BÁO CÁO ĐÁNH GIÁ TRẠNG THÁI HỆ THỐNG ĐIỆN (Power System State Estimation)

• Introduction. • Power System State Estimation. • Solution Methodologies. • Weighted Least Square State Estimator. • Bad Data Processing. • Conclusion. • References.

POWER SYSTEM STATE ESTIMATION

Presentation byAshwani Kumar Chandel

Associate Professor

NIT-Hamirpur

Presentation Outline

• Introduction

• Power System State Estimation

• Solution Methodologies

• Weighted Least Square State Estimator

• Bad Data Processing

• Conclusion

• References

• Transmission system is under stress.

 Generation and loading are constantly increasing.

 Capacity of transmission lines has not increased

proportionally.

 Therefore the transmission system must operate with ever

decreasing 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 typically provide.

 Information delivery needs to be sufficiently robust so that

it is available even if key measurements are missing.

• Interconnected power networks have become more complex.

• The task of securely operating the system has become more difficult.

Difficulties mitigated through use

of state estimation

• Variables of interest are indicative of:

 Margins to operating limits

 Health of equipment

 Required operator action

• State estimators allow the calculation of these variables

of interest with high confidence despite:

 measurements that are corrupted by noise

 measurements that may be missing or grosslyinaccurate

Objectives of State Estimation

• Objectives:

To provide a view of real-time power system conditions

 Real-time data primarily come from SCADA

 SE supplements SCADA data: filter, fill, smooth

 To provide a consistent representation for powersystem security analysis

• On-line dispatcher power flow

• Contingency Analysis

• Load Frequency Control

 To provide diagnostics for modeling & maintenance

Power System State Estimation

• To obtain the best estimate of the state of the systembased on a set of measurements of the model of thesystem

• The state estimator uses

 Set of measurements available from PMUs

 System configuration supplied by the topologicalprocessor,

 Network parameters such as line impedances asinput

 Execution parameters (dynamic adjustments…)

weight-Power System State Estimation (Cont.,)

• The state estimator provides

 Bus voltages, branch flows, …(state variables)

 Measurement error processing results

 Provide an estimate for all metered and unmetered quantities.

 Filter out small errors due to model approximations and measurement inaccuracies;

 Detect and identify discordant measurements, the called bad data.

Bad Data Processor

Network Observability Check

Topology Processor

V, θ

Power System State Estimation (Cont.,)

• The state (x) is defined as the voltage magnitude andangle at each bus

• All variables of interest can be calculated from the stateand the measurement mode z = h(x)

I12

P12

V1

Power System State Estimation (Cont.,)

• We generally cannot directly observe the state

 But we can infer it from measurements

 The measurements are noisy (gross measurementerrors, communication channels outage)

Ideal measurement:

H(x)

Noisy Measurements

z=h(x)+e

Measurement: z

Consider a Simple DC Load Flow Example

Three-bus DC Load Flow The only information we have about this system is provided by three MW power flow meters

Case with all meters have small errors

If we use only the M 13 and M 32 readings,

as before, then the phase angles will be:

This results in the system flows as shown in

Figure Note that the predicted flows match at

M 13 , and M 32 but the flow on line 1-2 does not

match the reading of 62 MW from M12

1

2

3

0.024rad0.0925rad0rad(still assumed to equal zero )

Power System State Estimation (Cont.,)

• The only thing we know about the power system comes to

us from the measurements so we must use themeasurements to estimate system conditions

• Measurements were used to calculate the angles atdifferent buses by which all unmeasured power flows,loads, and generations can be calculated

• We call voltage angles as the state variables for the bus system since knowing them allows all other quantities

three-to be calculated

• If we can use measurements to estimate the “states” ofthe power system, then we can go on to calculate anypower flows, generation, loads, and so forth that wedesire

State Estimation: determining our best guess at the state

• We need to generate the best guess for the state giventhe noisy measurements we have available

• This leads to the problem how to formulate a “best”estimate of the unknown parameters given the availablemeasurement

• The traditional methods most commonly encounteredcriteria are

 The Maximum likelihood criterion

 The weighted least-squares criterion

• Non traditional methods like

 Evolutionary optimization techniques like Genetic

Algorithms, Differential Evolution Algorithms etc.,

Solution Methodologies

Weighted Least Square (WLS)method:

 Minimizes the weighted sum of squares of the difference between measured and calculated values

 In weighted least square method, the objective function „f‟ to be minimized is given by

Iteratively Reweighted Least Square (IRLS)Weighted Least Absolute Value (WLAV)method:

 Minimizes the weighted sum of the absolute value of difference between measured and calculated values.

 The objective function to be minimized is given by

 The weights get updated in every iteration.

m

2 i 2

1 e

i

m

| p |

i 1

Least Absolute Value(LAV) method:

 Minimizes the objective function which is the sum of absolute

value of difference between measured and calculated values.

