4.2 State Estimation

8/14/2019 4.2 State Estimation

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4.2 State estimation

State

Estimation

Model

of

external system

Security

analysis

Security constraints:

met

not met

Network

equivalent

State vector

Location of

bad data

J(x), E{J(x)}

Network topology

Measurements

Variance of mea-

surement errorsNetwork parameter

Contingency set lines

transformers

generation units

Exeeding limits:

Ith

Vmin, Vmax

Imax

Real-time security analysis

4.2.1


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Definition of State Vector

=

k

k

V

V

V

x

M

M

i voltage angle

Vi voltage magnitude

is called the state vector of the network.

In a network with i = 1k nodes

the vector

4.2.2


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State vector in 4-node network

PG1

P14 P12

P23P43

PL3

1

24

3

V1,1 = 0 (Slack Node)

V2, 2< 0

V3, 3< 0

V4, 4< 0

[ ]

)sin(

)cos()sin()(

1

21

12

2112

21122121122112

2

12

12

2

12

12

++

=

X

VVP

RVVXVVRVXR

P

4.2.3


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Terms of state estimation

4.2.4

x

true value ofstate variables

h(x)

functional relation

between measurable

quantities and thestate variables

v

measurement

errors

Power System

x ; h(x)

Measuring

Instrumentation

State estimation

z

measured valuesz = h(x) + v

estimated value of

state variables

x

x


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Estimation by Weighted Least Squares:

Measuring equations:

z = h(x) + v

z vector of m measurements

x state vector with n variables (i, Vi)

h(x) vector of m measurement functions

v vector of m measurement errors

m n

4.2.5


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Assumptions in respect to measurements

v1.vm random variables with Gaussianprobability function

E{v} = 0 mean value of measurement

errors = 0

E { v vt } = R measurement covariance

matrix

no interdependence between

measurement errors

=2

m

2

1

0

0

R O

4.2.6


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Normal or Gaussian probability functionProbability density function (y) of a random variableY

estimated value of

the voltage vector

( )2

y

2

1

e2

1y

=

+


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Objective function J(x) and minimization

=

=

=m

1i 2i

2

ii

1t

))x(hz()x(J

))x(hz(R))x(hz()x(J

Necessary condition for the extremum of J(x):

=

=

==

n

m

1

m

n

1

1

1

1t

x

h

x

h

x

h

x

h

x

)x(hH

)1(0))x(hz(RH2

x

J

L

MM

L

H Jacobian matrix of measurement functions.

4.2.8


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Case A: h(x) linear function of x

vxAz +=

Necessary condition for extremum:

Objective function:

)xAz(R)xAz()x(J 1t =

0)xAz(RA2x

J 1t ==

zRA)ARA(x 1t11t =

Solution:

4.2.9


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Case B: h(x) nonlinear function of x

Introducing Taylor expansion in eq. (1) :

Taylor expansion of h(x) around a starting

point x0:

Iterative solution:

......)xx)(x(H)x(h)x(hooo

++=

[ ]

[ ] [ ])x(hzR)x(Hxx)x(HR)x(H

0)xx)(x(H)x(hzRH2

x

J

010t0010t

0001t

=

==

[ ] [ ]

[ ] [ ])x(hzR)x(H)x(HR)x(Hxx

)x(hzR)x(H)x(HR)x(Hxx

1t1

1t1

010t1010t01

+

+=

+=M

4.2.10


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0

0

0

0

0

0

0

0

0

0

0

0

State estimation: Numerical example

Given:

Find: Weighted least squares estimates of

bus voltage measurement

active power measurement of line flow

reactive power measurement of line flow

1 = 0

R = =

0

(0.1)2

0

0

0

0

(0.05)2

0

0

0

0

(0.05)2

(0.1)2

0

0

0

2

122

2

32

4

z2 = V2 = 1.0

z3 = P12 = 3.0

z4 = Q12 = 0.3

z1 = V1 = 1.02

V1 V2

1 2

2; V1; V2

4.2.11


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State estimation: Numerical exampleSolution:

Jacobian-Matrix

(Slack bus)g12=0; gs12=0; b12=-10; 1=0

bs12

=0;

=

3

4

2

4

1

4

3

3

2

3

1

3

3

2

2

2

1

2

3

1

2

1

1

1

);(

x

h

x

h

x

h

x

h

x

h

x

h

x

h

x

h

x

h

x

h

x

h

x

h

VH

h4(; V) = 10V12- 10V1V2cos 2

h4(; V) = -V12(b12+bs12) V1V2[g12sin(1- 2) b12cos(1- 2)]

h3(; V) = -10 V1 V2 sin 2

h3(; V) = V12(g12+gs12)-V1V2[g12cos(1- 2)+b12sin(1- 2)]

h2(; V) = V2h1(; V) = V1

4.2.12


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4.2.13

Iterative solution:

=;

