Power System Model Reduction Fall 2014 CURENT...

Power System Model Reduction

Fall 2014 CURENT Course

October 29, 2014

Joe H. Chow

Rensselaer Polytechnic Institute, Troy, New York, USA

www.ecse.rpi.edu/homepages/chowj

http://www.ecse.rpi.edu/homepages/chowj

Rensselaer Polytechnic Institute October 2014 JHC 2

Model Reduction of Large Power Systems

1. Simulation of power system dynamics for stability analysis on a digital computer: needs the most comprehensive power system model so that

i. the relevant dynamics can be accurately simulated given the computing resources

ii. the simulation can be completed in a reasonable amount of time.

2. One of the decisions is the geographic extent of the power system data set: disturbance always in the study region, but external system(s) also needed.


Model Reduction of Large Power Systems

1I

Study SystemExternal System

2I

kI

1bV

2bV

bkV

Gen + Controls

Gen + Controls

Gen + Controls

Gen + Controls

1P

Study SystemExternal System

1bV

2bV

Gen + Controls

Gen + Controls

Gen + Controls

Gen + Controls

2P

A

B

Internal

and

external

systems

Disturbance

Travelling

through the

external

system


Model Reduction Approaches

1. Coherency – some machines swinging together after a disturbancei. Time simulation (R. Podmore)ii. Modal coherency (R. Schleuter)iii. Slow coherency (J. Chow, et. al) – coherency with respect to

the slow interarea modesiv. Weak-link method – slow coherent areas are weakly

connected (N. Rao, J. Zaborzsky)

2. Aggregation – obtaining an equivalent power system modeli. Generator aggregation (R. Podmore)ii. Singular perturbations – provides improvements to

equivalent generator reactances (J. Chow, et. al)


Model Reduction Approaches

3. Linear model analysis of external systems

i. Modal truncation (J. Undrill, W. W. Price)

ii. Selective modal analysis (SMA) (G. Verghese, I. Peres-Arriaga, L. Rouco)

iii. Krylov method – moment matching (D. Chaniotis, M.A. Pai)

iv. Balanced truncaton – “best” reachibility (controllability) -observability (Shanshan Liu)

4. Artificial Neural Networks (F. Ma, V. Vittal)

5. Synchrophasor data (A. Chakrabortty, Chow)

1I

External System

2I

kI

1bV

2bV

bkV

Gen + Controls

Gen + Controls


Monographs

1982 2013


Topics

1. Slow coherency

2. Reduced order modeling of dominant transfer paths in large power systems using synchrophasor data (work with Dr. Aranya Chakrabortty)

Slow Coherency

A large power system usually consists of tightly connected controlregions with few interarea ties for power exchange and reservesharing

The oscillations between these groups of strongly connectedmachines are the interarea modes

These interarea modes are lower in frequency than local modesand intra-plant modes

Singular perturbations method can be used to show this time-scaleseparation

c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 2 / 29

Klein-Rogers-Kundur 2-Area System

Gen 1

Gen 2

Load 3

1

2

3Gen 11

Gen 12

1310 20

101120 110 11

12

Load 13

Interarea mode

Local

mode

Local

mode


Coherency in 2-Area SystemDisturbance: 3-phase fault at Bus 3, cleared by removal of 1 linebetween Buses 3 and 101

0 1 2 3 4 51

1.001

1.002

1.003

1.004

1.005

1.006

1.007

Time (sec)

Mac

hin

e S

pee

d (

pu

)

Gen 1Gen 2Gen 11Gen 12


Power System Model

An n-machine, N -bus power system with classical electromechanicalmodel and constant impedance loads:

miδi = Pmi − Pei = Pmi −EiVj sin(δi − θj)

x′di= fi(δ, V ) (1)

where

machine i modeled as a constant voltage Ei behind a transientreactance x′dimi = 2Hi/Ω, Hi = inertia of machine i

