IEEE Transactions on Power Delivery, Vol. 9, No. 4, October 1994
1936
A NEW NETWORK RECONFIGURATION TECHNIQUE FOR SERVICE
RESTORATION IN DISTRIBUTION NETWORKS
N.D.R. Sarma
Student Member
V.C. Prasad
K.S. Prakasa Rao
Senior Member
V. Sankar
Student Member
Department of Electrical Engineering
Indian Institute of Technology, Delhi
Hauz Khas, New Delhi-110016, INDIA
Service Restoration
ABSTRACT
Whenever a fault occurs in a particular
section of a distribution network and on
isolation of the fault some of the loads get
disconnected and are left unsupplied. Service
should be restored to these affected load points
as quickly as possible through a network
reconfiguration procedure. A new and efficient
technique is presented in this paper for this
purpose.
Network reduction and determination
of the interested trees of the reduced network
by a specially developed algorithm for finding
the required restorative procedures, are the
main contributions of this paper.
Keywords : Network reconfiguration,
restoration, Interested trees.
Due to the isolation of the faulted
section, some healthy sections may also get
disconnected and will be left without supply.
Service is to be restored to these affected
sections by closing and/or opening certain
switches in the network to see that every load
point is supplied through a radial line.
Fault Repair
The faulty section is to be repaired and
the time of repair would depend on various
factors like the type and location of fault,
availability of repair crew, etc..
Service
INTRODUCTION
The topology of a distribution system is
described by the branches of the system and each
branch is defined as a set of components in
series. The main system components are power
transformers, lines, cables, busbars, circuit
breakers, instrument transformers and isolators.
Though the various components are connected in
the form of a meshed network, the distribution
system is operated normally in a radial fashion.
A practical distribution system is mainly
divided into a primary subsystem consisting only
of system supply points and a number of secondary
subsystems. To ensure a reliable operation of
a distribution system, it is important to
restore power to all the customers rapidly on
the occurrence of a fault or an abnormality in
a certain section of the system.
Whenever a
fault occurs on some section of the system, the
following actions are required to be taken.
Fault Isolation
By
opening
the
appropriate
circuit
breakers/isolators, the faulted section is to
be isolated from the rest of the system as
quickly as possible.
91 SM 401-0 PWRD
A paper recommended and approved
by the IEEE Transmission and Distribution Committee
of the IEEE Power Engineering Society for presentation at the IEEE/PES 1991 Summer Meeting, San Diego,
California, July 28 - August 1, 1991. Manuscript
submitted January 21, 1991; made available for
printing July 1, 1991.
Restoration to Normal State
Once the fault is repaired, the system is
to be restored back to the normal state.
There has been a considerable interest in
the recent past in developing various algorithms
for
network
reconfiguration
for
service
restoration in distribution systems [1-12].
Castro et al [1] suggested algorithms which
determine fault location and generate switching
instructions based on tree searching techniques
utilizing switch tables that can be defined by
an operator. Castro Jr., et al [2] proposed a
method in which they combine two independent
processes: a service restoration to dark zones
and a load balance between feeders. Electrical
operating constraints are included in the
analysis using an appropriate fast decoupled
load flow.
Aoki et al [3] presented an
algorithm for load transfer by automatic
sectional izing switch operations in distribution
systems on a fault occurrence subject to the
transformer and Tine-capacity constraints. Aoki
et al [4] also developed a method in which loads
in an out-of-service area are transferred to the
transformers adjacent to the affected area based
on current and voltage constraints. Aoki et al
in [5] suggested a non-combinatorial algorithm
based on the effective gradient method. Aoki
et al in [6] presented yet another algorithm for
emergency service restoration in which load
restoration
priority
can
be
considered.
Stankovic
and
Calovic
[7]
developed
a
graph-oriented control
algorithm based on
linearized system model, where the synthesis of
corrective controls was
formulated
as
a
combinatorial
mixedinteger
programming
problem. Dialynas and Michos [8] suggested a
method based on graph theory which requires the
knowledge of all minimal paths leading to each
load point from all system service points under
various operating conditions.
