Energy and Power Engineering, 2010, 2, 283-290
doi:10.4236/epe.2010.24040 Published Online November 2010 (
Copyright © 2010 SciRes. EPE
Optimal Selection and Allocation of Sectionalizers in
Distribution Systems Using Fuzzy Dynamic Programming
Albornoz Esteban, Andreoni Alberto
Electrical Energy Institute, National University of San Juan, San Juan, Argentina
Received July 28, 2010; revised September 10, 2010; accepted October 12, 2010
This paper describes a calculation strategy that allows determining the optimal number and placement of
sectionalizing switches in MV radial distribution networks, in correspondence to technical, regulatory and
economical aspects. A formulation that takes into account the investment, maintenance and power interrup-
tion costs has been developed, seeking for a reduction in total costs while taking care of the regulatory and
technical aspects. A multicriteria optimization procedure allows incorporating in the calculating process
various quality indicators which can be either global or individual indexes. This way of formulation makes
the proposal flexible as well as applicable to allow including aspects that were not considered in previous
papers. The solution methodology is mainly based on dynamic programming, fuzzy logic, heuristics and
economic analysis techniques. Given its flexibility, the proposed tool is easily adapted to real distribution
systems, by considering the individual requirements of each network. The solutions obtained in simulations
are oriented to help decision-making for the operator.
Keywords: Distribution Systems, Protection, Fuzzy Logic, Reliability
1. Introduction
One of the critical requirements of any distribution sys-
tem is that related to the service-quality and reliability
requirements demanded by their users. For that, the dis-
tribution utilities must make significant investments to
meet these service-quality and reliability standards.
Therefore, network design engineers and system planners
focus their effort on the network design, the type of elec-
tric protection to choose from, and the placement of sec-
tionalizing switches.
The selection of a number of sectionalizing switches
and their correct placement in the MV distribution net-
work is a planning task that has not been solved entirely.
Most distribution utilities, however, resort to their own
practical experience, using the information from their
operators and clients and adopting solution methods that
are hardly optimal.
The purpose of the present work is to develop a calcu-
lus strategy that allows determining the placement and
optimal number of sectionalizing switches in an MV
radial distribution network, by linking the technical, reg-
ulatory and economic aspects.
2. State-of-the-Art
First, Many research papers have dealt with the problem
of optimal distribution protection design [1,2].
The problem of optimal placement of sectionalizing
devices in radial networks considering a balancing ap-
proach between investments and costs of Non-Supplied
Energy (NSE) has an exact solution developed by V.
Miranda [3] using dynamic programming. This approach
has been upgraded through the years, with the inclusion
of fuzzy values in the referential data, as shown in [4-6].
There are other solving techniques proposed. The ap-
plication of a general procedure of combinatory optimi-
zation known as “simulated annealing” is proposed en
[7], whereas [8] proposes using binary programming.
Likewise in [3], work [9] indicates applying a combina-
tion of dynamic programming with techniques to reduce
the data search-space.
References [7,10-12] resort to heuristic approaches.
Though these do not guarantee the precision of the ob-
tained results; besides, they demand long computing
times, which, in all, render them inadequate for real MV
Copyright © 2010 SciRes. EPE
After having reviewed the bibliography, we may con-
clude that there is no work that allows solving in an inte-
gral way the problem of sectionalizing switches alloca-
tion in the network. Not one considers the influence of
all variables involved, the uncertainty in the information,
while contemplating all technical constraints existing in a
distribution system.
3. General Formulation
The problems in which reliability is included as an opti-
mization criteria should be solved through a flexible
methodology. That must allow including in the formula-
tions the necessary reliability indexes according to the
specific requirements of the system, and those of the
customers as well. The problem thus stated will entail
complex mathematical formulations, with many objec-
tives to optimize (e.g., Total Costs, Frequency of Mo-
mentary Interruptions and NSE to the system, or to a
particular load, etc.). Besides, considering that these in-
dicators are not comparable one another and, therefore,
cannot be integrated into a single objective function, the
problem thus becomes one of multicriteria.
