a multistage value-based model for prioritization of distribution projects using a multiobjective...

J Control Autom Electr SystDOI 10.1007/s40313-013-0060-z

A Multistage Value-Based Model for Prioritization of DistributionProjects Using a Multiobjective Genetic Algorithm

Fernando L. R. Mussoi · Raimundo C. G. Teive

Received: 15 December 2011 / Revised: 11 June 2012 / Accepted: 6 September 2012© Brazilian Society for Automatics–SBA 2013

Abstract This article addresses the project prioritizationproblem as a part of the distribution system planning process.The projects must be selected and scheduled in a multistageplanning horizon, to fit the utility available budget. The pro-posed decision model is value based, and prioritizes the localprojects in a systemic way. The value attributes consider thefinancial impacts, number of consumers and some aspects ofthe distribution system operational performance and powerquality. An analytic model optimizes the project portfoliosby means of the NSGA-II multiobjective genetic algorithm.The optimization obtains a set of Pareto-optimal solutions,which are the trade-off project portfolios, with respect to theproblem objectives and constraints. The presented case stud-ies indicate some changes in the choice of priority projects.The proposed project selection methodology helps in rais-ing strategic investments and allows for better distributionsystem planning.

Keywords Distribution planning · Genetic algorithm ·NSGA-II · Pareto optimization · Power quality ·Project prioritizing

1 Introduction

The electric power distribution is characterized by the end-user direct service, the rigorous regulatory control, and thelarge investments in assets. The restructuring of the Brazilian

F. L. R. Mussoi (B)Instituto Federal de Santa Catarina, IFSC/DAELN,Florianópolis, SC, Brazile-mail: [email protected]

R. C. G. TeiveUniversidade do Vale do Itajaí, UNIVALI/LIA, Itajaí, SC, Brazile-mail: [email protected]

electric system raised new issues in the business of distribu-tion utilities. These companies have been under pressure toreduce costs and increase the return for their assets, and at thesame time, they are subjected to budget cuts and high capitalcosts on their investments (Brown and Marshall 2000).

From the regulation point of view, the electric power sup-ply has been considered as a product associated with ser-vices (ANEEL 2010). As a product, the power quality can bemonitored by means of some technical indexes related to thevoltage waveform (signal conformity). On the other hand, thepower distribution service can be evaluated through severalindexes associated with the continuity of supply. All theseindexes have an important role, especially regarding powerquality, because of the direct and significant impact in theload operation and in the relationship between the utility andtheir consumers and investors (Carvalho and Ferreira 2004;Choi et al. 2005; Kumar et al. 2008; Vuinovich et al. 2004).

The process of distribution system planning involves sev-eral steps, including mainly load forecasting, the current andfuture electrical network analysis, and the definition of thebest set of reinforcements projects to be carried out (Gönen1986). There are many analytic methods and computationaltools available for planning the reinforcement and expan-sion configurations for regional distribution systems (Gönenand Ramirez-Rosado 1986; Haffner et al. 2008; Lotero andContreras 2011; Ramírez-Rosado and Bernal-Agustín 1998;Vaziri et al. 2004; Wang et al. 2011; Zhou and Spare 2007).These approaches help us to define local projects consideringmany technical and economic criteria, but they usually arenot able to optimize the available resources by maximizingthe portfolio value. Thus, the proposed projects for regionaldistribution systems must be selected and scheduled from acorporative viewpoint, and in a systemic way. This processrequires methodologies that are able to hold the specific needsof power distribution business.

123

J Control Autom Electr Syst

The prioritization of projects is a combinatorial probleminvolving multiple objectives and constraints. Many mathe-matical techniques and meta-heuristics algorithms have beenused to solve this kind of problem (Choi et al. 2005; Fal-cão 2002; Huang et al. 2005; Stummer and Heidenberger2003; Zhou and Spare 2007), especially some multiobjec-tive genetic algorithms (Mishra et al. 2011; Wang et al. 2011).Among them, the Nondominated Sorting Genetic Algorithm(NSGA-II) has been widely used and has presented goodresults in solving diverse multiobjectives problems in thefield of power system planning (Deb et al. 2007; Dhanalak-shmi et al. 2011; Kannan et al. 2009; Kumar et al. 2008;Leite da Silva et al. 2011; Maghouli et al. 2009; Ramesh etal. 2012) and for project and investment selection (Mishra etal. 2011; Wang et al. 2005; Deb et al. 2010). However, theprioritization of power distribution projects, focused on theportfolio value optimization, has still not been adequatelyaddressed in the available literature.

This article addresses the project prioritization problem asa part of the distribution system planning process. The pro-posed methodology makes use of the Project Portfolio Man-agement (PPM) strategic viewpoint (Elton and Gruber 1991;Levine 2005) to select and schedule the available projects ina multistage planning horizon, in such a way to maximize theportfolio value. An analytic model maps into value attributesthe financial impacts of the available projects, the numberof benefited consumers, the power quality, and operationalperformance indexes. The Nondominated Sorting GeneticAlgorithm-II (NSGA-II) is used to optimize the project port-folios, according to the problem objectives and constraints.The projects are selected and scheduled along the planninghorizon, where each stage marks one-year time interval. Theoptimization process results in a set of trade-off portfolios,where the decision makers can choose the most strategic tothe utility goals. The presented case studies demonstrate howthe project scheduling changes in different optimization sce-narios, and show the proposed methodology’s potential tohelp decision makers in raising strategic investments.

Section 2 analyzes the distribution system planning con-text and the project prioritization problem. Section 3 presentsthe proposed decision model, which is formulated as a multi-objective optimization problem in Sect. 4. The optimizationprocess is discussed in Sect. 5, and the adopted multiobjectivegenetic algorithm is presented in Sect. 6. The proposed modelapplication is demonstrated in Sect. 7, and Sect. 8 presentsthe final remarks. Technical references are listed in Sect. 9.

2 Distribution System Planning

The power distribution planning is based on the analysis ofsystem performance, facing the forecasted demand growth,the regulatory requirements, and the utility planning crite-

ria. This task is supported by the supervisory system data,and by computer tools based on simulation models and onad-hoc techniques, such as demand forecasting, load flow,short-circuit analysis, etc. (Gönen 1986). When the analysisindicates inadequate system performance, some new configu-rations must be proposed and analyzed, considering specificmarket conditions, economic scenarios, available technicalsolutions, and other planning criteria. The proposed configu-rations involve projects for system improvement and expan-sion, which may include new assets and current infrastructurereinforcements (Lotero and Contreras 2011). These projectsmust fulfill the demand growth and improve system opera-tional performance and power quality, in compliance to theregulatory requirements, with the least social–environmentalimpacts (ANEEL 2010; Arrillaga et al. 2000; Carvalho andFerreira 2004; Vuinovich et al. 2004).

As shown in Fig. 1a, the distribution system planningprocess must be cyclic and interactive, in such a way to con-sider changes on consumer habits, to involve technologicaland methodological improvements, and to overcome possibledeficiencies of projects.This process is guided by the selec-tion of a specific set of projects (Choi et al. 2005; Haffner etal. 2008; Wang et al. 2011). The projects must be scheduledalong the considered planning horizon to compose the utilityproject portfolio, and to define the distribution system expan-sion plan. However, the planning scenario and the budgetconstraints require continuous costs reduction and efficientavailable resources allocation (Brown and Marshall 2000;Moreira et al. 2009). On the other hand, the growing pres-sure to improve, or even to maintain the service quality levelimpose additional costs (Arrillaga et al. 2000; Vuinovich etal. 2004). The investments must be enough to meet the plan-ning requirements, but should not exceed the utility budget.Therefore, in order to allocate the available resources effi-ciently, it is necessary to involve some operational perfor-mance aspects and some power quality indexes in the systemplanning and in the project prioritization process.

