The document proposes a self-healing approach for power distribution networks using bipartite graph modeling and reconfiguration algorithms. It describes modeling the network as a bipartite graph to represent switching possibilities abstractly. Two algorithms are presented: one for fault restoration that finds the best reconfiguration to restore power while maintaining topological feasibility, and one for overload mitigation that reconfigures the network to alleviate overloads. An example application to a real distribution network demonstrates initializing the bipartite graph and applying the algorithms to restore faults and reduce overloads through reconfiguration.
3. Concept of Self Healing
The capability to:
■ Identify
■ Diagnose
■ Recover
from system disruptions with the objective of maximizing:
■ System availability
■ Survivability
■ Maintainability
■ Reliability
4. Self Healing with respect to
Power Distribution Networks
Automatic fault
detection
Isolating faults
Restoring
power both
upstream &
downstream
the faults
5. Scope
■ Describes model properties and capabilities
■ Illustrated for a small real world medium voltage
distribution network.
■ Solutions for recovery of the system
■ By analyzing and automatically undertaking switching
operations to maximize restored load
■ With minimal human intervention
Focus
6. Approaches
Solutions may be based on:
1. Network optimization algorithms
Lacks speed of response due to inadequately
designed data structures and algorithms to handle
fault recovery
2. Data storage of pre-defined switching schemes
Lacks flexibility of response due to dynamic network
topology
7. Proposed Solution
■ New network data structures for which effective
reconfiguration algorithms can be designed.
■ Resolves problem of speed while maintaining flexibility
■ Based on Bipartite Graph Theory.
■ Represent the switching possibilities abstractly rather
than the physical network itself.
9. Graph terminology
■ In this paper vertices and edges
are referred to as nodes and arcs.
■ Vertices – represent Load,
Sub Stations and connection
points.
■ Edges – represent cables, lines,
busbars and switch devices in the
context of power system.
■ Graph - represents the Power
Distribution Network.
■ Bipartite Graph
■ Tree - graph that contains no
cycles, bipartite by its nature.
10. ■ Co-tree edges - represent
the edges of G that can be
closed to create a
mesh(cycle), so a co-tree
edge is not an edge of the
spanning tree T.
■ Fundamental cycle Y of G -
set of edges defined by a
tree edge e and the path in T
between the two end nodes
of e. In the distribution
context, the fundamental
cycle induced by a switching
ON operation represents the
corresponding mesh.
11. Matching Terminologies
■ Matching of G - set of edges
M if no two of its edges have
a common endpoint(vertex).
So the no. of vertices in M are
twice the no. of edges in M.
■ M-alternating - a path P is so
if it's edges are alternately in
M and E(G)/M (Edge set of
which contains edges of G
which are not in M).
■ M-Augmenting - an M
alternating path is so if it's
end vertices are not end
vertices of any of the edges in
13. The model is based on the idea of reconfiguring radial
networks by undertaking Switching Steps.
Pairs of switching operations consist of closing one
(arbitrary) branch of the network and opening another so
that the resulting configuration is also radial.
14. Definition Of Switching Steps:
Let G= (N, A) be a graph representing the distribution network.
Its operating configuration is a spanning tree, say T = (N, AT
𝐵
),
where AT
𝐵
∈ A.
Let YT
𝐵
be a fundamental cycle with respect to T, defined by a co-
tree arc element B.
A switching step {B, c} is then defined as an exchange of an arc
element c, that lies on the fundamental cycle YT
𝐵
, for a co-tree
arc element B, i.e.,
{B, c}| c ∈ YT
𝐵
∩ AT
𝐵
, B ∈ YT
𝐵
∩ (A AT
𝐵
).
15. Result 1: By proceeding in switching
steps, the network is guaranteed to be
topologically feasible, i.e., radial and
connected.
For instance see Fig. 1, where the
lower left square is a fundamental
cycle of G with respect to T defined by
the branch B. Switching OFF
operations that can be found on this
cycle are a, c and d.
16. Reconfiguration Dynamics
Network reconfiguration is a dynamic process.
Generally, switching steps are dependent, i.e. a switching step is
only valid with respect to the current network configuration.
After a switching step, the network configuration changes and
usually some of the operation pairs that were switching steps
before are disabled, while others become possible.
This requires a new search for feasible switching steps.
