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Content uploaded by Ying Zhang
Author content
All content in this area was uploaded by Ying Zhang on Apr 13, 2020
Content may be subject to copyright.
1
Abstract—Motivated by increasing penetration of distributed
generators (DGs) and fast development of micro-phasor
measurement units (μPMUs), this paper proposes a novel graph-
based faulted line identification algorithm using a limited number
of μPMUs in distribution networks. The core of the proposed
method is to apply advanced distribution system state estimation
(DSSE) techniques integrating μPMU data to the fault location.
We propose a distributed DSSE algorithm to efficiently restrict the
searching region for the fault source in the feeder between two
adjacent μPMUs. Based on the graph model of the feeder in the
reduced searching region, we further perform the DSSE in a
hierarchical structure and identify the location of the fault source.
Also, the proposed approach captures the impact of DGs on
distribution system operation and remains robust against high-
level noises in measurements. Numerical simulations verify the
accuracy and efficiency of the proposed method under various
fault scenarios covering multiple fault types and fault impedances.
Index Terms— Faulted line identification, distribution systems,
state estimation, phasor measurement units, micro-PMUs,
distributed generation, graph theory.
I. INTRODUCTION
AULTS are regarded as an important type of reliability
events, which may immensely affect normal system
operation. In the past decade, 22.2 million customers in
California experienced about 6000 outage hours resulted from
sustained faults [1]. Extensive studies on fault location are
developed in meshed transmission systems (e.g., [2] – [4]).
However, distribution systems are largely different from
transmission systems due to their radial topology and limited
real-time meters. Consequently, these existing fault location
methods in transmission systems cannot be applied to
distribution systems. On the other hand, increasingly pervasive
installation of distribution-level phasor measurement units, i.e.,
micro-PMUs (μPMUs), improves the system monitoring
significantly. Compared with conventional meters, μPMUs
provide more accurate measurements of voltage and current
phasors at a high resolution. Several emerging applications of
μPMUs include distribution system state estimation (DSSE),
fault detection, and faulted line location [5]. For instance, the
authors of [6] applied data-driven techniques with μPMU data
to detect the presence of a fault in distribution systems;
however, these detection algorithms cannot identify the
location of the faulted line.
Quick and accurate location of faults in distribution systems
helps the utilities to clear the faults and accelerate the system
restoration; however, this is a challenging task as the mal-trip
or fail-to-trip of the protection devices may lead to inaccurate
location of the fault. The chances of such unfavorable events
grow with the bidirectional power flow and the increasing
penetration of distributed generators (DGs) [7], [8]. The authors
of [9] pointed out that conventional protection devices such as
fault indicators may fail to clear a fault under the bidirectional
current flow conditions. Also, the overcurrent-based protection
devices may not be able to locate high-impedance faults in
distribution systems since it is difficult to identify the small
fault currents [10].
The existing fault location methods are classified into three
main types: 1) traveling wave-based, 2) training-based, and 3)
impedance-based. The traveling wave-based algorithms (e.g.,
[11] – [13]) locate a fault by utilizing the arrival time of the
original and reflected waves generated by the fault. These
methods require high-speed communication and high sampling
rate measurements that may not be prevalent in distribution
networks. Besides, the training-based fault location methods,
such as artificial neural network (ANN) [14] and support vector
machine (SVM) [15], require a large number of high-quality
measurements as training datasets and thus suffer from a high
computational burden in a training process.
Recent efforts are devoted to proposing impedance-based
location methods in distribution systems [9], [16] – [22]. For
instance, the authors of [18] and [19] proposed the fault location
methods focusing on single-phase to ground faults. However,
DGs are not considered in the test feeders of [18], while [19]
can only localize a faulted area, rather than yielding an exact
faulted line. Emerging works are applying μPMU
measurements to fault location by constructing generalized
impedance-based location methods [9], [20] – [22]. We
conclude that the search strategy in these works is to select each
bus or each line as the candidate fault source and then calculate
the values of the self-defined objective function for each
candidate. Then, the fault location is determined by minimizing
or maximizing these function values. Specifically, the authors
of [20] used a state estimation technique with sufficient μPMU
data to identify the fault at a distribution line. However, these
μPMUs are assumed available at each bus, which is impractical
due to economic and technical restrictions in distribution
systems. Further efforts are put into fault location with a fewer
number of μPMUs, such as [21] and [22]. The approach in [22]
requires equipping with μPMUs at all DGs. This arrangement
may not be practical due to a limited number of available
Graph-based Faulted Line Identification Using
Micro-PMU Data in Distribution Systems
Ying Zhang, Student Member, IEEE, Jianhui Wang, Senior Member, IEEE, and Mohammad Khodayar, Senior
Member, IEEE
F
Y. Zhang, J. Wang, and M. Khodayar are with the Department of
Electrical and Computer Engineering, Southern Methodist University,
Dallas, TX, USA 75205 (e-mail: yzhang1@smu.edu; jianhui@smu.edu;
mkhodayar@smu.edu).