A possibility is to formulate the problem by consider-
ing the objective function “Total Costs”, and including
the other criteria as restrictions. Therefore, an example of
a mathematical model for the problem of sectionalizing
equipment allocation can be expressed as follows:
Objective Function:
 (1)
Subject to:
CI = Annual Interruptions Costs [$]
CA = Annualized Investment Costs on Sectionalizing
devices and Installations [$]
COyM = Annual Operative and Maintenance Costs of
the Sectionalizing devices [$]
MAIFI = Mean Momentary Interruption Frequency of
the Feeder.
MAXIM = Maximum number of momentary interrup-
tions of the system.
MIFINi = Momentary Interruption Frequency Index at
node i.
MAXIMi = Maximum number of momentary interrup-
tions at node i.
NSEi = Non-Supplied Energy at node i [kWh-year]
MAXNSEi = Maximum ENS for a node i [kWh-year]
NIFIi = Node Interruption Frequency Index at node i.
MAXFIi = Maximum number of interruptions at node i.
Considering all the above analyzed aspects, and the
fact that the arising problems differ in nature, it is not
convenient to use a rigid methodology of solution. Ra-
ther, the approach must be flexible to allow choosing
“optimization criteria” according to the regulations and
client requirements.
As mentioned in 2 proposed solutions do not permit to
consider the problem before formulated, in an integral
way. Additionally, excessive calculation times make
them unsuitable for its application to real-size MV sys-
The methodology for the current proposal uses an op-
timization technique that works with fuzzy dynamic pro-
gramming which allows linking the regulatory and client
requirements to the technical and economical aspects of
the system operation.
4. The Proposed Algorithm
4.1. Fuzzy Decision and the Sectionalizing Device
The precursors of Fuzzy Logic, Bellman and Zadeh [13],
defined three Basic concepts: fuzzy objective, fuzzy con-
straints and fuzzy decision. They analyzed the applica-
tion of these concepts to the problems of decision-making
under uncertainty.
If X is a set of possible alternatives to solve a deci-
sion-making problem, where the objective function G is
a fuzzy set in X characterized by its membership func-
tion G(x)
, and restriction R is also a fuzzy set in X
with a membership function R(x)
, then it is necessary
that the objective and the restriction be satisfied simulta-
neously. Bellman and Zadeh defined a decision-making
fuzzy set D as a result from the intersection of set G with
set R.
Then, the membership function that characterizes the
decision-making set D can be defined as:
() min(),()
to all
The membership function ()
denotes the mem-
bership degree of a decision
X into the decision set
D. Then, ()
can be used as a criterion to choose the
optimal decision from the decision set D.
The optimal decision (x*) is such that it has the high-
est membership degree into the decision set D. This is
equivalent to maximizing the membership value of func-
tion ()
. Then, the decision maximization is defined
Copyright © 2010 SciRes. EPE
()max()maxmin(),( )
xX xX
 
In a general way, the fuzzy decision D resulting from
k objective functions G1,G2,....,Gk and m restrictions
R1.R2,....,Rm is therefore defined by:
12 12
.... ....
DG GGRRR (5)
And the corresponding maximization decision is:
()max ()
maxmin( ),..( ),(),..,( )
xx x
 
This means that, in the fuzzy decision defined by
Bellman and Zadeh, the fuzzy objectives and restrictions
take part like in the expression for the membership func-
tion of D. However, depending on the case at hand, and
when the fuzzy objectives and constraints are not equally
important, it would be possible to employ weighing co-
efficients to incorporate the relative importance between
The solving procedure in choosing the optimal alter-
native to locate the sectionalizing device can be stated as
an optimization problem. It considers modeling the ob-
jective function (which could be, for example, the total
costs) and the constraints (such as the indexes for per-
manent and momentary interruptions; the index of inter-
ruption duration, etc.) by means of the fuzzy sets theory.
The solving procedure in the decision-making set results
from the intersection of the fuzzy sets involved, such as:
D: Fuzzy set of Decisions
CT: Fuzzy Set of Total Costs
SAIFI: Fuzzy Set of Interruption Frequency of the Sys-
SAIDI: Fuzzy Set of Interruption Durations of the Sys-
The maximization decision will be:
Then, the problem thus stated can be interpreted as
one of decision making that involves choosing an alter-
native among the set of feasible ones that satisfies the
objectives and the constraint set simultaneously.
