM )

IEEE FBM was created by the IEEE Working Group on Subsynchronous Reso­nance in 1977 for the purpose of establishing a benchmark model which can be used as a test bench for the comparison of different methods of computer based analysis and simulation. The system consists of a single generator connected to an infinite bus through a single series compensated line as shown in Fig A .1.

XT R XACSYS INF BUS

Figure A.I. IEEE FBM system diagram

Table A.l gives the network impedances in per unit on the generator MVA base of 892.4 MVA.

Table A.l Network Impedances

Parameter Positive Sequence Zero Sequence

R 0.02 0.50 XT 0.14 0.14 XL 0.50 1.56

X SYS 0.06 0.06 Xc 0.35 0.35

Table A.2 gives the synchronous machine parameters.

240 ANALYSIS OF SUBSYNCHRONOUS RESONANCE IN POWER SYSTEMS

Table A.2 Synchronous machine parameters for IEEE FBM

I Reactance I Value I Time Constant I Value I Xaa 0.130 T~o 4.300 Xd 1.790 Til

dO 0.032 x' d 0.169 T~o 0.850 x" d 0.135 Til

qO 0.050 Xq 1.710 x' q 0.228 x" q 0.200

The reactances are in per unit on the generator base and the time constants are in seconds. From the specified open circuit time constants, the short circuit time constants can be derived from Eqs.(2.74) to (2.77) in chapter 2. These values are given below. T~ = 0.4000, T~' = 0.0259 T~ = 0.1073, T~' = 0.0463

The rotor model of the FBM is shown in Fig . A.2. This is typical of large

668BBB Figure A.2. Rotor model for FBM

turbine-generator which has several turbine sections modelled separately. The data are given in Table A.3.

Table A.3 Shaft inertias and spring constants for the FBM in per unit on the machine base

Inertia I Inertia I Shaft Section I Spring Constant Constant (H) K in p.u T /rad

HP turbine 0.092897 HP - IP 19.303 IP turbine 0.155589 IP - LPA 34.929

LPA turbine 0.858670 LPA - LPB 52.038 LPB turbine 0.884215 LPB - GEN 70.858

Generator 0.868495 GEN - EXE 2.82 Exciter 0.0342165

APPENDIX A: DATA ON IEEE BENCHMARK MODELS 241

The damping data is not provided as part of FBM data.The damping is neglected in the case studies given in chapter 4. Whenever the damping is considered, the following data are assumed. Self damping: DHP = DIP = DLA = DLB = 0.2 Mutual damping: DHI = DIA = DAB = DBG = 0.3, DGE 0.005

A.2 IEEE SECOND BENCHMARK MODEL ( SBM )

The system diagram for SBM is shown in Fig. A.3.

RI XLI Xc

Figure A.3. Second Benchmark Model system

INF BUS

The value of the capacitive reactance, Xc is not specified explicitly. It is specified as a variable taking on values from 10% to 90% of the series inductive reactance of the same line.The network impedance data are given in Table A.4.

Table A.4 Network Impedances in per unit based on 100 MVA for SBM

Parameter I Positive Sequence I Zero Sequence

RT 0.0002 0.0002 X T 0.0200 0.0200 RI 0.0074 0.0220 XLI 0.0800 0.2400 R2 0.0067 0.0186

XL2 0.0739 0.2100 Rsys 0.0014 0.0014 XSYS 0.0300 0.0300

All data are given on a 100 MVA base and the line impedances are on a 500 kV base. The generator is rated at 600 MVA.The reactances and time constants are given in Table A.5.

242 ANALYSIS OF SUBSYNCHRONOUS RESONANCE IN POWER SYSTEMS

Table A.5 Synchronous machine parameters for IEEE SBM

I Reactance I Value I Time Constant I Value I Ra 0.0045 T~o 4.500 Xaa 0.140 Til

dO 0.040 Xd 1.650 T~o 0.550 x' d 0.250 Til

qO 0.090 x" d 0.200 Xq 1.590 x' q 0.460 x" q 0.200

The rotor model is shown in Fig. A.4.

