Phadke A.G., Thorp J.S. Synchronized Phasor Measurements and Their Applications

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224 Chapter 10 Electromechanical Wave Propagation

It can be seen that there seems to be a delay between the frequencies at dif-

ferent points in the system. Similar effects were noticed in a variety of ex-

periments [1, 2]. The event described in [1] was a load rejection test in

Texas monitored with PMUs. The authors state ‘Note the delay detecting

the transient between the point closest to the plant (Venus) and the furthest

(Robinson). There is nearly a half second delay between the onset of the

frequency disturbance near the plant and its appearance at a similar level at

a remote site. The propagation phenomenon is not clear. It is not electrical

in nature because of the time lag. It appears to be related to the localized

electrical inertia in the system.’ A similar staged event in July, 1995

showed a delay of approximately a second between PMUs in Florida and

New York [3].

The second motivation was a desire to display the phasor measurements

obtained in the study described in Section 8.5.1 for the WECC 1994 dis-

turbance. The plot of Figure 10.2 was produced by locating the phasor

measurements geographically on a map of the WECC, making the z vari-

able at that point the angle measurement, and then fitting a smooth surface

to those points. By sampling the phase angles in time a movie of the sur-

face can be obtained. Constant contour lines are shown below on a map

showing state borders and the location of the two DC lines. Since to a first

approximation power flows down hill in angle, transmission lines should

be constructed perpendicular to the constant contour lines. Of course, the

contour map changes in response to system conditions

Fig. 10.2 Phase angle as a function of location.

10.2 The Model 225

The movies produced show wave-like motion of the surface when simu-

lated or measured phasor quantities are used to draw the surfaces. There

have been early attempts to describe this behavior even before phasor

measurement data were available. Over a 30-year period [4–7] a number of

authors have derived a wave equation for an idealized continuum power

system model. In [4–7] linearized swing equations are written with the DC

power-flow assumptions for a uniform, isotropic, and lossless network. A

different approach in [7] produced an elliptic partial differential equation

for a continuum load flow. In [7] the load flow equations are written under

the DC load flow assumptions but the uniformity and isotropy constraints

are relaxed. In [8–10] a combination of the two approaches produces a dif-

ferent model.

10.2 The Model

Consider the element at point (x, y) shown in Figure 10.3 with

Fig. 10.3 Incremental power system model.

V(x,y)

branch ‘i’

(x,y)

x - axis

xis

226 Chapter 10 Electromechanical Wave Propagation

+ jX

: p.u. line impedance

1/(R

+jX

) : p.u. generator admittance

+jB

: p.u. shunt admittance

|V|: voltage magnitude

φ(x,y): internal voltage phase angles

δ(x,y): external voltage phase angles

M: generator inertia

D: generator damping

P: mechanical power injection

: angle of the branch with respect to the x–y axis

The power flow at the external node (x, y) must satisfy the load flow equa-

tion in Eq. (10.1)

[]

22 2

1cos(δ(, ) δ(, ))

()

sin(δ(, ) δ(, ))

()

1cos(δ(, ) (, ))

()

sin(δ(, ) (, ))

()

yxxyy

xy x x y y

xy xy

−−+Δ+Δ

Δ+

+ − +Δ +Δ

Δ+

=−−

Δ+

+−

Δ+

∑

(10.1)

And the power flow at the internal node(x,y) must satisfy the swing equa-

tion in Eq. (10.2). The next step is to write Taylor series expansions

δ(, )

xy y±Δ ±Δ , to use trigonometric identities for small angles and

take the limit as

0→Δ . Two coupled partial differential equations result.

One is the swing equation in Eq. (10.3) where virtually everything is a

function of x and y, that is, m(x,y), d(x,y), p

(x,y), g

int

(x,y), b

int

(x,y), φ(x,y),

and δ(x,y). The other equation is the continuum load flow equation in Eq.