2
μPMUs. Also, this method does not consider high-impedance
faults, which are regarded as an untraceable fault type in system
operation. The authors of [22] defined a fault as a generalized
reliability event and presented an optimization model to locate
the event bus by μPMU data and pseudo-measurements
recorded at load/DG buses. Also, due to the presence of local
minimums in the objective function, the method needs to
compare all local minimums to obtain a global one. Further, the
global minimum of this function points to the final faulted bus.
However, this process may increase computational complexity
due to this traversal search strategy.
Various influence factors, such as fault types, fault
impedances, DG penetration, and measurement errors, may
degrade the effectiveness of the existing fault location methods
[22]. To mitigate these impacts, this paper proposes a graph-
based fault location method using advanced DSSE techniques
with μPMU data in unbalanced distribution systems. The core
of the proposed method is to determine the faulted line by
comparing the weighted measurement residuals (WMRs) of
DSSE in different topologies/graphs. This idea, as a typical
application of state estimation, is proposed in [2], [3], and [20],
where the power systems are observable by an adequate number
of PMUs. In comparison, the proposed method only requires a
limited number of μPMUs in distribution networks for such an
application. Specifically, we present an efficient distributed
DSSE algorithm to restrict the search region in a shorter feeder
between two adjacent μPMUs. Further, in the shorter feeder, the
fault source is identified at the exact line by applying the DSSE
methods to a hierarchical structure. The hierarchical structure
built on the graph theory is presented in Section III and captures
the graphs, subgraphs, and paths in the network.
We list the contributions of this method below:
The proposed algorithm uses data from limited μPMUs for
fault location. Compared with [20] and [21], this method
does not require installing μPMUs at each node or all DGs.
This method locates the fault with only during-fault μPMU
data, compared with [21] and [22] that require both pre-fault
and during-fault data. Also, unlike the traversal search
strategy with high computational complexity in [20]–[22],
this method running in a distributed manner has lower
computation cost and enables fast faulted line identification
within several tens of milliseconds.
Our approach considers the impact of DG penetration on
distribution system operation, and its location performance
is independent of fault types and fault impedances.
Furthermore, the proposed algorithm is robust against high-
level noises in measurements.
II. THEORETICAL BASIS
This section describes the theoretical basis for applying
DSSE to fault location. We introduce a classical state estimator
and further extend it to an advanced DSSE method using
measurements from a limited number of μPMUs.
The classical state estimator formulates the relationship
between measurements and state variables by
(1)
where ℝ denotes a state variable vector, and
ℝ is a measurement vector; is the measurement
function, and the measurement noise vector usually obeys
Gaussian distributions , where denotes a
covariance matrix, , and is the
variance of the th measurement noise, .
The weighted least square (WLS) criterion is used to
minimize the WMR:
(2)
where represents the weight matrix of these measurements,
and .
The estimated states are obtained iteratively by the Gauss-
Newton method until each component of at iteration is
sufficiently small. The process for updating the states is listed
as follows:
(3)
(4)
(5)
where denotes the Jacobian matrix of the measurement
function and is calculated by ; the symbol ∙
denotes the transpose of a matrix.
Developed from this classical estimator, the branch current
based DSSE method integrating μPMU data is regarded as a
computationally efficient method due to its constant and sparse-
structured Jacobian matrix, as reviewed in [23]. Therefore, this
paper uses the branch current based DSSE method proposed in
[24], [25] for the fault location task. Also, the voltage at the
slack node and branch currents are chosen as state variables,
and we express these states in a three-phase network as
𝑐
where and denote the real and imaginary parts
of the -phase slack node’s voltage, and ;
and denote the real and imaginary parts of the branch
current at branch , , and is the number of
branches. In the following, the phase index is suppressed for
simplicity.
Here, the measurement vector includes the μPMUs’ recorded
magnitudes and phase angles of voltages and currents as well
as power measurements from pseudo-measurements, and the
latter provides the historical or forecasting data with a low-level
accuracy of power consumption/production at loads/DGs [22],
[23]. We list the measurement functions for voltages, currents,
and powers in this estimator as follows:
(7)
(8)
(9)
3
where denotes measurement and is expressed as 1) the real
and imaginary parts of voltages, and , 2) the real and
imaginary parts of currents, and , or 3) the real and
reactive powers, and ; , , ,
, , and denote the corresponding
measurement functions; and are the sets of nodes and
branches with voltage/current measurements from limited
μPMUs installed in the distribution system, and is the set of
load/DG nodes; and are the indices of nodes and branches,
respectively. For , the pseudo-measurements at node
are further converted into equivalent currents in (9) by
(10)
where and are the real and imaginary parts of the
equivalent injection current at node ; as the voltage phasor
at the node is updated during the DSSE procedure, since the
μPMU measurement of is not available at each node; ∙
denotes the complex conjugate.
By the processing in (10), the Jacobian matrix is independent
of , i.e., [25]. The measurement functions of
(7), (8), and (10) and Jacobian elements of are listed below.