The objective function „g‟ to be minimized is given by g=

Subject to constraint zi= hi(x) + ei

Where, σ 2 = variance of the measurement

W=weight of the measurement (reciprocal of variance of the measurement)

i 1

| h (x)-z |

• The measurements are assumed to be in error: that is, thevalue obtained from the measurement device is close tothe true value of the parameter being measured but differs

by an unknown error

• If Zmeas be the value of a measurement as received from ameasurement device

• If Ztrue be the true value of the quantity being measured

• Finally, let η be the random measurement error.

Then mathematically it is expressed as

meas true

Gaussian distibution Actual

distribution

Weighted least Squares-State Estimator

• The problem of state estimation is to determine theestimate that best fits the measurement model

• The static-state of an M bus electric power network isdenoted by x, a vector of dimension n=2M-1, comprised of

M bus voltages and M-1 bus voltage angles (slack bus istaken as reference)

• The state estimation problem can be formulated as aminimization of the weighted least-squares (WLS)function problem

2

(z h (x))min J(x)=

• This represents the summation of the squares of themeasurement residuals weighted by their respectivemeasurement error covariance

• where, z is measurement vector

h(x) is measurement matrix

m is number of measurements

σ2 is the variance of measurement

x is a vector of unknown variables to be estimated

• The problem defined is solved as an unconstrainedminimization problem

• Efficient solution of unconstrained minimization problemsrelies heavily on Newton‟s method

• where, the Jacobian matrix H(x) is defined as:

• Then the linearized least-squares objective function isgiven by

h(x x) h(x) H(x) x

h(x) H(x)

2 m

1 J( x) (e(x) H(x) x) R (e(x) H(x) x)

2

H R H x H R e

G x H R e

Weighted Least Squares-Example

•

est 1 est

est 2

x

• To derive the [H] matrix, we need to write the measurements

as a function of the state variables These functionsare written in per unit as

0.41

0.25

•

1 1

-7

• We get

• From the estimated phase angles, we can calculate thepower flowing in each transmission line and the netgeneration or load at each bus

est 1 est 2

0.028571 0.094286

(0.62 (5 5 )) (0.06 (2.5 )) (0.37 (4 )) J( , )

2.14

Solution of the weighted least square example

Bad Data Processing

• One of the essential functions of a state estimator is todetect measurement errors, and to identify and eliminatethem if possible

• Measurements may contain errors due to

 Random errors usually exist in measurements due tothe finite accuracy of the meters

 Telecommunication medium

• Bad data may appear in several different ways dependingupon the type, location and number of measurements thatare in error They can be broadly classified as:

 Single bad data: Only one of the measurements inthe entire system will have a large error

• Multiple bad data: More than one measurement will be inerror

• Critical measurement: A critical measurement is the one whose elimination from the measurement set will result in an unobservable system The measurement residual of a critical measurement will always be zero.

• A system is said to be observable if all the state variables can be

calculated with available set of measurements.

measurement which is not critical Only redundant measurements may have nonzero measurement residuals.

• Critical pair: Two redundant measurements whose simultaneous

unobservable.

• When using the WLS estimation method, detection and identification of bad data are done only after the estimation process by processing the measurement residuals.

• The condition of optimality is that the gradient of J(x) vanishes

at the optimal solution x, i.e.,

• An estimate z of the measurement vector z is given by

• The vector of residuals is defined as e = z - Hx; an estimate of

Bad Data Detection and Identification

• Detection refers to the determination of whether or not themeasurement set contains any bad data

• Identification is the procedure of finding out which specificmeasurements actually contain bad data

• Detection and identification of bad data depends on theconfiguration of the overall measurement set in a givenpower system

• Bad data can be detected if removal of the correspondingmeasurement does not render the system unobservable

• A single measurement containing bad data can beidentified if and only if:

 it is not critical and

 it does not belong to a critical pair

Bad Data Detection

•

N

2 i

i 1

2 k

Y 

Chi-square probability density function

Chi-squares distribution table

• Then, f(x) will have a chisquare distribution with at most (m n) degrees of freedom.

-where, m is number of measurements.

n is number of state variables.

Steps to detection of bad data

Bad Data Identification

Steps to Bad Data Identification

R i=1,2, m

Bad Data Analysis-Example

•

• Measurement equations characterizing the meterreadings are found by adding errors terms to the systemmodel We obtain

• Forming the H matrix we get

0.625 0.125 0.125 0.625 H

0.375 0.125 0.125 0.375

100 0 0 0

0 100 0 0W

0 0 50 0

0 0 0 509.01

3.02z

6.985.01

2

V 16.0072V

8.0261VV

5.01070Vz

e e

R i=1,2, m

•

1 ' 11

2 ' 22 3 ' 33 4 ' 44

1.4178 (1 0.807) 0.01

R

3.5144 (1 0.807) 0.01

R

0.4695 (1 0.193) 0.02

R

3.8804 (1 0.193) 0.02

R

model,” IEEE Trans Power Apparatus and Systems, vol PAS-89, pp 120-125,

Jan 1970.

PAS-89, pp 345-352, Mar 1970.

pp.125-130, Jan 1970.

Power and Energy Systems, vol 12, Issue 2, pp 80-87, Apr 1990.

http://www.books.google.com.

http://www.books.google.com.