18. Feasibility of a sequence needs to be checked step by step,
which may be time-consuming as it employs fundamental cycle
search in the spanning tree.
This data model avoids the burden of repetitive checking
topology admissibility after every step. It captures all the
topological dynamics of the network in one simple Bipartite
graph. Searching this graph is efficient in several aspects:
Relevant information
Lossless compression
Topology feasibility
Algorithm design
19. Graph Initialisation
The set of switching steps that are feasible in T may be
represented by a bipartite graph B = (VAo,VAT, E).
VAo- denote the switches that are currently opened and
available to be closed.
VAT - the switches that are currently used and can be opened.
We refer to nodes of the bipartite graph as vertices and to its
arcs as edges.
20. Possible switching ON operations : VAo ⇔ A AT
Possible switching OFF operations: VAT ⇔ AT
E = {{vAo, vAT }|vAo ∈ VAo , vAT ∈ VAT }
21. Example:
VAo = {A, D}
VAT = {b, c, e, f, g}
Edges are formed where
pairs constitute valid
switching steps,
i.e. only pairs {D, c} and
{D, f} are excluded.
23. Graph Reconfiguration
Let B1 = (V1o , V1T , E1 ) be a bipartite graph representing the
current network configuration T.
Then doing a switching step {A, g} where A is a co-tree branch
of the current configuration T and g ∈ T sets up a new bipartite
graph B2 = (V2o , V2T , E2 ) such that:
V2o = {A, D}{A} + {G} = {G, D}
V2T = {b, c, e, f, g}+{a} {g} = {b, c, d, f, a}
24.
25. The new edge set E2 is then obtained in two steps. First, all
edges that were in E1 incident to A will be in E2 incident to G
and all edges previously incident to g will be in E2 incident to a.
Then, consider a matching M = {A, g} in B1 . For every M-
augmenting path in B1 , negate the existence of an edge
between its end vertices in E2 .
Thus, the changes to edge set E are: first exchanging the end
nodes A for G and g for a; and then deletion of edge D–b, edge
D–e and insertion of edge D–c and edge D–f.
27. Two Proposed Algorithms
Fault
restoration
•Here, the algorithm finds the best
to reconfigure the faulty circuit while keeping
in mind the topological feasibility.
Overload
mitigation
•Here, the algorithm tries to alleviate
overload by reconfiguring the network while
taking into account of topological feasibility.
28. Step-1
A fault occurred
and detected
Step-2
Bipartite graph is
inspected for
reconfiguration
Step-3
Decision making
is applied to
choose feasible
candidate
Step-4
Resolving
overloading
through
overload
mitigation
Fault Restoration
Flow Diagram
29. Fault Restoration – Example
Circuit with a spanning tree : {b, e, g,
f, c}
Co-tree arc : {A, D}
Assumption : Fault occurs in load g
Fault occurs and gets detected at load
g
Load g gets disconnected
Creates an island at the respective
position
Step - 1
30. Perform switching operation to
reconfigure it into topologically
feasible circuit
Step - 2
Implement bipartite graph of initial
circuit
(Rather than checking for fundamental
cycles repeatedly)
Inspect it for reconfiguration
31. Step -
4
Switching steps might cause
overloading
Overcome by using overload
mitigation
Step - 3
Reconfigure and make it radial and
connected
32. Step 1 : Occurrence And Detection Of The
Overload
The squares represent loads
present in the different branches.
The first figure shows the circuit
before application of the fault
restoration algorithm and the
second figure shows the circuit
afterwards.
We can see that load has
increased in the overall circuit
Overload Reconfiguration
33. STEP 2: Choose The Most Severe Overload
Since here we have the choice
between branch a and branch
b, both are overloaded, we pick
branch a which is more severe.
34. STEP 3: Inspect The Bipartite Graph To Choose Vertices
To Reconfigure The Graph And Decrease Load
Here we can replace branch a
by either branch G or by
branch D.
35. STEP 4: A Decision Making Process Is Applied To Choose The
Reconfiguration Operation That Solves The Overload Or Results
In The
Smallest Remaining Overload.
Here since we can not replace a by G
since it’s faulty,
We switch a by D in order to reduce
load.
36. Here is the final reconfigured topologically
feasible circuit after applying self-healing data
structure.
37. STEP 5:
If there are any overloads left, go to Step 2, otherwise stop
In case, that the overload can be solved right away, the
decision making process chooses the less loaded path.