2.1 Prioritization of Distribution Projects

In order to adequately supply the electric power to a largenumber of consumers in a wide geographic area, many util-ities make use of a decentralized structure to manage thesignificant contingent of assets and facilities. The regionalmanagement offices have some autonomy to propose projectsand to decide how to spend parts of the global budget in theiroperational areas. However, these projects usually are locallyselected without a formal and analytic method. Although thelocal judgments and expertise are important, if the corpora-tive objectives and the evaluation criteria are not clearly andglobally articulated, the project selection could not obtainthe best results (Martin and Merkhofer 2003). On the otherhand, budget apportions by area and based on historical

123


figures, does not ensure efficient and strategic resource allo-cation, since the impacts of each project to the portfolioglobal value are not considered. An important advantage ofthe analytic models focused on project prioritization is theportfolio optimization in a systemic level (Huang et al. 2005).In this level, the corporative global benefits are achieved byadjusting the local budgets, so that the resources are allocatedto the regional offices that have proposed the projects whichaggregate the greatest value to the portfolio.

Project prioritization is a multiobjective problem sub-jected to budget constraints, and involves significant invest-ment costs, public concerns, and regulatory requirements.The decision-making process must be strategic, since theresults can change the consumer’s satisfaction level anddefine the utility image and position in the market (Levine2005; Martin and Merkhofer 2003). An analytic model,incorporated to an optimization tool for maximizing the port-folio global value, can reduce the decision subjectivity andalign the local project evaluation criteria to the corporativeglobal objectives.

3 Proposed Decision Model

Figure 1 presents the proposed decision model to prioritizethe distribution system projects, which is structured in twosequential and integrated phases. The first phase, shown inFig. 1a is commonly accomplished by utilities, and is focusedon proposing the projects for reinforcement and expansion ofthe regional distribution systems. The distribution planningis aided by simulation tools to evaluate the system perfor-mance against the demand growth and to define some newconfigurations options. The interaction between the expertiseknowledge and the simulation data analysis allows plannersto propose projects for the regional distribution systems, todefine their goals and objectives, and to estimate their costsand expected outcomes.

3.1 Value Model

The priority of a project is measured by value attributesrelated to the criteria of concern, which reflect the projectstrategic relevance and its potential contribution to meet theestablished objectives (Martin and Merkhofer 2003). TheProject Portfolio Management (PPM) theory is based on theproject value attributes, and is focused on strategic resourceallocation. The PPM makes use of decision models and opti-mization techniques to efficiently prioritize projects, similarto some financial analysis and investment selection methods(Elton and Gruber 1991; Levine 2005).

A contribution of this article is the incorporation of thesecond phase in the decision model presented in Fig. 1b. Inthis phase, the proposed value-based model prioritizes the

Demand Forecasting

System Analisys

Set of Projects

New ConfigurationsHeuristics

Value Attributes

Multiobjective andMultistage Optimization

Trade-Off Portfolios

Decision MakingPreference

Criteria

Corporative Portfolio

Expertise Knowledge

Expertise Knowledge

Distribution System

(a) Planning Phase: Simulation Models

(b) Prioritization Phase:Value Model

Fig. 1 Decision model a planning phase; b prioritization phase

available projects in a systemic way, the objective of whichis to optimize the corporative portfolio. The proposed modeladopts the PPM principles and deals the available projectsas investment alternatives. The projects for local systems,previously defined by the utility regional offices in the firstphase, dispute equally for the available budget with all othercompeting projects, only as function of their value attributes.

Each project is defined to improve some performanceaspect of the distribution system. The operational condi-tion data and the impacts of projects are mapped into valueattributes. These attributes are related to project costs, finan-cial revenues, number of supplied consumers, system perfor-mance, and power quality indexes. The projects are selectedbased on their relative importance to the established techni-cal and financial objectives, and on their ability to contributeto the portfolio global value. The portfolio optimization is amultistage and multiobjective combinatorial problem, wherethe projects must be selected and scheduled along the plan-ning horizon stages. The projects can be anticipated, delayed,or not scheduled in the planning horizon, according to theavailable budget. To maximize the portfolio global value,

123


both technical and financial attributes of projects should beoptimized simultaneously, according to the problem con-straints.

In the proposed model, the portfolios are composed andoptimized by a multiobjective genetic algorithm (Sect. 6).This optimization technique does not find a unique and defin-itive solution, but a set of trade-off solutions between allproblem objectives. An advantage of this approach is to aidthe decision maker, suggesting a set of diverse and efficientportfolios from the optimization criteria viewpoint. Thus, thedecision on the most appropriate portfolio for the distributionsystem expansion plan can be made a posteriori, using someadditional preference criteria and according to the decision-makers expertise knowledge. The proposed model is focusedon the decision problem peculiarities and has been developedaround the characteristics of the power distribution system.The approach is adherent to the way the utilities make theirdecisions, involving its agents and does not impose changesin the company culture.

3.2 Value Attributes Tree

Figure 2 shows the structure of the value attributes treeadopted in the model for project selection. The value of eachportfolio is defined by the technical and financial attributesrelated to the set of selected projects and to the planning hori-zon stages in which they are scheduled. Portfolio costs aredetermined by the financial amounts required for execution ofthe selected projects. Portfolio financial benefits are referredto revenues from the power supply capacity and from thepotential cost reduction of power losses and penalties due toregulatory noncompliances. Technical attributes are relatedto the number of consumers and to the power quality andoperational condition indexes in the facilities involved bythe projects. The system operational condition indexes arethe loading level, voltage drop, and power losses. The powerquality indexes are related to service continuity and productconformity. Service continuity involves the supply interrup-tion frequency (SAIFI) and duration (SAIDI) indexes. Prod-uct conformity involves the voltage transgressions and powerfactor indexes.

Portfolio Value

Financial Attributes Technical Attributes

Project Costs

Power Quality

Product Service

Voltage Drop

Operational Conditions

LoadingPower Losses

Power Factor

SAIFI SAIDI

Losses Reduction

Penalties Reduction

Voltage Transgressions

Number of Consumers

Continuity Transgressions

Revenues

Power Supply

Fig. 2 Value attributes tree

3.3 Project Technical Attributes

The technical attributes for each project p are as follows:�V(p), voltage drop related to the rated voltage (%); S(p),loading level related to the rated power (%); PL(p), powerlosses level (%); DRP(p), relative duration of voltage trans-gressions to precarious level (%); DRC(p), relative dura-tion of voltage transgressions to critical level (%); � PF(p),power factor deviation (%); SAIFI(p), system average inter-ruption frequency index (interruptions/year); SAIDI(p), sys-tem average interruption duration index (h/year); NC(p),number of supplied consumers.

3.4 Project Financial Attributes

The financial attributes involve the required investment costsfor carrying out the projects, and the potential revenuesrelated to the planning horizon stages in which the projectsare scheduled. Technical improvements in the distributionsystem, obtained by project implementation, may increasethe supply capacity and may reduce the costs of power lossesand penalties because of transgressions of power qualityindexes. The potential cost reductions are considered finan-cial benefits of each project, and count in favor of its priori-tization.

The present value factor (PVF) is used to determine theequivalent financial amount of costs and revenues associatedto project p, incurred in a given stage k with a discount rated. In this article, each planning stage corresponds to one yearperiod. The financial cost c(p) to carry out a project is con-sidered in the scheduling stage k. The financial benefits b(p)

related to a project p are considered from the schedulingstage k by the end of planning horizon T . While the projectpostponement reduces the present value of implementationcost, the project anticipation increases the present value ofrevenues from its operation. Therefore, the portfolio financialvalue depends on the stages in which the selected projects arescheduled. The optimization process searches for an efficientproject scheduling along the planning horizon, to maximizethe portfolio net present value (NPV), according to the prob-lem constraints. The NPV of each project p scheduled instage k is given by (1):

NPVk(p) =(∑T

i=kbi (p) · PVFi

)− c(p) · PVFk . (1)

where bi (p) = ri (p) + CPi (p) + TCi (p) + TVi (p). Rev-enues ri (p) are related to the power supplied by the facili-ties associated with project p, and are considered from thescheduling stage k by the end of planning horizon T . Powerlosses decrease system efficiency and reduce utility revenues.Therefore, the power losses cost reduction is a financial ben-efit that adds value to the portfolio. The power losses costreduction CPi (p) is obtained by the average power capacity