Some heuristic
methods were also suggested in the papers
[9-12].
In all the above methods the total
distribution network after isolating the faulty
1937
section, has been used in restoring the supply
to all the affected consumers.
In this paper an attempt has been made to
reduce the network by suitably merging certain
set of nodes together and this reduced network
can easily be analyzed for finding alternate
paths of power supply to the affected load
points (nodes). This requires finding all the
trees called interested trees consisting of the
nodes corresponding to supply points and all
load points in the reduced network.
A new
algorithm is developed for finding all such
trees in a graph. The proposed method consists
of reducing the original distribution network
under faulty conditions, and finding a
restorative procedure satisfying voltage and
current constraints. This method is illustrated
in detail through a representative distribution
system example taken from [8].
Some of the terms used in this paper are
defined below:
Interested Nodes
Some of the nodes of the given graph which
are of specific interest are called as
interested nodes. For example, all the nodes
corresponding to supply points and all the load
points are called as interested nodes in the
reduced network. If a node is not an interested
node, it will be referred as a noninterested
node.
Interested Trees
An interested tree is a subgraph of the
given graph which satisfies the following
properties:
1.
It is connected.
2.
It contains all the interested nodes
3. No proper subset of it satisfies the
properties 1 and 2 above.
PROPOSED METHQP
The important steps of the proposed method
are:
a)
Network reduction (by coalescing some
nodes)
b)
Determination of all the interested trees
in the reduced network.
c)
To find the restorative procedure
satisfying the current and voltage constraints.
The above steps are explained in detail in
the following paragraphs:
a)
Network Reduction fbv coalescing some
nodes)
The system network under consideration may
be very large. It is possible to reduce the
original network into a smaller network for the
purpose of finding all the radial paths from
the source points to all the load points. This
reduced network can easily be analysed for
finding alternate paths of power supply to the
load points. After the occurrence of a fault
and isolation of the faulty components the
system will be divided into some groups of
connected components. All the elements of a
group which are connected together can be merged
together.
For example consider a sample
distribution system shown in Fig.1 in which L1 ,
L3, L7 represent supply points. L10, L11, L55,
L56, L57, L58, L59 represent the load points.
A fault is assumed on components 91 and 135.
Isolation of these faulty components from the
rest of the system leads to load points L55,
L56, L59 without supply.
Fig. 2 is the
graphical representation of the connection
diagram of Fig.1 under the above conditions.
The continuous lines indicate the elements
closed and dotted lines indicate the elements
which are open. The complete system will be
divided into five groups as shown in Fig.2 under
these faulty conditions. It may be noted that
for the fault under consideration, the loads in
the groups 4 and 5 are only affected.
Each
substation
configuration
employs
an
interlocking scheme which does not permit an
isolator to be closed if one of the isolators
in the same interlocking sequence is already
closed.
The elements which are in an
interlocking sequences in a group where the
loads are affected are to remain unaltered in
the process of network reduction.
Thus the
network shown in Fig.2 can be reduced to a
network shown in Fig. 3(a). In Fig. 3(a), node
ml, is the new node obtained after merging all
the nodes of group 1, m2 is obtained after
merging all the nodes of group 2 and m3 is the •
node obtained after merging all the nodes of
group 3. Since elements 136,137 and 138, 139
are in an interlocking scheme, Group 4 is shown
by nodes m4 and m5, retaining the element 136,
137, 138 unaltered. Node m4 is obtained after
merging the end nodes of the component 134.
Similarly the node m5 is obtained after merging
the. end nodes of components 140, 141 and 142.
This node corresponds to the load point L59 of
the original network.
Similarly group 5, in which the load points
are affected due to the isolation of faulty
components, is represented by the nodes m6, m7,
m8 and m9, retaining the elements which are in
interlocking schemes.