1) Voltages
The voltage measurement function of the μPMU at node
is expressed as:
(11)
where denotes a set of line segments from the slack node to
node , and and denote the 3×3 resistance and
reactance matrices of branch . Also, the complex variables
and are the
voltage phasor at the slack node and the current phasor at
branch . The Jacobian elements of (11) for are
constant, expressed as:
2) Currents
The current measurement function of the μPMU at branch
is shown as
(12)
where and denote the real and imaginary parts of the
current states, and thus the Jacobian elements are present at:
where denotes the branch index, and .
3) Power Injections
The power measurements presenting at node are
converted into equivalent current injection by (10), and then the
current measurement function is expressed as (13)
where 𝑙 and 𝑙as state variables denote the real and
imaginary inflow and outflowing currents at node , and
and denote the set of branches with the inflow and
outflow currents at node , respectively. The Jacobian elements
of (13) are calculated by
The complete DSSE procedure can be found in [24], and the
next section gives the details of the modified DSSE method for
faulted line identification.
III. GRAPH-BASED FAULTED LINE IDENTIFICATION METHOD
This section proposes a graph-based fault location method
that leverages the above DSSE method to narrow down the
searching area and then locate the faulted line.
We consider a distribution network as a graph ,
where and denote the sets of vertices (nodes) and the edges
(branches), respectively. A μPMU is installed at the substation,
and other μPMUs are installed at a limited number of nodes
along the feeder. Each of these μPMUs measures the nodal
voltage and the currents on the branches connected to that node
[24], [26]. We define a subgraph as the subset of
that connects two adjacent μPMUs, μPMUs and
, where and . Here, is the
number of μPMUs installed in the network. Fig.1 shows the
schematic diagram of the subgraphs. In the figure, is a
subgraph that includes the branches and nodes between μPMUs
1 and 2, while contains those between μPMUs 2 and 3.
We briefly describe the proposed fault location method:
1) Step One: Using a distributed DSSE algorithm, the
searching area for the fault is restricted to the feeder between
two adjacent μPMUs, i.e., a certain subgraph.
2) Step Two: The location of the fault is further identified as
the faulted line.
μPMU 3
μPMU 2
μPMU 1
G1G2
Substation
Fig. 1. A sample of radial distribution networks with three μPMUs. The dotted
lines with arrows at nodes denote the laterals (if present), and the μPMU
symbol is from [5].
4
Step One: Identifying the Faulted Subgraph
This step proposes an efficient DSSE algorithm in to
identify the subgraph that contains the faulted line, i.e., the
faulted subgraph. By the graph partition and the subsequent
network equivalence, the DSSE method leverages the μPMU
and pseudo-measurement data in and runs in parallel for
these subgraphs with shorter feeders, i.e., distributed DSSE [26].
1) Network Equivalence
In each subgraph, we suppose that the vertex of acts as
the root node of this subgraph. The lateral connected to
μPMU + is also included in , while the lateral at the root
node of is included in the last subgraph, i.e., . Fig. 2
shows the schematic diagram of in this design. At node
, one type of the following measurements exist and
: 1) Only μPMU data (i.e., the
measurements of μPMU at the root node), and let ;
2) μPMU data (i.e., measurements of μPMU +1) and pseudo-
measurements, and ; 3) Only pseudo-measurements,
and . To reduce the impact of the graph partition on the
power flow in the original network shown as Fig. 2(a), we do
the equivalent current calculation at node .
Specifically, the real and imaginary parts of the injected current
at node in , and , are equivalently calculated by
(14)
where and denote the real and imaginary parts of the
injection currents of pseudo-measurements obtained by (10),
and and are the real and imaginary parts of the current
to the downstream network measured by μPMU +1, shown
in Fig. 2(b). For simplicity, (14) does not show the
measurement noises.
At node , the measurement functions (10) –
(13) hold.
2) Identification Metric
We use the WMR in DSSE as the metric to determine the
faulted subgraph. In normal operation, assume measurement
noises follow Gaussian distribution, WMRs obey a Chi-square
distribution with at most degrees of freedom [27,
Chapter 5]. With a limited number of μPMUs installed in
distribution systems, the degree of freedom is low and equal to
the number of these μPMUs. Therefore, the values of a WMR
in each subgraph fluctuate within a limited range under the
impact of measurement noises, when no faults occur.
On the other hand, according to [2], a fault introduces an
additional unknown fault current injected to the ground or
other phases, while the DSSE equations are built on the
precondition . If a fault occurs in , the presence of
the fault violates the state estimation relationship and leads to a
high WMR in the faulted subgraph; The DSSE in normal
subgraphs have low WMRs even under the impact of
measurement noises [20]. Hence, the faulted subgraph is
determined by selecting the maximum of WMRs:
(15)
where denotes the WMR in subgraph calculated by (2).
Based on the state estimator in (1) – (5), we conclude the
procedure for identifying the faulted subgraph below:
ⅰ. Considering , the measurements in each
subgraph are collected to form the Jacobian and weight matrices,
i.e., and .
