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Guidelines for Authors This periodical is a publication of the Academic Publishing and Translation Directorate of Qassim University. Its purpose is to provide an opportunity for scholars to publish their original research.
Manuscripts will be published in on of the following platforms: i) Article: It should be original and has a significant contribution to the field in which the research was conducted. ii) Review Article: A critical synthesis of the current literature in a particular field, or a synthesis of the literature in a particular
field during an explicit period of time. iii) Brief Article (Technical Notes): A short article (note) having the same characteristics as an article. iv) Innovation and Invention Reports v) Forum: Letters to the Editor, comments and responses, preliminary results or findings, and miscellany. vi) Book Reviews The Editorial Board will consider manuscripts in the following fields:
• Electrical Engineering • Civil Engineering • Mechanical Engineering • Chemical Engineering • Computer Engineering
• Mining and Petroleum Engineering • Computer Science • Information Technology • Information Systems • Basic Engineering and Computer Sciences
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In The Name of ALLAH,
Most Gracious, Most Merciful
Journal of Engineering and Computer Sciences, Qassim University, PP 1-70, Vol. 1, No. 1
(January 2008/Muharram 1429H.)
II
Journal of Engineering and Computer Sciences
Qassim University, Vol. 7, No. 2, PP 73-199, for English (July 2014/Shaban 1435H.)
Qassim University Scientific Publications
(Refereed Journal)
Volume (7) – NO.(2)
Journal of
Engineering and Computer Sciences
July 2014 – Shaban 1435H
Scientific Publications & translation
Journal of Engineering and Computer Sciences, Qassim University, PP 1-70, Vol. 1, No. 1
(January 2008/Muharram 1429H.)
IV
Deposif: 1429/2023
EDITORIAL BOARD Prof. Mohamed A Abdel-halim (Editor in-Chief)
Prof. Saad Benmansour
Dr. Aboubekeur Hamdi-Cherif
Dr. Salem Dhau Nasri
Dr. Sherif M. ElKholy (Editorial Secretary)
Advisory Committee: Civil Engineering: Prof. Mahmoud Abu-Zeid, Egyptian Ex-Minister of Water Resources and Irrigation, President of the World Water Council, Professor of Water Resources, National Water Research Center, Egypt. Prof. Essam Sharaf, Professor of Transportation Engineering, Faculty of Engineering, Cairo University.
Prof. Abdullah Al-Mhaidib, Vice President for Research and higher studies, Professor of Geotechnical Engineering, College of Engineering, King Saud University, KSA.
Prof. Keven Lansey, Professor of Hydrology and Water Resources, College of Engineering, University of Arizona, Tucson, Arizona, USA.
Prof. Fathallah Elnahaas, Professor of Geotechnical/ Structure Engineering, Faculty of Engineering, Ein Shams University, Egypt.
Prof. Faisal Fouad Wafa, Professor of Civil Engineering, Editor in-chief, Journal of Engineering Sciences, King Abdul-Aziz University, KSA Prof. Tarek AlMusalam, Professor of Structural Engineering, College of Engineering, King Saud University, KSA.
Electrical Engineering : Prof. Farouk Ismael, President of Al-ahram Canadian University, Chairman of the Egyptian Parliament Committee for Education and Scientific Research, Professor of Electrical Machines, Faculty of Engineering Cairo University, Egypt. Prof. Houssain Anees, Professor of High Voltage, Faculty of Engineering, Cairo University. Prof. Metwally El-sharkawy, Professor of Electrical Power Systems, Faculty of Engineering, Ein Shams University, Egypt. Prof. Mohamed A. Badr, Dean of Engineering College, Future University, Professor of Electrical Machines, Faculty of Engineering, Ein Shams University, Egypt. Prof. Ali Rushdi, Professor of Electrical and Computer Engineering, College of Engineering, King Abdul-Aziz University, KSA. Prof. Abdul-rahman Alarainy, Professor of High Voltage, College of Engineering, King Saud University, KSA. Prof. Sami Tabane, Professor of Communications, National School of Communications, Tunisia.
Mechanical Engineering: Prof. Mohammed Alghatam, President of Bahrain Center for Research and Studies . Prof. Adel Khalil, Vice Dean, Professor of Mechanical Power, Faculty of Engineering, Cairo University, Egypt. Prof. Said Migahid, Professor of Production Engineering, Faculty of Engineering, Cairo University, Egypt. Prof. Abdul-malek Aljinaidi, Professor of Mechanical Engineering, Dean of Research and Consultation Institute, King Abdul-Aziz University, KSA
Computers and Information: Prof. Ahmad Sharf-Eldin, Professor of Information Systems, College of Computers and Information Systems, Helwan University, Egypt. Prof. Abdallah Shoshan, Professor of Computer Engineering,, College of Computer, Qassim University, Advisor for Saudi Minister of Higher Education, KSA . Prof. Maamar Bettayeb, Professor of Computer Engineering, AlSharkah University, UAE. Prof. Farouk Kamoun, Professor of Networking Engineering, Tunis University, Tunisia.
Qassim University, College of Engineering, P.O. Box: 6677 Buraidah 51431
Journal of Engineering and Computer Sciences
Qassim University, Vol. 7, No. 2, PP 73-199, for English (July 2014/Shaban 1435H.)
Contents Page
The Walsh Spectrum and the Real Transform of a Switching Function: A Review
with a Karnaugh-Map Perspective
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb .......... 73
Exploratory Study of Comparison between Ultimate Strength and Performance
Based Design for Reinforced Concrete Structures Subjected to Seismic Loading
Muhammad Saleem ....................................................................................... 113
Improving the Power Conversion Efficiency of the Solar Cells
Abou El-Maaty M. Aly .................................................................................. 135
Methodology for Minimizing Power Losses in Feeders of Large Distribution
Systems Using Mixed Integer Linear Programming
Hussein M. Khodr, E. E. El-Araby, and M. A. Abdel-halim ..................... 157
V
Journal of Engineering and Computer Sciences, Qassim University, PP 1-70, Vol. 1, No. 1
(January 2008/Muharram 1429H.)
VI
Journal of Engineering and Computer Sciences
Qassim University, Vol. 7, No. 2, pp. 73-112 (July 2014/Shaban 1435H)
73
The Walsh Spectrum and the Real Transform of a Switching Function: A
Review with a Karnaugh-Map Perspective
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb
Department of Electrical and Computer Engineering,
King Abdulaziz University,
P.O. Box 80204, Jeddah 21589, Saudi Arabia.
(Received 03/02/2014; Accepted for publication 24/04/2014)
Abstract. Using a Karnaugh-map perspective, this paper investigates the definitions, exposes the
properties, introduces new computational procedures, and discovers interrelationships between the Walsh spectrum and the real transform of a switching function. Appropriate Karnaugh maps explain the
computation of Walsh spectrum as defined in cryptology. An alternative definition of this spectrum
adopted in digital design and related areas is then presented together with procedures for its matrix computation. Then, the real transform of a switching function is defined as a real function of real
arguments. This definition is clearly distinguished from similar ones such as the multi-linear form or the
arithmetic transform. The real transform is visualized in terms of a particular version of the Karnaugh map called the probability map. Karnaugh maps are also used to demonstrate the computation of the
spectral coefficients adopted in digital design as the weight of the switching function and weights of its
subfunctions or restrictions. These maps match the earlier ones for the spectrum used in cryptology. Novel relations between the Walsh spectrum and the real transform are utilized in formulating two
simplified methods for computing the spectrum via the real transform with some aid offered by Karnaugh
maps. Finally, a solution is offered for the inverse problem of computing the real transform in terms of the Walsh spectrum.
Key Words. Switching functions, Walsh spectrum, Spectral coefficients, Real transform, Karnaugh maps,
Probability maps.
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 74
1. Introduction
The two-valued Boolean functions (switching functions) are the simplest interesting
multivariate functions [1]. They have a wide range of engineering applications,
including those of coding [1-3], cryptology [1-5], digital design [6-8], system
reliability [9-19], syllogistic reasoning [20-30], and operations research [31].
Switching functions have many useful representations which vary in their suitability
for handling different applications and in the kind of illumination they cast on
different functional properties. Notable among these presentations are:
(a) the Karnaugh map [32] which is a very powerful manual tool that
provides pictorial insight about the various functional properties and procedures
when the number of variables involved is small,
(b) the Walsh spectrum [33-44], which reveals information about the function
that is much more global in nature, and has prominent applications in cryptology as
well as digital design and related areas such as signal processing, information
transmission, function classification, and circuit analysis, design, synthesis and
testing, and
(c) the real or probability transform [37, 45-50], which is useful in many
areas such as that of system reliability.
The aim of this paper is to investigate the definitions, expose the properties,
introduce new computational procedures and discover interrelationships between the
Walsh spectrum and the real transform. This aim is achieved using a Karnaugh-map
perspective, which makes the exposition of the complex concepts encountered much
easier to understand. We strive to settle certain sources of confusion, such as (a) a
purported discrepancy between the definition of the Walsh transform used in
cryptology and that used in digital design, (b) the existence of different Walsh
spectra for the typical encoding {0,1} of the output of the switching function, or for
its decoding to polar encoding {+1, 1}, and (c) the nature of the domain and range
of the real transform and whether this transform is
or .
The organization of the rest of this paper is as follows. Section 2 presents
some preliminary definitions needed in subsequent sections. Section 3 exposes via
appropriate Karnaugh maps the computation of the Walsh spectrum as used in
cryptology. Section 4 presents the definition and matrix computation of the Walsh
transform as used by the digital-design community. Section 5 introduces the real
transform, explains its relation with the arithmetic transform, and visualizes it in
terms of a particular version of the Karnaugh map called the probability map.
Section 6 computes the first spectral coefficient, i.e., the first element in the Walsh
spectrum adopted in digital design, which is the weight of the switching function.
Section 7 utilizes the results of Section 6 in computing the rest of the spectral
coefficients as weights of subfunctions of the original function. These computations
are demonstrated on Karnaugh maps and are shown to match the earlier map
The Walsh Spectrum and the Real Transform of a Switching Function: … 75
computations in Section 3. Sections 8 and 9 present novel relations between the
Walsh spectrum and the real transform, and subsequently give two simplified
methods for computing the Walsh spectrum via the real transform with some aid
offered by Karnaugh maps. Section 10 discusses the inverse problem for those in
Sections 8 and 9, as it computes the real transform in terms of the Walsh spectrum.
Section 11 concludes the paper.
2. Preliminary Definitions
A switching function on n variables may be viewed as a mapping from {0, 1}n into
{0, 1}. We interpret a switching function as the
output column of its truth table f, i.e., a binary vector of length 2n
f = ..(1)
For switching functions , and of the same number of variables n ,
and of truth table vectors and , we denote by #( ) (respectively, #(
)) the number of places where the vectors and are equal (respectively,
unequal). The Hamming distance between the functions is denoted by
d( ), and given by
d( ) = #( ) . (2)
We also define the weight difference wd between the two functions
as
wd( ) = # ( ) #( ), (3)
. (4)
Also, the Hamming weight or simply the weight of a switching function ,
is the number of ones in its truth-table vector . This is denoted by wt( ). An n-
variable function f is said to be balanced if its output column in the truth table
contains equal numbers of 0's and 1's (i.e., Note that the
weight difference is not the difference between the weights of and .
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 76
A switching function can also be viewed as a function on the Boolean ring
[21], or as one over the simplest finite or Galois field, namely the binary field
GF(2). The addition operator over GF(2) is usually denoted by +. However, in this
paper, we will denote it by the symbol of the EXCLUSIVE OR operator in
switching algebra. We will retain the symbol + for its standard meaning of real
addition. In term of the operator, equations (2) and (4) can be rewritten as
(2a)
(4a)
An n-variable switching function
can be considered to be a
multivariate polynomial over GF(2). This polynomial can be expressed as a sum of
kth
-order products (0 k n) of distinct variables. More precisely,
can be written as [51]
(5)
or in expanded form as
a0 (a1 a2 an ) (a12 a13 (6)
,
3. Walsh Spectrum for Cryptographic Studies
The Walsh spectrum of an n-variable switching function is based on a set of
orthogonal functions defined by Walsh [52]. The spectrum is usually called the
Walsh-Rademacher spectrum, because the Walsh functions are an extension of a set
of functions defined by Rademacher [53]. The spectrum is also called the Walsh-
Hadamard spectrum, because among several orderings of the Walsh functions, the
most prominent one is the one due to Hadamard [42]. The Hadamard ordering is the
one to be adopted herein. Two equivalent (albeit apparently different) mathematical
descriptions of the Walsh transform appear in the literature. The description
typically adopted in cryptology is considered in this section, while the one typically
used in digital design and related areas is deferred to Section 4.
The Walsh Spectrum and the Real Transform of a Switching Function: … 77
Let and both belong to
{0,1}n, and let denote the linear switching function on the n variables
given by
. (7)
Let f( ) be a Switching function on the n variables . Then the Walsh
transform of f(X) is a real-valued function over {0, 1}n that is defined as
(8)
(9)
Note that f( ) = 0 when f( ) = ( for which
= 1), while f( ) = 1 when f( ) ( for which
= 1). Thanks to (4a) and (9), the Walsh spectrum can
be interpreted as the weight difference between and , i.e.,
. (10)
Example 1:
, (11)
The 2-out-of-3 function is represented by the Karnaugh map in Fig. 1(a) for
the usual {0, 1} encoding for its truth values. Figure 1(b) represents the same
function when the truth table values {0, 1} are recoded to {+1, 1}. Figure 1(c)
shows a Karnaugh map representation for the linear function of 3 variables.
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 78
(a) with values and disjoint
coverage. Typically, 0 entries are left blank. (b) with values recoded to {+1, 1}.
(C)
Fig. (1). Karnaugh-map representations for the 2-out-of-3 function with different encodings,
and the 3-variable linear function
Figures 2 is a Karnaugh map that represents as a
function of X. Entries of this map add to give the Walsh transform, namely
(12)
The Walsh Spectrum and the Real Transform of a Switching Function: … 79
Fig. (2). Particular values of for specific values of
Finally, Fig. 3 obtains the Walsh transform of the function in the form of
a Karnaugh map of map variables and . A temporary entry of each cell
of this map is a specific instant of the map in Fig. 2 with entries computed for
pertinent values of and . Each of these entries is either +1 or 1.
Initially, the temporary entry in the all-0 cell is the
map in Fig. 1(b). The temporary entries in other cells are derived from this initial
entry by switching each 1 to and each to 1 for values within the loops
shown. Note that this switching action is cumulative for overlapping loops. Now, the
actual intended entries of the large map in Fig. 3 result by adding the entries within
the smaller maps in each cell. Figure 3 therefore represents as a pseudo-
Boolean function of [31, 32]. This figure can now be read to give the following
expression for the Walsh transform
= (14a)
where
(13)
, (14b)
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 80
Fig. (3). A Karnaugh-map evaluation of the Walsh transform of f(X) in Fig. 1(b)
Fig. (4). The truth table of in (49), cast in the form of a probability map
The Walsh Spectrum and the Real Transform of a Switching Function: … 81
4. Walsh Spectrum for Digital Logic
The Hadamard ordering of the Walsh spectrum has a simple recursive structure of a
transform matrix, , which is called a Hadamard matrix. This matrix
relates the Walsh spectrum vector (which is a vector of spectral coefficients)
to the truth-table vector of , which is a vector of elements belonging to
that express the functional values of the minterms or discriminators of as
in (1). This is called the R-encoding in [42]. In an alternative formulation, the Walsh
spectrum is given by a vector which is multiplied by the polar truth-table
vector of , namely
, (14)
Where 1 is a vector of elements, each of value 1. Note that in , logic is
coded as and logic is coded as , an encoding called the S-encoding in
[42]. Figures 1(a) and 1(b) are examples of R-encoding and S-encoding for the 2-
out-of-3 function. In summary, we have:
, (15)
. (16)
The Hadamard matrix in (15) and (16) is a matrix with entries
belonging to and a recursive structure given by
, (17)
. (18)
The rows of are the set of n-variable Walsh functions [42]. The matrix is
symmetric ) , orthogonal and idempotent .
The matrix is also quasi-involutary (quasi self-inverse). It is equal to its own
inverse to within a multiplicative constant .
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 82
The matrix can be equivalently defined via Kronecker products [42, 51, 54-57]
as follows
, (19)
, (20)
where (20) could be rewritten as
. (21)
In the following, we will stress the S-encoding and use as our Walsh
spectrum, unless otherwise stated. This choice agrees with the definition used in
cryptography studies. The Walsh spectrum for the R-encoding is related to .
Thanks to (14) and its inverse relation
(22)
we have the following interrelationships between the elements of and those of
, (23)
(24)
(25)
. (26)
In passing, we note that the truth-table vector represents the function via a
basis vector , i.e.,
The Walsh Spectrum and the Real Transform of a Switching Function: … 83
, (27)
where is expressed recursively as
, (28a)
(28b)
5. The Real Transform of a Switching Function
Various forms of the real transform (also called the probability or arithmetic
transform) are discussed in [37, 45-50]. The following definition is taken from [37],
and the following exposition relies on results obtained in [18, 36, 37, 47, 48].
The real transform of a switching function ( ), denoted
by is defined to possess the following two properties:
a) R(p) is a multi-affine continuous real function of continuous real variables
i.e., R is a first-degree polynomial in each of its arguments .
b. R(p) has the same “truth table” as ( ), i.e.
, for j = 0, 1, …, ( 2n – 1 ), (29)
where tj is the jth input line of the truth table ; tj is an n-vector of binary
components such that
n
i 1
2n–i
tji = j, for j = 0, 1, …, ( 2n – 1 ). (30)
We stress that property (b) above does not suffice to produce a unique
and it must be supplemented by the requirement that be multiaffine to define
uniquely [47]. We also note that if the real transform and its arguments
are restricted to discrete binary values (i.e., if ) then
becomes the multilinear form of a switching function [58, 59], typically referred to
as the structure function [60, 61] in system reliability.
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 84
The definition above for implies that it is a function from the n-
dimensional real space to the real line . Though both R and p could
be free real values, they have a very interesting interpretations as probabilities, i.e.,
when restricted to the [0.0, 1.0] and [0.0, 1.0]n real intervals. An important property
of the real transform R(p) is that if its vector argument or input p is restricted to the
domain within the n-dimensional interval [ 0.0, 1.0 ]n, i.e. if 0.0 pi 1.0 for 1 i
n, then the image of R(p) will be restricted to the unit real interval [ 0.0, 1.0 ].
The probability transform is a bijective (one-to-one and onto) mapping from
the set of switching functions to the subset of multi-affine functions such that if the
function’s domain is the power binary set then its image belongs to the
binary set {0, 1}. Evidently, an R(p) restricted to binary values whenever its
arguments are restricted to binary values can produce the “truth table” that
completely specifies its inverse image ( ) via (29). On the other hand, a multi-
affine function of n variables is completely specified by independent conditions
[47], e.g., the ones in (29). In fact, such a function can be expressed by the finite
multivariable Taylor’s expansion [48]
R(p) =R() +
n
i 1
( R/pi )p= ( pi i ) + nji1
( 2R/pipj ) p= ( pi i ) ( pj j ) +
nkji1
( 3R/pipjpk ) p= ( pi i ) ( pj j ) ( pk k ) + ………..
+ (nR/p1p2…pn ) p= ( 1 ) ( 2 ) …….( n ). (31)
The expansion (31) has = 2n coefficients which are functions
of the expansion point p = . These coefficients can be viewed to form a spectrum
for the switching function = Note that the partial derivative of
in (31) w.r.t certain variables is independent of such variables, and therefore
this derivative is not affected by the assignments which are part of the
restriction . In particular, the nth-order derivative
is a constant, and its restriction via might be omitted.
The real transform as defined above is related to the vector of the
arithmetic transform in [49] as follows. The vector is simply a representation of
on a basis , namely
, (32)
The Walsh Spectrum and the Real Transform of a Switching Function: … 85
where is defined recursively as
, (33)
. (34)
For example,
,
,
In [49], the vector is obtained from the truth-table vector via
, (35)
where is defined recursively as
(36)
. (37)
Alternatively can be defined as a Kronecker product via:
, (38)
. (39)
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 86
The inverse of is defined by
,(40)
.(41)
or equivalently by
,(42)
.(43)
In the following, we show that the real transform of a switching function is
readily obtained via a disjoint sum-of-products expression of it. This observation in
very useful since there are literally hundreds of algorithms for producing such an
expression (see, e.g., [9-19]).