The nodes m8 and m9
correspond to the load points L55 and L56
respectively.
In addition, since one is interested in
finding paths from any of the sources to the
affected load points, the nodes corresponding
to the source points in the reduced network can
be merged together to form a single source node.
Accordingly in Fig. 3(b), the final reduced
network, the node S corresponds to the source
node and nodes m5, m8, m9 correspond to the load
points. These nodes are the interested nodes
of the reduced network.
b)
Determination of all the interested trees
of the reduced network
In the reduced network, it is required to
find all the interested trees. There are many
algorithms available in the literature for
finding all the spanning trees of a given graph
[14]. Such spanning trees contain all nodes of
the graph, However no algorithm seems to be
available for finding all interested trees in
a graph.
It is possible to generate all interested
1938
trees by first generating all the spanning trees of a
graph and manipulating each spanning tree according to
the definition of the interested tree. This is inefficient because an interested tree may be generated more
than once. Further generating all spanning trees and
manipulating them is time consuming.
proposed for generating a l l the interested trees
directly without repetition.
Algorithm to find a l l the interested trees in
a graph
Notation
In the next part of this paper, an efficient
algorithm "INTERESTED
TREES', is
L1
NODE
LINE
CO: TRANSFORMER
C.B (CLOSED)
w1 : SUPPLY POINTS
C.B (OPEN)
ISOLATOR (CLOSED)
ISOLATOR (OPEN)
17
L10
Lit 155 |09L56,]7 L57
L58
113
133
159
F i g . l Example system topology
G:
The given connected graph whose all the
interested trees are to be determined.
i:
current level of the stack.
Esi: Set of elements in which the first element
of this is either zero or a node number. Other
elements are the edges to be shorted in the
graph G. This is stored in the ith level of
the stack.
If the first element is zero, it
indicates that all the edges in Esi are to be
shorted. If it is N (a noninterested node), it
indicates that one more edge incident at N in
the original graph, has to be shorted along with
the other edges given in Esi.
Eoi: The set of edges to be opened in the graph
G. This is stored in the ith level of the stack.
Es': Set of all edges of current Esi
Eo': Set of all edges of current Eoi.
G': Resulting graph obtained after opening and
shorting the edges given in Eo' and Es'
respectively from G.
All edges corresponding
to self loops, if any are deleted. (Note that
G' may or may not contain all interested nodes)
EN: The set of edges incident at N in G.
TYPE:
A vector of 0's and 1's which denotes
the status of noninterested nodes. It is 1 if
only one edge incident at it is shorted. It is
zero if no edges or atleast two edges incident
at it are shorted.
Algorithm "INTERESTED TREES"
Step 1:
Set i=o, initialize the first element
of Esi as zero. Eoi = {*} (i.e., there are no
edges to be shorted and no edges to be opened).
Set G'=G, Eo'=Eoi, Es' = {*} • Set the type of
all noninterested nodes to zero i.e., set all
\\.7 .122123 124 125 j!26^ 128 130 131 132 L53
elements of vector TYPE to zero.
Step 2:
Check for the presence of a nonElements in
interested node in G'. If such a node exists
interlocking
go to step A.
scheme in
Step 3: Find all the spanning trees of G' using
groups 4 &. 5.
any algorithm which generates all spanning trees
136,137;138,139 without repetition. Each of these trees along
96,97; 98,99;
with the elements of Es' will give interested
104,105;
trees (Lemma 1 ) . Go to step 5.
Step 4:
Select a noninterested node N in G'.
Find the set of edges EN incident at N in G.
\ GROUP 3 * - /
1
Perform procedure ALL COMBINATIONS I with G',
Es', Eo', N, and EN as inputs and go to next
step.
Step 5:
Pop the stack.
If there are no
Fig. 2 Connection diagram of Fig. 1 after the
elements in the stack, go to step 12. Otherwise
isolation of faults on 91 and 135
m
Es' = all edges in Esi, Eo' = Eoi. Denote the
\ 136 136m 5
first element of Esi by x. i=i-1.