Ω = 2πfo = nominal system frequency in rad/s

damping D = 0

δ = n-vector of machine angles

Pmi = input mechanical power, Pei = output electrical power


Power System Model

the bus voltage

Vj =√

V 2jre + V 2

jim, θj = tan−1

(

Vjim

Vjre

)

(2)

Vjre and Vjim are the real and imaginary parts of the bus voltagephasor at Bus j, the terminal bus of Machine i

V = 2N -vector of real and imaginary parts of load bus voltages


Power Flow EquationsFor each load bus j, the active power flow balance

Pej − Real

N∑

k=1,k 6=j

(Vjre + jVjim − Vkre − jVkim)

(

Vjre + jVjim

RLjk + jXLjk

)∗

− V2

j Gj = g2j−1 = 0 (3)

and the reactive power flow balance

Qej −Imag

N∑

k=1,k 6=j

(Vjre + jVjim − Vkre − jVkim)

(

Vjre + jVjim

RLjk + jXLjk

)∗

−V2

j Bj +V2

j

BLjk

2= g2j = 0

(4)

RLjk, XLjk, and BLjk are the resistance, reactance, and linecharging, respectively, of the line j-k

Pej and Qej are generator active and reactive electrical outputpower, respectively, if bus j is a generator bus

Gj and Bj are the load conductance and susceptance at bus j

Note that j denotes the imaginary number if it is not used as an index.


Electromechanical Model

Mδ = f(δ, V ), 0 = g(δ, V ) (5)

M = diagonal machine inertia matrix

f = vector of acceleration torques

g = power flow equation


Linearized ModelLinearize (5) about a nominal power flow equilibrium (δo, Vo) to obtain

M∆δ =∂f(δ, V )

∂δ

∣

∣

∣

∣

δo,Vo

+∂f(δ, V )

∂V

∣

∣

∣

∣

δo,Vo

= K1∆δ +K2∆V (6)

0 =∂g(δ, V )

∂δ

∣

∣

∣

∣

δo,Vo

+∂g(δ, V )

∂V

∣

∣

∣

∣

δo,Vo

= K3∆δ +K4∆V (7)

∆δ = n-vector of machine angle deviations from δo

∆V = 2N -vector of the real and imaginary parts of the load busvoltage deviations from Vo

K1, K2, and K3 are partial derivatives of the power transferbetween machines and terminal buses, K1 is diagonal

K4 = network admittance matrix and nonsingular.

the sensitivity matrices Ki can be derived analytically or fromnumerical perturbations using the Power System Toolbox


Linearized Model

Solve (6) for∆V = −K−1

4 K3∆δ (8)

to obtainM∆δ = K1 −K2K

−14 K3∆δ = K∆δ (9)

where

Kij = EiEj (Bij cos(δi − δj)−Gij sin(δi − δj))|δo,Vo, i 6= j (10)

and Gij + jBij is the equivalent admittance between machine i and j.Furthermore,

Kii = −n∑

j=1,j 6=i

Kij (11)

Thus the row sum of K equal to zero. The entries Kij are known asthe synchronizing torque coefficients.


Slow Coherent Areas

Assume a power system has r slow coherent areas of machines and theload buses that interconnect these machines

Define

∆δαi = deviation of rotor angle of machine i in area α from itsequilibrium value

mαi = inertia of machine i in area α

Order the machines such that ∆δαi from the same coherent areasappears consecutively in ∆δ.


Weakly Coupled Areas

We attribute the slow coherency phenomenon to be primarily due tothe connections between the machines in the same coherent areas beingstiffer than those between different areas, which can be due to tworeasons:

The admittances of the external connections BEij much smaller

than the admittances of the internal connections BIpq

ε1 =BE

ij

BIpq

(12)

where E denotes external, I denotes internal, and i, j, p, q are busindices. This situation also includes heavily loaded high-voltage,long transmission lines between two coherent areas.


The number of external connections is much less than the numberof internal connections

ε2 =γE

γI(13)

where

γE = maxα

γEα , γI = minα

γIα, α = 1, ..., r (14)

γEα = (the number of external connections of area α)/Nα

γIα = (the number of internal connections of area α)/Nα

where Nα is the number of buses in area α.