ⅱ. For , the DSSE process in subgraph
is shown in the following steps:
a. Initialization–forward-backward sweep [28]: Set the
initial voltage at each node as the voltage of the root node ,
and calculate the current injections of power measurements by
(16)
(17)
where (16) is used for , and (17) holds at ;
comes from the voltage measurement from the μPMU at
the root node.
Then, obtain the initial branch currents by a backward
sweep method. Use and to calculate initial nodal
voltages by a forward sweep method.
b. Obtain using (11) – (13), and calculate and
update the new state variables by . Calculate
the latest voltages based on the new states by the forward
sweep.
c. If is less than a pre-set tolerance or reaches the
maximum iteration number, yield using (2) as the WMR of
; otherwise, use the latest to calculate injection currents
by (10) or (14), then go to step b.
ⅲ. Procure the faulted subgraph using (15).
Finding the faulted subgraph at this stage reduces the
computation burden associated with locating the faulted line in
subgraphs without faults.
Step Two: Locating the Faulted Line
Once we obtain the faulted subgraph by Step One, a
similar WMR metric based on the DSSE technique is developed
to identify the exact line that a fault lies at. Also, we use the
following definitions to present Step Two in .
GK
μPMU KμPMU K+1
Downstream
Upstream
μPMU KμPMU K+1
,
(a)
GK
(b)
Fig. 2. Subgraph (a) Embedded in the whole feeder (b) Decoupled with
other subgraphs. The reference directions of branch currents measured by two
μPMUs are shown.
5
Definition 1 (Paths in a subgraph). A path in a subgraph is a set
of interconnected edges that begins with the root node of the
subgraph. A path that a fault is located in is a corrupted path.
Definition 2 (Adjacent Paths and Boundary Edge). Two paths
denoted by and , ,…, , are defined as adjacent
paths, and if and , where is the
boundary edge that connects two vertices and ,
and 𝜇.
All paths in a faulted subgraph share a starting vertex (root),
and different paths are formed by radially expanding the
topology of . The paths in each subgraph are sorted by
their depth. The shortest path in the subgraph only includes
one edge, while the longest path is the whole subgraph .
In theory, the WMRs in two neighboring paths without fault
current injections should be close to each other; the WMR of
DSSE in a path is low if there is no fault in the path, while WMR
is significantly high once faults occur in the path. Therefore, we
convert the fault location problem into a problem of searching
for the corrupted path that includes the fault, and this corrupted
path is characterized by abnormally high WMR in DSSE. To
find the corrupted path, DSSE runs for each path in , and
the sending-end branch currents in the corresponding path are
chosen as state variables shown in Fig. 3. We apply the DSSE
algorithm in Step One for path and calculate the WMR by
[ ] (18)
where and denote the measurement vector and
measurement functions for path , and ,…, ;
denotes the diagonal weight matrix for this path.
According to Definition 2, if a fault occurs at the boundary
edge , we have
(19)
where and are the WMRs in paths and
calculated by (2), respectively. To find the faulted boundary
edge, set the user-defined identification thresholds to quantize
the relationship in (19):
(20)
where denotes the identification threshold for evaluating the
abnormally high WMR.
We consider that various fault conditions may occur, and
they are unpredictable for system operators. As a result,
although a proper identification threshold is beneficial for fault
location, the specific value of this threshold is difficult to
determine when the fault location, fault impedance, and fault
type are unknown. Similar to [2] and [29], the identification
threshold could be properly selected by using historical or
simulation data of different faults to enforce (20). Also, it is
efficient to run the efficient DSSE method for verifying the
relationship in (20), since the Jacobian matrix for path is
sparse and independent of state variables.
Illustrative Example: To clarify the procedure of the
proposed method, let us consider a 5-node subgraph shown in
Fig. 3, where , and a lateral is connected to
node 3. There are three paths: 1-2, 1-3, and 1-5 in this subgraph.
Path 1-3 is the set of branches from node 1 to node 3, including
the lateral 3-4. Table Ⅰ lists the state variables and
measurements used in these paths, and we show for path
1, 2, 3, which are marked by three block matrices,
respectively. In Fig. 3(a), the fault is located at the boundary
edge between paths 2 and 3 by the proposed method, i.e., branch
Graph-based Faulted Line Identification Algorithm
Input: System model and mesurement data
While The presence of a fault is detected, and its location
is unknown
Step One: Run the distributed DSSE algorithm for in
parellel, and obtain by (15).
Step Two:
Let , and obtain and Then, calculate
by (18).
If
The faulted line is identified as the first branch
in .
Else
For
If and
The boundary edge between paths and is
located as the faulted line.
End if
End for
End if
Output: The faulted line
μPMU 1 μPMU 2
123
4
5
i1i2i4
i3i5
×
μPMU 1 μPMU 2
123
4
5
i1i2i4
i3
i5
(a)
(b)
Fig. 3. Sample network of a 5-node subgraph, and a lateral is connected to
node 3 shown as a dotted line. (a) A fault occurs at branch 3-5. (b) A fault
occurs at branch 3-4.
