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Ordinal Optimization for DynamicNetwork ReconfigurationR. El Ramli a , M. Awad a & R. A. Jabr aa Department of Electrical and Computer Engineering, AmericanUniversity of Beirut, Beirut, Lebanon

Available online: 31 Oct 2011

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Electric Power Components and Systems, 39:1845–1857, 2011

Copyright © Taylor & Francis Group, LLC

ISSN: 1532-5008 print/1532-5016 online

DOI: 10.1080/15325008.2011.615801

Ordinal Optimization for Dynamic

Network Reconfiguration

R. EL RAMLI,1 M. AWAD,1 and R. A. JABR1

1Department of Electrical and Computer Engineering, American University of

Beirut, Beirut, Lebanon

Abstract Motivated by the challenge of efficiently reconfiguring distribution net-

works for power loss reduction, this study presents an approach for finding a minimumloss radial configuration for a power network using ordinal optimization. Ordinal

optimization relies on order comparison and goal softening to make the problemsolution easier and the computation more efficient. The successful application of

ordinal optimization to such a complex optimization problem required the investigationof several algorithmic parameters. The solution algorithm was implemented in a

software package, where an acceptable solution is considered good enough if it isin the top m% of the solutions with a probability P. Testing it on 33- and 136-bus

systems, minimal power loss results were obtained on the 33-bus system that are in thetop 0.03% of the search space. Comparing the experimental results with other recently

published methods showed the effectiveness of ordinal optimization for minimum losscalculations and motivated further studies in smart-grid-like scenarios, where the

results obtained for different load levels were in the top 0.13% of the search space.

Keywords network reconfiguration, optimal power flow, optimization, radial distri-bution networks

1. Introduction

In an era when energy efficiency is becoming of concern, several studies are being

conducted by researchers in order to obtain the most efficient and least expensive means

of generation, transmission, and distribution of electric power with special emphasis on

distribution, since studies have shown that at peak operation up to 5% of the generated

power goes in line losses of distribution networks [1]. One of the most effective methods

of loss reduction in distribution systems is network reconfiguration, which allows load

transfer from heavily loaded feeders to lightly loaded ones, with, of course, the funda-

mental research question remaining as how to effectively identify the optimal network

configuration that results in minimal power loss.

Among the early research on network reconfiguration, Merlin and Back [2] proposed

an optimization and heuristic method to obtain an exact solution for the minimum loss

spanning tree. Although all system constraints were neglected, the problem solution was

still demanding from a time perspective. Shirmohammadi and Hong in [3] also introduced

a heuristic algorithm where, first, all switches were closed, and then a sequential switch

Received 10 February 2011; accepted 16 August 2011.Address correspondence to Prof. Mariette Awad, Department of Electrical and Computer

Engineering, American University of Beirut, P.O. Box 11-0236, Riad El-Solh, Beirut 1107 2020,Lebanon. E-mail: mariette.awad@aub.edu.lb

1845

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1846 R. El Ramli et al.

was implemented in such a way that optimal flow was obtained while accounting for

voltage and current constraints. To maintain a network radial configuration, Civanlar

et al. [4] used an empirical formula to find losses and a branch exchange method in

which the opening of one switch required the closing of another. This method achieved

a reduction in losses but not necessarily overall loss minimization. Improving further on

Civanlar’s work, Baran and Wu [5] and Castro and Watanabe [6] implemented variations

in the power flow computation, guaranteeing only local optimal solutions. Brute force was

used in [7], and a global optimal solution was always achieved. Being a rather exhaustive

approach, it is limited to moderate-sized networks because the number of configurations

increases exponentially with the number of network branches. A different method was

introduced in [8], where a distance measurement technique was adopted that first found

a loop and then determined a switching operation to improve load balance; however, this

technique still resulted in a near-optimal solution.

Many researchers used a genetic algorithm (GA) for solving the network recon-

figuration problem. For instance, Nara et al. [9] suggested a simple GA that encodes

the switching status in strings and adopts the system losses as a fitness function. They

obtained a minimal loss solution, but the algorithm run-time was long. In [10], a core

schema genetic shortest-path algorithm was suggested for large-scale distribution net-

works. Claiming convergence to a global solution by searching among the local optima to

limit the working space, the voltage constraints were considered in the searching process

but were not verified after obtaining the solution. In [11], a combination of GA and

load balance index (LBI) was used, where the GA obtained the number of reconfigured

networks and the LBI selected the network with the least LBI as the optimal solution.

