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Smart Reconfiguration of Electric PowerDistribution Networks for Power Loss

Minimization and Voltage Profile Optimization

Alberto Landeros

Thesis of 60 ECTS creditsMaster of Science (M.Sc.) in Sustainable Energy

Engineering

May 2018

Smart Reconfiguration of Electric Power DistributionNetworks for Power Loss Minimization and Voltage Profile

Optimization

by

Alberto Landeros

Thesis of 60 ECTS creditssubmitted to the School of Science and Engineeringat Reykjavík University in partial fulfillment

of the requirements for the degree ofMaster of Science (M.Sc.) in Sustainable Energy Engineering

May 2018

Supervisor:

Superior A. Mohamed Abdelfattah, SupervisorAssistant Professor, Reykjavík University, Iceland

Superior B. Slawomir Koziel, SupervisorProfessor, Reykjavík University, Iceland

Examiner:

Eyjólfur Ingi Ásgeirsson, ExaminerAssociate Professor, Háskólinn í Reykjavík, Iceland

CopyrightAlberto Landeros

May 2018

iv


Optimization

Alberto Landeros

May 2018

Abstract

Distribution network reconfugration DNR can significantly reduce power losses, improve thevoltage profile, and increase the power quality. DNR studies require implementation of thepower flow analysis and complex optimization procedures capable of handling large combina-torial problems. The size of the distribution network influences the type of the optimizationmethod to be applied. In particular, straight forward approaches can be computationally expen-sive or even prohibitive whereas heuristic or meta-heuristic approaches can yield acceptableresults with less computation cost. In this thesis work, a customized evolutionary algorithmhas been introduced and applied to power distribution network reconiguration. The recombina-tion operators of the algorithm are designed to preserve feasibility of solutions here, the radialstructure of the network thus considerably reducing the size of the search space. Consequently,an improved repeatability of results as well as lower overall computational complexity of theoptimization process have been achieved. The proposed technique is referred to as feasibil-ity preserving evolutionary optimization FPEO. Another approach is adopted to solve DNR.The method is based on sequential stochastic optimization that utilizes mechanisms adoptedfrom simulated annealing to avoid getting stuck in local minima), and customized networkmodication procedures that aim at improving the cost function while maintaining the radialarchitecture of the distribution system. The proposed technique is referred to as feasibility-preserving simulated annealing(FPSA). Both, FPEO and FPSA are comprehensively validatedusing three IEEE test cases, 33, 69 and 119-bus systems. At last, a novel algorithm for powerloss reduction through distribution network reconguration(DNR) and optimization based allo-cation of distributed generation (DG) sources is reported. Here, DNR is solved simultaneouslywith DG allocation. The problem at hand is a complex mixed-integer task. A customizedevolutionary algorithm has been developed with recombination operators preserving a radialstructure of the network, integer-based operators for DG placement, and floating point opera-tors for handling their power output capacities. Comprehensive numerical validation performedon standard IEEE 33- and 69-bus systems indicates that our methodology outperforms state ofthe art algorithms available in the literature in terms of the obtained power loss reduction. Fur-thermore, it features good repeatability of results as demonstrated through statistical analysisof multiple algorithm runs.

Snjall endurskipulagning raforkuflutningsneta fyrir minnkunorkutaps og spennu bestun

Alberto Landeros

maí 2018

Útdráttur

Þessi ritgerð fjallar um þróun erfðafræðilegs reiknirits fyrir endurskipulagningu raforkudreifi-kerfa. Markmiðið með endurskipulagninguna er sú að draga úr orku tapi í dreifikerfinu. Aðauki er hefðbundið bestunarreiknirit metið þar sem reikniritð hermir simulated annealing að-ferðina aðallega til að koma í veg fyrir að lenda í staðbundri bestun. Að lokum er reikniritiðaðlagað til að framkvæma samtímis endurskipulagningu dreifikerfis og ákjósanlegasta úthlutundistributed generation í raforkudreifikerfinu. Allar niðurstöður úr þessum þremur aðferðum eruborin saman við niðurstöðurnar úr nýlegum greinum.

vi


Optimization

Alberto Landeros

Thesis of 60 ECTS credits submitted to the School of Science and Engineeringat Reykjavík University in partial fulfillment of

the requirements for the degree ofMaster of Science (M.Sc.) in Sustainable Energy Engineering

May 2018

Student:

Alberto Landeros

Supervisor:

Superior A. Mohamed Abdelfattah

Superior B. Slawomir Koziel

Examiner:

Eyjólfur Ingi Ásgeirsson

The undersigned hereby grants permission to the Reykjavík University Library to reproducesingle copies of this Thesis entitled Smart Reconfiguration of Electric Power DistributionNetworks for Power Loss Minimization and Voltage Profile Optimization and to lend orsell such copies for private, scholarly or scientific research purposes only.The author reserves all other publication and other rights in association with the copyright inthe Thesis, and except as herein before provided, neither the Thesis nor any substantial portionthereof may be printed or otherwise reproduced in any material form whatsoever without theauthor’s prior written permission.

date

Alberto LanderosMaster of Science

To my sister Alejandra, to all of those in the battle against cancer and to all of those whomhave lost a loved one to cancer. I would also like to dedicate this to my family; thank you for

your sweet love.-Alberto Landeros

Acknowledgements

I would like to thank my supervisor Mohamed Abdelfattah, professor at Reykjavik University,for the opportunity to work on this project and all the information learned during the Smartgridscourse he lectured and I was part of as a student. Thanks to my advisor Slawomir Koziel for allhis guidance through out this project, all the optimization I learned during his engineering op-timization course and during this thesis work. Special thanks to professor Ragnar Kristjánssonfor the power system lessons. Finally I would like to thank my friend Lin Zeng for his initialguidance on understanding the nature of distribution reconfiguration.

xv

Contents

Acknowledgements xiii

Contents xv

List of Figures xvii

List of Tables xix

List of Abbreviations xxi

List of Symbols xxiii

1 Introduction 11.1 The Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Electric Power Distribution System . . . . . . . . . . . . . . . . . . . 31.1.2 Radial Distribution Networks . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Distribution Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.4 Voltage Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.5 Power Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.6 Line Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.7 Power Flow Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.8 Distribution Network Reconfiguration . . . . . . . . . . . . . . . . . . 81.1.9 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Literature Review 112.1 Power Flow Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Distribution Network Reconfiguration for Power Loss Minimization and Volt-

age Profile Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Distribution Network Reconfiguration and Optimal Distributed Generation Al-

location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Methodology 173.1 Power Distribution System Modeling . . . . . . . . . . . . . . . . . . . . . . . 173.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Feasibility Preserving Evolutionary Optimization (FPEO) Applied to the Dis-

tribution Network Reconfiguration . . . . . . . . . . . . . . . . . . . . . . . . 203.3.1 Algorithm Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.2 Feasibility-Preserving Mutation Operator . . . . . . . . . . . . . . . . 203.3.3 Feasibility-Preserving Crossover Operator . . . . . . . . . . . . . . . . 21

xvi

3.4 Feasibility Preserving Simulated Annealing Applied to the Distribution Net-work Reconfiguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4.1 Algorithm Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4.2 Function random_radial_configuration . . . . . . . . . . . . . . . . . . 233.4.3 Function local_search . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4.4 Function random_change . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Feasibility-Preserving Evolutionary Search Applied to DNR and Optimal Allo-cation of Distributed Generation . . . . . . . . . . . . . . . . . . . . . . . . . 263.5.1 Data Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.5.2 Recombination Operators for Distributed Generator Allocation . . . . . 263.5.3 Recombination operators for Distributed Generator capacities . . . . . 27

4 Test Results 294.1 Distribution Network Reconfiguration By Means Of Feasibility-Preserving Evo-

lutionary Optimization for Power Loss Reduction . . . . . . . . . . . . . . . . 294.1.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Journal Manuscript Annexations 35

Bibliography 61

A Distribution Test Networks Information 69

xvii

List of Figures

1.1 Power system subsystems and basic components. . . . . . . . . . . . . . . . . . . 31.2 Single line diagram or a radial distribution network. . . . . . . . . . . . . . . . . . 51.3 Single line diagram representation of a power line segment . . . . . . . . . . . . . 51.4 Magnetic fields present in conductor lines . . . . . . . . . . . . . . . . . . . . . . 7

3.1 MATLAB commands access the main simulation engine through the COM inter-face. Some of the commands include switch operations, power flow execution,results extraction, etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Operation of feasibility-preserving mutation operator: (a) part of the radial networkwith one of the switches randomly selected to be closed and create connection(marked thick dashed line), (b) a loop created by closing the switch (thick solidline), (c) one of the connections along the loop randomly selected to be opened,thus retaining the radial configuration of the network.. . . . . . . . . . . . . . . . . 21