Internal and External ConnectionsFor a large power system, the weak connections between coherent areascan be represented by the small parameter

ε = ε1ε2 (15)

Separate the network admittance matrix into

K4 = KI4 + ǫKE

4 (16)

where KI4 = internal connections and KE

4 = external connections.The synchronizing torque or connection matrix K is

K = K1 −K2(KI4 + ε(KI

4 ))−1K3

= K1 −K2(KI4 (I + ε(KI

4 )−1KE

4 ))−1K3 = KI + ǫKE (17)

whereKI = K1 −K2(K

I4 )

−1K3, KE = −K2KE4εK3 (18)

In the separation (17), the property that each row of KI sums to zerois preserved.c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 14 / 29

Slow Variables

Define for each area an inertia weighted aggregate variable

yα =

nα∑

i=1

mαi ∆δαi /m

α, α = 1, 2, . . . , r (19)

where mαi = inertia of machine i in area α, nα = number of machines

in area α, and

mα =

nα∑

i=1

mαi , α = 1, 2, . . . , r (20)

is the aggregate inertia of area α.


Denote by y = the r-vector whose αth entry is yα.The matrix form of (19) is

y = C∆δ = M−1a UTM∆δ (21)

whereU = blockdiag(u1, u2, . . . , ur) (22)

is the grouping matrix with nα × 1 column vectors

uα =[

1 1 . . . 1]T

, α = 1, 2, . . . , r (23)

Ma = diag(m1,m2, . . . ,mr) = UTMU (24)

is the r × r diagonal aggregate inertia matrix.


Fast VariablesSelect in each area a reference machine, say the first machine, anddefine the motions of the other machines in the same area relative tothis reference machine by the local variables

zαi−1 = ∆δαi −∆δα1 , i = 2, 3, . . . , nα, α = 1, 2, . . . , r (25)

Denote by zα the (nα − 1)-vector of zαi and conside zα as the αthsubvector of the (n− r)-vector z. Eqn. (25) in matrix form is

z = G∆δ = blockdiag(G1, G2, . . . , Gr)∆δ (26)

where Gα is the (nα − 1)× nα matrix

Gα =

−1 1 0 . 0

−1 0 1 . 0

. . . . .

−1 0 0 . 1

(27)


Slow and Fast Variable Transformation

A transformation of the original state ∆δ into the aggregate variable yand the local variable z

[

y

z

]

=

[

C

G

]

∆δ (28)

The inverse of this transformation is

∆δ =(

U G+)

[

y

z

]

(29)

whereG+ = GT (GGT )−1 (30)

is block-diagonal.


Slow SubsystemApply the transformation (28) to the model (9), (17)

May = ǫKay + ǫKadz

Mdz = ǫKday + (Kd + ǫKdd)z (31)

where

Md = (GM−1GT )−1, Ka = UTKEU

Kda = UTKEM−1GTMd, Kda = MdGM−1KEU

Kd = MdGM−1KIM−1GTMd,Kdd = MdGM−1KEM−1GTMd (32)

Ka, Kad, and Kda are independent of KI

System (31) is in the standard singularly perturbed form

ε is both the weak connection parameter and the singularperturbation parameter

Neglecting the fast dynamics, the slow subsystem is

May = εKay (33)


Formulation including Power NetworkApply (28) to the model (6)-(7)

May = K11y +K12z +K13∆V

Mdz = K21y +K22z +K23∆V

0 = K31y +K32z + (KI4 + εKE

4 )∆V (34)

where

K11 = UTK1U, K12 = UTK1G+, K13 = UTK2, K21 = (G+)TK1U

K22 = (G+)TK1G+, K23 = (G+)TK2, K31 = K3U, K32 = K3G

+(35)

Eliminating the fast variables, the slow subsystem is

May = K11y +K13∆V

0 = K31y +K4∆V (36)

This is the inertial aggregate model which is equivalent to linking theinternal nodes of the coherent machines by infinite admittances.c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 20 / 29