A chromosome coding for optimal reconfiguration was proposed in [12]. Selecting a

minimum length chromosome, this method maintained network radiality and proved to

be effective for large distribution networks. Another meta-heuristic method that was

recently introduced [13] is a tabu search algorithm consisting of a dynamic tabu list

of variable size and a random multiplicative move in order to enhance convergence to

the global solution. According to the test performed on a single-substation system, this

method was successfully applied to different load levels, and it improved node voltages

and avoided local optima.

The researchers in [14] recently solved the reconfiguration and capacitor allocation

problem using mixed-integer non-linear programming and a primal-dual interior-point

method. The Lagrange multipliers were used as sensitivity indices for reconfiguration.

The mixed-integer solution time can, however, be prohibitive for large systems due to its

combinatory nature that requires a large number of simulations. Sivanagaraju et al. [15]

showed an improvement in the LBI as compared to GA with their proposed discrete

particle swarm optimization technique (DPSO) for loss reduction and load balancing.

To improve on the solution convergence time that is affected by the large number of

non-radial cases, the DPSO algorithm was modified in [16] by adding a new variable

expression that resulted in faster convergence and less memory requirement.

In [17], ordinal optimization (OO) was applied for system reconfiguration for a 16-

node system that was expanded by dividing each branch to obtain systems of 32, 48,

64, 80, and 96 branches. Unlike the uniform sampling adopted in this article, the set

of chosen designs in [17] was randomly sampled from the entire space. The ordered

performance curve (OPC) obtained was U-shaped because the author considered all

possible configurations, even the non-radial ones, which is also different than what is

proposed in this work. Thus, to overcome the possibility of accepting an unfeasible case

(non-radial configuration), a correction in the final step was employed in [17], which

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OO for Dynamic Network Reconfiguration 1847

checks the topology of the configuration to make sure the system is radial. The error

between the exact and the OO solution ranged between 0.46% (for a search space of size

4960) and 4.18% (for a search space of size 82,160).

The goal of this study is to identify the configuration of an arbitrary distribution

network that results in minimum system losses while considering the physical and

technical constraints of the power network.

1.1. Contribution of this Article

There are many contributions that this article makes to the field of reconfigurable power

networks. First, a computationally efficient algorithm was implemented for generating

all the spanning trees. Thus, all radial reconfigurations were considered in this study.

Second, a computationally fast crude model was proposed for estimating network power

losses based on the B-matrix and Sherman Woodbury formulas. Third, a computationally

efficient software module for OO was developed. The successful application of OO to

the reconfiguration problem required from a thorough investigation of several algorithmic

choices, such as the level of error associated with the crude model, the OPC shape, and the

size s of the selected subset of choices, each of which is considered an essential research

output that contributes to the success of the OO methodology for network reconfiguration.

The rest of this paper is organized as follows. Section 2 gives a general problem

formulation, and Section 3 describes the OO technique. In Section 4, the spanning tree

generation algorithm using elementary branch exchange is introduced, while Section 5 de-

scribes the proposed crude model. The basic algorithmic choices for OO are discussed in

Section 6, and experimental results are compared with previous published techniques

in Section 7. While Section 8 discusses the potential application of OO for dynamic

network reconfiguration in the presence of variable loads, Section 9 concludes the study

with planned future work.

2. Problem Formulation

The reconfiguration problem in a distribution system consists of finding the best network

configuration, by determining the switching states of ties that are normally opened and

sectionalizing switches that are normally closed, such that power losses are minimized.

Based on the concept of soft-computing, where a good-enough solution is acceptable in

computationally complex situations, OO could be applied to solve the power network

reconfiguration problem. For illustration purposes, consider a moderate system, such

as the 33-node system consisting of 33 nodes and only 5 ties [5]. Such a network has

(according to Eq. (2) in Section 4) 50,751 possible radial configurations or spanning trees,

which would make exact power loss calculations for all these scenarios computationally

demanding. In practice, distribution systems can be much larger than 33 nodes, thus

resulting in a larger number of possible configurations. The computational complexity of

the problem motivates the use of the OO technique and shows the need to implement a fast

and efficient method for this task. OO is based on the idea that the relative order (instead

of the cardinal value) of the performance of different alternatives in a decision problem is

robust with respect to estimation noise. OO narrows the search for optimum performance

to a good enough subset in the design space instead of estimating the accurate values of

the system performance. This implies that if a set of alternative designs is approximately

evaluated and ordered according to a crude model, then there is a high probability that

the actual good alternatives can be found in the top s estimated choices. As an example,