3.3 Flow diagram of feasibility-preserving crossover operator. . . . . . . . . . . . . . 223.4 Flow diagram of the feasibility-preserving simulated annealing (FPSA) algorithm. . 25

4.1 Flow diagram of the feasibility-preserving simulated annealing (FPSA) algorithm. . 314.2 Number of objective function evaluation required by FPEO to obtain perfect re-

peatability (i.e., optimum solution at every algorithm run) versus feasible searchspace size. The solid line plot obtained for the test cases considered in the testcases reported in Section 5. The dashed line is a rough estimation for a potentiallarger problem of 300-400 buses . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Network voltage profile before and after reconfiguration/optimization . . . . . . . 324.4 Network power losses before and after reconfiguration/optimization . . . . . . . . 33

A.1 33-bus test distribution network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69A.2 69-bus test distribution network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.3 119-bus test distribution network . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

xix

List of Tables

2.1 Descriptive table of optimization approaches for distribution reconfiguration avail-able in the literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Comparative table of metaheristic optimization algorithms applied to optimal allo-cation of DG and DNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1 Initial condition of the test systems . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Control parameter setup vs algorithm performance . . . . . . . . . . . . . . . . . . 304.3 Statistics of the optimization results of the 69-bus test network . . . . . . . . . . . 304.4 Distribution system characteristics after reconfiguration . . . . . . . . . . . . . . . 31

A.1 33-bus distribution test network [80]. . . . . . . . . . . . . . . . . . . . . . . . . . 69A.2 69-bus distribution test network[79] . . . . . . . . . . . . . . . . . . . . . . . . . 71A.3 119-bus distribution test network [76] . . . . . . . . . . . . . . . . . . . . . . . . 73

xxi

List of Abbreviations

DNR Distribution Network ReconfigurationDG Distributed GenerationGA Genetic AlgorithmPSO Particle Swarm OptimizationPLI Power Loss IndexVSI Voltage Stability IndexLSF Loss Sensitivity FactorVSF Voltage Sensitivity FactorsFWA Fireworks AlgorithmAC Alternating CurrentPQ Active/Reactive powerDMS Distribution Management SystemNR Newton RaphsonCGA Continuous Genetic AlgorithmRGA Refined Genetic AlgorithmITS Improved Tabu SearchHSA Harmony Search AlgorithmIAICA Improved Adaptive Imperialist Competitive AlgorithmCSA Cuckoo Search AlgorithmNDE Node Depth EncodingFPEO Feasibility Preserving Evolutionary AlgorithmFPSA Feasibility Preserving Simulated AnnealingCOM Component Object ModelFPES Feasibility Preserving Evolutionary Search

xxiii

List of Symbols

Symbol Description Value/UnitsPF Power of feeder bus WPn Power demand at bus n WPloss Power loss of line segments WVi Voltage at bus i (complex) p.u/V~V(i+1) Voltage at bus (complex) i+1p.u/V~I(i,i+1) Current flow from bus i to bus i+1 ARi Resistance of line i ΩjXi Reactance of line i Ω∆V Voltage drop p.u/VPloss(i,i+1) Power loss of i line segment kWPi Active power load at bus i kWQi Reactive power load at bus i kVarP netloss Net power loss of the system kVAVimax Maximum bus voltage p.u/VVimin Minimum bus voltage p.u/VIimax Maximum current capacity of line i Agj Network configurationGradial Set of radial configurationsNs Total number of switch controlsNsec Number of sectionalizing switchesNts Number of tie switches∆P ratio

loss Power loss ratioP recloss Power loss after reconfiguration kWP

(rec/DG)loss Power loss after reconfiguration and DG kWP initloss Power loss before reconfiguration kWVD Voltage deviation IndexPminDG Minimum DG source power capacity kWPmaxDG Maximum DG source power capacity kWPDG Power output of DG kWNbr Number of line segments(G) Set of spanning treesG(x) Network graph configuration Binary stringFbest Best found objective function valueFcurrent Current objective function valueF (x(0)) Initial objective function valueF (xrand) Objective function value of random configurationF (xlocal) Objective function value of local search

xxiv

F (x(i+ 1)) Objective function value of following configurationimax Maximum number of iterationsT0 Initial system temperatureT Current temperaturesk Open switchessk.jv Adjacent switch to skFk.j Objective function of resulting radial network x(k.j)x(k.j) Resulting radial networkx.L Binary string of zeros (open switches)x.G Vector of integers (bus location)x.P Vector of floating point numbers (DG power) kWdmax Maximum distanceMg Global mutationxandy Parent individualsz Offspringr Random vector between 0 and 1pk Power of DG component k kW∆pk Power variation of DG component k kWpk,max Upper bound of the kth DG elementpk,min Lower bound of the kth DG elementβ Mutation rateNs Number of switchesNedg Number of edges

