1.pdf
DESCRIPTION
Load flow analysisTRANSCRIPT
This article was downloaded by: [National Institute of Technology - Kurukshetra]On: 24 November 2011, At: 01:05Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Electric Power Components and SystemsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/uemp20
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
To cite this article: R. El Ramli, M. Awad & R. A. Jabr (2011): Ordinal Optimization for DynamicNetwork Reconfiguration, Electric Power Components and Systems, 39:16, 1845-1857
To link to this article: http://dx.doi.org/10.1080/15325008.2011.615801
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae, and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.
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: [email protected]
1845
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11
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
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11
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,
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11
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 ;
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11
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)
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11
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.
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11
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
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11
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.
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11
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
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11
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
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11
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
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11
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).
References
1. Bunch, J. B., Miller, R. D., and Wheeler, J. E., “Distribution system integrated voltage and
reactive power control,” IEEE Trans. Power Apparatus Syst., Vol. 101, No. 2, pp. 284–289,
February 1982.
2. Merlin, A., and Back, H., “Search for a minimal-loss operating spanning tree configuration in
an urban power distribution system,” Proceedings of the Fifth Power Systems Computations
Conference, pp. 1–18, United Kingdom, 1–5 September 1975.
3. Shirmohammadi, D., and Hong, H. W., “Reconfiguration of electric distribution networks for
resistive line loss reduction,” IEEE Trans. Power Delivery, Vol. 4, No. 2, pp. 1492–1498, April
1989.
4. Civanlar, S., Grainger, J. J., and Lee, S. H., “Distribution feeder reconfiguration for loss
reduction,” IEEE Trans. Power Delivery, Vol. 3, No. 3, pp. 1217–1223, July 1988.
5. Baran, M. E., and Wu, F. F., “Network reconfiguration in distribution systems for loss reduction
and load balancing,” IEEE Trans. Power Delivery, Vol. 4, No. 4, pp. 101–102, April 1989.
6. Castro, C. A., and Watanabe, A. A., “An efficient reconfiguration algorithm for loss reduction
of distribution systems,” Elect. Power Syst. Res., Vol. 19, No. 2, pp. 137–144, January 1990.
7. Morton, A. B., and Mareels, I. M. Y., “An efficient brute-force solution to the network
reconfiguration problem,” IEEE Trans. Power Delivery, Vol. 15, No. 3, pp. 996–1000, July
2000.
8. Kashem, M. A., Ganapathy, V., and Jasmon, G. B., “Network reconfiguration for load balancing
in distribution networks,” IEEE Proc. Generat. Transm. Distribut., Vol. 146, No. 6, pp. 563–
567, November 1999.
9. Nara, K., Shiose, A., Kitagawa, M., and Ishihara, T., “Implementation of genetic algorithm for
distribution systems loss minimum reconfiguration,” IEEE Trans. Power Syst., Vol. 7, No. 3,
pp. 1044–1051, August 1992.
10. Yu, Y., and Wu, J., “Loads combination method based core schema genetic shortest-path
algorithm for distribution network reconfiguration,” Int. Conf. Power Syst. Technol., Vol. 3,
pp. 1729–1733, December 2002.
11. Ravibabu, P., Venkatesh, K., and Kumar, C. S., “Implementation of genetic algorithm for
optimal network reconfiguration in distributions for load balancing,” IEEE Region 8 Interna-
tional Conference on Computational Technologies in Electrical and Electronics Engineering,
pp. 124–128, Novosibirsk, Russia, 21–25 July 2008.
12. Sivanagaraju, S., Srikanth, E., and Zagadish Babu, E., “An efficient genetic algorithm for loss
minimum distribution system reconfiguration,” Elect. Power Compon. Syst., Vol. 34, No. 3,
pp. 249–258, March 2006.
13. Abdelaziz, A. Y., Mohamed, F. M., Mekhamer, S. F., and Badr, M. A. L., “Distribution system
reconfiguration using a modified tabu search algorithm,” Elect. Power Syst. Res., Vol. 80, No. 8,
pp. 943–953, August 2010.
14. De Oliveira Jr., L. W. S. C., De Oliveira, E. J., Pereira, J. L. R., Silva Jr., I. C., and Costa,
J. S., “Optimal reconfiguration and capacitor allocation in radial distribution systems for energy
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11
OO for Dynamic Network Reconfiguration 1857
losses minimization ” Int. J. Elect. Power Energy Syst., Vol. 32, No. 8, pp. 840–848, October
2010.
