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International Journal of

Electrical Power and Energy Systems

Manuscript Draft

Manuscript Number: IJEPES-D-13-01721

Title: Service Restoration for Unbalanced Distribution Networks Using aCombination Two Heuristic Methods

Article Type: Research Paper

Keywords: service restoration, unbalanced distribution network, switches

index, graph-based, three phase load flow

Abstract: In this paper, two heuristic methods are proposed to find

effective and fast solution in unbalanced three phase distribution

networks. Switch selection indices based on analytically approach and

practicable heuristic graph-based method are proposed for solving the

service restoration problem in unbalanced distribution networks. The

problem formulation proposed, consists of three different objectivefunctions: First, minimizing the de-energized customers' load, second,

minimizing the number of switching operation, and finally, customer's

priority. A suitable assignment of switch indices to all tie switches

(ts) in networks are used to find best solution and decrease number of

switching operation. New graph-based approach for finding best

sectionalizes switch (ss) and minimizing voltage drop's amount is

utilized. The validity of these approaches has been tested on the two

unbalanced three phase distribution networks. Results have been presented

for modified IEEE 13-node and IEEE 37-node test case. The fastness and

effectiveness convergence of these approaches helps finding best solution

for service restoration problem.

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Service Restoration for Unbalanced Distribution Networks Using a Combination Two

Heuristic Methods

Meysam Gholami

a,*

, Jamal Moshtagh

b

a,* ,Department of electrical engineering, university of Kurdistan, sanandaj, Iran.

b, Department of electrical engineering, university of Kurdistan, sanandaj, Iran.

(a,m.gholami.67.86@gmail.com,tel:+989181706749)

(b

,moshtagh79@yahoo.com,tel:+988716660073)

ABSTRACT

In this paper, two heuristic methods are proposed to find effective and fast solution in

unbalanced three phase distribution networks. Switch selection indices based on analytically

approach and practicable heuristic graph-based method are proposed for solving the service

restoration problem in unbalanced distribution networks. The problem formulation proposed,

consists of three different objective functions: First, minimizing the de-energized customers

load, second, minimizing the number of switching operation, and finally, customers priority.

A suitable assignment of switch indices to all tie switches (ts) in networks are used to find

best solution and decrease number of switching operation. New graph-based approach for

finding best sectionalizes switch (ss) and minimizing voltage drops amountis utilized. The

validity of these approaches has been tested on the two unbalanced three phase distribution

networks. Results have been presented for modified IEEE 13-node and IEEE 37-node test

case. The fastness and effectiveness convergence of these approaches helps finding best

solution for service restoration problem.

Key wordsservice restoration, unbalanced distribution network, switches index, graph-

based, three phase load flow.

anuscript

ck here to view linked References

mailto:m.gholami.67.86@gmail.commailto:m.gholami.67.86@gmail.commailto:m.gholami.67.86@gmail.commailto:moshtagh79@yahoo.commailto:moshtagh79@yahoo.commailto:moshtagh79@yahoo.comhttp://ees.elsevier.com/ijepes/viewRCResults.aspx?pdf=1&docID=8830&rev=0&fileID=176957&msid={DE9CFA2E-68AA-4381-A3AF-1359C2266D85}http://ees.elsevier.com/ijepes/viewRCResults.aspx?pdf=1&docID=8830&rev=0&fileID=176957&msid={DE9CFA2E-68AA-4381-A3AF-1359C2266D85}mailto:moshtagh79@yahoo.commailto:m.gholami.67.86@gmail.com

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1. Introduction

With significant extension of the modern power distribution networks in the world, the

likelihood of occurrence of fault and then blackout for one or more area will increase.

