# Efficient harmony search algorithm for multi-stages scheduling problem for power systems degradation

**ABSTRACT** Usually power energy demand increases randomly with time. To enhance system performance, expansion-planning to adapt the power

system capacity to the demand is predicted. This paper uses a harmony search meta-heuristic optimization method to solve the

multi-stage expansion problem for multi-state series-parallel power systems. The study horizon is divided into several periods.

At each period the demand distribution is forecasted in the form of a cumulative demand curve. A multiple-choice of additional

components from a list of available products can be chosen and included into any subsystem component at any stage to improve

the system performance. The components are characterized by their cost, performance (capacity), and availability. The objective

is to minimize each investment over its study period while satisfying availability or performance constraints. A universal

generating function technique is applied to evaluate power system availability. The harmony search approach is required to

identify the optimal combination of adding components with different parameters to be allocated in parallel at each stage.

KeywordsExpansion-planning-Harmony search-Redundancy optimization-Power system-Universal generating moment function

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**ABSTRACT:**Many methods have been developed and are in use for structural size optimization problems, in which the cross-sectional areas or sizing variables are usually assumed to be continuous. In most practical structural engineering design problems, however, the design variables are discrete. This paper proposes an efficient optimization method for structures with discrete-sized variables based on the harmony search (HS) heuristic algorithm. The recently developed HS algorithm was conceptualized using the musical process of searching for a perfect state of harmony. It uses a stochastic random search instead of a gradient search so that derivative information is unnecessary. In this article, a discrete search strategy using the HS algorithm is presented in detail and its effectiveness and robustness, as compared to current discrete optimization methods, are demonstrated through several standard truss examples. The numerical results reveal that the proposed method is a powerful search and design optimization tool for structures with discrete-sized members, and may yield better solutions than those obtained using current methods.Engineering Optimization - ENG OPTIMIZ. 01/2005; 37(7):663-684. -
##### Article: Optimal multistage modernization of power system subject to reliability and capacity requirements

[show abstract] [hide abstract]

**ABSTRACT:**This paper addresses the multistage modernization problem for power systems with series-parallel structure. The study period is divided into several stages according to demand forecast. At each stage the consumer demand distribution is predicted in the form of a cumulative load curve. Different actions, such as modifying existing equipment, changing maintenance policy, or adding new elements, may be undertaken in any system component at any stage to increase the total system capacity and/or reliability. The objective is to minimize the sum of costs of system modernization actions over the study period while satisfying reliability constraints at each stage. In order to solve the problem, a genetic algorithm is used as an optimization tool. The solution encoding technique developed allows the genetic algorithm to manipulate integer strings representing multistage system modernization planes. A solution quality index is comprised of both reliability and cost estimations. The procedure based on the universal generating function is used for evaluation of the availability of multistate series-parallel power systems. This technique allows one to estimate the effect on the entire system availability when both capacity and availability of system elements are varied. An illustrative example is presented in which the optimal expansion plan is found for a coal transportation system of a power station.Electric Power Systems Research. 01/1999; - [show abstract] [hide abstract]

**ABSTRACT:**The problem of the optimization of the structure of a power system where redundant elements are included in order to provide a desired level of reliability is considered. A procedure which determines the minimal cost series-parallel system configuration is proposed. In this procedure, system elements are chosen from a list of products available on the market and the number of such elements is determined for each system component. The elements are characterized by their capacity, availability and cost. System reliability is defined as the ability to satisfy consumer demand which is represented as a piecewise cumulative load curve. To evaluate system reliability, a fast procedure is developed which is based on a universal generating function. A genetic algorithm is used as an optimization technique. An example of the redundancy optimization of a power station coal feeding system is presented.Electric Power Systems Research. 01/1996;

Page 1

Electr Eng (2010) 92:87–97

DOI 10.1007/s00202-010-0165-3

ORIGINAL PAPER

Efficient harmony search algorithm for multi-stages scheduling

problem for power systems degradation

A. Zeblah · S. Hadjeri · E. Chatelet · Y. Massim

Received: 8 April 2009 / Accepted: 13 June 2010 / Published online: 7 July 2010

© The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract Usually power energy demand increases ran-

domly with time. To enhance system performance, expan-

sion-planning to adapt the power system capacity to the

demandispredicted.Thispaperusesaharmonysearchmeta-

heuristicoptimizationmethodtosolvethemulti-stageexpan-

sion problem for multi-state series-parallel power systems.

The study horizon is divided into several periods. At each

period the demand distribution is forecasted in the form of

a cumulative demand curve. A multiple-choice of additional

components from a list of available products can be chosen

and included into any subsystem component at any stage to

improve the system performance. The components are char-

acterized by their cost, performance (capacity), and avail-

ability. The objective is to minimize each investment over

its study period while satisfying availability or performance

constraints. A universal generating function technique is

applied to evaluate power system availability. The harmony

search approach is required to identify the optimal combina-

tion of adding components with different parameters to be

allocated in parallel at each stage.