Theorem 1:
Let the switching function be expressed by the disjoint sum-of-products
(s-o-p) form
(44)
Where
Di Dj = 0 i, j , (44a)
k, (44b)
and none of the products Dk has any redundant literal (redundant literals can be
eliminated via idempotency of the AND operator Here,
and are the sets of indices for uncomplemented literals and complemented
literals in the product . Now, we let the expression
The Walsh Spectrum and the Real Transform of a Switching Function: … 87
(45)
Where
(46)
be obtained from (44) by replacing the AND operator by the multiplication
operator, the OR operator by the addition operator, each un-complemented variable
Xi by pi, and each complemented variable by (1 pi ). Then
Proof: For j = 1, 2,.. ., n, the degree of pj in T(Dk) for k = 1, 2, ..., m is at
most 1. Hence, its degree in T(f) is also at most 1, i.e. T(f) is a first-degree
polynomial in pj. This means that T(f) is a multi-affine function of p . Furthermore,
for each line of the “truth table” X = p = tj , T(Dk) has the same value (0 or 1) as Dk.
If f(tj) = 0, then all products Dk are 0, all T(Dk) are 0, and T(f) is 0. If f(tj) = l , then
exactly one product Dk is 1 since the products are disjoint. Only the transformed
product T(Dk) originating from this particular Dk is 1, while all other transformed
products are 0. Hence, T(f) is 1. Thus, f and T(f) have the same truth table. Since T(f)
is a multi-affine function with the same truth table as f, it is equal to
Example 2:
Let us consider a switching function
. (47)
which represents the failure of a 3-out-of-4:F (2-out-of-4:G) system [62, 63].
This function in a disjoint s-o-p form is [62]:
(48)
The real transform of f(X) is obtained by replacing the AND operations and
the OR operations in (48) by their algebraic counterparts of addition and
multiplications, and replacing variables Xi and by their expectations pi and
(1 pi), via:
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 88
R(p) = (1 p1 ) (1 p2 ) (1 p3 ) + (1 p1 ) (1 p2 ) p3 (1 p4 )
+ (1 p1 ) p2 (1 p3 ) (1 p4 ) + p1(1 p2 ) (1 p3 ) (1 p4 )
= 1 ( p1p2 + p1p3 + p1p4 + p2p3 + p2p4 + p3p4 ) +
2( p1p2p3 + p1p2p4 + p1p3p4 + p2p3p4) – 3p1p2p3p4 (49)
The “truth values” in the “truth table” of R(p) are necessary and sufficient for
determining R(p). The “truth table” of R(p) is similar to that of f(X) and is given by
the following special version of a Karnaugh map called probability map [11], shown
in Fig. 4
6. Computation of The Weight of a Switching Function
Theorem 2:
The weight of the switching function is given in terms of its real
transform as
(50)
where R(p) denotes the real transform of f(X), and means a vector of n
elements each of which is
Theorem 3:
Let the switching function f(X) be expressed by the disjoint sum-of-products
(s-o-p) form (44), then the weight of f(X) is given by
(51)
where is the number of irredundant literals in the product Dk, e. g.
The logical value 0 is not
considered a product . But anyhow we assume that so were we to
have it would contribute nothing to The minterm expansion of f(X)
is a special case of (44). for which (Dk) = n, k, and (51) produces the correct
result for this special case.
The Walsh Spectrum and the Real Transform of a Switching Function: … 89
The polarized weight of a switching function (wp (f)) is the sum of its truth
table entries when its truth table is recoded from {0, 1} to {+1, 1}, The value of
wp(f) is given by
wp (f) = ( No. of Off values of f ) * ( 1 )0 + ( No. of On values of f ) * ( 1 )
1
(52)
In particular, the polarized weight is given in terms of the real transform as
(53)
Example 3:
The 2-out-of-3 function discussed earlier in Example 1, is shown with a
disjoint coverage of non-overlapping loops in Fig. 1(a), and hence is expressed by
the disjoint s-o-p expression
(11a)
Therefore, its weight and polarized weight are obtained via (51) and (52) as
wt (f) = + + = 2 + 1 + 1 = 4,
in agreement with what can be deduced from Figs. 1(a) and 1(b).
7. The Walsh Spectrum in Terms of Subfunction Weights
Each row of can be viewed as an encoding {1 0, 11} of a particular odd
parity function , which involves
the (possibly empty) subset { } of the set of n elements of X. For a
given row, the variables involved are those corresponding to l’s in the binary
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 90
expansion of the row index, with X1 corresponding to the most significant bit [37].
The spectral coefficients are distinguished using subscripts identifying the variables
involved in the corresponding row. Such a subscript identification is useful, since
measures the ‘correlation’ between and the odd parity function
. In fact, equals the number of input vectors for which
(54a)
minus the number of input vectors for which
(54b)
The first spectral coefficient is denoted by r0 and measures the ‘correlation’
of f(X) to , and hence equals the number of input vectors for which f(X)
= 1, i.e. equals the weight wt(f(X)), namely:
(55)
There is a set of a ‘first-order’ spectral coefficients ri, i =1, 2,..., n, each of which
measures the correlation of f(X) to (Xi)= Xi , and hence equals
ri = wt(f(X | (Xi) = 0) – wt(f(X | (Xi) = 1)). (56)
Here, the notation denotes the subfunction or
restriction of when where This result is generalized into
the following theorem.
Theorem 4:
The spectral coefficients of a switching function are given by:
= wt(f(X | ) = 0)) wt(f(X |
= 1)). (57)
The Walsh Spectrum and the Real Transform of a Switching Function: … 91
Equations (55)-(57) are for the R-encoding. Note that for m = 0, we need
i.e., the odd-parity function of 0 variables, which is 0, and (57) reduces to
(55) if we understand that . The counterparts of (55)-(57) for
the S-encoding use polarized weights and {+1, 1} encoding for the odd-parity
functions. They are given by
, (55a)
(56a)
(57a)
Example 4:
Figure 5 demonstrates the matrix computation of the S spectrum as the matrix
product , where for convenience the column vector F is not placed to the right
of but instead the transpose of F (a row vector) is placed above . This is a
well-known trick used frequently [64] to enhance the readability of matrix
multiplication. Figure 6 illustrates Karnaugh maps for F with superimposed loops
representing pertinent odd parity functions. Each of these maps is a demonstration of
equations (55a)-(57a), since it gives the pertinent spectral coefficient as a sum of un-
circled entries (for which the pertinent ) minus the sum of encircled
entries (for which the pertinent ). For convenience, we indicate below
each map the odd-party function used, and the value of the corresponding spectral
coefficient. Figures 6 and 3 are in total agreement. In Fig. 6, all the maps have
identical entries and we take the difference of the sum of un-circled entries and that
of encircled ones, and in Fig. 3, each map has its own entries that we simply add. In
other words, we observe that we could redraw the Karnaugh maps in Fig. 6 without
loops by simply negating all the entries encircled by the loops therein, and then
calculating the spectral coefficients simply as the sums of each map entries. This
observation brings us exactly to the situation depicted in Fig. 3. In passing, we note
that since the 2-out-of-3 function is a totally symmetric function, its spectral
coefficients of equal numbers of subscripts are the same, i.e.,
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 92
(58a)
(58b)
Fig. (5). Matrix computation of the spectrum S for the 2-out-of-3 function
The Walsh Spectrum and the Real Transform of a Switching Function: … 93
Fig. (6). Computation of the Walsh Spectrum (S-encoding) for the 2-out-of-3 function
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 94
8. The Walsh Spectrum in Terms of The Real Transform
Equation (57a) can be rewritten for in the form
(59)
where is an (n m) variable function obtained as a subfunction or a
restriction of f(X) through one particular assignment of the m variables
which constitute a subset of X. Since the vectors X, Y and
are of dimensions n, m, and , respectively, there are and
such vectors, respectively. The set of vectors for which the number of l’s
in the components of the m-tuple is even (odd) is denoted by Em (Om). The
cardinality (number of elements) of each of the sets Em and Om (for ) is
. Due to (53), equation (59) can be reduced to
(60)
where R is the real transform of f(X). Since R is a multi-affine function,
mathematical induction can be used to reduce the expression for further
into
= ( 1)m+1
(61)
Equation (61) means that an mth
-order spectral coefficient is proportional to
the mth
derivative of the real transform of the switching function with respect to the
pertinent variables. That derivative is independent of these variables; an immediate
consequence of the multi-affine nature of R. As a corollary of (61), if the switching
function is vacuous in any input variable , i.e., is independent of , then
the spectral coefficients that contain in their subscript identification will be
zero valued Also if is partially symmetric in and , (and hence, and
The Walsh Spectrum and the Real Transform of a Switching Function: … 95
are interchangeable in , then and moreover any two spectral
coefficients that share the same subscripts and differ only in the replacement of by
are also equal. We stress that (59)-(61) are used for . The first spectral
coefficient is given by (55a), possibly combined with (53).
Example 5:
The real transform of the 2-out-of-3 function in Examples 1 and 3 can be
obtained from its disjoint s-o-p expression (54) as
= p1 p2 +(1 p1) p2 p3 + p1(1 p2) p3= p1 p2 + p2 p3+ p1p3 – 2 p1 p2 p3 (62)
The first spectral coefficient is obtained as
= 23 – 2
4 * R = 0,
while the spectral coefficients , , and can be obtained via (60) as
= [R (0 , , ) – R ( l, , )] = ,
= [R ( 0 , 0 , ) + R ( 1 , 1 , ) – R ( 0 , 1 , ) – R ( l , 0 , )]
= ,
= [R ( 0 , 0 , 0 ) + R ( l , l , 0 ) + R ( l , 0 , l ) + R ( 0 , l , 1 )
R ( l , 0 , 0 ) – R ( 0 , 1 , 0 ) – R ( 0 , 0 , l ) – R ( l , l , 1 )]
= .
Alternatively, these spectral coefficients can be found via (61) as
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 96
.
Example 6:
With a little abuse of notation, we use and to denote the real
transform and the two types of the spectral coefficients of the complement f ( X )
of f (X). The real transform is given by
R (p) = 1 – R (p). (63)
Equations (55)-(57) and (55a)–(57a) can be used to express the spectrum of f ( X )
as
, (64a)
, for m ≥ 1 (64b)
, (65a)
, for m ≥ 1 (65b)
It is interesting to note that the Dotson and Gobien algorithm [10] for
producing a disjoint s-o-p expression of f, also yields a disjoint s-o-p expression for
f as an offshoot output. The more compact expression among the disjoint ones for
f and f is to be chosen, and the corresponding spectrum is to be obtained. If the
spectrum of is obtained, the conversion to that of is straight via (64), or (65), and
vice versa.
The Walsh Spectrum and the Real Transform of a Switching Function: … 97
Example 7:
The basic spectrum for an n-variable function f(X) that expresses a single
literal Xi is obtained as follows:
f(X) = Xi , R(p) = pi
= 2n – 2
n+1 R( ) = 0,
= 2n ( R / pi) = 2
n,
while all the remaining spectral coefficients are 0’s.
Example 8:
The first spectral coefficient of a single product is obtained from (52)
and (55a) as follows:
( ) = . (66)
The real transform of is obtained from (46) as
(46a)
Thanks to (61), the higher-order spectral coefficient of is
proportional to the mth
-order derivative of w.r.t. . Now,
we consider three cases:
Case a: is not a subset of i.e., if one or more
of the variables does not appear in , and hence does not
appear in , then according to (61)
(67a)
Case b: If is equal to then according to (61):
(67b)
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 98
where is the cardinality of the set or simply the
number of complemented literals in the product whose literal subscripts are
considered in .
Case c: If is a proper subset of
(67c)
Note that (67c) includes (67b) when the cardinality
For example consider the product when viewed as a function of the four
variables According to (66), and (67) all its spectral coefficients are
0’s, except
9. Walsh Spectrum Computation Revisited
If the switching function f (X) is given by the disjoint s-o-p expression (44), then its
real transform is given by
(68)
The first spectral coefficient is obtained from (52), (55a), and (66) as
(69)
The Walsh Spectrum and the Real Transform of a Switching Function: … 99
Thanks to (61), (68), and the fact that the two linear operators of
differentiation and finite summation are interchangeable, the higher-order spectral
coefficients are obtained as
(70)
In (68) and (69), the coefficient and ( ) represent the
spectrum of single product Dk, and can be obtained via equations (66) and (67). The
set of equations (66), (67), (69), and (70) constitute a procedure for spectrum
computation. This procedure can be viewed as a formal development of the method
outlined in [6, pp. 35-40] and [34].
As a special case, f(X) can be expressed by its minterm expansion
(71)
where K is the set of indices for the true minterms of f. In this case, the spectrum of f
is given by
(72)
Here, can be obtained from (67c), or via (25), (26) and the fact that
( ) = +1 ( 1) if the number of uncomplemented variables among
in the minterm is even (odd). This observation can be used to
explain the technique of [65], and is equivalent to the basic definition of the
spectrum (equation (16)).
Example 9:
Consider the 4-variable function
(73)
The disjoint s-o-p representations of f and f are obtained in Fig. 7(a) and (b) as
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 100
, (73a)
. (73b)
Computation of the spectra of f and f via (66), (67), (69), and (70) are
illustrated on the spectral - coefficients maps of Figs. 7(c) and 7(d). The spectra
obtained satisfy (16) and (65).
The Walsh Spectrum and the Real Transform of a Switching Function: … 101
Fig. (7). Disjoint representation of (a) and (b) and spectral-coefficient maps for computing the
spectra of (c) and (d)
10. The Real Transform in Terms of The Spectrum
The real transform R(p) can be expanded about an arbitrary point p = t via a finite
multivariable Taylor’s expansion of 2n coefficients [36, equation (8)]. If t is chosen
as , and the conditions of equation (61) are invoked, then R(p) is given by
r13 = 3 - 3 = 0
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 102
(74)
Equation (74) expresses the real transform in terms of the spectral
coefficients. Since a switching function is uniquely defined by its real transform, eq.
(74) can be viewed as a computational method for the inverse spectrum, i.e. for
determining the switching function in terms of its spectral coefficients.
Example 10:
We know that the spectral coefficients of the 2-out-of-3 function are
and .
Hence, its real transform is
R( , ) = [ – (0)] + ( 4 (p1 – ) + 4 (p2 – )) + 4 (p3 – ) 0
+ ( )( 4) (p1 – ) (p2 – ) (p3 – )
(75)
Since f and R share the same truth table, the 2-out-of-3 function is easily
retrieved from (75).
Example 11:
Figure 8 summarizes some of the properties discussed herein for the sixteen
binary switching functions of the form
. (76)
The functions are displayed within the cells of a Karnaugh map of arguments
. Each map cell represents the name of the pertinent function, its
Boolean expression its real transform , its truth-table encodings
and and its corresponding Walsh spectra and , where
(77)
The Walsh Spectrum and the Real Transform of a Switching Function: … 103
(78)
, (79)
(80)
(81)
In passing, we note that Fig. 8 has a wealth of useful information that
facilitates easy reference, and which is even richer then the contents of a full paper
[66] that rediscovers the arithmetic transform as an arithmetic version of Boolean
algebra. The arrangement of the 16 binary switching functions on the Karnaugh map
of Fig. 8 is equivalent to their arrangement on a Hasse diagram (see, e.g., [67, 68]).
With such an arrangement, Fig. 8 can be used to reproduce Figs. 1 and 2 of [69] in
which the real transform is rediscovered under the disguised name of the
Generalized Boolean polynomial (GBP). Another reference which rediscovers the
real transform is [70] which labels it as a measure on a finite free Boolean algebra,
and uses it in the evaluation of sizes of queries applied to binary tables in relational
databases and to the identification of frequent sets of items and to association rules
in data mining.
Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 104
Fig. (8). The sixteen binary switching functions.
The Walsh Spectrum and the Real Transform of a Switching Function: … 105
11. Conclusions
The pictorial insight provided by the Karnaugh map makes the discussion of
many topics in combinatorics from a Karnaugh-map perspective a lucid and
appealing task. An earlier example supporting this point is the comparative study of
methods of system reliability analysis in a Karnaugh-map perspective [71].
The current paper is a clear demonstration of the utility of the Karnaugh map
as a pedagogical tool not only for its conventional usage in solving various coverage
problems for switching functions, but also for novel non-conventional uses in
explaining a variety of complex concepts. The task addressed herein is the
exposition of two famous representations for switching functions, namely, the Walsh
spectrum and the real transform. The literature on these two representations is both
complex and confusing due to the existence of different definitions used by different
communities of researchers. This paper is a serious attempt for a unified and
simplified treatment of the subject fully addressing sources of ambiguity or
discrepancy. The interrelationship between the Walsh spectrum and real transform is
thoroughly and explicitly expressed and utilized as a basis for computational
procedures of one representation in terms of the other.
Acknowledgement
The authors are deeply indebted and sincerely grateful to Dr. Ahmad Ali Rushdi of
ALFARABI Consulting Group, California, USA for the help he offered in the
preparation of this manuscript.
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Ali Muhammad Ali Rushdi and Fares Ahmad Muhammad Ghaleb 112
قسم اهلندسة الكهربائية وهندسة احلاسبات، كلية اهلندسة،
، اململكة العربية السعودية21589، جامعة امللك عبد العزيز، جدة، 80204. .ص. ب
م(24/4/2014م ؛ قبل للنشر بتاريخ 3/2/2014)قدم للنشر بتاريخ
تستخدم ورقة البحث هذه منظور خريطة كارنوه يف استقصاء التعريفات وشرح اخلصائص
ديدة واكتشاف التزاوجات بني طيف والش والتحويل احلقيقي لدالة تبديلية. واستحداث إجراءات حسابية ج
يتم استخدام خرائط كارنوه مناسبة يف شرح كيفية حساب طيف والش كما يعرف يف علم التعمية. يلي ذلك
تقديم تعريف بديل هلذا الطيف مستخدم يف التصميم الرقمي واجملاالت اللصيقة به، ويردف ذلك
ءات املصفوفية الالزمة حلسابه. ثم يتم تعريف التحويل احلقيقي لدالة تبديلية كدالة حقيقية يف باإلجرا
متغريات حقيقية، وجيري التمييز بوضوح بني هذا التعريف وتعريفات شبيهة به مثل الصيغة عديدة اخلطية أو
خلريطة كارنوه تدعى التحويل احلسابي. كما يتم شرح التحويل احلقيقي تصويريا من خالل صيغة خاصة
خريطة االحتماالت. تستعمل خرائط كارنوه أيضا لبيان كيفية حساب املعامالت الطيفية املستخدمة يف
التصميم الرقمي كوزن للدالة التبديلية وكأوزان للدوال الفرعية الناشئة من تقييد هذه الدالة. تتواءم هذه
يف علم التعمية. يستفاد من التزاوجات املستحدثة بني طيف اخلرائط مع اخلرائط السابقة للطيف املستعمل
والش والتحويل احلقيقي يف صياغة طريقتني مبسطتني حلساب الطيف بداللة التحويل احلقيقي من خالل
بعض املعاونة من خرائط كارنوه. تختتم الورقة حبل املسألة العكسية املعنية حبساب التحويل احلقيقي بداللة
طيف والش.
Journal of Engineering and Computer Sciences
Qassim University, Vol. 7, No. 2, pp. 113-133 (July 2014/Shaban 1435H)
113
Exploratory Study of Comparison between Ultimate Strength
and Performance Based Design for Reinforced Concrete Structures
Subjected to Seismic Loading
Muhammad Saleem
Department of Basic Engineering, College of Engineering, University of Dammam,
Kingdom of Saudi Arabia
(Received 30/08/2014; Accepted for publication 31/01/2015)
Abstract. Structural design of RC buildings has been going through a transition from allowable stress
design approach (ASD) to the ultimate strength design approach (USD) leading to the present
performance based design philosophy. PBD ensures that the members and structure as a whole reach a desired demand level and includes service, strength and capacity requirements. In the presented study a
nonlinear analysis and design has been conducted on an intermediate rise RC building considering all
seismic zones as specified by Unified Building Code in order to compare the USD and PBD approaches. Furthermore a comparison of different shear wall modeling techniques is presented along with
modification to design methodology to make the structural design more economical. For the USD, UBC-
97 and for PBD, ATC-40, FEMA-273 and FEMA-356 guidelines are used. The presented results reveal that the USD approach is unable to predict the soft story mechanism and shear failure governed the final
failure in all seismic zones, whereas the flexural design was found to be adequate as inelastic
deformations remained within the acceptable limits. However, capacity analyses revealed a non uniform damage throughout the building in contrast to PBD approach.