139.
m2
133 1• \ ini
136 138m 5
143
405
^
9 9 srn 5 97 1 104
.
109
m8
F i g . 3 ( a ) Reduced network F i g . 3 ( b ) F i n a l reduced
o t Fl
<3-*
network of Fig. 2
Step 6:
Open the edges given in Eo' from G.
Denote the resulting graph by G'. If all edges
at a noninterested node are opened, delete that
particular node from G'.
Step 7: Check for the connectivity of G'. If
G' is not connected go to step 5.
Step 8:
If x is not equal to zero, go to step
10.
Step 9: (i) Set the status of all nodes of G in
the vector TYPE to zero.
9.(ii)If there are no edges in Es',go to step
2. Let e = first edge in Es'.
1939
9.(iii):
If e is a
self loop,go to step 5. Otherwise let v1 and
v2 be the end nodes of the edge e in the
original graph G.
If v1 is a noninterested
node, change its status in the vector TYPE (If
its status is 0 make it 1. On the other hand
if it is 1 make it 0 ) . Similarly, if v2 is a
noninterested node, change its status also in
the vector TYPE.
9.(iv): Short the edge e in G'.
Denote the
resulting graph by G'.
9.(v):lf there are no more edges in Es' go to
step 9.(vi). Otherwise take the next edge in
Es'. Denote this as e and go to step 9.(iii).
9.(vi): If none of the noninterested nodes has
status 1 in the vector TYPE, go to step 2.
Otherwise let v be a nonintrested node whose
status is 1 in TYPE.
9.(vii):1=1+1. Store v as the first element in
Esi. Append Es' to Esi. Eoi = Eo'. Go to steps.
Step 10 (i): Find all the edges incident at
node x in the original graph G. Let this set
be Ex.
10.(ii):Delete the edges from Ex which are also
present in Eo'.
10.(iii ):Short the edges given in Es' as follows:
Let e = first edge in Es'.
1O.(iv): If e is a self loop go to step 5.
Otherwise short it in G'. Delete this edge from
Ex, if it is present in Ex. The resulting graph
is G'.
10.(v) : If there are no more edges in Es', go
to step 11. Otherwise take the next edge in
Es'. Denote this as e and go to step 10(1v).
Step 11: Perform procedure ALL COMBINATIONS II
with G', Ex, Es' and Eo' and x as inputs and go
to step 5.
Step 12: End.
Procedure ALL COMBINATIONS I
{This procedure generates the combinations
of edges to be shorted and edges to be opened
at a noninterested node which has a status 0.
The inputs to this procedure are G', Es', Eo',
N and EN}.
Step a:
Delete the edges from EN which are
also present in Eo'.
Step b:
Delete the edges from EN which are
also present in Es'.
Step c: If the number of edges in EN is less
than two, go to step (h). Otherwise choose e1,
e2 the first two edges in EN (Lemma 2 shows why
two edges are enough).
Step d: i = i+1. Store zero as first element
in Es. Append e1, e2 and Es' to Esi. Eoi=Eo'.
Step e:
i=i+i. Store N as first element in
Esi.
Append e1 and Es' to Esi.
Eoi=Eo'.
Append e2 to Eoi.
step f:
i=i+i. Store N as first element in
Esi.
Append e2 and Es' to Esi.
Eoi=Eo'.
Append e1 to Eoi.
Step a: i=i+i. Store zero as first element in
Esi. Append Es' to Esi. Eoi=Eo'. Append e1
and e2 to Eoi. Go to step i.
Step h:
if the set EN is null, go to step i.
Otherwise i=i+1. Store zero as first element
in Esi. Append Es' to Esi.
Eoi=Eo'.
Append the single element present in EN to Eoi.
Step i: End.
Procedure ALL COMBINATIONS II
{This procedure generates the combinations
of edges to be shorted and edges to be opened
at a noninterested node which has a status of
1}.