2-Area System ExampleConnection matrix

K =

−9.4574 8.0159 0.5063 0.9351

8.7238 −11.3978 0.9268 1.7472

0.6739 0.9520 −9.6175 7.9917

1.3644 1.9325 8.1747 −11.4716

(37)

Decompose K into internal and external connections

KI =

−8.0159 8.0159 0 0

8.7238 −8.7238 0 0

0 0 −7.9917 7.9917

0 0 8.1747 −8.1747

(38)

εKE =

−1.4414 0 0.5063 0.9351

0 −2.6739 0.9268 1.7472

0.6739 0.9520 −1.6258 0

1.3644 1.9325 0 −3.2969

(39)


EM Modes in 2-Area System

λ(M−1K) = 0,−14.2787,−60.7554,−62.2531 (40)

with the corresponding eigenvector vectors

v1 =

0.5

0.5

0.5

0.5

, v2 =

0.4878

0.4031

−0.5672

−0.5271

, v3 =

0.6333

−0.7446

0.1924

−0.0863

, v4 =

0.1102

−0.1494

−0.8098

0.5566

(41)

Interarea mode:√−14.279 = ±j3.779 rad/s

Local modes:√−60.755 = ±j7.795 rad/s and√

−62.253 = ±j7.890 rad/s


Slow and Fast Subsystems of 2-Area System

Slow dynamics:

Ma =1

2π × 60

[

234 0

0 234

]

, εKa =

[

−4.1154 4.1154

4.9227 −4.9227

]

(42)

The eigenvalues of M−1a Ka are 0 and −14.561 ⇒ an interarea mode

frequency of√−14.561 = ±j3.816 rad/s.

Fast local dynamics:

Md =1

2π × 60

[

58.500 0

0 55.611

]

, Kd =

[

−8.3699 0

0 −8.0628

]

(43)The eigenvalues of M−1

d Kd are −53.939 and −54.660 ⇒ local modes of±j7.3443 and ±j7.3932 rad/s.


Finding Coherent Groups of Machines

1 Compute the electromechanical modes of an N -machine power

2 Select the (interarea) modes with frequencies less than 1 Hz

3 Compute the eigenvectors (mode shapes) of these slower modes

4 Group the machines with similar mode shapes into slow coherentgroups

5 These slow coherent groups have weak or sparse connectionsbetween them


Grouping AlgorithmPlot rows of the slow eigenvector Vs of the interarea modes

1

2

0

1

2

0 1

1

Use Gaussian elimination to select the r most separated rows asreference vectors and group them into Vs1 and reorder Vs as

Vs =

[

Vs1

Vs2

]

⇒[

Vs1

Vs2

]

V −1s1 =

[

I

L

]

=

[

I

Lg

]

+

[

0

O(ε)

]

(44)

Use Vs1 to form a new coordinate systemc©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 25 / 29

Coherent Groups in 2-Area System

Vs =

0.5 0.4878

0.5 0.4031

0.5 −0.5672

0.5 −0.5271

Gen 1

Gen 2

Gen 11

Gen 12

, Vs1 =

[

0.5 0.4878

0.5 −0.5672

]

Gen 1

Gen 11

(45)Then

V ′sV

−1s1 =

1 0

0 1

0.9198 0.0802

0.0380 0.9620

Gen 1

Gen 11

Gen 2

Gen 12

(46)


Coherency for Load Buses

Vθ =

0.5 0.4283

0.5 0.3535

0.5 0.2556

0.5 0.3844

0.5 −0.5018

0.5 −0.4667

0.5 −0.3556

0.5 0.3128

0.5 −0.0523

0.5 −0.4671

0.5 −0.4125

Bus 1

Bus 2

Bus 3

Bus 10

Bus 11

Bus 12

Bus 13

Bus 20

Bus 101

Bus 110

Bus 120

VθV−1

s1 =

0.9436 0.0564

0.8727 0.1273

0.7800 0.2200

0.9020 0.0980

0.0620 0.9380

0.0953 0.9047

0.2006 0.7994

0.8342 0.1658

0.4880 0.5120

0.0949 0.9051

0.1466 0.8534

(47)