When applying the fuzzy set theory to the decision
making formulation, it can be noted that both, the objec-
tives and the constraints, are mapped into the same fussy
space [0,1], as opposed to a traditional optimization
problem, where the objective function values are used
Stated this way, the solution method will find an al-
ternative for locating the sectionalizing device based on a
set of criteria, in which the regulatory aspects can also be
modelled. Besides, the other alternatives can also be
made available ordered according to the fuzzy decision
membership value.
To some constraints (SAIDI, SAIFI, etc.), the current
quality regulations state a limit value from which the
utilities start being penalized. The above suggests that
they can be treated properly from the modelling of the
membership function.
There are a number of alternatives to define the mem-
bership functions that are valid to define fuzzy sets as
well. They should be selected carefully, though, regard-
ing the characteristics to be represented and the problem
to be solved, as well as its logical structure in order to
define its associated linguistic value.
Finally, the objectives and constraints may show dif-
ferent importance degrees. The planner or decision-
maker shall ponder this importance among objectives
and constraints, according to his preference or judgment
criterion. Some pondering methods used are detailed in
references [13,14,15]. This paper uses the method pro-
posed by T.L. Saaty [14].
Exponential pondering will be used in evaluating the
fuzzy sets involved in the decision-making process. Then,
the decision making fuzzy set will be:
pp p
 
 (9)
This model is adequate to transform an idiomatic
quantifier into quantifying measurements. In general,
values of p > 1 decrease the membership degree, and
values of 0 < p < 1 increase the membership degree of
each element into the set.
4.2. Fuzzy Dynamic Programming (FDP)
The FDP method requires a set of discrete-variables
whose values, represented by their respective member-
ship functions, are mapped onto the decision making
fuzzy set, which incorporates both the objectives and the
constraints. The result involves evolving along an opti-
mal trajectory formed by following the criteria that every
optimal state stems from the variant that is linked to the
maximum membership value of the decision fuzzy set.
The following terms will be used to define the FDP
model proposed by Bellman and Zadeh applied to the
problem of the sectionalizing device allocation.
()max( )maxmin( ),(),( ),..,( ),( )
xX xX
 
Copyright © 2010 SciRes. EPE
si: State Variables (rigid), i = 1, 2, ... M, where
is the set of values allowed by the state
variables (dominion).
xi: Decision variables (rigid), i = 1, 2, ... M, where
1,..., M
is the set of possible decisions.
N : Number of stages
M : Number of possible states
For each stage k, k = 1,,..., N, the following is defined:
A fuzzy constraint k
C, that bounds the decision-
making space. It is featured by its membership function
A fuzzy objective G characterized by its membership
function G
The problem to be solved is that of finding a maximi-
zation decision:
* 0,...,
Ddi M
 for a given state “s” (10)
The model for establishing the decision set will be that
of the “intersection” between the restrictions and the
objective functions. That is:
 (11)
with R = number of restrictions
Once the membership functions of the objective func-
tion and the constraint are defined, the value of the
membership decision function can be determined, for
each one of the possible links proposed by the FDP search
process Thus, using the operator min (intersection)1 to
aggregate the fuzzy restriction and objective function,
the membership function of the decision fuzzy set is:
,..,,..,( ,,,(1,))
CCGjiD i
ks x
ink sxks
 
 (12)
Where (k-1, si) denotes the optimal decision for state si
in stage k-1.
The optimal decision in FDP is that one presenting the
highest membership value to the decision set. The mem-
bership function of the maximized function for stage si
of stage k is:
( ,)maxmin,..,,..,( ,,,(1,))
DjC CGjiDi
ksks xks
The membership function of the maximized decision for stage k is:
j=1,..M i=1,..M
*D(k) denotes the optimal decision for stage k.
The complete solution will be determined recursively. The
“backward” technique of dynamic programming is used.