123


PLi (p) released by project p implementation, consideringthe time period H (in hours) of each planning stage i and thecost of energy CE(p) in (2):

CPi (p) = PLi (p) · H · CE(p). (2)

The financial benefit T Ci (p) by avoiding transgressions onregulatory limits of service continuity in each planning stagei , after project p implementation, is given by (3) (ANEEL2010):

TCi (p) =[(

SAIDIi (p)

SAIDIM− 1

)· q1 +

(SAIFIi (p)

SAIFIM− 1

)· q2

]·

SAIDIM · CE(p)

H· δ(p) · NC(p). (3)

where SAIDIi (p) and SAIFIi (p) are, respectively, the sys-tem average interruption duration and frequency indexes instage i ; SAIDIM and SAIFIM are the respective maximumregulatory limits; NC(p) is the number of consumers con-nected to the facilities involved by project p; δ(p) is the volt-age coefficient (LV = 15; MV = 20; HV = 27); q1 = 0,if SAIDIi (p) < SAIFIM and q1 = 1, if SAIDIi (p) ≥SAIDIM ; q2 = 0, if SAIFIi (p) < SAIFIM and q2 = 1,if SAIFIi (p) ≥ SAIFIM .

The financial benefit TVi (p) by avoiding transgressionson regulatory limits of voltage levels in each planning stagei , after project p implementation, is given by (4) (ANEEL2010):

TVi(p) =[(

DRPi (p) − DR PM

100

)· f1(p)

+(

DRCi (p) − DRCM

100

)· f2(p)

]· EC(p). (4)

where DRPi (p) and DRCi (p) are, respectively, the relativedurations of voltage transgressions to critical and precariouslevel indexes in the facilities involved by project p in stage i ;DRPM = 3 % and DRCM = 0, 5 % are the respective maxi-mum regulatory limits; f1(p) = 0, if DRPi (p) ≤ DRPM

and f1(p) = 3, if DRPi (p) > DRPM ; f2(p) = 0, ifDRCi (p) ≤ DRCM ; f2(p) = 7 (LV), f2(p) = 5 (MV)or f2(p) = 3 (HV), if DRCi (p) > DRCM .

4 Optimization Model

In the proposed model, the multistage project prioritizationand portfolio value maximization are formulated by meansof a multiobjective optimization problem. The optimizationprocess selects and schedules the projects in the planningstages, composing portfolios the objective of which is tomaximize the global values of both technical and finan-cial attributes, according to the problem constraints. Theanalytic model is formulated in (5) and (6) as a binary

integer programming problem, where NPVk(p),�V (p),

S(p), PE(p), DRPC(p),�PF(p), SAIFI(p), SAIDI(p),

and NC(p) are the optimization attributes related to eachproject p; B (k) is the available budget for each stage K ofthe planning horizon T , in present value; and N is the numberof available projects. The binary variable xk(p) representsthe decision of including (1) or not including (0) project p instage K for each portfolio alternative.

Max. :

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

T∑k=1

N∑p=1

[NPVk(p) · xk(p)] ;N∑

p=1

[�V (p) · xk(p)] /

N∑p=1

�V (p)

N∑p=1

[S(p) · xk(p)] /

N∑p=1

S(p)

N∑p=1

[PL(p) · xk(p)] /

N∑p=1

PL(p)

N∑p=1

[DRPC(p) · xk(p)] /

N∑p=1

DRPC(p)

N∑p=1

[�PF(p) · xk(p)] /

N∑p=1

�PF(p)

N∑p=1

[SAIFI(p) · xk(p)] /

N∑p=1

SAIFI(p)

N∑p=1

[SAIDI(p) · xk(p)] /

N∑p=1

SAIDI(p)

N∑p=1

[NC(p) · xk(p)]

(5)

s.t.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

T∑k=1

N∑p=1

[PVFk · c(p) · xk(p)] ≤ B (k) .

xk(p) ∈ {0, 1} ;p ∈ {1, 2, · · · , N } ;k ∈ {1, 2, · · · , T } ;

T∑k=1

xk(p) ≤ 1

DRPC(p) = DRP(p) + DRC(p)

(6)

The global value of a technical attribute is the sum of the localvalues aggregated in this attribute by the selected projects,which compose each portfolio alternative, normalized by themaximum global value possible in this attribute (for all Navailable projects). This procedure allows measuring the rel-ative contribution of each selected project to the best value inthat attribute. The global net present value is obtained by thesum of the local net present values of each selected project,which compose the portfolio alternative, and depends on thestages in which the projects are allocated. The selection and

123


allocation of projects must match the problem constraints andthe available budget for each stage of the planning horizon.

Exclusive Projects a subset of mutually exclusive projectsE ⊂ [1, . . . , N], which represent different (or competing)alternatives for similar results, can be handled by the addi-tional constraint (7) in the model:

s.t. :{∑

p∈Exk(p) ≤ 1, ∀k = [1, 2, · · · , T ] . (7)

Conditional Projects when the selection of project a is con-ditioned to scheduling project b, the additional constraint (8)is used in the model:

s.t. :{

[xk (a) − xk (b)] ≤ 0k (a) ≥ k (b)

. (8)

Constraint (8) ensures that project a must be scheduled inthe same or in subsequent scheduling stage of project b. Thescheduling stage of project b is independent of project a.

5 Multiobjective Optimization

Multiobjective optimization problems have two or moreobjective functions simultaneously. In some approaches, cri-teria and weights are used to aggregate the multiple func-tions in a single objective function. Thus, the problem opti-mal solution is found according to the a priori-providedinformation. Mathematical programming and heuristic algo-rithms are techniques commonly used to solve such prob-lems (Stummer and Heidenberger 2003; Leite da Silva et al.2011). However, these approaches rely on subjective judg-ments and are poorly suited to handle financial and nonfi-nancial attributes simultaneously. In other approaches, eachobjective function is optimized sequentially, where the othersare treated as problem constraints (Tamaki et al. 1996).

In many multiobjective problems, the notion of optimal-ity changes, because the result may not be a global optimum,but a set of trade-off solutions among all problem objec-tives (Coello Coello 2006; Ngatchou et al. 2005). In suchproblems, the most appropriate solution to a given particularsituation can be chosen according to some criteria defineda posteriori (Rivas-Dávalos et al. 2007). The trade-off solu-tions can be obtained adopting the optimality notion intro-duced by Francis Ysidro Edgeworth (1881), and generalizedby Vilfredo Pareto (1896), which is based on the conceptof solution dominance. This notion states that, in a maxi-mization problem, solution x1 dominates solution x2 if twoconditions (9) are satisfied:

Maximization :{

I : ∀i : fi (x1) ≥ fi (x2) ;I I : ∃i : fi (x1) > fi (x2) .

(9)

Therefore, a solution is called Pareto Optimal if there is noother feasible solution that could improve some attributewithout degrading another attribute simultaneously. This

concept generally does not lead to a single solution, but to aset of nondominated solutions in the search space S, whichis called Pareto-Optimal Set P (10):

P = {xi ∈ S|�x j ∈ S : xi is dominated by x j

}. (10)

The objective functions chart, nondominated solutions ofwhich are in the Pareto-Optimal Set, is called Pareto frontier(11):

PF = {F (x) = ( f1 (x) , f2 (x) , · · · , fm (x)) |x ∈ P } . (11)

Specific mathematical techniques and computer tools arenecessary to determine the Pareto-Optimal solutions. Manymeta-heuristic algorithms have been developed specially formultiobjective optimization based on Pareto optimality, suchas the multiobjective genetic algorithms (Coello Coello 2006;Rivas-Dávalos et al. 2007; Mishra et al. 2011). These meta-heuristic algorithms are suitable for combinatorial optimiza-tion problems, especially when it is not possible to obtainan analytic model to be solved by usual mathematical tech-niques (Leite da Silva et al. 2011).

