Download BRANCH CURRENT BASED STATE ESTIMATION

Anais do XX Congresso Brasileiro de Automática
Belo Horizonte, MG, 20 a 24 de Setembro de 2014
BRANCH CURRENT BASED STATE ESTIMATION: EQUALITY-CONSTRAINED WLS AND
AUGMENTED MATRIX APPROACHES
WESLEY PERES, EDIMAR J. OLIVEIRA, JOÃO A. P. FILHO, JOSÉ L. R. PEREIRA,GUILHERME O. ALVES
Department of Electrical Engineering, Federal University of Juiz de Fora (UFJF), Juiz de Fora, MG, Brazil
E-mails: [email protected]
Abstract This paper presents a comparison study among three methods for solving the Branch Current Based Distribution
State Estimation (BCSE) problem: normal equations method, normal equations with constraints and augmented matrix approach.
Different types of measurements are considered. The methods are evaluated taking into account their numerical stability, computational requirements and implementation issues. For this purpose, three systems have been used to perform the simulations.
Keywords Distribution system state estimation, smart distribution system, branch current state estimation, phasor measurements units.
ments (Baran, et al., 2009) and Advanced Metering
Infrastructure (Baran & McDermott, 2009). In
(Baran, et al., 2009) is proposed a methodology for
topology errors processing and bad data identification using the BCSE method. In (Wang & Schulz,
2004) the BCSE method was modified to consider
the polar representation of branch currents as state
variables.
The performance of SE methods can be strongly
improved by using the new measurements technologies that are currently available. As an example, the
Phasor Measurements Units (PMUs) are becoming
increasingly widespread in transmission systems
(Phadke, et al., 2009). PMUs can directly measure
the bus voltage and the phase angles, because they
are synchronized by the GPS universal clock. PMUs
can also measure the current phasors through all
incident branches. At the distribution level, the use of
PMUs in the SE process looks very promising and it
has been subject of recent papers. In (Pau, et al.,
2013) the BCSE method has been modified to include phasor measurements.
The solution of the WLS SE problem using
Normal Equations (NE) can almost always be successfully carried out (Abur & Expósito, 2004). However, it is well known that under certain circumstances that are likely to occur in actual systems, the NE
will become prone to numerical instabilities. As a
result, the SE process may diverge.
In order to compare some solutions for the numerical ill-conditioning problem associated with the
rectangular nodal voltage based state estimators,
(Holten, et al., 1988) have conducted a study with
different methods: (i) normal equations method; (ii)
orthogonal transformation method; (iii) hybrid method; (iv) normal equations with constraints and (v)
augmented matrix approach (so called Hatchel's
formulation). According to the authors, the methods
(iii) and (v) have presented a good compromise between numerical stability and computational efficiency. It should be stressed that this comparison
study was performed taking into account the single
phase state estimators, that are used at the transmission level.
Despite the study presented in (Holten, et al.,
1988), investigations concerning the numerical sta-
1 Introduction
The operation of smart distribution systems is a
recognized challenge to be faced by distribution
operators in the next few years. The high penetration
level of Distributed Generators (DG) will call for
new tools for real time monitoring. In this context,
Distribution System State Estimation (DSSE) together with the availability of new measurement technologies will play an important role in the Distribution
Management System (DMS).
State Estimation (SE) for real time monitoring of
transmission systems has been well established over
the past four decades (Abur & Expósito, 2004).
However, the research on DSSE dates back to the
early 1990s (Baran & Kelley, 1994). Nonetheless, a
very limited number of practical implementations
have been performed, probably due the lack of proper infrastructure (Huang, et al., 2012).
Distribution and transmission systems are very
different from each other. The major differences can
be summarized taking into account the distribution
systems characteristics: (i) the number of measurements is very limited, (ii) in order to achieve the
observability, the historical load data (that have limited accuracy) is used as pseudomeasurements; (iii)
high R/X ratio; (iv) radial and weakly meshed operation; (v) unbalanced operation, etc. Therefore, the
state estimators available for monitoring transmission
systems are not suitable for distribution systems.