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1848 R. El Ramli et al.

consider the limiting case, where the estimation noise associated with an approximate

evaluation has infinite variance; i.e., the top s alternatives are randomly picked. Moreover,

assume that the search space has N D 1000 alternatives and the actual good-enough

alternatives are considered to be the top 50 (g D 50). By blindly picking s D 86 samples

from the search space, the alignment probability that at least one good-enough alternative

(k D 1) is in the 86 samples is given by Eq. (1) [18]:

AP.k D 1/ D

min.g;s/X

iDk

g

i

!

N � g

s � i

!

N

s

! D

50X

iD1

50

i

!

950

86 � i

!

1000

86

! Š 0:99: (1)

This implies that if any alternative in the top 50 is considered satisfactory, there is no

need to do an exhaustive search to identify the good solution. The previous example

demonstrates that more than a ten-fold reduction in the search space is achieved using

OO. Moreover, the crude model is usually constructed such that the ordering according

to the approximate evaluation is biased in favor of the actual good alternatives. Therefore,

the number of samples (s D 86) of the previous example is an upper bound on the size

of the selected subset that contains at least one good-enough alternative with a 99%

chance.

In this study, given the initial network configuration and the number of tie-lines, all

the possible radial configurations or spanning trees will be generated using an elementary

tree transformation algorithm. Then with 1000 trees uniformly sampled from the search

space, the performance of these configurations will be estimated using a crude model.

The power losses for these top s designs identified by the crude model will be accurately

evaluated to finally choose the design with the least power loss.

3. OO

OO is based on two tenets stating that the optimization of complex problems can be

made much easier by order comparison and goal softening [19]:

Order comparison: “Order” is much more robust against estimation noise

as compared to “value.” That is, it is much easier to estimate whether one

design is better than another than to find the differential performance of the

two designs. The error in selecting a design as superior to another using noisy

estimates of design performances decrease rapidly as the difference between

their true performances increases.

Goal softening: For many practical problems, it is enough to settle for a

“good-enough” solution instead of insisting on finding the “best.” In fact, ex-

act optimization is computationally too expensive for many real life problems.

The following terms are defined:

‚ is the search space of optimization variables;

‚N is the set of N chosen designs;

N is the number of designs uniformly chosen from ‚;

G is the good-enough subset, usually the true top g designs in ‚N ;

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OO for Dynamic Network Reconfiguration 1849

S is the selected subset, usually the estimated top s designs in ‚N ;

G \ S is the set of truly good-enough designs in S ;

AP is the alignment probability = PrŒjG \ S j � k�, the probability that there are actually

k truly good-enough designs in S ; and

k is the alignment level.

The procedure for the practical application of OO to complex optimization problems

becomes as follows.

Step 1: Uniformly sample N designs from ‚ to form ‚N .

Step 2: Estimate the performance of the designs in ‚N using a crude and compu-

tationally fast model.

Step 3: Estimate the error level in the crude model as large, moderate, or small.

Step 4: Order and plot the OPC based on the crude model evaluation. Note that

there can only be five types of this monotonically increasing curve: flat,

U-shaped, neutral, bell, and steep.

Step 5: Choose the size of the good-enough subset g, the required alignment level k,

and the corresponding alignment probability AP .

Step 6: Based on the choices in Steps 3–5, use the universal alignment probability

table [18] to determine the size of the selected subset s. If N > 1000, the

analytical results in [20] are also required to estimate s.

Step 7: Select the estimated top s designs from Step 2 to form the selected subset S .

Step 8: OO theory ensures that S contains at least k truly good-enough designs

with a probability level no less than AP . Evaluate the designs in S using an

accurate model to determine the good-enough solution by picking the best

solution from the set S .

In [19], a theoretical foundation of the OO method was provided by showing that the

alignment probability converges exponentially with respect to the number of replications

and with respect to the sizes of the good-enough and selected subsets.

4. Generation of All Spanning Trees

In order to obtain all the possible network configurations that are needed by the OO

technique, all the possible trees were generated from the given graph of a network.