15. Sivanagaraju, S., Viswanatha Rao, J., and Sangameswara Raju, P., “Discrete particle swarm
optimization to network reconfiguration and load balancing,” Elect. Power Compon. Syst.,
Vol. 36, No. 5, pp. 513–524, May 2008.
16. Abdelaziz, A. Y., Mekhamer, S. F., Badr, M. A. L., Mohamed, F. M., and El-Saadany, E. F.,
“A modified particle swarm algorithm for distribution systems reconfiguration,” IEEE Power
and Energy Society Meeting, Calgary, Alberta, Canada, 26–30 July 2009.
17. Li, F., “Application of ordinal optimization for distribution system reconfiguration,” Power
Systems Conference and Exposition, pp. 1–8, Seattle, Washington, 15–18 March 2009.
18. Lau, T. W. E., and Ho, Y. C., “Universal alignment probabilities and subset selection for
ordinal optimization,” J. Optim. Theory Appl., Vol. 93, No. 3, pp. 455–489, June 1997.
19. Ho, Y. C., Zhao, Q. C., and Jia, Q. S., Ordinal Optimization: Soft Optimization for Hard
Problems, New York: Springer, pp. 7–50, 2007.
20. Ho, Y. C., and Deng, M., “The problem of large search space in stochastic optimization,” Proc.
33rd IEEE Conf. Decision Control, Vol. 2, pp. 1470–1475, December 1994.
21. Mayeda, W., and Sechu, S., “Generation of trees without duplications,” IEEE Trans. Circuit
Theory, Vol. 12, No. 2, pp. 181–185, June 1965.
22. Braverman, M., Kulkarni, R., and Roy, S., “Parity problems in planar graphs,” 22nd Annual
IEEE Conference on Computational Complexity, pp. 225–235, San Diego, CA, 13–16 June
2007.
23. Dogrusoz, U., and Krishnamoorthy, M. S., “VLSI algorithms for finding a fundamental set of
cycles and a fundamental set of cutsets of a graph,” Int. Conf. Comput. Inform., Vol. 6, pp.
46–65, April 1994.
24. Chiang, H., and Jean-Jumeau, R., “Optimal network reconfigurations in distribution systems,”
IEEE Trans. Power Delivery, Vol. 5, No. 3, pp. 1568–1574, July 1990.
25. Debs, A. S., Modern Power System Control and Operation, Boston: Kluwer Academic Pub-
lishers, pp. 180–188, 1987.
26. Sherman, J., and Morrison, W. J., “Adjustment of an inverse matrix corresponding to a change
in one element of a given matrix,” Ann. Math. Statist., Vol. 21, No. 1, pp. 124–127, 1950.
27. Ho, Y. C., and Sreenivas, R. S., “Ordinal optimization of discrete event dynamic systems,” J.
DEDS, Vol. 2, No. 1, pp. 61–88, 1992.
28. Goswami, S. K., and Basu, S. K., “A new algorithm for the reconfiguration of distribution
feeders for loss minimization,” IEEE Trans. Power Delivery, Vol. 7, No. 3, pp. 1484–1491,
July 1992.
29. Mantovani, J. R. S., Casari, F., and Romero, R. A., “Reconfiguration of radial distribution
systems using the voltage drop criteria,” Sociedade Brasileira de Automática, Vol. 11, No. 3,
pp. 150–159, December 2000.
30. Carreno, E. M., Romero, R., and Feltrin, A. P., “An efficient codification to solve distribution
network reconfiguration for loss reduction problem,” IEEE Trans. Power Syst., Vol. 23, No. 4,
pp. 1542–1551, November 2008.
31. Swarnkar, A., Gupta, N., and Niazi, K. R., “A novel codification for meta-heuristic techniques
used in distribution network reconfiguration,” Elect. Power Syst. Res., Vol. 81, No. 7, pp. 1619–
1626, July 2011.
32. Kachem, M. A., Jasmon, G. B., and Ganapathy, V., “A new approach of distribution system
reconfiguration for loss minimization,” Elect. Power Energy Syst., Vol. 22, No. 4, pp. 269–276,
May 2000.
Dow
nloa
ded
by [
Nat
iona
l Ins
titut
e of
Tec
hnol
ogy
- K
uruk
shet
ra]
at 0
1:05
24
Nov
embe
r 20
11