Therefore, customers satisfaction and service reliability are the important topic where most

of the paper localizes in this issue. Revenue earned by the Power Distribution Companies and

customerssatisfaction is closely depending on reliability in distribution networks. In order to

satisfy users demand and maintain profit of power Supply Company, it is necessary to

restoring power service as soon as possible [1]. Due to a high number of switches, feeders

and branches in typical distribution systems, it is not easy to restore an out-of-service area

solely depending on the past experiences of human operators [2]. Therefore, with the advent

of quick computers and changing technology, to reduce the out of service area as efficiently

as possible, a computer aided decision supports assist the operators. How to arriving a fast

and effective service restoration in power distribution networks (PDNs), considering

unbalanced distribution network is of major concern in this paper. Protection devices in

network detect the fault location, when a fault is occurred in the PDN. After isolating the

fault by operation line switches, the PDN is divided to three sections: First, the upstream

section that is supplied from same feeder, second, the downstream un-faulted section that are

transferred to neighboring feeders according to presented approaches in this paper, and

finally, damage buses and lines that are isolated from network. In service restoration problem,

several issues must also be considered that are described as follow:

Service restoration plan must be restored maximum safe out-of-service loads.

Service restoration plan is implemented by changing switches state in PDNs,

therefore, the time taken by the service restoration depends on the number of

switching operation. Therefore, the number of switching operation should be kept to

minimum as possible.

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Hospital, police station, firehouse, etc, are the highest priority in PDNs. This issue

must be considered in service restoration plan that the supply must be restored to

highest priority customers.

In each PDN, most important constraint is radial structure due to various reasons,

such as ease of fault location detection, fault isolation and protective devices

coordination. When the structure of the network is changed during the service

restoration, this constraint must be kept on.

Buses voltage, lines current and elements loading also changes during the service

restoration plan. Therefore, it is important that these constraints dont cross their

respective operational limits.

Customers satisfaction and reliability ofdistribution networks are closely dependent

on interruption frequency and duration. Therefore, the restoration plan runtime must

be minimized for finding a quick solution.

In past years, many methods have been proposed to find solution for restoration problem

from different perspective. Considering complexity PDNs, analytic method for solving the

restoration problem can hardly be applied. Therefore, heuristic search method [3-8] or expert

system approach [9] have been adopted. In [10] G-net inference mechanism with operation

rules is applied. In [11] Petri Net combined with a rule-based expert systems have been

applied to implement the service restoration. In [12] fuzzy cause-effect networks are used to

model the heuristic knowledge inference involved in the restoration plan. In [13] a fuzzy

decision-making approach has been applied to determine the most desirable restoration plan

with consideration different practical factors, but fuzzy method doesnt guarantee the optimal

solution. In [14] non- dominated sorting genetic algorithm-II (NSGA-II) for solving the

service restoration problem is presented and to reduce the software runtime, a faster version

of NSGA-II has been implemented. In [15] mathematical programming is presented to

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reconfigure the network to restore un-faulted section of the system. In [16, 17] combination

methods are applied. In [16] objective functions are modeled with fuzzy sets, and then

optimization problem is solved by the Genetic Algorithm (GA). [17] consist of two stages:

First, the fuzzy multi criteria evaluation, and afterward, the Grey relational analysis. In [18],

service restoration with Load curtailment of in-service customers via direct load control has

been implemented. In [19], reliability assessment of complex radial distribution systems is

presented, however this paper simplifies the restoration plan. In [20, 21], restoration problem

in distribution network with dispersed generation is implemented. In this paper, a fast and

effective methods based on two new heuristic algorithms for service restoration in

unbalanced PDNs ispresented. Unbalanced distribution network, customers priority, buses

voltage, lines current, equipments loading and minimum software runtime consideration, are

the main features of the proposed method.

This paper is organized as follows: Section 2 describes the problem formulation of a

typical restoration problem. In section 3, indices for ranking the networks switch are

described. In section 4, graph-based method is described. Section 5 reviews two new heuristic

algorithms for service restoration. Section 6 briefly describes three-phase load flow program

for fast response to the network change inducted by system reconfiguration. Section 7 shows

a numerical example to demonstrate the fastness and effectiveness of the proposed methods

and the conclusion are drawn in section 8 finally.

2. Problem formulation

Service restoration in unbalanced distribution networks considering customers priority

are formulated as multi-constraint and multi-objective optimization problem. In this paper,

three different objective functions are presented. Maximizing total load to be restored,

minimizing the number of the switching operations and maximizing priority load restored are

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these objective functions. Besides, important constraints consists of network radial structure,

buses voltage, lines current, equipments loading have also been considered in this paper.

