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    POWER SYSTEMSTATE ESTIMATION

    Presentation by

     Ashwani Kumar Chandel

     Associate Professor 

    NIT-Hamirpur 

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    Presentation Outline

    • Introduction

    • Power System State Estimation

    • Solution Methodologies

    • Weighted Least Square State Estimator • Bad Data Processing

    • Conclusion

    • References

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    Introduction• Transmission system is under stress.

      Generation and loading are constantly increasing.

      Capacity of transmission lines has not increasedproportionally.

      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 certainvariables of interest than direct measurement can typicallyprovide.

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

    • Interconnected power networks have become more complex.

    • The task of securely operating the system has become moredifficult.

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    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 grossly

    inaccurate

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    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 power 

    system security analysis

    • On-line dispatcher power flow

    • Contingency Analysis• Load Frequency Control

    To provide diagnostics for modeling & maintenance

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    Power System State Estimation

    • To obtain the best estimate of the state of the system

    based on a set of measurements of the model of the

    system.

    • The state estimator uses

     Set of measurements available from PMUs

     System configuration supplied by the topological

    processor,

      Network parameters such as line impedances asinput.

     Execution parameters (dynamic weight-

    adjustments…)

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    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 unmeteredquantities.

     Filter out small errors due to model approximations andmeasurement inaccuracies;

      Detect and identify discordant measurements, the so-called bad data.

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    State Estimation

     Analog Measurements

    Pi , Qi, Pf  , Qf  , V, I, θkm

    Circuit Breaker Status

    State

    Estimator 

    Bad Data

    Processor 

    Network

    Observability

    Check

    Topology

    Processor 

    V, θ

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    Power System State Estimation (Cont.,)

    • The state (x) is defined as the voltage magnitude and

    angle at each bus

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

    i ji iV Ve

    1 2 n 1 bx [V ,V ,...,V , ,..., ]

    Measurement

    Model: h(x)

    I12

    P12

    V1

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    Power System State Estimation (Cont.,)

    • We generally cannot directly observe the state

    But we can infer it from measurements

    The measurements are noisy (gross measurement

    errors, communication channels outage)

    Ideal

    measurement:H(x)

    Noisy

    Measurementsz=h(x)+e

    Measurement: z

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    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

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    (Cont.,) Only two of these meter readings are required to calculate the bus

    phase angles and all load and generation values fully

    Now calculating the angles, considering third bus as swing bus we get

    13M 5MW 0.05pu

    32M 40MW 0.40pu

    13 1 3 13

    13

    32 3 2 32

    23

    1f ( ) M 0.05pux

    1f ( ) M 0.40pu

    x

    1

    2

    0.02rad

    0.10rad

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    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

    0.024rad

    0.0925rad

    0rad(still assumed to equal zero  )

    12

    13

    32

    M 62MW 0.62pu

    M 6MW 0.06pu

    M 37MW 0.37pu

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    Power System State Estimation (Cont.,)

    • The only thing we know about the power system comes tous from the measurements so we must use themeasurements to estimate system conditions.

    • Measurements were used to calculate the angles at

    different buses by which all unmeasured power flows,loads, and generations can be calculated.

    • We call voltage angles as the state variables for the three-bus system since knowing them allows all other quantitiesto be calculated

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

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    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 available

    measurement.• The traditional methods most commonly encountered

    criteria are

      The Maximum likelihood criterion

      The weighted least-squares criterion.

    • Non traditional methods like

      Evolutionary optimization techniques like Genetic Algorithms, Differential Evolution Algorithms etc.,

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    Solution MethodologiesWeighted Least Square (WLS)method:

    Minimizes the weighted sum of squares of the difference betweenmeasured 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

    i2i 1   i

    1e

    i

    m| p |

    i 1

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    (Cont.,)Least Absolute Value(LAV) method:

     Minimizes the objective function which is the sum of absolutevalue 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, σ2 = variance of the measurement

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

    measurement)

    ei = zi-hi(x), i=1, 2, 3 ….m.

    h(x) = Measurement function, x = state variables and Z= MeasuredValue

    m=number of measurements

    mW

    ii 1

    | h (x)-z |i i

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    (Cont.,)

    • The measurements are assumed to be in error: that is, the

    value obtained from the measurement device is close to

    the true value of the parameter being measured but differs

    by an unknown error.