In order to take into account the characteristics
aforementioned, a new algorithm for DSSE was
proposed in (Baran & Kelley, 1995). In this formulation, the real and imaginary parts of branch currents
are used as state variables. The methodology is based
on the Weighted Least Squares (WLS) and the normal equations. The voltage measurements have not
been considered and simplifications have been done
to obtain a constant gain matrix. This method is referred in the literature as Branch Current based State
Estimation (BCSE).
The BCSE method has been found to be faster
than the nodal voltage based state estimators. Because of this, many papers have proposed improvements on BCSE method to include voltage measure-
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bility of the BCSE method for distribution systems
should be conducted. This analysis has not been
conducted before and it is a contribution of this paper. As previously mentioned, the future distribution
systems will call for new tools for a reliable monitoring. In this context, it is important to investigate the
performance of different approaches for solving the
BCSE method under ill-conditioning problems.
This paper presents a comparison study among
three methods for solving the BCSE method: (i)
normal equations method (NE); (ii) normal equations
with constraints (NE/C) and (iii) augmented matrix
approach (AM). This paper is a preliminary study
conducted by the authors. Future works will investigate the performance of other methods for solving
the BCSE method, such as those presented in
(Holten, et al., 1988).
Different types of measurements are considered.
The methods are evaluated taking into account their
numerical stability, computational requirements and
implementation issues.
It should be stressed that if any PMU is not installed in the system, the substation voltage (which is
the slack node) cannot be estimated, due to a lack of
a reference angle for the estimation process. In this
case, the state variable vector will be composed only
by branch currents, as defined in equation (3).
(3)
In order to obtain the estimated state
, a WLS
problem is performed to minimize the weighted sum
of the squares of the residuals:
(4)
The term
in equation (4) corresponds to the
inverse of the weighting matrix
associated with
measurements. Generally, the elements of
correspond to the variances of each measurement.
The solution of the optimization problem presented in equation (4) can be obtained through an iterative process. Thus, if
is the value of the estimated state at the iteration , the value at the next iteration will be:
2 Branch Current based Distribution System
State Estimation including PMUs
In this section, the methodology presented in
(Pau, et al., 2013) is revised. The normal equation
method is used to solve the SE problem.
+
where
is the Jacobian matrix and
Gain matrix:
2.1 Mathematical Formulation
(5)
is the
(6)
Consider a vector
composed of a set of
measurements, such as: real and reactive power flow,
real and reactive power injection, current magnitude,
voltage magnitude, voltage phasor and current
phasor. The measurements are a nonlinear function
of the state variables
:
(7)
It is important to emphasize that the phasor measurements (voltage and currents) are included in the
Jacobian matrix through their rectangular representation (real and imaginary parts). However, PMUs
provide these types of measurements in terms of
magnitude and phase angles. Therefore, it is necessary to perform a coordinate transformation (polar to
rectangular) as proposed in (Singh, et al., 2009) and
(Korres & Manousakis, 2011).
The mathematical expressions associated with the
Jacobian matrix can be obtained in (Baran & Kelley,
1995) and (Pau, et al., 2013).
(1)
where
is the noise vector assumed to be composed by independent zero mean Gaussian variables,
with
diagonal
covariance
matrix
.
When at least one PMU is installed in the system,
the substation voltage can be estimated. Therefore,
the state variable vector
will be composed by the
real
and imaginary
parts of branch currents
(
branches) together with the real
and imaginary
parts of substation voltage:
2.2 Algorithm
The solution of the BCSE method can be separated
in two steps that should be executed at every iteration:
 estimate the branch current by using the
equation (5);
 calculate the nodal voltages through a
forward step starting from the substation
bus. Because of this, the separation of the
layers in the radial system is required. For
additional information on layers separa-
(2)
The state variable vector defined in equation (2), is
composed by (
) elements, where
is the
number of branches. In this case, (
) is associated with the number of current variables (real and
imaginary parts for each branch). The other two
elements correspond to the real and imaginary parts
of substation voltage.