Figure A.4. Rotor model for SSM

It has four masses including the exciter. The data are given in Table A.6.

Table A.6 SBM rotor model data

I Mass I Inertia I Damping I Ibm - ft 2 Ibf-ft-sec/rad

Shaft Section

Spring Constant Ibf-ft/ rad

EXC 1383 4.3 EXC-GEN 4.39 x 106

GEN 176204 547.9 LP-GEN 97.97x 106

LP 310729 966.2 HP-LP 50.12x 106

HP 49912 155.2

The rotor mode shapes are given in Table A.7 and the computed modal quantities are given in Table A.8.

Table A.7 Rotor mode shapes for SBM

I Rotor I Mode 1 I Mode 2 I Mode 3 I EXC 1.307 1.683 -102.600 GEN 1.000 1.000 1.000 LP -0.354 -1.345 -0.1180 HP -1.365 4.813 0.0544

APPENDIX A: DATA ON IEEE BENCHMARK MODELS 243

Table A,8 Computed modal quantities for SBM

I Mode fk Uk Hk

Hz rad/s seconds

1 24,65 0,05 1.55 2 32,39 0,05 9,39 3 51.10 0,05 74,80

Appendix B Calculation of Initial Conditions

In general,the inital conditions (equilibrium values) ofthe system state variables are obtained by solving the algebraic equations

(B.I)

where Ue is the input vector when the system is in equilibrium.lt includes parameters such as input mechanical torque (Tm ) ,infinite bus voltage (Eb), voltage reference to the AVR ( Vre ! ). In power sysem studies,the operating point is established by conducting a power flow analysis;the output from which gives the active power (P), reactive power (Q),voltage magnitude (V) and angle (0) at each bus including the generator bus. This is the starting point for the calculation of the initial conditions of the state variables. Synchronous Generator 1. The armature current (fa ) is calculated from

r = f LA. = Pg - jQg a a 'f' V L-O

9 9

(the subscript 'g' refers to the generator terminal bus) 2. Compute Eq and J from

where Eq is the voltage behind Xq

3. Compute id ,iq , Vd , and Vq from

id = -fa sin(J - c/J)

iq = fa cos(J - c/J) Vd = -Vg sin(J - Og)

Vq = Vgcos(J - Og)

(B.2)

(B.3)

(BA) (B.5) (B.6)

(B.7)

246 ANALYSIS OF SUBSYNCHRONOUS RESONANCE IN POWER SYSTEMS

4. Compute Efd ,E~ ,E~ from

EJd = Eq - (Xd - Xq)id

E~ = EJd + (Xd - X~)id

E~ = -(xq - x~)iq

5. Compute t/Jd and t/Jq from

t/Jd = Xdid + EJd

t/Jq = xqiq

6. Compute t/JJ ,t/Jh ,t/Jg and t/Jk from

X' t/Jf = t/Jd + ( d I)EJd

Xd - xd

7. Calculate VreJ and Tm from

t/Jh = t/Jd

t/Jg = t/Jq

t/Jk = t/Jq

(B.8)

(B.9)

(B.lO)

(B.ll)

(B.12)

(B.13)

(B.14)

(B.15)

(B.16)

(B.17)

(B.18)

For the rotor system,the slips of the various masses (Sm, SHP, SIP etc) cannot be calculated from the equations describing them.They are assumed to be zero if the rotor speed (w o ) is same as the base speed (w B)' The equilibrium values of the shaft torques can be calculated from the state equations for the rotor system.For example,for the four mass system shown in Fig.2.l6 (chapter 2),the initial condtions of the shaft torques are given by

TLG = Te, TIL = TLG - FLpTm

THI = TIL - F[pTm = FHpTm

(B.19)

(B.20)

If, instead of shaft torques,the modal angles are used as state variables, their initial conditions can be calculated from the state equations. For example,for the system shown in Fig.2.l6, we have,

<11 = (q~ FTm - Te)/ J{1

<12 = (q~FTm - Te)/ J{2

<13 = (q~FTm - Te)/ J{3

<10 = <I - <11 - <12 - <13

(B.2l)

(B.22)

