censcir presentation 20070328

7/30/2019 CenSCIR Presentation 20070328

1/40

1

Static State Estimation in

Electric Power Systems:

Prevalent methods and improvements using

measurements from multiple scans

Presented by: Ellery Blood Advisors: Marija Ili and Bruce Krogh

CenSCIR - Graduate Student Collaboration

Presentation: 28 March 2007


2/40

2

Presentation Outline

Introduction

The Static State Estimation Process

General In Power Systems

Dynamic Estimation of Static State

Power System Pseudo-Dynamics

Algorithms


3/40

3

Difficulties in power transmissionsystem operation

Transmission system is under stress. Generation and loading are constantly increasing.

The transmission capacity has not increased proportionally.

Therefore the transmission system must operate with everdecreasing margin from its maximum capacity.

Operators need reliable information to operate. Need to have more confidence in the values of certain

variables of interest than direct measurement can typicallyprovide.

Information delivery needs to be sufficiently robust so that itis available even if key measurements are missing.


4/40

4

Difficulties mitigated through useof state estimation

Variables of interest are indicative of: Margins to operating limits

Health of equipment

Required operator action

State estimators allow the calculation of thesevariables of interest with high confidencedespite:

measurements that are corrupted by noise measurements that may be missing or grossly

inaccurate


5/40

5

State estimators in existingsystems

The state estimator is anintegral part of the overallmonitoring and controlsystems for transmissionnetworks.

Information from thestate estimator flows to

control centers anddatabase servers acrossthe network.

Monitoring

& State

Estimation

Parameter

Data Base

Data

SCADA

Archive

Load

Power delivery

control

Power market

Data

Acquisition

Control

Parameter Data

Operator

Displays

Power grid

CommandsArchive Data

Alarms

EventsMeasurements


6/40

6


Introduction The Static State Estimation Process

General In Power Systems


Power System Pseudo-Dynamics

Algorithms


7/40

7

Electric power system state

The state (x) is defined as the complex voltage magnitudeand angle at each bus

All variables of interest can be calculated from the state and

the measurement model

[ ]Tnn VVV LL 2121 =xij

ii eVV=

~

)(xhz =

V3

State: x

SystemP47

I12

Measurement

Model: hi(x)


8/40

8

Problems in finding system state

We generally cannot directly observe

the state

State:x

System


9/40

9


the state But we can infer it from measurements

Measurements:z

System


State:x


10/40

10


the state But we can infer it from measurements

However, the measurements are noisy

(How can we see behind the noise?)

Ideal

Measurements

h(x)

State:xNoisy

Measurements

z=h(x)+v

Noise

System



11/40

11

State estimation: determining ourbest guess at the state

We need to generate the best guess for thestate given the noisy measurements we haveavailable.

The best guess is that which generates the

minimum weighted squared error Find a value for the state that makes the

measurement model output match themeasurements as closely as possible in the

minimum squared error sense. Weight the errors by the uncertainties in therespective measurements (weighted leastsquares)


12/40

12

State: x Measurements: zi=h

i(x)+v

iUncertainty: Ri=E[(vi)

2]

Cost function:

= =m

iiii RxhzxC

1

2

/))(()(

State Estimator

(minimize C(x))

State

Estimate

Noisy

Measurements

State estimation equations (1)


13/40

13


C(x) is minimized when

where

0))(()()(

)( 1 ==

= xhzRxHx

xxg TC

)(

)( xhx

xH

=


14/40

14


If the measurement model is linear, then

h(x)=Hx and

If h(x) is nonlinear, then there is generally not ananalytic solution. However, if we approximate h(x)as h(xk)+H(xk){x-xk}, then

and we can set up the following iteration: [1]

.][ 111 zRHHRHx = TT

)).(()]()([ 1111 kTkkTkk xhzRHxHRxHxx += +

( ) 0)}){()(()()( 1 =+= kkkkT xxxHxhzRxHxg

[1] Ali Abur and Antonio Gomez Exposito, Power System State Estimation: Theory and Implementation, Chapter

2: Weighted Least Squares State Estimation,, 2004, Marcel Dekker, Inc (ISBN: 0-8247-5570-7), pp 9-36


15/40

15



General

In Power Systems


Power System Pseudo-Dynamics Algorithms


16/40

16

Electric power system state

The state (x) is quasi-static Generally operates at steady state AC where the

time constants for transients are faster than therate of measurement collection (or analysis).

Analyzed as a series of static states The estimation is based on a snapshot or

scan of measurements (assumed to be

synchronized)


17/40

17

Reduced state unambiguous

measurement model

Power flows and injections are strongly dependent ondifference in the phase angles, not the phase angles

themselves. The system of equations composing our measurement

model does not have a unique solution. Thus, state estimation equations are poorly conditioned

unless the ambiguity in angle is removed.

One angle is identified to be the reference angle andis set to zero. All other angles are now angle differences with respect to

the reference angle. Thus, x is reduced by one element and

the new state vector is defined to be:

The ambiguity is removed.