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

    measurement 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 trueZ Z

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    (Cont.,)

    2 21PDF( ) exp( / 2 )2

    20

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    20

    Probability Distribution of Measurement Errors

    3

    f(x)

    x

    0

    Gaussian

    distibution Actual

    distribution

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    Weighted least Squares-State Estimator 

    • The problem of state estimation is to determine the

    estimate that best fits the measurement model .

    • The static-state of an M bus electric power network is

    denoted by x, a vector of dimension n=2M-1, comprised of 

    M bus voltages and M-1 bus voltage angles (slack bus is

    taken as reference).

    • The state estimation problem can be formulated as a

    minimization of the weighted least-squares (WLS)

    function problem.

    •2m

    i i

    2i 1 i

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

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    (Cont.,)

    • 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.

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    (Cont.,)

    • The type of  Newton‟s method of most interest here is the

    Gauss-Newton method.

    •  In this method the nonlinear vector function is linearized

    using Taylor series expansion

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

    • Then the linearized least-squares objective function is

    given by

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

    h(x)H(x)

    x

    T 11J( x) (z h(x) H(x) x) R (z h(x) H(x) x)

    2

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    (Cont.,)

    • where, R is a weighting matrix whose diagonal elements

    are often chosen as measurement error variance, i.e.,

    • where, e=z-h(x) is the residual vector.

    2

    1

    2

    m

    R   

    T 11J( x) (e(x) H(x) x) R (e(x) H(x) x)2

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    (Cont.,)

    T 1J( x) H R (e H x) 0x

    T 1 T 1H R H x H R e

    T 1G x H R e

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    Weighted Least Squares-Example

    est

    1est

    est

    2

    x

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    (Cont.,)

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

    as a function of the state variables . These functions

    are written in per unit as1 2and 

    12 12 1 2 1 2

    13 13 1 3 1

    32 32 3 2 2

    1M f ( ) 5 50.2

    1M f ( ) 2.5

    0.4

    1M f ( ) 4

    0.25

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    (Cont.,)

    5 5

    [H]   2.5 0

    0 4

     

    2 2

    M12 M122 2

    M13 M13

    2 2

    M32 M32

    0.0001

    R 0.0001

    0.0001

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    (Cont.,)

    11

    est

    1

    est

    2

    1

    0.0001 5 55 2.5 0 0.0001 2.5 05 0 4

    0.0001 0 4

    0.0001 0.625 2.5 0

    0.0001 0.065 0 4

    0.0001 0.3

     - -

     

    - -7

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    (Cont.,)

    • We get

    • From the estimated phase angles, we can calculate the

    power flowing in each transmission line and the net

    generation or load at each bus.

    est

    1

    est

    2

    0.028571

    0.094286

     

    2 2 2

    1 2 1 21 2

    (0.62 (5 5 )) (0.06 (2.5 )) (0.37 (4 ))J( , )0.0001 0.0001 0.0001

    2.14 

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    Solution of the weighted least square example

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    Bad Data Processing

    • One of the essential functions of a state estimator is to

    detect measurement errors, and to identify and eliminatethem if possible.

    • Measurements may contain errors due to

     Random errors usually exist in measurements due to

    the 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 

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    (Cont.,)• Critical measurement: A critical measurement is the one whose

    elimination from the measurement set will result in an unobservablesystem. The measurement residual of a critical measurement willalways be zero.

    • A system is said to be observable if all the state variables can becalculated with available set of measurements.

    • Redundant measurement: A redundant measurement is ameasurement which is not critical. Only redundant measurementsmay have nonzero measurement residuals.

    • Critical pair: Two redundant measurements whose simultaneousremoval from the measurement set will make the systemunobservable.

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    (Cont.,)• When using the WLS estimation method, detection andidentification 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 

    e is given by

    ( h( ) 'J x z x) W z h(x)

    1GX H WZ 0   1 1 X G H WZ

      Z HX

    e z h(x)

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    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 

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    Bad Data Detection

     N2

    ii 1Y X

    2

    k Y

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    Chi-square probability density function

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    Chi-squares distribution table

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    (Cont.,)

    • The degrees of freedom k, represents the number of independent variables in the sum of squares.

    • Now, let us consider the function f(x), written in terms of themeasurement errors:

    • where e is the measurement error, Rii  is the diagonal entry of the measurement error covariance matrix and m is the totalnumber of measurements.