(10.4). The conductance G and susceptance B are 2 × 2 tensor fields, func-

tions of (x,y) which capture the nonuniformity and anisotropy in the net-

work

10.2 The Model 227

[]

22 2

1cos((,)δ(, ))

()

sin( ( , ) δ(, ))

()

MDP

yxy

xy xy

φφ

∂∂

+=−

∂∂

−−

Δ+

−

Δ+

(10.2)

[]

int int

1cos( δ)] sin( δ)

mdpg b

φφ

∂∂

=− − − − −

∂∂

(10.3)

[

]

[][]

int int s

() δδ

cos(δ )1 sin(δ )gbg

φφ

−∇• ∇ + ∇ • •∇ =

−

−− − −

(10.4)

Equations (10.3) and (10.4) are coupled by the angle difference

− .

Equation (10.4) is a continuum load flow with no time dependence. In

principle, given a

),( yx

, equation (10.4) can be solved for δ(, )

y . Al-

ternately, given

),( yx

equation (10.3) can be solved for δ(, ,)

yt ; that

is, equation (10.3) is a swing equation for each (x,y).

The phase angle gradient field

∇

is imaged to the power-flow field P

through

( δ)P

−∇B

(10.5)

Several things can be noticed. The first is that the power injected at a point

(x,y) in the system is

(, ) ( δ) δδpxy

⎡⎤

=−∇• ∇ +∇ • •∇

⎣⎦

(10.6)

The nonlinearity is in the second term and is due to the electrical losses.

The lossless models in [4, 5, 6, 7] had linear wave behavior because they

had no losses. It must be observed that the equation in Eqs. (10.3) and

(10.4) describe an electromechanical system, not an electromagnetic sys-

tem. It may be disquieting that linear electrical losses produce a nonlinear

effect in the electromechanical system. Further, the net power loss in a re-

gion R in the system is given by

228 Chapter 10 Electromechanical Wave Propagation

( δδ)d d

loss

Pxy=∇••∇

∫∫

(10.7)

A linear analogy can be made between the electromechanical model and

the electromagnetic wave propagation model. It is summarized in Table

10.1

Table 10.1 Correspondance between linear electromagnetic and electromechanical

systems.

Electromagnetism Electromechanical

Quantity Relationship Quantity Relationship

Electric

potential

VE ∇−=

Voltage phase

angle

−

∇

Electric field

Intensity

Phase angle

gradient field

Permittivity

tensor

ED ε=

Susceptance

tensor

P Φ

Electric flux

density

D∇−=

Power flow

density

Pp •∇−=

Charge

density

Power injection

density

V( ∇•∇−=ρ ε )

)(p δ∇•∇−= B

Example 10.1 A ring system.

As an example consider a ring system made up of 64 identical generators

connected in a ring with identical transmission lines connecting the gen-

erator.

If the equilibrium is chosen as a 2π increase in angle in the counter-

clockwise direction around the ring then power flows in the same direction

(Figure 10.4). The discrete model has 64 differential and algebraic equa-

tions in the form of Eq. (10.8) or the continuum form is Eq. (10.9):

[]

[][]

int

11int

sin( δ )

2cos(δδ)cos(δδ)

sin(δδ)sin(δδ)sin(δ )

ii ii

ii ii ii

MPb

+−

=− −

−−−−

+−+−= −

(10.8)

10.2 The Model 229

Fig. 10.4 Uniform power flow in a counterclockwise direction.

[]

int

() sin(() δ())

mx b x x

ϕπ

∂

⎛⎞

=− −

⎜⎟

∂

⎝⎠

(10.9)

[]

int

δδ

sin( ( ) δ())bb xx

∂∂

⎛⎞

−= −

⎜⎟

∂∂

⎝⎠

(10.10)

Fig. 10.5 The 64 angles from Example 10.1 versus time.

Time

(t) n=1,2…64

2π

Time

(t) n=1,2…64

2π

230 Chapter 10 Electromechanical Wave Propagation

The model has no source admittance and hence no dispersion. The 64 an-

gles from the discrete model are shown with δ

at the bottom and δ

at the

top. Figure 10.5 shows an initial perturbation from the equilibrium about a

quarter of the way around the ring propagating in both directions. Re-

markably it grows in the counterclockwise direction but decays in the

clockwise direction.