The elementary tree transformation method (replacement of one branch) proposed by

Mayeda and Sechu [21] was used because it does not produce any non-tree (that is

non-radial) configuration, nor does it generate duplicate configurations. Once the space

of spanning trees is obtained, OO was used to identify the minimal loss spanning tree,

which represents the optimal network reconfiguration.

The total number of trees that can be generated from a given graph can be calculated

using the determinant of a Laplacian matrix [22]. As shown in Eq. (3), the Laplacian

matrix is equal to the degree matrix minus the adjacency matrix. One row and one column

from this matrix is removed, and the determinant is calculated, which represents the total

number of spanning trees.

number of all spanning trees D determinant (reduced Laplacian), (2)

where

Laplacian matrix D degree matrix � adjacency matrix. (3)

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1850 R. El Ramli et al.

The following steps outline how all possible trees were generated.

Step 1: Find an initial tree t0 from the graph H , where H is the graph obtained by

closing all the ties of the network.

Step 2: Find the fundamental cut-sets [23] of all branches with respect to t0. The

fundamental cut-sets are obtained by finding the transitive closure using

the Warshall algorithm and comparing it with graph H .

Step 3: Form a set of trees T1 by replacing a branch from t0 by a branch from the

fundamental cut-set of the corresponding branch. All the trees in T1 are of

distance one from t0.

Step 4: Find the intersection of the fundamental cut-sets of all branches of all the

trees in T1 with the fundamental cut-set of the branch to be replaced and do

the replacement as in Step 3 to obtain a set of trees T2.

Step 5: Go to Step 4, as long as there is intersection between the fundamental

cut-sets.

Step 6: All possible trees in H are those in the sets T1; T2; T3; T4; T5; : : : :

For larger networks, the total number of possible spanning trees is very high, for

example, the 69-node system of [24] has in the order of 400,000 possible spanning trees.

Therefore, the algorithm of generating all spanning trees discussed above was modified

in order to uniformly sample trees during the tree generation phase; this resulted in a

more efficient computing approach and less storage requirements.

5. Crude Model

As stated before, because the OO procedure requires a crude and computationally fast

model to estimate power losses, a model based on the B-matrix loss formula is proposed,

as shown in Eq. (4):

B D A�1M T LC MA�1; (4)

where

PG is the generated power at every node of the system,

PD is the load demand at every node of the system,

M is the line–bus incidence matrix,

Lc is the diagonal matrix of the lines’ conductance,

Ybus is the matrix of nodal admittances of the buses, and

A is �imag.Ybus/.

The B-matrix loss formula estimates the losses as function of the transmission network,

distribution of loads, and generation levels [25]:

PL D P TD BPD � 2P T

D BPG C P TG BPG : (5)

The system losses depend strongly on matrix B , which is known as the B-coefficient

and is itself strongly dependent on the system topology.

Calculation of the B-matrix involves the evaluation of A�1. However, for large

space problems, it is not efficient to calculate the inverse of matrix A every time in a

dynamic environment. Therefore, the matrix inversion lemma [26] (also known as the

Sherman-Morrison formula) was used to find the inverse of A.

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OO for Dynamic Network Reconfiguration 1851

The matrix inversion lemma states that if matrix A can be written in the form

A D E C CDT ; (6)

then the inverse of A is given by

A�1 D E�1 � E�1C�

Im C DT E�1C�

�1DT E�1; (7)

where E is an n � n non-singular matrix whose inverse is known. In this case,

� E is the imaginary part of Ybus of a previous configuration network, and n

represents the number of nodes in the network;

� A is an n � n non-singular matrix whose inverse is to be calculated;

� CDT is the incremental change in the Ybus matrix due to a branch exchange; and

� Im is an identity matrix of size m, where m is the number of branch exchanges

performed to obtain a new configuration from its parent one.

The main reason for using the matrix inversion lemma is because for the same network,

matrix A will not differ much from one system reconfiguration to another. Only the

inverse of the initial matrix A corresponding to the first configuration is to be calculated,

while the other inverse matrices for the different system configurations will be computed

iteratively using the inversion lemma.

6. Algorithmic Choices

Because the successful application of OO to a complex optimization problem requires

the investigation of several algorithmic choices, this study’s selection is detailed in the

following subsections.

