This document discusses state estimation in power systems. It begins by defining state estimation as assigning values to unknown system state variables based on measurements according to some criteria. It then discusses that the most commonly used criterion is the weighted least squares method. It provides an example of using measurements to estimate voltage angles as state variables and calculate other power flows. Finally, it discusses the weighted least squares state estimation technique in detail including developing the measurement function matrix and solving the weighted least squares optimization.
2. State Estimation
assigning a value to an unknown system state variable based on
measurements from that system according to some criteria.
The process involves imperfect measurements that are redundant and the
process of estimating the system states is based on a statistical criterion that
estimates the true value of the state variables to minimize or maximize the
selected criterion.
Most Commonly used criterion for State Estimator in Power System is the
Weighted Least Square Criteria.
3. State & Estimate
What’s A State?
The complete “solution” of the power system is known
if all voltages and angles are identified at each bus.
These quantities are the “state variables” of the system.
Why Estimate?
Meters aren’t perfect.
Meters aren’t everywhere.
Very few phase measurements?
SE suppresses bad measurements and uses the
measurement set to the fullest extent. 2-3
4. State variable & Input to estimator
In the Power System, The State Variables are the voltage Magnitudes and
Relative Phase Angles at the System Nodes.
The inputs to an estimator are imperfect power system measurements of
voltage magnitude and power, VAR, or ampere flow quantities.
The Estimator is designed to produce the “best estimate” of the system
voltage and phase angles, recognizing that there are errors in the measured
quantities and that they may be redundant measurements.
5. State Estimator output
Bus voltages, branch flows, …(state variables)
Measurement error processing results
Provide an estimate for all metered quantities
Filter out small errors due to model approximations and
measurement inaccuracies.
Detect and identify discordant measurements, the
so- called bad data.
6. Case 1 Suppose we use M13 and M32 and further suppose that
M13 and M32 gives us perfect readings of the flows on their
respective transmission lines.
M13=5 MW=0.05pu
M32 =40 MW=0.40pu
f13=1/x13*(1- 3 )=M13 = 0.05
f32=1/x32*(3- 2)=M32 = 0.40
Since 3=0 rad
1/0.4*(1- 0 )= 0.05
1/0.25*(0- 2) = 0.40
1 =0.02 rad
2 =-0.10 rad
7. Case1-Measurement with accurate meters)
Bus1
Only two of these meter
readings are required to
calculate the bus phase
angles and all load and
generation values fully.
Bus2
Bus3
60 MW
40 MW
65 MW
100 MW
Per unit Reactances
(100 MVA Base):
X12=0.2
X13=0.4
X23=0.25
M12
M13
M32
Meter Location 5 MW
35 MW
8. Case-2: use only M12 and M32.
M12=62 MW=0.62pu
M32 =37 MW=0.37pu
f12=1/x12*(1- 2 )=M12 = 0.62
f32=1/x32*(3- 2)=M32 = 0.37
Since 3=0 rad
1/0.2*(1- 2 )= 0.62
1/0.25*(0- 2) = 0.37
1 =0.0315 rad
2 =-0.0925 rad
9. Bus1
Bus2
Bus3
62 MW
37 MW
65 MW
100 MW
Per unit Reactances
(100 MVA Base):
X12=0.2
X13=0.4
X23=0.25
M12
M13
M32
6 MW (7.875MW)
Meter Location
35 MW
Case2-result of system flow. (M12 & M32)
Mismatch
10. Analysis of example
• use the measurements to estimate system conditions.
• Measurements of were used to calculate the angles at different
buses by which all unmeasured power flows, loads, and
generations can be calculated.
• voltage angles as the state variables for the three- bus 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.
11. Solution Methodologies
Weighted Least Square (WLS)method:
Minimizes the weighted sum of squares of the difference between
measured and calculated values .