A. Zeblah (B ) · S. Hadjeri

Engineering Faculty, University of Sidi Bel Abbes, Sidi Jilali,

22000 Sidi Bel Abbes, Algeria

e-mail: azeblah@yahoo.fr

S. Hadjeri

e-mail: Shadjeri2@yahoo.fr

E. Chatelet

Institut Charles Delaunay (CNRS FRE 2848), LM2S,

University of Technology of Troyes, 12 rue Marie Curie,

Troyes Cedex, France

e-mail: Eric.chatelet@utt.fr

Y. Massim

Faculty of Science, University of Sidi Bel Abbes, Gumbeta,

BP 89, 22000 Sidi Bel Abbes, Algeria

e-mail: yamanimassim@yahoo.fr

Keywords

Redundancy optimization · Power system · Universal

generating moment function

Expansion-planning · Harmony search ·

1 Introduction

In many industrial systems, expansion planning system is

considered as an important problem in design, e.g., in

power systems, water distribution and in manufacturing sys-

tems. Such as modifying existing structure, designing a new

structure and adding r(k)∗or retrieving r(k)∗∗components

belonging to the redundancy optimization problem as sug-

gested in [1]. This latter is a well known combinatorial opti-

mization problem where adding components is achieved by

numerous discrete choice made from components available

on the market. Based on the cost, availability and perfor-

mance, the objective function is to minimize the investment-

costs over each study period within the planning horizon for

a certain availability or (reliability) requirement. Figure 1

shows the typical series-parallel power structure. However,

the capacity of many production systems is defined by mul-

tiple, heterogeneous units. In this situation the system can

have a several-range level of performance depending from

perfect working to total failure; in this case it is considered

as a multi-state system (MSS).

The MSS system consists of n subsystems Ci(i =

1,2,...,n) in series arrangement. Each subsystem Cican

contain several components of type i connected in parallel

from various versions, which are proposed by the suppliers

on the market. Each version in turn can contain one or more

identical components in parallel. Components are character-

ized by their cost, availability and performance according to

their version. Different versions of components may be cho-

sen for any given subsystem component. Thus, our study is

123

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88Electr Eng (2010) 92:87–97

°°°

Element 1

° ° °

Element 2

Element 3

Element K1

Element 1

° ° °

Element 2

Element 3

Element K2

Component C1

Component C2

Element 1

Element 2

Element 3

Element Kn

Component Cn

Fig. 1 Series-parallel power system structure

not limited to the case where only the homogeneous compo-

nents are used. Beside a lot of alternatives lead to change the

performance and reliability as in series-parallel replacement

and modernization problems [2,3]. The simplest method in

this work to help system performance to face to the increase

demand is its expansion.

1.1 Literature review

Theclassicalreliabilitytheoryisbasedonthebinaryassump-

tionthatthesystemiseitherworkingperfectlyorcompletely

failed. However, in many real life situations we are actu-

ally able to distinguish among various levels of performance

for both subsystem and its components. In this case, it is

importanttodevelopMSSreliabilitytheory.Mostofresearch

works in MSS reliability (availability) analysis extend the

results in binary state system theory to the multi-state case.

A good and extensive recent review of the literature can

be found for example in [4] or [5]. Generally, the methods

of MSS reliability assessment are based on many different

approaches such as: the structure function approach, the sto-

chastic process (mainly Markov) approach, the simulation

technique (Monte-Carlo), and the universal moment gener-

ating function (UMGF) approach.

The problem of total investment-cost minimization, sub-

ject to reliability or availability constraints, is well known

as the redundancy optimization problem (ROP). The ROP

for series parallel systems is NP-hard [6] and has been stud-

ied in many different forms as summarized in [7] and more

recently in [8]. In most existing works on ROP, it is usually

assumed that a system may experience only two possible

states.TheROPforthemulti-statereliabilitywas introduced

in [9]. In [10] and [1], genetic algorithms were used to find

theoptimalornearlyoptimalpowersystemstructure.In[11]

the multi-stage expansion-planning problem for multi-state

series-parallel system is solved by using a genetic algorithm

as an optimization tool.

This paper uses a harmony search optimization to solve

the multi-stage expansion-planning problem for multi-state

series-parallel electrical power system. The idea of employ-

ing a harmony search to solve combinatorial optimization

problems was recently proposed in [12]. It has been recently

adaptedforthereliabilitydesignofbinarystatesystems[13].

The harmony search method has not yet been used for the

multi-stages expansion problem.

1.2 Approach and outline

As the problem formulated in this paper is a complicated

combinatorial optimization one, an exhaustive examination

ofallpossiblesolutionsisnotrealisticconsideringreasonable

timelimitations.Becauseofthis,theharmonysearchoptimi-

zation (HSO) will be adapted to the multi-stage expansion-

planning problem to find optimal or nearly optimal solutions

tobeobtainedinashorttime.Theharmonysearchisinspired

by the behavior of real musicians based on natural musical

performance processes that occur when a musician searches

for a better state of harmony, such as during jazz improvi-

sation. Jazz improvisation seeks to find musically pleasing

harmony (a perfect state) as determined by an aesthetic stan-

dard, just as the optimization process seeks to find a global

solution (a perfect state) as determined by an objective func-

tion. The pitch of each musical instrument determines the

aesthetic quality, just as the objective function value is deter-

mined by the set of values assigned to each design variable

[14].

To evaluate the availability of a given selected structure

of the series-parallel system, a fast procedure of availabil-

ity estimation is developed. This procedure is based on a

modernmathematicaltechnique:thez-transformoruniversal

moment generating function which was introduced in [15].

It was proven to be very effective for high-dimension com-

binatorial problems: see e.g. [4,5] and references therein.

The universal moment generating function is an extension of

theordinarymomentgeneratingfunction.Themethoddevel-

opedinthispaperallowstheavailabilityfunctionofreparable

series-parallel MSS to be obtained using a straightforward

numerical procedure.

The remainder of this paper is outlined as follows: in

Sect. 2, the expansion-planning problem EPP is formulated.