Bus 101


17-Area Partition of NPCC 48-Machine System


EQUIV and AGGREG Functions for Power System

Toolbox

L group: grouping algorithm

coh-map, ex group: tolerance-based grouping algorithm

Podmore: R. Pormore’s algorithm of aggregating generators atterminal buses

i agg: inertial aggregation at generator internal buses

slow coh: slow-coherency aggregation, with additional impedancecorrections

These functions use the same power system loadflow input data files asthe Power System Toolbox.



Power Transfer Paths/Interfaces



1. The integrity of the power transfer interfaces is crucial for a large interconnected system to function properly. A disruption of a major transfer path may have a “spillover” effect into parallel power transfer paths in neighboring regions.

2. Questions:a. How can PMU data be used to monitor the integrity of

the power transfer interfaces, for both internal and external visibility?

b. How many PMUs do we need and where should they be located?

c. Are there analytical concepts and approaches to extract stability margin information from PMU data?



P1

Zone A

Exports Power

Zone B

Imports Power

PMU 1 PMU 2

PMU 3

P2

P3

Power transfer between two areas with multiple lines and PMU monitoring

Some US Interconnection interfaces may consist of

• a limited number of transmission lines (CA, NY, NE)

• many transmission lines


Inter-area Model Estimation

2-Area Power System 2-Machine Equivalent

• Problem: Given synchronized voltage phasors (magnitude and phase) at Buses 3 and 13 and current phasor between the two buses in the 2-area power system, develop an equivalent 2-machine model to represent the power transfer between the two areas.

• Variations: • Voltage phasors at Bus 101 also available.• Voltage phasor at Bus 3, the current phasor going from Bus 3 to Bus 101, and the

impedance between Buses 3 and 13. • Difficulty is how to look outward into the importing and exporting areas. • Need dynamic data, i.e., can’t do it with State Estimator solution. In particular, need

interarea mode oscillations in the voltage and current phasors.

3 13

3V 13V

I

2

e er jx

1jx 2jx

2 2,E 1 1,E

~ ~

2

e er jx101V~

101

1aG 2aG1

1G

2G

3G

4G

10

2 1220 120

3 13101

4 14

11011

4L 14L



• Measurement-based model reduction approach• Use bus voltage and line current phasors• Develop new concepts and calculation methods:

Exact extrapolation equations can be developed for an ideal 2-machine system – Dynamic Model Estimation (DME) algorithm

Extend DME to 2-area system – Interarea Model Estimation (IME) algorithm: verify with disturbances

• Additional research and customization needed: transfer paths with intermediate voltage support, transfer interfaces with multiple transmission lines, impact of disturbances, …


Dynamic Model Estimation

Two machine power system

Classical model representation

zTi = transformer impedance

zi = transformer impedance + direct-axis

transient reactance

lossem PPPH

2(i =1, 2)

sm

e

sm

loss

s

mm

m

H

HH

z

EEP

H

EHEH

z

rP

H

PHPHP

)cos()cos( 1221

2

21

2

12

2112

where

• Two generators G1 and G2 with

inertias H1 and H2 are connected

to Buses 1 and 2

• G1 supplies power to G2 (load)

• Dynamic model – swing equations

DME Problem Formulation for a 2-machine system

1 2

1V 2V

2 2r jx

~ ~

2z

1 1 1E E 2 2 2E E

1 1r jx

1z I

ez

e er jx~ ~

1G1 2

1V 2VI

ez

1 1T Tr jx

1E

2 2T Tr jx

2G

2E~ ~

~ ~

1Tz

e er jx

2Tz



• Assume PMUs are located at Buses 1 and 2

• Available Phasor Variables –

DME Problem : Given the available phasor variables, that exhibit a few cycles of

oscillations, and assuming E1 and E2 to be constant, compute

to completely characterize the dynamic behavior of the 2-machine system.