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Telemetered and phasor measurements have low
standard deviation , while the pseudomeasurements
are assigned with a higher to highlight the lower
confidence given to such quantities. The standard
deviations (squared) appear in the covariance matrix
, as shown in equation (4). In order to emphasize
the accuracy of virtual measurements, high weighting
factors are used (inverse of square of the standard
deviation) (Holten, et al., 1988).
According to (Abur & Expósito, 2004), the following specific sources of ill-conditioning have been
described in the literature:
 very large weighting factors used to enforce
virtual measurements;
 short and long lines simultaneously present
at the same bus;
 a large proportion of injection measurements.
It should be stressed that in the distribution systems, these sources of ill-conditioning can appear in
a stronger way. Additionally, it is important to remember that the pseudomeasurements are modeled
as power injection measurements. However, since
the accuracy of pseudomeasurements is lower than
telemetered measurements, low weights are assigned
to pseudomeasurements (Lu, et al., 1995).
In order to solve the ill-conditioning problems,
different approaches have been proposed in the literature (Holten, et al., 1988) and (Abur & Expósito,
2004).
In this paper, only two methods are analyzed to
solve the ill-conditioning problem: (i) normal equations with equality constraints and (ii) augmented
matrix approach.
tion, the reader is referred to (de Araujo,
et al., 2010).
In this paper a backward approach is carried out to
get the initial value of current. The initial value of
voltage at every node is set to be a per unit value.
Thereafter, using the injected power at every node,
the value of branch currents is obtained.
The following steps are performed:
Step 1) Initialize the values of voltages and currents;
Step 2) Calculate the updates of the state variables
by using (5);
Step 3) Calculate the nodal voltages through a
forward step starting from the substation node;
Step 4) If the increment
is
smaller than a convergence tolerance, stop. Otherwise:
1. if the number of iteration is smaller than a pre-specified maximum
iteration number, go to step 2.
2. if it is not, the iterative process
has not achieved the convergence,
stop.
2.3 Weakly meshed distribution feeders
Distribution systems operate with radial configuration due to protection issues (Blackburn & Domin,
2014).
However, under some circumstances, distribution
systems operate with loops created by closing normally open tie-switches. In order to deal with weakly
meshed distribution feeders, a strategy has been
taken into account in (Baran & Kelley, 1995).
For each loop, an equality constraint is included,
representing the Kirchhoff Voltage Law (KVL). The
KVL says that the sum of voltage drops around the
loop is equal to zero.
Weakly meshed distribution feeders will not be
considered in this paper. For additional information,
the reader is referred to (Baran & Kelley, 1995).
3.1 Normal Equations with Constraints (NE/C)
The use of large weights for modeling very accurate virtual measurements leads to ill-conditioning
of the gain matrix
. One way to avoid the use of
large weights, is to model these measurements as
explicit constraints in the WLS BCSE (Abur &
Expósito, 2004):
(8)
st
3 Alternative Formulations of the BCSE method
This optimization problem can be solved by the
Lagrangian method and leads to the following linear
system to be solved at every iteration:
Beyond the telemetered and phasor measurements, there are two additional types of measurements (Abur & Expósito, 2004) and (Holten, et al.,
1988) :
 Pseudomeasurements: historical load data
used to improve the observability in the DSSE problem;
 Virtual measurements: are kind of information that does not require metering, such as zero
injection buses.
(9)
(10)
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4 Performance Indexes
where is a parameter used to control the numerical stability of the problem and
are the Lagrange multipliers associated with the equality constraints.
In order to compare the three methods under
analysis (normal equation method, normal equation
with equality constraints and augmented matrix approach), two performance indexes are used: (i) condition number of the coefficient matrix and (ii)
sparsity degree. This section describes these indexes.
3.2 Augmented Matrix Approach (AM)
Similar to the virtual measurements, regular
measurements (telemetered and phasor) and
pseudomeasurements equations can be written as
equality constraints if the associated residuals are
retained as explicit variables (Abur & Expósito,
2004):
st
4.1 Condition Number
(11)
According to (Holten, et al., 1988), the condition
number of the coefficient matrix can be used to
measure the ill-conditioning of a problem.