Keywords: Intermediate rise, Reinforced concrete structures, Performance based design, Non-linear analysis, Ultimate strength design.
Muhammad Saleem 114
1. Introduction
The kingdom of Saudi Arabia has been blessed with vast natural resources which
have resulted in an immense construction boom in the past few decades. Owing to
the construction boom a large number of reinforced concrete structures have been
constructed, preferring them over the steel construction, as they are cheaper to
construct, require less maintenance and can be constructed using locally available
resources and unskilled manpower. Many of these reinforced concrete buildings are
categorized as intermediate rise buildings. However in light of the rapid boom in
construction industry many buildings in the kingdom have been constructed using
the old design approach of the USD which lacks in providing real feedback to
engineers and may suffer large damage in the event of dynamic loading such as
earthquake, wind and blast loading. Furthermore the old design techniques are
unrealistic as they lack to take into account the full potential of the material
performance and lead to an uneconomical design. For the last several decades
research has been going on in the field of structural design for reinforced concrete
buildings. RC structures are particularly vulnerable to earthquakes. Earthquakes are
a common phenomenon throughout the world and many of the world's most
populated regions have been situated in areas of high seismic activities. Hence in
order to better protect life and infrastructure governments all over the world have
been encouraging research in the field of structural dynamics, structural retrofitting,
strengthening and rehabilitation. Many new methods have been invented along with
the refinements of old methods [1-6] which can be applied to the structure to make
them safe against earthquake. Since earthquakes are unpredictable and random in
nature, the engineering tools need to be developed for analyzing structural response
under the action of such forces.
Performance based design has been gaining a new dimension in the seismic
evaluation philosophy wherein the ground acceleration is considered. The
approaches towards designing of structural members and in-particular the cross-
sections have been changing over time in the past few decades. There has been an
ongoing shift from allowable stress design approach (ASD) to ultimate strength
design approach (USD) leading to the present strength and performance based
design (PBD) in light of achieving economy by more realistic material and structural
performance. The numerical and analytical methodologies used in each of these
design approaches are independent of each other. This is necessary in certain
aspects, but while considering the flexural design of reinforced concrete members,
an integrated approach is needed that satisfies the requirements of serviceability,
strength and performance. For lateral loads such as earthquake and high speed wind,
performance is generally of main concern. Serviceability design ensures that
deflections and vibrations for service loads are within limits. Although it checks the
maximum stresses in the materials at design load levels but it is oblivious to the
strength requirements and capacity ratio. The USD on the other hand ensures that a
certain load factor against overload is available within a member but is silent on the
topic of structural performance once the load exceeds the design level or in case the
Exploratory Study of Comparison between Ultimate Strength: … 115
load is less than the assumed overload value. The PBD however, ensures that the
member and structure as a whole reaches a desired demand level and included
service, strength and capacity based design requirements.
Hence with the advent of new design approach such a PBD it is imperative
that the performance evaluation of the old design method be conducted to evaluate
its short coming and suggest modifications for the practicing engineers. In this
respect the paper presents an exploratory study to evaluate to the performance of
ultimate strength based designed structure using Performance Based Design, the
moment-rotation relationship for plastic hinges adopted for the presented work is
shown in Fig. (1). Rozman et. al [7] presented the research in which a three stores
RC frame structure SPEAR is modeled as different variants to compare the seismic
response of these variants. The first variant was considering the older building
construction, only designed for vertical loading without considering the seismic
loading while the 2nd
and 3rd
building models are designed according to Eurocode-8.
They concluded that the structures modeled as per new standard EC8 code are safer
than the old designed structures. Proper detailing of reinforcement which ensures
suitable plastic mechanism results in greater global and local ductility of structures.
Fahjan et. al [8] and Ioannis et. al [9] worked for modeling of shear walls in
buildings for non-linear analysis and presented mid-pier approach and smeared
approach for shear wall modeling. They showed that these shear wall modeling
techniques are suitable in simulating the nonlinear behavior of shear walls. Chung
and Kadid et. al [10,11] presented a technique for the modeling of the parameters of
plastic hinge properties (PHP) for structure containing RC wall in the pushover
analysis introduced by Anil et. al [12,13].
Fig. (1). Moment-rotation relation for plastic hinges
Response-2000 and Membrane-2000 codes were used to calculate the
nonlinear relationship between the lateral shear force and lateral deformation of RC
wall. The PHP values of each parameter were a product of two parameters α and β
Muhammad Saleem 116
for the pushover analysis in SAP2000. Experimentation was conducted to confirm
the accuracy of this newly proposed method. It revealed that the method could be
used professionally to help engineers to conduct the performance based design of
structures containing RC shear wall using the SAP2000. Previous research work
[14-18] has shown that shear failure might be the governing cause of failure for
older designed structures which might prohibit them from achieving their
serviceability objectives. In SAP2000 hinges can be assigned to frame elements at
any location along its length. The idea of the presented work is to widen the
application of this method to the RC structures containing shear wall. The advantage
of modeling RC wall as a wide and flat column, frame elements, is that SAP2000
[20] can not only considers steel reinforcements exactly, but could also assign the
hinges based on FEMA-356, hence the new modification could help professional
engineers in conducting a more realistic and economic analysis.
2. Objectives
The presented research work aims to focus on comparing the design evaluation of
USD approach and the PBD approach for intermediate rise reinforced concrete
building subjected to seismic loading under all zoning categories as suggested in
Unified Building Code, objectives of this research study are as explained below:
1. To compare the design framework of both the USD and PBD and to evaluate
and compare the performance level of the structure after earthquake.
2. To modify the design methodology for seismic design based on the
numerical results.
3. To highlight shortcomings in both of the design approaches.
3. Numerical Modeling Description
Fig. (2) presents the isometric view of the RC building selected for the proposed
study. The building is a ten story structure which consists of moment resisting
frame, shear walls at the four corners of the building and two lift wells. Fig. (3)
depicts the typical plan of the building with the details of the moment resisting
frame are mentioned in Table (1). The work was carried out in two stages. In stage-I,
the analysis and design was carried out in accordance with UBC-97 [19] for all the
zones. In stage-II, the performance of UBC-97 designed building was investigated in
detail. The selected building has a combination of RC moment resisting frames and
shear walls. Gravity loads were carried by moment resisting frame, whereas lateral
loads were carried by the concrete shear wall and lift wells. Beams and columns
were modeled as frame elements with the centerlines joined at the nodes. The load
of slab was transferred to the beams depending upon the slab width support
approach. Roof slab was 150 mm thick whereas the typical floor was 200 mm thick.
For shear wall modeling, mid-pier approach was used. In mid-pier approach, shear
Exploratory Study of Comparison between Ultimate Strength: … 117
wall were modeled as a frame element having the same cross-sectional stiffness
properties as that of shear wall and the beams framing into shear wall were
connected to mid-pier with the help of rigid beam offset having rigid zone factor
equals to 1. It was a factor used to define the percentage of the zone specified
through end offsets to be taken as fully rigid. Lateral load resistance was provided
by concrete shear walls and lift-well walls in two perpendicular directions.
According to their relative stiffness and rigidity, total base shear due to seismic load
was resisted by shear walls and lift-well walls. L-shaped shear walls were present at
the four corners of the building. Lift-wells were eccentrically situated with respect to
the building geometric center. The ground floor columns were assumed to be fixed
at the base. Building was analyzed and designed for all seismic zones of Pakistan as
defined using UBC-97 i.e., zone-1, zone-2A, zone-2B, zone-3 and zone-4. The
stiffness for cracking of the members was taken as per ACI 318-08 and was 0.7EIg
for columns and shear walls (mid pier), 0.35EIg for beams and 0.25EIg for slab.
Fig. (2). Isometric view of the structure
Muhammad Saleem 118
Fig. (3). Typical story plan of the structure
Table (1). Description of the moment-resisting frame
Sr. No. Description Values
1 Number of stories 10
2 Number of bays along X-direction 3
3 Number of bays along Y-direction 3
4 Story height 3m
5 Bay width along X-direction 8m
6 Bay width along Y-direction 8m
4. Load and Material Modeling Details
Table (2) provides the description of various types of loading and material inputs used
for modeling the RC building. All the vertical loads due to structural and non-structural
components of building e.g. self-weight, masonry walls, partitions, floors and roof
finishes along with all other permanent construction were grouped as dead load. Loads
due to occupants, moveable machineries and equipment, vehicular load and impact
loadings were treated under the category of live loads. Zone factor, occupancy category,
type of moment resisting frame, respective R-values and soil profile types were
considered as seismic parameters. Reinforcement yield strength was considered as 414
MPa and concrete strength for different elements is provided as below;
APPENDIX-A GRAPHICAL DESCRIPTION OF THE BUILDING
3 Bays @ 8 m c/c = 24 m 2
7'-
6''
8 m
8
m
8 m
3 B
ays @
8 m
c/c
= 2
4 m
8 m 8 m 8 m
Exploratory Study of Comparison between Ultimate Strength: … 119
Table (2). Loading and material input details
Sr. No. Description Values
DEAD LOADS
1 Finishes 1.44 kN/m2
2 Partitions 0.961 kN/m2
3 Roofing 0.96 kN/m2
4 Plaster 0.48 kN/m2
LIVE LOADS
5 Typical Floor 4.8 kN/m2
6 Roof Floor 1.44 kN/m2
SEISMIC INPUT
Description zone-1 zone-2A zone-2B zone-3 zone-4
7 Units mm-KN mm-KN mm-KN mm-KN mm-KN
8 Direction X & Y X & Y X & Y X & Y X & Y
9 Occupancy "I" 1 1 1 1 1
10 Moment Resisting
Frame OMRF IMRF IMRF SMRF SMRF
11 Peak Ground
Acceleration "g" 0.05-0.08 0.08-0.16 0.16-0.24 0.24-0.32 > 0.32
12 R value 5.5 6.5 6.5 8.5 8.5
13 Soil Profile Type SD SD SD SD SD
14 Z 0.075 0.15 0.2 0.3 0.4
15 Ca 0.12 0.22 0.28 0.36 0.44
16 Cv 0.18 0.32 0.4 0.54 0.64
17 hn 30 m 30 m 30 m 30 m 30 m
18 Ct 0.03 0.03 0.03 0.03 0.03
19 Eccentricity 5% 5% 5% 5% 5%
20 Eccentricity Overrides NO NO NO NO NO
21 Period Calculation Auto Auto Auto Auto Auto
22 Top Story Roof Roof Roof Roof Roof
23 Bottom Story Ground Ground Ground Ground Ground
MATERIAL INPUT
24 Concrete Strength Columns Shear Wall Beams
28 MPa 28 MPa 21 MPa
Muhammad Saleem 120
Ultimate strength approach was based on UBC-97 for seismic design,
whereas, for performance based approach, pushover analysis was used. To check the
performance objective, Capacity Spectrum Method (CSM) as per ATC-40 was
applied. For pushover analysis, concentrated plastic hinges as defined in FEMA-356
were used for beams, flexure (M3) and shear (V2) hinges, for columns and mid piers
(shear walls), bi-axial bending (P-M2-M3) and shear (V2 and V3) hinges are
employed. For performance objective, the zone-wise demand spectrum as per UBC-
97 for SD soil profile type was used in the analyses. Different computer models in
software were generated for the study, while the seismic values were calculated as
shown below in Table (3).
𝑇𝑎 = 𝐶𝑡 ℎ𝑛3/4
𝑉 = 𝐶𝑣 𝐼 𝑊𝑅 𝑇⁄
If T ≤ 0.7 sec, then Ft = 0
If T > 0.7 sec, then Ft = 0.07 T V
Where,
Ft ≤ 0.25 V
Table (3). Seismic Values for Model Input
Description Zone-1 Zone -2A Zone -2B Zone -3 Zone -4
Ta 0.9375 0.9375 0.9375 0.9375 0.9375
T (sec) 1.3124 1.3124 1.3124 1.3124 1.2187
W (kN) 77562.439 79054.435 78405.268 78396.927 79488.159
V (kN) 1934.115 2965.404 3676.316 3794.852 4910.991
Ft (kN) 177.688 272.434 337.746 348.636 418.949
5. Shear wall Modeling Techniques
Fig. (4) shows the two techniques used for modeling of shear walls namely the
multilayered shell element approach and the mid-pier approach. In multilayered
shell element approach, shell elements were used in which non-linear smeared layers
of concrete and steel were defined. Whereas, in mid pier modeling approach, frame
elements were used to model shear walls having same stiffness as that of shear wall.
Rigid beam offsets having rigidity equals to 1 were defined to connect mid pier with
the beams framing into wall. From the plastic hinge distribution results presented in
Fig. (5); it is evident that in mid pier modeling approach, the hinge formed at the
base at 243 mm top displacement. Similarly, from the concrete and steel stress
Exploratory Study of Comparison between Ultimate Strength: … 121
distribution in as shown in Fig. (6), in multilayered shell element modeling, steel
stresses at the bottom of shear wall crossed the yield value at 273 mm top
displacement which indicated that the hinge is formed at the base of the shear wall.
Hence, both the approaches were in close agreement and were reasonable for
simulating non-linear behavior of shear walls with mid-pier approach resulting in a
more conservative approach. However in the presented work the mid pier modeling
technique was applied in modeling for its conservative approach and since FEMA-
356 guidelines were used for predicting non-linear behavior of frame elements.
Multilayered Shell Element Approach Mid-Pier Approach
Fig. (4). Shear wall modeling techniques
Muhammad Saleem 122
Fig. (5). Plastic hinge distribution
Fig. (6). Concrete and steel stress distribution
Exploratory Study of Comparison between Ultimate Strength: … 123
6. Performance Evaluation and Discussion
6.1 Zone-1
The building was designed as per UBC-97 in zone-1 and then pushed by applying
mode-1 based on lateral load distribution; displacements were monitored
accordingly in global X-direction. For nonlinearity, M3 hinges were assigned to
beams, P-M2-M3, V2 and V3 hinges were assigned to columns and shear walls.
Capacity curves for building designed for zone-1 is shown in Fig. (7). Firstly the
columns and shear walls of the building were modeled with bi-axial flexure and
shear hinges, in this case shear failure starts first. It can be seen from Fig. (7); that
the hinges could withstand 5000 KN of base shear at the top displacement of
545mm. Then, the columns and shear walls of the building were modeled with only
bi-axial flexure hinges. After the structure was analyzed it was seen that it could
withstand more than 6000 KN of base shear at the top displacement of 725mm. The
nominal base shear calculated based on UBC-97 equivalent lateral load method was
1935 KN. The capacity curve indicated that the ultimate design capacity was over
designed and the building could resist more than 6000 KN of base shear without any
excessive damage. It was evident from the analysis that due to bi-axial flexure
hinges only, no excessive damage occurred in the shear walls and columns while
some flexure hinges in the beams reached their collapse stage. Damage degree due
to the presence of shear hinges in the model showed that excessive damage had
occurred in shear walls and columns at the 5th
story level. Flexure hinges in beams
remained within the acceptable limit of immediate occupancy (IO). Failure was
concentrated in columns and shear-walls at 5th
story and showed the extent of soft
story mechanism. Furthermore, it was evaluated that at the performance point no
excessive damage and yielding occurred in the building.
Fig. (7). Capacity curve for zone-1
Muhammad Saleem 124
6.2 Zone-2A
For analysis in zone-2A the building was designed as per UBC-97
requirements and then pushed by applying mode-1 lateral load. For non-linearity,
M3 hinges were assigned to beams, P-M2-M3 and V2 or V3 hinges were assigned to
columns and shear walls. Capacity curve for the building is shown in Fig. (8). Firstly
the columns and shear walls of the building were modeled with bi-axial flexure and
shear hinges, this resulted in shear hinge failure. It was seen that it could sustain
5900 KN of base shear at the top displacement of 580 mm. Then, the columns and
shear walls of the building were modeled with only bi-axial flexure hinges. It was
seen that the building could sustain more than 7500 KN of base shear at the top
displacement of 800 mm. The nominal base shear calculated by UBC-97 equivalent
lateral load distribution method was 2965 KN which indicated that the lateral load
capacity of the structure was much greater than the required value and the building
could resist more than 7500 KN of base shear without any excessive damage. It was
evident from the result evaluation that no excessive damage occurred in shear walls
and columns in case of only bi-axial flexure hinge. Some flexure hinges in beams
reached the collapse stage. However damage degree due to both bi-axial flexure and
shear hinges showed that excessive damage had occurred in mid piers and columns
at the 4th
story level. Failure was concentrated in columns and shear-walls at 4th
story and showed the extent of soft story mechanism. Furthermore at the
performance point, no excessive damage and yielding has appeared in the structure
and the building was well within the serviceable range.
Fig. (8). Capacity curve for zone-2A
Exploratory Study of Comparison between Ultimate Strength: … 125
6.3 Zone-2B
The building was designed and analyzed using the input of zone-2B as
described in Table (2). The non-linearity of the model was considered. Firstly the
columns and shear walls of the building were modeled with bi-axial flexure and
shear hinges in which case the shear hinge triggers failure first and the capacity
curve is shown in Fig. (9). It can be seen that the structure was able to withstand
7400 KN of base shear at the top displacement of 600mm. Then, the columns and
shear wall of the building were modeled with only bi-axial flexure hinges and it was
seen that the capacity increased up to 9000 KN of base shear at the top displacement
of 950 mm. The nominal base shear calculated by UBC-97 equivalent lateral load
distribution was 3675 KN and which indicated that the ultimate design capacity was
large. It was evident that due to only bi-axial flexure hinge, no excessive damage
occurred. Damage degree due to the presence of shear hinges in the model showed
that excessive damage occurred in the shear walls and columns of the 6th
story level.
Flexure hinges in beams remained within the acceptable limit. Failure was
concentrated in columns and mid piers at 6th
story and at the performance point no
excessive damage or yielding appeared in the building.
Fig. (9). Capacity curve for zone-2B
6.4 Zone-3
Capacity curves for building designed for zone-3 is shown in Fig. (10). As
the columns and shear walls of the building were modeled with bi-axial flexure and
shear hinges as before the shear hinge triggered failure first at the base shear level of
6900 KN and the top displacement of 500 mm. Afterwards, the columns and shear-
Muhammad Saleem 126
walls of the building were modeled with only bi-axial flexure hinges and It could
withstand base shear upto7775 KN at the top displacement of 650mm. The nominal
base shear calculated by UBC-97 equivalent lateral load distribution was 3795 KN
which was much lower than the performance level achieved by the building. Some
flexure hinges in beams reached the life safety stage and the damage degree due to
the presence of shear hinges in the model showed that excessive damage occurred in
shear walls and columns at the 6th
story level. Flexure hinges in beams remains
within the acceptable limit of life safety (LS) while the failure was concentrated in
columns and shear walls at 6th
story and lead to a soft story mechanism, see Fig.
(11). However, Fig. (12) showed that at the performance point, no excessive damage
or yielding appeared in the building.
Fig. (10). Capacity curve for zone-3
Exploratory Study of Comparison between Ultimate Strength: … 127
Fig. (11). Damage degree for flexure and shear
hinges (P-M2-M3, M2, M3, V2 and V3)
Fig. (12). Damage degree at performance
point zone-3
6.5 Zone-4
Zone-4 was the most sever zone with regards to the performance
requirements. Hence in order to ascertain accuracy the building was modeled with
and without slab. For non-linearity in beams M3 hinges were assigned, P-M2-M3, V2
and V3 hinges were assigned to columns and shear walls. Firstly the columns and
shear walls of the building were modeled with bi-axial flexure and shear hinges, the
capacity curve is as shown in Fig. (13). It can be concluded that the structure could
withstand 5900 KN of base shear at the top displacement of 490 mm. Then, the
columns and shear walls of the building were modeled with only bi-axial flexure
hinges. It resulted in more than 12000 KN of base shear at the top displacement of
1700 mm. The nominal base shear calculated by UBC-97 using the equivalent
lateral load distribution method was 4915 KN which indicates that the lateral load
capacity of the building is additional. It was evident from the Fig. (13); that slab
played its role as a rigid diaphragm, when the building was modeled with slab as
shell elements it adds stiffness to the building and better capacity curve was
achieved with and without shear hinges in columns and shear walls. It was evident
from the Fig. (14); that due to only bi-axial flexure hinge, no excessive damage
occurred. However, most of the beams of the exterior frame may have reached the
collapse stage and showed that excessive damage occurred in shear walls and
columns at the 5th
story level. Flexure hinges in beams remained within the
acceptable limit and failure was concentrated in columns and shear walls at 5th
story.
Fig. (15) showed that excessive damage or yielding due to shear failure had
appeared in the building even at the performance point.
