Study of Subsynchronous Resonance in

Power Systems

A project report submitted by

Apelagunta Kama Naveen
EE10B004

In partial fullment of the requirements

for the degree of Bachelor of Technology

May, 2014

Department of Electrical Engineering,

Indian Institute of Technology, Madras
Contents
1 Introduction 6
1.1 Introduction to SSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Report outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Equipment Modeling 8
2.1 Generator modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Modeling rotating masses . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Network equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Fault modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Small disturbance stability analysis 14
3.1 Eigenvalues of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Variation of eigenvalues with change in capacitance . . . . . . . . . . . . . 18
4 System Simulation 20
4.1 Fault simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2
Abstract
This report presents a study of subsynchronous resonance. The ieee rst benchmark
model [1] is used for this purpose. The study includes modeling of synchronous generator,
turbine-shaft rotating elements and network elements. The stability of this system is
studied by determining the eigenvalues. The system stability is studied for dierent
values of the series capacitor. Three phase fault is modeled in the network and the
system is simulated to determine the transient response.

3
Notations
ωB base angular frequency
Ψk ux linkage along the kth axis after transformation and normalization k =
d, q, F, 1D, 1Q, 2Q

ωi angular frequency of the ith mass

vd , v q , v 0 transformed and normalized generator voltages
Td , Td , Tq , Tq short circuit time constants
0 00 0 00

Xd , Xd , Xd , Xq , Xq , Xq generator reactances
0 00 0 00

excitation
0
Ef

va , vb, vc phase voltages at generator terminals

Hi inertia constant of the ith mass
δi position of the ith mass
electrical torque
0
Te

Kij spring constant

Ti mechanical torque of the ith turbine section
ratio of the steady state mechanical torque of the ith turbine section to Tm
0
ri

Total mechanical torque

0
Tm

id , i q currents owing in the network

vc voltage across the capacitor
vt voltage of the innite bus
XL sum of reactance of transformer, transmission line and the innite bus
XL1 sum of reactance of transformer and transmission line
X2 reactance of innite bus
XC series capacitor reactance

4
R total resistance
D damping factor in pu torque per pu speed deviation,
DHP ,DIP ,DLA ,DLB self damping coecients

DHI ,DIA ,DAB ,DBG ,DGE mutual damping coecients

P power generated by the generator

V voltage at the generator bus
I current owing out of generator terminals
φ phase angle of voltage at generator
ψ phase angle of current

5
1 Introduction
Subsynchronous resonance (SSR) is a case where the electric network exchanges signi-
cant amount of power with the mechanical system. This phenomenon arises as a result
of the interaction between a xed series capacitor, used for compensating transmission
lines and the turbine generator shaft. This results is excessively high oscillatory torque
on machine shaft causing fatigue and damage. Since the two shaft failures at Mohave
station in Navada in 1970 and 1971, subsynchronous resonance has become topic of inter-
est in utilities where this phenomenon is a problem, and the determination of conditions
that excite these SSR oscillations are important to those who design and operate these
power systems [2-3].

1.1 Introduction to SSR

The formal denition of SSR is provided by IEEE [4] to be,

Subsynchronous resonance is an electric power system condition where the electric net-
work exchanges energy with a turbine generator at one or more of the natural fre-
quencies of the combined system below the synchronous frequency of the system.
Subsynchronous resonance can exist in a power system wherein the network has natural
frequencies that fall below the nominal frequency of the network voltages. Currents
owing in the ac network have two components: one component at the frequency of
the driving voltages and another sinusoidal component at a frequency that depends
entirely on the elements of the network. Park's transformation makes the 50/60Hz
component of current appear, as viewed from the rotor, as a dc current in the steady
state, but the currents of frequency that depends on the network elements are transformed
into currents of frequencies containing the sum and dierence of the two frequencies.
The dierence frequencies are called subsynchronous frequencies. These subsynchronous
currents produce shaft torques on the turbine-generator rotor that cause the rotor to
oscillate at subsynchronous frequencies. The presence of subsynchronous torques on
the rotor causes concern because the turbine-generator shaft itself has natural modes of
oscillation. It happens that the shaft oscillatory modes are at subsynchronous frequencies.
If the induced subsynchronous torques coincide with one of the shaft natural modes of
oscillation, the shaft will oscillate at this natural frequency, with a high amplitude. This
is called subsynchronous resonance, which causes shaft fatigue and possible damage or
failure[5].