The inputs to this procedure are G', Es',
Eo', x and Ex.
Step a:
If Ex is null, go to step d.
Otherwise let e1 be the first edge in Ex.
Step b: i=i+1. Store zero as the first element
in Esi. Append e1 and Es' to Esi. Eoi=Eo'.
Step c: i=i+1. Store x as the first element of
Esi. Append Es' to Esi. Eoi=Eo'. Append e1
to Eoi.
Step d:
End.
The proof of this algorithm is given in
appendix I and an illustration is given in
appendix II.
c)
Findina the restorative procedure
satisfying the current and voltage constraints
The above algorithm is used to find all
the interested trees of the reduced network.
If any interested tree contains all the elements
which are in an interlocking sequence, then it
is not a valid interested tree since both the
elements which are in interlocking sequence
cannot be closed simultaneously.
All such
invalid interested trees are to be eliminated.
Henceforth the remaining interested trees are
only considered. Once the interested trees are
determined, the elements to be closed and the
elements to be opened for each tree can be found
out. This ensures that all the load points are
connected to the source point in a radial
fashion.
The total number of switching
operations (the sum of number of elements to be
closed and the number of elements to be opened)
forms the weight of a tree. Thus the weights
of all
the
interested
trees
(possible
restorative procedures) are obtained. These
trees are then arranged in the ascending order
of their weights.
To start with, the interested tree
(possible restorative procedure) with the
minimum weight is chosen. For this procedure,
a complete path from the actual source point to
the actual load points in the original network
is traced. (This is obtained by replacing the
merged nodes in the path by the elements which
have been shorted in the network reduction
process). This constitutes the reconfigured
network
under
system
restoration.
A
Distribution Load Flow (DISTLF) is run for the
resulting
network with this
restorative
procedure.
Different distribution load flow
methodologies are available in the literature.
In the present work the method given in [13] is
used. The DISTLF gives us the branch currents
and node voltages for a given source node
voltage and specified loads at different load
points of the above network.
If the branch
currents and the node voltages in the entire
network are within their tolerable limits, the
present restorative procedure can be implemented
by closing and/or opening of some components,
as given by the procedure explained earlier.
If the present restorative procedure does not
1940
result in acceptable branch currents and node
voltages in the network, the restorative
procedure with next minimum weight is chosen and
check for branch current and node voltage limits
are carried out as above.
This procedure is
carried out till a feasible interested tree
(restorative procedure) is obtained satisfying
both the current and voltage constraints.
If
none of the above interested trees satisfy the
current and voltage constraints, then some
partial load shedding is to be resorted to for
satisfying the current and voltage constraints
in the reconfigured network.
CONCLUSIONS
In this paper a new network reconfiguration
technique
for
service
restoration
in
distribution networks is proposed. The proposed
method is based on network reduction. All the
interested trees of the reduced network will
directly
give
all
possible
restorative
procedures.
For this purpose a new algorithm
'INTERESTED TREES' is also developed which
directly gives all the interested trees of a
graph without repetition.
This algorithm has
been justified by relevent theorms and Lemmas
which have also been proved.
ILLUSTRATION
REFERENCES
The proposed method is illustrated with a
sample system shown in Fig. 1, which is a part
of a distribution system [8]. This is the same
example chosen to explain the proposed method
in the above section.
In Fig. 2 the dotted
lines indicate the open components and complete
lines indicate the closed components, before
the fault.
The list of possible restorative
procedures along with the components to be
closed and opened for a fault on components 91
and 135 is given in Table 1.
For procedure
number 1 in Table 1, DISTLF solution has been
obtained and it has been observed that none of
the current and voltage containts are violated
in the reconfigured network.
Hence this
procedure can be implemented by closing the
components 139,103
and opening the component
97.
Table 1: Possible restorative procedures for a
fault on components 91 and 135.
S. Components
Components Components Weight
No. in the path
to be
to be
of the
in the reduclosed
opened
path
ced network
1.