Muhammad Saleem 128
Fig. (13). Capacity curve for zone-3
Fig. (14). Damage degree for flexure and shear
hinges (P-M2-M3, M2, M3, V2 and V3)
Fig. (15). Damage degree at performance
point zone-4
7. Design Methodology Modification
The building designed in zone-4 showed excessive damage degree owing to shear
failure. Hence it was decided to carry out a detailed performance based design to
ascertain the performance of building against realistic demand. Therefore, the
Exploratory Study of Comparison between Ultimate Strength: … 129
building was re-analyzed and re-designed by selecting the performance objective of
Life Safety (LS). Performance based design facilitates the engineers in deciding the
performance objective and demand for the building. The members which cross the
life safety performance limit were revised by increasing the stiffness. This process
was continued until the desired performance objective was achieved. From the
capacity curve shown in Fig. (16), it is evident that by revising the cross-sectional
properties and corresponding design of the members crossing the Life Safety
performance objective, better performance of the building was achieved for the same
demand. Before deciding the higher performance objective, the building could resist
5900 KN of base shear with top displacement of 490 mm. While, after revising the
performance objective of LS, the building could resist 7400 KN of base shear with
top displacement of 800 mm. By comparing the ATC-40 capacity spectrum before
and after deciding the performance objective as shown in Fig. (17) and Fig. (18), it
was observed that the performance point after revising the stiffness of the members
improved from base shear of 3197.845 KN and displacement of 466.806 mm to base
shear of 6654.527 KN and displacement of 598.238 mm. Revised analysis and
design of the building in zone-4 revealed that performance based design is the more
suitable design methodology in high seismic regions as ultimate strength design
remained ambiguous about the performance of discrete elements of the building
when the base shear calculated as per UBC-97 equivalent lateral load distribution
was exceeded.
Fig. (16). Comparison of capacity curve in zone-4
Muhammad Saleem 130
Fig. (17). ATC-40 Capacity spectrum before
deciding the performance objective
Fig. (18). ATC-40 Capacity spectrum after
deciding the performance objective of LS
8. Conclusions
An exploratory study regarding the design approaches of USD and PBD is presented
for a ten story reinforced concrete building analyzed under various seismic zones as
defined by UBC-97. From the detailed analysis, design and assessment of the
building the following conclusions can be drawn;
1. It is evident from the presented results that for intermediate rise buildings
designed using USD; shear failure governs the final failure mode in all seismic
zones.
2. Lowest performance point occurred in zone-4 owing to the higher
performance requirements. Hence the structures designed should be made more
ductile and inelastic deformations should be allowed to occur at critical sections to
dissipate energy.
3. USD based design was found to be adequate as inelastic deformations
remained within the acceptable limits, however the approach lead to an
uneconomical design since the realistic material capacity were not considered.
4. It was evident from the damage degree observation that the building
designed using USD showed soft story failure while the building designed using
PBD showed uniform damage throughout the building. Hence the soft story failure
mechanism can be avoided by adopting PBD approach.
5. Comparison of mid pier and multilayered shear wall approach reveals that
in mid pier approach, plastic hinge was formed at the base of mid pier at the
displacement of 242 mm. Similarly in multilayered shell modeling, steel crosses
yield stress at the base at the displacement of 273 mm. Hence it can be stated that
these techniques are in reasonable agreement, while mid pier approach being more
conservative, these approaches can be used for simulating nonlinear response of
shear walls.
Exploratory Study of Comparison between Ultimate Strength: … 131
6. In mid pier approach, rigid zone factor should be selected with much care
as it effects the moment redistribution in frame elements. The rigid beam offset with
rigidity equals to 1 effect the moment distribution within a member up to 3 to 5%.
9. References
[1] Ishibashi, T. and Tsukishima, D., "Seismic damage of and seismic
rehabilitation techniques for railway reinforced concrete structures", Journal
of Advanced Concrete Technology, Vol 7, No. 3, (2009), pp. 287-296.
[2] Cook, R.A., Doerr, G.T. and Klingner, R.E., "Bond stress model for design of
adhesive anchors", ACI Structural Journal, Vol. 90, No. 5, (1993), pp. 514–
24.
[3] Cook, R.A., Kunz J., Fuchs W. and Konz, R.C., "Behavior and design of
single adhesive anchors under tensile load in uncracked concrete", ACI
Structural Journal, Vol. 95, No. 1, (1998), pp. 9–26.
[4] Zamora, N.A., Cook, R.A., Konz, R.C. and Consolazio, G.R., "Behavior and
design of single, headed and unheaded, grouted anchors under tensile load",
ACI Structural Journal, Vol. 100, No. 2, (2003), pp. 222–30.
[5] Saleem, M. and Tsubaki, T., "Multi-layer model for pull-out behavior of post-
installed anchor", Proc. FRAMCOS-7, Fracture Mechanics of Concrete
Structures, AEDIFICATIO publishers, Germany, Vol II, (2010), pp. 823-830.
[6] Ashraf, M., "Development of low-cost and efficient retrofitting technique for
unreinforced masonry buildings", Ph.D. Dissertation, University of
Engineering & Technology, Peshawar, Pakistan, (2010).
[7] Matez, R. and Peter, F., "Seismic Response of a RC Frame Building Designed
According To Old and Modern Practices", Bull Earthquake Engineering,
Vol. II, (2009), pp. 779-799.
[8] Fahjan, Y. M., Kubin, J. and Tan, M. T.,: "Nonlinear Analysis Methods for
Reinforced Concrete Buildings with Shear Walls”, Proc. of 14th
International
European Conference on Earthquake Engineering, Macedonia, (2010).
[9] Ioannis, P. G., "Seismic Assessment Of A RC Building According to FEMA
356 And Eurocode 8”, 16th
International Conference on Concrete, Paphos,
Cyprus, Vol. 10, (2009), pp. 21-23.
[10] Chung, Y. W. and Shaing, Y. H., "Pushover Analysis For Structure
Containing RC walls", The 2nd
International Conference on Urban Disaster
Reduction, Taipei, Taiwan (2007), pp. 27-29.
[11] Kadid, A. and Boumrkik, A., "Pushover Analysis of Reinforced Concrete
Frame Structures”, Asian Journal of Civil Engineering - Building and
Housing, Vol. 9, (2008), pp. 75-83.
Muhammad Saleem 132
[12] Anil, K. and Rakesh K., "A Modal Pushover Analysis Procedure For
Estimating Seismic Demands For Buildings", Earthquake Engineering And
Structural Dynamics (2002), pp. 561-582.
[13] Anil, K. and Rakesh K., "A Modal Pushover Analysis Procedure To Estimate
Seismic Demands For Unsymmetric-Plan Buildings", Earthquake
Engineering And Structural Dynamics (2004), pp. 903-927.
[14] Rahul, R., Limin, J. and Atila, Z., "Pushover Analysis of A 19 Story Concrete
Shear Wall Building", 13th
World Conference On Earthquake Engineering,
Vancouver, B.C., Canada (2004).
[15] Federal Emergency Management Agency, "Pre-standard and Commentary
for Seismic Rehabilitation of Buildings", FEMA 356 (2000).
[16] Peter, F., "Capacity Spectrum Method Based On Inelastic Demand Spectra",
Earthquake Engineering and Structural Dynamics (1999), pp. 979-993.
[17] Peter, F. and Eeri, M., "A Nonlinear Analysis Method for Performance Based
Seismic Design", Earthquake Spectra, Vol. 16, (2000), pp. 573-592.
[18] Fahjan, Y. M., Kubin, J. and Tan, M. T., "Comparison Of Practical
Approaches For Modelling With Shear Walls In Structural Analysis Of
Buildings", 14th
World Conference On Earthquake Engineering, Beijing,
China (2008).
[19] UBC-97, Uniform Building Code, Earthquake Design (1997), Vol. 2, No. IV.
[20] CSI, SAP2000 V-12, Integrated finite element analysis and design of structures
basic analysis reference manual, Berkeley, Computers and Structures Inc,
(2008).
Exploratory Study of Comparison between Ultimate Strength: … 133
اهلندسة , جامعة الدمام ,اململكة العربية السعوديةقسم اهلندسة األساسية , كلية [email protected]
(31/1/2015؛ وقبل للنشر يف 30/08/2014)قدم للنشر يف
مازال مستمر من خالل االنتقال من أسلوب تصميم اإلجهاد التصميم اإلنشائي للمباني
مما يؤدي إىل فلسفة التصميم (USD) نهاية املطاف إىل اسلوب تصميم القوة القصوى يف (ASD) املسموح به
يضمن ان القوام واهلياكل اإلنشائية تصل إىل (PBDاألداء على اساس التصميم ) .القائمة على األداء احلالي
املستوى املطلوب وتشمل متطلبات اخلدمة, والقوة والقدرة. يف الدراسة اليت قدمت مت إجراء التصميم و
طي على ارتفاع متوسط للمبنى اإلنشائي مع االعتبار يف مجيع املناطق الزلزالية على النحو التحليل الغري خ
واألداء على اساس التصميم (USD) الذي حيدده قانون البناء املوحد من أجل مقارنة تصميم القوى القصوى
(PBD) جن مع بع وعالوة على ذلك يتم تقديم مقارنة يف جدار القص بني خمتلف التقنيات جنبا إىل
,PBD ,ATC-40وUSD , UBC-97التعديالت لتصميم املنهجية جلعل التصميم اإلنشائي أكثر اقتصادا.
FEMA-273 و FEMA-356 تستخدم كمبادئ توجيهية. تظهر النتائج املقدمة أن منهجية تصميم القوى
الذي حيكم االنهيار النهائي يف مجيع غري قادر على التنبؤ على آلية الليونة وانهيار القص (USD) القصوى
املناطق الزلزالية, يف حني مت العثور على تصميم االحنناءات لتكون كافية حيث ظلت التشوهات الغري مرنة
ضمن احلدود املقبولة. ومع ذلك, كشفت التحليالت قدرة الضرر الغري موحدة يف مجيع أحناء املبنى على
(PBDلتصميم)النقي من نهج األداء على اساس ا
Muhammad Saleem 134
Journal of Engineering and Computer Sciences
Qassim University, Vol. 7, No. 2, pp. 135-156 (July 2014/Shaban 1435H)
135
Improving the Power Conversion Efficiency of the Solar Cells
Abou El-Maaty M. Aly 1,2
1-Power Electronics and Energy Conversion Department, ERI, NRCB, Egypt
2-College of Computer, Qassim University, P.O.B. 6688, Buryadah 51453, KSA
Tel: +202-3310500, +966-63800050 Fax: +202-3351631, +966-505901562
[email protected] & [email protected]
(Received 27/05/2014; Accepted for publication 12/10/2014)
Abstract. In this paper, low-dimensional nanostructures are used to enhance the power conversion
efficiency of solar cell (SC). Quantum dots intermediate band (QDIB) is used for this purpose. This idea
maintains a large open-circuit voltage and increases the produced photocurrent. The balance efficiency analysis of SC is performed by using the Blackbody spectrum as well as more realistic spectra at AM0
and AM1.5. A comparison between conventional solar cell (CSC), intermediate band solar cell (IBSC),
and quantum dots intermediate band solar cell (QDIBSC) is performed. The Schrödinger equation is used to calculate the sub-bandgap energy transition in the quantum dots (QDs) by using the effective mass of
charge carriers. Also, the influences of sub-bandgap energy transition, the location of intermediate band
(IB), the QDs width size (QDW), and the barrier thickness (BT) between dots are studied on the power conversion efficiency of SC. The results show that the efficiency of SC is significantly improved when it
is based on nanostructures.
Keywords: Quantum dots, Intermediate Band, Efficiency, Solar Cell, Intermediate Band Solar Cell
Abou El-Maaty M. Aly 136
1. Introduction
In the past decades, the whole world commenced to feel the severity of the
deficiency in the fossil fuel. Therefore a large number of researchers around the
world began to look for and study the alternative energy to compensate conventional
energy loss. The solar energy is one of the important topics for alternative energy
and improving this technology can clearly reduce the problem of energy in the world
[1 - 3]. Therefore many researchers at Nanopower Research Laboratories focus on
the enhancement of SC power conversion efficiency by using nanostructures with
suitable adjusted properties [4 - 6]. The new proposed concepts are called third-
generation solar cell such as hot carrier cells, tandem cells, and intermediate-band
cells [2, 5, 7 - 10]. The intermediate energy bands are established within a forbidden
gap of a bulk semiconductor material when a superlattice low-dimensional
nanostructure (quantum well, wire, dots) of other semiconductor material injected
within a bulk semiconductor material [11 - 13]. This structure is called
heterostructure and it allows for the proper operation of the IBSC. For quantum well
and wire heterostructures, the photogenerated charge carriers will lose energy
through thermalization due to the high density of states available in the in-plane and
longitudinal direction respectively [14]. Also, the Fermi level will not be split into
the number of bands but it will be split into two levels similar to the CSC. Therefore
the lowest intermediate band will act as the conduction band (CB) and the
heterostructure will behave like the CSC [15, 16]. For QDs heterostructure, it will
provide zero density of states between the excited bands due to the discrete energy
spectrum therefore the thermalization is reduced. Each band is thermally isolated
therefore the Fermi energy level will be split into the number of bands and charge
concentration in each band is described by its own chemical potential as required for
improving IBSC operation [14, 17]. Figure (1a) shows the QDIBSC with the barrier
material surrounding the QDs. Where the QDs are made of the materials such as Ge,
InAs, InAsN…..etc. which have smaller bandgap energy than the barrier materials
such as Si, GaAs, GaAsN….etc. The superlattice structure is an array of closely
spaced QDs therefore a sufficient amount of wavevectors overlapped to form the
intermediate energy bands [6, 10], Fig. (1b). These bands are very important in the
advanced solar cell because they allow the absorption of low energy photons
addition to the normal absorbed photons processes, and the efficiency is improved
via increasing the generated photocurrent [18]. In this paper, the theoretical study for
CSC, IBSC, and QDIBSC will be discussed and the balance efficiency is studied
versus structural and different locations of intermediate-band in IBSC to optimize
the efficiency. The organization of this paper includes the related work in section 2.
The balanced efficiency analysis in section 3, the conventional solar cell and one
intermediate band solar cells (one-IBSCs) are studied in sections 4 and 5. The
quantum dots one intermediate band solar cell (one-QDIBSC) with ternary materials
structure and different QDs width sizes and barrier thickness is discussed in section
6. Finally, a conclusion of the results and future works are summarized in section 7.
Improving the Power Conversion Efficiency of the Solar Cells 137
Fig. (1). (a) Diagram of the QDs inside a barrier semiconductor material. (b) Intermediate energy
bands of an array of QDs
2. Related Work
Conventional solar cells have demonstrated the ability to achieve power conversion
efficiency values near the maximum theoretical limit, i.e., Shockley–Queisser
limit of 31% - 41% [19, 20]. However, the maximum power conversion limitation
for CSCs is lower than desired, due to energy loss for solar photons with energy
exceeding the bandgap energy and absence of SC response to solar photons with
energy below the bandgap energy. Power conversion efficiency has been improved
for tandem and multi-junction solar cells (54.93% for 1 sun and 67.23% for 1000
suns), but these devices are more complex and are accompanied by higher
manufacturing costs [21]. In recent years, IBSC have been proposed to exceed
efficiency limitations of CSCs [19, 22]. Previous calculations have suggested a
theoretical 63.2% efficiency limitation [22] for one-IBSCs, motivating experimental
efforts to realize these promising solar cell devices. Several approaches have been
proposed to practically realize an IBSC, including quantum dots [23 - 27], impurity-
band SCs [28], and dilute semiconductor alloys [29, 30]. The present work will
concentrate on the theoretical study and comparison of CSC and IBSC to improve
the power conversion efficiency. Thereafter, alloys for the bulk and injected
semiconductor materials will be selected to achieve the concept of IBSC by using
quantum dots with varying the size and the distance among them.
Barrier material
QDs array
IB
s
(a) (b)
Wavevectors
Abou El-Maaty M. Aly 138
3. Balance Efficiency Analysis
The principle of balance efficiency to describe the operation of a solar cell is
considered that the sun a Blackbody has a surface temperature of 6000 K. The
generalization of Planck’s radiation law for Blackbody and luminescent radiation is:
𝑆𝑅 =2𝜋ℎ𝑐2
𝜆5
1
𝑒((ℎ𝑐 𝜆−𝜇)/𝑘𝑇⁄ )−1 (1)
where SR is solar spectrum, 𝜆 is wavelength of light, h is Planck’s constant, c
is the light speed, µ is the chemical potential, k is Boltzmann’s constant, and T is the
temperature of the Blackbody. From the Roosbroeck-Shockley equation, the flux of
photons (𝜑) absorbed by the semiconductor or emitted from the semiconductor is
[17]:
𝜑(𝜆𝑎, 𝜆𝑏 , 𝑇, 𝜇) = ∫ 𝑆𝑅𝜆𝑏
𝜆𝑎𝑑𝜆 (2)
Where𝜆𝑎 and 𝜆𝑏 are the lower and upper light wavelength at the energy
limits of the photo flux for the corresponding transitions, respectively. When
treating the sun as an ideal Blackbody and the solar absorption of a semiconductor is
limited by the bandgap energy (Eg), the flux of photons absorbed at the sun’s surface
(𝜑𝑠) is:
𝜑𝑠(𝜆𝑎, 𝜆𝑔, 𝑇𝑠, 0) = ∫2𝜋ℎ𝑐2
𝜆5
1
𝑒(ℎ𝑐 𝜆𝑘𝑇𝑠⁄ )−1
𝜆𝑔
𝜆𝑎=0𝑑𝜆 (3)
Where λg is the light wavelength at the bandgap energy of the solar
semiconductor, Ts is the temperature of the sun. The photon flux absorbed by the
solar cell 𝜑𝑠𝑠𝑐 is:
𝜑𝑠𝑠𝑐 = 𝜑𝑠 × 𝑓𝑠 (4)
Where fs = 2.1646×10-5
is called geometric parameter [14].
Improving the Power Conversion Efficiency of the Solar Cells 139
Similarly, the solar cell may be considered as an ideal Blackbody at
temperature Tc = 300 K. Therefore, a photon emission from solar cell with a flux
𝜑𝑠𝑐 is emitted when an electron-hole pairs is direct recombined.
𝜑𝑠𝑐(𝜆𝑎, 𝜆𝑔, 𝑇𝑐, 0) = ∫2𝜋ℎ𝑐2
𝜆5
1
𝑒((ℎ𝑐 𝜆𝑘𝑇𝑐⁄ )−1
𝜆𝑔
𝜆𝑎=0𝑑𝜆 (5)
According to the Shockley-Queisser model, the current density, j, of the solar
cell device can be written as [31]:
𝑗 = 𝑞[𝑐𝑠𝑓𝑠𝜑𝑠(0, 𝜆𝑔, 𝑇𝑠, 0) + (1 − 𝑐𝑠𝑓𝑠)𝜑𝑠𝑐(0, 𝜆𝑔, 𝑇𝑐 , 0) − 𝜑𝑠𝑐(0, 𝜆𝑔, 𝑇𝑐 , 𝜇)] (6)
Where𝑐𝑠 is the light intensity on a solar cell, it is called the number of suns,
i.e., 𝑐𝑠 = 1 at the surface of the Earth’s atmosphere and 𝑐𝑠 = 1 𝑓𝑠⁄ at the surface
of the sun’s. The chemical potential of luminescent photons , 𝜇, is equal to the
applied bias, qV, which it is equal to the amount of splitting in the quasi-Fermi
energy levels. This equation represents the balance formula; it gives directly the
current-voltage characteristics of a solar cell and the solar power conversion
efficiency can be directly calculated.
4. Conventional Solar Cell
To illustrate the tradeoffs in material selection for CSC, the balance performance I-
V characteristics of some elemental (Ge and Si) and binary (GaAs, AlSb, and GaP)
semiconductors are shown in Fig. (2). This Figure illustrates the main problem of
the CSCs which is when the output current increases the output voltage decreases
and vice versa. Therefore, the optimum solar cell bandgap energy matches the
maximum efficiency can be calculated from maximum solar cell output power
which it is clearly shown in P-V characteristic, Fig. (3).
Abou El-Maaty M. Aly 140
Fig. (2). I-V characteristics of some semiconductors solar cell
Fig. (3). P-V characteristics of some semiconductors solar cell
Improving the Power Conversion Efficiency of the Solar Cells 141
To determine the optimum bandgap energy of the CSC corresponding to the
maximum efficiency, the solar cell efficiencies are analyzed with changing the
bandgap energies at the Blackbody, AM0, and AM1.5 solar spectra and several
different solar concentrations. The Blackbody, AM0, and AM1.5 spectra are plotted
in Fig. (4). The Blackbody spectrum is calculated and used in theoretical analysis.