The optimization model in (5) and (6) is a combinato-rial constrained multiobjective problem with several vari-ables, and involves a large search space. The NondominatedSearching Genetic Algorithm-II (NSGA-II), which is dis-cussed in Sect. 6, is efficient in providing a set of Pareto-Optimal solutions in a single run, and is used to solve theproposed optimization model. The obtained solutions repre-sent the efficient project portfolios, since they are the besttrade-offs among all the considered technical and financialattributes, according to the problem constraints.

6 Multiobjective Genetic Algorithms

Genetic algorithms are efficient search mechanisms inspiredby the evolution of species (Holland 1975). They use aset of coded parameters and probabilistic transition rules(crossover and mutation operators) to perform a simultaneoussearch in the solution space. Current and historical informa-tion (fitness function, selection procedure, and elitism mech-anism) guide and determine the next search state, to explorethe most prosperous regions of the solution space. This pro-cedure is well suited to combinatorial problems.

However, the genetic algorithms performance depends onthe adopted selection procedure and on the crossover andmutation operators. The fitness function must incorporate allpossible knowledge about the problem to be solved. Sincethey are probabilistic search techniques, genetic algorithmscan find slightly different solutions in each run, even usingthe same set of parameters and starting from the same initialpopulation. Although the genetic algorithms do not ensureobtaining the problem global optimum, they provide goodquality solutions (Coello Coello 2006; Goldberg 1989).

123


There are several genetic algorithms, for both mono-and multiobjective problems. In multiobjective genetic algo-rithms based on Pareto Optimality, the selection proceduremust assess the dominance degree of individuals in termsof the problem objectives and optimization attributes. Thisis the main difference between common genetic algorithms,since the crossover and mutation operators remain virtuallythe same. To improve the algorithm convergence capacity,the reproduction procedure must include elitist mechanismsto preserve some potentially efficient solutions along the evo-lutionary process.

An important advantage of multiobjective genetic algo-rithms is the ability to find multiple nondominated solu-tions in a single run. Efficient multiobjective genetic algo-ritms maintain a good diversity and distribution of solu-tions in the Pareto-Optimal frontier (Coello Coello 2006;Zitzler et al. 2000). They are also less susceptible to theshape or continuity of the Pareto frontier. Since Goldberg(1989) has suggested to incorporate the Pareto-optimalityconcepts in an evolutionary algorithm, several multiobjec-tive genetic algorithms have been proposed (Srinivas andDeb 1994; Horn et al. 1994; Fonseca and Fleming 1993; Zit-zler and Thiele 1998; Knowles and Corne 2000; Zitzler et al.2001).

6.1 NSGA-II

Among several multiobjective genetic algorithms available,the Nondominated Sorting Genetic Algorithm-II (NSGA-II),presented by Deb et al. (2002), was chosen to solve the pro-posed model, since it has good performance and overcomessome difficulties presented by other algorithms, such as com-putational complexity for large populations, deficiency ofelitist mechanisms, diversity maintenance and problem con-straints handling.

The NSGA-II incorporates a fast nondominated sortingprocedure, where each solution is associated with a set ofdominated solutions and with a counter of solutions thatdominate them. In this procedure, the solutions are rankedaccording to the nondominated front in which they are. Thus,the first frontier (rank 1) is composed by solutions which arenondominated by any other in the population. The secondfrontier (rank 2) is composed by solutions which are domi-nated only by the first frontier solutions, and so forth. Thisprocess reduces the number of comparisons between solu-tions, and improves the algorithm performance (Deb et al.2002).

Many multiobjective genetic algorithms use a sharingfunction, where a parameter is responsible for maintain-ing the population diversity (Deb and Goldberg 1989). Tomaintain good diversity and distribution of solutions in thePareto-Optimal frontier, NSGA-II makes use of a crowded-comparison operator that does not require parametrization.

This operator calculates the crowding distance, which isestimated by the average distance between two neighboringpoints on both sides, for each problem objective (Deb et al.2002). Parent and offspring individuals are combinated in anintermediate population, and the binary tournament is used toselect the best individuals for reproduction, according to theirfitness rank and crowding distance. The elite nondominatedsolutions are favored by the crowded-comparison operator inthe tournament selection and in composing the new popula-tion.

A change in the dominance definition simplifies andimproves the constraints handling in multiobjective prob-lems. In the constrained-domination principle, any feasiblesolution gets a better dominance rank than any infeasiblesolution. All feasible solutions are ranked according to theirnondomination level, based on the objective-function values.Among two infeasible solutions, the one with the lowest over-all constraint violation gets a better dominance rank. Accord-ing to Deb et al. (2002), the constrained-domination principledoes not change the computational complexity of NSGA-IIand the other algorithm procedures remain the same. Avail-able studies show that NSGA-II is able to obtain a goodsolution distribution and a convergence closer to the truePareto-Optimal frontier (Coello Coello 2006; Mishra et al.2011).

6.1.1 Controlled Elitism

The meaning of an elite individual is different in the contextof multiobjective optimization, because all solutions belong-ing to the best nondominated frontier are the best individualsin the population and are equally important. These elite solu-tions may be the majority in the current population and, inprinciple, all must be preserved. In this case, not many newsolutions can be accepted in next populations, and the searchmay stagnate or prematurely converge to a set of sub-optimalsolutions.

The Controlled NSGA-II (Deb and Goel 2001) makes useof an additional parameter to reduce the effect of elitism, sothat the new population contains only a portion of the bestnondominated solutions found. A reduction rate exponen-tially restricts the number of individuals from each nondom-inated frontier to be included in the new population, allow-ing the coexistence of individuals from various nondomi-nated frontiers (lateral diversity). This parameter balancesthe current best solutions exploitation and the new solutionsexploration. The controlled elitism increases the populationdiversity and helps the crossover operator in creating differ-ent solutions to advance toward the Pareto-Optimal frontier,improving the algorithm convergence ability, especially indiscrete search spaces (Dhanalakshmi et al. 2011; Ramesh etal. 2012).

123


Yes

No

Start

Project Data Acquisition

Project Value Attributes

Initial Population of Portfolios

Global Value Attributes

Portfolios Fitness Ranking

Nondominated Sorting Procedure

Binary Tournament Selection

Crossover and Mutation Operators

Offspring Portfolios

Nondominated Sorting Procedure

Crowding Distance Sorting

Global Value Attributes

New Population of Portfolios

Portfolios Fitness Ranking

Stop Criteria

Set of Pareto-Optimal Portfolios

Intermediate Population:Parent and Offspring Individuals

Fig. 3 Multiobjective genetic algorithm flowchart

6.2 Optimization Algorithm

Figure 3 shows the flowchart of the multiobjective geneticalgorithm based on NSGA-II, and used to optimize theproject portfolios. The algorithm starts with acquisition ofdata related to the projects proposed for the regional distrib-ution systems. From these data, the local value attributes aremapped. Then the algorithm generates a number of randomgenetic sequences (chromosomes) to compose the initial pop-ulation of portfolios, to which are calculated the respectiveglobal value attributes. The initial population is ranked usingthe constrained nondominated sorting procedure of NSGA-II (Deb et al. 2002). Each portfolio fitness is related to therank of nondominance, where the lower the rank, the betterthe solution.

In the main loop, binary tournament selection, doublepoint crossover and uniform mutation operators are used tocreate the first offspring population of portfolios. An inter-mediate population is composed combining parent and off-spring individuals. The elitist mechanism is applied to this

population, where offspring individuals are compared to thebest individuals previously found, using the constrained non-dominated sorting procedure. A fitness value, correspondingto the nondominance rank, is assigned to the individuals ofthe intermediate population. Then, the crowding distancesare calculated to the neighboring solutions in each nondom-inated frontier.

The new population is composed by adding individualsin order of their ranks, from the first nondominated fron-tier until the number of individuals exceeds the populationsize. In order to have the expected number of individuals,the members of the last nondominated frontier included inthe new population are sorted by the crowding distance, indescending order. Thus, the individuals with greater crowd-ing distances complete the new population. In this procedure,the solutions in the less dense regions of the search spacehave greater importance in choosing the individuals that willcomplete the new population.

