POWER SYSTEM STATE ESTIMATION Presentation by Ashwani Kumar Kumar Chandel Associate Associat e Professor NIT-Hamirpur
Presentation Outline • Introduction
• Power System State Estimation • Solution Methodologies • Weighted Least Square State Estimator • Bad Data Processing • Conclusion • References
Presentation Outline • Introduction
• Power System State Estimation • Solution Methodologies • Weighted Least Square State Estimator • Bad Data Processing • Conclusion • References
Introduction • 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. capacity. information to operate. • Operators need reliable information Need Ne ed to ha have ve more more con confi fide denc nce e in the value values s of certai certain n 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 nterconnected power networks have become more complex. complex. • I • 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 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
Objectives of State Estimation • Objectives: To
provide a view of real-time power system conditions Real-time data primarily primarily come from SCADA
SE
supplements SCADA data: filter, fill, smooth.
To
provi pro vide de a co cons nsiiste ten nt re rep pre res sent nta ati tion on fo forr po pow wer system 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 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 as
input.
Execution parameters adjustments…)
(dynamic
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 socalled bad data.
State Estimation Analog Measurements Pi , Qi, Pf , Qf , V, I, θkm
State Estimator
Topology Processor
Network Observability Check
Circuit Breaker Status
V, θ
Bad Data Processor
Power System State Estimation (Cont.,) • The state (x) is defined as the voltage magnitude and
angle at each bus j Vi Vi e i
x
[V1 , V2 ,..., Vn , 1,...,
b
]
• All variables of interest can be calculated from the state
and the measurement mode. z = h(x) I12 Measurement Model: h(x)
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 measurement errors, communication channels outage)
Ideal measurement: H(x)
Noisy Measurement: z Measurements z=h(x)+e
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
(Cont.,)
Only two of these meter readings are required to calculate the bus phase angles and all load and generation values fully
M13 M32 f13 f 32
5MW
0.05pu
40MW
1 x13 1 x 23
( (
1
3
3
0.40pu
)
M13
)
M32
2
0.05pu 0.40pu
Now calculating the angles, considering third bus as swing bus we get 1 2
0.02rad 0.10rad
Case with all meters have small errors M12
62MW
M13
6MW
M 32
37MW
0.62pu 0.06pu 0.37pu
If we use only the M 13 and M 32 readings, as before, then the phase angles will be: 1 2 3
0.024rad 0.0925rad 0rad(still assumed to equal zero)
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 M 12.
Power System State Estimation (Cont.,) • The only thing we know about the power system comes to
us from the measurements so we must use the measurements 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 threebus system since knowing them allows all other quantities to be calculated • If we can use measurements to estimate the “states” of the power system, then we can go on to calculate any power flows, generation, loads, and so forth that we desire.
State Estimation: determining our best guess at the state • We need to generate the best guess for the state given
the 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.,
Solution Methodologies Weighted
Least Square (WLS)method:
Minimizes
the weighted sum of squares of the difference between measured and calculated values .
weighted least square method, the objective function „f‟ minimized is given by m 1 2 e i 2
In
i 1
i
Iteratively
Reweighted Least Square Value (WLAV)method:
(IRLS)Weighted Least Absolute
Minimizes the weighted sum of the absolute value of between measured and calculated values.
The
objective function to be minimized is given by
m
| pi | i 1 The
to be
weights get updated in every iteration.
difference
(Cont.,) 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=
m i 1
W | h (x)-z | i i i
Subject to constraint z i= hi(x) + ei Where, σ 2 = 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= Measured Value m=number of measurements
(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
Zmeas
Z true
(Cont.,) •
PDF( )
1 2
exp(
2
/2
2
)
20
Probability Distribution of Measurement Errors f(x) Gaussian distibution
Actual distribution
0
3
x
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 function problem. •
weighted m
min J(x)= i 1
(z i
least-squares
h i (x)) 2 2 i
(WLS)
(Cont.,) • This represents the summation of the squares of the
measurement residuals weighted by their respective measurement 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 unconstrained minimization problem. • Efficient solution of unconstrained minimization problems relies heavily on Newton‟s method.
(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 h(x
x)
h(x)
H(x) x
• where, the Jacobian matrix H(x) is defined as:
H(x)
h(x) x
• Then the linearized least-squares objective function is
given by J( x)
1 2
(z h(x) H(x) x) T R 1 (z
h(x) H(x) x)
(Cont.,) • where, R is a weighting matrix whose diagonal elements
are often chosen as measurement error variance, i.e., 2 1
R
2 m
J( x)
1 2
(e(x) H(x) x) T R 1 (e(x) H(x) x)
• where, e=z-h(x) is the residual vector.