TABLE Ⅰ
STATE VARIABLES AND MEASUREMENTS IN A 5-NODE SUBGRAPH
State Variables
Measurements
1
1-2
,
, ,
2
1-3
, , ,
, , , ,
3
1-5
, , , ,
, , , , , , ,
6
3-5; the source of the fault in Fig. 3(b) is located at the boundary
edges between paths 1 and 2, i.e., branches 2-3 and 3-4, and
there are two boundary edges due to the existence of a lateral.
In the case that a fault occurs at a lateral as Fig. 3(b), the fault
source could be located at the lateral or the only upstream
branch connected to it, even when there is no μPMU installed
at the lateral. Granting complete observability on laterals may
not be of economic interest, and therefore, in many practical
cases, it is sufficient to identify the faulted laterals [22].
1
1
1
-1
1
-1
1
-1
-1
=
1
-1
-1
1
1
1
-1
1
-1
1
1
We summarize the proposed algorithm in the pseudo-code.
Owing to the hierarchical graph-subgraph-path structure in the
proposed method and the adaptation of the advanced DSSE
method, the search along the faulted subgraph is more efficient,
compared with the ones considering the whole distribution
feeder.
Micro-PMU Placement
The identification accuracy of the proposed method relies on
the number and locations of μPMUs. As the number of μPMUs
increases, the size of a partitioned graph in the distribution
network will be smaller. This would increase the measurement
redundancy defined as the ratio of the number of measurements
to that of states. The minimum number of μPMUs to be installed
in a distribution feeder is two, similar to the requirement
proposed in [22]. Moreover, with more μPMUs installed, the
location performance and computational efficiency of the
proposed method can be improved.
To guarantee the observability for faulted lines, the following
conditions presented in earlier research about meter placement
are considered:
1) μPMU measurements are available at a substation and
nodes in the main feeders that have many downstream nodes.
Such design is suggested in [26] and [28] to provide improved
observability with a limited number of μPMU measurements.
2) μPMUs can be installed at the ends of feeders or long
laterals to ensure the observability and identify the faulted
lateral if necessary [21], [22].
To maximize the location accuracy using a certain number of
μPMUs, an optimal μPMU placement method presented in [30]
can be implemented by considering the probability of various
fault types’ occurrence. However, the optimal meter placement
is a complicated multi-objective optimization problem,
involving multiple impact factors, such as the installation cost,
estimation accuracy, and faulted line observability, etc. We
adopt a meter placement scheme with a low measurement
redundancy following the above-mentioned conditions, which
illustrates the potential of the proposed method for faulted line
identification with limited μPMU installations.
IV. CASE STUDY
We test the proposed algorithm on a three-phase unbalanced
24.9kV, IEEE 34-node test feeder [31]. The test system is
modified by adding three DGs (two synchronous generators and
one PV [21]) into the original systems and simulated in
PSCAD, and the proposed location method runs in MATLAB.
The detailed models of these DGs can be found in [33]. Fig. 3
shows the μPMU and DG placement in the system, where five
μPMUs are installed and the nameplate capacity of DGs is
500kVA. The procedure adopted by μPMUs to obtain voltage
or current phasor measurements is described in the appendix.
Illustrated as Table Ⅱ, the propose graph partition in Section Ⅲ
divides the system into four subgraphs, i.e., and =
1, 2, 3, 4.
Assume that measurement noises obey Gaussian
distributions. Moreover, the maximum errors of magnitudes
1234
5
6789
10
11
12
13
14
15 16 17
18
19 20
21
22
23
24 25
26
27
28
29
30 31
32
33 34
μPMU 2 μPMU 4
μPMU 3 μPMU 5
μPMU 1
DG
DG
DG
Fig. 4. Single-line diagram for the three-phase 34-node test feeder
TABLE Ⅱ
NETWORK INFORMATION OF SUBGRAPHS
subgraph
μPMU
μPMU
Nodes in
1
2
1-12
2
3
9-18
3
4
17-29
4
5
25-34
7
and phase angles for μPMU data are 1% of the true values and
0.01 rads [32], respectively, while the maximum errors for the
powers recorded by pseudo-measurements at load/DG nodes
are 20% of the true values [27]. Also, smart meters can be
installed at DGs for accurately monitoring power outputs, and
the maximum errors of these outputs are 3%. By collecting
measurements at the DG nodes, distribution system operators
(DSOs) do not require the specific DG models. Moreover, DG
operators may not share these detailed models and control
policies with DSOs due to a lack of agreements between them.
However, DSOs can still monitor their power dispatch by the
measurement data [7], [33].
