Monocriteria and Multicriteria Optimization of Network
Configuration in Distribution Systems
R.C. BERREDO
E.S. CRUZ
Department of Engineering and Coordination of Distribution Expansion Planning
Minas Gerais State Energy Company
Ave. Barbacena, 1200, 30190-131 - Belo Horizonte - MG
BRAZIL
http://www.cemig.com.br
P.Ya. EKEL
M.F.D. JUNGES
M.M. GONTIJO
J.G. PEREIRA Jr.
Graduate Program in Electrical Engineering
Pontifical Catholic University of Minas Gerais
Ave. Dom Jose Gaspar, 500, 30535-610 - Belo Horizonte - MG
BRAZIL
http://www.ppgee.pucminas.br
V.A. POPOV
Graduate Program in Electrical Engineering
Federal University of Santa Maria
Camobi, University Campos, 97105-900 – Santa Maria - RS
BRAZIL
http://www.ufsm.br/ppgee
Abstract: - This paper presents results related to the project aimed at developing models, methods, and a
computing system associated with problems of optimizing network configuration in distribution systems. Three
lines of improving the validity and efficiency in solving these problems are considered. The first line is directed
at taking into account a power system reaction while minimizing power and energy losses. The second line is
related to formulating and solving the problems within the framework of multicriteria optimization models to
consider and to optimize diverse indices reflecting reliability, service quality, and economic feasibility of energy
supply. The third line is associated with taking into account the uncertainty of initial data. Adequate and
computationally effective approaches to considering the factors indicated above are considered in the paper.
Key-Words: - Distribution Systems, Network Reconfiguration, Power System Reaction, Multicriteria
Optimization, Bellman-Zadeh Approach, Uncertainty Factor, Fuzzy Preference Relations.
1 Introduction
The problems of optimizing network configuration
(network reconfiguration) are associated with altering
distribution system topological structures by
changing the state of their switches (in other words,
by changing locations of their disconnections). These
problems are traditionally solved in long- and shortterm planning, dispatcher operation, and can be
applied in design studies. An increased interest to the
network reconfiguration problems is associated with
wide automation of distribution systems whose
switches are remotely monitored and controlled, that
permits one to solve these problems as the on-line,
real time problems. Many works have been dedicated
to their solution on the basis of diverse approaches:
search methods, branch and bound type techniques,
heuristic procedures, expert systems, genetic
algorithms, neural networks, etc. (for example, [1-4]).
These works "compete" in aspiration for obtaining
"more optimal" solutions. However, this aspiration,
considering that combination of the information
uncertainty (which is not considered in the papers [14]) and relative stability of optimal solutions
produces decision uncertainty regions [5,8], is not
convincing. At the same time, the results of [1-4] as
well as other works in this field do not permit one to
consider a power system reaction when optimizing
network configuration in distribution systems.
The operating conditions of distribution and power
systems are related: changes of distribution network
configuration lead to redistributing substations loads
and, consequently, to changing losses in the power
system. The lack of considering the change of power
system losses may result not only to deterioration in
network reconfiguration efficiency, but may result to
the negative effect as well. It demands to minimize
total losses in the distribution and power systems.
This statement serves for real increasing the adequacy
of models and, as a result, factual efficiency of
solutions, which can be obtained on their basis.
Moreover, the works [1-4] as well as other works in
this field are directed to solving the problems on a
monocriteria basis. At the same time, the problems of
network reconfiguration are multicriteria in nature
because have an impact on power and energy losses,
reliability, and service quality.
Taking the above into account, the work reported
in this paper is dedicated to solving problems of
monocriteria and multicriteria optimization of
network configuration in distribution systems with
taking into account the power system reaction and the
uncertainty factor.
2 CONSIDERATION OF POWER
SYSTEM REACTION
Direct taking into account the power system reaction
is hampered because of a large volume of information
reflecting parameters and operating modes of
distribution and power systems. Considering this, the
paper is devoted to building functional equivalents to
evaluate power system reaction at any step of
distribution system optimization (under transference
or an attempt of transference of any disconnection
location for an arbitrary distribution network loop).
The active power losses in the power system for
an arbitrary step of bus load curves
 J p ,1 
 J q ,1 




 ... 
 ... 
 J p ,l 
 J q ,l 




J  J p  jJ q   ...   j  ... 