Then, the selection, crossover and mutation operators areapplied to the new population. The binary tournament selec-tion is now based on the individuals nondominance rank andon crowding distance. The evolutionary process is repeatediteratively until a stopping criterium is satisfied. The indi-viduals from the best nondominated frontier found by thealgorithm run compose the set of Pareto-optimal portfolios,representing the optimization problem solution. In the casestudies presented in Sect. 7, the genetic algorithm showedgood performance and fast convergence in several runs, whenhandling a population with 390 individuals and using thecrossover probability pc = 0.8 and mutation probabilitypm = 0.015, with the number of generations limited to 400.

6.2.1 Chromosomes of Multistage Portfolios

In genetic algorithms, the optimization problem parametersare encoded in a manner analogous to the biological chromo-somes, where the set of genes characterizes each individual.The genetic encoding of optimization variables is critical tothe algorithm performance. The multistage project selectionand scheduling involves decision variables. The binary stringgenetic encoding is strongly adherent to the problem formu-lation, since it allows to include all necessary information forrepresenting the possible solutions in a single chromosome,facilitating its handling and decoding.

In the proposed model, each portfolio alternative is repre-sented by a chromosome, which is encoded in a binary stringof concatenated genetic blocks. The number of genetic blocksN in each chromosome corresponds to the number of avail-able projects to be analyzed. The genetic block is a set of bitsthat encode the decision variables of including a project inthe portfolio and the planning stage in which it is scheduled.The number of bits b required to encode each project geneticblock depends on the number of stages T of the planning

123


1 1 0 0 0 1

GB1 GB2 GB3

Project 1:Stage 3

Project 2:Not Scheduled

Project 3:Stage 1

1 0

GB4

Project 4:Stage 2

Fig. 4 Example of a portfolio bynary chromosome

horizon, where (T + 1) ≤ 2b. Each project genetic blockGBp is represented by (12):[

GBp] = {

yp1; · · · ; ypb}∀ypj ∈ {0, 1} ; p ∈

{1, · · · , N } ; j ∈ {1, · · · , b} ; p, j ∈ Z. (12)

Figure 4 depicts the decision variables of a binary chromo-some example, which is encoded by four genetic blocks withtwo bits, representing a portfolio composed by four projectsand three planning stages.

The genetic block binary-to-decimal decoding providesinformation on two variables: x(p), the decision of includ-ing (1) or not including (0) project p in the portfolio; andk(p), the planning stage k in which project p is scheduled.If the genetic block decimal value is zero, then x(p) = 0and k(p) = 0, representing that project p is not included inthe portfolio. On the other hand, if the genetic block decimalvalue is not zero, then x(p) = 1, representing that project pis selected to the portfolio, where the scheduling stage k(p) isgiven by the corresponding decimal value of

[GBp

]. There-

fore, each portfolio is a binary chromosome that correspondsto a point in the search space, feasibility of which is verifiedby the algorithm procedure, depending on the problem para-meters and constraints.

7 Case-Studies

The proposed model is applied to a distribution utility deci-sion problem, which involves 13 projects to be prioritizedfor the improvement and expansion of the regional distri-bution systems (Moreira et al. 2009). The planning horizoninvolves three stages (years). The available budget for projectfunding is U$550,000.00 per year (in present value). Theproject selection and scheduling must be based on the valueattributes related to the project costs, expected annual rev-enues from the power supply, cost reduction of power losses,and penalties from regulatory limits transgressions, numberof benefited consumers, feeders loading, voltage drops, andsystem interruption indexes. The project-related data are pre-sented in Table 1. In addition, the utility considers projects 2and 7 mutually exclusive, and project 12 scheduling is con-ditioned to project 10. The adopted discount rate is 10 % peryear.

The problem goal is to find the Pareto-Optimal portfolios,where projects are selected and scheduled in the planning

horizon considering the optimization criteria in each casestudy, according to the available budget and to the condi-tional and mutually exclusive constraints. In each case study,the genetic algorithm performance is analyzed for 10 runs,starting from different random genetic seeds (initial popula-tions). The presented results correspond to the best Pareto-Optimal frontiers found. To verify the algorithm conver-gence, some reference solutions are obtained by an iterativemono-objective genetic algorithm, where the objectives areaggregated by an additive function using different weights in40 runs.

Case 1 Portfolio cost minimization and number of selectedprojects maximization

This case study goal is to search for trade-off solu-tions between the number of selected projects and the port-folio total cost, taking into account the available budgetand considering the conditional and mutual exclusive con-straints. The genetic algorithm optimizes two objective func-tions simultaneously: the number of selected projects max-imization and the portfolio total cost minimization. Table 2presents the trade-off portfolios found, where the geneticalgorithm processing time is 16.8 min. There is no projectscheduled in the first stage. The projects are postponedbecause of cost minimization in present value. The lowestcost portfolio (with only one project) has scheduled project5 in the third planning stage. The highest cost portfolio hasscheduled 11 projects, where eight projects are scheduled inthe third stage and three projects in the second stage. Accord-ing to the mutual exclusive projects constraint, only project7 was scheduled, since its cost is lower than project 2 cost.The conditional projects constraint is also satisfied, sinceproject 12 is not scheduled earlier than project 10. Project1 is not scheduled in any portfolio because of its highercost.

Figure 5 shows the nondominated solutions in the Paretofrontier, where the average solution dispersion is 1.957.NSGA-II converges to the same solution points obtainedfor the reference frontier. By analyzing the efficient frontier,decision makers can check the project scheduling in relationto the portfolio impact on the available budget. However, it isnot possible to verify the technical and financial benefits pro-vided by the set of scheduled projects in each portfolio option.

Case 2 Cost minimization and revenues maximizationThis case study goal is to maximize the expected total rev-

enue and minimize the portfolio total cost simultaneously,considering the available budget and the conditional andmutual exclusive projects constraints. The genetic algorithmoperates with two simultaneous objectives, and provides64 efficient portfolios in 13.78 min running time. Table 3presents 15 portfolios with the greatest revenues in this case.Figure 6 shows that NSGA-II solutions converge to the samepoints in the reference frontier. The average dispersion ofPareto-optimal solutions is 0.9501.

123


Table 1 Distribution system expansion and improvement projects

Nr Feeder Project description Cost (U$) Revenue (U$/stage) Number ofconsumers

Loading (%) �V (%) SAIFI

1 ALA2 New feeder construction (CL) toimprove system reliability

675,224.75 205,315.52 2,350 67.52 8.1 12.0

2 ALB3 Feeder cables replacement (CL) toimprove system reliability

106,036.89 351,174.32 4,170 40.71 4.6 9.5

3 ALC1 Feeder extension to improvesystem reliability

51,033.06 1,815,486.71 1,789 71.25 0.7 0.7

4 ALD5 Feeders interconnection forloading transfer and reliabilityimprovement

24,179.50 983,553,52 2,350 82.35 8.7 1.5

5 ALE9 Feeders interconnection forloading transfer and reliabilityimprovement

10,027.17 1,063,592.05 1,350 72.22 4.9 1.8

6 ALA4 New feeder construction (CL) toimprove system reliability

262,990.10 1,462,765.38 3,742 75.00 2.3 10.9

7 ALB5 Feeder cables replacement forloading reduction and reliabilityimprovement

101,898.51 842,294.94 1,896 67.38 5.1 0.3

8 ALB1 Feeders cables replacement (CL)for reliability improvement andregulatory requirementscompliance

210,616.43 295,482.59 6,524 42.03 2.8 14.0

9 ALE4 Feeders interconnection forloading transfer to avoidinterruptions and to improvesystem reliability

75,354.96 1,354,331.20 19,381 74.90 9.7 0.5

10 ALF6 Interconnection for loadingtransfer between feeders ALF6and ALF1 and power qualityimprovement

24,679.90 429,261.80 13,354 68.14 13.4 1.9

11 ALG3 Feeder cable replacement (IL) toreduce voltage drop and improveloading level and power quality

134,516.14 1,239,967.79 15,131 99.42 4.6 10.3

12 ALF4 Feeder cable replacement to reduceoperational problems caused by alarge consumer

13,498.24 479,250.94 7,032 68.91 6.0 6.0

13 ALH6 Feeder interconnection (CL) forloading transfer to avoidinterruptions and to improvesystem reliability

376,963.93 638,103.18 38,762 76.84 5.0 9.5

CL Compact distribution line with protected cables, IL distribution line with insulated cables, �V voltage drop, SAIFI interruptions/year

The great number of solutions is due to the different finan-cial attributes values in each possible project scheduling com-bination. In this case study, there is a tendency to anticipateprojects to maximize the portfolio expected revenue over theplanning horizon. Project 1 is not scheduled in any port-folio option because it has the highest cost and the lowestexpected revenue among the available projects. With respectto the mutual exclusive projects constraint, Project 7 is sched-uled because of its lower cost and higher expected revenuethan Project 2. Project 13 is scheduled only in four port-folios, because of its high cost and low expected revenue.Project 8 is scheduled only in three portfolios because of itsrelatively high cost and low expected revenue. On the otherhand, Project 5 is scheduled in all portfolios, since it presentsattractive cost and expected revenues. For the same reason,

Projects 3, 4, and 10 are scheduled in the majority of portfo-lios.

