basics of state estimation - unlv department of electrical ...eebag/state estimation.pdf · used in...
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Introduction to State Estimation
of Power Systems
ECG 740
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Introduction (cont.)
• Electric utilities have installed extensive supervisory control
and data acquisition (SCADA) throughout the network to
support computer-based systems.
• The data is used for numerous applications (e.g., system
monitoring, economic system operation, security assessment,
control of generation, etc…)
• Before any assessment is made or control action is taken, a
reliable estimate of the existing state of the system must be
determined.
• For this purpose, the number of physical measurements cannot
be restricted to those quantities required to support power flow
calculations.
• Moreover, errors in one or more of the input quantities can
lead to useless results.
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Utility SCADA
Meas.
(P,.Q.
V,I..)
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Introduction (cont.)• In practice, other conveniently measured quantities
(such as P&Q line flows) are available, but cannot be
used in power flow calculations.
• The unavoidable errors in the measurements are
assigned statistical properties.
• Such limitations are removed by state estimation based
on weighted least-squares calculations.
• Gross errors detected in the course of state estimation
are filtered out.
• A State Estimator allow the calculation of the 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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Introduction (cont.)
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Illustration A: 3 –Bus DC Power Flow
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What we Need ?
• A procedure that uses the information available from
all the three meters to produce the best estimate of
the actual angles, line flows, and bus load and
generation.
• We have three meters providing us with a set of
redundant readings with which to estimate the two
states 1 and 2.
• We say that the readings are redundant since, as we saw
earlier, only two readings are necessary to calculate 1 and
2 the other reading is always “extra”.
• The “extra” reading does carry useful information and
ought not to be discarded.
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2-10
Method of Least Squares
• The acquired data always contains inaccuracies during
measurement and/or transmission. The best estimates
are chosen as those which minimize the weighted sum
of the squares of the measurement errors.
• Mathematically, let Z = h( x ) + e
where,
Z = Measurement Vector
h = System model relating state vector to the
measurement set
x = State vector (voltage magnitudes and
angles)
e = Error vector associated with the
measurement set
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Normal Gaussian distribution function
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2-12
SE Problem Development
• Classical Approach: Weighted Least Squares…
• In case of a linear system, i.e., h(x) = Hx, the weighted
least square estimate of x is
xest =G-1 HTWz
where the gain matrix G =HTWH
Minimize: J(x) = [z - h(x)] t. W. [z - h(x)]
where,
J = Weighted least squares
W = Weighting matrix = reciprocal of error variances
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Special Cases
• Fully Determined Case: When the number of
measurements is equal to the number of state
variables,
xest = H-1z
• Underdetermined case: When the number of
measurements is smaller than the number of state
variables (unobservable case), minimize the sum of
the squares of the solution values,
xest = [HTHHT]-1 z
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Back to Illustration A
• Assume that all the three meters have the following
characteristics:
– Meter full scale value: 100 MW
– Meter Accuracy: ± 3 MW
• This is interpreted to mean that the meters will give a
reading within ± 3 MW of the true value being
measured for approximately 99 % of time.
• Mathematically, we say that the errors are distributed
according to a normal probability density function with
a standard deviation ,, i.e., 3 = ± 3 MW. Hence, the
metering standard deviation = 1 MW = 0.01 pu.
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Illustration A (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 = 1/0.2 x(1 - 2) =5 1 - 52
– M13= f13 = 1/0.4 x(1 - 3) =2.5 1
– M32 = f32 = 1/0.25 x(3 - 2) =-4 2
• Error covariance matrix
2
1,
40
05.2
55
][
xH
0001.00
00001.0
000001.
W
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Illustration B
Variance of amp-meters = 1/100, variance of volt-meters = 1/50
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Solution
(^) represent estimated values
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How Good are the Estimates?
• What criterion for acceptance is reasonable?
• If a grossly erroneous meter reading is present,
can we detect that fact and identify the bad
measurement?
• These questions can be answered within a
quantifiable level of confidence by attaching a
statistical meaning to the measurement errors
in the least square calculations.
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Test for Bad Data• Each estimated error is a Gaussian variable with zero mean.
• The weighted sum of the squares of these has a Chi-square
distribution where k is the degree f freedom.
• Hypothesis Testing: Probability that J(x) > tj = α
• Where
• J(x): measurement residual
• α: significance level
(prob. of false alarm)
• tj: test threshold
2
k
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Hypothesis Testing Parametersk
α
tj
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Continuing with Illustration B
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Bad measurements can be identified by computing normalized
residual errors and removing the largest ones > 3, one at a time.
[0.06228 0.15439 0.05965 0.49298]T
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Covariance matrix: R’ = W-1 – HG-1HT
With 3 measurements, መ𝑓= .0435, and Chi-square value = 6.64.
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Power System State Estimation
• State variables: voltage magnitudes and their phase angles.
• Two kinds of inputs: data ( e.g., P&Q measurements), and
status information (e.g. on/off status of switching devices).
• Number of actual measurements is far greater than
required.
• Unlike the earlier DC examples, the measurement
equations h(x) are nonlinear.
• Common technique: calculate the gradient of J(x) and force
it to zero using Newton’s method.
• See algorithm in next slide
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State Estimation Solution Algorithm
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Example 6-Bus Power System
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Assumption:
Base case is assumed true, but
Impossible to know in practice
PDF with zero mean, and the
Following standard deviations:
P&Q meters: σ = 3 MW/MVAR
V meters: σ = 3.83 kV
Random number generator was
to produces random errors which
are added to the base values.
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State Estimation algorithm
results shown in right column.
Number of variables 11:
No. of degrees of freedom: 51
After 4 iterations, the residual
converged to 56.3
Residual threshold with 99%
Confidence: 63.4
→ No bad data detected
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Simulation of bad data:
Reverse the reading of M12
After 5 iterations, the residual
Converged to 6,455.
Recall threshold value: 63
→ Presence of bad data.
Largest normalized residual
Occurred at meter M12 = 76.9
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State Estimation after removal
of bad data:
No. degrees of freedom: k = 49
After 3 iterations, the residual
Converged to 37.5
Bad data Threshold: 61
→ Bad data no longer present
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Phasor Measurement Units (PMU)
PMUs improve the accuracy of state estimation since they
eliminate time differences and provide additional
measurements of voltage magnitudes and their phases
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Summary
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References
1. Yih-Fang Huang, Stefan Werner, Jing Huang,
Neelabh Kashyap, and Vijay Gupta, “State
Estimation in Electric Power Grids”, IEEE Signal
Processing Magazine, Sept. 2012.
2. G. Ortiz, D.G. Colome and J.J.Q. Puma, “State
estimation of power system based on SCADA and
PMU measurements”, IEEE ANDESCON, 2016
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Assignment
• Solve Problem 9.4 of Chapter 9 (pp. 465-466)