Objective function briefly:

(1)

(2)

(3)

Where

Energized loads in network;

Total buses are restorable;

Buses with high priority those are restorable;

Number of switching operation;

Constraints:

1) Radial network structure should be maintained.

2)

Bus voltage limits (for all buses):

(4)

Where

Minimum acceptable bus voltage;

Voltage at bus k, phase p;

Maximum acceptable bus voltage.

3) Line current limits (for all lines):

(5)

Where

Minimum acceptable line current;

Current in line j, phase p;

Maximum acceptable line current.

tNk

kLmax

HPNk

kLmax

opNmin

kL

tN

HPN

opN

maxmin

k

p

kk VVV

min

kV

p

kV

max

kV

maxmin

j

p

jj III

p

jI

max

jI

min

jI

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4) Equipment loading limits (for transformers):

(6)

Where

Loading for i transformer;

Rated loading for i transformer.

Operational constraint can be obtained from three phase load flow calculation. In actual

practice, the minimum limit of the line current should not be taken and, in fact, this limit has

been taken as zero [14]. In this study, we are used two heuristic approaches based on switch

indices and graph-based for finding best solution to restore maximum total customers in de-

energized area. In most restoration plan, there are several plans available for a restoration

problem, however, how to select best plan, two methods must describe. These methods are

presented in next sections.

3. Switch ranking

In this paper, two switch indices for best selection have been used. the base of the

proposed algorithm for these indices is voltage drop. A first and most important index is VD

that is proportionate with voltage drop between substation bus and primary side of each tie

switch (ts). For each ts, VD is defined as follow:

(7)

Where

Sum of active loads between substation bus and primary side of tie switch i, for each

three phase;

Sum of reactive loads between substation bus and primary side of tie switch i, for

each three phase;

Sum of real segment of positive impedance sequence of lines between substation bus

max

itr

itr

max

ii trtr

cbapV

XQRPVD i

p

ii

p

i ,,

p

iP

p

iQ

iR

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and primary side of i tie switch;

Sum of imaginary segment of positive impedance sequence of lines between

substation bus and primary side of i tie switch;

and V is substation voltage.

This index is shown in Fig.1. Suppose that one fault took place at point A. Therefore,

area1 is downstream un-faulted area and ts1 is one candidate switch for service restoration

implementation. For ts1, VD is obtained from node number 5, 6, that is proportionate with

direction1 (dir1). ts3 is another candidate switch for service restoration implementation. For

ts3, VD is obtained from node number 10, 12 and 13 that is proportionate with direction2

(dir2). A second index (Zpath) is direction impedance (per-unit) for lines lying in the path

between the secondary side of each ts and end buses in network. For each ts, this index is

defined as follow:

(8)

Where Zbis positive impedance sequence of branch b and Nbris lines lying in the path

between the secondary side of each ts and end buses in network. This index is shown in Fig.1.

Suppose that one fault took place at point B. Therefore, area2 is downstream un-faulted area

and ts2 is one candidate switch for service restoration implementation. For ts2, Zpathis

impedance of direction3 and direction4 (dir3 and dir4). These utilized indices, help both

reducing the solution search space and ranking switches to find best solution for service

restoration plan. Indeed, when search space is reduced then runtime software will decrease

and this issue makes guarantee to fast service restoration implementation. In next section

graph-based method is described and how to utilize these methods is described in section 5.

4. Graph-based method

brNb

bpa th ZZ

iX

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Before description this method for finding the minimum voltage drop in network (tree or

graph) we must define three groups. For any bus (node) belongs to the graph, the following

definition is valid:

Sub-graph D, which consists of nodes that have already been added to the graph.

Sub-graph UD, which consists of nodes that have not been added to the graph.

Sub-graph B, which consists of branches that can connect nodes from sub-graph D.

When one branch from tree with minimal weights is selected in each iteration, one node

(node S) of each branch is in sub-graph D and the other node (node R) is in sub-graph UD.