This case allows decision makers to check the portfolios’expected revenues sensitivity against project scheduling, andthe respective impacts on the utility budget. However, in thiscase, it is not possible to check the technical benefits providedby the scheduled projects in each portfolio option.

Case 3 Portfolio global value maximizationThe aim of this case study is to maximize the portfolio

global value, considering the available budget and the condi-tional and mutual exclusive projects constraints. Therefore,the optimization objective is to schedule projects focusedon improvements in facilities which currently operate withthe greatest number of consumers and with the highest load-ing levels, voltage drops and interruption frequency indexes,

123


Table 2 Portfolios of Case 1

Project schedulingin each portfolio

Total cost (U$,present value)

Numberof projects

1 2 3 4 5 6 7 8 9 10 11 12 13

A 0 0 0 0 0 0 0 0 0 0 0 0 0 0.00 0

B 0 0 0 0 3 0 0 0 0 0 0 0 0 7,533.56 1

C 0 0 0 3 3 0 0 0 0 0 0 0 0 25,699.98 2

D 0 0 0 0 3 0 0 0 0 3 0 3 0 36,217.36 3

E 0 0 0 3 3 0 0 0 0 3 0 3 0 54,383.78 4

F 0 0 3 3 3 0 0 0 0 3 0 3 0 92,725.67 5

G 0 0 3 3 3 0 0 0 3 3 0 3 0 149,340.97 6

H 0 0 3 3 3 0 3 0 3 3 0 3 0 225,898.83 7

I 0 0 3 3 3 0 3 0 3 3 3 3 0 326,962.79 8

J 0 0 3 3 3 0 3 3 3 3 3 3 0 485,202.04 9

K 0 0 3 3 3 3 2 3 2 3 3 3 0 696,107.71 10

L 0 0 3 3 3 3 2 3 2 3 3 3 2 1,007,648.14 11

0

2

4

6

8

10

12

0.00 0.20 0.40 0.60 0.80 1.00 1.20

Num

ber

of s

ched

uled

pro

ject

s

Portfolio cost in present value (U$, million)

NSGA-II Reference Frontier

Fig. 5 Pareto-optimal and reference frontiers in Case 1

in such a way to maximize the portfolio’s expected NPV.The genetic algorithm operates simultaneously with fiveobjective functions and finds eight efficient portfolios, pre-sented in Table 4. The algorithm running time is 16.12 min,and the average dispersion of Pareto-optimal solutions is1.0791.

In this case, Project 1 is scheduled in six portfolios becauseof the technical attributes, such as the high voltage dropand the interruption frequency index in the related facil-ities. Despite the cost, Project 1 is prioritized because itis focused on improvements in feeder power quality andoperational conditions. According to the budget availabil-ity, Project 1 is scheduled in the third planning stage. Themutual exclusive projects (2 and 7) are differently sched-uled in each portfolio option. Although Project 7 presentsbetter financial attributes (lower cost and higher expectedrevenue), Project 2 is scheduled in three portfolios becauseof the number of consumers and to the interruption frequencyindex in the related feeder, what requires attention by deci-

sion makers. The conditional projects (10 and 12) are bothscheduled in the same planning stage, in all portfolios. Themost attractive projects in cost (Project 5), revenue (Project3), loading level (Project 11), and voltage drop (Project 10)attributes are scheduled in all portfolios. Project 13 has themost attractive number of consumers and is scheduled inseven portfolio options. Project 8 has the highest interrup-tion frequency index and is scheduled in six portfolio options.The obtained project scheduling is because the multiobjec-tive genetic algorithm checks the nondominance of solutionsin terms of the portfolio global values, instead of the projectlocal value attributes.

This case involves a multidimensional optimization, andit is not possible to visualize the Pareto frontier. Figure 7gives an idea of the solution distribution presenting the rela-tion between the NPV and each portfolio total score, whichis calculated by summing the normalized values of techni-cal attributes. The NSGA-II solutions converge to the bestreference points. The lowest score portfolio (3.1387) is theunique where Project 13 is not scheduled. Owing to theequal attribute weights considered in summing the normal-ized scores, this solution is dominated by others and couldbe eliminated from the decision-making process. However,if decision makers assign different weights for each attribute,then the solution distribution in Fig. 7 may change. Thisprocess can be used to support decision makers in choos-ing the most strategic portfolio option.

The presented case studies show the proposed multiob-jective approach, and point out the importance of includ-ing the distribution system operational condition and powerquality indexes as value attributes in the optimization model,and in the decision-making process. The adopted multistageapproach allows verifying the impacts of project schedul-

123


Table 3 Greatest revenue portfolios of Case 2

Project scheduling in each portfolio Total cost(U$, presentvalue)

Total revenues(U$, presentvalue)

NPV (U$)

1 2 3 4 5 6 7 8 9 10 11 12 13

A 0 0 1 1 1 1 1 0 1 1 0 2 0 511,303.93 20,529,813.91 20,018,509.97

B 0 0 1 1 1 1 1 0 1 1 0 1 0 512,419.49 20,965,496.58 20,453,077.09

C 0 0 1 1 1 1 0 0 1 2 1 2 0 538,916.66 21,128,529.43 20,589,612.76

D 0 0 1 1 1 1 0 0 1 1 1 2 0 540,956.32 21,518,767.43 20,977,811.10

E 0 0 1 1 1 1 0 0 1 1 1 1 0 542,071.88 21,954,450.10 21,412,378.22

F 0 0 1 1 1 2 1 0 1 1 1 2 0 611,856.61 22,283,643.57 21,671,786.95

G 0 0 1 1 1 2 1 0 1 1 1 1 0 612,972.17 22,719,326.24 22,106,354.07

H 0 0 1 1 1 1 1 0 1 1 2 1 0 623,589.85 22,921,869.50 22,298,279.65

I 0 0 1 1 1 1 2 0 1 1 1 1 0 626,285.53 23,283,390.27 22,657,104.75

J 0 0 1 1 1 1 2 3 1 1 1 1 0 784,524.77 23,505,390.72 22,720,865.95

K 0 0 1 1 1 1 2 2 1 1 1 1 0 800,348.69 23,749,591.21 22,949,242.51

L 0 0 1 1 1 1 2 0 1 1 1 1 3 909,504.11 23,762,806.64 22,853,302.53

M 0 0 1 1 1 1 1 0 1 1 2 1 2 935,130.29 23,928,643.87 22,993,513.57

N 0 0 1 1 1 1 2 0 1 1 1 1 2 937,825.96 24,290,164.64 23,352,338.67

O 0 0 1 1 1 1 2 3 1 1 1 1 2 1,096,065.21 24,512,165.08 23,416,099.88

0.00

5.00

10.00

15.00

20.00

25.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20

Port

folio

NPV

(U$,

mill

ion)

Portfolio cost in present value (U$, million)



ing in the portfolio value attributes over the planning hori-zon. The results demonstrate that the proposed method is apotential tool to support decision makers with important andstrategic information to aid the distribution system planningprocess.