Example 10.2

A more realistic power flow is around both sides of the ring in the same

direction. Let the equilibrium angles be

0.1( 15.5)

δ / 32, 1, 2,...32

δ (64 ) / 32, 33,34,...64

with δ

−−

=− =

(10.11)

Then the waves are as shown in Figure 10.6

Fig. 10.6. Phase angles versus time for Example 10.2.

(t) k=1,2,…32

(t) k=33,34 …64

(t) k=1,2,…32

(t) k=33,34 …64

10.3 Electromechanical telegrapher’s equation 231

Vertical cross-sections of the phase angle for Example 10.2 are shown in

Figure 10.7. Each of the eight figures is the angle at a fixed instant in time

plotted versus position around the ring δ(

,t), where

is the position on

the ring as shown for the first cross-section. The pulse separates into two

pulses in the second cross-section, spreads in the next two cross-sections,

crosses and returns to the center in the last two cross-sections.

Fig. 10.7 Vertical cross sections of phase angle.

10.3 Electromechanical telegrapher’s equation

If we open up the loop in Example 10.1 so we have a line rather than a

ring, a system such as that shown in Figure 10.8 is produced.

Fig. 10.8 A one-dimensional line.

If we assume small internal impedances and R<<X the equations become

,)(

)(

∂

−=

∂

−=

∂

(10.12)

………

2 3 4 spreading

5 crossing

7, 8 recombining

δ(θ,o)

−π π

2 3 4 spreading

5 crossing

7, 8 recombining

5 crossing

7, 8 recombining

δ(θ,o)

−π π

232 Chapter 10 Electromechanical Wave Propagation

where

P = –b

δδ

∂

(10.13)

The result is a form of the Telegraphers equation in angle and power rather

than voltage and currents.

δδ

() () () ,

mx p x bx

∂∂

(10.14)

which has

()

velocity , 1/ ( ) ( )

()

bxmx

==.

(10.15)

Fig. 10.9 Time snapshots of angular velocity versus position.

0 20 40 60

-5

5x 10

-4

0 Seconds

0 20 40 60

-5

5x 10

-4

3.7 Seconds

0 20 40 60

-5

5x 10

-4

14.5 Seconds

0 20 40 60

-5

5x10

-4

22.9 Seconds

0 20 40 60

-5

5x10

-4

41 Seconds

40 60

-5

5x 10

-4

49.4 Seconds

In other words the one-dimensional line has a characteristic velocity of

propagation and a characteristic ‘impedance’. The impedance is not in

ohms, since the ratio is angle divided by power rather than volts divided by

amps. The velocity of propagation measured in the power system is far less

than the velocity of light and varies in different parts of the country.

Speeds of hundreds of miles a second to a thousand miles per second have

been observed in the frequency measurement network (FNET) system [11]

where the measured waves are in frequency rather than angle. They are, of

course, a different manifestation of the same waves.

10.4 Continuum voltage magnitude 233

Determining the onset of a wave can be complicated with electromechani-

cal waves just as in electromagnetic waves. The use of discriminant func-

tions as in traveling wave relays [12] is possible

−

=−=

=+=

ωω

pZD

(10.16)

Typical plots of angular velocity versus position along a uniform line are

shown in Figure 10.9 at a sequence of times. Although the waves begin

crisply, they become distorted as time goes on. The forward and reverse

discriminant functions are shown in Figure 10.10

Fig. 10.10 Forward and reverse discriminate functions for the example.

10.4 Continuum voltage magnitude

The development of Section 10.2 assumed constant voltages. The load

flow equation can be re-derived with the external bus voltage magnitude

depending on (

x,y), that is,

.),(

),( yxj

eyxE

(10.17)

The resulting two equations are a real power equation

0 10 20 30 40 50 60

-2

10x 10

-4

Forward discriminan

Reverse discriminant

(

seconds

)