Therefore, node R is added to the sub-graph D and deleted from sub-graph UD. This method

is shown in Fig. 2. This approach is utilized for finding best sectionalizes switch (ss) after

finding best tie switches (ts). The evaluation of weighting coefficient has been described in

follow:

(9)

Where

X is branch state (0 for close branches and inf for open branches), Zbis positive sequence

impedance of branch b, Nbris branches lying in the path between the secondary node (node

R) of branch is in sub-graph UD and substation node. The starting state and the major steps in

the first iteration has been described as follows:

Starting state:

1, 2 are branches between sub-graph D and sub-graph UD.

First iteration (selection of branch with minimum weight, (a) in Fig. 2):

Weighting coefficient for branch 1 and 2:

W1= X1+Z1 and W2= X2+Z2

Nbrb

bZXW

B

UD

D

2,1

12,11,10,9,8,7,6,5,4,3,2

1

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Therefore, branch with minimum of W is selected (suppose this is W1).

Updating sub-graphs:

Second iteration (selection of branch with minimum weight, (b) in Fig. 2):

Weighting coefficient for branch 2, 3 and 4:

W2= X2+Z2 , W3=X3+Z1+ Z3 and W4= X4+Z1+ Z4

Therefore, branch with minimum of W is selected (suppose this is W2).

Updating sub-graphs:

Updating D, UD and B will be done for next iteration. When sub-graph UD was emptied,

the iteration procedure is finished and after last iteration, all branches belong sub-graph B

must be opened. This state that is final structure is shown in (f), Fig. 2.

5. Service restoration algorithm

To describe the problem of multi-objective service restoration in distribution networks,

the faults scenario must be described. When a short-circuitsfault is occurred on the feeder,

circuit breaker at the outset of feeder is operated to clear the fault. All boundary line switches

are operating to isolate the faulted area. The feeders circuit breaker is then closing to restore

the upstream customers. For the downstream area, best switch indices for best switches

selection based on first heuristic approach is implemented. This approach is described in this

section. The proposed approach is calculated fast and implemented using remotely controlled

switches in unbalanced distribution networks. In this paper, three objective functions are

prioritizing as: 1) maximizing the amount of total load to be restored, 2) minimizing the

B

UD

D

4,3,2

12,11,10,9,8,7,6,5,4,3

2,1

B

UD

D

5,4,3

12,11,10,9,8,7,6,5,4

3,2,1

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number of the switching operation and 3) customers priority consideration. Proposed

algorithm considering five steps are described in follow.

Step 1) isolation the fault;

Step 2) creation the candidate tie switches list;

Step3)selection one ts due to first proposed algorithm, three phase unbalanced load flow

implementation, and network constraints survey. If no constraints violation

exists, go to step 5;

Step4)selection next ts due to first proposed algorithm, selection respective sectionalize

switch (ss) due to graph-based method, three phase unbalanced load flow

implementation and network constraints survey. If no constraints violation exists

go to step 5, else repeat this step;

Step 5) return best service restoration plan.

Step 1) isolating the fault

When a fault is occurring in each PDN, faults line, sending and receiving bus sides for

this line are detected. The adjacent buses and lines are wended in each direction sequentially

and to clear the fault, first circuit breaker in these directions is founded and operated.

Therefore, feeder that fault has been occurred on it and feeders circuit breaker is detected.

For fault isolation, the adjacent buses and lines are wended in each direction sequentially.

The first switches in each direction is found and operated. Therefore three areas are formed in

network: First, the upstream out-of-service area that is first restored by closing the feeders

circuit breaker, second, the damage area that must been repaired, and finally, the downstream

un-faulted area then is transferred to the neighboring feeders according to the proposed

algorithms.

Step 2) creation candidate switch lists

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Candidate tie switches are identified from energized feeder that can connect directly into

the out-of-service area. Candidate sectionalizes switches (ss) that are located in the out-of-

service area and identified from graph-based method. For each candidate ts, VD and Zpathare

obtained. In this section, new weighting factor is utilized to converts of these two indices into

an equivalent single index. Final index has been described as follow:

(10)

Where and are two weight factors, that have two continues amounts between 0 and 1

(0< and

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4.4) if an overload or voltage violation exists, open previously ts, close previously ss and

repeat step 4.1, 4.2, 4.3, until iteration procedure is finished. If no restoration plan exists, go

to step 4.5.