In Sect. 3, availability evaluation method is developed. In

Sect. 4, the harmony search optimization method is adapted

to solve the expansion-planning problem. In Sect. 5, illustra-

tive examples and numerical results are presented in which

the optimal choice of components in a subsystem is found.

Conclusions are drawn in Sect. 6.

2 Expansion-planning problem

There has been much interest in production scheduling

models, where expansion-planning (EPP) is considered. To

123

Page 3

Electr Eng (2010) 92:87–9789

SubsystemCi

⎪ ⎪⎩

⎩

⎪⎨

⎧

⎪⎨

⎪⎩

⎪⎨

⎪⎩

⎧

⎪⎨

⎧

⎧

k1vi

°°°

°°°

°°°

Ai1,Σi1, Ci1

k11

k12

°°°

Aivi, Σivi, Civi

Aivi, Σivi, Civi

Ai2, Σi2, Ci2

Ai2, Σi2, Ci2

Ai1, Σi1, Ci1

⎪⎩

⎪⎨

⎪⎩

⎧

⎪⎨

⎧

Fig. 2 Detailed structure of a given subsystem

formulate the expansion-planning problem of power system,

let us consider a series-parallel power system containing

n subsystems Ci(i = 1,2,...,n) in series arrangement

as sketched in Fig. 1. Each component Ci in turn con-

tains a number of different components connected in par-

allel. All components of any given subsystem component

belonging to different version v. Components are character-

ized by their availability (Aiv), cost (Civ) and performance

(?iv) according to their version. The structure of subsys-

tem component i can be defined by the numbers of par-

allel components (of each version) kiv for 1 ≤ v ≤ Vi,

where Vi is a number of versions available for component

of type i. Figure 2 illustrates these notations for a given

component i. Each version kivcontains m-identical compo-

nents which are also connected in parallel. The entire sys-

tem can therefore be defined by the set of triplets k0 =

{Aiv,?

k0, the total cost of the initial power system structure can be

calculated as:

iv,Civ}(1 ≤ i ≤ n,1 ≤ v ≤ Vi). Where k0: rep-

resent the initial system structure. In fact, for given triplet

C0=

n

?

i=1

Vi

?

v=1

kivCiv

(1)

2.1 Partial optimal design problem

The multi-state EPP of electrical power system at the ini-

tial period can be formulated as follows: find the minimal

cost system configuration k1,k2,...,kn, such that the cor-

responding availability exceeds or equal the specified avail-

ability A0. That is,

Minimize

C0=

n

?

i=1

Vi

?

v=1

kivCiv

(2)

subject to

A(k1,k2,...,kn,D,T) ≥ A0

(3)

2.2 Determination of the expansion plan

In the cases when the production system is not able to satisfy

the consumer demand. The existing system structure must

be reinforced. This is way to enhance system performance

and/or reliability to be improved. This solution is more real-

istic, practical and rather than designing a new structure.

Indeed, the system should be expanded at different stage of

the study period Y, in which the load curve demand varies

from stage to stage. Each stage θ begins τ(θ) years after the

initial stage (i.e., initial stage θ). To provide the desired level

of productivity and reliability, systematically, the addition or

retrieval by replacement of components can be chosen from

different versions of any given subsystem component. To

distinguish between the components’ characteristics of the

existing structure and the adding components the notation of

component versions is introduced. For each subsystem com-

ponent i there are several component versions available in

the market. A set of parameters contain the nominal capacity

(?ih), availability (Aih) and cost (Cih) is specified for each

version h of component of type i. Therefore, component can

be defined by rih(θ)(1 ≤ h ≤ Hl) where Hldefine the total

number versions of the expanding components.

The entire expansion system may be defined by the vec-

tors ri(θ) = {rih}(1 ≤ i ≤ n,1 ≤ h ≤ Hl) at stage (θ).

For a given set of r1(θ),r2(θ),...,rn(θ), the total cost of

the system expansion at stage (θ) can be calculated at the

present value as:

C(θ) =

n

?

i=1

Hl

?

h=1

rih(θ)Cih±

1

(1 + IP)τ(θ)C0

(4)

where IP represents the interest rate.

For given expansion structure k defined by the vectors

k = {r1(θ),r2(θ),...,rn(θ)}, the total cost investment dur-

ing all the study period Y can be calculated as follows

⎛

i=1

C(k) =

Y

?

θ=1

⎝

n

?

Hl

?

h=1

rih(θ)Cih±

1

(1 + IP)τ(θ)C0

⎞

⎠

(5)

2.3 Total optimal design problem

The multi-state system redundancy expansion optimization

canbeformulatedasfollows:findtheminimalcostofsystem

expansion structure k0,k,D(θ),T(θ)) that meets or exceeds

123

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90Electr Eng (2010) 92:87–97

the required availability A0. That is,

Minimize

C(k) =

Y

?

θ=1

⎛

⎝

n

?

1

i=1

Hl

?

h=1

rih(θ)Cih

±

(1 + IP)τ(θ)

n

?

i=1

Vi

?

v=1

kivCiv

⎞

⎠

(6)

Subject to

A(k0,k,D(θ),T(θ)) ≥ A0

(7)

To estimate the availability index at each stage of study, it

is necessary to calculate the overall probability that the load

demand corresponding to this stage is not meet. This method

is referred to the availability of reparable multi-state system

that is developed in the next section.

2.4 Availability of reparable multi-state systems

Theconsideredsystemisaseries-parallelpowersystemcom-

posedofanumber offailureproneunits,suchthatthefailure

of some components leads only to a degradation of the sys-

tem performance. This system is considered to have a range

of performance levels from perfect functioning to complete

failure.Infact,thesystemfailurecanleadtodecreasedcapa-

bility to accomplish a given task, but not to complete failure.