7.1 Algorithm Performance

In the presented case studies, the multiobjective genetic algo-rithm optimizes up to five attributes simultaneously. How-ever, the proposed decision model is formulated to han-dle simultaneously all optimization attributes presented inSect. 4. The use of each optimization attribute depends onthe characteristics and objectives of the addressed problem,and on the adequate data availability. Other value attributes,defined by decision makers as important to evaluate andselect the proposed projects for distribution system improve-

ment and expansion, can also be incorporated into the opti-mization model.

The NSGA-II obtains a set of Pareto-Optimal solutions ina single run, and does not require the optimization attributeweights to be defineda priori. The algorithm processing timeis considered adequate for this kind of planning problem. Onthe other hand, the mono-objective genetic algorithm usesan additive function and finds a single solution each run. Toobtain several points on the reference frontier, it demands along time iterative process. In addition, the obtained resultsare highly dependent on the attributes weights, which mustbe provideda priori.

A greater number of projects and planning stages, thanthose presented in this article, can also be handled by theNSGA-II in the proposed model. The algorithm performanceis satisfactory in a study carried out involving 40 projects, fiveoptimization attributues, and seven planning stages. How-ever, the greater the number of projects and planning stages,the greater the number of bits required to encode the chro-mosomes, and the larger the search space and the numberof feasible solutions. In this case, the processing time mayincrease substantially and limit the algorithm convergenceability.

Table 5 presents a comparison between the presented casestudies and other two cases carried out with a larger numberof projects and planning stages. Case 4 involves the sameoptimization attributes of Case 2, and Case 5 involves thesame optimization attributes of Case 3. In all cases, NSGA-II have converged to the best reference points obtained by aniterative genetic algorithm using an additive function withrandom weights. It is observed that when the optimization

123


Table 4 Portfolios of Case 3

Project scheduling in each portfolio NPV (U$) Numberof consumers

Portfolio scores Total score(normalized)

1 2 3 4 5 6 7 8 9 10 11 12 13 Loading SAIFI �V

A 3 1 1 1 1 1 0 2 2 3 1 3 2 20,027,904.98 115,935 0.9257 0.9962 0.9328 3.8386

B 3 0 1 1 1 1 1 2 2 3 1 3 2 21,253,011.44 113,661 0.9551 0.8796 0.9394 3.7387

C 3 0 1 1 1 1 0 2 1 1 1 1 2 22,046,700.15 111,765 0.8808 0.8758 0.8722 3.5773

D 3 2 1 1 1 1 0 0 1 1 1 1 2 22,220,997.76 109,411 0.8793 0.8188 0.8959 3.5225

E 3 0 1 1 1 1 2 2 1 1 1 1 0 22,596,192.75 74,899 0.8703 0.7592 0.8735 3.1387

F 0 2 1 1 1 1 0 3 1 1 1 1 2 22,637,808.73 113,585 0.8512 0.8441 0.8261 3.4854

G 3 0 1 1 1 1 2 0 1 1 1 1 2 22,999,288.91 107,137 0.9087 0.7022 0.9025 3.4226

H 0 0 1 1 1 1 2 3 1 1 1 1 2 23,416,099.88 111,311 0.8806 0.7275 0.8327 3.3855

3.003.103.203.303.403.503.603.703.803.90

15.00 16.00 17.00 18.00 19.00 20.00 21.00 22.00 23.00 24.00

Tot

al s

core

s (n

orm

aliz

ed)

Porfolio NPV (U$, million)



involves both financial and technical attributes (Cases 3 and5), the decision model is able to find efficient portfolios witha greater number of scheduled projects.

8 Final Remarks

Efficient resource allocation in power systems is a strategicprocedure to fulfill regulatory requirements, to ensure con-sumer satisfaction, and to improve utility business results.An analytic model that incorporates an efficient tool to opti-

mize the portfolio value can reduce subjectivity in decision-making process, selecting, and scheduling projects that jus-tify costs in relation to the technical and financial benefits,according to the available budget.

This article has addressed the project prioritization prob-lem in a multistage planning horizon. The projects are pro-posed for expansion and improvement of regional distrib-ution systems and dispute the utility available budget. Thedeveloped decision model is based on value attributes toselect and schedule projects over the planning horizon. Amultiobjective approach allows address several technical andfinancial attributes simultaneously. The NSGA-II is used tooptimize the decision model. The solution is a Pareto-optimalfrontier composed by a set of trade-off portfolios, accord-ing to the problem objectives and constraints. The stragegicportfolio can be chosen a posteriori, by means of judgmentsbased on the decision makers’ expertise.

The multistage approach allows to verify the projectscheduling sensitivity in relation to the value attributes overthe planning horizon. The presented case studies demon-strate the importance of considering power quality and opera-tional performance indexes as value attributes in the decisionmodel, since they change the project scheduling and improvethe portfolio value. The decision model can be used to prior-

Table 5 Case studiescomparison

a Computer CPU Intel� CoreTM

2 Duo, 2 GHz, RAM 3 GB

Case 1 Case 2 Case 3 Case 4 Case 5

Available projects 13 13 13 40 40

Planning stages 3 3 3 3 7

Chromosome bits 26 26 26 80 120

Objectives 2 2 5 2 5

Solutions 12 64 8 222 102

Scheduled projects 11 11 12 16 28

Dispersion of solutions 1.957 0.9501 1.0791 0.9630 0.6848

Processing time (min)a 16.8 13.8 16.1 163.32 392.4

123


itize projects for regional distribution systems in a systemicway, using available data and the current planning proce-dures. The proposed method is a potential tool to supportdecision makers with important and strategic information toaid the distribution system planning process.

References

ANEEL (2010). PRODIST: Distribution System Procedures “Proced-imentos de distribuição de energia elétrica no Sistema ElétricoNacional”. http://www.aneel.gov.br. Accessed 10 Dec 2011.

Arrillaga, J., Bollen, M. H. J., & Watson, N. R. (2000). Power qual-ity following deregulation. Proceedings of the IEEE: Invited Paper,88(2), 246–261.

Brown, R. E., & Marshall, M. (2000). Budget constrained planningto optimize power system reliability. IEEE Transactions on PowerSystems, 15(2), 887–892.

Carvalho, P. M. S., & Ferreira, L. A. F. M. (2004). Urban distributionnetwork investment criteria for reliability adequacy. IEEE Transac-tions on Power Systems, 19(2), 1216–1222.

Choi, J. H., Park, C. H., Chae, W. K., Jang, S. I., Kim, K. H., & Park, J.K. (2005). An improved investment priority decision method for theelectrical facilities considering reliability of distribution networks.Proceedings of the IEEE Power Engineering Society General Meet-ing, Vol. 3 (pp. 2185–2191).

Coello Coello, C. A. (2006). Evolutionary multi-objective optimiza-tion: A historical view of the Field. IEEE Computational IntelligenceMagazine, 1(1), 28–36.

Deb, K., Goldberg, D. E. (1989). An investigation of niche and speciesformation on genetic function optimization. In J. D. Schaeffer (Ed.),Proceedings of the 3rd International Conference on Genetic Algo-rithms (pp. 42–50). San Mateo, CA: Morgan Kauffman.

Deb, K., & Goel, T. (2001). Controlled elitist non-dominated sortinggenetic algorithms for better convergence. Lecture Notes in Com-puter Science: Evolutionary Multi-Criterion Optimization (pp. 67–81). Berlin: Springer.

Deb, K., Pratap, A., Agrawal, S., & Meyarivan, T. (2002). A fast andelitist multiobjective genetic algorithm: NSGA-II. IEEE Transac-tions on Evolutionary Computation, 6(2), 182–197.

Deb, K., Sinha, A., Korhonen, P. J., & Wallenius, J. (2010). An interac-tive evolutionary multiobjective optimization method based on pro-gressively approximated value functions. IEEE Transactions on Evo-lutionary Computation, 14(5), 723–739.

Deb, K., Rao, U. B., & Karthik, S. (2007). Dynamic multi-objectiveoptimization and decision-making using modified NSGA-II: A casestudy on hydro-thermal power scheduling. Evolutionary Multi-Criterion Optimization, 4403, 803–817.