4.5) first switch pair (ts, ss) in the step 4 is operated and for restoration procedure

continuance, this step (step 4) is repeated.

Step 5) restoration plan (final step).

In this step sequence operations for ts and ss that have been selected is described. First, all

ss that have been selected must be opened and afterward, all of ts that have been selected

must be closed. To illustrate the fastness and effectiveness of the proposed algorithm, two

unbalanced distribution networks consists of: modified IEEE 13-node and IEEE 37-node

have been tested and presented in section 7.

6. Fast load flow technique

After network restoration, the three phase unbalanced distribution load flow has to be

calculated to examine the voltage, current and capacity constraints for feeders, lines and

elements with additional of new load points. In this paper, we are used fast load flow

technique for fast service restoration. For receipt more information about this technique,

please refers to [22]. In this section, summary of this technique [22] is described. The

fundamental idea discussed here is how to obtain the power flow solution by using the

elements of a unique quasi-symmetric matrix called TRX in the iterative process. The

proposed TRX matrix constitutes a complete database by including information of network

topology structure as well as branch impedances of the distribution feeder. The method is

described in six steps: data preparation, initialization, current and voltage calculations, quasi-

symmetric matrix calculation, and convergence process. This formulation is given including

three phase line shunt-admittances and loads are modeled as constant power. The input data

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is given by three-phase per-unit node-branch oriented information. The basic data required is:

three-phase injected powers and sending and receiving nodes of a given line impedance. The

branch impedances are given as a rectangular 3nx3 phase impedance matrix Zabc.

(11)

Where is the 3-phase matrix impedance corresponding to ij line section:

(12)

Shunt admittances modeled by a rectangular 3nx3 matrix Yabc:

(13)

Fig. 3 shows a radial distribution network with n+1 nodes, and n branches and a single

voltage source at the root node 0. Under the unbalanced approach, nodal power injection

vector S is given per node and per phase.

(14)

At given iteration k, the relationship between injected currents Ikand branch currents Jkis

set through an upper triangular matrix T accomplishing the Kirchhoff Current Laws (KCL) as

follows:

(15)

(16)

Update voltage:

(17)

][ )1(01 nnabc

ij

abcabcabc ZZZZ

ij

abcZ

ij

cc

ij

cb

ij

ca

ij

bc

ij

bb

ij

ba

ij

ac

ij

ab

ij

aa

ij

ab c

ZZZ

ZZZ

ZZZ

Z

][ )1(01 nnabc

ij

abcabcabc YYYY

cbap

jQP

jQP

jQP

S

S

S

S

T

npnp

ipip

pp

T

np

iP

p

,,

11

1

k

abc

k

abc ITJ .

cbap

k

ip

ij

ppk

ip

ipk

ip VYV

SI

,,

*

kabcabcabc

Tabcabc

kabc ITZTVV ...01

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Where, Vabc-0 is initial voltage vector.

Convergence check-in and final calculations:

(18)

7. Numerical example

To illustrate effectiveness of proposed algorithms, two different unbalanced distribution

networks have been considered. These methods have been coded in MATLAB software. this

section, is demonstrated performance of method. When a fault is occurred on network,

protection devices are operated immediately for detecting and isolating the fault. In this

paper, total numbers of switching operations for isolating the fault and service restoration are

obtained. The impedance of lines, phase impedance matrix and phase admittance matrix are

calculated from data of these networks in [23]. In some distribution networks, some loads are

modeled as distributed load; therefore, if every load point is modeled as a node, then systems

will have a large number of nodes. Thus, these loads are modeled as spot load in this paper.

In these test cases, one, two, or three phase loads with wye or delta connections can exist. In

this paper some assumptions for unbalanced distribution networks have been considered that

are described as follow:

1) All load are modeled based on constant power model;

2) The Regulator and Capacitors components is removed from networks;

3) For all branches in networks, one switch in send side of branch is considered;

4) Some tie switches are introduced in the networks for illustrating restoration plan.

5)

Amount of loads are modified in order to performance restoration plane and regard

networks constraints.

