lecture 15 stateestimation

8/17/2019 Lecture 15 StateEstimation

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Lecture 15

Power system state estimation

Kun Zhu2013-05-02


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Outline

•

State estimation

- What

- Why

• Weighted Leaset Square (WLS) algorithms

- Mathmatics

- Concepts

• Operation challenges


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Course road map


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What is state estimation?


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How could the operator know thesystem?

62 MW

6 MW


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The truth is out here!

Courtesy of ABB Ventryx


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Power system states: x

• The power system states are those parameters that can beused to determine all other parameters of the power system

•

Node voltage phasor

- Voltage magnitude V k-

Phase angle !k• Transformer turn ratios

- Turn ratio magnitude t kn- Phase shift angle "kn

•

Complex power flow

- Active power flow P kn , Pn

k- Reactive power flow Qkn , Qnk


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Analog measurements

•

Voltage magnitude

• Current flow magnitude & injection

• Active & reactive power

- Branches & groups of branches

- Injection at buses

-

In switches

- In zero impedance branches

- In branches of unknown impedance

• Transformers

- Magnitude of turns ratio

-

Phase shift angle of transformer• Synchronized phasors from

Phasor Measurement Unit


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Network topology processing

Bus breaker model Bus branch model


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Power system measurements: z

z=ztrue+e

z=ztrue+e

z=ztrue+e

ztrue: power system truthztrue=h(x)

e: measurement errore=esystematic+erandom


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Measurement model

• How to determine the states (x) given a set ofmeasurements (z)?

z j = h j ( x ) + e j

known unknown unknown

where

- x is the true state vector [V 1 ,V 2 ,…V k , !1 , !2 ,… !k ]

- z j is the jth measurement- h j relates the jth measurement to states

- e j is the measurement error


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State estimation process


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Why do we need state estimation?


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Measurements correctness

• Imperfections in- Current & Voltage transformer

-

Transducers

• A/D conversions

•

Tuning-

RTU/IED Data storage

- Rounding in calculations

- Communication links

• Result in uncertainties in the measurements


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Measurement timeliness

• Due to imperfections in SCADA system themeasurements will be collected at different pointsin time, time skew.

• If several measurements are missing how long towait for them?

• Fortunately, not a problem during quansi-steadystate.

•

State estimation is used for off-line applications

t

Pab

Pbc

Pcb Pdc


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How can the states be estimated?


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Approaches

•

Minimum variance method- Minimize the sum of the squares of the weighted

deviations of the state calculated based onmeasurements from the true state

• Maximum likelihood method

-

Maximizing the probability that the estimate equalsto the true state vector x

• Weighted least square method (WLS)

- Minimize the sum of the weighted squares of theestimated measurements from the true state

2

1

( ( ))( )

ii

m

i i

i

z h x J x

R=

!="


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Least square (Wiki)

• "Least squares" means that the overall solutionminimizes the sum of the squares of the errors madein the results of every single equation.

• The method of least squares is a standard approach

to the approximate solution of over determinedsystem, i.e., sets of equations in which there are moreequations than unknowns.

• The most important application is in data fitting.

• Carl Friedrich Gauss is credited with developing the

fundamentals of the basis for least-squares analysis in1795.


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WLS state estimation

• Fred Schweppe introduced state estimation to powersystems in 1968.

• He defined the state estimator as “a data processingalgorithm for converting redundant meter readings

and other available information into an estimate of thestate of an electric power system”.

• Today, state estimation is an essential part in almostevery energy management system throughout theworld.

Felix F. Wu, “Power system state estimation: a survey”,International Journal of Electrical Power & EnergySystems, Volume 12, Issue 2, April 1990, Pages 8


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WLS state estimation model

z j = h j ( x ) + e jknown unknown unknown


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Error characteristics

• The errors in the measurements is the sum ofseveral stochastic variables

- CT/VT, Transducer, RTU, Communication…

• The errors is assumed as a Gaussian Distribution

with known deviations

- Expected value E[e j ] =0

- Known deviation # j • The errors are also assumed to be independent

- E [eie j ] =0


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WLS objective function

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

)( 1

12

2

xh z R xh z xh z

x J T

m

i i

i !!=!

=

!

=

"#

wherei=1,2,…m

Solution to above is iterative using newton methods

R = diag{! 12,! 2

2,! 3

2,...! m

2} =Cov(e) = E [e !e

T ]


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Newton iteration

• At the minimum, the first-order optimality conditionswill have to be satisfied

0)]([)()()( 1 =!="

"=

! xh z R x H x x J x g T

])(

[)( x

xh x H

T

!