An important MSS measure is related to the ability of the

system to satisfy a given demand.

Whenappliedtoelectricpowersystems,reliabilityiscon-

sidered as a measure of the ability of the system to meet the

load demand (D), i.e., to provide an adequate supply of elec-

trical energy (?). This definition of the reliability index is

widely used for power systems: see e.g., [16–18] and [1].

The loss of load probability (LOLP) index is usually used

to estimate the reliability index [19]. This index is the over-

all probability that the load demand will not be met. Thus,

we can write R = Prob(? ≥ D) or R = 1 − LOLP with

LOLP = Probab(? < D). This reliability index depends on

consumer demand D.

For reparable MSS, a multi-state steady-state availability

E is used as Prob(? ≥ D) after enough time has passed for

this probability to become constant [18]. In the steady-state

the distribution of states probabilities is given by Eq. (8),

while the multi-state stationary availability is formulated by

Eq. (9):

?Prob??(t) = ?j

E =

?j≥D

At each operation period of stage (θ) the demand distribu-

tion represented by the cumulative curve is predicted. If the

periodofstage(θ)isdividedinto Mθintervalswithduration’s

Tj(θ)(1 ≤ j ≤ Mθ)andeachintervalhasarequireddemand

level Dj(θ).WedenotebyD(θ)andT(θ)thevectors{Dj(θ)}

Pj= lim

t→∞

?

??

(8)

Pj

(9)

and {Tj(θ)}(1 ≤ j ≤ Mθ), respectively. Some assumptions

are taken into account, the intervals for the stages are equal

and the redundant intervals Tj(θ) = 0(M ≤ θ ≤ Y). Then

the generalized MSS availability index A at each stage can

be calculated as:

A =

1

?M

j=1Tj(θ)

M

?

j=1

Prob??T(θ) ≥ Dj

?Tj(θ)

(10)

where ?T(θ) define the production of system containing the

initial structure and the additional components presented at

stage (θ). As the availability A is a function of k0,k,D(θ)

and T(θ), it will be written A(k0,k,D(θ),T(θ)). In the case

of a power system, the vectors D and T define the cumula-

tive load curve (consumer demand). In general, this curve is

known for every power system.

3 Multi-state system availability estimation based

on UMGF method

The procedure used in this paper is based on the universal

z-transform, which is a modern mathematical technique

introduced in [15]. This method, convenient for numerical

implementation, is proved to be very effective for high-

dimension combinatorial problems. In the literature, the

universal z-transform is also called universal moment gener-

atingfunction(UMGF)orsimplyu-functionoru-transform.

Inthispaper,wemainlyusetheacronymUMGF.TheUMGF

extendsthewidelyknownordinarymomentgeneratingfunc-

tion.

3.1 Definition

The UMGF of a discrete random variable?is defined as a

polynomial:

u(z) =

J

?

j=1

Pjz?j

(11)

where the variable ? has J possible values and Pj is the

probability that ? is equal to ?j.

Theprobabilisticcharacteristicsoftherandomvariable?

can be found using the function u(z). In particular, if the

discrete random variable ? is the MSS stationary output

performance, the availability E is given by the probability

Prob(? ≥ D) which can be defined as follows:

?

Probab(? ≥ D) = ?

u(z)z−D?

(12)

123

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Electr Eng (2010) 92:87–9791

where ? is a distributive operator defined by expressions

(13) and (14):

?

⎛

j=1

It can be easily shown that Eqs. (13) and (14) meet condition

Prob(? ≥ D) =?

with ?j ≥ D, and the probability that ? is not less than

some arbitrary value D is systematically obtained.

Consider single components with total failures and each

componenti hasnominalperformance?iandavailability Ai.

Then, Prob(? = ?i) = Aiand Probab(? = ˜) = 1 − Ai.

The UMGF of such a component has only two terms and can

be defined as:

?

Pzσ−D?

J

?

=

?P, if σ ≥ D

⎞

j=1

0,

if σ < D

(13)

?

⎝

Pjz?j−D

⎠=

J

?

?

?

Pjz?j−D?

(14)

?j≥DPj. By using the operator ?, the

coefficients of polynomial u(z) are summed for every term

ui(z) = (1 − Ai)z0+ Aiz?i= (1 − Ai) + Aiz?i

To evaluate the MSS availability of a series-parallel system,

two basic composition operators are introduced. These oper-

ators determine the polynomial u(z) for a group of compo-

nents.

(15)

3.2 Parallel components

Let us consider a system component m containing Jmcom-

ponents connected in parallel. As the performance measure

is related to the systemproductivity, the totalperformance of

the parallel system is the sum of performances of all its com-

ponents. In power systems engineering, the term capacity is

usually used to indicate the quantitative performance mea-

sureofacomponent[6].Thecapacityofacomponentcanbe

measuredasapercentageofnominaltotalsystemcapacity.In

power system, components are generators units, transformer

and transmission lines. Therefore, the total performance of

the parallel unit is the sum of performances.

The u-function of MSS component m containing Jmpar-

allel components can be calculated by using the ? operator:

up(z) = ?(u1(z),u2(z),...,un(z)),

where ?(g1,g2,...,gn) =

n

?

i=1

gi.

Therefore for a pair of components connected in parallel:

⎛

i=1

Parametersaiandbjarephysicallyinterpretedastherespec-

tive performances of the two components. n and m are num-

bers of possible performance levels for these components.

?(u1(z),u2(z)) = ?

