radman ttu simulation wide area frequency 20090604

8/9/2019 Radman Ttu Simulation Wide Area Frequency 20090604

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Simulation of Wide Area Frequency

Measurements fromPhasor Measurement Units (PMUs)

or

Frequency Disturbance Recorders (FDRs)

Ghadir Radman and Nick Hodges



Outline

• Motivation for the Study• What is Meant by Power System Frequency at Certain

Bus/Location?• Frequency Measured by PMUs/FDRs• Simulation of Frequency Measured by PMUs/FDRs

• Proposed Methods for Simulation of Frequency Measured byPMUs/FDRs

• Some Comments: – Frequency Wave Propagation – Bus Frequency Calculated by PSS/E

– Equivalent Inertial Center Frequency• Five-Bus Example System• Other Test Systems• Discussion

2



Motivation for the Study

• It is anticipated that PMUs and FDRs will be integrated with powersystem for real time control: – Power System Stabilizer (PSS) – Automatic Load Frequency Control (ALFC)

• It would be desired to simulate the system performance before theactual implementation: – In actual implementation we use the measurements from FDRs

and/or PMUs – What should we use for those measurements in simulation

phase?

• We propose a method of estimating the system frequency atcertain bus / location that simulates the measured frequency byPMUs / FDRs

3



What is Meant by Power System Frequency at

Certain Bus/Location? (Cont.)

• The voltage at a generic bus is contributed by all

generators in the power system. Thus this voltage

involves the frequencies of all generators

• Question:

– What is meant by power system frequency at certain

location/bus?

– Which frequency is measured by PMUs / FDRs?

5





• Answer:

– In Steady State all generators have the same frequency

( f ss Hz). The ith machine may be considered as a voltagesource of:

– Thus, the voltage at all points of the power system will

have the same frequency ( f ss) and this is the frequencymeasured by PMUs and/or FDRs.

( )issii

t f E t e δ π += 2cos)(

6





• Upon a disturbance the power system enters into a dynamicstate and different generators will run with differentfrequencies that vary with respect to time.

In this case, a generator may be considered as a voltagesource of:

where, E i , f i , and δi are slow varying functions of time. Thesetime functions may be found by solving the DifferentialAlgebraic Equations (DAE) representing the power system.

( )iiii

t f E t e δ π += 2cos)(

7





At a given time t, for a small interval or window of time

(Δt = n cycles), E i , f i, and δi may be assumed constants.

During this interval, the power system can be represented as:

Transmission

Network

SERIES

DEVICES

[ ]Y-matrix

LO A

D

LO A

D

LO A

D

.

.

.

. . . .

.

.

.

SHUN

T

D

EVI

CE

.

SHUN

T

DEVI

C

E

. . . . . . jXE

I G

8



Now we have a circuit with several voltage sources of differentfrequencies. The actual voltage at Bus i, using superposition theorembecomes:

where, represents the voltage at Bus i due to generator j .

∑=

=++=G

G

N

j ji N ii

act

i t vt vt vt v

1,,1,

)()()()(



( ) ( )GGG N i N N iii

t f V t f V ,,1,11,

2cos2cos θ π θ π ++++=

( )∑=

+=G

N

j ji j ji

t f V 1

,, 2cos θ π

9

)(, t v

ji





The equation shows that the voltage at Bus i is a multi-frequency voltage. These frequencies are all close to 60 Hz.

Now, even though the voltage at Bus i involves many

frequencies, we approximate this voltage to an estimatedsingle-frequency voltage (this approximation is acceptable fordynamic studies of power systems) defined as follows:

The frequency f iest represents the frequency of the system at

this location / bus and equals the frequency measured byPMUs / FDRs.

( )est

i

est

i

est

i

est

i t f V t v θ π += 2cos)(

10



Frequency Measured by PMUs/FDRs

• PMUs and FDRs implicitly assume that the estimatedvoltage signal is equal to the actual signal:

or

It should be noted that, the above equality is notpossible and it is only an approximation

∑=

==G N

j ji

act

i

est

i t vt vt v

1, )()()(

( ) ( )∑=

+=+G N

j ji j ji

est

i

est

i

est

i t f V t f V

1,,

2cos2cos θ π θ π

11



Frequency Measured by PMUs/FDRs (Cont.)

The FDRs/PMUs do not know the individual terms in the right

hand side of the equation; they only have access to thesampled value of the total (or actual) signal, vi

act (t), whichthey use to calculate f i

est .

The measurements for a certain interval (or window) of time,(Δt = n cycles) are used to find the value of f i

est for the entireduration of Δt .

12

∑=

==G N

j ji

act

i

est

i t vt vt v

1, )()()(



Simulation of Frequency Measured by

PMUs/FDRs

Simulation of the PMUs/FDRs measurements is alsobased on the same equation as used by PMUs/FDRs:

Unlike the real world measurements, in simulationdomain we know every term in the right hand side of theequation using which we try to find the best value of theestimated frequency, f i

est .

∑=

==G N

j ji

act

i

est

i t vt vt v

1, )()()(

or

( ) ( )∑=

+=+G N

j ji j ji

est

i

est

i

est

i t f V t f V

1,,


13



Proposed Methods for Simulation of Frequency

Measured by PMUs/FDRs

Method 1- Brute Force: – Assuming the actual voltage is identical to the estimated

voltage as follows:

then we have:

– The above equation can be used for several instants of timeduring the interval of interest. The average will taken andchosen to be the estimated frequency

−

= − est

iest

i

act

iest

i

V

t v

t f θ

π

)(cos

2

1 1

15

( )est

i

est

i

est

i

act

i t f V t v θ π += 2cos)(




Measured by PMUs/FDRs (Cont.)