• Because and ,

the problem reduces to the estimation of

, , , , 1 1 2 2 , IV V I

, , , , , , , 1 1 2 2 1 2 1 2, eE E z z z H H

IVVze

~/)

~~( 21 IjzVE

~~~111 IjzVE

~~~222

, , 1 2 1 2, z z H H

DME Problem Formulation for a 2-machine system (Contd.)

1 2

1V 2V

2 2r jx

~ ~

2z

1 1 1E E 2 2 2E E

1 1r jx

1z I

ez

e er jx~ ~



• Key idea : Amplitude of voltage oscillation at any point on the transfer path is a

function of its electrical distance from the two fixed voltage sources.

]))sin()cos(([ ]))sin()cos(()1([),(~

2112 bEabEjbaEaErxV

where

22 ''

''

ee

ee

xr

xxrra

22 ''

''

ee

ee

xr

rxxrb

• Voltage Magnitude at B

)sin()cos()(2|),(~

|),(| 22

21 bbaaEEcrxVrxV

)())1(( 222

1

222

2 abEabEc where

Reactance Extrapolation

1 1,E I V x~

2 2, 0E

A B Cz




• Linearize model about (δ0, ω0 = 0, Vss) :

),,()( 0baJxV

where the Jacobian is

)]cos()sin()[(),,(

2),,(:),,( 00

22

0

210

0

0

bbaabaV

EEbaVbaJ

• Assume uniform impedance along transfer path → and ex

xa 0b

)1( ),,(

)sin(2),0,(

0

0210 aa

baV

EEaJ

(*)

• Note : J(x,δ0) in (*) has a numerator varying

with x, and a denominator equal to steady

state voltage magnitude at B.

(*)



• J(x,δ0) in (*) can be used to estimate x1 and x2 if voltage oscillations are measured

or calculated at an additional intermediate bus between Bus 1 and 2

• Following a perturbation, let

iV

iV

iss

im

Busat Voltage state-Steady

Busat Swing Voltage of Amplitude

• From (*) , with321 , )1( , , iaaAVVV iiissimin

Computed from

PMU measurements

1 22 sin( oA E E

• The reactances x1 and x2, and the unknown constant A can be solved

numerically (3 nonlinear equations with 3 unknowns)


1 2

1V 2V

1ez1z

2 2,E 1 1,E

~ ~3V

~

3

2ez2z



Inertia Estimation

• From linearized model

21

21

HH

HHH

)0( er

where f s is the measured swing frequency and is the aggregate

Inertia

• For a second equation in H1 and H2, use conservation of angular momentum

0 )()(222 221122112211 dtPPPPdtHHHH emem

1

2

2

1

H

H

• However, ω1 and ω2 are not available from PMU data,

• Estimate ω1 and ω2 from the measured frequencies ξ1 and ξ2 at Buses 1 and 2

)(2

)cos(

2

1

21

021

xxxH

EEf

e

s



• Again use linearization to derive bus frequency and machine speed expressions

12111

2121211111

)cos(2

)cos()(

cba

cba

22122

2221212122

)cos(2

)cos()(

cba

cba

where 22

221

22

1 ),1( ,)1( iiiiiii rEcrrEEbrEa

exxx

xr

21

11

e

e

xxx

xxr

21

1

2

(i=1, 2)

• ξ1 and ξ2 are measured, and ai, bi, ci are

known from reactance extrapolation

• Hence, we can calculate ω1/ω2 and solve

for H1 and H2

and ,

0 0.2 0.4 0.6 0.8 1

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

1

2

1

2

Normalized Reactance r

Fre

qu

en

cy O

scill

atio

n M

ag

nitu

de

(r/

s)

Inertia Estimation



Example

• Illustrate DME on classical 2-machine model (re = 0)

• Disturbance is applied to the system and the response simulated in MATLAB

Voltage Oscillations at 3 buses

0376.0 0326.0 0301.0

0136.1 0317.1 0320.1

0371.0 0316.0 0292.0

321

321

321

nnn

ssssss

mmm

VVV

VVV

VVV

DME Algorithm

x1 = 0.3382 pu

x2 = 0.3880 pu

• Exact values: x1 = 0.34 pu, x2 = 0.39 pu



Example (cont.)