1

Chapter 1

Introduction

As smart grid technologies mature and deploy, the gap for developing efficient operations meth-ods is created [1]. In electric power distribution systems, it is desirable to take advantage ofcomputational intelligence and automation provided by the Smart Grid technologies which in aunified fashion create an efficient, reliable, clean, secure, and sustainable electric power supply[2]. Some of the inconveniences in distribution systems are system overloading at different pe-riods of the day and load capacity increase; these inconveniences cause excessive power lossesand voltage levels variations beyond standard operational levels, consequently lowering powerquality or even interrupt the power supply. Reduction of power losses and improvement ofvoltage profile in power distribution networks has been often approached by methodologiessuch as distribution network reconfiguration (DNR), capacitor bank placement and Volt/Varcontrol [3], [4]. Recently, optimal allocation of distributed generation (DG) has been takeninto consideration as a method for grid support due to its attributions towards frequency dropstabilization [5], power loss reduction and voltage profile improvement [6]–[13]. Combinationof multiple methodologies such as distribution reconfiguration and optimal allocation of DGshave been proposed to increase the benefits; however, the practical complexity of implementingsuch combination of methodologies increases significantly.

Radial distribution networks often feature sectionalizing switches and tie switches mainlyused for fault isolation, power supply recovery and system reconfiguration. These switchesallow for reconfiguring the topology of the network, with the objectives being reduction ofpower losses, load balancing, voltage profile improvement and increasing reliability of the sys-tem. Reference [14] presents a comprehensive overview of reconfiguration benefits and someoptimization algorithms applied to the DNR problem. A large search space resulting fromall possible switch configurations make the network reconfiguration task a complex non-linearcombinatorial problem, with the complexity dramatically increasing with the number of de-grees of freedom, i.e., both the network size and the number of switches. Due to the nature ofthe DNR problem, artificial intelligence and meta-heuristic optimization techniques are oftenapplied. In [15] the authors implement an algorithm for network reconfiguration for a realisticdistribution network based on a genetic algorithm (GA), taking as objective power loss mini-mization and load balancing index. In [16], an optimization algorithm based on the parasiticbehavior of the Cuckoo bird is proposed and applied to 33-, 69- and 119-bus test networkstaking as objective function active power loss minimization and voltage profile improvement;the results are then compared with other methods applied to the same networks. In [17], theauthors propose a method for solving the DNR problem based on the Hopfield neural network,having as objective power loss reduction and reliability improvement.

2 CHAPTER 1. INTRODUCTION

Although the majority of modern DNR solution methods exploit population-based algo-rithms, sequential stochastic methods such as simulated annealing [18] may also be potentiallyadvantageous. In [19], a feasible approach to DNR with multiple feeding points is introduced;The approach adopts a hybrid optimization method involving simulated annealing and the Tabusearch to avoid solution unfeasibility; DNR is implemented on a small distribution networkconsisting on 3 power nodes and 16 load nodes.