The weighted least-squares criterion, where the objective is to minimize
the sum of the squares of the weighted deviations of the estimated measurements, z,
from the actual measurements, z.
m
1
e2
2 i
i 1 i
Iteratively Reweighted Least Square
Value (WLAV)method:
(IRLS)Weighted Least Absolute
Minimizes the weighted sum of the absolute value of difference
between measured and calculated values.
The objective function to be minimized is given by
m
| pi|
i 1
The weights get updated in every iteration.
12. (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
푧푚푒푎푠 = 푧푡푟푢푒 + 휂
13. Probability density function
The random number, η, serves to model the uncertainty in
the measurements.
If the measurement error is unbiased, the probability
density function of η is usually chosen as a normal
distribution with zero mean.
푃퐷퐹 휂 =
1
휎 2휋
∗ 푒(−휂 2 2휎2)
where σ is called the standard deviation and σ^2 is called
the variance of the random number.
If σ is large, the measurement is relatively inaccurate (i.e.,
a poor-quality measurement device), whereas a small value
of σ denotes a small error spread (i.e., a higher-quality
measurement device).
15. Weighted least Squares-State Estimator
min 퐽 푥 =
푁푚
푖=1
푧푖
푚푒푎푠 − 푓푖 푥 2
휎푖
2
where
fi = function that is used to calculate the value being
measured by the ith measurement
휎푖
2 = variance for the ith measurement
J(x) = measurement residual
Nm = number of independent measurements
zmeas= ith measured quantity.
Note: that above equation may be expressed in per unit or
in physical units such as MW, MVAR, or kV.
17. In vector form f(x)
푓 푥 =
푓1 푥
푓2 푥
.
.
푓푁푚(푥)
= [퐻 푥 ]
Where
[H]=an Nm by Ns matrix containing the coefficients of the linear function
[H]=measurement function coefficients matrix
Nm= number of measurements
Ns=number of unknown parameters being estimated
푍푚푒푎푠 =
푧푚푒푎푠
1
푧푚푒푎푠
2
푧푚푒푎푠
푁푚
18. min 퐽 푥 = [푧푚푒푎푠 − 푓 푥 ]푇 푅−1 [푧푚푒푎푠
− 푓 푥 ]
푅 =
2 0 0
0 ⋱ 0
0 0 휎푁푚
휎1
2
[R]=covariance matrix of measurement error
min J x = {zmeas R−1 zmeas
− xT H T R−1 zmeas − zmeasT
R−1 H x
+ xT H T R−1 H x}
19. The gradient of 훻퐽 푥
훻퐽 푥 = −2 퐻 푇 푅−1 푧푚푒푎푠 + 2 퐻 푇 푅−1 퐻 푥
훻퐽 푥 = 0
푥푒푠푡 = [ 퐻 푇 푅−1 퐻 ]−1[퐻]푇 [푅−1]푧푚푒푎푠
Case Ns<Nm over determined
xest = [ H T R−1 H ]−1[H]T[R−1]zmeas
Case Ns=Nm completely determined
xest = [H]−1zmeas
Case Ns>Nm underdetermined
xest = [H]T[ H H T]−1zmeas
21. (Cont.,)
• To derive the [H] matrix, we need to write the measurements
as a function of the state variables
are written in per unit as
1 2 . These functions and
1
M f ( ) 5 5
12 12 1 2 1 2
0.2
1
M f ( ) 2.5
13 13 1 3 1
0.4
1
M f ( ) 4
32 32 3 2 2
0.25
24. (Cont.,)
• We get
est
1
est
2
0.028571
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.
5 ))2 (0.06 (2.5 ))2 (0.37 (4 ))2
1 2
0.0001
2.14
1
0.0001
2
0.0001
J( 1, 2 )
(0.62 (5
26. Application of State Estimation
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
27. References
Power generation operation and control by Allen J.Wood &
Bruce F.Wollenberg
Operation and control in Power systems by P.S.R. Murthy