6
1.2 Report outline

Chapter 2 gives the equations to model synchronous generator, turbine shaft system, and
the network. It includes a discussion on Park's transformation to convert the network
equations into Park variables. Chapter 2 also presents fault analysis. In Chapter 3, small
disturbance stability is evaluated while varying the value of series capacitor reactance.
Chapter 4 describes the results of the simulation.

7
2 Equipment Modeling
The IEEE rst benchmark model is used for the study of subsynchronous resonance [1].
Fig. 2.1 shows the single line diagram of this sysytem.

Figure 2.1: network for sub synchronous resonance studies

2.1 Generator modeling

The following generator equations were taken from [6]

dΨd
= −ω2 Ψq − ωB vd (2.1)
dt
dΨq
= ω2 Ψd − ωB vq (2.2)
dt
dΨ0
= −ωB v0 (2.3)
dt
0
dΨF 1 X
(2.4)
0
= 0 (−ΨF + Ψd − 0 d Ef )
dt Td Xd − Xd
dΨ1D 1
= 00 (−Ψ1D + Ψd ) (2.5)
dt Td
dΨ1Q 1
= 0 (−Ψ1Q + Ψq ) (2.6)
dt Tq

8
 dΨ2Q 1
= 00 (−Ψ2Q + Ψq ) (2.7)
dt Tq

Ψ0 = X0 i0 (2.8)
1 1 1 1 1
id = 00 Ψd + ( − 0 )ΨF + ( 0 − 00 )Ψ1D (2.9)
Xd Xd Xd Xd Xd
1 1 1 1 1
iq = 00 Ψq + ( − 0 )Ψ1Q + ( 0 − 00 )Ψ2Q (2.10)
Xq Xq Xq Xq Xq

Neglect the zero component terms Ψ0 , v0

2.2 Modeling rotating masses

The following equations are used to model the rotating masses [6]

Figure 2.2: rotating masses

The masses corresponding to i = 1, 2, 3, 4, 5, 6 are

mass 1 : exciter
mass 2 : generator
mass 3 : low pressure B
mass 4 : low pressure A
mass 5 : intermediate pressure
mass 6 : high pressure

dδ1
= ω1 − ωo (2.11)
dt
dδ2
= ω2 − ωo (2.12)
dt
dδ3
= ω3 − ωo (2.13)
dt
dδ4
= ω4 − ωo (2.14)
dt

9
 dδ5
= ω5 − ωo (2.15)
dt
dδ6
= ω6 − ωo (2.16)
dt
dω1 ωB
= [K12 (δ2 − δ1 )] (2.17)
dt 2H1
dω2 ωB
(2.18)
0
= [K23 (δ3 − δ2 ) − k̄12 (δ2 − δ1 ) − Te ]
dt 2H2
dω3 ωB
= [T3 + K34 (δ4 − δ3 ) − K23 (δ3 − δ2 )] (2.19)
dt 2H3
dω4 ωB
= [T4 + K45 (δ5 − δ4 ) − K34 (δ4 − δ3 )] (2.20)
dt 2H4
dω5 ωB
= [T5 + K56 (δ6 − δ5 ) − K45 (δ5 − δ4 )] (2.21)
dt 2H5
dω6 ωB
= [T6 − K56 (δ6 − δ5 )] (2.22)
dt 2H6

Use Te = Ψd iq − Ψq id and re-write equation (2.18) as

0

dω2 ωB
= [K23 (δ3 − δ2 ) − K12 (δ2 − δ1 ) − (Ψd iq − Ψq id )] (2.23)
dt 2H2

Including damping in the analysis modies the equations of the rotating masses to the
following [7]
dω1 ωB ω1 − ω2
= [K12 (δ2 − δ1 ) − DGE ] (2.24)
dt 2H1 ωo

dω2 ωB 1 1 1
= [K23 (δ3 − δ2 ) − K12 (δ2 − δ1 ) − (Ψd ( 00 Ψq + ( − 0 )Ψ1Q
dt 2H2 Xq Xq Xq
1 1 1 1 1
+( 0 − 00 )Ψ2Q ) − Ψq ( 00 Ψd + ( − 0 )ΨF
Xq Xq Xd Xd Xd
1 1 ω2 − ω1 ω2 − ω3
+( 0 − 00 )Ψ1D )) − DGE − DBG ] (2.25)
Xd Xd ω o ωo

dω3 ωB
= [T3 + K34 (δ4 − δ3 ) − K23 (δ3 − δ2 ) −
dt 2H3
ω3 − ω2 ω3 − ω4 ω3 − ωo
DBG − DAB − DLB ] (2.26)
ωo ωo ωo