Dhanalakshmi, S., Kannan, S., Mahadevan, K., & Baskar, S. (2011).Application of modified NSGA-II algorithm to combined economicand emission dispatch problem. International Journal of ElectricalPower and Energy Systems, 33(4), 992–1002.

Edgeworth, F. Y. (1881). Mathematical physics. London: P. Keagan.Elton, E. J., & Gruber, M. J. (1991). Modern portfolio theory and invest-

ment management (4th ed.). New York: Wiley.Falcão, D. M. (2002). Genetic algorithms applications in electrical dis-

tribution systems. Proceedings of the 2002 Congress on EvolutionaryComputation (pp. 1063–1068).

Fonseca, C. M., Fleming, J. P. (1993). Genetic algorithms for multi-objective optimization: formulation, discussion and generalization.Proceedings of the 5th International Conference on Genetic Algo-rithms (p. 8), California: San Mateo.

Goldberg, D. E. (1989). Genetic algorithms in search optimizationand machine learning (1st ed.). Massachusetts: Addison WesleyPublishing.

Gönen, T., & Ramirez-Rosado, I. J. (1986). Review of distribution sys-tem planning models: A model for optimal multistage planning. IEEProceedings: Generation Transmission and Distribution, 133(7),397–408.

Gönen, T. (1986). Electric power distribution system engineering. NewYork: McGraw Hill.

Haffner, S., Pereira, L. F. A., Pereira, L. A., & Barreto, L. S. (2008).Multistage model for distribution expansion planning with distrib-uted generation, Part I: Problem formulation. IEEE Transactions onPower Delivery, 23(2), 915–923.

Holland, J. H. (1975). Adaptation in natural and artificial systems. AnnArbor: University of Michigan Press.

Horn, J., Nafpliotis, N., & Goldberg, D. E. (1994). Niched pareto geneticAlgorithm for multiobjective optimization. IEEE World Congress onComputational Intelligence, Vol. 1 (pp. 82–87).

Huang, Y. S., Zhou, J. H., Qi, J. X. (2005). Optimization of technologicalinnovation projects based on value engineering and fuzzy compre-hensive evaluation. Proceeding of the 4th International Conferenceon Machine Learning and Cybernetics (pp. 2736–2741), Guangzhou.

Kannan, S., Baskar, S., McCalley, J. D., & Murugan, P. (2009). Applica-tion of NSGA-II algorithm to generation expansion planning. IEEETransactions on Power Systems, 24(1), 454–461.

Knowles, J. D., & Corne, D. W. (2000). Approximating the nondomi-nated front using the Pareto achived evolution strategy. EvolutionaryComputation, 8(2), 149–172.

Kumar, Y., Das, B., & Sharma, J. (2008). Multiobjective, multiconstraintservice restoration of electric power distribution system with prioritycostumers. IEEE Transactions on Power Delivery, 23(1), 261–270.

Leite da Silva, A. M., Rezende, L. S., Honório, L. M., & Manso, L.A. F. (2011). Performance comparison of metaheuristics to solve themulti-stage transmission expansion planning problem. IET Genera-tion, Transmission and Distribution, 5(3), 360–367.

Levine, H. A. (2005). Project portfolio management: A practical guideto selecting projects, managing portfolios, and maximizing benefits.San Francisco: Wiley.

Lotero, R. C., & Contreras, J. (2011). Distribution system planning withreliability. IEEE Transactions on Power Delivery, 26(4), 2552–2562.

Maghouli, P., Hosseini, S. H., Buygi, M. O., & Shahidehpour, M. (2009).A multi-objective framework for transmission planning in deregu-lated environments. IEEE Transactions on Power Systems, 24(2),1051–1061.

Martin, E. M., Merkhofer, L. (2003). Lesson learned: resource alloca-tion based on multi-objective decision analysis. Proceedings of the1st Power Delivery Asset Management. Workshop, New York.

Mishra, S. K., Panda, G., Meher, S., Majhi, R., Singh, M. (2011). Port-folio management assessment by four multiobjective optimizationalgorithm. IEEE Recent Advances in Intelligent Computational Sys-tems (pp. 326–331).

Moreira, W. S. C., Mussoi, F. L. R., Teive, R. C. G. (2009). Investmentprioritizing in distribution systems based on multi-objective geneticalgorithm. Proceedings of the 15th IEEE International Conferenceon Intelligent System Applicatons to Power Systems (ISAP’09) (p. 6)Brazil: Curitiba.

Ngatchou, P., Zarei, A., El-Sharkawi, M. A. (2005). Pareto multi objec-tive optimization. Proceedings of the 13th International Conferenceon Intelligent Systems Application to Power Systems (ISAP) (pp. 84–91).

Pareto, V. (1896). Cours d’economie politique. Paris: Rouge Lausanne.Ramesh, S., Kannan, S., & Baskar, S. (2012). Application of modi-

fied NSGA-II algorithm to multi-objective reactive power planning.Journal of Applied Soft Computing, 12(2), 741–753.

Ramírez-Rosado, I. J., & Bernal-Agustín, J. L. (1998). Genetic algo-rithms applied to the design of large power distribution systems.IEEE Transactions on Power Systems, 13(2), 696–703.

Rivas-Dávalos F., Moreno-Goytia E., Gutiérrerz-Alacaraz G., Tovar-Hernandez T. (2007). Evolutionary multi-objective optimization in

123

http://www.aneel.gov.br


power systems: state-of-the-art. Proceedings of the IEEE PowerTechnology, Vol. 1 (pp. 2093–2098), Lausanne.

Srinivas, M., & Deb, K. (1994). Multiobjective optimization using non-dominated sorting in genetic algorithms. Journal of EvolutionaryComputation, 2(3), 221–248.

Stummer, C., & Heidenberger, K. (2003). Interactive R&D portfolioanalysis with project interdependencies and time profiles of multipleobjectives. IEEE Transactions on Engineering Management, 50(2),175–183.

Tamaki, H., Kita, H., Kobayashi, S. (1996). Multi-objective optimizationby genetic algorithms: a review. Proceedings of the IEEE Interna-tional Conference on Evolutionary Computation (pp. 517–521).

Vaziri, M., Tomsovic, K., & Bose, A. (2004). A directed graph for-mulation of the multistage distribution expansion problem. IEEETransactions on Power Delivery, 19(3), 1335–1341.

Vuinovich, F., Sannino, A., Ippolito, M. G., & Morana, G. (2004). Con-sidering power quality in expansion planning of distribution systems.Proceedings of the IEEE Power Engineering Society General Meet-ing, Vol. 1 (pp. 674–679).

Wang, D. T. C., Ochoa, L. F., & Harrison, G. P. (2011). Modified GAand data envelopment analysis for multistage distribution networkexpansion planning under uncertainty. IEEE Transactions on PowerSystems, 26(2), 897–904.

Wang, H., Lin, D., Li, M.Q. (2005). A competitive genetic algorithm forresource-constrained project scheduling problem. Proceedings of the4th International Conference on Machine Learning and, Cybernetics(pp. 2945–2949).

Zhou, Y., Spare, J. H. (2007). Optimizing reliability project portfoliosfor electric distribution companies. Proceedings of the IEEE PowerEngineering Society General Meeting (pp. 1–5).

Zitzler, E., Thiele, L. (1998). An evolutionary algorithm for multiobjec-tive optimization: The strength Pareto approach. Technical Reportn. 43, Computer Engineering and Communication Networks Lab,Swiss Federal Institute of Technology, Zurich.

Zitzler, E., Deb, K., & Thiele, L. (2000). Comparison of multiobjectiveevolutionary algorithms: Empirical results. Evolutionary Computa-tion, 8(2), 173–195.

Zitzler, E., Laumanns, M., Thiele, L. (2001). SPEA2: Improving thestrength Pareto evolutionary algorithm. Technical Report nr. 102,Computer Engineering and Communication Networks Lab: TIK,Swiss Federal Institute of Technology: ETH, Zurich.

123

a multistage value-based model for prioritization of distribution projects using a multiobjective...

Documents