In the other case, the solution that leaves the smallest
remaining overload is chosen.
Since here we have maximum overload in branch d now,
we can replace it by branch A or G.
But replacing it by branch A would result in more load and
would make the previous change redundant and branch G
is faulty, so we stop.
39. As we saw earlier that we apply 2 algorithms under
gradual steps in accordance with the requirement for
removal of the fault load and to reduce the overload
caused by reconfiguration of switches.
40. Visiting a Real Distribution Network
Let us Consider 10 kV distribution network where dashed
lines represent branches with open switch devices; triangles
represent loads; squares represent sectionalizing
substations; and the circle represents a 60/10 kV substation.
42. The network consists of three feeders. Let us consider
having named the branches so that Feeder 1 consists of
branches 1–3; Feeder 2 consists of branches 4–10 and
Feeder 3 consists of the remaining branches, 11–16. In
order to read easily, the open branches are denoted by
capital letters A, B, C, D, E, F and Z.
43. Working of Algorithm
Let us consider a fault in each switch at once and find the
best candidate from the open switches.
This means that when a fault occurred in any load there was
no other switch at fault and hence we obtain the best
alternative for each faulted switch. The obtained vector is
s0 = [𝑍𝑍𝐸𝐵𝐵𝐵𝐵𝐸𝐸𝐹𝑍𝐶𝐷𝐹𝑍𝐴]
44. Working of Algorithm
This vector is read as s0 = [𝑍𝑍𝐸𝐵𝐵𝐵𝐵𝐸𝐸𝐹𝑍𝐶𝐷𝐹𝑍𝐴]
■ Switch Z is best to be closed in case to restore fault in
branch 1,2, 11,15
■ Switch E is best to be closed in case to restore faults in
3,8,9 and so on.
■ A reconfiguration under such case would be a normal
reconfiguration.
45. Working of Algorithm
s0 = [𝑍𝑍𝐸𝐵𝐵𝐵𝐵𝐸𝐸𝐹𝑍𝐶𝐷𝐹𝑍𝐴]
Once we apply the switching it is possible especially in peak
load condition that some may yield an overload condition
to resolve which we apply the second algorithm which
provides best alternative switching to reduce the load and
attain reconfiguration.
46. Working of Algorithm
Let us take into consideration another situation where we
already have a faulty branch and we are taking best switch
to get another configuration and another branch which has
the same switch as its best alternative becomes faulty.
47. Working of Algorithm
■ For Example the Best alternative for branch 1 and 2 is
switch Z.
s0 = [𝑍𝑍𝐸𝐵𝐵𝐵𝐵𝐸𝐸𝐹𝑍𝐶𝐷𝐹𝑍𝐴]
■ Suppose that branch 1 has failed and was restored,
according to s0, by Z. Suppose that branch 2 fails before
the fault in 1 is cleared.
48. Working of Algorithm
■ Now, 1 is unavailable and Z (that corresponds to a predefined
post-fault switch for 2 – see the second entry in s0) is already in
use. We apply A1 again to find out that E has to be operated in
order to restore a failure in 2 after 1 has failed and was restored
by Z.
■ Taking it given that branch 1 is already at fault the vector of
best alternatives is given by
S1 = [EEBBBBEEFBCDFBA]
■ Considering Similar way we can define S2, S3, etc. where Si,
stands for ith branch being already at fault.
49. Table of all Vectors
Here ith represents every Si i.e. every vector which represents
the list of best alternatives when the ith branch is already at
fault.
50. The Bipartite Graph
The table was obtained by
repeatedly applying the
above mentioned
algorithms after modeling
the network by a bipartite
graph as shown
51. The Part II
Looking for the overloads after restoring the two faults in
peak load conditions. Depending on the table of Vectors
we can find the branches which have an extra overload
over them we can use the Algorithm A2 to find the
switching steps that alleviate the overload by
reconfiguring the network.
52. Part II
■ Based on the Table of Vectors
Mentioned above the following
figure shows the load after
reconfiguration to resolve the
fault.
■ The grey colour intensity
depends on the load severity
after optimization. For ex. see
row 5 column 9, where both the
faults occur in Feeder2.
■ Three reconfiguration steps are
needed to alleviate the overload
53. Table of all Vectors
Exclamation mark in the Table of vectors represents that the
Faults could no more be restored.