Faulted-subgraph Identification
This section shows the identification performance of the
proposed method in Step One for faulted subgraphs. We test the
proposed algorithm with single-phase LG faults, which are set
at three branches in each subgraph, e.g., branches 3-4, 7-8, and
10-11 in . Moreover, these faults are placed at the beginning
(0.25 ), in the middle (0.5 ), and at the end (0.75 ) of the
lines, and denotes the corresponding line length. In each
fault location, fifty sets of measurements are generated by
Monte Carlo simulations. Also, considering nine fault locations
for each subgraph, 9×50=450 fault scenarios for two influence
factors (fault locations and measurement noises) in each
subgraph are tested. In all tests, the subgraphs with the highest
values of the identification function correctly point to those
faulted subgraphs. The values of across = 1, 2, 3, 4 for
these faults are shown in Fig. 5, where we average the WMRs
of each subgraph for conciseness. As discussed in Section Ⅲ,
WMR greatly increases in the faulted subgraph, indicating that
the fault is located at that subgraph. For example, when a fault
occurs in , is abnormally higher than , , and . This
leads to an immediate conclusion that the fault is located in .
Furthermore, we test the two-phase line-to-line faults with 50
Ω fault impedance in , and Fig. 6 depicts in all
subgraphs. We observe that the maximum of correctly
indicates the location of the faulted subgraph. Also, the
identification performance for the faulted subgraph is not
influenced by fault types and fault impedances.
Faulted-line Location
We test various fault scenarios to evaluate the location
performance of the proposed method. Fig.7 shows the WMRs
for different paths in the faulted subgraph for LL faults on
phases B and C on branch 3-4 in , where we run 100 Monte
Carlo simulations for random combinations of measurement
noises. In this figure, the WMRs of the normal paths are much
lower than those for the corrupted paths. Also, with the radial
expansion of paths, the WMRs of the corrupted path that the
boundary edge 3-4 lies in have high values. Consequently,
branch 3-4 is identified as the faulted line.
Fig. 7. Location results in , when LL faults with 10Ω fault impedance
occur at branch 3-4. The secondary y-axis shows the WMRs at the
corrupted path in 100 Monte Carlo simulations.
TABLE Ⅲ
PERFORMANCE WITH DIFFERENT FAULT TYPES (50Ω IMPEDANCE)
Fault Type
Max Error
LG
94.50%
4.67%
0.83%
2 branches
LL
95.75%
4.25%
0%
1 branch
LLG
96.83%
3.17%
0%
1 branch
LLL
95.67%
4.33%
0%
1 branch
Fig. 5. Identification results in different faulted subgraphs, where we set
LG faults on phase A with 100 Ω impedance in .
Fig. 6. Identification results in different faulted subgraphs. LL faults on
phases B and C with 50 Ω fault impedance.
8
We further evaluate the location accuracy of the proposed
method by calculating the probabilities of two types of test
results: 1) the faulted branch is correctly located and 2) an
immediate neighboring branch of the faulted branch is
determined as a faulted one [22]:
(21)
(22)
where and denote the number of the tests in these two
cases, respectively, and is the total number of the tests; also,
is used to calculate the probability of other results,
i.e., other branches are determined as a faulted line.
We calculate these accuracy indices and in scenarios
with various fault types and fault impedances, where
is set to obtain statistical results in each scenario, and here
the identification threshold .
1) Fault Type
The impacts of various fault types on the location accuracy
of the proposed algorithm are investigated. Four fault types
denoted as LG, LL, LLG, and LLL, are tested. We list the
location results of these fault types in Table Ⅲ. It is shown that
the proposed method enables correct faulted line location with
various fault types and reaches 94% and higher accuracy.
2) Fault Impedance
We test the impacts of fault impedances on the accuracy of
the proposed algorithm. We set different fault impedances at
each branch of , and Table Ⅳ shows the accuracies of this
method to locate faults with these impedances. Especially, the
proposed method enables accurate locations of bolted faults,
owing to the existence of the fault injection currents with high
magnitudes. The results in Table Ⅳ show that the proposed
method enables correct fault-line location with multiple fault
impedances.
The proposed method is tested with high-impedance LG
faults (100, 200, 500, 800, and 1000 Ω) at branch 3-4 to show
the sensitivity towards magnitudes of fault currents. Fig.8
shows the probabilities of the correct location of the faulted
branch, where Monte Carlo simulations with 400 samples of
measurements are used. The location probabilities are higher
than 88% under these various current injections. The reason is
that the measurement errors of voltages and currents are
proportional to the measurement values, while the measurement
weights are inversely proportional to them. While smaller fault
current injections occur, as the weights of measurements are
higher in this case, the WMR will be high. We conclude that the
proposed method works effectively when the fault impedance
is not higher than 1000 Ω in the test system. Once the fault
impedance exceeds about 2000 Ω, the proposed approach may
not observe the small fault injection at long branches in the 34-
node system.
Robustness and Sensitivity Analysis
We investigate the robustness and sensitivity of the proposed
method against various measurement noises and identification
thresholds.