Incentivized by factors such as liberalization of the energy market, environmental con-cerns, and technology development, incorporation of DG to the power systems has recentlybeen increasing [20], [21], particularly at the distribution level [8]. DG interconnections maybring multiple technical benefits, i.e. active/reactive power loss reduction, voltage profile im-provement, reliability enhancement as well as overall power delivery efficiency. However, ifmisallocated, DG can actually lead to degradation of the network performance (in particular,an increase of the power losses, and increase of the voltage levels above the operational stan-dards) and contribute to instability [9]. Consequently, it is important to consider optimal sittingand sizing of DG sources. Several optimization methods have been proposed for solving theoptimal DG allocation problem. In [10], a Particle Swarm Optimization (PSO) algorithm hasbeen applied to the optimum sitting and sizing of two DGs on a 13-bus network, however, thebasis for selecting the optimum location of DG is not provided. In [8], the authors determineoptimal location of DG based on the Power Loss Index (PLI) and sizing is performed usingthe GA taking into account three objective functions, power loss reduction, voltage profile im-provement and voltage stability index (VSI). In [11], the authors determine candidate busesfor DG allocation based on the loss sensitivity (LSF) and voltage sensitivity factors (VSF); theoptimum DG size is then determined with a hybrid PSOGSA optimization algorithm. WhileDNR and optimal allocation of DG can be performed individually, their benefits can be fur-ther enhanced when performed collectively. In [12], DNR and optimum sitting and sizing ofDG based on the fireworks algorithm (FWA) under three different loading scenarios has beenproposed. The buses with the lowest VSI are selected as candidate buses followed by optimalDG sizing determined by the FWA; when both methods are performed simultaneously, lossesare reduced by up to 84 percent. In [13], the authors apply GA to a 16-bus network takingas objective minimization of power costs; here, all buses are candidate for DG placement andDG sizes are generated randomly. Although all methods mentioned above solve the reconfig-uration and optimal DG allocation problem, no statistical data is presented, which is essentialfor measuring quality of results obtained by the algorithm. Also, in majority of cases, verysmall networks are considered, which does not permit conclusive assessment of the proposedmethodologies in terms of their prospective performance when applied to more realistic cases.

1.1 The Power SystemWhen performing power distribution system studies, it is convenient to be familiarized withthe basic structure and basic physics behind power system. This chapter presents the elementalsubsystems comprising the power system, the basic structure of the power distribution systemand some essential electrical characteristics related to power lines and power flow. Specificdetail on power flow is presented with further detail in chapter 3.

The electrical power system is a set of subsystems composed of generating sources andmultiple layers of electric power transmission networks [22]. These together make up the elec-trical power system which aims to supply several loads at different voltage levels according tothe end user operational standards. The subsystems of the electric power system are presented

1.1. THE POWER SYSTEM 3

in Fig.1.1. The generation subsystem is typically a collection of power generating units, wheresome form of energy (e.g. coal, hydro, nuclear, wind, etc.) is transformed to electric energywith certain characteristics (e.g. voltage level, frequency). Typically, the voltage of electricitygenerated by high capacity generators is increased via the use of transformers to high and superhigh voltage levels for long distance transmission, although the modern electric power systemmay integrate small power generation at distribution voltage levels. Eventually, the high volt-age power reaches the subtransmission network where voltage is often reduced significantly inorder to supply industrial loads. The distribution network represents the final stage of powertransfer, where individual customers (i.e. commercial and residential) are supplied. At thisstage, voltage has been reduced from substransmission levels to lower levels ranging from 34kV-120 V.

Figure 1.1: Power system subsystems and basic components.

Power systems are often comprised of the following basic characteristics:

• Most generation, transmission, and distribution is carried out under the three-phase, alter-nating current (AC) method. Components of the electric power system, including, but notlimited to, industrial scale motors, transformers and generators fall under the three-phasecategory.

• The power system often transfers electric power over long distances, supplies a largevariety of customers at different voltage levels and covers large geographical areas.

• The power system is comprised of subsystems which facilitate the transmission and de-livery of electric power to the existing interconnected loads.

1.1.1 Electric Power Distribution SystemThe distribution system is a subsystem of the electrical power system representing the last stageof power transfer. Distribution systems are fed by a set of voltage reduction and protection de-


vices named “substation”. At the substation, high voltage of transmission lines is reduced bytransformers in order to supply customers connected at the distribution layer. It is essentialto understand the arrangement and components of the distribution system for its analysis isfunction of the elements that make up the power distribution system. Power flow studies fun-damentally require the known topology of the distribution system and its manipulation is basedon the components such as switches and power lines

1.1.2 Radial Distribution NetworksTraditionally, utility companies have designed and developed power distribution systems asradial network topology[23]. When performing distribution system studies, it is convenient torepresent the distribution network as a graph, in strict terms, as a tree graph (as referred to ingraph theory studies), where a bus can be represented as a node and a power line as an edge(branch). Based on [24], the following characteristics can describe a tree4:

• A tree is a connected graph that contains no loops (cycles); for instance, in a tree, an edgenever splits and rejoins.

• Let us represent G(n) as a graph. G(n) can represent a tree if and only if the graph isconnected, it has no loops and contains n− 1 edges (cf. Eq. 1.1) where n represents thenumber of nodes.