• Jacobian curve fit for reactance extrapolation



Example (cont.)

• Inertia estimation

• Find the bus frequencies by passing the bus voltage angles through a derivative

filter with bandwidth higher than the inter-area mode frequency (0.912 Hz)

102.0)(

s

ssG

DME Algorithm

H1 = 6.48 pu, H2 = 9.49 pu

• Exact values: H1 = 6.5 pu, H2 = 9.5 pu



Example (cont.)

• Frequency curve fit for inertia estimation



• Consider inter-area dynamics across the transfer interface of a multi-

machine power system

• Extend the DME algorithm to estimate an equivalent two-machine model

• Assumption: Interface exhibits a single dominant mode of inter-area

oscillation

• A new approach to power system model reduction based on PMU

measurements


IME Problem : Given the phasor measurements that exhibit a

few cycles of inter-area oscillations, and assuming E1 and E2 to be constant, compute

to completely characterize the dynamic behavior of the reduced 2-machine system.

, , , , 3 3 13 13 , IV V I

, , , , , , , 1 1 2 2 1 2 1 2, e a aE E z z z H H

• Similar derivations as DME – but in this case local modes are present in system

response

• Apply a modal estimation method such as ERA to isolate inter-area mode


2-Area Power System 2-Machine Equivalent

1

1G

2G

3G

4G

10

2 1220 120

3 13101

4 14

11011

4L 14L

3 13

3V 13V

I

2

e er jx

1jx 2jx

2 2,E 1 1,E

~ ~

2

e er jx101V~

101

1aG 2aG


• Disturbance applied to a 2-area system

Bus 3 Voltage Bus 13 Voltage

Bus 101 Voltage


• With 300 MW power transfer the inter-area oscillation

frequency is 0.574 Hz, and the local mode

frequencies are 1.293 Hz and 1.308 Hz.

• IME estimates: x1 = 0.5069 pu, x2 = 0.0618 pu

Ha1 = 18.98 pu, Ha2 = 12.55 pu


• Disturbance applied to a 2-area system


Frequency at Bus 3 Frequency at Bus 13



0 0.2 0.4 0.6 0.8 1-3

-2

-1

0

1

2

Normalized Reactance rF

req

ue

nc

y O

sc

illa

tio

n M

ag

nit

ud

e (

r/s

)

x1

x2

xe

Reactance Curve Fit Frequency Curve Fit


Inter-area Model Estimation• Disturbance plots comparing full model to reduced models

Voltage Oscillations at Bus 3 Voltage Oscillations at Bus 13

Angular Difference Speed Difference



Some important observations• Inter-area model estimation is independent of the severity of

the disturbance event resulting in the same post-fault condition (robustness)

• Inter-area model is dependent on the amount of inter-area power transfer. In the 2-area system example, larger amount of power transfers results in:

i. Higher angular separation between the sending end and the receiving end

ii. Slower frequency of oscillationiii. Smaller reactance at the sending end and larger reactance

at the receiving area iv. Equivalent inertias of the two areas also change


Conclusions and Future Research

• New techniques in using synchronized phasor data to identify power transfer paths/interfaces and hence for PMU siting.

• Establish power-angle curves and energy function analysis on each power transfer path. Monitor the power-angle curves as the system evolves through a sequence of disturbances.

• Monitor energy levels in the transfer paths to determine stability (loss of synchronism or negative damping) and to determine which transfer path is under the most stress.

• Improve system transmission planning and operations planning studies Different types of studies Different ways to see and evaluate results

• Extend calculations to dynamic voltage stability.


This work was supported primarily by the ERC Program of the

National Science Foundation and DOE under NSF Award Number

EEC-1041877.

Other US federal and state government agencies and industrial

sponsors of CURENT research are also gratefully acknowledged.

Acknowledgements

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