Dynamic Phasors for Small Signal Stability AnalysisInteractions Between Nearby HVDC Converters A simple Network for model Validation Two HVDC lines, ac lters, ac transmission line,

IntroductionDynamic Phasors

ApplicationsCurrent Research Work

Dynamic Phasors for Small Signal Stability

Analysis

Udaya Annakkage (University of Manitoba)

Chandana Karawita (Transgrid Solutions)

Udaya Annakkage (University of Manitoba) Dynamic Phasors for Small Signal Stability Analysis



Outline

1 IntroductionSimulation and Analysis TechniquesTypical OutputsModelling of Components

2 Dynamic Phasors

3 ApplicationsInteractions Between Nearby HVDC ConvertersTorsional Interactions

4 Current Research Work




Simulation and Analysis TechniquesTypical OutputsModelling of Components

Power System Simulation and Analysis

Electromagnetic Transient Simulation - Time DomainTechnique

Transient Stability Simulation - Time Domain Technique

Small Signal Stability Analysis - Frequency DomainTechnique





Typical Output of an EMT Simulation

Sample responses of a four-generator power system after athree phase fault

The amplitude of the 60 Hz voltage waveform is modulated bythe low frequency of oscillations of the rotor.





Typical Output of a TS Simulation

Sample responses of a four-generator power system after athree phase fault

Rotor angle of generator 2 and the rms voltage of Bus 2 showlow frequency oscillations around 1 Hz.

Transient Simulation

Time(s) 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 3.8 4.71.2 f

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

Gen

erat

or S

peed

Cha

nge

(Rad

/s)

-0.084 0.181 0.265

Min -1.998 Max 1.754 Diff 3.752

W2

25

50

75

100

125

150

175

200

225

250

Bus

Volta

ge (k

V)

227.277 230.320 3.043

Min 195.244 Max 230.363 Diff 35.120

V4





Typical Output of a Small Signal Analysis

Structural Information

Oscillation modes (frequencies and correspondingdamping).

Mode Shapes of oscillation frequencies.

Participation of state variables in oscillation modes.

Observability of oscillation modes.

Controllability of oscillation modes.

Residues for input-output pairs.





Typical Output of a Small Signal Analysis:

Participation Factors

Participation Factors show the relative participation of statevariables when a mode is excited.

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

100

110

State Variables

Perc

enta

ge P

artic

ipat

ion

HVDC1 Z1

Z2 F1 and F2 HVDC2 Z3

Z4 F3 and F4Tie

Generator





Typical Output of a Small Signal Analysis: Mode

Shape

Mode Shape shows whether the state variables are oscillatingtogether or not.

Mode 6 Mode 7

Mode 8 Mode 9

30

210

240

270

300

150

330

180 0

60

90

120

Idcr1

Vcap1Idcr2

Idci2

Vcap2

30

210

240

270

300

150

330

180 0

60

90

120

Idcr1

Idci1

Vcap1

Idcr2

Idci2

Vcap2

X r2 X r1Idci1

30

210

60

240

90

270

120

300

150

330

180 0

Idcr1

X r1

Vcap1Idcr2

V

X r2

cap2

30

210

60

240

90

270

120

300

150

330

180 0Idcr1

X r1

Idci2

Idcr2

Idci1

X r2





Machine Models

Transient Stability and Small Signal Stability

Rotor fluxes are modelled as state variables.Stator fluxes are NOT modelled as state variables.

Electromagnetic Transient Simulation

Rotor fluxes and stator fluxes are modelled as statevariables.

Common to both

Dynamics of the rotor and that of auxiliary controllers aremodelled using differential equations.





Transmission Line Models

Transient Stability and Small Signal Stability

Series inductance and shunt capacitance are modelled asconstant impedances (admittances) calculated at thenominal frequency ω0.

Electromagnetic Transient Simulation

Transmission line is modelled using differential equations(telegraphic equations).





Small Signal Stability: Frequency domain technique

Only accurate in the vicinity of nominal frequency.

Structural Information relevant to the system is available.

Transient Stability: Time domain technique

Only accurate in the vicinity of nominal frequency.