In this work, all existing line segmens (power lines) are assumed to have a switching device,therefore the number of edgesNedg must be equal to the number of switching switching devicesNs (c.f. Eq 1.2), each acting upon one and only one line segment.

Nedg = n− 1 (1.1)

Nedg = Ns (1.2)

1.1.3 Distribution NetworkDistribution networks are often characterized by their radial structure, meant to carry electricpower from the substation to all the existing loads interconnected to the network. Fig.1.2 showsa single line diagram of a 16-bus radial distribution network. Based on the radial topology of thedistribution network, the feeder bus must be able to withstand the summation of load demand atthe subsequent nodes and the power losses in the line segments. In strict terms, as representedby Eq(1.1) the loading of the feeder bus can be described as follows:

PF =N∑n=1

Pn + Ploss (1.3)

Where PF is the power fed to the network by the feeder bus,N is the number of nodes, Pn isthe load power demand at bus n and Ploss are the power losses in the lines. During distributionsystem analysis, it is common practice to consider the feeder bus as the source node, and loads,generator or other components are assumed to be connected at the subsequent nodes.


Figure 1.2: Single line diagram or a radial distribution network.

1.1.4 Voltage DropVoltage drop calculations are essential when performing distribution system analysis and de-sign. Particularly, when performing distribution reconfiguration studies, one of the objectivefunctions is often voltage drop minimization across the power lines. Voltage drop directly af-fects power losses given in the line segments and other components of the distribution systemwhen assuming constant PQ loads. Taking as reference the line segment shown in Fig.1.3 andassuming voltage V ∗

i corresponding to node i is known, the receiving end voltage V ∗(i+1) at node

i+ 1, is determined as:

~Vi+1 = ~Vi − ~Ii,i+1(Ri + jXi) (1.4)

where ~Ii,i+1 represent the current flowing from bus i to bus i + 1, Ri and jXi are the lineresistance and reactance of the ith line respectively.

Upon determining both, receiving voltage and sending voltages, the voltage drop across apower line segment can be determined as:

∆V = ~Vi − ~Vi+1 (1.5)

Figure 1.3: Single line diagram representation of a power line segment

1.1.5 Power LossMost of the power losses present in power systems are due to resistive losses (I2R) given in thepower lines, although losses given in other equipment (e.g. transformers) can be considerable


[25]. Power losses in distribution systems can account for up to 70 percent of the total powerlosses [26], consequently, utility companies often search for solutions to minimize operationalcosts related to electric power delivery and power losses are no exception. Recent developmentsof Smartgrid technology and distributed generation represent an incentive for power engineersto develop methodologies applied to the reduction of power losses which directly minimizeoperational costs of power delivery.

1.1.6 Line ImpedanceLine impedance is a critical parameter to be determined for it is necessary when modelingpower systems. Impedance is a phenomena present in AC circuits and it is the measure ofopposition that a circuit presents to a current when voltage is applied. One particular parameterwhen working with power lines is series impedance which consists of conductor resistanceand self and mutual inductive reactances as a result of magnetic fields around conductors [23].Conductor manufacturers typically provide conductor resistance, however, reactance is to bedetermined for it is case specific. Fig.1.4 shows conductors 1 to n with respective magneticfluxes created by current flow in the conductor. Based on the assumption that current flows outthe page the sum of all the currents in all lines is equal to zero as expressed by:

I1 + I2 + ...+ Ik + ...+ In = 0 (1.6)

The total flux linking conductor k is:

λk = N ·φ = 2 ·10−7 ·(I1 ·ln(1

Dk1

)+I2 ·ln(1

Dk2

)+ ...IK ·ln(1

GMRk

)+ ...In ·ln(1

Dkn

)) (1.7)

where:N is the number of times the flux surrounds the conductor

Dkn is the distance between conductor i and conductor kGMRk is the geometric mean radius of conductor k

Conductor inductance is composed of self inductance and mutual inductance of conductork and the rest k-1 conductors. Self inductance and mutual inductance units are henry per meter[H/m] and express by equations 1.8 and 1.9 respectively.