6.1. Size of the Reduced Search Space

For illustration purposes, the same 33-node system of [5] was considered. As mentioned

earlier, this system has 37 lines. By running the algorithm for generating all the spanning

trees, a total of 50,751 trees were obtained. An exact power flow method was used

to obtain power losses for every tree. Then the power losses were ordered, the values

were normalized, and they were plotted versus the number of trees. The graph obtained

in Figure 1 shows that the normalized OPC of the complete search space is flat. This

indicates that any solution in the top 1% is good enough. If N D 1000 uniformly sampled

trees are chosen, then the probability that at least one of the sampled solutions is in the

top 1% is

1 � .1 � 0:01/1000 � 0:99999:

6.2. Type of OPC

Obtaining the shape of the OPC of the whole search space required an exact calculation

of the power losses for every tree in this space. However according to OO, the OPC of the

reduced search space must be obtained, i.e., for N D 1000. At this stage, the OPC should

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1852 R. El Ramli et al.

Figure 1. Normalized OPC for the 33-node system considering all possible reconfigurations.

be obtained from the crude model. For comparison and validation purposes, Figure 2

shows the OPC obtained for the reduced search space from (a) the exact model and

(b) the crude model for the 33-node system. The results verify that this problem, when

evaluated over the reduced search space, also has a flat OPC, thus indicating a large

number of good solutions.

Figure 2. Normalized OPC for the 33-node system using: (a) exact model and (b) crude model.

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OO for Dynamic Network Reconfiguration 1853

Table 1

Error statistics for the 33-node system

Standard deviation of error 1.79

Standard deviation of signal 3.15

�signal=�noise 1.76

6.3. Error Level in the Crude Model

The B-matrix loss formula discussed earlier is accurate for networks with high X=R

ratios, but since distribution networks have high R=X ratio, it is expected that the B-

matrix loss formula would be a good candidate for the OO crude model. According

to [20], the error of the crude model is considered moderate if the ratio of the standard

deviations of the signal to noise (�signal=�noise) is greater than one, where the signal in the

current case represents the power loss from the exact model. Defining the error (noise)

as the minimum power loss from the exact model minus the minimum power loss from

the crude model, it is found that the error level for the 33-bus network is moderate, as

shown in Table 1.

z1, z2, z3, and z4 can be obtained, according to [27], as z1 D 8:4299, z2 D 0:7844,

z3 D �1:1795, and z4 D 2. For k D 1, the size s for the top estimated designs set is

s D ez1 kz2 gz3 C z4 D 53: (8)

7. Comparison with Other Techniques

The proposed algorithm was implemented into a software package in MATLAB 7.1 (The

MathWorks, Natick, Massachusetts, USA) running on an Intel® coreTM i5 CPU (Intel,

Santa Clara, California, USA) PC (Sony, Minato, Tokyo, Japan) with 8 GB RAM. The

approach was tested on 33-, 69-, and 136-node systems. The time for obtaining the

optimal solution using OO for each system was 2.13, 5.69, and 26.28 sec, respectively.

In order to illustrate the fast performance of OO, Table 2 compares the run-time of OO

Table 2

Comparison of CPU run-times

Reconfiguration approach

CPU

run-time

System size

(number

of buses) Processor memory

Proposed OO 2.13 sec 33 2.4 Ghz Intel core, 8 GB

5.69 sec 69

26.28 sec 136

Genetic shortest path 57.27 min 360 Pentium 2, 566 MHz

algorithm [10]

Modified tabu search [13] 5 sec 16 Pentium 3,700 MHz, 128 MB

25 min 69

5 hr 119

Meta-heuristic and GA [31] 403.83 sec 136 Pentium Duo, 2.2 GHz, 1 GB

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1854 R. El Ramli et al.

versus other previous methods for the different system sizes. In general, OO shows faster

performance, but it should be noted that this comparison is more qualitative, because

each of the previous works used different processor speeds and memory, and the proposed

algorithms could not be publicly found in order to implement them on the same processor

for comparison.

7.1. Thirty-three-node System

This test system is a hypothetical 12.66-kV network with 33 nodes, 37 lines, and only

1 energy input at node 33, which is considered the slack node. The system consists of

five tie-lines and sectionalizing switches on every branch of the system. The complete

system data can be found in [5]. The total system loads are 3715 kW and 2300 kVAR.

The real power loss for the initial reconfiguration is 204.14 kW.