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

    where, m is number of measurements.

    n is number of state variables.

    thi

    2m m m

    21 2 Ni

    ii i i

    i 1 i 1 i 1ii

    e

    f (x) R e eR 

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    Steps to detection of bad data

    m2 2

    i i

     j 1f e / .

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    Bad Data Identification

    '

    i i ii(z z ) / R  

    ' 1 T 1

    iiR (I HG H R )R  

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    Steps to Bad Data Identification

     N i

    i 'ii

    ee

    R  i=1,2,...m

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    Bad Data Analysis-Example

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    Cont.,

    • Measurement equations characterizing the meter 

    readings are found by adding errors terms to the system

    model. We obtain

    1 1 2 1

    2 1 2 2

    3 1 2 3

    4 1 2 4

    5 1z x x e

    8 81 8

    z x x e8 8

    3 1z x x e

    8 81 3

    z x x e8 8

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    (Cont.,)

    • Forming the H matrix we get

    0.625 0.125

    0.125 0.625H

    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

    6.98

    5.01

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    (Cont.,)

    • Solving for state estimates i.e.,

    • We get

    11 T

    2

    VG H Wz

    V

    1

    2

    V 16.0072V

      8.0261VV

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    (Const.,)•

    1

    2

    3

    4

    z 9.00123A

    z 3.01544A

    7.00596Vz

    5.01070Vz

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    (Cont.,)

    1

    2

    3

    4

    e 9.01 9.00123 0.00877A

    e 3.02 3.01544 0.00456A

    6.98 7.00596 0.02596Ve

    5.01 5.01070 0.00070Ve

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    (Cont.,)

    42 2 2 2 2 2

    i i

     j 1

    f e / 100(0.00877) 100(0.00456) 50(0.02596) 50(0.00070)

    0.043507

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    (Cont.,)

    T T

    1 2 3 4[z z z z ] [9.01A 3.02A 6.98V 4.40V]

    T T

    1 2 3 4[e e e e ] [9.01A 3.02A 6.98V 4.40V]

    42 2 2 2 2 2

    i i

     j 1

    f e / 100(0.06228) 100(0.15439) 50(0.05965) 50(0.49298)

    15.1009

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    (Cont.,)

    ' 1 T 1

    iiR (I HG H R )R  

    i N

    i'

    ii

    e

    eR 

     i=1,2,...m

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    (Cont.,)•

    1

    '

    11

    2

    '

    22

    3

    '

    33

    4

    '

    44

    e   0.06228 1.4178(1 0.807) 0.01R 

    e   0.154393.5144

    (1 0.807) 0.01R 

    e   0.05965 0.4695(1 0.193) 0.02R 

    e   0.492983.8804

    (1 0.193) 0.02R 

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    Conclusion

    • Real time monitoring and control of power systems isextremely important for an efficient and reliable operationof a power system.

    • Sate estimation forms the backbone for the real timemonitoring and control functions.

    • In this environment, a real-time model is extracted atintervals from snapshots of real-time measurements.

    • Estimate the nodal voltage magnitudes and phase anglestogether with the parameters of the lines.

    • State estimation results can be improved by usingaccurate measurements like phasor measurement units.

    • Traditional state estimation and bad data processing isreviewed.

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    References

    •  F. C. Schweppe and J. Wildes,  “Power system static state estimation, part I: exactmodel,”   IEEE Trans. Power Apparatus and Systems, vol. PAS-89, pp. 120-125,Jan. 1970.

    •  R. E. Tinney W. F. Tinney, and J. Peschon,   “State estimation in power systems,part i: theory and   feasibility,”   IEEE Trans. Power Apparatus and Systems, vol.PAS-89, pp. 345-352, Mar. 1970.

    •  F. C. Schweppe and D. B. Rom,   “Power  system static-state estimation, part ii:approximate   model,”   IEEE Trans. Power Apparatus and Systems, vol. PAS-89,pp.125-130, Jan. 1970.

    •  F. F. Wu,   “Power  System State   Estimation,”   International Journal of Electrical Power and Energy Systems, vol. 12, Issue. 2, pp. 80-87, Apr. 1990.

    •  Ali Abur and Antonio Gomez Exposito. (2004, April). Power System State

    Estimation Theory and Implementation (1st

    ed.) [Online]. Available:http://www.books.google.com.

    •  Allen J wood and Bruce F Wollenberg. (1996, February 6). Power Generation,Operation, and Control (2nd ed.) [Online]. Available:http://www.books.google.com.