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.
Power System Analysis was a core subject for Electrical & Electronics Engineering, Based On Anna University Syllabus. The Whole Subject was there in this document.
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These slides are all about Phasor Measurement Units (PMUs). An introduction to PMU is presented as a preliminary knowledge for the course 'Distribution Generation and Smart Grid'. Your valuable suggestions are welcome.
Classification Of Power System StabilityAravind Shaji
The Slide Deals With Power System Stability. Contents Include
Power System Stability Overview
Power System Stability: A Proposed Definition
Need of Stability Classification
Classification of stability
Power System Stability Classification
Rotor Angle Stability
Voltage Stability
Frequency Stability
Rotor Angle Stability vs. Voltage Stability
References
Power System Analysis was a core subject for Electrical & Electronics Engineering, Based On Anna University Syllabus. The Whole Subject was there in this document.
Share with it ur friends & Follow me for more updates.!
These slides are all about Phasor Measurement Units (PMUs). An introduction to PMU is presented as a preliminary knowledge for the course 'Distribution Generation and Smart Grid'. Your valuable suggestions are welcome.
Classification Of Power System StabilityAravind Shaji
The Slide Deals With Power System Stability. Contents Include
Power System Stability Overview
Power System Stability: A Proposed Definition
Need of Stability Classification
Classification of stability
Power System Stability Classification
Rotor Angle Stability
Voltage Stability
Frequency Stability
Rotor Angle Stability vs. Voltage Stability
References
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The obtained results are compared with the results obtained by conventional WLS method
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that allow the manipulation mechanism for each axis independently performed. The implementation was carried
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6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
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The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
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