It should also be considered into the formulation the
possibility of installing equipment with reclosing capa-
bilities. This means that, at each stage, there will be two
alternatives, namely: locate equipments capable of re-
closing and locate equipment without capability of re-
closing. Therefore, in the optimal decision, this aspect
shall be incorporated. The modified expression to define
the optimal decision for stage k, *D(k), with this variant,
is shown below:
j=1,..M i=1,..M
where ei(j) will be 1 if the equipment has reclosing capa-
bility; and 0 when lacking it.
Finally, expression (15) will allow including as many
optimization criteria (objectives and constraints) as nec-
essary, without modifying the general structure of the
algorithm. This brings enough flexibility to solve the
various problems arising from regulations and client de-
mands. It should be kept in mind that every criterion
shall have a different weight as regards its relative im-
portance. The algorithm calculus sequence along with its
main features is presented in Figure 1.
The calculation procedure is as follows:
1) The procedure starts (k = 0) by considering that
there is not any sectionalizing device in the feeder, ex-
cepting the feeder circuit breaker.
2) The value of the total costs and the reliability in-
dexes (SAIFI, MAIFI, SAIDI, etc.) are computed, alto-
gether with their respective membership values. The
membership value of the decision μD(0) is found by us-
ing the operator min (intersection).
3) The stage value is increased (k = 1). All reliability
indexes are computed, with their respective membership
values associated to each state j. That is, the location of
sectionalizing switches is simulated, one at a time. Then,
applying the operator min, the membership value of the
decision μD(j,1)2 is found for each possible state.
4) According to the maximum value of the μD(j,1), the
1In the theory of fuzzy sets, the operation “intersection” between two
fuzzy sets A and B, with their respective membership functions μAand
μB , is defined as: CAB
; and its membership function is: μC =
min(μA , μB). That is, the membership value of the resulting fussy set
is the minimum value of the membership functions of fuzzy sets Aand
2μD(j,k) represents the membership function of the decision for state j
and stage k.
Copyright © 2010 SciRes. EPE
0,00 2,00 4,00 6,00 8,0010,00
Membership function for SAIFI
0,005,0010,00 15,00 20,00
Membership function for MIFINI
0,00 1,00 2,00 3,00 4,00
Membership function for MAIFI
19.900 29.900 39.900 49.900 59.900
Total Costs [US$]
Membership function for "Total
Figure 1. Membership functions used in FDP for the
real system.
*(1) are determined as the best alternative for stage 1.
5) Stage (k = k + 1) is increased.
6) At each state j, and for all possible links with stage
k-1, the reliability indexes are computed together with
their respective decision membership values for each
possible link by applying the operator min. That is, the
location of the k-th sectionalizing equipment is simulated,
considering all the links with the previous stage.
7) For each state j, considering the maximum value of
the membership functions resulting from each link with
stage k-1, the value μD(j, k) is established.
8) According to the maximum value of the μD(j, k), the
*(k) is found as the best alternative for stage k; besides,
the link with the associated stage k-1 is determined as
9) Steps 5, 6, 7 and 8 are then repeated until the bene-
fits resulting from the best alternative of stage k + 1 be-
come lower than those reached by the best alternative of
stage k.
10) It is made N = k , and N is defined as the stage of
optimal benefit.
11) According to the values of μ(j,N) (fuzzy decision
for state j of stage N), the optimal solution is found.
From it, and following the backward technique and ac-
cording to the optimal decisions, the optimal trajectory is
rebuilt to determine in which branch the sectionalizing
switch will be installed. Besides, it is possible to
re-construct all the paths for each state j of stage N.
5. Application
In order to show the practical application of the proposed
methodology, a number of simulations were performed
using a real MV feeder that supplies energy to a region
of the Province of Azuay, south of the Republic of Ec-
uador. In the first part of its layout, it supplies a typically
urban sector; then, it supplies power to several rural ar-
eas. In addition, the line serves several industrial cus-
Table 1 shows the feeder most important characteristics
and parameters (according to information obtained in
The historical data recorded by the distribution utility
allow establishing the interruption times and failure rates
presented in Table 2.
Some considerations were made for the simulations.
First, at the feeder head there is a circuit breaker with
protective relays and automatic reclosing relay.