139,104,98,103
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
104,98,143,117
139,117,104,98
103,104,98,143
103,104,99,143
139,103,104,109
139,103,104,99
103,98,143,117
139,103,98,117
105,103,98,143
103,98,109,143
139,105,103,98
139,103,98,109
103,104,109,143
139,105,99
139,105,109
105,99,143
105,109,143
139,99,109
139,99,117
139,109,117
99,109,143
99,143,117
109,143,117
139,103
143,117
139,137
103,143
103,99,143
139,103,109
139,103,99
103,143,117
139,103,117
105,103,143
103,109,143
139,105,103
139,103,109
103,109,143
139,105,99
139,105,109
105,99,143
105,109,143
139,99,109
139,99,117
139,109,117
99,109,143
99,143,117
109,143,117
97
97
97
97
97,98
97,98
97,98
97,104
97,104
97,104
97,104
97,104
97,104
97,98
97,104,98
97,104,98
97,104,98
97,104,98
97,104,98
97,104,98
97,104,98
97,104,98
97,104,98
97,104,98
3
3
3
3
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
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M.
Topka,
"Generalized
Alrogithms
for
distribution
Feeder
Deployment
and
Sectionalizing, " IEEE Trans, on Power Apparatus
and Systems, Vol. PAS-99, No.2, PP. 549-557,
March/April 1980.
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Algorithm Including Operating Constraints",
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APPENDIX I
PROOFS OF THE ALGORITHM 'INTERESTED TREES'
Theorm I: Let Gs be a tree of a graph G such
that it contains all interested nodes. Then Gs
is an interested tree, if and only if every
end node of Gs is an interested node.
Proof:
Only if: let Gs be an interested tree.
To prove that every end node of Gs is an
interested node. For if this is not true, let
noninterested node v be an end node of Gs. Then
the edge connected to v can be removed resulting
in another interested tree.
This violates
property 3 of the interested tree showing that
any end node of an interested tree is an
interested node.
If:
Let Gs be a tree of G containing all
interested nodes such that each end node of Gs
is an interested node. To prove that Gs is an
interested tree.
It is obvious that Gs satisfies properties
1 and 2 in the definition of an interested tree.
If property 3 is also true, then Gs is an
interested tree. Thus if Gs violates property
3, then let Gs' be a subgraph of Gs satifying
all the three properties of the definition.
That is at least one edge of Gs has to be
removed to get Gs'. This edge has to be an edge
incident at an end node. For this is not true
Gs' will be disconnected. But if this edge is
removed an interested node is lost showing that
Gs' is not an interested tree. This proves that
Gs satisfies the property 3 of the definition.
Hence the result.
Corollary: A noninterested node cannot be an
end node of an interested tree.
Proof:
This is only a restatement of the
'only if part of the theorm I.
Lemma 1:
Let Gs be a spanning tree of the
reduced graph G' in step 3 of the algorithm.
Let Es' be the set of edges shorted to get the
reduced graph.
Then the set of edges
consisting of the edges Es' and Gs is an
interested tree T.
Proof:
In the algorithm Es 1 contains at least
two edges from each noninterested
node
(procedures I and II). Thus no end node of T
can be a noninterested node. T contains all
interested nodes. For if this is not true, let
v be an interested node missing from T. i.e.,
no edge incident at v is present in T. i.e.,
in Es' as well as in G'. This is not possible
unless all the edges incident at v are
eventually eliminated as self loops. But a self
loop cannot be formed unless all the edges in
a loop except the edge appearing in the self
loop are shorted. This requires an edge at v
to be shorted to create a self loop.
This
implies that there is an edge in Es' which is
incident at v. Thus if v is not present in G',
then it is present as a node of some edge in
Es'. If an edge is not eliminated as self loop,
then it is anyway present in G'. In either case
T contains v. Thus T contains all interested
nodes. Further T is a tree. Therefore from
Theorm I,T is an interested tree.