[ ]Tnn VVV LL 2111312 =x


18/40

18



General

In Power Systems




19/40

19

Power system pseudo-dynamics:motivation

State is quasi-static

Time constants are sufficiently fast so that system dynamicsdecay away quickly (with respect to measurementfrequency).

The system appears to be progressing through a sequence

of static states. Analysis of the sequence of static states indicates

slower dynamics, driven by: Changes in load profile

Other slowly varying effects Integrating information from multiple measurement

snapshots may improve state estimation


20/40

20

Power system pseudo-dynamics:approach

Define a discrete-time pseudo-dynamicsystem to model the transitions fromone static state to the next.

Model the slower dynamics Factor in more information to filter out

noise more effectively

Use Kalman filter to generate optimalestimator gain.


21/40

21



General

In Power Systems




22/40

22

Dynamic state estimationalgorithms

A discrete time dynamic model,,

provides a prediction of what the next state

should be prior to integrating measurements A[t], B[t], and u[t] are dependent on the

dynamic model used.

w[t] represents process noise and modelerror

][][][][][]1[ tttttt wuBxAx ++=+


23/40

23

Base-case dynamic stateestimation algorithm

To compare static vs. dynamic state estimators,

consider the model proposed by Debs and Larson[2]. Dynamic Model:

Assumes that the state does not greatly change fromsnapshot to snapshot.

Any changes that occur, are the result of perturbation of thestate by zero-mean Gaussian noise

Tested on 8-bus decoupled model subject to a rampload increase (worst case load dynamic).

Showed smaller error than static estimator if processnoise (w[t]) was characterized appropriately.

][][]1[ ttt wxx +=+

[2] A.S. Debs, R. E. Larson,A dynamic estimator for tracking the state of a power system, IEEE Transactions onPower Apparatus and Systems, vol PAS-89, iss. 7, pp. 1670-1678, Sept-Oct 1970


24/40

24

From Debs-Larson, 1970

Comparative performance of

static and base-case dynamic

estimator at differentsampling periods and process

noise coefficients.

Dynamic performance isdependent on

characterization of w(),which can be tuned to give

the minimum error.

Performance of base-case

dynamic estimator


25/40

25

Performance of base-case dynamic

estimator vs. sampling time

Plotting C(x) for the static

estimator and the optimizedbase-case dynamic estimator for

various sampling periods, we

see:

1) Optimized dynamic estimator

consistently achieves smaller

C(x),

2) C(x) decreases monotonically

with sampling period.


26/40

26

Improved dynamic stateestimation algorithms (1)

Shih-Huang[3]

Improves over the base-case dynamic model

by utilizing a more detailed dynamic model. Enhances the Kalman filter by adjusting the

weights of suspect measurements to filter out

grossly bad data.

][][][]1[ tttt wBxAx ++=+

[3] Kuang-Rong Shih and Shyh-Jier Huang,Application of a Robust Algorithm for Dynamic State Estimation of a Power System,

IEEE Transactions on Power Systems, Vol. 17, No. 1, February 2002

|))(|exp(~ xhzww ii =


27/40

27

Effect of weighting modification

The effect of exponentially

decreasing the weigh ofmeasurements which are far from

predicted measurement model

output results in noticeable

improvement over the basicKalman filter.


28/40

28

Improved dynamic stateestimation algorithms (2)

Blood-Ilic-Krogh-Ilic[4]

Assumes state change is based on load

change B[t]u[t] is based linearization of Load-

Flow equations to determine the

relationship between the respectiveincremental changes of u[t] and x[t]

][][][][]1[ ttttt wuBxx ++=+

[4] Ellery Blood, Marija Ilic, Bruce Krogh, and Jovan Ilic, A Kalman Filter Approach to Static State

Estimation in Electric Power Systems,North American Power Symposium 2006 Conference Proceedings


29/40

29

Incremental change in state

corresponds to an incremental

change in power injections

The Load-Flowequations describethe relationship

between the powerinjections and thestate of a powernetwork

At steady state, theequation is balanced


30/40

30

Load-Flow equations( )

)()1()()1(

)(~

)(

)()1(

)()1()1(

~

0~

0

),~

(~

~

),~

(

0~~

),~

(

1

2

1

tttt

tVt

tt

ttt

V

V

V

V

V

V

jQPYVVV

f

f

f

ii

k

ikki

wuJxx

x

QQ

PPu

eQ

PJ

e

Q

PIJ

e

Q

P

Q

P

Q

Pf

Q

Pf

Q

Pf

+++=+

=

+

+=+

+

=

=+

=+

+

=+=

Starting with the load-flow equations,

we lineralize around an operating

point and look at the incremental

changes to state and injections.

Solving for the incremental change in

the state, we see that it related to the

injections through the power

injection Jacobian.

We identify the incremental change

in injections as the input, lump thelinearization error (ef) in with the

process noise, and we have our

dynamic model.