Lkk =λkkIk

= 2 · 10−7 · ln(1

GMRk

)[H/m] (1.8)

Lkn =λknIn

= 2 · 10−7 · ln(1

Dkn

)[H/m] (1.9)

where:Dkn is the distance between conductor k and conductor n. Assuming a three phase balancedsystem and power lines transposed, self and mutual inductance can be determined [27] as fol-lows:

Lk = 2 · 10−7 · ln(Deq

GMRk

)[H/m] (1.10)


Figure 1.4: Magnetic fields present in conductor lines

where:Deq is the equivalent distances between phases a,b and c and expressed as:

Deq = 3√Dab ·Dbc ·Dca (1.11)

Given operational frequency, the inductive reactance is given as:

Xi = ω · Li = ω · ln(Deq

GMRk

)[Ω/(m, km,mile)] (1.12)

Typically, power distribution systems operate with untransposed power lines and unbal-anced phases [23], therefore other methods considering these factors in addition to ground andneutral line influences have been developed such as Carson’s equations. Carson’s equationswere not used due to the high amount of calculations involved which was considered tediousuntil the introduction of digital computers. Computers have allowed for the wide use of Car-son’s equations and modified equations.

1.1.7 Power Flow AnalysisThe analysis of a power distribution network under steady state operation lays essentially on apower flow simulation [23] . Power flow simulations in distribution systems may intuitively bethough similar to those of transmission networks and that is true at a certain extent, however,their network topology differ significantly. In radial distribution systems, methods used intransmission systems are not applied "because of poor convergence characteristics" [23]. Inmeshed and weakly meshed distribution networks, other techniques may be applied [28], [29].Chapter two and three present detailed information on power flow methods applied to powerdistribution systems. Power flow analysis often determine the following parameters:

• Apparent, real, and reactive power flows

• Instant voltage magnitudes and per unit values at the buses of the network


• Power Losses given across lines, transformers, capacitors, etc.

• Voltage and current angle deviations

1.1.8 Distribution Network ReconfigurationDNR attempts to change the configuration of a a distribution network by altering the status ofexisting tie and sectionalizing switches with the intention of minimizing or maximizing certainoperational characteristics such as power losses. Minimum bus voltage level maximization isoften one of the objective functions of major importance [14] when DNR is implemented:

max(minVi) (1.13)

where V i is the voltage level at bus i.Some of the constraints when applying DNR are:

1. Bus voltage Constraint. Here, operational voltage standards should always be met whenreconfiguring the system, in strict terms, voltage values at all buses of the system shouldnot be lower than the minimum voltage Vmin of greater than the maximum voltage valueVmax:

V mini ≤ Vi ≤ V max

i (1.14)

2. After reconfiguration, all power lines must be operating within their thermal limits:

Ii ≤ Imaxi (1.15)

Where Ii is the current carried by line segment i and Iimax is the maximum currentcarrying capacity of line segment i.

3. Typically, power distribution networks adopt a radial topology. When performing DNR,the network topology of the distribution system must be kept at all times to be considereda feasible solution:

gi ∈ Gradial (1.16)

Where gj is the network configuration obtained after DNR and Gradial represents a set ofall possible radial configurations corresponding to a given system.

4. When working with large distribution networks, some exhaustive methods might be pro-hibitive due to computational power limits. DNR not only requires the computation ofcomplex DNR algorithms, but to compute power flow analysis, which can be consider-able [28].


1.1.9 ObjectiveBased on the high power losses and voltage drops given in power distribution systems, Dis-tribution network reconfiguration (DNR) is proposed as an alternative solution to the afore-mentioned problems.The objective of this thesis work is to develop a generalized approach toperforming reconfigurations on radial power distribution networks and optimal allocation ofdistributed generation sources, taking as objective function power loss minimization and volt-age profile improvement for both, power losses and voltage losses relate to each other. In orderto achieve this main objective, two other sub objectives are necessary:

• Power flow simulation: An accurate, reliable and fast power flow method must be im-plemented based on the system topology in order to obtain power losses at every linesegment, system power losses and voltage values at every node on the network.

• Select and implement an efficient optimization algorithm capable of being extrapolatedto DNR

11

Chapter 2

Literature Review

2.1 Power Flow Analysis

Power flow calculation is an essential numerical (often nonlinear) analysis needed when makingpower systems studies [23]. The practical application of proposed methods on the improvementof the power distribution network (e.g. power loss reduction, increase in reliability levels, volt-age profile improvement) [28], [30] typically require many power flow analysis algorithm runs.In areas like distribution management systems (DMS), a tool for the management of the SmartGrid, distribution power flow analysis will be implemented on more frequent basis, thereupon,power flow methods must be as efficient as possible. The power flow methodologies applied todistribution systems will have to adapt to the nature of the distribution network, in strict terms,they will be demanded to include single phase and three phase unbalanced system analysis, aswell as the influence of distributed generation interconnection. In [31], two industry expertsreport a comprehensive review on the evolution of the distribution system, the importance ofDMS and the core importance of robust power flow analysis tools. In Distribution system re-configuration, some of the objectives are to minimize power losses and enhance voltage profile,consequently, power flow analysis becomes an essential procedure to be performed in order toobtain steady state electrical characteristics of the distribution system as consequence of recon-figuration.