Lemma 2: In the algorithm INTERESTED TREES,
only two edges have to be shorted at any
noninterested node to generate an interested
tree containing that node.
Proof: Let N be the noninterested node under
consideration.
Let e1, e2 be the two edges
selected for shorting at N. Let T be an
interested tree containing at least three edges
say e1, e2, e3 incident at N. Some of the edges
of this tree are obtained from shorting. Let
Es' be the set of edges falling in this
category.
Let G' be the reduced graph
containing only interested nodes whose spanning
trees are generated in the step 3 of the
algorithm. If G' contains e3, then all spanning
trees containing e3 are generated in step 3 of
the algorithm.
Thus T is also generated.
Similarly if e3 is considered for shorting at
another node then it is not present in G', but
it will be present in Es' of the algorithm and
so it will be present in T.
Thus T is notgenerated only if e3 is eliminated as a self
loop. This implies that e3 forms a loop with
some of the edges in Es'.
But this is not
possible if e3 is present in T (as a tree does
not have loops). Thus e3 cannot be eliminated
as self loop if it is present in T. Hence the
result.
Theorm IJ:
Algorithm INTERESTED TREES
generates all interested trees of the given
graph without duplication.
Proof:
That the algorithm generates
interested trees is proved in Lemma 1. It is
required to prove that it generates all
interested trees without duplication. This is
done by considering two cases as follows:
Case (i): First assume that all the nodes of
the given graph are interested nodes only. In
this case step 3 of the algorithm is executed.
Since an interested tree is a spanning tree
also, all interested trees are generated without
duplication.
Case (ii): Consider a noninterested node say N
of the graph. Let e1, e2,...ep be the set of
edges E incident at N. Any interested tree that
contains N will have at least two edges of E in
it. Thus all interested trees of the graph can
be expressed as union of subsets of trees as
follows:
{T} = e,e2 {T,} U 6,63(62) {T2} V...
...U e1ep(e2>e3,...ep.1) {Tp_,}
Ue e
2 3 ( e l } { Tp+1 } U
e e
i 2"- e p ) { T } •
A.I
1942
where the edges shown in the brackets should be
opened and other edges shown should be shorted
to get a graph which gives interested trees
denoted in the flower backets.
For example, e^p (e 2> e 3
eJ>.1){Tp.1} means
that e^ep are the edges to be shorted and e2,
e3...ep_1 are the edges to be opened to get a
graph which gives the interested trees denoted
by { V , } .
Similarly (e,,e2,e3,.. .ep){T'} means e,,
e2
ep are the edges to be opened to get a
graph
whose interested trees are denoted by
1
{T }. These interested trees do not contain
the noninterested node N.
It may be noted that in the above equation
A.1, the edges to be opened as indicated is
required to avoid repetition.
Let T be any
interested tree of the graph G. If it does not
contain N it falls uniquely in to the group
{T'}. If it contains N, let e M I e12.. . e,. be
the set of edges incident at N and present in
T. Without loss of generality let im > im-1 >
... e> i2 > i1. To generate this tree, the edges
e
n> i2 have to be shorted (Other edges e j3 to
e,. of T need not be shorted in view of Lemma 2)
There is only one term in the equation
corresponding to this. Thus subsets of trees
corresponding to the terms in the equation are
disjoint i.e., no tree is generated more than
once. Further since T is any tree, this shows
that all trees are generated.
Thus the
algorithm generates all trees without repetition
provided it implements the equation A.1
properly.
To save memory,equation A.1 is implemented
as follows:
{T}F = e^CT,} U xe/ej.HT.,} U xe2 (e,) {T3}
U (e,e2) {T4}
{T}x = e, {T,}
A.2
U x (e,)
{T2}
A.3
Equations A.2 and A.3 are implemented in
the algorithm in procedures I
and II
respectively.
Noninterested node 2 from G' is selected in
step 4.
Therefore N=2, EN={1,3,4}.