10
 dω4 ωB
= [T4 + K45 (δ5 − δ4 ) − K34 (δ4 − δ3 ) −
dt 2H
ω4 − ω3 ω4 − ωo ω4 − ω5
DAB − DLA − DIA ] (2.27)
ωo ωo ωo

dω5 ωB
= [T5 + K56 (δ6 − δ5 ) − K45 (δ5 − δ4 ) −
dt 2H5
ω5 − ω4 ω5 − ωo ω5 − ω6
DIA − DIP − DHI ] (2.28)
ωo ωo ωo

dω6 ωB ω6 − ω5 ω6 − ωo
= [T6 − K56 (δ6 − δ5 ) − DHI − DHP ] (2.29)
dt 2H6 ωo ωo

2.3 Network equations

The following simpliedRLC circuit is used to model the network

Figure 2.3: network for modeling the RLC elements

Writing network equations in a,b,c phases, we get two sets of normalized equations

   dia         
va dt ia vca vta 0
1
−  vb  + XL  dib
dt
 + R  ib  +  vcb  +  vtb  =  0  (2.30)
ωB dic
vc dt
ic vcc vtc 0
   dvca 
ia dt
1
 ib  =  dvcb
dt
 (2.31)
ωB XC dvcc
ic dt

11
The above set of equations are transformed to
ω2 XL did
−vd + XL iq + + Rid + vcd + vtd = 0 (2.32)
ωB ωB dt
ω2 XL diq
−vq − XL id + + Riq + vcq + vtq = 0 (2.33)
ωB ωB dt
dvcd
= id ωB XC − ω2 vcq (2.34)
dt
dvcq
= iq ωB XC + ω2 vcd (2.35)
dt

Transform the Park parameters vtd and vtq into Kron parameters.

vtq + jvtd = (VtQ + jVtD )e−jδ2

vtd = VtD cos(δ2 ) − VtQ sin(δ2 ) (2.36)

vtq = VtQ cos(δ2 ) + VtD sin(δ2 ) (2.37)

2.4 Fault modeling

The following circuit model is used for deriving network equations for simulating transient
response during fault [1]
Writing equations in a,b,c phases before the fault is cleared from any of the three phases:

Figure 2.4: network for fault analysis

12
      dia   
0 va dt ia
1
 0  = −  vb  + XL1  didtb  + R  ib  +
ωB dic
0 vc dt
ic
   di2a   
vca dt vta
 vcb  + 1 X2  di2b  +  vtb  (2.38)
ωB dt
vcc di2c vtc
dt
   di1a   di2a   
0 dt dt vta
 0  = − 1 XF  di1b  + 1 X2  di2b  +  vtb  (2.39)
ωB dt ωB dt
0 di1c di2c vtc
dt dt
   dvca 
ia
1  dvdtcb 
 ib  =
dt
(2.40)
ωB XC dvcc
ic dt
Using the relation
i2abc = iabc − i1abc
And doing Park's Transformation we get,

ω2 X2 X2 1 X2 X2 did
0 = −vd + (XL1 + X2 − )iq + (XL1 + X2 − ) +
ωB XF + X2 ωB XF + X2 dt
X2
Rid + vcd + vtd (1 − ) (2.41)
XF + X2

ω2 X2 X2 1 X2 X2 diq
0 = −vq − (XL1 + X2 − )id + (XL1 + X2 − ) +
ωB XF + X2 ωB XF + X2 dt
X2
Riq + vcq + vtq (1 − ) (2.42)
XF + X2

dvcd
= id ωB XC − ω2 vcq (2.43)
dt
dvcq
= iq ωB XC + ω2 vcd (2.44)
dt

Use equation (2.41) to equation (2.44) to model the network during the fault

13
3 Small disturbance stability analysis
Use
P = V I cos(φ − ψ) (3.1)
P
I=
V cos(φ − ψ)
and then ψ can be found using
P
ψ = φ − cos−1 ( ) (3.2)
VI