Method 2--Four-Point Total Square Error Minimization: – In this method, f i

est is found by minimizing the sum of thesquare error at four points. These points are t = T/4, t = T/2, t =3T/2, t = 2T , where T = 1/60. Where the Square Error (SE) at agiven time, t k , is found by:

– Thus the Total Square Error (TSE) is given by the following:

– The minimum total square error is found by taking the firstderivate of TSE(t) , setting it equal to zero, and solving for f i

est .

( )2))()( k est ik

act ik t vt vt SE −=

)()()()()( 4321 t SE t SE t SE t SE t TSE +++=

16





Method 3- Proportional Voltage-Frequency:

– In this method, f iest is estimated using the

following equation:

– It is assumed that the contribution of each of the

components on the right hand side to f iest isproportional to its magnitude:

∑=

−=

G N

j jest

i

jiest i jiest

i f V

V f

1

,, cos θ θ

17

( ) ( )∑=

+=+G N

j ji j ji

est

i

est

i

est

i t f V t f V

1,,






Comparison of the Methods:

The table shows a comparison of different proposed

methods in terms of Integral of Squared Error (ISE) and

Normalized Error (NE) :

18

Frequency Estimation

Method

Integral of Squared

Error (ISE)Normalized Error

Brute Force 5.946E-16 1.8888E-07

Four-Point Total

Square Error

Minimization

6.2675E-16 1.9392E-07

Proportional Voltage-

Frequency 6.2985E-16 1.944E-07



Some Comments

Frequency Wave Propagation:

– The frequency measurement from PMUs/FDRs has thecharacteristics of a traveling wave which shows theimpacts of a disturbance travels with a certain speed. Thisspeed (although less than the speed of light) is large

enough for the impacts to be considered instantaneous fordynamic behavior studies of power systems.

– In simulation domain, the impact of any disturbance in thesystem is felt by any part of the system instantly. This is

due to the fact that the transmission system is modeled asa static system by its Y-matrix.The magnitude of the impactmay be different for different locations at different times,similar to PMUs/FDRs measurements.

19



Some Comments (Cont.)

Bus Frequency Calculated by PSS/E:

– PSS/E also calculates a frequency for each bus.

– When plotted for the simulations we performed, it

seems that the frequencies of bus-voltages

estimated by PSS/E are unrealistically different

than the frequencies of the generators which areresponsible for those voltages.

20



Some Comments (Cont)

Equivalent Inertial Center Frequency:

The equivalent inertial center frequency of a system is a singlevalue representing (approximately) the frequency of the entire system.

It is calculated by taking a weighted average of machine frequenciesand machine inertias as follows:

The center frequency is only a function of time (not location /bus) and is, thus, the same for all buses in the system considered.

∑=

= G N

ii

ii

c

H

f H f

1

21

c f



Five-Bus Example System

Bus 4

Bus 5

Bus 1 Bus 3

Bus 2

22

A 5-cycle, three-phase short-circuit fault is applied

to Bus 3



Five-Bus Example System (Cont.)

• Bus Voltage Magnitude from Simulation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

Time (s)

B u s

V o l t a g e M a g n i t u d e s ( p . u . )

V1

V2

V3

V4

V5

23





• Bus Voltage Angle from Simulation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-20

-15

-10

-5

0

5

10

15

Time (s)

B u s

V o l t a g e

A n g l e s

( d e g )

th1

th2

th3

th4

th5

24





• Frequency Simulation for Bus 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 559.95

60

60.05

60.1

60.15

Time (s)

F r e q u e n c y

( H z )

Machine Freq 1

Machine Freq 2

Bus Freq 1/PSSE

Bus Freq 1/Proposed Method

Center Frequency

25






0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 559.95

60

60.05

60.1

60.15

60.2

Time (s)

F r e q u e n c y

( H z )

Machine Freq 1

Machine Freq 2

Bus Freq 2/PSSE


Center Frequency

26






0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

60

60.2

60.4

60.6

60.8

61

61.2

Time (s)

F r e q u e n c y ( H z )

Machine Freq 1

Machine Freq 2

Bus Freq 3/PSSE


Center Frequency

27






0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 559.95

60

60.05

60.1

60.15

60.2

Time (s)

F r e q u e n c y

( H z )

Machine Freq 1

Machine Freq 2

Bus Freq 4/PSSE


Center Frequency

28






0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

59.95

60

60.05

60.1

60.15

60.2

Time (s)

F r e q u e n c y

( H z )

Machine Freq 1

Machine Freq 2

Bus Freq 5/PSSE


Center Frequency

29




30

Other Test Systems

IEEE 57-bus System: A 5-cycle, three-phase

short-circuit fault is applied to Bus 18



Other Test Systems

31

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

59.8

59.9

60

60.1

60.2

60.3

Machine Freq 1

Machine Freq 6

Machine Freq 9

Bus Freq 18/PSSE


Center Freq

Frequency Simulation for Bus 18



Discussion

?

32

radman ttu simulation wide area frequency 20090604

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