21221221

2122221

cos10cos1020sin10

sin10sin10cos10

100

010

)(

VVVVV

VVVVVH

=

=

0.1

0.1

0

2

1

2

3

2

1

o

o

o

o

o

o

V

V

x

x

x

=

=

3.0

0.3

0

02.0

0

0

0.1

0.1

3.0

0.3

0.1

02.1

)V;(hz o

=

10100

0010

100

010

);( oVH


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4.2.14

Ht R-1 (z-h) =

0

0

1

-10

0

0

0

10

-10

0

1

0

0

100

0

0

0

0

400

0

0

0

0

400

100

0

0

0

0.02

0

3.0

0.3

Ht R-1 (z-h) =-12000

1202

-1200

Ht R-1 H = 1040

4.01

-4.0

0

-4.0

4.01

4.0

0

0

(Ht R-1 H) -1 = 10-40

50.0624

49.9376

0

49.9376

50.0624

0.2500

0

0

x1 = x0 + (Ht R-1 H) -1 Ht R-1 (z-h)


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4.2.15

Result of the first iteration step:

0

1.0

1.0

x11

x21

x31

= + 10-4

-0.3000

1.0250

0.9950

x11

x21

x31

=

-0.3000

0.0250

-0.0050

0

1.0

1.0

x11

x21

x31

= +

-0.3000

1.0250

0.9950

21 =

V11 =

V21 =

0

50.0624

49.9376

0

49.9376

50.0624

0.2500

0

0

-12000

1202

-1200


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iteration

= 2

= 3

start

= 1

= 4

- 0.298

- 0.298

2

- 0.300

- 0.298

0

1.004

1.003

V1

1.025

1.003

1.0

1.018

1.018

V2

0.995

1.018

1.0

0.018

0.018

max|z-h(;V)|

0.463

0.018

3.000

Final result after = 4 iterations:

-0.298

1.003

1.018

24 =V1

4 =

V24 =

2 =V1 =

V2 =

^^

^

4.2.16


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State estimation:

- Observability test

- WLS-algorithm

- Identification of bad data

- Suppression of bad data

Network topology

Transformer tap

settings

Measurements

Pseudo-measurements

Variance of

measurement errors

network parameters

Estimated value of state

vector

Other quantities derived

from state vector

Observed value

Expected value

Location of bad data

)x(J

))x(J(E

Input data and results of state estimation

4.2.17


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Observability1

6 5

432

Slack

6 (Pij, Qij) measurements

nodes , not observable4 5

7 (Pij, Qij) measurements

all nodes observable

1 6 5

432

Slack

4.2.17 /1


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Pi = 0

Qi = 0node i : passive

i

Pi 0Qi 0

node i : non passive

i

Pseudo-measurements

4.2.17 /2


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Y1, Y2, , Yk independent random variables; with normal distribution;

mean value zero; variance one;

2 = Y12 + Y22 + + Yk2

Mean value ; Variancek2 = k22

2 =

20 806040 2

0.1

0

(2)

k = 20

k = 50

k degree of freedom

2

, CHI-SQUARE DISTRIBUTION

4.2.17 /3


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Under the assumptions concerning the statistics of measurementerrors, J(x) is a random variable with 2 distribution and degree offreedom k = (m n)

Mean value J = (m n)

Variance J2 = 2(m n)

JJ - 3J J + 3J

(J(x) )

J(x)

4.2.17 /4


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Input data of state estimation

On-Off status quantities of circuit breakers and isolators (network topology)

Transformer tap settings

Measurements (bus voltage magnitudes, active and reactive power flows,

active and reactive power injections)

Pseudo measurements of passive nodes (Pi = 0, Qi = 0)

Information regarding the measuring instrumentation (specification and

location of measurements, variance of measurement errors)

Network parameters (lines: R, X, G, B; transformers: R, X, G, B, t)

4.2.18


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Estimated value of the state vector of all busses

(voltage magnitude and voltage phase angle )

Other quantities derived from the state vector

( )

Observed value of the minimization function

Expected value of the minimization function

Location of bad data

Covariance matrix of the state vector

Results of state estimation

x

x

i

iV

)x(J

))x(J(E

vviiiikikik Q,P,Q,P,I,Q,P,I

4.2.19


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First implementations

1973 USA: American Electric Power

765 kV and 345 kV network

16 busses, 25 branches

measurements: V1, Pij, Qij

1976 Germany: RWE

380 kV and 220 kV network

150 busses, 250 branches

measurements: Vi, Pij, Qij, Pi, Qi

4.2.20


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Size of network

- 280 busses

- 360 transmission lines

- 40 transformers

Total number of measurements

- 1010 measurements of active and reactive power flow

- 210 measurements of active and reactive power injections

- 140 measurements of active and reactive power injections

(pseudo measurements)

- 60 measurements of voltage mangnitude

Redundancy:

54.1n

nmr =

=

4.2.21

Data for the application of state estimation (RWE network)


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Application of state estimation (RWE network)

4.2.22


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Wednesday, July, 19th 11:30 a.m. Wednesday, January 17th 3:00 a.m.

State vector

380 kV State vector

380 kV

4.2.23

Application of state estimation (RWE network)


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State estimation: Sources of errors Interruption of data transmission

(measurements, on-off status quantities, transformer tap settings)

Bad measurement data (defect of measuring instrumentation, wrong measurementrange, false sign of measurement)

Network topology (incomplete representation of on-off status quantities)

Variance of measurement errors (instrument transformer, measuring instrument, A/D-

converter)

Network parameter (line length, transformer model)

Delays in the transmission of data

4.2.24

Documents

4.2 State Estimation