Large integration time step is used ⇒ simulation is fast.

Electromagnetic Transient Simulation: Time domain technique

Accurate over a wide frequency range.

Integration time step is small ⇒ simulation is slow.




Dynamic Phasors

Instantaneous Current Waveform

iac = Amejφe jω0t = [Am cos(φ) + jAm sin(φ)]e jω0t

Am is the magnitude of the current , φ is the phase of thecurrent, and ω0 is the nominal system frequency.

In Rectangular Coordinates

iac = (IR + jII )ejω0t




Modelling a Transmission Line using Dynamic

Phasors

Series Branch

Series R-L circuit connected between nodes 1 and 2.

v12 = Ldi12dt

+ Ri12

Using the Complex rotating phasor relationships

(VR + jVI )ejω0t = L

d(IR + jII )ejω0t

dt+ R(IR + jII )e

jω0t




Assuming that the nominal system frequency (ω0) is constant

VR + jVI = Ld(IR + jII )

dt+ (R + jω0L)(IR + jII )

Since L is in pu, (ω0/L) terms appear instead of (1/L)

[∆IR∆II

]=

[ −Rω0

Lω0

−ω0−Rω0

L

] [∆IR∆II

]+

ω0

L0 −ω0

L0

0 ω0

L0 −ω0

L

∆V 1R

∆V 1I

∆V 2R

∆V 2I




Modelling a Transmission Line using Dynamic

Phasors

Parallel Branch

[∆V1R

∆V1I

]=

[ −ω0

RCω0

−ω0−ω0

RC

] [∆V1R

∆V1I

]+

ω0

C0

0 ω0

C

[∆IR∆II

]




Other Interpretations of Dynamic Phasors

d-q Components of Network Voltages and Currents

Network voltages and currents are represented by their d-qcomponents which are modelled as state variables.

Fourier Components of Network Voltages and Currents

Network voltages and currents are represented by their Fouriercomponents which are modelled as state variables.




Power System Signals as Amplitude Modulated

Signals

If R and I components are constants

The instantaneous waveforms are sinusoidal.

If R and I components are oscillating at frequency ω

The instantaneous waveforms are amplitude modulatedwaveforms with carrier frequency ω0. This results in twosidebands of ω0 − ω and ωo + ω




Power System Signals as Amplitude Modulated

Signals

Example

If f0 = 60 Hz and f = 5 Hz, the two sideband frequencies aref1 = 55 Hz and f2 = 65 Hz. Both are close to 60 Hz and theconstant admittance representation of transmission network isacceptable.

Example

If f0 = 60 Hz and f = 25 Hz, the two sideband frequencies aref1 = 35 Hz and f2 = 85 Hz . Both are significantly different to60 Hz and the constant admittance representation oftransmission network is NOT acceptable.




Interactions Between Nearby HVDC ConvertersTorsional Interactions

Interactions Between Nearby HVDC Converters

A simple Network for model Validation

Two HVDC lines, ac filters, ac transmission line, and agenerator.

A pulse of magnitude of 5 % and duration 0.3s wasapplied to the rectifier current controller input.





Comparison of EMT, SS-traditional, and

SS-Dynamic Phasor Approach.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0

0.02

0.04

0.06

0.08

time(s)

curre

nt (k

A)

(a) Change in Idcr of HVDC1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

−0.01

0

0.01

0.02

time(s)

curre

nt (k

A)

(b) Change in Idcr of HVDC2

PSCAD/EMTDCModel−1Model−2






Rotor Oscillations: SS-traditional and SS-Dynamic

Phasors give same results

0 0.5 1 1.5 2 2.5 3 3.5 4

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10−3

time(s)

spee

d (p

u)

Change in Generator Speed






Frequency Response of the Model – EMT Vs

SS-Dynamic Phasor

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

frequency (Hz)

% c

hang

e

Magnitude (input: , output: Vcap)

PSCAD/EMTDCSSS model

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

frequency (Hz)

phas

e an

gle

(Deg

.)

Phase (input: , output: Vcap)






Response to a 200 Hz signal

Changes in Rectifier side DC currents for a 5 %, 200Hzsinusoidal change of the HVDC1 rectifier side AC sourcevoltage (VS1).