The AM0 and AM1.5 spectra are measured by the American Society for Testing and
Materials (ASTM) and represent the solar spectra at outside the Earth’s atmosphere
and at the surface of the Earth’s respectively [32].
Fig. (4). Solar spectrum of Blackbody at 6000 K and more realistic solar spectra
Figure (5) illustrates this study, which it shows the balance efficiency with
different material bandgap energies under Blackbody spectrum and more realistic
AM0 and AM1.5 solar spectra. The effect of Blackbody and AM0 is approximately
similar although the AM0 spectrum is not a smooth function. The AM1.5 is
significantly different than the Blackbody and AM0 by presence the irregularity of
the AM1.5 spectrum. The improve efficiencies for the AM1.5 are due to smaller
value of the solar constants corresponding to this spectrum [14]. Table (1)
summarizes the maximum efficiencies and optimum bandgap energies at 1 sun and
maximum solar concentration for the three standard spectra. These data are plotted
over the entire concentration range in Fig. (6). It is clear that the optimum design
under one spectrum is not under another spectrum.
Abou El-Maaty M. Aly 142
Fig. (5). Comparison of the maximum efficiencies of CSC for different bandgap energies and
spectra at 1 sun (sold lines) and maximum solar concentration (dashed lines)
Table (1). Summarizes the maximum efficiencies and optimum bandgap energies at 1 sun and
maximum concentration for the three standard spectra
Solar Spectra
1 Sun Concentration Max. Concentration
Max. Efficiency
(%)
Opt. Bandgap
(eV)
Max. Efficiency
(%)
Opt. Bandgap
(eV)
Blackbody 30.93 1.31 40.71 1.10
AM0 30.17 1.28 40.43 1.00
AM1.5 33.58 1.35 44.83 0.96
Improving the Power Conversion Efficiency of the Solar Cells 143
Fig. (6). Comparison between the three standard spectra with maximum efficiencies and optimum
bandgap energies for CSC.
5. Intermediate Band Solar Cell
The conventional solar cell would only be able to absorb photons with energy equal
to or greater than the energy between CB and valence band (VB), ECV. For the
nanostructure solar cell, a new band is located at an intermediate band level, EI,
between CB and VB. This band increases the absorb photons which leads to the
increased performance of this design when compared to the CSC. To study a balance
analysis of this type, the following assumptions are necessary [3, 10, 33]: (1)There is
no overlap of energy transitions for a given photon energy, i.e., if a photon may
energetically induce a transition from one band to another then all photons of the
same energy will only cause that specific transition. (2) Electrons enter the IB only
pumped from the VB and they leave only to the CB. (3) The difference in chemical
potentials between any two bands is simply the difference between the quasi-Fermi
levels of these bands.
According to these assumptions, the IBSC have three transitions of electrons
between bands. The standard effect of an electronic transition across the bandgap
energy, ECV, gives jCV as in previous analysis in section 4. The effect of the IB to CB
transition, ECI, produces jCI which it must be equal to jIV which it produces from the
transition between the VB and IB, EIV, according to the above assumptions. The two
intermediate band transitions ECI and EIV are independent of each other whoever the
conventional bandgap energy ECV is a function of these bands. Therefore, the three
generated currents are:
Abou El-Maaty M. Aly 144
𝑗𝐶𝑉 = 𝑞[𝑐𝑠𝑓𝑠𝜑𝑠(0, 𝜆𝐶𝑉 , 𝑇𝑠, 0) + (1 − 𝑐𝑠𝑓𝑠)𝜑𝑠𝑐(0, 𝜆𝐶𝑉 , 𝑇𝑐 , 0) − 𝜑𝑠𝑐(0, 𝜆𝐶𝑉 , 𝑇𝑐 , 𝜇)]
𝑗𝐶𝐼 = 𝑞[𝑐𝑠𝑓𝑠𝜑𝑠(𝜆𝐶𝐼 , 𝜆𝐼𝑉 , 𝑇𝑠, 0) + (1 − 𝑐𝑠𝑓𝑠)𝜑𝑠𝑐(𝜆𝐶𝐼 , 𝜆𝐼𝑉 , 𝑇𝑐 , 0) − 𝜑𝑠𝑐(𝜆𝐶𝐼 , 𝜆𝐼𝑉 , 𝑇𝑐 , 𝜇𝐶𝐼)] (7)
𝑗𝐼𝑉 = 𝑞[𝑐𝑠𝑓𝑠𝜑𝑠(𝜆𝐼𝑉 , 𝜆𝐶𝑉 , 𝑇𝑠 , 0) + (1 − 𝑐𝑠𝑓𝑠)𝜑𝑠𝑐(𝜆𝐼𝑉 , 𝜆𝐶𝑉 , 𝑇𝑐 , 0)− 𝜑𝑠𝑐(𝜆𝐼𝑉 , 𝜆𝐶𝑉 , 𝑇𝑐 , 𝜇𝐼𝑉)]
Since the currents jCI and jIV must be equal, therefore the total current jT = jCV
+ jCI. As in previous analysis, the chemical potential of photons is equal to the
applied bias, µ = µCI + µIV = qV, which it is equal to the amount of splitting in the
quasi-Fermi energy levels. By following the same methodology in CSC, the balance
efficiency is analyzed under different solar spectra at 1 sun and maximum solar
concentration, and changing the IB location to give the maximum efficiency.
Figures (7 - 9) and Table (2) show the balance efficiency, optimum bandgap
energies, and the location of IB of IBSC at 1 sun and maximum solar concentration
for the three standard spectra.
(a) (b)
Fig. (7). Balance efficiency of IBSC as it varies with the position of IB between the CB and VB at
Blackbody spectrum under: (a) 1 sun, (b) Maximum solar concentration
Improving the Power Conversion Efficiency of the Solar Cells 145
(a) (b)
Fig. (8). Balance efficiency of IBSC as it varies with the position of IB between the CB and VB at
AM0 spectrum under: (a) 1 sun, (b) Maximum solar concentration
(a) (b)
Fig. (9). Balance efficiency of IBSC as it varies with the position of IB between the CB and VB at
AM1.5 spectrum under: (a) 1 sun, (b) Maximum solar concentration
Abou El-Maaty M. Aly 146
Table (2). Summarizes the maximum efficiencies and optimum bandgap energies and IB of IBSC at
1 sun and maximum solar concentration for the three standard spectra
Solar Spectra
1 Sun Concentration Max. Concentration
Max.
Efficiency (%)
Opt. Bandgap (eV) Max.
Efficiency (%)
Opt. Bandgap (eV)
ECI EIV ECV ECI EIV ECV
Blackbody 46.74 1.50 0.93 2.43 63.13 1.25 0.72 1.97
AM0 45.70 1.39 0.86 2.25 63.56 1.12 0.66 1.78
AM1.5 49.35 1.49 0.93 2.42 67.60 1.23 0.69 1.92
Figure (7a) shows the balance efficiency under Blackbody spectrum at 1 sun as the maximum efficiency is 46.74% which it corresponds to IBs of EIV = 0.93 eV and ECI = 1.5 eV; the total bandgap energy ECV = 2.43 eV. When the solar concentration is increased to its maximum value as shown in Fig.(7b), the efficiency is increased to 63.13% and the optimum values of both EIV and ECI decrease to 0.72 eV and 1.25 eV respectively. Also, the total bandgap energy is decreased to 1.97 eV. The efficiency of the IBSC does not depend on the order of the energy bands ECI and EIV. Efficiency depends only on the energy transition themselves. For example, the theoretical efficiency is equal to 63.13% under maximum solar concentration when ECI = 1.25 eV and EIV = 0.72 eV. However, the same result is also achieved when ECI = 0.72 eV and EIV = 1.25 eV, therefore the order of the energy transition is unimportant. Comparing the behavior of IBSC with Fig.(6) and Table (1) for the CSC, note that both of them demonstrate the same behavior where as the concentration of solar increases the efficiency increases and the large bandgap energy decreases. Also, it is clear that for the maximum solar concentration, the matching between jCI and JIV is difficult to obtain, it explains the absence of more possibilities for efficiency in Fig. (7b). Therefore, the location of IB is limited for maximum solar concentration.
Similar for CSC, the balance efficiency of the IBSC is more realistic when analyzed with the actual AM0 and AM1.5 solar spectra. The results are shown in Figs. (8) and (9). From Fig.(8), it is clear that the balance efficiency is similar due to the similarity between the Blackbody and AM0 spectra. Some aliasing is seen in the AM0 contours due to the irregularity of the AM0 spectrum. Also for AM1.5 in Fig.(9), the contours of the efficiencies are a much more irregular when compared with the Blackbody and AM0 analysis. Due to the large of irregularity in the AM1.5 spectrum, the several local maxima are present and thus the efficiency is increased.
The summary of the above discussion are plotted over the entire solar concentration
range in Fig. (10). It is clear that the optimum design under one spectrum is not
under another spectrum as in the CSC approximately.
Improving the Power Conversion Efficiency of the Solar Cells 147
Fig. (10). Comparison between the three standard spectra with maximum efficiencies and optimum
bandgap energies for IBSC
6. Quantum Dot Intermediate Band Solar Cell
For this type of solar cell, the size, shape, and spacing of the quantum dots in the
barrier material are very important to improve the efficiency of the device.
Whenever the quantum dots are arranged uniform in the barrier material, the IBs are
well-placed and the recombination is reduced. Various techniques such as Metal
Organic Chemical Vapor Deposition (MOCVP) and Molecular Beam Epitaxy
(MBE) are available to produce highly uniform QD arrays in barrier material [13,
34]. In this paper, we assume that the geometry of the QD is cubic and therefore it is
characterized by one dimension. Also, to simplify the analysis, we assume that no
valance band offset (VBO) and the only confining potential occurs at the conduction
band offset (CBO). Under these assumptions, the time-independent Schrödinger
Equation and the Krong-Penny model are used to calculate the energy bands in the
QDs using the effective mass of charge carriers [8, 34]. In this work, Al1-xGaxAs/In1-
yGayAs QDIBSC is analyzed theoretically. Where Al1-xGaxAs is the barrier ternary
alloy and In1-yGayAs is the QD ternary alloy. The bandgap energies, 𝐸𝑔, and
effective masses of these alloys, 𝑚∗, are calculated by the following chemical
formulas [5, 35]:
Abou El-Maaty M. Aly 148
For barrier ternary alloy material
𝐸𝑔(𝐴𝑙1−𝑥𝐺𝑎𝑥𝐴𝑠) = 3.099(1 − 𝑥) + 1.519𝑥 − 0.2𝑥(1 − 𝑥),
𝑚(𝐴𝑙1−𝑥𝐺𝑎𝑥𝐴𝑠)∗ = (1 − 𝑥)𝑚(𝐴𝑙𝐴𝑠)
∗ + 𝑥𝑚(𝐺𝑎𝐴𝑠)∗
For QD ternary alloy material
𝐸𝑔(𝐼𝑛𝑦𝐺𝑎1−𝑦𝐴𝑠) = 1.519(1 − 𝑦) + 0.417𝑦 − 0.588𝑦(1 − 𝑦),
𝑚(𝐼𝑛𝑦𝐺𝑎1−𝑦𝐴𝑠)∗ = (1 − 𝑦)𝑚(𝐺𝑎𝐴𝑠)
∗ + 𝑦𝑚(𝐼𝑛𝐴𝑠)∗
Where x and y are limited between 0 and 1. They are called the molar
concentration for GaAs and InAs in barrier and QD ternary alloys materials,
respectively.
The compound semiconductors which are used in this study for QDIBSC are
Al0.4Ga0.6As/In0.42Ga0.58As [36, 37]. The bandgap of ternary alloy In0.42Ga0.58As
(0.913 eV) dot array material is smaller than the bandgap of ternary alloy
Al0.4Ga0.6As (2.103 eV) barrier or bulk material to induce the IB by coupling of
confined electronic states in the In0.42Ga0.58As conduction band. The bandgap
energies of the structure of Al0.4Ga0.6As/In0.42Ga0.58As QDIBSC are approximately
close to the ideal values determined from Table (2) in the pervious section. The
balance efficiency for this structure under the above assumptions is plotted in Figs
(11 - 13) and summarized in Table (3). From the first point of view when comparing
the Figs (11 - 13) and Table (3) with the results in Figs (7 - 9) and Table (2), it is
clear that the proposed model is approximately ideal because it is based on the
difficult of technology for ternary semiconductor materials.
Improving the Power Conversion Efficiency of the Solar Cells 149
(a) (b)
Fig. (11). Balance efficiency of QDIBSC as it varies with the BT and QDW at Blackbody spectrum
under: (a) 1 sun, (b) Maximum sola concentration
(a) (b)
Fig. (12). Balance efficiency of QDIBSC as it varies with the BT and QDW at AM0 spectrum
under: (a) 1 sun, (b) Maximum solar concentration
Abou El-Maaty M. Aly 150
(a) (b)
Fig. (13). Balance efficiency of QDIBSC as it varies with the BT and QDW at AM1.5 spectrum
under: (a) 1 sun, (b) Maximum solar concentration
Table (3). Summarizes the maximum efficiencies and optimum BT and QDW of QDIBSC at 1 sun
and maximum solar concentration for the three standard spectra
Solar
Spectra
1 Sun Concentration Max. Concentration
Max.
Efficiency
(%)
Opt. Barrier
thickness, BT
& QD width,
QDW (nm)
IB
Width
(eV)
Max.
Efficienc
y (%)
Opt. Barrier
thickness, BT &
QD width,
QDW (nm)
IB
Width
(eV)
BT QDW BT QDW
Blackbod
y 45.32 1.98 1.60 0.0696 62.81 1.94 1.64 0.0684
AM0 45.13 1.98 1.68 0.0612 62.66 1.78 1.68 0.0756
AM1.5 48.06 1.98 1.52 0.0792 66.91 1.98 1.52 0.0792
The results in Figs (11 - 13) show the effect of QDW and BT in the
efficiency of QDIBSC at 1 sun and maximum solar concentration under different
spectra. Before discuss the results in Figs (11 – 13), Fig (14) shows changing of the
first energy bandwidth of IB when changing the BT and QDW. It is clear that the
larger width of QDs and BT tend to decrease the first energy bandwidth of the IB
and it moves toward the VB therefore Eci increases and Eiv decreases.
Improving the Power Conversion Efficiency of the Solar Cells 151
Fig. (14). Width of IB energy (eV) when changing BT and QDW (nm)
This work studied QDIBSC with only one IB therefore the QDW and BT in
this type of ternary materials are limited with the values in Fig. (14), i.e., 1 nm ≤
QDW ≤ 3 nm and 0.5 nm ≤ BT ≤ 2 nm. From the results in Figs (11 – 13), the
maximum energy conversion efficiency of Al0.4Ga0.6As/In0.42Ga0.58As QDIBSC as a
function of QDW and BT under different solar spectra is changed from 45.32% to
48.06% at 1 sun, and from 62.66% to 66.91% at the maximum solar concentration.
These curves illustrate that the efficiency of this structure is not available for all
limits of QDW and BT that are mentioned above. When the QDW decreases toward
low limit; the BT increases to upper limit. Another note for these curves is that the
overall QDIBSC realizes current-matching easily for 1 sun than the maximum solar
concentration which it is very difficult. Table (3) summarizes the maximum
efficiencies and optimum BT and QDW of QDIBSC at 1 sun and maximum solar
concentration for the three standard spectra.
7. Conclusions and Future Works
In this paper, the balance efficiency of conventional and nanostructure solar cell
dependent on Shockley and Queisser model has been studied under more realistic
solar spectra, in addition to the standard Blackbody spectrum. For the nanostructure
or QDIBSC, the balance analysis showed that it varies with the location of the
intermediate band within the bulk semiconductor bandgap energy. Therefore, the
optimum location of intermediate band which gives the maximum energy
conversion efficiency can be determined. Accordingly, the choice of materials for
Abou El-Maaty M. Aly 152
solar cell implementation to achieve these results is easy. In this work, the
parameters of QDIBSC structure have been determined based on the balance
efficiency by using the Schrödinger Equation and the Krong-penny model. The
results have shown that both the location of the IB and the energy conversion
efficiency strongly depend on the width of the QD and the barrier thickness. The
maximum efficiency is changed from 62.66% to 66.91% under the maximum solar
concentration at different solar spectra for only one IB of Al0.4Ga0.6As/In0.42Ga0.58As
QDIBSC, although it is very difficult to obtain the current-matching in the QDIBSC.
In future works, additional energy levels in QDIBSC to increase the
efficiency will consider and when searching for the optimally energy bands placed
must take into account the bands overlaps with each other or with the CB and VB.
This condition is necessary because the bandgap of the barrier material is reduced
therefore the operating voltage and output power of the cell’s are decreased. Also,
the searching for new and more promising materials with significant sub-bandgap
absorption, high mobility, and suppressed non-radiative processes is crucial for the
success of intermediate band devices. This might include alloys and quantum
confined structures. The theoretical and modeling effort should focus on explanation
of the low efficiencies observed in real devices, revealing the underlying physical
reasons, and provides feasible solutions to the problems. In addition, continued
effort on theoretical work can predict/propose novel concepts to either exceed
current efficiency limits or more practically.
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Abou El-Maaty M. Aly 156
,
قسم الكرتونيات الطاقة العالية وحتويل الطاقة، معهد حبوث اإللكرتونيات،
1مبنى املركز القومى للبحوث، مجهورية مصر العربية
2عربية السعودية قسم هندسة احلاسب، كلية احلاسب، جامعة القصيم، بريدة، اململكة ال
[email protected] & [email protected]
م( 12/10/2014م ، وقبل للنشر يف 27/5/2014)قدم للنشر يف
هذا البحث ابعاد النانو استخدمة لتحسني كفاءة حتويل الطاقة من اخلاليا الشمسية. الكم يف
أستخدم هلذا الغرض. هذه الفكرة حتافظ على القيمة العالية للجهد (QDIB)النقطي لنطاق الطاقة الواسطي
اخللية يفمع زيادة التيار املتولد (Open Circuit Voltage) الدائرة واليت تسمى يفعندما اليوجد تيار مير
الشمسية عند سقط ضوء الشمس عليها. سوف يتم حتليل كفاءة اخلاليا الشمسية بستخدام طيف اجلسم
. مت عمل مقارنة من ناحية AM1.5و AM0وكذلك أطياف أكثر واقعية مثل (Blackbody) االسود
الكفاءة بني اخلاليا الشمسية التقليدية، اخلاليا الشمسية ذات النطاق الطاقة الواسطي، اخلاليا الشمسية ذات
ات الطاقة الفرعية النطاق الطاقة الواسطي باستخدام الكم النقطي. أستخدمة معادلة شرودجنر حلساب نطاق
نقل الطاقة وذلك عن طريق الكم النقطي وحجم الشحنات الناقلة للطاقة. كذلك مت دراسة يفاملستخدمة
نقل الطاقة، مكان نطاق الطاقة الواسطي، حجم الكم يفتأثري كل من نطاقات الطاقة الفرعية املستخدمة
قة من اخلاليا الشمسية. النتائج أوضحت أن كفاءة النقطي، املسافة بني الكم النقطي على كفاءة حتويل الطا
حتويل الطاقة من اخلاليا الشمسية حتسنت بشكل ملحوظ عند استخدام االبعاد النانونية.
Journal of Engineering and Computer Sciences
Qassim University, Vol. 7, No. 2, pp. 157-199 (July 2014/Shaban 1435H)
157
Methodology for Minimizing Power Losses in Feeders of Large Distribution
Systems Using Mixed Integer Linear Programming
Hussein M. Khodr(1), E. E. El-Araby(2)(*), and M. A. Abdel-halim(3)
(1)Aramco Company, SA, [email protected]
(2)College of Engineering, Qassim University, [email protected]
(3) College of Engineering, Qassim University, [email protected]
(Received 12/10/2014 ; accepted for publication 18/12/2014)
Abstract. This research work presents a reconfiguration experience of a real case power distribution
networks aiming to minimize the active power losses and consequently to improve the voltage profile of the overall grid. The grid is modelled as a mixed integer linear programming problem whose objective
function is to minimize the active power losses subject to technical constraints, which are: the power
balance in each node of the power grid, the capacity limits of the lines and the substations, maintaining
the radial operation of the network. The model results are the open branches on each circuit. The
technique has been successfully applied to a large power distribution networks with 2344 nodes at a
13.8kV level, located in real city resulting in an 8% and 6% losses reduction in terms of active power losses and energy respectively, and a substantial improvement in the voltage profiles regarding the current
state of grid with very acceptable convergence times.