(B.23)

(B.24)

where ql, q2 and q3 are the columns of the [QJ matrix corresponding to torsional modes 1,2 and 3. [FJ is a row vector defined by

F= [FHP ,FIP ,FLP ,OJ (B.25)

APPENDIX 8: CALCULATION OF INITIAL CONDITIONS 247

f{l ,f{2 and f{3 are the modal spring constants. Electrical Network The network equations are given by

(B.26)

If the matrix [AN] is nonsingular,the inital conditions of the network state variables are given by

(B.27)

The input vector UN includes variables such as D-Q components of the gener­ator (armature) currents and infinite bus voltages. For a particular case of a simple network made up of a series R-L-C network con­nected between a genearator and an infinite bus (with no shunt branch),there are only two state variables VCD and vCQ .The initial values of those are com­puted from the complex equation

(VCQ + JVCD) = -jXc(iQ + jiD) (B.28)

where Xc = --L-c and Wo

(B.29)

Appendix C Abbreviations

ac : alternating current AGC : automatic generation control AVR : automatic voltage regulator CA : constant angle CACV : constant ac voltage (control) CC : constant current (control) CDCV : constant dc voltage (control) CEA : constant extinction angle (control) dc : direct current EPC : equidistant pulse control ESS : excitation system stabilizer FACTS: flexible ac transmission system FCL : fault current limiter GPU : gate pulse unit GTO : gate turn-off thyristor H P : high pass (filter) HP : high pressure (turbine) HVDC : high voltage direct current Hz : hertz IGBT : insulated gate bipolar transistor IGE : induction generator effect IP : intermediate pressure (turbine) IPC : individual phase control KVL : Kirchoff's voltage law LP : low pass (filter) LP : low pressure (turbine) MCT : metal oxide semiconductor controlled thyristor MOV : metal - oxide varistor PLL : phase-locked loop PSDC : power swing damping controller

250 ANALYSIS OF SUBSYNCHRONOUS RESONANCE IN POWER SYSTEMS

PSS : power system stabilizer PWM : pulse width modulation rad : radian rms : root mean square s: second SC : synchronous condenser SCR : short circuit ratio SEDC : supplementary excitation damping controller SMC : supplementary modulation controller SR : susceptance regulator SSO : subsynchronous oscillation SSR : subsynchronous resonance SSSC : static synchronous series compensator STATCOM : static synchronous compensator STATCON : static condenser SVC : static var compensator SVR ; synchronous voltage reversal TCBR : thyristor controlled braking resistor TCPAR : thyristor controlled phase angle regulator TCR : thyristor controlled reactor TCSC : thyristor controlled series compensator T-G : turbine generator TI : torsional interaction TSC : thyristor switched capacitor UPFC : unified power flow controller VDCOL : voltage dependent current order limiter VSC : voltage source converter VST : voltage setting terminal

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Index

Acceleration signal, 131 Analysis of Graetz Bridge, 138 Analysis of Induction Generator

Effect(IGE),83 Analysis of TCSC, 208 Analysis of Torsional Interaction(TI), 87 Antiphase mode, 120 Apparent impedance of TCSC, 225 Automatic Generation Control (AGC), 7 Automatic Voltage Regulators (AVR), 7 Back To Back (BTB) links, 145 Band reject filter, 131 Bypass Damping Filter, 113 Bypass, 213 Calculation of initial conditions, 245 Clarke's transformation, 68 Combined system equations, 103 Common mode, 120 Commutation voltage, 141 Composite signal, 134 Constant AC Voltage (CACV), 145 Constant Angle (CA) control, 214 Constant Current (CC), 145 Constant DC Voltage (CDCV), 145 Constant Extinction Angle (CEA), 145 Control of SSSC, 231 Converter control, 144 Countermeasures for SSR, 111 D-axis equivalent circuit, 37 D-Q Axes impedances, 77 D-Q components, 65 D-Q reference frame, 193 Damping torque analysis, 14 Delay angle, 141 Design of PSS, 126 Dynamic stabilizer, 118 Eigenvalue analysis, 96 Electrical analogue for the torsional system,