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.04

0.02

0

0.02

0.04

time(s)

curre

nt (k

A)

(a) Change in Idcr of HVDC1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.01

0.005

0

0.005

0.01

time(s)

curre

nt (k

A)

(b) Change in Idcr of HVDC2

PSCAD/EMTDC Model 2 Model 1

PSCAD/EMTDC Model 2 Model 1





Participation Factors ⇒ presence of an interaction

between the two HVDC converters

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

100

110

State Variables

Perc

enta

ge P

artic

ipat

ion

HVDC1 Z1

Z2 F1 and F2 HVDC2 Z3

Z4 F3 and F4Tie

Generator





Mode Shape ⇒ state variables of the two

converters oscillate against each other

Mode 6 Mode 7

Mode 8 Mode 9

30

210

240

270

300

150

330

180 0

60

90

120

Idcr1

Vcap1Idcr2

Idci2

Vcap2

30

210

240

270

300

150

330

180 0

60

90

120

Idcr1

Idci1

Vcap1

Idcr2

Idci2

Vcap2

X r2 X r1Idci1

30

210

60

240

90

270

120

300

150

330

180 0

Idcr1

X r1

Vcap1Idcr2

V

X r2

cap2

30

210

60

240

90

270

120

300

150

330

180 0Idcr1

X r1

Idci2

Idcr2

Idci1

X r2





Torsional Interactions

The CIGRE benchmark HVDC test system with somemodifications.

A synchronous generator is connected at rectifier side ACbus to supply half of the P-Q requirement of rectifier.

S2Rectifier InverterS1 DC Line

Idcr IdciVcap

F1

F2

Z1 Z2

G

Generator with a multi-mass turbine





Comparison of EMT and SS-Dynamic Phasor

SS-Dynamic-Phasor provides accurate results in the frequencyrange of interest

10 % change in rectifier current reference for 10 ms

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

time(s)

curre

nt (k

A)

Change in Rectifier side DC current






Torsional Interaction Modes

Mode Freq. D Major Participants(Hz) (%)

A 16.24 -0.03 HVDC-Generator-TurbineB 16.36 1.05 HVDC-Generator-Turbine





Participating states are identified using

Participation Factors

0

20

40

60

80

100

0

20

40

60

80

100

Gen−Turbine HVDC

Mode−B

Mode−A

!LPB!LPA !IP !HP!gen XCCC XCEA Idcr Idci Vcap





Publications

1 C. Karawita, U.D. Annakkage, A Hybrid Network Model forSmall Signal Stability Analysis of Power Systems, IEEETransactions on Power Systems, Vol 25, 2010, Page(s):443–451.

2 C. Karawita, U.D. Annakkage, Multi-Infeed HVDC InteractionStudies Using Small-Signal Stability Assessment, IEEETransactions on Power Delivery, Vol 24, 2009, Page(s):910–918.

3 C. Karawita, U.D. Annakkage, HVDC-Generator-TurbineTorsional Interaction Studies Using A Linearized Model WithDynamic Network Representation, International Conference onPower System Transients (IPST), Kyoto, Japan, June 2009.




Current Research Work

SSR between DFIG based Wind Power Plant and seriescompensated transmission lines (Hiranya).

SSI between nearby LCC-HVDC and VSC-HVDCterminals (Kevin – MH).

SSR mitigation using FACTS controllers (TGS).

Transient Stability Simulation using Dynamic Phasors(Rae – MH).

Chandana has developed an SSR–Small Signal AnalysisProgram




Acknowledgements

Research Funding – HVDC Interactions – NSERC,University of Manitoba, and Province of Manitoba.

Research Funding – Wind Power Plant SSR Studies –NSERC and Manitoba Hydro.

Valuable Feedback – Bret Davis, Ioni Fernando, Ani Gole,Shaahin Filizadeh, and Garth Irwin.


Documents

Dynamic Phasors for Small Signal Stability AnalysisInteractions Between Nearby HVDC Converters A simple Network for model Validation Two HVDC lines, ac lters, ac transmission line,