6) For conversing distributed loads to spot loads, virtual nodes 2, 3 are introduced in

IEEE 13-node network.

cbapniVVk

ip

k

ip ,,,...,11

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Fig. 4 and 5 are shows the one-line diagram of modified IEEE 13-node and IEEE 37-node

networks respectively. Introduced tie lines (or tie switches) for service restoration, actual and

with loss are considered. In this paper, two weighting (, ) are considered 0.8 and 0.2

respectively. Voltage magnitude range must be within limits of 0.95 and 1.05 per-unit. For

demonstrates validity of service restoration algorithm, fault simulation in several locations

are considered.

7.1.Service restoration results

Table 1 and table 2 displays the restoration algorithms results for IEEE 13-node and IEEE

37-node unbalanced distribution networks respectively. In these tables, following information

is provided:

Total number of switching operations for isolating the fault and service restoration;

Runtime software for service restoration plan.

When fault is occurred on the branch 2-3 in IEEE 13-node network, two switch

operations have been required to isolate the fault and three switch operations have been

required to full service restoration implementation. Bus number 3 is damaged bus that cant

be restored. Bus numbers 4, 5, 10, 11, 12, 13 and 14 are de-energized bus that must be

restored from neighboring tie lines (ts 6-11 and 9-12). For removing loop in network, one

switch must be opened; therefore switch number 4-10 is obtained from graph-based method.

Voltage magnitude per-unit for fault on this branch is shown in Fig. 6. This Fig shows that

voltage for all buses are in definition limits. When fault is occurred on the branch 4-12 in

IEEE 13-node network, three switch operations have been required to isolate fault. In this

case, bus number 12, 13 cantbe restored.

When fault is occurred on the branch 3-4 in IEEE 37-node network, two switch

operations have been required to isolate the fault and three switch operations have been

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required to full service restoration implementation. Voltage magnitude per-unit for all bus for

this case is shown in Fig. 7.

8. Conclusions

In this paper, fast service restoration in unbalanced PDNs, with consideration to priority

customers as multiple objective functions consists of: 1) maximizing the amount of total load

to be restored, 2) minimizing the number of the switching operation, 3) customers priority

considering are implemented. For this work, we are used two new heuristic algorithm based

on two important indices and graph-based method. Core of the proposed first algorithm is

voltage drop between candidate ts and substation bus and second algorithm is based on

graph-based method to minimize voltage dropping in networks. Fast load flow technique

based on a real quasi-matrix [22] has been utilized. Finally, the proposed algorithm has been

implemented and tested on two unbalanced distribution networks and results that have been

obtained, summarized as follow:

1)

Total number of switching operation for isolating the fault and service restoration;

2) Buses that not restored.

3) Sequence operation for selected ts and ss.

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[23] Radial distribution test feeders.

Table 1: Restoration Results for Modified IEEE 13-Node Network.

casefault

location

switch operation to fault

isolation

switches operation to

SR

runtime

(S)

1 4-10 4-10, 10-11 6-11 0.35

2 2-3 2-3, 3-4 6-11, 9-12,4-10 0.52

3 4-12 4-12, 12-14, 12-13 5-14 0.3

4 1-8 1-8, 8-9 9-12 0.39

Table 2: Restoration Results for Modified IEEE 37-Node Network.

casefault

location

switch operation to fault

isolation

switches operation to

SR

runtime

(S)

1 2-26 2-26, 26-27 36-27 0.45

2 27-30 27-30, 30-31, 30-33 26-35 0.43

3 2-23 2-23, 23-24, 23-25 20-25 0.35

4 3-4 3-4, 4-5 27-36, 22-16, 7-8 1.12

5 5-6 5-6, 6-14, 6-7 22-16, 36-12, 9-10 0.96

6 7-8 7-8, 8-9, 8-15 36-12, 22-16 0.86

7 3-19 3-19, 19-20 25-20 0.52

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Fig.1. A 16-bus distribution network

.

Fig. 2. Example of graph-based method.

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Fig.3. The branch and node numbering of a radial distribution network.

Fig.4. modified IEEE 13-node network.

Fig.5. modified IEEE 37-node network.

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Fig.6. voltage after restoration for modified IEEE 13-node network.

Fig.7. voltage after restoration for modified IEEE 37-node network.