Second, it is assumed that all sectionalizing and pro-
tection equipments are 100% reliable, and that the supply
alternatives are always available. Third, capacity con-
straints will be considered for transformers and line for
the cases of load transference. Finally, to compute the
NSE, the average demand value of the feeder loads will
be used.
Table 1. Feeder characteristics.
Type (Urban-Rural)
Installed load [kVA] 9,560
Number of MV/LV transformers 240
Total circuit length [km] 48.7
Number of branches 358
Voltage level [kV] 22.0
Peak load [kW] 3,106
Energy [kWh-year] 14,992,638
Load factor 0.551
Average load [kW] 1,711
Number of customers 4,553
NSE [kWh-year] 29,908
SAIFI [Interruptions/Customer-year] 7.54
SAIDI [Hours/Customer-year] 15.24
MAIFI [Hours/Customer-year] 0.00
Table 2. Time and failure rates for the feeder.
Isolation Time (ts) 1 hour
Transference time ( tt ) 1 hour
Time for failure fixing ( tr ) 5 hour
Urban permanent-failure rate ( pu) 0.181 failure/yearkm
Rural permanent-failure rate ( pr) 0.198 failure/yearkm
Urban momentary-failure rate ( tu) 0.289 failure/yearkm
Rural momentary-failure rate ( tr) 0.317 failure/yearkm
Copyright © 2010 SciRes. EPE
The protection device cost, its useful life, the discount
rate, the unitary cost of NSE and the operation and main-
tenance costs are all detailed in Table 3.
5.1. Case 1: Using Conventional Dynamic
With the above data and considerations, the optimal lo-
cation of the feeder’s reclosing devices will be deter-
mined using Conventional Dynamic Programming (DP)
with the minimization objective function: “Total Costs”.
Table 4 shows the detailed results of the best two possi-
ble alternatives.
Alternative 1 (3 reclosing and 1 interconnection device)
is the best solution, with an annual total benefit of US$
13,730. Alternative 2, with two interconnection and 3
reclosing devices is also a good solution. The SAIFI,
SAIDI and MAIFI values are practically the same for
each alternative, and the NSE is slightly lower for the
second one. But, because this alternative has one addi-
tional interconnection device, the benefit results about
12% smaller than with Alternative 1. This means that, for
this feeder, to count with more than one tie point does
not bring along a greater benefit; even though it does
improve the quality indexes, this improvement is minimal.
5.2. Case 2: Using Fuzzy Dynamic Programming
In order to illustrate the advantage of applying FDP to
Table 3. Annual costs and financial costs.
Useful life of equipment and installations 15 years
Annual discount rate 12%
Cost of sectionalizing equipment US$ 10.000
Cost of NSE kWh
Residential Customers
Commercial Customers
Industrial Customers
US$ 1.0 kWh
US$ 1.0 kWh
US$ 2.0 kWh
O&M Costs 2% of investment costs
Table 4. Results using DP.
Alt. 1 Alt. 2
# of interconnection devices 1 2
# of sectionalizing devices 3 3
NSE 9,505 9,485
Benefits [US$] 13,730 12,082
SAIFI 1.57 1.57
SAIDI 5.79 5.78
MAIFI 5.07 5.07
the previous network, in addition to looking for mini-
mizing the total costs, it will be sought to minimize the
momentary interruptions to an industrial customer whose
load is significantly sensible to such interruptions. Be-
sides, and as a third criterion, the feeder’s interruption
frequency will be minimized. The membership functions
of the various adopted criteria are shown in Figure 1.
The values for the criteria weights employed in this
case were computed according to what is stated in 4.1.
Then, the decision making problem, considering the
weights value, results as follows:
2.238 0.68240.998
The results are presented in Table 5, which shows the
total costs for the best solution alternatives, and the val-
ues for the feeder indexes NSE, SAIFI, SAIDI and
MAIFI, and the momentary interruption index (MIFINi)
at the node that supplies power to the industrial cus-
Moreover, Table 5 specifies, for both alternatives, the
number of equipments to be installed, making a differ-
ence between equipments that must (or not) have reclos-
ing capacity.