 J p,m 
 J q,m 
 ... 
 ... 




 J p , n 
 J q , n 
may be calculated in the following form [9]:
P  3(J tp RJ p  J tq RJ q )  10 3 ,
J   J p  jJ q
J p ,1
J q ,1








...
...




 J p ,l  J p ,lm 
 J q ,l  J q ,lm 





...
...
(3)
 j





 J p ,m  J p ,lm 
 J q ,m  J q ,lm 




...
...




J p ,n
J q ,n




produced by transferring a location of disconnection
of the distribution network loop connecting the buses
l and m. This redistribution leads to an increment of
power losses which may be estimated as follows:
(Plm )  3(J pt RJ p  J qt RJ q
 J tp RJ p  J tq RJ q )  10 3 .
(4)
However, taking into account the properties of the
matrices R and the fact that in the process of
optimization we have to calculate the increments of
losses ( Plm ) not for all combinations of the power
system buses l (1  l  n) and m (1  m  n) (it is
enough to consider pairs of substation buses (feed
buses), which are connected by the distribution
system), this approach is not sufficiently effective.
An alternative approach is based on constructing
functional equivalents to calculate ( Plm ) only for
combinations of feed buses connected by the
distribution system. In particular, the transformation
of (4) with taking into account (1) and (3) leads to
the following expression:
(Plm )  3{(I p2,lm  I q2,lm )(Rll  Rmm  2Rlm )
 2[(J p,lm J p,l  J q,lm J q,l )( Rll  Rlm )
 (J p,lm J p,m  J q,lm J q,m )(Rmm  Rlm )
 J p ,lm
n
J
p ,i ( Rli
 Rmi )
q ,i ( Rli
 Rmi )]}  10 3 ,
i l
i l ,m
(1)
 J q ,lm
n
J
(5)
i l
i l ,m
(2)
where t is the transpose symbol; R is the bus
resistance matrix, which can be obtained from the
corresponding bus impedance matrix Y by its
inversion [7].
Let us suppose that we have load redistribution
where Rll and R mm are proper resistances of the
buses l and m; Rlm , Rli , and Rmi are mutual
resistances of the buses l and m, l and i, m and i,
respectively.
This approach needs separate elements of the
matrices R , which can be obtained without inversion
of the matrices Y , using distribution coefficients and
influence resistances [8].
The load homogeneity that may take place in
distribution systems permits one to simplify (5):
2
(Plm )  3{J lm
( Rll  Rmm  2Rlm )
 2J lm [ J l ( Rll  Rlm )  J m ( Rmm  Rlm )
n

 J (R
i
li
 Rmi )]}  10 3 .
(6)
i 1
i l ,m
Besides, the lack of reliable information in some
cases allows one to be oriented only to using (6).
Thus, the expressions (5) and (6) can serve for
constructing exact equivalents to estimate the power
system reaction to transferring the location of
disconnection of a distribution network loop.
Our experience shows that the use of experimental
design [9] allows one to build functionally oriented
models (their structure may be prompted, for
example, by the structure of (5) or (6)) in rational
way: the goal of experimental design is to organize
experiments with a system or its model so as to
maximize the amount of information obtained from a
minimal
number
of
experiments
while
simultaneously allowing a statistical evaluation of the
result reliability. This evaluation provides:
- the possibility to eliminate from consideration
such factors, which have no essential influence on
( Plm ) to reduce the dimension of equivalents;
- the possibility to test the adequacy of built
equivalents and, if necessary, to change intervals of
parameter varying to obtain adequate models.
The experimental design technique is based on
varying factors on a limited number of levels. In
particular, a full factorial experiment is associated
with conducting experiments for all combinations of
factor levels. It is common to use the full factorial
experiment with varying factors on two levels, that
demands the performance of N  2 s experiments to
construct models of the following type:
y  b0 
s