⎝

n

?

Pizai,

m

?

j=1

Qjzbj

⎞

⎠.

Piand Qjare steady-state probabilities of possible perfor-

mancelevelsforcomponents.Onecanseethatthe? operator

is simply a product of the individual u-functions. Thus, the

component UMGF is:

up(z) =

Jm

?

j=1

uj(z).

Given the individual UMGF of components defined in

Eq. (15), we have:

up(z) =

Jm

?

j=1

?1 − Aj+ Ajz?i?.

3.3 Series components

When the components are connected in series, the compo-

nent with the least performance becomes the bottleneck of

the system. This component therefore defines the total sys-

tem productivity. To calculate theu-function for system con-

taining n components connected in series, the operator η

should be used: us(z) = η(u1(z),u2(z),...,um(z)), where

η(g1,g2,...,gm) = min{g1,g2,...,gm} so that

⎛

i=1

n

?

η(u1(z),u2(z)) = η

⎝

n

?

m

?

Pizai,

m

?

j=1

Qjzbj

⎞

⎠

=

i=1

j=1

PiQjzmin{ai,bj}

Applying composition operators ? and η consecutively, one

can obtain the UMGF of the entire series-parallel system. To

dothiswemustfirstdeterminetheindividualUMGFofeach

component. The next section presents the harmony search

meta-heuristic optimization method to solve the redundancy

optimization problem for multi-state expansion-planning

systems.

4 The harmony search optimization approach

The problem formulated in this paper is a complicated com-

binatorial optimization problem. The total number of differ-

ent solutions to be examined is very large, even for rather

smallproblems.Anexhaustive examination oftheenormous

number of possible solutions is not feasible given reasonable

time limitations. Thus, because of the search space size of

the ROP for MSS, a new meta-heuristic is developed in this

section. This meta-heuristic consists in an adaptation of the

harmony search optimization method.

123

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92Electr Eng (2010) 92:87–97

4.1 The HS principle

Recently, [20] introduced a new approach to optimization

problems derived from the study of musician, called “Har-

monySearchAlgorithm”.TheHSalgorithmmimicsmusical

improvisationprocesswherethemusicianstrytofindabetter

harmony. A musician always desires to reach the best har-

mony, which can be obtained by numerous practices. The

pitches of the instruments are adjusted after each practice.

The HS will be explained in the following, similar to the

work in [21]. Figure 3 demonstrates the analogy between

HM and power system design. HM is the most important

part of HS, which is shown in Fig. 1. Jazz improvisation is

the best example for clarifying the HM. Many jazz of five

music instruments consist of a guitarist, double bassist, and

pianist. Each musician in the trio has different pitches: gui-

tarist (Sol, Si, Re, Fa, Mi); double bassist (Fa, Mi, Re, La,

Si); pianist (Si, Re, Mi, Do, La). Let the guitarist randomly

play Sol out of its pitches (Sol, Si, Re, Fa, Mi), double bass-

ist Si out of (Fa, Mi, Re, La, Si) and pianist Re (Si, Re, Mi,

Do, La). Therefore, the new harmony (Sol, Si, Re) becomes

another harmony (musically G-chord). If the new harmony

is better than existing worst harmony in the HM, new har-

mony is included in the HM and the existing worst harmony

is excluded from the HM. The process is repeated until the

best harmony is obtained.

We consider a power system design process to built,

which consists of five different design variables: First design

variable is selected from generators subsystem: x1

{x1

tem x1

25}, third from THT lines subsystem

x1

34},fourth from HT/MT transformers sub-

system x1

45}, and the end from the MT lines

subsystem x1

solution is selected randomly from each subsystem one and

/or more components are selected. For this stage the process

starting to optimize the power designs.

Ifthenewselectedpowerdesignisbetterthantheexisting

worstpowerdesign,theworstpowerdesignpresentthehigh-

est objective function value, the new design is included, and

1

→

11,...,x1

2→ {x1

3→ {x1

18}, second from MT/HT transformers subsys-

21,...,x1

31,...,x1

4→ {x1

5→ {x1

41,...,x1

51,...,x1

55}. At initial stage a vector

Fig. 3 Analogy between HS and power system design

the worst design is excluded from vector solution of power

design process. This procedure is repeated until the termi-

nating criterion is satisfied.

Ananalogyispresentedbetweenthemusicimprovisation

process [20] and the optimum electrical power design can be

established in the following way: The harmony denotes the

power design vector; whereas the different harmonies dur-

ing the improvisation represent the different power design

vectors throughout the optimum power design process. Each

musicalinstrumentdenotesthesubsystempowerdesignvari-

ables (set of electrical components) of objective function.

Thepitchesoftheinstrumentsrepresentthedesignvariable’s

values (components technology with no position). A better

harmony represents local optimum, and the best harmony is

the global optimum.

4.2 HS-based initial solution approach

The optimum design algorithm developed for power design

based on harmony search method treats the sequence num-

berofthecomponentstechnologyinthelistselectedforeach

subsystemcomponentsasadesignvariable.Onceasequence

numberoftechnologyisselectedthenthesubsystemcompo-

nents designation and properties of this latter become a new

available for the algorithm. Hence the design vector consists

of the integer numbers that are the sequence number of tech-

nologies in the discrete set. Since each row of the harmony

memorymatrixcorrespondstoacandidateforpowerdesign,

the components of this matrix represent the order number of

components technologies. The design algorithm consists of

the following steps.

The optimum power design algorithm using HS is

sketched basically as shown in Fig. 4.