Keywords. Distribution system, Optimal Reconfiguration, Mixed-Integer Linear Programming, Optimization.
* On leave from EE Department, College of Engineering, Port Said University, Egypt
Hussein M. Khodr et.al. 158
1. Introduction
Reconfiguration is an attractive alternative for active power losses reduction in radial
power distribution systems. The reconfiguration procedure consists in minimizing
losses by determining the switches states open or closed till to obtain a radial and
connected distribution networks topology. Nowadays the distributions grids require a
constant adaptation in terms of its operation, allowing reducing the loss rates, and thus
obtain energy savings that will benefit consumers and suppliers companies. In the
literature, there are many methodologies for optimization of distribution systems [1]-
[12] requiring large computational effort to solving large problems.
Due to vaster references number on this subject of optimal reconfiguration of
distribution networks, only the works published on the last decade are considered in
this paper aiming to highlight the most recent advances on this important subject.
In [1, 18] a systematic feeder reconfiguration technique that develops an
optimal switching scheme is presented, maximizing the losses reduction in a
distribution network. Meanwhile, in [2] an improved method based on a refined
genetic algorithm to study the distribution network reconfiguration is proposed. In
[3], a control strategy of open-closed status of the tie-switches is proposed. In this
control strategy, the tie-switches are installed at the MV/LV substations, in this way;
each line can be sectionalized at both ends, where the substations are located.
Detailed features of this method can be seen in this reference. In [4] a method for
feeder reconfiguration for load balancing among distribution feeders is proposed.
The G-nets inference mechanism with operation rules is applied to derive the
optimal switching operation decision to perform the optimal load transfer among
feeders after the fault has been identified and isolated. In this method, the customer
information systems and the transformers in outage management are used to
determine the daily load patterns of service areas, sectionalizing switches,
distribution feeders and main transformers. In [5] proposes a method that
maximizes a fuzzy index using a maximum loadability index (MLI) that gives a
measure of the proximity of the present state of a line in the radial distribution
systems (RDS) to maximum loadability.
The MLI is computed at each bus of RDS. The value of this index, close to 1.0,
indicates that the feeder is overloaded and would be unable to supply more apparent
power. The overloaded buses may be identified by using this index, and adequate
action may be used for improving the optimal reconfiguration scheme. Ref. [6] applied
Genetic–Tabu algorithm (GTA) that is a tabu search combined with Genetic algorithm
to find a global solution for optimal service restoration and reconfiguration of
distribution networks. Ref. [7] presents an algorithm based on heuristic strategy that
starts with the system in a meshed status with all switches closed.
The choice of the switches to be opened is based on the calculation of the
minimum total network losses using a load–flow program. Ref. [8] introduces an ant
colony search algorithm to solve the optimal network reconfiguration for power losses
reduction. Ref. [9] proposes both a deterministic branch and bound algorithm included
Methodology for Minimizing Power Losses in Feeders... 159
in the CPLEX optimization package and a genetic algorithm to solve the large scale
mixed-integer-programming formulation of the distribution reconfiguration problem.
Ref. [10] describes a genetic–algorithm for reconfiguration of a distribution network in
order to minimize financial losses due to voltage sags.
The proposed method starts with a selected number of switches that generate
various topologies. Load–flows are then performed to evaluate the feasibility of
these topologies. For each of these topologies, fault analysis is performed to first
calculate voltage sags at different buses and then, to compute financial losses
incurred by voltage sags at buses with sensitive industrial process. The developed
software-based methodology has been applied to a 295-buses generic distribution
system. Ref. [11] contributes such as a technique at low–voltage and medium–
voltage levels of a distribution network simultaneously with reconfiguration at both
levels. This Reference introduces a heuristic method for the phase balancing/loss
minimization problem. A comparison of this heuristic with the neural networks is
also performed. The application in conjunction of the neural network with the
heuristic method that enables different reconfiguration switches to be turned on/off
and connected consumers to be switched between different phases to keep them
balanced is also carried out.
Ref. [13] proposes an improved Tabu Search algorithm for loss-minimum
reconfiguration in apparently large-scale distribution systems. This algorithm take
advantages of the features of the network structure itself and make Tabu Search
algorithm be more suitable to solve the reconfiguration problem. ITS algorithm with
mutation operation and high quality candidate neighbourhood has been developed in
this work. This methodology has been tested to an 11 kV distribution system with
118 sectionalizing switches and 15 tie switches.
Ref. [14] presents an application of the ant colony optimization concepts to
the optimal reconfiguration of distribution systems. The considered objective is to
minimize the power losses at distribution systems. The methodology has firstly been
tested in a distribution network 33 nodes the Baran and Wu case, where, the
obtained active power losses on the base case are 176.4 kW and after application of
the methodology the obtained active power losses are 127.4 kW.
These reported value are not true at all, please see reference [15] where this
example has been studied by many others researchers around the world. The values
reported are 202 kW in the actual state of the network, and after reconfiguration the
active power losses are 139.5 kW. The methodology has secondly been applied to a
531 nodes distribution network. The characteristics and data of this distribution
networks are omitted.
Ref. [16] proposes a rule based comprehensive approach to study distribution
network reconfiguration. The proposed algorithm consists of a modified heuristic
solution and the rules base. The objective is minimizing the system power loss. The
methodology has been tested on a small distribution system of 33 nodes.
Hussein M. Khodr et.al. 160
Ref. [17] presents a (PSO) algorithm to solve the optimal reconfiguration
problem minimizing the power loss. The PSO algorithm is introduced with some
modifications such as using an inertia that decreases linearly during the simulation
process, which allows the PSO to explore a large area at the start of simulation. This
methodology bas been tested on two test distribution systems of 32 nodes and 69
respectively.
Ref. [15] presents a deterministic methodology to solve the distribution
network reconfiguration problem. It is worth to mention that the formulation stated
in this reference is the complete one, because it takes into consideration not only the
switching status opened/closed, but also the transformer taps, the distributed
generation and the capacitors bank steps. This formulation has been solved by
benders decomposition technique that consists on dividing the problem in master
and slave. The first one is to determine the distribution networks topology that must
be sent to the slave problem for adjust the system parameters. This methodology has
been tested on 201 nodes real distribution networks.
As can be seen there are widely used models dealing with the distribution
reconfiguration problem. Many techniques have also been used to solve the
reconfiguration problem, deterministic, evolutionary and inspired algorithms, but all
the tested and published methods have a limited horizon regarding to tested
distribution networks. No large distribution systems that consist of more than 600
nodes have been tested. The new contribution of this paper is the application and
testingthe proposed methodology to real large distribution systems consisting of
2344 nodes, 3 substations and 10 primary distribution feeders.
The paper is organized as follows: Section 2 explains the reconfiguration
problem formulation of power distribution networks. Section 3 identifies the
optimization technique to solve the proposed model. Section 4 explains the analysis
and validating the results. Section 5 exposes the mathematical model of the case
under study. Section 6 analyses the real case study and finally Section 7 highlights
the main conclusions of this research work.
2. Formulation of The Problem of Distribution Networks Reconfiguration
The problem of distribution networks reconfiguration can be formulated as an MILP
type optimization problem. Thus, it is necessary to develop a type of methodology that
can solve the problem under study. In order to formulate the problem, it is necessary to
define a mathematical model, so the annual costs of active power losses within the
network can be minimized (small investments). Active power losses have a square
expression, according to the power rate which circulates through the lines. Thus, it is
important to conduct approximations and use linear optimization techniques which
lead to best mathematical solutions and reduced computer time calculation.
In order to investigate distribution networks reconfiguration, a mathematical
model is proposed to minimize annual costs of power losses on the lines. In the
Methodology for Minimizing Power Losses in Feeders... 161
case the company does not want to invest on build new line sections nor new
substations. The model can include many aims as per the operator/planner
considerations.
Active power losses have a square expression according to the power which
circulates on the lines. In this work, it is estimated that conducting approximations and
using linear methods which offer almost exact mathematical solutions will be used.
The developed mathematical model has the following components:
Variables declaration: Includes two variables groups
a) Representation of the powers which circulate through early network line
segments. Powers occupy the first column (Li) on the coefficient matrix of the
mathematical model’s variables and are real variables:
𝑋𝑖 ≥ 0; 𝑖 = 1, … , 𝐿𝑖 (1)
b) The variables for the decision of opening/closing the line selected, which
are attached to correspondent powers, at a 1 to 1 ratio, in other words, powers which
circulate through the lines if there are any. Integer variables occupy the following
columns within the coefficient matrix:
𝑋𝑖; 𝑖 = 𝐿𝑖 + 1, … ,2𝐿𝑖 (2)
These are binary variables which indicate when a certain line is opening/closing by
means of a switch.
c) Integer variables which represent the need to expand or increase
substations, for instance, installing L expansion capacities on a p substation, are
represented by the binary variableYP,L. If there are Nt feeding points to expand the
network and each of them must evaluate several capacities of transformers
substations to select the most suitable capacity (analysing technical-economic
factors), the quantity will be different for each Substation. If it is Nc(p), there
are ∑ NcNtp=1 (p) integer variables, occupying the columns:
𝑖 = 2 𝐿𝑖 + 1, … , 2𝐿𝑖 + 1 + ∑ 𝑁𝑐𝑁𝑡𝑝=1 (𝑝) (3)
Hussein M. Khodr et.al. 162
2.1 Objective function
The problem of electric power distribution networks reconfiguration can be
formulated as a linear programming problem with binary variables. Its solution
includes a selection of, among other possible configurations, the one which results in
less active power losses and subjected to a set of constraints:
Radial configuration of the distribution system;
Acceptable voltage levels;
Reliability, among others.
In this section, authors are trying to minimize the costs of energy losses by
using loss curves piecewise linearization to linear functions (straight small sections).
Generally, this function can be written as follows:
Min F = Z1 + Z2 + Z3 (4)
Objective function components are:
a) Cost of losses within the feeder braches or lines (Z1)
In Nd load nodes, the set of lines which enter the nods coincide with the total
number of paths on the initial network. Considering the linearized expression of
power losses to the maximum load condition [46] the following expression was
obtained:
𝑍1 = ∑ ∑ 𝐶𝐼𝑁[𝑖,𝐾𝑙(𝑖)]𝑘=𝐼𝑁(𝑖,1) ∙
𝑁𝑑𝑖=1
[𝑆(𝑘)]2 ∙ 𝑊(𝑖, 𝑘) (5)
where:
IN(i, k): Line reference number 𝑘 which enters the i node; k = 1, … , Kl(i)
Kl(i): Quantity of lines which enter the 𝑖 node
S(k): Is the apparent power which circulates through k lines (kVA); k = 1, … , Li
C : Is the coefficient of the cost of losses due to the power which circulates through
the lines [€/(kVA.year)]
𝐶 = 𝐶1 If there is a three-phase section branch and otherwise 𝐶 = 𝐶3
𝐶1 = 𝜌∙𝐽𝑒𝑐𝑜∙(12∙𝐶𝑑+𝐶𝑒∙𝐹𝑝𝑑∙8760)∙𝐹𝑉𝐴∙𝐷𝑖𝑗
√3∙𝑈𝑙 (6)
Methodology for Minimizing Power Losses in Feeders... 163
𝐶3 = 4∙𝜌∙𝐽𝑒𝑐𝑜∙(12∙𝐶𝑑+𝐶𝑒∙𝐹𝑝𝑑∙8760)∙𝐹𝑉𝐴∙𝐷𝑖𝑗
√3∙𝑈𝑙 (7)
𝐹𝑉𝐴 = (1 + 𝑅𝐼)𝑡; 𝑡 = 1, … , 𝑁 (8)
being:
FVA : Is the conversion factor for the current value of the losses cost, this is because
the investments in several variants are made in different years. It is necessary to
reduce costs in the first year [19]
: Resistivity of the conductors or cable material (Ω-mm2 km⁄ )
𝐶𝑑 : Contracted power rate (€/kVA/month); is this type of tariff is applied
𝐶𝑒 : Cost rate (€/kVA.h). It is supposed to be a uniform power factor in the network
Fpd : Loss factor
Ul: Line-to-line voltage (V)
𝑅𝑙: Capital growth rate (%/ year)
N : Lifetime of the project (year)
𝐷𝑖𝑗 = 𝐿 : Distance between two nodes (m)
The linearization of the objective function was conducted according to the
Venikov criterion [24]. This approximation supposes that the facilities lines
conductors are working close to the rated current which is the economic density
(𝐽𝑒𝑐𝑜).
I = Jeco ∙ Sec (9)
where:
Jeco: Current economic density (A/mm2)
𝑆𝑒𝑐: Conductor’s cross section area (mm2)
The active power losses are:
∆𝑃 = 𝑘 , ∙ 𝑅 ∙ 𝐼2 = 𝑘 , ∙ 𝑅 ∙ 𝐼 ∙ 𝐼 (10)
Hussein M. Khodr et.al. 164
replacing:
∆𝑃 = 𝑘 , ∙ 𝑅 ∙ 𝐽𝑒𝑐𝑜 ∙ 𝑆𝑒𝑐 ∙ 𝐼 (11)
and therefore:
𝐼 = 𝑘 ,, ∙𝑆
𝑈𝑙; 𝑅 =
𝜌∙𝐿
𝑆𝑒𝑐 (12)
Where:
𝑘 , and 𝑘 ,, are constants which depend on the type of service (single-phase or three-phase).
𝑆 : Load power (kVA)
𝑅 : Conductor’s resistance (Ω km⁄ )
𝐼 : Current passing through the conductor (A)
Substituting:
∆𝑃 = 𝑘∙𝜌∙𝐿∙𝐽𝑒𝑐𝑜
𝑈𝑙∙ 𝑆 (13)
It is a linear expression is function of the apparent power.
b) Investment cost on the network lines (Z2) in the case of expanding the
network
The expression used to this end was:
𝑍2 = ∑ ∑ 𝐾𝑒𝑙𝐼𝑁[𝑖,𝐾𝑙(𝑖)]𝑘=𝐼𝑁(𝑖,1) ∙
𝑁𝑑𝑖=1 𝑊(𝑖,𝑘) (14)
Using the formula:
𝐾𝑒𝑙 = (𝐸𝑛 + 𝑁𝑒𝑙) ∙ 𝐾𝑖𝑙 (15)
Methodology for Minimizing Power Losses in Feeders... 165
where:
𝐸𝑛: Economic effectiveness ratio (1/year).
𝑁𝑒𝑙 = (𝑁𝑎𝑙 + 𝑁𝑜𝑒) (16)
Nel: Scanning the lines rule or utility norm (1/year)
𝑁𝑎𝑙: Amortizing the lines rule or amortizing norm according to Company Standards
(1/year)
Noe: Company’s fixed costs rule according to Company Standards (1/year)
Kil: Line investment costs (€/km)
( , )i kW : Integer of installing a line from the k to the i node
c) Transformers investment cost (Z3)
There are 𝑁𝑡 feeding points with cN p evaluated capacities in each point.
𝑃 = 1, … , 𝑁𝑡
then:
𝑍3 = ∑ ∑ 𝐾𝑒𝑡𝑁𝑐(𝑝)𝑙=1 ∙
𝑁𝑡𝑝=1 𝑌(𝑝,𝑙) (17)
The 𝐾𝑒𝑡 parameter includes a capital inflation rate, a fixed exploration costs rule
and costs that come from iron losses in the transformers.
where:
𝐾𝑒𝑡 = (𝐸𝑛 + 𝑁𝑒𝑡) ∙ 𝐾𝑖𝑡 + 𝐾Δ𝐹𝑒 (18)
where:
𝐾𝑖𝑡: Total investment cost in the transformers (€)
Hussein M. Khodr et.al. 166
𝑁𝑒𝑡 = (𝑁𝑎𝑡 + 𝑁𝑜𝑒) (19)
𝑁𝑒𝑡: Fixed exploration costs rule or exploration norm according to Company
Standards (1/year)
𝑁𝑎𝑡: Transformers amortization rule or amortizing norm for transformer according
to Company Standards (1/year)
𝐾Δ𝐹𝑒: Cost that come from iron losses in the transformers (€/year)
𝑌(p,L): Binary variable for the decision of installing a 𝐿 capacity on point 𝑃
The objective function includes copper loss costs in the transformers, taking another
author [19] into consideration. Indicating the reasons for this cost:
- Load losses are expressed considering the square function.
∆𝑃 = 𝑃𝑘 ∙ (𝑆 𝑆𝑛⁄ )2 (20)
where:
p : Copper losses in the transformers (kW)
𝑃𝑘: Power losses with loads (kW)
𝑆 : Apparent load power (kVA)
𝑆𝑛: Apparent transformer nominal/rated power
In the objective function it is necessary to linearize the copper’s loss
expression, introducing small errors into the final results.
Transformers are limited to work between a maximum and minimum
apparent power interval 𝑆 (kVA), 𝑆𝑚𝑖𝑛(𝑙) ≤ 𝑆 ≤ 𝑆𝑚𝑎𝑥(𝑙) which is usually between
the transformer’s 70% and 100% rated or nominal capacity.
where:
𝑙 : Capacity of the installed transformer in each substation (kVA)
Total loss costs have a square variation, and so the variable cost component
can be used both in nominal capacity 𝑆 ≤ 𝑆𝑛 and smaller loads. This is a small
component when compared to the fixed cost, concluding that:
(𝑆 𝑆𝑛⁄ )2 < (𝑆 𝑆𝑛⁄ ) < 1 (21)
Methodology for Minimizing Power Losses in Feeders... 167
and so, acceptable intervals to work on the variable cost difference,
comparing several capacities, are small. When despised, there are no serious
mistakes.
2.2 Technical constraints
a) Power balance in the nodes (Kirchhoff's First Law for apparent power):
Includes all load nodes and is expressed according to the apparent power in (kVA),
supposing that there is a single power factor to the network.
∑ 𝑆[𝐼𝑁(𝑖, 𝑘)]𝐾1(𝑖)𝑘=1 − ∑ 𝑆[𝐼𝑇(𝑖, 𝑘)]𝐾2(𝑖)
𝑘=1 = 𝑆(𝑖); 𝑖 = 1, … 𝑁𝑑; 𝑘 = 1, … , 𝐾2(𝑖) (22)
where:
𝐼𝑇(𝑖,𝑘): Reference number for lines leaving the 𝑖 node
𝑆 : Apparent power circulating through the line (kVA)
𝐾2(𝑖): Quantity of line leaving the i node
𝐾1(𝑖): Quantity of line entering the i node
𝑆(𝑖): Apparent load power of the i node (kVA)
b) Capacity selection in each substation:
In each p substation point, only one transformer already existent on the initial
network will be installed (if necessary, in case of expansion) through the following
expression:
∑ 𝑌(𝑝, 𝑙) ≤ 1; 𝑝 = 1,𝑁𝑐(𝑝)𝑖=1 … , 𝑁𝑡; 𝑙 = 1, … , 𝑁𝑐(𝑝) (23)
c) Eliminating minimum load or underutilized substations:
This restriction helps eliminating disabled substations
∑ 𝑆[𝐼𝑇(𝑖, 𝑘)]𝐾2(𝑝)𝑘=1 ≥ 𝑆𝑚𝑖𝑛(𝑙) ∙ 𝑌(𝑝, 𝑙); 𝑝 = 1, … , 𝑁𝑡; 𝑙 = 1, … , 𝑁𝑐(𝑝) (24)
Hussein M. Khodr et.al. 168
where:
𝑆𝑚𝑖𝑛(𝑙): Minimum acceptable load limit imposed by the rules of related institutions
(kVA) or electricity company utility.
d) Maximum load limitation or overload in substations:
This constraint enables a maximum load limitation in each substation, this
avoiding their overload.
∑ 𝑆[𝑠, 𝐼𝑇(𝑖, 𝑘)]𝐾2(𝑝)𝑘=1 ≤ 𝑆𝑚𝑎𝑥(𝑙) ∙ 𝑌(𝑝, 𝑙); 𝑝 = 1, … , 𝑁𝑡; 𝑙 = 1, … , 𝑁𝑐(𝑝) (25)
where:
𝑆𝑚𝑎𝑥(𝑙): Maximum acceptable load limit (kVA). This value is usually defined by
the electric utility.
e) Thermal limit in lines:
Power circulating in each line cannot exceed the maximum limit imposed
by itself.