(Cont.,) •
J( x) x
H T R 1 (e H x)
H T R 1H x G x
H T R 1e
H T R 1e
0
Weighted Least Squares-Example •
x est
est 1 est 2
(Cont.,) • To derive the [H] matrix, we need to write the measurements as a function of the state variables 1 and 2 . These functions
are written in per unit as M12
f12
M13
f13
M 32
f 32
1 0.2 1 0.4 1
(
1
2
)
5
(
1
3
)
2.5
0.25
(
3
2
)
5
1
1
4
2
2
(Cont.,) •
[H]
2 M12
R
5
5
2.5
0
0
4
2 M12 2 M13
0.0001 2 M13
2 M32
0.0001 2 M32
0.0001
(Cont.,) •
est 1
5
2.5 0
est 2
-5
0 -4
5
2.5 0
-5
0 -4
1
1
0.0001 0.0001 0.0001
1
0.0001 0.0001
5
5
2.5
0
0
4
0.62 0.06
0.0001
0.37
(Cont.,) • We get est 1
0.028571
est 2
0.094286
• From the estimated phase angles, we can calculate the
power flowing in each transmission line and the net generation or load at each bus. J( 1 ,
2)
(0.62 (5
1
0.0001 2.14
5 2 ))2
(0.06 (2.5 1 )) 2 0.0001
(0.37 (4
2 )) 2
0.0001
Solution of the weighted least square example
Bad Data Processing • One of the essential functions of a state estimator is to
detect measurement errors, and to identify and eliminate them 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 depending upon the type, location and number of measurements that are in error. They can be broadly classified as: Single bad data: Only one of the measurements in the entire system will have a large error • Multiple bad data: More than one measurement will be in error
(Cont.,) • 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.
• Redundant
measurement: A redundant measurement is a measurement which is not critical. Only redundant measurements may have nonzero measurement residuals.
• Critical pair: Two redundant measurements whose simultaneous
removal from unobservable.
the
measurement
set
will
make
the
system
(Cont.,) • When
using the WLS estimation method, detection and identification of bad data are done only after the estimation process by processing the measurement residuals.
J x
(z
h( x)) ' W z
h(x)
• The condition of optimality is that the gradient of J(x) vanishes
at the optimal solution x, i.e., GX
1
H WZ 0
X
G 1H1WZ
• An estimate z of the measurement vector z is given by
Z
HX
• The vector of residuals is defined as e = z - Hx; an estimate of
e is given by
e
z
h(x)
Bad Data Detection and Identification • Detection refers to the determination of whether or not the
measurement set contains any bad data. • Identification is the procedure of finding out which specific measurements actually contain bad data. • Detection and identification of bad data depends on the configuration of the overall measurement set in a given power system. • Bad data can be detected if removal of the corresponding measurement does not render the system unobservable. • A single measurement containing bad data can be identified if and only if: it is not critical and it does not belong to a critical pair
Bad Data Detection •
N
X i2
Y i 1
Y
2 k
Chi-square probability density function
Chi-squares distribution table
(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 the measurement errors: 2 m m m 2 e 1 2 N i f (x) R ii ei ei R ii i 1 i 1 i 1 • where e is the i th measurement error, Rii is the diagonal entry of
the measurement error covariance matrix and m is the total number 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.
Steps to detection of bad data •
m
ei2 /
f j 1
2 i
.
Bad Data Identification •
(zi
R ii'
z i ) / R ii'
(I HG 1HT R 1 )R
Steps to Bad Data Identification •
N i
e
ei ' ii
R
i=1,2,...m
Bad Data Analysis-Example •
Cont., • Measurement
equations characterizing the meter readings are found by adding errors terms to the system model. We obtain z1
5 8
1
z2 z3 z4
x1
8 3 8 1 8
1 8
8
x1
x1 x1
x2
8 1 8 3 8
x2
e1 e2
x2
e3
x2
e4
(Cont.,) • Forming the H matrix we get
0.625 H
0.125
100
0.125 0.625
0.375
0.125
0.125
0.375
W
9.01 z
3.02 6.98 5.01
0
0
0
0
100 0
0
0
0
50 0
0
0
0 50
(Cont.,) • Solving for state estimates i.e.,
V1
G 1H TW z
V2 • We get
V1
16.0072V
V2
8.0261V
(Const.,) •
z1
9.00123A
z2
3.01544A
z3
7.00596V
z4
5.01070V
(Cont.,) •
e1
9.01
9.00123
0.00877A
e2
3.02
3.01544
0.00456A
e3
6.98
7.00596
0.02596V
5.01
5.01070
0.00070V
e4
(Cont.,) •
4 2
f
ei /
2 i
j 1
0.043507
100(0.00877)
2
100(0.00456)
2
50(0.02596)
2
50(0.00070)
2
(Cont.,) •
[z1 z 2 z 3 z 4 ]T
[9.01A 3.02A 6.98V 4.40V]T
[e1 e 2 e3 e 4 ]T
[9.01A 3.02A 6.98V 4.40V] T
4
ei2 /
f j 1
15.1009
2 i
100(0.06228) 2 100(0.15439) 2
50(0.05965) 2
50(0.49298) 2
(Cont.,) •
R ii'
(I HG 1HT R 1 )R
ei e N i
' ii
R
i=1,2,...m
(Cont.,) • e1
' 11
(1 0.807) 0.01
R e2
' 22
' 33
' 44
R
0.05965 (1 0.193) 0.02
R e4
0.15439 (1 0.807) 0.01
R e3
0.06228
0.49298 (1 0.193) 0.02
1.4178
3.5144
0.4695
3.8804
Conclusion • Real time monitoring and control of power systems is
extremely important for an efficient and reliable operation of a power system. • Sate estimation forms the backbone for the real time monitoring and control functions. • In this environment, a real-time model is extracted at intervals from snapshots of real-time measurements. • Estimate the nodal voltage magnitudes and phase angles together with the parameters of the lines. • State estimation results can be improved by using accurate measurements like phasor measurement units. • Traditional state estimation and bad data processing is reviewed.