1) Measurement Errors
We conduct robustness analysis concerning higher
measurement noises. We set the measurement noises of μPMUs
as 2% in magnitudes and 0.02 rads in phase angles, while
considering the maximum errors of pseudo-measurements as
10%, 30%, and 50%. Table Ⅴ lists the accuracy of the proposed
algorithm with these measurement noises. As shown, even with
high pseudo-measurement errors up to 50%, either the correct
line or its immediately neighboring line is identified. It implies
that such high-level noises do not degrade the location
performance since DSSE takes the weights of measurement
noises into full account. Also, the location performance of this
algorithm is robust against measurement errors.
2) Identification Threshold
We test the location performance of the proposed method
with various identification thresholds . Fig. 7 shows that the
WMRs in the corrupted paths are much higher than those for
the normal paths. Further, different thresholds are set in the
cases of Section Ⅳ-B, and the location accuracy with these
thresholds is calculated and listed in Table Ⅵ. We conclude that
the identification threshold could be properly selected to
maintain a desirable identification sensitivity.
TABLE Ⅳ
PERFORMANCE WITH DIFFERENT FAULT IMPEDANCES (LL FAULTS)
Fault Impedance
Max Error
0 Ω
100%
0%
0 branch
10 Ω
94.67%
5.33%
1 branch
50 Ω
94.83%
5.17%
1 branch
100 Ω
100%
0%
0 branch
200 Ω
95.08%
4.92%
1 barnch
Fig. 8. Performance for high-impedance LG faults at branch 3-4
TABLE Ⅴ
PERFORMANCE WITH HIGHER MEASUREMENT ERRORS (LG, 50 Ω)
Max Error of
μPMU Data
Max Error of
Pseudo-meas.
Max Error
2%, 0.02 rads
10%
94.67%
1 branch
30%
94.50%
1 branch
50%
94.42%
1 branch
TABLE Ⅵ
IMPACT OF IDENTIFICATION THRESHOLDS
Threshold ϵ
Max Error
100
88.69%
10.35%
0.96%
2 branches
500
95.92%
3.91%
0.17%
2 branches
1000
97.08%
2.92%
0%
1 branch
2000
97.25%
2.75%
0%
1 branch
9
Impact of Line Parameters
Line parameters in distribution systems are subject to
changes with environmental conditions. Considering this
uncertainty, the range of line parameters is generally set within
±5% of their nominal values [34]. Therefore, we consider the
variation in line parameters to evaluate the accuracy of the
proposed method.
We use Monte Carlo simulations to generate 400 test
scenarios, where imprecise line parameters are assumed to obey
Gaussian distribution with various maximum deviations and
zero means. Table Ⅶ lists the location accuracy of the
proposed algorithm, and the maximum errors are 2%, 5%, and
10% of true values of line parameters. Also, LG faults with 100
Ω are set on different branches in . With 5% deviation in the
line parameters, either the correct faulted line or its immediate
neighboring branch is identified.
We conclude that inaccurate line parameters degrade the
location accuracy of the proposed method, and hence line
calibration in power systems is necessary periodically.
Performance in a Larger-scale System
To show the scalability of the proposed graph-based
algorithm, we test the proposed method on the modified IEEE
123-node distribution system, where the nodes at the main
feeders are renumbered by [22]. The details of the 123-node
system can be found in [31]. Fig.9 depicts the DG installations
and meter placement, and μPMUs are placed according to the
conditions in Section III-C. The distribution network graph is
divided into six subgraphs as shown in Table VIII. As the faults
on branch 5-22 are directly observable by two-end μPMUs, and
thus this branch is not included in the subgraphs in Table Ⅷ.
Table Ⅸ lists the accuracy of fault location for various fault
types in subgraphs , , and , where the fault impedance
is set as 50 Ω. Set for each fault type to calculate
and by (21) and (22), and set . Table IX shows that
the proposed method can correctly identify the faulted lines
with accuracy higher than 93%. We conclude that the proposed
algorithm uses the distributed DSSE technique to work in this
larger-scale distribution system.
Computational Efficiency
Numerical experiments for different faults are performed to
demonstrate the computational efficiency of the proposed
algorithm. We run this method on a PC with 2.6 GHz i5, and
8GB RAM using MATLAB 2017b.
Table Ⅹ lists the average CPU time of the proposed method,
including two steps, for faulted line identification in these test
systems. It shows that once the measurement data are collected,
this algorithm locates the faults within 15 milliseconds in the
34-node distribution system. Compared to the traversal search
strategy for a whole feeder in [20] and [22], the proposed
method runs in parallel for feeders with a reduced size, which
improves the computational efficiency for application to the
larger-scale networks. Hence, our location method has a low
computational cost, which is also verified in the 123-node test
system. It should be noted that owing to the increase in the
number of nodes in subgraphs of this larger-scale system, the
proposed method takes a longer CPU time, i.e., about 20 ms,
for faulted line identification. With more μPMUs installed, the
number of nodes in a subgraph decrease, and the computational
efficiency of this method can be further improved.