The OO method as described before was applied to this system. Note that the total

search space is equal to 50,751. After sampling, the power losses were estimated using

the B-loss formula. The top 53 designs obtained from the crude model computation

were evaluated using the exact model, and the minimum power losses were obtained to

be 126.01 kW; this solution has a rank of 15 out of the 50,751 trees obtained. Thus,

the solution is in the top 0.03% of the search space. The optimal configuration has

switches 7, 11, 14, 28, and 36 opened. The results obtained were compared with the

works of [3, 5, 27, 28]. As Table 3 shows, OO gave superior results compared to other

methods. The minimum voltage obtained was 0.97 p.u., and the maximum voltage was

1.06 p.u., implying that voltages are in an acceptable range.

7.2. One-hundred-thirty-six-node System

The 136-node system is a 13.8-kV real distribution system located in Tres Lagoas, Mato

Grosso State, Brazil. It consists of 136 nodes, 156 branches, and 21 tie-lines. The original

configuration has a power loss of 320.17 kW, and the complete data can be found in [29].

The same procedure as for the 33-node system was applied on this system. The

optimal configuration obtained has the following switches set as open: 38, 51, 53, 90,

96, 106, 119, 126, 136, 137, 138, 144–148, 150–152, 155, and 156, and it has a total

power loss of 281.72 kW. The OO result is slightly better or comparable to [29, 30], as

shown in Table 4.

Table 3

Power loss for the 33-node system

Method Power loss (kW)

Exact minimum power loss 122.9

Baran and Wu M1 [5] 167.0

Baran and Wu M2 [5] 178.4

Baran and Wu M3 [5] 178.4

Shirmohammadi and Hong [3] 158.2

Goswami and Basu [28] 157.5

Kachem [32] (minimal tree search) 132.8

OO 126.0

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OO for Dynamic Network Reconfiguration 1855

Table 4

Power loss for the 136-node system

Method Power loss (kW)

Heuristic [29] 285.5

GA [30] 280.17

OO 281.72

8. Dynamic Network Reconfiguration in the Presence ofVariable Loads

Because smart grids are expected to be the future electric power systems, where real-

time power demand and generation fluctuations can be used to initiate dynamic network

reconfiguration, this study attempted to show the effectiveness of the OO in an emulated

environment of variable load levels. Several simulations were performed on the 33-node

network, as given in [5], while varying the original loads. The new loads were obtained

by taking a normally distributed random variable centered at the original loads levels and

having a standard deviation of 10, 20, 40, 50, and 60% of the load values. The results

are summarized in Table 5, where seven different cases are considered (two cases with

a standard deviation equal to 10%, two cases with standard deviation equal to 20%, and

one case for each of 40, 50, and 60%). For each case, N D 1000 uniformly sampled trees

were used. The minimum power loss was computed using an exact, and thus exhaustive,

model and compared to the OO approach. Results obtained show that the OO solution is

within 0.13% of the global minimum loss solution.

9. Conclusion

This study proposed applying the OO technique to achieve dynamic reconfiguration

of distribution networks. The algorithmic solution first generates all possible network

configurations and then uniformly and dynamically samples 1000 of them. Using a crude

Table 5

Power losses for different load levels (33-node system)

Different load variations

Exact

power

loss

(p.u.)

Estimated

power loss

using OO

(p.u.)

Maximum

power

demand

(p.u.)

Minimum

power

demand

(p.u.)

Case 1 (initial case) 1.23 1.26 4.2 0.45

Case 2 (standard deviation D 10%) 1.16 1.18 4.43 0.43

Case 3 (standard deviation D 10%) 1.24 1.29 5.01 0.53

Case 4 (standard deviation D 20%) 1.38 1.43 5.23 0.37

Case 5 (standard deviation D 30%) 1.27 1.34 5.49 0.39

Case 6 (standard deviation D 40%) 0.97 0.99 2.95 0.17

Case 7 (standard deviation D 50%) 1.35 1.40 4.91 0.05

Case 8 (standard deviation D 60%) 1.17 1.19 3.66 0.17

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1856 R. El Ramli et al.

model for power loss estimation, a selected subset from these 1000 possible configu-

rations is identified and further evaluated using an exact model. Comparing simulation

results with other previously implemented methods showed superior performance for the

proposed OO approach, motivating further investigation to adapt it for real-time dynamic

reconfiguration in a smart grid environment.

Acknowledgment

This work was supported by the Lebanese National Center for Scientific Research (NCSR;

grant 01-08-10).

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