𝑆(𝑘) ≤ 𝑆𝐿(𝑘)𝑚𝑎𝑥 ∙ 𝑊(𝑖, 𝑘); 𝑘 = 𝐼𝑁(𝑖, 𝑘), … , 𝐼𝑁[𝑖, 𝐾1(𝑖)]; 𝑙 = 1, … , 𝑁𝑑 (26)
where:
𝑆𝐿𝑚𝑎𝑥: Maximum apparent power allowed by the line capacity (kVA)
𝑊(𝑖, 𝑘): Binary variable of line decision
f) Radial configuration condition:
Each load node receives energy from one single line.
∑ 𝑊(𝑖, 𝑘) = 1; 𝑖 = 1, … , 𝑁𝑑𝐼𝑁[𝑖,𝐾1(𝑖)]𝑘=𝐼𝑁(𝑖,1) (27)
Methodology for Minimizing Power Losses in Feeders... 169
g) General load power limitation according to the capacities of substations
or electric power generation unities. General power balance of the distribution
system.
The total load power at a given substation point must be lower or equal to the sum
of the maximum capacity of each substation.
∑ 𝑆(𝑘) ≤ 𝑆𝑚𝑎𝑥𝐼𝑇[𝑝,𝐾2(𝑝)]𝑘=𝐼𝑇(𝑝,1)
[𝑁𝑐(𝑝)] ∙ 𝑌[𝑝, 𝑁𝑐(𝑝)]; 𝑝 = 1, … , 𝑁𝑡 (28)
where:
𝑆(𝑘): Apparent power sent in each output line capacity (kVA)
h) Eliminating two-way power flow directions on the lines:
The line should have one way only.
If the 𝑙𝑘 = 𝐼𝑇 (𝑖, 𝑘); 𝑘 < 𝐾1(𝑖) and 𝐾1 = 𝐼𝑁(𝑖, 𝑘); 𝑘 < 𝐾2(𝑖)
then:
𝑊(𝑖, 𝑙𝑘) + 𝑊(𝑖, 𝑘𝑙) ≤ 1; 𝑗 = 1, … , 𝑁𝑑 (29)
where:
𝑖 : Node references
𝑘𝑙 : Line references
The constraint on the maximum voltage drop is not directly established in
this model. Initially, the network’s configuration is not known, such as, for instance,
the number of network branch lines. A branch line is a section that starts from the
transformer rand finishes on the last connected load. It should be pointed out that
when network losses rise, voltage drops rise, and so the cost of power losses in the
objective function demands a punishment in the cost minimization process [20-23].
One way of confirming the demands of voltage drops is when the network is
relatively large, is through the power flow which circulates through the radial
system after finishing the optimization process. According to the results of the
Hussein M. Khodr et.al. 170
power flow, it is possible to confirm that the values have violated the capacity of
line sections. Constraints f and h might be considered redundant, but in practice it is
possible to observe that they help increasing the speed while searching for an
optimal solution. The quantity of variables considers slack and artificial variables
through the following expression:
𝑄𝑉 = 2 ∙ 𝐿𝑖 + ∑ 𝑁𝑐(𝑝)𝑁𝑡𝑝=1 (30)
where 𝑄𝑉 : Quantity of mathematical model variables.
This quantity depends on the total number of lines (𝐿𝑖), the quantity of substation
points (𝑁𝑡) and suggested capacities in each point [𝑁𝑐(𝑝)].
Amount of expressed restrictions:
𝑄𝑅 = 𝐿𝑖 + 2 ∙ 𝑁𝑑 + 2 ∙ 𝑁𝑡 + 𝑁𝐸 + 2 ∙ ∑ 𝑁𝑐(𝑝)𝑁𝑡𝑝=1 (31)
where:
𝑄𝑅: Quantity of restrictions on the mathematical model
𝑁𝐸: Number of equations resulting from two-way flow lines
3. Identification of Optimization Techniques to Solve The Proposed Model
The problem of the distribution networks reconfiguration can be formulated as a MILP
type problem to which the solution would be the optimal topology of the network. All
technical constraints must be satisfied, such as the power balance in all the nodes and
the feeders, minimum and maximum substations load limits, thermal limit in the lines
and the radial configuration condition of resulting networks. This is a combinatorial
optimization programming problem and has two types of variables: binary or
complicated variable, as they are usually called which are integer and continuous
variables and represent the powers circulating through the branches of the network,
and integer variables of the substations which may be installed into the substations
points. When the number of these binary variables is increased, execution time
becomes exponential function according to the time of computer calculation.
This type of problem is difficult to handle because of its combinatory nature,
in addition to the difficulty in creating a mathematical formulation of certain
constraints such as the radial configuration of resulting networks. There is also the
problem with integer and continuous variables, which makes it a MILP problem
even more difficult to solve.
Methodology for Minimizing Power Losses in Feeders... 171
The network on Fig. (1) has 4 load nodes and 2 nods as substation points.
The network is made of two branch lines which are represented, in the figure, with
thick lines. The dotted lines are open lines in normal operation conditions and their
switches are represented on figure (1). The conductor parameters and technical data
are shown on Table (1). Each circuit is fed through a 100 kVA transformer. Loads
are shown in Fig. (1).
Fig. (1). Current state distribution network.
It is necessary to know the current state of the network and, in order to obtain
that information, a power flow for radial distribution networks is executed. In this
case, it is very simple to know the power loss values. These values are shown on
table (2) and, when they are analyzed, it is possible to deduce that this network is not
operating in an optimal condition. In order to determine the state of the optimal
operation of this network, it is necessary to formulate the optimization problem.
Table (1). Network data.
Branch Initial
Node
Final
Node
Distance
(m)
Conductor
Section
(mm2)
R
(Ω/km)
X
(Ω/km)
IMAX
(A) JECO (A/mm2)
1 5 1 300 3x400 AL 0.102 0.096 515 0.5114
2 5 2 400 3x400 AL 0.102 0.096 515 0.5114
3 2 1 150 3x150 AL 0.265 0.101 285 0.8631
4 1 2 150 3x150 AL 0.265 0.101 285 0.8631
5 3 1 600 3x150 AL 0.265 0.101 285 0.8631
6 1 3 600 3x150 AL 0.265 0.101 285 0.8631
Hussein M. Khodr et.al. 172
Continu table (1)
Branc
h
Initia
l Node
Final
Node
Distanc
e (m)
Conductor
Section (mm2) R (Ω/km) X (Ω/km) IMAX (A) JECO (A/mm2)
7 4 2 500 3x150 AL 0.265 0.101 285 0.8631
8 2 4 500 3x150 AL 0.265 0.101 285 0.8631
9 3 4 150 3x150 AL 0.265 0.101 285 0.8631
10 4 3 150 3x150 AL 0.265 0.101 285 0.8631
11 6 3 200 3x400 AL 0.102 0.096 515 0.5114
12 6 4 350 3x400 AL 0.102 0.096 515 0.5114
Table (2). Current state network losses.
Current state of the network
Branch Power losses (kW) Power losses (kVA)
5-1 1.48 1.69
1-3 3.58 3.89
2-4 1.39 2.22
4-6 1.29 1.70
Total 7.74 9.50
4. Analyzing and Validating The Results
The optimization problem is based on the initial network shown on Fig. (2). In this
figure, a single power flow sense is shown when there is no doubt about its sense,
for instance, between nodes 5-1, 5-2 and 6-3 and 6-4. Logically, power circulates
from the source nodes until loading nodes where these powers will be consumed.
When flow senses are not obvious, then they are represented with double opposite
senses in order to give node 1 the possibility to, for instance, receive energy from
the node 5 transformer or any other way through node 6 which represent another
power source.
In order to formulate the optimization problem, powers are represented by the
letter S and directions are represented by the numbers of involved nodes
respectively. For instance, S51 means that the apparent power circulates from node 5
to node 1. This power is apparent and its unit is kVA.
Methodology for Minimizing Power Losses in Feeders... 173
In order to find the optimal operational configuration of the radial network, it
is necessary to minimize power losses in this case, because circuits are built and so,
the investment cost is null, and it is not necessary to build new branches.
The network in figure (2) has 12 lines and 2 transformers installed in nodes 5
and 6 respectively. Our aim is to minimize power losses which are the square
function of the apparent power. It is necessary to calculate objective function
coefficients. These values are shown in table (3).
Fig. (2). Initial considered network
Table (3). Technical features of the conductors.
Branch Initial Node
Final Node
Conductor
Section
(mm2) Thermal capacity per
phase (kVA)
Loss coefficient * 10-3
(kVA)
1 5 1 3x400 AL 118.45 19.21
2 5 2 3x400 AL 118.45 25.61
3 2 1 3x150 AL 65.55 16.21
4 1 2 3x150 AL 65.55 16.21
5 3 1 3x150 AL 65.55 64.85
6 1 3 3x150 AL 65.55 64.85
7 4 2 3x150 AL 65.55 54.04
8 2 4 3x150 AL 65.55 54.04
Hussein M. Khodr et.al. 174
Continu table (3).
Branch Initial
Node
Final
Node
Conductor
Section (mm2)
Thermal capacity per
phase (kVA)
Loss coefficient * 10-3
(kVA)
9 3 4 3x150 AL 65.55 16.21
10 4 3 3x150 AL 65.55 16.21
11 6 3 3x400 AL 118.45 12.81
12 6 4 3x400 AL 118.45 22.41
5. Mathematical Model of The Case Under Study
5.1 Objective function of the case under study
In order to formulate the objective function to minimize line power losses, it is
necessary to know how many lines are there in Fig. (2). When observing this, it is
possible to confirm that it has 12 lines and so, the objective function will have 12
terms as it is possible to confirm on the following equation:
MIN F = 10-3
* (19.21 * S51 + 25.61 * S52 + 16.21 * S21 + 16.21 * S12 + 64.85 *
S31 + 64.85 * S13 + 54.04 * S42 + 54.04 * S24 + 16.21 * S34 + 16.21 * S43 +
12.81 * S63 + 22.41 * S64); (32)
5.2 Technical constraints of the case study
1. Power balance in the nodes (Kirchhoff's First Law):
It includes all loading nodes and is expressed according to the apparent
power in (kVA), assuming a single power factor to the network. In this network,
there are 4 loading nodes, which lead to the following 4 equations:
S31 + S21 + S51 - S13 - S12 = 28;
S52 + S12 + S42 - S21 - S24 = 41;
S63 + S13 + S43 - S31 - S34 = 60;
S64 + S34 + S24 - S42 - S43 = 35;
(33)
Methodology for Minimizing Power Losses in Feeders... 175
2. Selection of the capacity of each substation.
In this network there are 2 nodes with existing substation points, so the
binary variables are fixed with the optimization process.
Y1 = 1;
Y2 = 1;
(34)
3. Eliminating minimum load or underused substations.
In the example there are 2 nodes with substation points, so there will be 2
equations:
S51 + S52 - 50 * Y1 >= 0;
S63 + S64 - 50 * Y2 >= 0;
(35)
4. Maximum load limitation or overload in substations:
In the example there are 2 substation nodes, so there will be two equations:
S51 + S52 - 95 * Y1<= 0;
S63 + S64 - 95 * Y2 <= 0;
(36)
Hussein M. Khodr et.al. 176
5. Thermal limit in lines.
The network has 12 lines, so there will be 12 equations.
S51 – 118.45 * X51 <= 0;
S52 – 118.45 * X52 <= 0;
S21 – 65.55 * X21 <= 0;
S12 – 65.55 * X12 <= 0;
S31 – 65.55 * X31 <= 0;
S13 – 65.55 * X13 <= 0;
S42 – 65.55 * X42 <= 0;
S24 – 65.55 * X24 <= 0;
S34 – 65.55 * X34 <= 0;
S43 – 65.55 * X43 <= 0;
S63 – 118.45 * X63 <= 0;
S64 – 118.45 * X64 <= 0;
(36)
6. Radial condition of operation of the network.
In this example the network has 4 load nodes, so there will be 4 equations:
X51 + X21 + X31 = 1;
X52 + X12 + X42 = 1;
X13 + X63 + X43 = 1;
X34 + X24 + X64 = 1;
(37)
7. General power limitation according to the capacities of substations or
electric power generation unites. General power balance of the distribution system.
All the loads must be covered by the available generated power plus the power
losses.
Each example has 1 general network power balance equation:
S51 + S52 + S64 + S63 - 95 * Y1 - 95 * Y2 <= 0;
(38)
Methodology for Minimizing Power Losses in Feeders... 177
8. Eliminating two-sense power flow on the lines. Between both paths there
should be only one sense.
In this example there are 4 two-sense arches, so there will be 4 equations:
X31 + X13 <= 1;
X24 + X42 <= 1;
X21 + X12 <= 1;
X34 + X43 <= 1;
(39)
Binary variables are: X51, X52, X21, X12, X31, X13, X24, X42, X34, X43, X63, X64, Y1, Y2.
The necessary format to be executed in general optimization software
LINGO is that shown in figure (3).
! Objective function of the 4 load nodes, substation nodes and twelve lines network;
MIN = (19.21 * S51 + 25.61 * S52 +
16.21 * S21 + 16.21 * S12 + 64.85 *
S31 + 64.85 * S13 + 54.04 * S42 + 54.04
* S24 + 16.21 * S34 + 16.21 * S43 +
12.81 * S63 + 22.41 * S64) * 10^-3 + 0 * y1 + 0 * y2;
! Restrictions;
! 1. Power balance in the nodes: Kirchhoff's First Law;
S31 + S21 + S51 - S13 - S12 = 28;
S52 + S12 + S42 - S21 - S24 = 41;
S63 + S13 + S43 - S31 - S34 = 60;
S64 + S34 + S24 - S42 - S43 = 35;
! 2. Capacity selection in each substation;
y1 = 1;
y2 = 1;
! 3. Eliminating minimum loaded substation or underused substations;
S51 + S52 - 50 * y1 >= 0;
S63 + S64 - 50 * y2 >= 0;
! 4. Maximum loaded substation limitation or overused substations;
S51 + S52 - 95 * y1 <= 0;
S63 + S64 - 95 * y2 <= 0;
Hussein M. Khodr et.al. 178
! Thermal limit of the lines;
S51 - 118.45 * X51 <= 0;
S52 - 118.45 * X52 <= 0;
S21 - 65.55 * X21 <= 0;
S12 - 65.55 * X12 <= 0;
S31 - 65.55 * X31 <= 0;
S13 - 65.55 * X13 <= 0;
S42 - 65.55 * X42 <= 0;
S24 - 65.55 * X24 <= 0;
S34 - 65.55 * X34 <= 0;
S43 - 65.55 * X43 <= 0;
S63 - 118.45 * X63 <= 0;
S64 - 118.45 * X64 <= 0;
! Radial conditions of the final distribution network configuration;
X51 + X21 + X31 = 1;
X52 + X12 + X42 = 1;
X13 + X63 + x43 = 1;
X34 + X24 + X64 = 1;
! General load power limitation according to the capacities of substations or electric power generation
unities. General power balance of the distribution system;
S51 + S52 + S64 + S63 - 95 * y1 - 95 * y2 <= 0;
! Eliminating two-way power flow on the lines. Between both paths there should be only one way!;
X31 + X13 <= 1;
X24 + X42 <= 1;
x21 + X12 <= 1;
X34 + x43 <= 1;
! Defining the binary variables or the complexity variables;
@BIN( X51);
@BIN( X52);
@BIN( X21);
@BIN( X12);
@BIN( X31);
@BIN( X13);
@BIN( X24);
@BIN( X42);
@BIN( X34);
Methodology for Minimizing Power Losses in Feeders... 179
@BIN( X43);
@BIN( X63);
@BIN( X64);
@BIN( y1);
@BIN( y2);
!;
Fig. (3). Problem formulation code in LINGO software format.
Software results (LINGO) just as they can be seen in figure (4).
Fig. (4). Execution of the mathematical model on LINGO Software.
LINGO software results can be seen in table (4).
Table (4). Results of the general optimization program.
Global optimal solution found at iteration: 9
Objective value: 3.140840
Variable Value Reduced Cost
Hussein M. Khodr et.al. 180
S51 28.00000 0.000000
S52 41.00000 0.000000
S21 0.000000 0.2261000E-01
S12 0.000000 0.9810000E-02
S31 0.000000 0.5845000E-01
S13 0.000000 0.7125000E-01
S42 0.000000 0.5084000E-01
S24 0.000000 0.5724000E-01
S34 0.000000 0.6610000E-02
S43 0.000000 0.2581000E-01
S63 60.00000 0.000000
S64 35.00000 0.000000
Y1 1.000000 0.000000
Y2 1.000000 0.000000
X51 1.000000 0.000000
X52 1.000000 0.000000
X21 0.000000 0.000000
X12 0.000000 0.000000
X31 0.000000 0.000000
X13 0.000000 0.000000
X42 0.000000 0.000000
X24 0.000000 0.000000
X34 0.000000 0.000000
X43 0.000000 0.000000
X63 1.000000 0.000000
X64 1.000000 0.000000
Row Slack or Surplus Dual Price
1 3.140840 -1.000000
2 0.000000 -0.1921000E-01
3 0.000000 -0.2561000E-01
4 0.000000 -0.1281000E-01
5 0.000000 -0.2241000E-01
6 0.000000 0.000000
7 0.000000 0.000000
8 19.00000 0.000000
Methodology for Minimizing Power Losses in Feeders... 181
9 45.00000 0.000000
10 26.00000 0.000000
11 0.000000 0.000000
12 90.45000 0.000000
13 77.45000 0.000000
14 0.000000 0.000000
15 0.000000 0.000000
16 0.000000 0.000000
17 0.000000 0.000000
18 0.000000 0.000000
19 0.000000 0.000000
20 0.000000 0.000000
21 0.000000 0.000000
22 58.45000 0.000000
23 83.45000 0.000000
24 0.000000 0.000000
25 0.000000 0.000000
26 0.000000 0.000000
27 0.000000 0.000000
28 26.00000 0.000000
29 1.000000 0.000000
30 1.000000 0.000000
31 1.000000 0.000000
32 1.000000 0.000000
5.3 Interpretation of the results of the case under study
The values to be interpreted are in table (5) and the meaning of each of the
variables is shown in figure (5). In this figure, there are 2 branch lines with different
configurations from the current state because of the result of the optimization
process. These branch lines have fewer power losses as shown in table (6).
In order to compare the optimization results, it is necessary to calculate
branch lines power losses or circuits arising from the optimization, so that this might
be accessed. As this is a simple network, then it is very simple to know how many
power losses there are. These values are shown in table (6).
Hussein M. Khodr et.al. 182
Table (5). LINGO software results.
Global optimal solution found at iteration: 9
Objective value: 3.140840
Variable Value Reduced Cost
S51 28.00000 0.000000
S52 41.00000 0.000000
S63 60.00000 0.000000
S64 35.00000 0.000000
Y1 1.000000 0.000000
Y2 1.000000 0.000000
X51 1.000000 0.000000
X52 1.000000 0.000000
X63 1.000000 0.000000
X64 1.000000 0.000000
Fig. (5). Network topology after reconfiguration.
Methodology for Minimizing Power Losses in Feeders... 183
Table (6). Network power losses after the reconfiguration.
Branch After reconfiguration
Initial Node Final Node Losses (kW) Losses (kVA)
5 1 0.15 0.54
5 2 0.43 1.05
6 3 0.46 0.77
6 4 0.27 0.78
Total 1.31 3.14
Comparing the results from the point of view of network power losses before
reconfiguration and after this process, which are shown in table (7) (it contains
losses values of apparent power in kVA and active power in kW), that is, of the
current network state and the reconfiguration state. To sum up, network
reconfiguration allows us to conclude that this method has many benefits, saving
almost 66% of apparent power, allowing us to conclude that this method has great
advantages when applied in distribution networks.
Table (7). Comparing losses in the current state and after reconfiguration.
Branch Current State Reconfiguration
Losses (kW) Losses (kVA) Losses (kW) Losses (kVA)
5-1 1.48 1.69 0.15 0.54
1-3 3.58 3.89 - -
2-4 1.39 2.22 - -
4-6 1.29 1.70 0.27 0.78
5-2 - - 0.43 1.05
6-3 - - 0.46 0.77
Total 7.74 9.50 1.31 3.14
Reduction - - 83.1 % 66.9%
Hussein M. Khodr et.al. 184
6. Real Study Case
This real case study consists of 3 substations. The general data is shown in tables (8,
9 and 10).
Table (8). General Substation Data I (S/E).