V. CONCLUSION AND OUTLOOK
This paper proposes a graph-based faulted line identification
algorithm using μPMU data in distribution systems. We present
a distributed DSSE algorithm to identify the faulted subgraph
efficiently, and this method significantly reduces the searching
scale and speeds up the subsequent fault location procedure.
Further, we conveniently determine a faulted line by applying a
TABLE Ⅶ
LOCATION ACCURACY WITH UNCERTAINTY IN LINE PARAMETERS
Maximum Errors
of Line Parameters
Max Error
2%
100%
0%
0 branch
5%
97.5%
2.5%
1 branch
10%
95.5%
3.25%
2 branches
123457815
μPMU 2
μPMU 1
DG
30
22
25
DG
19 18
24
31
32
33
36
10
9
35 37 44
43
42
38 39
13
12
17
14
μPMU 3
20
21
μPMU 7
23
μPMU 8
DG
16
μPMU 5
6
11
41
40
26
27 28 29
34
μPMU 6
μPMU 4
DG
DG
Fig. 9. Diagram of the modified 123-node distribution system
TABLE Ⅷ
SUBGRAPH INFORMATION IN 123-NODE SYSTEM
subgraph
Nodes in
1-5 between μPMUs 1 and 2
5-13 between μPMUs 2 and 3
13-21 between μPMUs 3 and 4
13-44 between μPMUs 3 and 5
22-29 between μPMUs 6 and 7
22-39 between μPMUs 6 and 8
TABLE Ⅸ
LOCATION ACCURACY IN THE 123-NODE SYSTEM
Fault Type
Max Error
LG
93.67%
4.67%
2 branches
LL
95.67%
4.33%
1 branch
LLG
96.33%
3.67%
1 branch
LLL
96.0%
4.0%
1 branch
10
hierarchical graph-subgraph-path structure to the DSSE method.
In the case of inadequate μPMU and DGs installed at the
distribution level, the proposed method enables accurate
faulted-line location. Extensive simulations verify the accuracy
and efficiency of this method under various fault scenarios.
The proposed method is suitable for radial distribution
systems and can be updated by incorporating the linear
measurement functions proposed in [35] for weakly meshed
distribution systems. Also, we assume that there is no bad data
in a measurement dataset, and robust state estimation can be a
promising solution against potential bad data [27]; hence, our
future work focuses on the application of robust state estimation
to faulted line identification in distribution systems. The
proposed method requires prior knowledge of the network
topology to identify the faulted line in the distribution network.
For networks with partially known topology, efficient topology
identification methods that use measurement data under normal
conditions, such as [34], should be adopted.
APPENDIX
The proposed method leverages post-fault phasors from
μPMUs. Fig.10 shows the diagram of a waveform for fault
currents and the latency of the proposed method for faulted line
identification. According to [20], considering the duration of
transients, i.e., , there is a short latency between 0 and 20 ms
before measuring the steady-state phasors by μPMUs. Later, to
obtain accurate post-fault synchrophasors, the discrete Fourier
transform (DFT) method is used to process a dataset of raw-
sampled waveforms [2], [37]. The time window for μPMUs
to get post-fault phasors is about two to three periods derived
by a fundamental frequency, e.g., 30 ms. Moreover, according
to IEEE Standard C37.118, the shortest length of the
observation window can reach up to 17 ms [32].
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TABLE Ⅹ
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#System
Faulted Subgraph
Average CPU Time [ms]
34-node System
14.28
13.50
14.72
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123-node System
21.37
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t
Fig. 10. The diagram of a waveform of fault current and the latency of the
proposed method in identifying a fault [19].
11
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Ying Zhang (S’18) received the B.S. and
M.S. degrees in electrical engineering from
Shandong University, Jinan, China, in 2014
and 2017. She is currently pursuing the
Ph.D. degree in Department of Electrical
and Computer Engineering at Southern
Methodist University, Dallas, Texas, USA.
Her research interests include distribution
system state estimation for distribution
system moniroting and control via optimization and machine
learning. She serves as a peer reviewer in IEEE Transactions on
Smart Grid, IEEE PES letters, and Journal of Modern Power
Systems and Clean Energy.
Jianhui Wang (M’07-SM’12) received the
Ph.D. degree in electrical engineering from
Illinois Institute of Technology, Chicago,
Illinois, USA, in 2007. Presently, he is an
Associate Professor with the Department of
Electrical and Computer Engineering at
Southern Methodist University, Dallas,
Texas, USA. Prior to joining SMU, Dr.
Wang had an eleven-year stint at Argonne
National Laboratory with the last appointment as Section Lead
– Advanced Grid Modeling. Dr. Wang is the secretary of the
IEEE Power & Energy Society (PES) Power System Operations,
Planning & Economics Committee. He has held visiting
positions in Europe, Australia and Hong Kong including a
VELUX Visiting Professorship at the Technical University of
Denmark (DTU). Dr. Wang was the Editor-in-Chief of the
IEEE Transactions on Smart Grid and an IEEE PES
Distinguished Lecturer. He is also a Clarivate Analytics highly
cited researcher for 2018.