Fig. 4 Basic flowchart diagram For HS algorithm

123

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Electr Eng (2010) 92:87–9793

4.2.1 Initialize the harmony search parameters

The HS algorithm parameters are attained in this step. These

are harmony memory size (HMS), harmony memory con-

sideration rate (HMCR), pitch adjustment ratio (PAR), and

stopping criteria (number of improvisation). These parame-

ters are selected depending on the problem.

4.2.2 Initialize harmony memory

The HM matrix is filled with randomly generated designs as

the HMS.

⎡

⎢

⎢

⎢

HM=

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

x1

11

x2

21

x1

12

x2

22

......x1

1k−1

x1

1k

x2

2k

......x2

2k−1

◦

xHMS−1

g−11

xHMS

g1

◦

xHMS−1

g−12

xHMS

g2

◦

......xHMS−1

◦

g−1k−1xHMS−1

......xHMS

gk−1

g−1k

xHMS

gk

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

→ f

→ f

?

?

?

?

x1?

x2?

xHMS−1?

xHMS?

→ f

→ f

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(16)

Each vector denotes a power design vectors solutions in the

HMx1

gk

andthecorrespondingunconstrained

objective function value, respectively f (x1),..., f (xHMS).

The power system designs are stored by the unconstrained

objective function values f (x1) ≺ f (x2),...,≺ f (xHMS).

The aim of using HM is to preserve better power systems

designs in the search process.

1k,x2

2k,...,xHMS

Calculate the initial cost power design at the initial stage: C0

Calculate the availability of the initial power design at the

initial stage: A0.

4.2.3 Improvise a new harmony

A new harmony [xnυ] = {xnυ

from either the HM or entire technologies from each subsys-

tem’s list. Three rules are used for the generation of a new

harmony.TheseareHMconsideration,pitchadjustment,and

random generation. In the HM consideration, the value of

the first power design variable [xnυ

chosen from any value HM (i.e. {x1

availabletechnologyinthemarketdenotedtheavailabletech-

nologies by [YAT]2. HMCR is applied as follows:

⎧

⎩

interval [0,1] and generated. If the ran number is less than

the HMCR value, ith power design variable of new power

design [xnυ

ith column of HM. Otherwise the ith power design variable

11,xnυ

12,...,xnυ

gk} is improvised

g−1,1] for a new harmony is

1k,...,xHMS

gk

} ) or entire

⎨

(ran) is the random number uniformly distributed over the

xnυ

gi∈

xnυ

gi∈ YT A

?

x1

gi,x2

gi,...,xHMS

gi

?

if ran ≤ HMCR

if ran ? HMCR

(17)

gk] is selected from the current values stored in the

of new power design [xnυ

able technologies list [YAT].

Any power system design variable of the new harmony,

[xnυ] = {xnυ

consideration is examined to determine whether it is pitch-

adjustedornot.PitchadjustmentismadebyPAR.PARinves-

tigates better power system design in the neighboring of the

current power system design. PAR is applied as follows:

gk] is selected from the entire avail-

11,xnυ

12,...,xnυ

gk}, obtained by the memory

PAD for xnυ

gi

?Yes if ran1≤ PAR

Noif ran1? PAR

(18)

(ran1) is the random number uniformly distributed over the

interval [0,1] and generated. If this random number is less

than the PAR, xnυ

giis replaced with its neighbor technology

components list. If this random number is not less than PAR,

xnυ

giremains the same.

4.2.4 Update the harmony memory

If the new harmony [xnυ] = {xnυ

thantheworstpowerdesignintheHM,thenewpowerdesign

is included in the HM, and the existing worst harmony is

excluded from the HM. In this process, it should be noted

that HM matrix is sorted again by unconstrained objective

function and the same power design is not permitted in the

HM more than once.

11,xnυ

12,...,xnυ

gk} is better

4.2.5 Stopping criterion

The steps in Sects. 4.2.3 and 4.2.4 are repeated until a termi-

nation criterion is satisfied or reaching the number of impro-

visation (NI).

Calculate the initial optimal cost power design at the initial

stage: τ(0) = 0 → C0

Calculate the availability of the optimal power design at the

initial stage: τ(0) = 0 → A.

Considering now the case when the power demand increases

or decreases randomly. For any stage τ(θ) ?= 0, the process

starts by proposing the following method.

4.3 HS-based solution approach for EPP

The design algorithm of EPP consists of the following steps.

The optimum power design algorithm using HS is sketched

basically as shown in Fig. 5.

123

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94Electr Eng (2010) 92:87–97

Fig. 5 Basic flowchart diagram Of HS algorithm For EPP

4.3.1 Using the filled harmony memory

The HM matrix is filled with the first optimal solution at the

initial stage τ(0) = 0.

HMτ(0)

=

?

xτ(0)1

11

xτ(0)1

12

...

xτ(0)1

1k−1

xτ(0)1

1k

?

→ f

?

xτ(0)1??

(19)

ThenbegintofilltheHMτ(θ)matrixwithrandomlygenerated

designs as the HMSτ(θ)for the next stage τ(0) ?= 0.

HMτ(θ)

⎡

⎢

⎢

g1

g2

gk−1

=

⎢

⎢

⎣

⎢

⎢

⎢

⎢

xτ(θ)1

11

xτ(θ)2

21

xτ(θ)1

12

xτ(θ)2

22

...xτ(θ)1

1k−1

...xτ(θ)2

2k−1

•

...xτ(θ)HMS

xτ(θ)1

1k

xτ(θ)2

2k

•

xτ(θ)HMS

•

xτ(θ)HMS

•

xτ(θ)HMS

gk

⎤

⎥

⎥

⎥

⎥

⎦

⎥

⎥

⎥

⎥

→ f

→ f

?