VD , ID , VQ , IQ are found using the following equations

VD = V sin φ (3.3)

VQ = V cos φ (3.4)

ID = I sin ψ (3.5)

IQ = I cos ψ (3.6)

Consider equation (2.1) to equation (2.7), the two algebraic equations (2.9) and (2.10)
and equation (2.11) to equation (2.16). Equate the derivatives to zero. Use subscript
o to denote initial value of the state variables

0 = −ω2o Ψqo − ωB vdo (3.7)

0 = ω2 Ψd − ωB vq (3.8)
0
1 X
(3.9)
0
0 = 0 (−ΨF + Ψd − 0 d Ef )
Td Xd − Xd
1
0= 00 (−Ψ1D + Ψd ) (3.10)
Td
1
0= (−Ψ1Q + Ψq ) (3.11)
Tq0

14
 1
0= (−Ψ2Q + Ψq ) (3.12)
Tq00

0 = ω1 − ωo (3.13)

0 = ω2 − ωo (3.14)

0 = ω3 − ωo (3.15)

0 = ω4 − ωo (3.16)

0 = ω5 − ωo (3.17)

0 = ω6 − ωo (3.18)
and
1 1 1 1 1
id = 00 Ψd + ( − 0 )ΨF + ( 0 − 00 )Ψ1D
Xd Xd Xd Xd Xd

1 1 1 1 1
iq = 00 Ψq + ( − 0 )Ψ1Q + ( 0 − 00 )Ψ2Q
Xq Xq Xq Xq Xq

we get the following relations:

ω1o = ω2o = ω3o = ω4o = ω5o = ω6o = ωo (3.19)
we assume that
ωo = ωB
therefore,

Ψqo = −vdo (3.20)

Ψdo = vqo (3.21)

0
Xd
(3.22)
0
ΨF o = Ψdo − E
Xd − Xd f o
0

Ψ1Do = Ψdo (3.23)

Ψ1Qo = Ψqo (3.24)

15
 Ψ2Qo = Ψqo (3.25)
0
1 1 1 X 0 1 1
ido = 00 Ψdo + ( − 0 )(Ψdo − 0 d Ef o ) + ( 0 − 00 )Ψdo
Xd Xd Xd Xd − Xd Xd Xd

which gets simplied to

0 0
(Ψdo − Ef o ) (vqo − Ef o )
ido = = (3.26)
Xd Xd

which is equal to
1 1 1 1 1
iqo = Ψqo + ( − 0 )Ψqo + ( 0 − 00 )Ψqo
Xq00 Xq Xq Xq Xq

which gets simplied to

Ψqo −vdo
iqo = =
Xq Xq

To get δo ,
0
vqo + jvdo = Xd ido − jXq iqo + Ef o

0
vqo + jvdo + jXq (iqo + jido ) = Xd ido + Ef o − Xq ido

0
(VQo + jVDo ) + jXq (IQo + jIDo ) = (Xd ido + Ef o − Xq ido )ejδo

δo = ∠(VQo + jVDo ) + jXq (IQo + jIDo ) (3.27)

also,

vqo = Re{e−jδo (VQo + jVDo )} (3.28)

vdo = Im{e−jδo (VQo + jVDo )} (3.29)

iqo = Re{e−jδo (IQo + jIDo )} (3.30)

ido = Im{e−jδo (IQo + jIDo )} (3.31)

Therefore, Ψqo , Ψdo , ΨF o , Ψ1Do , Ψ1Qo , Ψ2Qo , Ef o can be found
0

Now consider equation (2.26) to equation (2.29) and equate their derivatives to zero

16
 ωB ω2
0= (vd − XL iq − Rid − vcd − vtd ) (3.32)
XL ωB
ωB ω2
0= (vq + XL id − Riq − vcq − vtq ) (3.33)
XL ωB

0 = id ωB XC − ω2 vcq (3.34)

0 = iq ωB XC + ω2 vcd (3.35)

From equation (3.34), we get

vcqo = XC ido (3.36)

And from equation (3.35), we get

vcdo = −XC iqo (3.37)

We can substitute for vcqo and vcdo and nd vtdo and vtqo
VtD and VtQ are found using

VtD = Im{ejδ2 (vq + jvd )} (3.38)

VtQ = Re{ejδ2 (vq + jvd )} (3.39)

