Guidelines for Authors…جلة... · 2020-02-23 · Guidelines for Authors This periodical is a...

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.

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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



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

[email protected]

mailto:[email protected]


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



VI


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.

[email protected]

(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 .


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.


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.


(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)


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)


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


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 .


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.,


, (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.


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)


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)


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


(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:


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 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


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)


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.,


(58a)

(58b)

Fig. (5). Matrix computation of the spectrum S for the 2-out-of-3 function


Fig. (6). Computation of the Walsh Spectrum (S-encoding) for the 2-out-of-3 function


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


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


.

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.


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)


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)


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


, (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).


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


(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)


(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.


Fig. (8). The sixteen binary switching functions.


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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57-95, (May 1983) (In Arabic, with an English abstract).


قسم اهلندسة الكهربائية وهندسة احلاسبات، كلية اهلندسة،

، اململكة العربية السعودية21589، جامعة امللك عبد العزيز، جدة، 80204. .ص. ب

[email protected]

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.



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

[email protected]


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


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


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


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


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


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. (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. (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


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.


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).


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



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]


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


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].


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).


Fig. (2). I-V characteristics of some semiconductors solar cell

Fig. (3). P-V characteristics of some semiconductors solar cell


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.


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


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:


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


(a) (b)


AM0 spectrum under: (a) 1 sun, (b) Maximum solar concentration

(a) (b)


AM1.5 spectrum under: (a) 1 sun, (b) Maximum solar concentration


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


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.


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]:


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.


(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


(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


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.


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


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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,

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.




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.


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


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)


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)


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)


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)


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 (€)


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)


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)


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)


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


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.


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


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.


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


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)


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)


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)


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;


! 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);


@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


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


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).


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.


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%


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.


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:


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


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.


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


Continu table (14).


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.



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


Continu table (16).



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.


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


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


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


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


Continu table (21).

Total losses


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


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).


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


Continu table (24).

Reduction of power losses


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


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.


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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“Distribution systems reconfiguration using a modified particle swarm

optimization algorithm”, Electric Power Systems Research 79 (2009) 1521-

1530.

[18] Zhou Q., “Distribution Feeder Reconfiguration for Operation Cost Reduction”.

IEEE Trans. Power Systems, Vol. 12, No. 2, May 1997, pp. 730-735.

[19] Kashem M., “A Geometrical Approach for Network Reconfiguration Based

Loss Minimization in Distribution Systems”. Electrical Power and Energy

Systems 23, 2001, pp. 295-304.

[20] Fattal E., Armas A., Capri L., “Modelo para la optimización de redes

secundarias de distribución utilizando programación Entera-Mixta”, V

Congreso de distribución de energía eléctrica, Venezuela, 1994.

[21] Gönen T., Foote B.L., “Distribution system planning using Mixed-Integer

Programming”, IEE Proceedings, Part C, vol. 128, no. 2, March 1981, pp.

70-79.

[22] Gönen T., Ramirez-Rosado I.J., “Optimal multi-stage planning of power

distribution system”, IEEE Transactions on Power Delivery, vol.

PWRD-2, no. 2, Apr. 1987, pp. 512-519.

[23] Hindi K.S., Brameller A., “Design of Low Voltage Distribution Networks: A

Mathematical Programming Method”, Proceedings IEE, vol. 124, no.1, pp.

54-58, Jan. 1977.

[24] Venikov V.A., “Kibernetika Elektriczeskich Sistiem”, Moskowa, 1974


[email protected]

[email protected]

[email protected]

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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

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