31/40

31

Load-flow based dynamics vs.

base-case

Simulation on a decoupled 4-bus system showed that the load-

flow based dynamic state estimator (blue) generated

consistently smaller squared error than the base-case dynamic

estimator (red).


32/40

32

Conclusion

Static state estimation in electric power

systems provides useful and reliableinformation to operators.

State estimation results can be improved

through Kalman filtering with a pseudo-dynamics of the system.

Basic dynamic state estimators can be

improved through careful selection of thedynamic model and handling of suspect data.


33/40

33

Questions?


34/40

34

Measurement Model: h(x)

( )( )

i

ijijij

ijijijijjijsiiiij

ijijijijjijsiiiij

jijijijijjii

j

ijijijijjii

V

QPI

bgVbbVVQ

bgVggVVP

BGVVQ

BGVVP

22

)cossin()(

)sincos()(

)cossin(

)sincos(

+=

+=

++=

=

+=

ij=i-j Gij+jBij=Yij Bus admittance matrix element

gij

+jbij

=series branch admittance between bus i and j

gsi+jbsi=shunt branch admittance at bus i


35/40

35

Measurement Jacobian: HPower Injections

( )

( )

( )

( )

( )

( )

( )

( )ijijijiji

j

i

j

iiiijijijijj

i

i

ijijijijji

j

i

j

iiiijijijijji

i

i

ijijijiji

j

i

j

iiiijijijijj

i

i

ijijijijji

j

i

j

iiiijijijijji

i

i

BGVV

Q

BVBGV

V

Q

BGVVQ

GVBGVVQ

BGVV

P

GVBGV

V

P

BGVVP

BVBGVVP

cossin

cossin

sincos

sincos

cossin

sincos

sincos

cossin

=

+=

=

+=

+=

++=

=

+=


36/40

36

Measurement Jacobian: HPower Flows

( )

( )

( )

( )

( )

( )

( )

( )ijijijiji

j

ij

siijiijijijijj

i

ij

ijijijijji

j

ij

ijijijijji

i

ij

ijijijiji

j

ij

siijiijijijijj

i

ij

ijijijijji

j

ij

ijijijijji

i

ij

bgVV

Q

bbVbgVV

Q

bgVVQ

bgVVQ

bgVV

P

ggVbgVV

P

bgVVP

bgVVP

cossin

)(2cossin

sincos

sincos

sincos

)(2sincos

cossin

cossin

=

+=

+=

+=

+=

+++=

=

=


37/40

37

Measurement Jacobian: HVoltage and Current Magnitudes

( )

( )ijijij

ijij

j

ij

ijji

ij

ijij

i

ij

ijji

ij

ijij

j

ij

ijjiij

ijij

i

ij

j

i

i

i

j

ij

i

ij

VVI

bgVI

VVI

bg

V

I

VVI

bgI

VVI

bgI

V

V

V

V

P

P

cos

cos

sin

sin

0

1

0

0

22

22

22

22

+=

+

=

+=

+

=

=

=

=

=


38/40

38

Appendix 1:Least Squares Eqs

+= )(xhz

[ ] [ ]TC )()(21)( 1 xhzRxhzx =

[ ] 0xhzRxHx

x

xx

==

=

)()()( 1

C

x

xhH

=

)(

[ ] [ ])()()()( 111 xhzRxHxxxHRxH = +

Tii

T

[ ])()),(( 1 xhzRxHxx +=+ pinvii

{ } 111 )()()),(( = RxHxHRHRxH TTpinv


39/40

39

Appendix 2:Prediction (Pseudo-Dynamic) Eqs( ) ii

k

ikki jQPYVV +=~~

[ ]Tnn QQQPPP LL 2121=Inj

( ) ( )0),( =+=

iik

ik

j

k

j

iijQPYeVeVg ki

Injx

[ ][

]Tn

n

T

QP

gimaggimag

grealgreal

)),(()),((

)),(()),((

),(),(),(

1

1

InjxInjx

InjxInjx

InjxfInjxfInjxf

L

L

==

0),(),(

=+

+

feInj

u

Injxfx

x

Injxf

Jff

ff

x

Injxf =

=

V

V

QQ

PP

),(

II

I

Q

f

P

fQ

f

P

f

u

Injxf=

=

=

0

0),(

QQ

PP

[ ] )()()()1( 1 tttt wuJxx ++=+

ij

ii eVV=

~

[ ]Tnn VVV LL 2121 =x

)1()()( = ttt InjInju


40/40

40

Appendix 3:Kalman Filter Eqs

[ ])()1()()1(1

tttt uuJxx++=+

( )[ ] [ ][ ]{ } 11111 )()()1( +++=+ RHJWJSHHJWJSL TTTT ttt

{ }))1(()1()1()1()1( +++++=+ ttttt xhzLxx

[ ] [ ] [ ])1()1(

)1()()1()1( 11

+++

+++=+

tt

tttt

T

TT

RLL

HLIJWJSHLIS

censcir presentation 20070328

Documents