To nd δ and torque (T ) values in steady state, consider equation (2.17) to equation
(2.22)
ωB
0= [K12 (δ2 − δ1 )] (3.40)
2H1
ωB
(3.41)
0
0= [K23 (δ3 − δ2 ) − K12 (δ2 − δ1 ) − Te ]
2H2
ωB
0= [T3 + K34 (δ4 − δ3 ) − K23 (δ3 − δ2 )] (3.42)
2H3
ωB
0= [T4 + K45 (δ5 − δ4 ) − K34 (δ4 − δ3 )] (3.43)
2H4
ωB
0= [T5 + K56 (δ6 − δ5 ) − K45 (δ5 − δ4 )] (3.44)
2H5
ωB
0= [T6 − K56 (δ6 − δ5 )] (3.45)
2H6

Te = P ; T3 = r1 Te ; T4 = r2 Te ; T5 = r3 Te ; T6 = r4 Te ; r1 , r2 , r3 , r4 values are provided

0 0 0 0 0

17
 δ = δ2o = δo (3.46)
0
Te
δ3o = + δ2o (3.47)
K23
0
T − T3
δ4o = e + δ3o (3.48)
K34
0
T − T3 − T4
δ5o = e + δ4o (3.49)
K45
0
Te − T3 − T4 − T5
δ6o = + δ5o (3.50)
K56

3.1 Eigenvalues of the system

Table 3.1: Computed Eigenvalues for the First Benchmark Model with Damping with
XC = .35
Eigenvalue number Real Part, (s−1 ) Imaginary Part,(rad/s)
1,2 -4.7121744130 +
616.6209083590
3,4 -3.7008370093 +
298.1489949776
5,6 -0.7304785937 +
202.7832149399
7,8 -1.2872748318 +
160.2733094920
9,10 -2.9058597402 +
136.7709702242
11,12 -0.3706471041 +
127.2403263027
13,14 -0.3999191450 +
99.8004681100
15 -32.9865535860
16 -20.3727594459
17,18 -0.7519406710 +
10.2114921725
19 -3.8483197170
20 -0.3151077171

3.2 Variation of eigenvalues with change in capacitance

The series capacitor reactance is varied from 10% to 100% of transmission line reactance,
and the real parts of eigenvalues are plotted.

18
Figure 3.1: graph of eigenvalues with dierent values of capacitance

19
4 System Simulation
Parameter values for simulation taken from [1] are presented
Generator power output P = 0.9 pu
Generator power factor PF = 0.9 pu (lagging)
Fault reactance XF = 0.04 pu
Series capacitor reactance XC = .350 pu

4.1 Fault simulation

A three phase fault is simulated using the equations described in Section 2.4, and capac-
itor voltage, generator current and generator electrical torque are plotted.

Figure 4.1: response curves for transient case

20
References
[1] IEEE Transactions on Power Apparatus and Systems, Vol. PAS-96, no.5,
September/October 1977, First Benchmark Model For Computer Simulation
of Subsynchronous Resonance
[2] IEEE Subsynchronous Resonance Working Group of the System Dynamic
Performance Subcommittee, "Terms, Denitions And Symbols For Subsyn-
chronous Oscillations," IEEE Transactions on Power Apparatus and Systems,
vol. PAS-104, No. 6, pp. 1326- 1333, June 1985.
[3] Iman Mohammad Hoseiny Naveh and Elyas Rakhshani,"Mitigation of Detri-
mental Subsynchronous Oscillations by Linear Optimal Control with Pre-
scribed Degree of Stability," 2009 Third IEEE International Conference on
Power Systems,Kharagpur, INDIA December 27-29
[4] IEEE SSR Working Group, "Proposed Terms and Denitions for Subsyn-
chronous Resonance," IEEE Symposium on Countermeasures for Subsyn-
chronous Resonance, IEEE Pub. 81TH0086-9-PWR, 1981,p 92-97.
[5] Paul M. Anderson, B. L. Agrawal, J. E. Van Ness, Subsynchronous Resonance
in Power Systems, Wiley-IEEE Press,January 1999
[6] K.R. Padiyar, Analysis of Subsynchronous Resonance in Power Systems,Springer,
1998.
[7] P. Kundur, Power SystemControl and Stability, McGraw-Hill, Inc.,1993.

21