In accordance with (Abur & Expósito, 2004), the
condition number can be calculated as shown in
equation (13).
(13)
This optimization problem can also be solved by
the Lagrangian method and leads to the following
linear system to be solved at every iteration:
(12)
where is a parameter used to control the numerical stability of the problem,
are the Lagrange
multipliers associated with the first set of equality
constraints and are the Lagrange multipliers associated with the second set of equality constraints.
3.3 Implementations Issues
The first comparison among the methodologies
under analysis is about the method used for factorization. According to the literature (Holten, et al., 1988),
the coefficient matrices of the linear systems associated with NE/C and AM methods (equations (9) and
(12)) are no longer positive definite.
For a symmetric positive definite matrix, numerical stability is guaranteed when the pivots are taken
from the diagonal in any order. Therefore, optimal
ordering can be performed using only sparsity criterion. It is true for the Gain matrix of the NE method.
For the NE/C and AM methods, row pivoting to
preserve numerical stability must be combined with
sparsity-oriented techniques during LU factorization
(Abur & Expósito, 2004).
In this paper, the solution of the linear systems
was carried out by using the routines available in the
MATLAB environment. These routines were found
to be very robust.
Regarding to the size of the linear system, it
should be stressed that the solution of the NE/C and
AM methods is not particularly expensive, since the
coefficient matrices are very sparse.
where
represents a given matrix norm. One
approximation which yields a good estimation of the
condition number is the ratio
, where
and
are the largest and the smallest absolute eigenvalues of the coefficient matrix.
In solving the equation
, because of the
errors introduced in and during the solution process, it is needed to know by how much does the
results
or
differ from the true
solution
, where the matrix and the vector
represent errors. If they differ greatly we say the
matrix is ill-conditioned.
The condition number is equal to unity for identity matrices and tends to infinity for matrices approaching singularity. The more singular the coefficient matrix is, the more ill-conditioned the linear
system will be (Abur & Expósito, 2004). Therefore,
lower values for condition numbers are desirable.
The condition number of the Gain matrix in the
normal equation method is the square of the condition number of the Jacobian matrix.
4.2 Sparsity Degree
As previously mentioned, for the normal equation with constraints and augmented matrix methods,
the dimension of the linear system is increased.
In order to evaluate the sparsity of the coefficient matrices, the sparsity degree
is defined in
equation (14). This index yields the percentage of
null elements with respect to the total of elements.
(14)
where
is total of elements of coefficient matrix (square of the number of state variables) and
is the number of nonzero elements.
The sparser a matrix is, the lower the number of
elements stored will be.
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Other parameters are used to compare the methods, such as: the number of iteration and the computational time required by each one.
Table 1 summarizes the measurement planning
adopted for the simulations. It should be emphasized
that no PMU is installed in the system. Therefore, the
substation voltage is not included in the state variable
vector and the total of state variables is equal to the
double of the number of branches.
5 Results
Table 1. Measurements Set of the 18-bus Test System.
In this section the results obtained by using the
three methods under analysis are presented:
 normal equations (NE);
 normal equations with constraints
(NE/C);
 augmented matrix approach (AM).
The MATLAB environment was used to perform the simulations. The simulations were carried
out using an Intel Core i7 2.93 GHz computer with 8
GB of RAM and Windows 7 64-bit operating system.
The study has been conducted by using three
distribution test systems from the literature: 18-bus,
69-bus and 83-bus.
The measurement set for each system was obtained by adding a Gaussian error to the power flow
solution.
The following standard deviations were considered for telemetered and phasor measurements:
 voltage magnitude: 8e-3;
 current magnitude: 4e-3;
 phasor measurements: 4e-5;
 power injection: 1e-2.
Pseudomeasurements were supposed to contain a
maximum error of 50% with respect to the nominal
values (Gaussian distribution). The weights of these
measurements were set to be 1. Since the
pseudomeasurements are not accurate, their weights
will be kept constant. In order to ensure the
observability, all load buses are treated as
pseudomeasurement buses.