Alternative 1 with 3 protection devices (2 with reclos-
ing capabilities and 1 without them) and 1 interconnec-
tion device, is the best solution.
By comparing these figures with the results obtained
with DP in Table 4, it can be seen that, even though the
values for NSE, SAIDI, SAIFI are slightly greater, the
MAIFI value is smaller and, above all, the MIFINi index
at the client’s node is substantially lower: 2.23. Com-
pared with the best alternative using DP, i.e. 6.23, the
FDP solution is 64.2% lower.
On the other hand, the NSE value has increased
scarcely in 0.44%, and the benefits have decreased about
US$ 639 a year –a negligible value when contrasted with
Table 5. Results using fuzzy dynamic programming.
Alt. 1 Alt. 2
Interconnection Equipment 1 2
Sectionalizing Equipment with
reclosing capabilities 2 2
Sectionalizing equipment without
reclosing capabilities 1 1
NSE [kWh] 9,547 9,485
Benefits [US$] 13,091 12,082
SAIFI 1.57 1.57
SAIDI 5.81 5.78
MAIFI 4.07 4.07
MIFINi 2.23 2.28
Copyright © 2010 SciRes. EPE
the upgraded quality service rendered to the customers
and, specifically, to the industrial one.
From the above analysis, it can be concluded that the
solution alternatives with similar costs present as well
very differing MAIFI and MIFINi values. The method-
ology with FDP allows identifying solutions that have
very convenient characteristics for certain reliability in-
dicators, without having to jeopardize the economical
aspects. It is here where the proposal of the work marks
its advantage over other approaches.
It is thus shown that the method of FDP has an addi-
tional advantage over the conventional approach because
it allows considering in the optimization process various
decision-making criteria. In this case, considering the
hypotheses of the example, three fundamental criteria
were employed: Total Costs, MAIFI and MIFINi.
Finally, because this is a combinatory-type problem,
the computer times are significant and depend, mostly,
on the network size under study, and the computing
equipment available, i.e., processing capacity and speed.
When making computations on a 128 Mb RAM, 2GHz
Pentium IV, the processing takes little less than 4 min-
utes. An important part of this time is used in computing
the power flow needed to verify the technical constrains
(voltage drop and loadbility of the elements)
On the grounds that the proposed methodology is
meant to be applied in the planning processes, the com-
puter times are adequate enough for this kind of studies.
6. Conclusions
The problem of locating the sectionalizing equip-
ment in MV distribution networks have certain
properties that allow using FDP to find its solution,
without having to make an exhaustive search. In-
deed, the approach of analyzing all possible alter-
natives in real distribution networks is practically
impossible to do, because it is a combinational-
type problem.
Considering that the problems always differ in
some point, because the feeders present different
characteristics, the rigid-solution methodologies
turn to be impractical. Instead, the proposed solv-
ing approach allows choosing “optimization crite-
ria” according to the network type, client require-
ments and service regulations, which renders it in a
very flexible algorithm.
The chance of stating the problem of sectionalizing
equipment placement based on a set of criteria,
with which the regulatory aspects can be modeled,
besides incorporating specific conditions with the
help of tools and concepts of fuzzy programming,
allows considering, into a single framework, vari-
ous hypotheses and specific requirements for opti-
The proposed methodology regards a static net-
work; that is, it does not implicitly consider varia-
tions in the input data. But this fact does not pre-
vent from including or interacting with long-term
studies, where demand-growth, cost variations and
regulatory changes have to be considered, because
the proposed approach can be used with every
variant in each year, within the long-term planning
The work has shown the effectiveness of the pro-
posal to find the number and location of sectional-
izing and protection devices for MV networks, re-
garding all specific conditions and constraints.
From the comparative analysis made for two dif-
ferent scenarios –the first one considering only the
total costs, and the second regarding additional op-
timization criteria through FDP- it was shown that
using the algorithm adequately , very similar solu-
tions are found, with comparable costs, though
there are advantages with the latter solution be-
cause it allows considering other aspects. This is
particularly useful for feeders presenting heteroge-
neous, sensible and/or sizable loads.
Because the proposed methodology is meant to be
applied in the planning processes, the computer
times are adequate enough for this kind of studies.
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