bp x p 
p 1

the replacements (g) defines the 2 s g experimental
design. For example, to build the model
~ ~
~
~
(8)
y b b ~
x b ~
x b ~
x ,
0
1 1
2 2
3 3
we have to perform eight experiments, although it is
enough to perform four experiments in accordance
with the 2 2 full factorial design taking x 3  x1 x 2 if
this interaction is insignificant.
To illustrate the construction of exact equivalents
as well as approximate equivalents (based on the use
of experimental design) we consider the 138 kV
power system shown in Fig. 1.
Line and transformer (13813.8 kV) data are listed
in Table 1. The load curves of feed system buses are
given in Table 2 that also includes information about
power ratings of the corresponding transformers.
Let us consider the construction of (P2,6 ) .
The elements of the matrix R necessary to
construct (P2,6 ) on the basis of (5) or (6) are the
R2,1  1.85  ,
R2,2 = 5.83  ,
R2,3
 1.33  , R2, 4  3.82  , R2,5  3.82  , R2,6  R6,2
 1.33  ,
R6,1  0.99  ,
R6,3  1.96  ,
R6,4
 1.33  , R6,5  1.33  , and R6,6  8.74  . Then, on
the basis of (6), for example, we can obtain
following:
(P2,6 )  (35.74 J 22,6  27.00 J 2,6  44.46 J 2,6 J 6
 5.16J 2,6 J1  3.78J 2,6 J 3  14.94J 2,6 J 4
 14.94 J 2,6 J 5 )  10 3 .
1
2
1´
2´
(9)
5
4
4´
5´
s
b
pq x p x q
0
pq
s
b
interaction effects of little significance (for example,
x p x q x r in (7)) by new parameters. The number of
pqr x p x q x r
 ... .
(7)
3´
pqr
Considering that 2  s  1, data obtained in the
full experiment have excessiveness, which allows
one to construct models on the basis of so-called
fractional experiment. Its matrices are parts of the
corresponding full experiment matrices, and a
number of experiments, which must be performed, is
less than a number of points in a factorial space.
The fractional experiment matrices [9] may be
obtained with reducing a number of experiments of
the full experiment in two, four, etc. times, replacing
6´
s
3
6
Fig 1. Power system
While constructing (P2,6 ) on the basis of
experimental design, the following factors, defined
by the structure of (6), are taken into account: J 2,6 ,
J 2,6 J 2 , J 2,6 J 6 , J 2,6 J 1 , J 2,6 J 3 , J 2,6 J 4 , and
J 2,6 J 5 . Considering this, it is possible to apply the
2 74
fractional
design
[9]
x1  J 2,6 ,
 J 2,6 J 1 , x1 x3
taking
x2  J 2,6 J 2 , x3  J 2,6 J 6 , x1 x 2
 J 2,6 J 3 , x 2 x 3  J 2,6 J 4 , and x1x2 x3  J 2,6 J5.
When analyzing models of multicriteria optimization,
a vector F ( X ) = {F1 ( X ),..., Fq ( X )} of objective
Table 1. Line and transformer data
Line/Transformer
0 - 1´
0 - 3´
1´- 2´
1´- 3´
2´- 3´
2´- 4´
4´- 5´
3´- 6´
1´- 1
2´- 2
3´- 3
4´- 4
5´ - 5
6´ - 6
R ( )
3.24
2.65
3.34
5.73
6.11
3.63
3.82
4.77
2.01
2.01
13.74
4.71
4.71
2.01
X ( )
14.97
6.89
8.61
14.76
15.74
9.35
9.84
12.30
67.15
67.15
228.37
188.86
188.86
67.15
functions is considered, and the problem consists in
simultaneous optimizing all objective functions, i. e.,
Fp ( X )  min , p  1,..., q ,
X L
Table 2. Load curves of feed buses
Transformer
Bus Power Rating
(MVA)
0-4 4-8
1
41
50 60
2
41
50 60
3
10
20 12