46 Electrical system equations, 101

Electro-hydraulic governor, 56 Electromagnetic Transients Program, 59 Electromagnetic transients, 78 Energy neutral, 234 Equivalent circuits, 35 Excitation control system, 41 Exciter mode, 129 Extinction angle, 143 FACTS (Flexible AC Transmission System),

8 Fault Current Limiter, 11 Firing angle control, 147 Gain supervisor, 182 Gate Pulse Unit (GPU), 179 Graetz bridge, 138 GTO thyristors, 190 HVDC converter station, 138 HVDC transmission, 137 IEEE First Benchmark Model (FBM), 239 IEEE Second Benchmark Model (SBM), 241 IGBT,190 Immittance functions, 94 Impedance functions, 75 Implicit algorithm, 79 Induction Generator Effect(IGE), 5 Interarea mode, 121 Interface between AC and DC system, 148 Intraplant mode, 121 Kron's transformation, 65 Local swing mode, 121 Lumped multi mass model, 44 Master control, 146 MCT,190 Mechanical system equations, 101 Mechanically switched capacitors (MSC),

169 Metal Oxide Varistor (MOV), 214 Mitigation of SSR, 223 Modal damping, 49 Modal inertia, 48

262 ANALYSIS OF SUBSYNCHRONOUS RESONANCE IN POWER SYSTEMS

Modal spring constant, 48 Mode coupling, 13 Mode shapes, 51 Modelling of TCSC, 216 Modelling of transmission network, 64 Module control functions, 213 NGH Damping Scheme, 116 NGH damping, 8 Numerical integration, 78 Operational impedances, 35 Overlap angle, 141 Park's transformation, 22 Per unit quantities, 30 Phase-Locked Loop(PLL)., 147 Pole control, 146 Power angle curve, 174 Power invariant transformation, 28 Power scheduling control, 214 Power Swing Damping Control (PSDC), 215 Power System Stabilizers(PSS), 4, 122 Protective bypass, 213 Pulse Width Modulation (PWM), 190 Q-axis equivalent circuit, 37 Self excitation, 4 Sequence networks

or and (3,69 Series capacitor protection, 112 Short Circuit Ratio (SCR), 145 Speed governor, 55 SSR neutral, 234 Static Blocking Filter (SBF), 113 Static Compensator (STATCOM), 170 Static Condenser (STATCON), 170, 10 Static Synchronous Compensator

(STATCOM), 10 Static Synchronous Series Compensator

(SSSC), 10, 229 Static Var Compensator (SVC), 10, 169,4 Steam turbine, 55 Subsynchronous Damping Control (SSDC),

215 Subsynchronous Damping Controller

(SSDC),167

Subsynchronous oscillations, 3 SubSynchronous Resonance(SSR), 2 Supplementary Excitation Damping

Control(SEDC), 114 Supplementary Modulation Controller

(SMC),170 Susceptance Regulator (SR), 183 SVC Controller, 177 Synchronizing and damping torque, 88 Synchronizing circuit, 151 Synchronous Condenser (SC), 189 Synchronous machine model, 18 Synchronous Voltage Reversal (SVR), 220 System control hierarchy, 146 Thyristor Controlled Phase Angle Regulator

(TCPAR),10 Thyristor Controlled Reactor (TCR), 8, 170 Thyristor controlled series capacitor, 206 Thyristor Controlled Series Compensator

(TCSC), I, 205 Thyristor Switched Capacitor (TSC), 171 Torque angle loop, 15 Torsional filter, 130 Torsional interaction with PSS, 130 Torsional interaction with voltage controller,

200 Torsional interaction, 5 Torsional interactions with SSSC, 234 Transient Stability Control (TSC), 215 Transient torques, 5 Transmission lines, 64 Trapezoidal rule, 79 TSR mode, 213 Turbine generator mechanical system, 43 Unified Power Flow Controller (UPFC), 10,

231 Valve group control, 146 Vernier control, 208 Voltage Dependent Current Order Limiter

(VDCOL), 147 Voltage Source Converters (VSC), 189 Waiting mode, 208