?

?

xτ(θ)1?

xτ(θ)2?

xτ(θ)HMS?

→ f

⎫

⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(20)

Each vector denotes a power design vectors solutions in

the HMτ(θ)xτ(θ)1

1k

2k

ing unconstrained objective function value, respectively

f (xτ(θ)1),..., f (xτ(θ)HMS). The power system designs are

stored by the unconstrained objective function values

f (xτ(θ)1) ≺ f (xτ(θ)2),...,≺ f (xτ(θ)HMS). The aim of

using HMτ(θ)is to preserve better power systems designs

in the search process.

,xτ(θ)2

,...,xτ(θ)HMS

gk

and the correspond-

Calculate the initial cost power design at stage τ(0) ?= 0 →

C(k)

Calculate the availability of the initial power design at the

stage τ(0) ?= 0 → A0.

4.3.2 Improvise a new harmony

A new harmony [xτ(θ)nυ] = {xτ(θ)nυ

is improvised from either the HMτ(θ)or entire technolo-

gies from each subsystem’s list. Three rules are used for

the generation of a new harmony. These are HMτ(θ)con-

sideration, pitch adjustment, and random generation. In the

11

,xτ(θ)nυ

12

,...,xτ(θ)nυ

gk

}

HMτ(θ)consideration,thevalueofthefirstpowerdesignvar-

iable [xτ(θ)nυ

HMτ(θ)(i.e.{xτ(θ)1

nology in the market denoted the available technologies by

[Yτ(θ)

⎧

⎪⎩

g−1,1] for a new harmony is chosen from any value

1kgk

,...,xτ(θ)HMS

})orentireavailabletech-

AT]2. HMCRτ(θ)is applied as follows:

xτ(θ)nυ

gi

∈

xτ(θ)nυ

giT A

⎪⎨

?

xτ(θ)1

gi

,xτ(θ)2

gi

,...,xτ(θ)HMS

gi

?

if ranτ(θ)≤HMCRτ(θ)

∈ Yτ(θ)

If ranτ(θ)? HMCRτ(θ)

(21)

(ran)τ(θ)is the random number uniformly distributed over

the interval [0,1] and generated. If the ranτ(θ)number is less

than the HMCRτ(θ)value, ith power design variable of new

power design [xτ(θ)nυ

stored in the ith column of HMτ(θ). Otherwise the ith power

design variable of new power design [xτ(θ)nυ

from the entire available technologies list [Yτ(θ)

Any power system design variable of the new harmony,

[xτ(θ)nυ] = {xτ(θ)nυ

memoryconsiderationisexaminedtodeterminewhetheritis

pitch-adjusted or not. Pitch adjustment is made by PARτ(θ).

PARτ(θ)investigatesbetterpowersystemdesignintheneigh-

boringofthecurrentpowersystemdesign.PARτ(θ)isapplied

as follows:

?

No If ranτ(θ)

gk

] is selected from the current values

gk

] is selected

AT].

11

,xτ(θ)nυ

12

,...,xτ(θ)nυ

gk

}, obtained by the

PADτ(θ)

for xτ(θ)nυ

gi

YesIf ranτ(θ)

1

≤ PARτ(θ)

? PARτ(θ)

1

(22)

(ran1)τ(θ)is the random number uniformly distributed over

the interval [0,1] and generated. If this random number is

less than the PARτ(θ),xτ(θ)nυ

gi

technology components list. If this random number is not

less than PARτ(θ),xτ(θ)nυ

gi

remains the same.

is replaced with its neighbor

4.3.3 Update the harmony memory

If the new harmony [xτ(θ)nυ] = {xτ(θ)nυ

xτ(θ)nυ

gk

} is better than the worst power design in the HMτ(θ),

thenewpowerdesignisincludedintheHMτ(θ),andtheexist-

ing worst harmony is excluded from the HMτ(θ). In this pro-

cess,itshouldbenotedthatHMτ(θ)matrixissortedagainby

unconstrained objective function and the same power design

is not permitted in the HMτ(θ)more than once.

11

,xτ(θ)nυ

12

,...,

4.3.4 Stopping criterion

The steps in Sects. 4.3.2 and 4.3.3 are repeated until a termi-

nation criterion is satisfied or reaching the number of impro-

visation (NI).

Calculate the optimal cost power design at the stage:

τ(θ) → C(k)

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Electr Eng (2010) 92:87–9795

Table 1 Data of available different power components technologies

Subsystem #Data components # (%)Technologies #

# 1 # 2# 3# 4# 5# 6# 7# 8# 9

1. GeneratorsReliability

Cost

Performance

Reliability

Cost

Performance

Reliability

Cost

Performance

Reliability

Cost

Performance

Reliability

Cost

Performance

0.890

0.590

120

0.995

0.205

100

0.971

7.525

100

0.977

0.180

115

0.984

0.986

128

0.977

0.535

100

0.996

0.189

92

0.973

4.720

60

0.978

0.160

100

0.983

0.825

100

0.982

0.470

85

0.997

0.091

53

0.971

3.590

40

0.978

0.150

91

0.987

0.490

60

0.978

0.420

85

0.997

0.056

28

0.976

2.420

20

0.983

0.121

72

0.981

0.475

51

0.983

0.400

48

0.998

0.042

21

/

/

/

0.981

0.102

72

/

/

/

0.920

0.180

31

/

/

/

/

/

/

0.971

0.096

72

/

/

/

0.984

0.220

26

/

/

/

/

/

/

0.983

0.071

55

/

/

/

/

/

/

/

/

/

/

/

/

0.982

0.049

25

/

/

/

/

/

/

/

/

/

/

/

/

0.977

0.044

25

/

/

/

2. MT/HT transformers

3. Lignes HT

4. HT/MT transformers

5. Lignes MT

Calculate the Availability of the optimal power design at the

stage: τ(θ) → A(k).