Circuit Length (m) Transformers number Installed kVA
1 14181.1 243 11157.5
2 15401.7 376 17732.5
3 10026.7 258 13574
4 11505.0 307 16164
5 16008.2 179 10062.5
6 30021.7 352 15042.6
7 7361.0 12 197.5
8 2393.8 5 85
Substation 1 has an installed capacity in power transformers of 84.02 MVA, 8
circuits at 13.8 kV and has a total length of 95.4 km.
Table (9). General Substation Data II.
Circuit Length (m) Transformers Installed kVA
9 12829.2 169 9980
10 10376.1 105 8836.5
11 15245.33 226 8902.5
12 10386.9 230 9100
13 15895.34 104 9692.5
Substation II has an installed capacity in power transformers of 46.5 MVA, 5
circuits at 13.8 kV, and has a total length of 64.7 km.
Methodology for Minimizing Power Losses in Feeders... 185
Table (10). General Substation Data III.
Circuit Length (m) Transformers Installed kVA
14 20926 169 4902.5
15 3683 6 117.5
16 3712 13 600
Substation III has an installed capacity in power transformers of 5.6 MVA, 3
circuits at 13.8 kV, and has a total length of 28.3 km.
A database was created from these plants in which necessary values and
parameters are as follows: 2799 nodes for the 14 circuits in 13.8 kV with the
following data:
1. Nominal or rated voltage level in kV;
2. Average measured voltage level in kV;
3. Installed kvar in each circuit;
4. Type of dominating pole;
5. Load curve in each circuit in kVA and kW;
6. Sending node and ending node in each branch section after defining the
nodes;
7. Length of each section in meters;
8. Conductor cross section area in each section;
9. Phase number;
10. Total number of kVA installed in each node.
The process of defining nodes was conducted through the plants provided in
a digital format and in which the following points of interest were considered:
1. Transformers banks;
2. Conductor’s cross section area or size;
3. Switches and isolators;
4. Taps, transformers or capacitor banks;
5. Any other interest point in the circuit.
Measurements were also conducted in each circuit, more precisely on
substations outputs, at the start of each circuit so that it would be possible to
calculate the factors which define each of them.
Figure (6) represents an example of a load curve:
Hussein M. Khodr et.al. 186
Fig. (6). Typical load curve in a circuit.
6.1 Evaluation of the current state in each circuit
In order to know the state of each circuit, it is necessary to carry out a power
flow to radial distribution networks. The power factor was measured in each
substation and the average value is 0.85. After evaluating the current state in each
circuit, power flow results and operation features in each circuit are shown in tables
11, 12, 13, 14, 15, 16, 17 and 18:
Table (11). Maximum load in S/E I circuits.
Circuit Installed kVA Max kW Capacity Factor
1 11157.5 9911 1.00
2 17732.5 15073 1.00
3 13574 11506 1.00
4 16164 13739 1.00
5 10062.5 8553.1 1.00
6 15042.6 12786 1.00
7 197.5 167.9 1.00
8 85 72.3 1.00
Methodology for Minimizing Power Losses in Feeders... 187
Table (12). Active power losses in S/E I circuits.
Circuit Line Losses Transformer Losses Total Losses
kW In % kW In % kW In %
1 691.66 6.98 168.626 1.70 860.286 8.68
2 879.86 5.84 255.465 1.69 1135.325 7.53
3 616.17 5.36 175.857 1.53 792.027 6.89
4 618.70 4.50 231.628 1.69 850.328 6.19
5 540.98 6.32 147.796 1.73 688.776 8.05
6 724.83 5.67 221.189 1.73 946.019 7.40
7 0.21 0.12 3.311 1.97 3.521 2.09
8 0.01 0.01 1.405 1.94 1.415 1.95
Table (13). Voltage drop in S/E I circuits.
Circuit Length (m) Worst Node U (kV) ΔU (%)
1 14181.1 187 11.65 11.71
2 15401.7 231 11.56 12.42
3 10026.7 212 11.72 11.22
4 11505 .0 155 12.10 8.33
5 16008.2 155 11.79 10.68
6 30021.7 198 11.57 12.37
7 7361.0 7 13.17 0.22
8 2393.8 9 13.20 0.03
Table (14). Energy losses in S/E I circuits.
Circuit Line Losses Transformer Losses Total Losses
MWh/year In % MWh/year In % MWh/year In %
1 3535.84 5.52 862.03 1.35 4397.87 6.87
2 4497.67 4.61 1305.89 1.34 5803.55 5.95
3 3149.92 4.23 899.00 1.21 4048.92 5.44
Hussein M. Khodr et.al. 188
Continu table (14).
Circuit Line Losses Transformer Losses Total Losses
MWh/year In % MWh/year In % MWh/year In %
4 3163.04 3.56 1184.17 1.33 4347.22 4.89
5 2765.54 5.00 755.55 1.37 3521.09 6.37
6 3705.48 4.48 1130.76 1.37 4836.24 5.85
7 1.07 0.10 16.92 1.58 17.99 1.68
8 0.05 0.01 7.17 1.41 7.22 1.42
Table (15). Maximum load in S/E II and III circuits.
Circuit Installed kVA max kW Capacity Factor
9 9980 7144.3 1.00
10 8836.5 7457.9 1.00
11 8902.5 6795.8 1.00
12 9100 7586.3 1.00
13 9692.5 8174.5 1.00
14 4902.5 4081.7 1.00
15 117.5 99.87 1.00
16 600 510 1.00
Table (16). Active losses in S/E II and III circuits.
Circuit Line Losses Transformer Losses Total Losses
kW In % kW In % kW In %
9 282.32 3.95 119.218 1.67 401.538 5.62
10 308.13 4.13 98.639 1.32 406.769 5.45
11 138.39 2.04 134.234 1.98 272.624 4.02
12 156.22 2.06 130.573 1.72 286.793 3.78
13 743.35 9.09 136.392 1.67 879.742 10.76
14 54.95 1.35 73.275 1.80 128.225 3.15
Methodology for Minimizing Power Losses in Feeders... 189
Continu table (16).
Circuit Line Losses Transformer Losses Total Losses
kW In % kW In % kW In %
15 0.04 0.04 2.257 2.26 2.297 2.30
16 0.31 0.06 8.842 1.73 9.152 1.79
Table (17). Voltage drops in S/E II and III circuits.
Circuit Length (m) Worst Node U (kV) ΔU (%)
9 12829.2 136 12.33 6.56
10 10376.1 128 12.29 6.91
11 15245.33 171 12.67 4.01
12 10386.9 169 12.65 4.14
13 15895.34 84 11.21 15.06
14 20926 137 12.78 3.16
15 3683 4 13.19 0.06
16 3712 17 13.18 0.15
Table (18). Energy Losses in S/E II and III circuits.
Circuit Line Losses Transformer Losses Total Losses
MWh/year % MWh/year % MWh/year %
9 23867.95 3.52 5818.98 0.86 29686.93 4.38
10 30080.86 3.67 8733.90 1.07 38814.75 4.74
11 21273.63 1.82 6071.57 0.52 27345.20 2.34
12 21362.55 1.84 7997.68 0.69 29360.23 2.53
13 18667.70 8.11 5100.02 2.22 23767.72 10.33
14 24812.61 1.2 7571.81 0.37 32384.42 1.57
15 7.25 0.04 114.27 0.63 121.51 0.67
16 0.34 0.05 48.44 7.03 48.78 7.08
Hussein M. Khodr et.al. 190
6.2 Network reconfiguration
The model which should be used in the network reconfiguration includes the
following circuits:
1. Circuit 1; 2. Circuit 2;
3. Circuit 3; 4. Circuit 4;
5. Circuit 5; 6. Circuito 6;
7. Circuit 9; 8. Circuito 10;
9. Circuit 11; 10. Circuito 12.
Other circuits were not included in the reconfiguration study because they
didn’t have any connection or were close to other circuits/substations and had a low
loss rate.
In order to carry out modification works in the network, a meeting with
operators from the electric company was conducted in order to define possible
connections or positions (opening/closing) of switches and a shorter connection
distance of 55 meters was established because of the companies’ economic
restrictions. The file of possible connections is shown in table (19).
Table (19). Archive of possible connections agreed with the electric company.
Sending Node Ending Node Length (m) Conductor size Initial Circuit Final Circuit
35 143 1 4/0 1 2
56 181 44 2 1 2
66 249 1 1/0 1 2
77 146 43 4 1 2
136 151 1 2/0 1 9
139 195 24 2/0 1 2
213 124 1 1/0 2 10
217 98 34 2 2 12
231 104 39 4 2 3
244 153 1 4/0 2 10
29 8 35 1/0 3 4
42 53 43 2 3 4
55 98 41 4 3 4
Methodology for Minimizing Power Losses in Feeders... 191
Continu table (19).
Sending Node Ending Node Length (m) Conductor size Initial Circuit Final Circuit
40 83 1 4/0 1 3
90 132 37 4 3 4
70 97 1 2/0 1 3
113 74 45 2/0 3 12
118 10 1 2/0 3 11
119 68 1 2/0 3 12
133 121 1 2/0 3 10
140 110 1 2/0 3 10
146 105 5 2 3 10
191 97 1 4 3 10
210 131 1 2/0 3 10
24 128 1 2/0 4 5
40 123 23 4 4 5
186 95 1 4/0 4 12
189 84 1 1/0 4 12
202 134 1 2 4 11
199 130 43 2 4 11
13 37 13 1/0 5 6
176 196 1 2 5 11
10 263 1 1/0 5 6
25 38 1 4/0 9 10
27 22 10 1/0 9 10
40 26 1 1/0 9 10
60 72 26 1/0 9 10
80 83 10 1/0 9 10
106 87 41 2 9 10
112 146 26 4 11 11
135 138 52 2 11 12
Hussein M. Khodr et.al. 192
After creating a database (2344 circuit nodes were included in the reconfiguration
process), an optimization program was executed of which the results are that the
switches and sectionalizing should be open. These results are shown in table (20).
Table (20). Circuits should be open.
Nodes Circuit Nodes Circuit
348 4 91 5
143 4 168 11
260 5 259 6
32 9 33 10
31 9 28 10
37 9 38 9
98 10 99 11
114 11 118 11
35 1 143 2
41 4 122 4
189 12 84 12
60 9 72 10
After the reconfiguration, the circuit database was changed mainly in order to study
these circuits which are a result of the optimization process. In order to know the
features and parameters of optimized circuits, a power flow was executed in radial
networks to calculate losses in branches and voltage drops in busbars. The results of
the configured network are shown in tables (21-23).
Table (21). Active power losses in reconfigured circuits.
Total losses
Circuit Current State Reconfigured State
kW In % kW In %
1 860.286 8.68 503.22 6.48
2 1135.325 7.53 658.00 5.47
3 792.027 6.89 785.00 5.63
4 850.328 6.19 556.91 5.33
Methodology for Minimizing Power Losses in Feeders... 193
Continu table (21).
Total losses
Circuit Current State Reconfigured State
kW In % kW In %
5 688.776 8.05 358.84 5.55
6 946.019 7.40 571.49 6.08
9 401.538 5.62 353.53 4.95
10 406.769 5.45 408.75 5.48
11 272.624 4.02 402.91 5.92
12 286.793 3.78 345.12 4.55
Table (22). Energy losses in reconfigured circuits.
Total losses
Circuit Current State Reconfigured State
In MWh/year In % MWh/year In %
1 4397.87 6.87 2572.48 5.13
2 5803.55 5.95 3363.70 4.33
3 4048.92 5.44 4012.76 4.45
4 4347.22 4.89 2846.85 4.21
5 3521.09 6.37 1834.38 4.38
6 4836.24 5.85 2921.46 4.81
9 2710.50 5.01 2750.24 4.63
10 2720.51 4.84 1926.24 4.52
11 1841.22 3.59 6383.39 5.12
12 1936.94 3.38 3781.12 4.54
In order to evaluate operational features in the circuits after its reconfiguration and
to quantify power and energy saving, as well as the improvement on voltages, it is
necessary to compare the values before and after the reconfiguration. Results
concerning power and energy saving are shown in tables (24& 25).
Hussein M. Khodr et.al. 194
Table (23). Voltage drops in reconfigured circuits.
Voltage Drop
Circuit Current State Reconfigured State Reduction
U (kV) in the worst node % U (kV) in the worst node In % U (kV) In %
1 11.65 11.71 12.25 7.16 0.60 4.55
2 11.56 12.42 11.92 9.70 0.36 2.72
3 11.72 11.22 12.26 7.10 0.54 4.12
4 12.10 8.33 12.35 6.46 0.25 1.87
5 11.79 10.68 12.34 6.50 0.55 4.18
6 11.57 12.37 11.9 9.86 0.33 2.51
9 12.33 6.56 12.39 6.17 0.06 0.39
10 12.29 6.91 12.38 6.18 0.09 0.73
11 12.67 4.01 12.68 3.96 0.01 0.05
12 12.65 4.14 12.6 4.52 -0.05 -0.38
It is necessary to explain that during the reconfiguration process, some circuits with
heavier load will lose some of it. This load will be transferred to other circuits with a
lighter load. Thus, it is possible to conclude that the first case, circuits will decrease
its losses and, consequently, voltage in busbars will improve. On the second case,
circuits will increase its losses and its voltage will worsen. Generally, total losses
will lessen and voltage will comply with the equipment’s acceptable limits.
Table (24). Power saving.
Reduction of power losses
Circuit Total Reduction
In kW In %
1 -357.062 -2.20
2 -477.325 -2.06
3 -7.028 -1.26
4 -293.421 -0.86
5 -329.937 -2.50
6 -374.528 -1.32
9 -48.004 -0.67
Methodology for Minimizing Power Losses in Feeders... 195
Continu table (24).
Reduction of power losses
Circuit Total Reduction
In kW In %
10 +1.983 +0.03
11 +130.287 +1.90
12 +58.324 +0.77
Total -1696.711 -8.17
Table (25). Energy saving.
s
Circuit Total Reduction
MWh/year In %
1 -1825.387 -1.74
2 -2439.858 -1.62
3 -36.163 -0.99
4 -1500.364 -0.68
5 -1686.707 -1.99
6 -1914.776 -1.04
9 +39.740 -0.38
10 -794.268 -0.32
11 +4542.167 +1.53
12 +1844.173 +1.16
Total -3771.444 -6.07
In tables (24 and 25), the signs (+) and (-) mean increase and decrease
respectively. In these tables it is possible to confirm that, generally, reconfiguration
leads to an approximately 6% to 8% saving in energy and power respectively
without investment.
Hussein M. Khodr et.al. 196
7. Conclusions
The main conclusions to be taken from this work are:
Application of the formulation in real cases of electric power distribution
networks, controlling all variables through a MILP type optimization model. Its
solution is based on the Branch and Bound method. It may also be applied in other
systems such as gas, water and communication systems;
Introduction of mathematical simplifications on a non-linear problem in
order to linearize it, so that linear optimization techniques can be applied;
This model can be used by the distribution system operator, in the SCADA
system to solve the system overload or to isolate the fault. Also it can be used by the
SCADA and with the automated switches to take intelligent decision in the SCADA
making the distribution networks more intelligent, acting as a smart network.
The advantages of applying suggested methodology in the real case were an
energy saving of about 3771.444 MWh/year, allowing a significant economic saving
which is obtained by only opening/closing switches devices without any
investments.
Acknowledgment
The authors would like to express their sincere thanks to the Scientific Research
Deanship of Qassim University for financial support of the research.
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Methodology for Minimizing Power Losses in Feeders... 199
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القدرة مع احلفاظ على احلدود املقبولة جلهود قضبان التوزيع و حتسني جودة القدرة.
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أثبتت اليتمدينة ةقيقية و يفكيلو فولت تقع 13.8نقطة عند مستوى جهد 2344تتكون من ةقيقية كبرية
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Hussein M. Khodr et.al. 200
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ب
ج
2العدد – 7المجلد
م2014 يوليو –هـ 1435 شعبان
النشر العلمي والرتمجة
د
هيئةالتحريررئيسالتحريرأ.د./حممدعبدالسميععبداحلليم .1
ساعدبنمنصورأ.د./ .2
بكرحامدشريفد./أبو .3
يد./ساملنصر .4
سكرتريالتحريريد./شريفحممدعبدالفتاحاخلول .5
ةللمجل ةاالستشاري ةأعضاء اهليئ
اهلندسة املدنية
–لبحتو امليتاهيللمياهوأستتا املتوارداملائيتةبتاملرلقالقتوميورئيساجمللسالعلمسابقاياملصريوزيراملوارداملائيةوالر–أ.د./حممودأبوزيد .1
مصر.
مصر.–جامعةالقاهرة–أستا هندسةالنقلبكليةاهلندسة–أ.د./عصامشرف .2
–جامعتةامللتكستعود–وأستتا اهلندستةاويوتكنيكيتةبكليتةاهلندستةيوليلجامعةامللكسعودلشئونالبحثالعلم–أ.د./عبداهللاملهيدب .3
اململكةالعربيةالسعودية.
.ةالوالياتاملتحد-جامعةأريقونا–لليةاهلندسة–ةقسماهلندسةاملدني–ةواملوارداملائيأستا اهليدروليكا–أ.د./ليفنالنذى .4
مصر.–جامعةعنيمشس-واإلنشائيةبكليةاهلندسةةأستا اهلندسةاويوتكنيكي–أ.د./فتحاهللالنحاس .5
.ةالسعوديةالعربيةاململك–العقيقجبامعةامللكعبدةعلوماهلندسيورئيسحتريرجملةالةأستا اهلندسةاملدني–أ.د./فيصلفؤادوفا .6
.ةالسعوديةالعربيةاململك-جبامعةامللكسعودةنشائيأستا اهلندسةاإل–أ.د./طارقاملسلم .7
اهلندسة الكهربائية
ةيئاالتالكهربوأستا هندسةاآليمبجلسالشورىاملصريرئيسجامعةاألهرامالكنديةورئيسونةالتعليموالبحثالعلم–أ.د./فاروقإمساعيل .8
مصر.–جامعةالقاهرة–بكليةاهلندسة
مصر.–جامعةالقاهرة–بكليةاهلندسةيأستا هندسةاوهدالعال–أ.د./حسنيإبراهيمأنيس .9
مصر.–جامعةعنيمشس–ئيةبكليةاهلندسةاالتالكهربجامعةاملستقبلوأستا هندسةاآل–عميدلليةاهلندسة–أ.د./حممدعبدالرحيمبدر .11
.مصر–جامعةعنيمشس–أستا القوىالكهربائيةبكليةاهلندسة–يالشرقاويأ.د./متول .11
اململكةالعربيةالسعودية.–جامعةامللكعبدالعقيق–واحلاسببكليةاهلندسةائيةأستا اهلندسةالكهرب–يأ.د./علىحممدرشد .12
اململكةالعربيةالسعودية.–جامعةامللكسعود–بكليةاهلندسةيأستا هندسةاوهدالعال–أ.د./عبدالرمحنالعريين .13
س.تون–أستا االتصاالتباملدرسةالوطنيةلالتصاالت-أ.د./ساميتابان .14
اهلندسة امليكانيكية
رئيسمرلقالبحرينللدراساتوالبحو .–أ.د./حممدالغتم .15
مصر.–جامعةالقاهرة-وأستا القوىامليكانيكيةوليللليةاهلندسة–أ.د./عادلخليل .16
مصر.-جامعةالقاهرة–بكليةاهلندسة–نتاجأستا هندسةوميكانيكااإل-أ.د./سعيدجماهد .17
ةالعربيةاململك–جامعةامللكعبدالعقيق-وعميدمعهدالبحو واالستشاراتبكليةاهلندسةةأستا اهلندسةامليكانيكي–يديأ.د./عبدامللكاون .18
.ةالسعودي
احلاسبات واملعلومات
مصر.–جامعةحلوان–أستا نظماملعلوماتبكليةاحلاسباتواملعلومات–أ.د./أمحدشرفالدين .19
جامعةالقصيمومستشاروزيرالتعليمالعاىلوالبحثالعلمتىباململكت –أستا هندسةاحلاسببكليةاحلاسباآلىل–أ.د./عبداهللالشوشان .21
العربي السعودي .
األماراتالعربي املتحده.–امعةالشارق األهلي جب–أستا هندسةاحلاسب–أ.د./معمربطيب .21
تونس.-جامعةتونس–لليةعلوماحلاسب–أستا الشبكات–أ.د./فاروقلمون .22
ه
توياتمحلا
هت
(هت1435شعبان/م2114يوليوباإلجنليقية)73199(،صص1(،العدد)7اجمللد)القصيم
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