Virtual measurements (zero injection buses): the
weighting factors were varied in a range of values.
This procedure was adopted in (Holten, et al., 1988).
A convergence tolerance was set to 1e-4.
The scaling factor used to control the numerical stability of NE/C and AM methods was set to 1.
Number of Measurements
Regular
Virtual
Pseudo
4
14
20
Total Number
States Measu.
34
38
Redundancy
Level
1.11
Regular Measurements
, ,
,
In order to evaluate the performance of the methodologies, the weighting factors associated with the
virtual measurements were varied. The condition
numbers of the coefficient matrix are presented in
Table 2.
Table 2. Condition Number of the Coefficient Matrix.
Method
NE
NE/C
AM
Weighting factors (Virtual Measurements)
1e6
1e8
1e10
1e12
4.960e7
4.959e9 4.959e11 4.983e13
9.603e5
4.179e4
From the results of Table 2 it is possible to see
that the AM method presented better numerical stability than the other methods. In addition, the condition number of coefficient matrix remained constant
for NE/C and AM methods. On the other hand, the
condition number of the NE method increases with
the weighting factors. An additional simulation was
executed (weighting factors equal to 1e14 and 1e16)
and the NE method has not converged.
Although the condition numbers of the coefficient
matrix was found to be sensitive to the method used,
the number of iterations is the same and the average
execution times are close, as shown in Table 3.
Table 3. Number of Iterations and Average Execution Time (sec).
Method
NE
NE/C
AM
Iterations
7
7
7
Execution time
0.25
0.29
0.28
Finally, Table 4 presents the sparsity degree of
each method. As previously mentioned, the solution
of the NE/C and AM methods is not particularly
expensive, due to their higher sparsity degree.
5.1 18-bus test system
This system has a radial configuration and its data can be obtained in (Ahmadi & Green, 2009). The
original version without distributed generation was
used and the on line diagram is shown in Figure 1.
Table 4. Sparsity Degree.
Method
NE
NE/C
AM
Figure 1. On line diagram of the 18-bus test system
3202
Number of
elements
(34)2
(48)2
(72)2
Nonzero
elements
224
270
316
Sparsity
degree (%)
80.6
88.3
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of NE/C method (Holten, et al., 1988). This procedure aims at obtaining better solutions.
5.2 69-bus test system
It is a radial system and its data can be obtained
in (da Silva, et al., 2008).
This system was modified through the inclusion
of a distributed generator as proposed in (Dias, et al.,
2012). A DG of 1870 kW and 1159 kVar was installed at bus 61.
Figure 2 shows the on line diagram.
Table 6. Condition Number of the Coefficient Matrix.
NE
NE/C
AM
Table 7 shows that the iteration number was the
same for both methods. In addition, the average execution time is very close.
1
2
3
4
36
29
47
5
37
30
48
6
38
31
49
7
39
32
50
8
28
33
34
53
35
54
11
55
12
Table 7. Number of Iterations and Average Execution Time (sec).
Method
NE
NE/C
AM
40
9
51
41
10
52
42
13
57
14
58
15
66
44
67
45
Iterations
3
3
3
Execution time
0.30
0.34
0.33
Finally, Table 8 presents the sparsity degree of
each method. As previously mentioned, the solution
of the NE/C and AM methods is not particularly
expensive due to their high sparsity degree.
43
56
Weighting factors (Virtual Measurements)
1e6
1e8
1e10
1e12
3.846e12 3.857e12 5.724e12 5.709e14
3.8623e12
1.731e5
Method
46
68
Table 8. Sparsity Degree.
69
59
16
60
17
61
18
62
19
63
20
64
21
65
22
Number of
elements
(138)2
(178)2
(290)2
Method
NE
NE/C
AM
Nonzero
elements
4269
4536
1492
Sparsity
degree (%)
77.6
85.7
98.2
5.3 83-bus test system
23
24
The 83-bus system of the Taiwan Power Corporation (TPC) has been used for optimal reconfiguration and capacitor allocation studies (de Oliveira, et
al., 2010). In this paper, this system is used for evaluating the state estimation methodologies under
analysis. Its data can be obtained in (Chiou, et al.,
2005).