!= is the measurement Jacobian matrix


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0.....))(()()( =+!+= k k k x x xG x g x g

Newton iteration cont’d

• Expanding the g(x) into its Taylor series around statevector

k x

)()(

)(

)(

1 k k T k

k

x H R x H x

x g

xG

!=

"

"=

where


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Newton iteration cont’d

• Neglecting the higher order terms leads to an iterativesolutions scheme know as the Gauss-Newton methodas :

)()]([ 11 k k k k

x g xG x x !"= "+

k is the iteration index,is the solution vector at iteration k k x


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Newton iteration IV

• Convergence

• If not, update

Go back to the previous step

! "# )max( k x

1

1

+=

!+=+

k k

x x x k k k


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Why the measurements areweighted?

),,,(

),,,(

ji jiij

ji jiij

vv f Q

vv f P

! !

! !

=

=

voltage mag

complex power flow


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Weight

• Weight is introduced to emphasize the trustedmeasurement while de-emphasize the lesstrusted ones.

•

WLS

2

1

i

iW

!

=


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Observability

• Based on system topologyand location ofmeasurements parts of the

power system may beunobservable.

• Unobservable parts of the

system can be madeobservable via data

exchange (CIM), pseudo

measurements, etc…


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Observability example

Pij = f (vi,v j ,! i,! j )

Qij = f (vi,v j ,! i,! j )

voltage mag

complex power flow


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Observability criterion

Necessary but not sufficient condition

m: Number of measurements

n: Number of states

Is the system’s oservability guranteed in this

case?

nm !


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Observability example II

),,,(),,,(

ji jiij

ji jiij

vv f Qvv f P

! !

! !

=

=

voltage mag

complex power flow


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n of m measurements has to be independent

),,,(

),,,(

ji jiij

ji jiij

vv f Q

vv f P

! !

! !

=

=

voltage mag

complex power flow


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Summary of assumptions

• Quansi-steady system

- No large variations of states over time

- State estimator will be suspended when largedisturbance happens.

• Errors in measurements are

- Gaussian in nature with known deviation

- Independent

•

Strong assumption- Power system topology model is correct


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Refinements

• Refinements of method for State Estimation hasbeen the objective of much research.

• Reduce the numerical calculation complexity in

order to speed up the execution.- 3000 bus node in 1-2 seconds

• Improve estimation robustness

- less affected by erroneous input


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Bad Data Detection


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Data quality

• Analog measurement error

• Parameter error

• Topological error

- Discrete measurement error

- Model error


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Bad Data Detection (Analog)

• Putting the measurements up to a set oflogical test (Kirchhoff's laws) before they areinput into the State Estimator

• Calculating J(x) and comparing with a pre-determined limit. If the value exceeds the limit,we can assume there is “something” wrong in themeasurements (Chi-square)


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Bad Data Detection (Analog)

• Once the system state has been identified, i.e. wehave an estimate of x

• We can use the estimate to calculate

• If there is any residual in this calculation that “stands-out” this is an indication that particularmeasurement is incorrect

)ˆ( xh z r iii !

=


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Largest normalized residual

• The normalized residual follows the standardnormal distribution

i

i N

i

xh z r

!

)ˆ("

=


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Limitations

• The critical measurements have zero residuals,hereby they can not be detected

!

!!!!!

"

#

$

$$$$$

%

&

=

t

t

t t t t t

S

0000

....

.0...

00000

Sz z K z z r

Kz z R H H H H z T T

ii

=!=!=

==

!

)1(ˆ

)(ˆ 1


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Critical measurement example

),,,(),,,(

ji jiij

ji jiij

vv f Qvv f P

! !

! !

=

=

voltage mag

complex power flow


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Parameter and structural processing

•

Similar theory can be applied

iiiiii e p f z e xh z +=!+= )()(


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State estimation functions

1. Bad data processing and elimination givenredundant measurements

2. Topology processing:

create bus/branch model (similar to Y matrix)

3. Observebility analysis:

all the states in the observable islands haveunique solutions

4. Parameter and structural processing


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Challenges

• Perception of the process is from themeasurements whose quality is, to largeextent, out of our control

•

The quality of estimates relies on the inputwhose uncertainty is highly dependent on theICT infrastructure

• Power system model can contain errors

lecture 15 stateestimation

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