After
performing the procedure ALL COMBINATIONS I,
and when i=4 is selected, Es4 = {0}, Eo4 =
{1,3,4}. This leads to the combinations Es4 =
{0,2,5}, Eo4 = {1,3,4}; Es5 = {4,2}, Eo5 =
{1,3,4,5}; Es6 = {4,5}, Eo6 = {1,3,4,2}; Es7 =
{0}, Eo7 = {1,3,4,2,5}. Now i = 7 and i = 6
results the graphs G' which are disconnected.
For i=5, x=4, Es' = {2}, Eo' = {1,3,4,5}. The
resulting graph G' obtained after opening the
edges of Eo' from G is shown in Fig. A2.3.
Since all the edges incident at 2 in G are
opened, node 2 is also deleted from G'. Ex =
{2,3,5,7}.
Executing the step 10 of the
algorithm results in Ex = {7} and G' as shown
in Fig. A2.4. Executing step 11 results in Es5
= {0,7,2}, Eo5 = {1,3,4,5}; Es6 = {4,2}, Eo6 =
Fig.A.2.1
Graph G
Fig.A 2.2
F1g.A2.3 Fig.A2.4
Fig.A 2.5
{1,3,4,5,7}. For 1=6, when all edges of Eo6
are opened from G, the resulting graph is
disconnected. For i=5, x=0, Es' = {7,2}, Eo'
= {1,3,4,5}. Executing steps 6,7,8 and 9 of
the algorithm leads to the graph G' shown in
Fig. A2.5.
It has only one edge 6 and no
noninterested nodes of status 1 are present.
So step 2 is executed resulting one spanning
tree of G' which has 6 as its edge. Therefore
the interested tree T1 = {7,2,6} is generated.
Thus proceeding further according to the
algorithm,
all
the
following
remaining
interested trees of the graph G are obtained :
T2 = {2,5,7}, T3 = {2,5,6}, T4 = {2,3,4,6}, T5
= {2,3,4,7}, T6 = {1,4,6}, T2 = {1,4,5,7}, T8
= {1,2,4,7}, T9 = {1,3,6,7}, T10 = {1,3,4,7},
T11 = {1,3,5,7}, T12 = {1,3,5,6}.
APPENDIX-II
ILLUSTRATION OF THE ALGORITHM "INTERESTED TREES"
Consider the example graph given in Fig.
A2.1 in which nodes 1,3,5 are the interested
nodes. This graph is denoted by G. Select the
noninterested node 2 in step 2 of the algorithm.
Therefore N=2, EN={1,3,4}. Next when step 4 is
executed the following combinations are stored
in the stack. Es1 = {0,1,3}, Eo1 = {0}; Es2 =
{2,1}, Eo2 = {3}; Es3 = {2,3}, Eo3 = {1}; Es4
= {0}; Eo4 = {1,3}.
Now in step 5 i=4 is
chosen. Therefore Es' = {<(>}, Eo' = {1,3}, x =
0.
Opening the edges in Eo' from G, G' is
obtained as shown in Fig. A2.2.
G' is
connected.
Since x=0, Step 9 is executed.
Since
Es'={$},
step
2
is
executed.
N.D.R.Sarma is a research scholar working for
his Ph.D. degree in the Deptt., of Elect. Engg.
in IIT Delhi, India. His area of interest is
Graph Theory Applications to Power Systems.
V.C.Prasad is a professor in the Deptt. of
Elect. Engg., in IIT Delhi.
His areas of
interests are nonlinear circuits, CAD of
networks and VLSI, Reliability and Graph Theory.
K.S.Prakasa Rao is a professor in Deptt. of
Elect. Engg., IIT Delhi. His areas of interests
are Power System Planning and Operation Studies,
and Power System Reliability.
V. Sankar is working as lecturer in Elect. Engg.
at J.N.T.U., College of, Engg., . Ananthapur,
India. Presently he is on leave and working
for his Ph.D. His areas of interests are Power
systems, Reliability studies.