5 Illustrative example

To illustrate the proposed harmony search algorithm, a

numerical example is solved by use of the data given in

Table 1. Each component of the subsystem is considered

as a unit with total failures. Figure 6 contains the data of

cumulative load demand. The maximum numbers of elec-

trical components kmaxin parallel are set to (7, 8, 4, 9, 4).

The numbers of music instruments are equal to the power

subsystems. The simulation results depend greatly on the

HS algorithm parameters values: HMCR = 0.7; PAR = 0.5;

HMS = 5; and NI = 75 for the first design. Several simu-

lations are made for the expansion-planning power system

where 0.7 ≤ HMCRτ(1)≤ 0.80.5 ≤ PARτ(1)≤ 0.65;

HMSτ(1)= 0.5;HMSτ(θ)andNIτ(1)= 100thebestsolution

is obtained in 81 improvisations. The simulation was imple-

mented to a real example taken from the Algerian network.

Table 2 presents the obtained optimal electrical configura-

tion.

5.1 Description of the system to be optimized

The electrical power station system which supplies the con-

sumers is designed with five basic subsystems (stations) as

depicted in Fig. 7. Figure 7 shows the detailed process of

the electrical power station system distribution. The process

of electrical power system distribution follows as: The elec-

Fig. 6 Data of cumulative load–demand curves

trical power is generated from the station units (subsystem

1). Then transformed for high voltage (HT) by the HT trans-

formers (subsystem 2) and carried by the HT lines mashed

electricalnetwork(subsystem3).Asecondtransformationin

HT/MT transformers (subsystem 4) which supplies the MT

load by the MT mashed electrical network (subsystem 5).

5.1.1 Solution obtained by harmony search optimization

algorithm

See Table 2.

5.1.2 Optimization result and discussion

Much work in the field of reliability optimization analysis

has been devoted using meta-heuristics methods. So, few

of them treat the problem of expansion-planning EPP. The

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96Electr Eng (2010) 92:87–97

Table 2 Optimal solutions for expansion-planning problem EPP

A0

StructureStage 0: current year (cost = 13.60)

Ak= 0.9927

4–4–6–7–4

4–4-4–4–4–4–4

1–4

7–7–7–9

4–4–4

Stage 1: 2 years (cost = 17.703)

Ak= 0.9935

4–4–6–7–4; 1(4)∗–1(7)∗

4–4–4–4–4–4–4

1–4; 2(4)∗

7–7–7–9; 1(8)∗

4–4–4; 2(4)∗

Stage 2: 3 years (cost = 12.997)

Ak= 0.997

4–4–6–7–4; 1(4)∗∗

4–4–4–4–4–4; 2(4)∗∗

1–4

7–7–7–9; 1(7)∗∗

4–4–4

0.991Subsystem 1

Subsystem 2

Subsystem 3

Subsystem 4

Subsystem 5

Fig. 7 Synoptic of the detailed

electrical network system

Fig. 8 Graph of different stage

for EPP

proposed harmony search method seeks system configura-

tion that provides the optimal combination of components

costwithaddingorretrievingnewcomponentfromtheexist-

ing design structure under reliability constraint. Usually a

gain in performance and reliability can only be obtained at

an extra—trade-off between components (Fig. 8). To rein-

force the existing design adding or retrieving components

by replacement depend greatly on increasing or decreasing

demand levels.

The natural objective function is to define the minimal

investment design configuration under given reliability con-

straints at each stage. The whole of the results obtained by

the proposed harmony search for given values of A0is illus-

trated in Table 2. The latter shows the best initial optimal

power design and the optimal power design at each stage for

the EPP for one desired reliability levels A0(0.991). Table 2

illustrates the computed cost and reliability index of the cor-

responding power design. The choice of these values affects

stronglythesolution.Aqualitysolutionmeasureisproposed

taken from the NN method and the best solution is selected

from the lower NN coefficient α =Objective Function(%)

Since it is a meta- heuristic method only near optimal solu-

tions can be obtained.

Constraint(%)

.

6 Conclusion

In this paper, we solve the electrical expansion-planning

power design optimization which is a very interesting

123

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Electr Eng (2010) 92:87–9797

problem often reencountered in energy industry. It is

formulated as redundancy optimization problem. The

resolution of this problem uses a developing harmony search

algorithm. This new algorithm for choosing an optimal

series-parallel electrical power system design for EPP is

proposed which minimizes total investment cost subject to

reliability constraints. This algorithm seeks electrical com-

ponents technologies according to their availability, nom-

inal capacity (perfor-mance) and cost. Also defines the

number and the kind of series-parallel electrical power com-

ponents to put in each subsystem when consumers’ demand

changes by selects adding/or retrieving from initial power

design. The proposed method allows a practical way to solve

wide instances of reliability optimization problem of multi-

state systems without limitation on the diversity of electrical

components technologies put in series-parallel. A combina-

tion is used in this algorithm based on the universal moment

generating function UMGF and an HS algorithm.

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noncommercial use, distribution, and reproduction in any medium,

provided the original author(s) and source are credited.

ThisarticleisdistributedunderthetermsoftheCreative

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