4
15
20 20
5
15
20 35
6
41
40 45
Load Curve (A)
8-12 12-16 16-20 20-24
110
100
120
70
100
90
100
50
15
20
30
30
30
55
50
25
50
30
50
40
90
90
80
50
To cover daily ranges of feed bus load changes
with the sufficient margin (20%), the intervals:
40.0 J 1 144.0, 40.0 J 2 120.0, 9.6 J 3 36.0,
16.0 J 4 66.0, 16.0 J 5 60.0, and 32.0 J 6 108.0
have been determined. Besides, it has been taken
0.1 J 2,6 2.1. These load levels have served for
constructing N  8 (in accordance with the 2 74
fractional design) combinations of 0.10 x1 2.10,
4.00 x 2 252.00, 3.20 x 3 226.80, 4.00 x1 x 2 
302.40, 0.96 x1 x3 75.60, 1.60 x 2 x 3 138.60, and
1.60 x1 x 2 x 3 126.00 to realize the corresponding
calculations of power system losses. As a result, the
following model has been obtained:
(P2,6 )  (6.87  78.60 J 2,6  26.98J 2,6 J 2
 44.44J 2,6 J 6  5.16J 2,6 J1  3.78J 2,6 J 3
 14.94 J 2,6 J 4  14.94 J 2,6 J 5 ) 10 3 .
3 MULTICRITERIA
OPTIMIZATION OF NETWORK
CONFIGURATION
(10)
The use of the models (9) and (10) leads to
practically identical results. However, the way of
constructing the power system reaction equivalents
should be defined at every concrete case.
(11)
where L is a feasible region in R n .
The lack of clarity in the concept of "optimal
solution" is the fundamental difficulty in solving
multicriteria problems. When applying the BellmanZadeh approach [10], this concept is defined with
reasonable validity: the maximum degree of
implementing all goals serves as a criterion of
optimality. This conforms to the principle of
guaranteed result and provides a constructive line in
obtaining harmonious solutions from the Pareto set
[11,12]. Besides, the approach permits one to realize
a computationally effective method of analyzing
models [11]. Finally, the approach allows one to
preserve a natural measure of uncertainly in the
process of decision making and consider indices,
criteria, and constraints of qualitative (semantic,
contextual) character based on experience,
knowledge, and intuition of a decision maker (DM).
Taking this into account, the approach has found
wide applications in power industry problems [13].
When using the Bellman-Zadeh approach, each
objective function F p ( X ), X  L, p  1,..., q is to be
replaced by a fuzzy objective function or a fuzzy set
A p  {X ,  Ap ( X )}, X  L,
p  1,..., q ,
(12)
where  Ap (X ) is a membership function of Ap
[10].
A fuzzy solution D with setting up the fuzzy sets
(12) is turned out as a result of the intersection
q
D
A
p
with a membership function
p 1
 D ( X )  min  A p ( X ),
p 1,.., q
X L .
(13)
With the use of (13) it is possible to obtain the
solution X 0 providing the maximum degree of
belonging to the fuzzy solution D
max  D ( X )  max min  A p ( X )
(14)
X L p 1,.., q
D and reduced the problem (11) to
X 0  arg max min  A p ( X ) .
X L p 1,..., q
(15)
To obtain (15), it is necessary to construct
membership functions  Ap (X ), p  1,..., q reflecting
a degree of achieving "own" optima by F p (X ),
X  L , p  1,..., q . This condition is satisfied by the
use of membership functions