Table 9 summarizes the measurement planning
adopted for the simulations. A PMU is installed at
the substation node (node 84). In this case the substation voltage is directly measured.
25
26
27
Figure 2. Diagram of the 69-bus test system
Table 5 summarizes the measurement planning
adopted for the simulations. It is possible to notice
that one PMU was installed at bus 61. Therefore, the
voltage substation is considered to be a state variable
in the SE problem.
Table 5. Measurements Set of the 69-bus Test System.
Number of Measurements
Regular
Virtual
Pseudo
16
40
96
,
,
,
,
Total Number
States Measu.
138
152
Regular Measurements
,
,
,
,
,
,
,
,
Table 9. Measurements Set of the 83-bus Test System.
Redundancy
Level
Number of Measurements
Regular
Virtual
Pseudo
33
34
132
1.10
,
,
,
,
,
The condition numbers associated with the coefficient matrix of each method are presented in Table
6. It is interesting to notice that the condition number
of NE/C method is larger than those associated with
NE method in some cases. It has already been noticed in the literature and a solution would be optimizing the scaling factor
in the coefficient matrix
,
,
,
,
Total Number
States Measu.
168
199
Regular Measurements
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
Redundancy
Level
1.18
,
,
,
,
,
,
The condition numbers associated with the coefficient matrix of each method are presented in Table
10.
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Table 10. Condition Number of the Coefficient Matrix.
Method
NE
NE/C
AM
Regarding to the numerical stability, the NE
method was found to be prone to convergence problems depending on the weighting factors assigned to
the virtual measurements.
Finally, the AM method presented the best
sparsity degree. In general, the NE/C and AM methods are not expensive to solve with respect to the
effort required by the NE method.
Concluding, although the NE was found to be
numerically unstable when high weighting factors
are used, it is possible to conclude that all methods
can present a satisfactory performance in practical
applications.
Concerning future studies, the analysis of other
systems will be evaluated as well as other solution
techniques, such as the orthogonal transformation.
Moreover, other issues will be evaluated, such as
observability analysis and bad data processing techniques.
Weighting factors (Virtual Measurements)
1e6
1e8
1e10
1e12
2.5516e10 2.6860e10 4.3899e11 4.3692e13
2.5869e10
4.9024e5
Table 11 shows that the iteration number was the
same for both methods. In addition, the average execution time is very similar.
Table 11. Number of Iterations and Average Execution Time (sec).
Method
NE
NE/C
AM
Iterations
3
3
3
Execution time
0.33
0.38
0.38
Finally, Table 12 presents the sparsity degree of
each method.
Table 12. Sparsity Degree.
Method
NE
NE/C
AM
Number of
elements
(168)2
(202)2
(367)2
Nonzero
elements
1794
1940
1649
Sparsity
degree (%)
93.64
95.25
98.78
Acknowledgment
The authors wish to thank CAPES,
CNPq/INERGE, FAPEMIG and CEPEL for supporting this research. The first author thanks CAPES by
the CAPES-REUNI scholarship.
For the 83-bus system, the same previous conclusions arise: the AM method is the most stable
among the methods considered. All methods considered present a high sparsity degree, providing an
efficient performance. However the AM approach
yields the largest sparsity degree. Concerning computational effort, both methods presented similar
performance.
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5 Conclusions
This paper presented a comparison study among
three different methods for solving the branch current
based distribution system state estimation:
 normal equation (NE);
 normal equation with constraints
(NE/C);
 matrix augmented approach (AM).
The comparison study has been conducted in order to evaluate the impact of virtual measurement
weighting factors on the convergence behavior of the
branch current based estimation method. For this
class of state estimators, this study had not been
conducted before and it is a contribution of this paper. Three distribution systems were used to perform
the simulations.
With respect to implementation issues, the NE/C
and AM methods require more sophisticated factorization and ordering techniques. In this paper, the
MATLAB routines were used and were found to be
very effective.
Both methodologies spent the same number of
iterations and average execution time was very close.
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