 max F ( X )  F ( X )  p
p
p

 Ap ( X )  
. (16)
 max F p ( X )  min F p ( X ) 
X L
 X L

In specific cases it is possible to use
 min F p ( X ) 

 A p ( X )   X L
 F p ( X ) 
p
.
(17)
The peculiarities of solving the problem (14) with
the use of the algorithms indicated above consists in
the following. If X (m) is a current point, the
transition to a point X ( m1) is expedient if
(p  1,..., q) :  Ap ( X ( m1) )  min  Ap ( X ( m) ) . (21)
1 p  q
In contrast, if
(p  1,..., q) :  Ap ( X ( m1) )  min  Ap ( X ( m) ) , (22)
1 p  q
( m 1)
the transition to X
is not expedient. This way of
evaluating the expediency of the transition to the next
point X ( m1) leads to the solution (15) that is Pareto,
if all inexpedient transitions are rejected.
In (16) and (17)  p , p  1,..., q are objective
function importance factors. Their forming and
correcting on the basis of a procedure convenient for
the DM is considered in [13].
The construction of (16) demands to solve the
following problems:
(18)
Fp ( X )  min ,
X L
F p ( X )  max ,
(19)
X L
providing
X 0p  arg min F p ( X )
X L
and
X 00
p
 arg max F p ( X ), respectively.
XL
At the same time, the construction of (17)
demands to solve only the problem (18).
Hence, the solution of the problem (11) demands
analysis of 2q+1 monocritéria problems (18), (19),
and (14), respectively, or q+1 monocriteria problems
(18) and (14), respectively.
Since the solution X 0 is to belong to the Pareto
set   L , it is necessary to build
 D ( X )  min { min  A ( X ),   ( X )} ,
p 1,.., q
p
(20)
where   ( X )  1 if X   and   ( X )  0 if
X  .
When analyzing problems of optimizing network
configuration in multicriteria statement (as objective
functions may be considered power losses, energy
losses, undersupply energy, poor energy quality
consumption, integrated overload of network
elements, etc. in diverse combinations), several
modifications of the univariate method [14] are used
to solve the problems (18), (19), and (14). The
corresponding algorithms (simple search, search for
an effective coordinate, search for an effective step,
etc.) are flexible and easily adapted to different
practical strategies in the problem solution.
4 CONSIDERATION OF
UNCERTAINTY FACTOR
The consideration of the uncertainty factor is
associated with constructing sets of rational solutions
(solutions which belong to decision uncertainty
regions or solutions which cannot be distinguished
from the point of view of the considered objective
function or considered objective functions) and their
subsequent analysis to choose a final solution.
Considering available initial information (loads,
load curves, reliability indices, etc.), it is possible to
form some representative combinations of initial data
(so called "nature states") applying techniques
considered in [15] or LPτ consequence techniques
[16]. Resolving the corresponding monocriteria or
multicriteria problem for each "nature state", it is
possible to form the decision uncertainty region.
The first step in analyzing the decision uncertainty
regions is associated with applying the approach [15]
based on elements of game theory. This approach
consists in constructing and analyzing so called
"payment matrices", which reflect effects obtained
for different solution alternatives in accordance with
formed representative combinations of initial data.
The analysis of the decision uncertainty regions is
based on applying special criteria [15].
The second step is associated with constructing
and analyzing so called  X , R  models, where
R  {R1 ,..., Rt } is the vector fuzzy preference relation
[11,17]. In this case, we have
Rs  [ X  X ,  Rs ( X k , X l )] , s  1,..., t , X k , X l  X ,
(23)
where  Rs ( X k , X l ) is a membership function of
fuzzy preference relation.
In (23), R s (also called a nonstrict fuzzy
preference relation, fuzzy week preference relation,
and fuzzy binary relation in literature) is defined as a
fuzzy set of all pairs of the Cartesian product X  X ,
such that the membership function  Rs ( X k , X l )
represents the degree to which X k weakly dominates
X l , i.e., the degree to which X k is at least as good
as X l ( X k is not worse than X l ) for the sth
criterion. In a somewhat loose sense [18],
 Rs ( X k , X l ) also represents the degree of truth of
the statement " X k is preferred over X l ".
Information given in the form (23) permits one to
realize the evaluation, comparison, choice, and/or
ordering of alternatives on the basis of criteria of
quantitative as well as qualitative ("flexibility of
operation", "comfort of maintenance", etc.) character.
The questions of constructing and analyzing
 X , R  models are considered in [11,19].
[6]
[7]
[8]
[9]
[10]
[11]
5 Conclusion
The results of research related to improving the
validity and efficiency in solving problems of
network reconfiguration in distribution systems have
been presented. Techniques for building power
system functional equivalents have been considered
and illustrated. Taking into account multicriteria
nature of the problems, they have been formulated
and solved within the framework of multicriteria
optimization models. Two approaches based on
elements of game theory and fuzzy preference
relations have been described to consider the
uncertainty of information in solving the problems.
[12]
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[15]
[13]
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[16]
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[18]
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