Often, power flow analysis methodology applied to transmission systems is mainly com-prised by the Gauss-Seidel, Newton-Raphson (NR) and its decoupled version [32], [33]. Thesepower flow methods are typically used assuming a balanced system, consequently using asingle-phase representation of a three phase system. Due to the particular characteristics of dis-tribution systems (i.e. unbalanced loads, radial topology, often untransposed lines, distributedgeneration) the assumptions made in the analysis of transmission systems fail to be extrap-olated to the distribution system. Thereupon, a power flow method considering three phaseunbalanced networks must be considered for its application in distribution systems. Reference[30] presents a comprehensive review and comparative analysis of power flow methods appliedto the distribution system in phase frame and sequence frame references. In the study, the for-ward/backward methods (a popular power flow method applied to distribution systems) suchas the ones presented in [34]–[40] are capable of performing power flow analysis assuming un-balanced phase systems, however, it is limited to radial networks and does not have the abilityto consider the influence of distributed generation, given in both, phase frame and sequenceframe references. On the other hand, Newton based methods [41]–[44] typically can deal withany topology type (i.e. radial, weakly meshed and meshed) and can consider the influence ofdistributed generation, in both, phase frame and sequence frame references. As expressed by

12 CHAPTER 2. LITERATURE REVIEW

[41], the main attribution to the success of the modified Newton method in distribution systemsis the representation of the Jacobian matrix which is created based on the nature of the networktopology.

Open DSS is an efficient and fast simulation tool for distribution network analysis; althoughit evolved as a harmonic flow analysis tool [45], it can solve power flow analysis, being powerflow one of the largest applications of the program. It handles not only radial networks butmeshed, multi-phase unbalanced networks. OpenDSS provides the capability of modeling awide variety of power system elements (e.g. multiphase power lines, loads, generators, capac-itors, storage, etc.). The power flow solving methods implemented by OpenDSS are based onthe “current injection mode” and the “Newton” making OpenDSS a fast and robust power flowanalysis tool. In [46], authors present the solution algorithm of the current injection power flowmethod and the characteristics of the Newton method which considers multi-phase real andimaginary parts. In addition, authors present a listing of all the applications given to OpenDSS.In [47], distribution reconfiguration for power loss reduction is implemented based on a radialnetwork. The optimization is based on a genetic algorithm, here, the fitness function (this beingline segment power loss) is provided by OpenDSS; authors also consider the influence of threePV generators and three wind energy sources all rated at 50 kW. In [48], OpenDSS is used toobtain power flow results as consequence of distribution network reconfiguration. Authors alsoperform a study on distributed energy sources penetration.

2.2 Distribution Network Reconfiguration for Power LossMinimization and Voltage Profile Improvement

The electric power distribution network is an elemental component of the power distributionsystem. It deserves particular attention for the most significant challenges, both design- andoperational-wise, emerge at this level [49]. One particular area of interest for power engineersworking on distribution systems is finding feasible solutions for power loss reduction; powerlosses in distribution systems can account for up to 70 percent of the total power losses [26],consequently increasing the operational costs of the system [50]. Introduction of the smart grid(SG) concept that aims at supporting the transition into a safe, efficient and sustainable powersystem [51], requires the use of computational intelligence methods to meet the aforemen-tioned objectives. Distribution network reconfiguration (DNR) has been shown to be a feasibleapproach, often involving computational intelligence algorithms to optimize power delivery byreducing power losses, balancing loads, increasing power quality and improve reliability [52].The radial distribution systems often feature sectionalizing switches and tie switches mainlyused for fault isolation, power supply recovery and system reconfiguration. The radial networktopology analysis may be a complex task by itself. Due to a large number of possible networkarchitectures (increasing exponentially with the size of the system), as well as multi-modalnature of the problem (i.e., the presence of many local optima), DNR is a highly complexcombinatorial problem. Its practical solution requires implementation of efficient optimizationalgorithms with global search capability.

Optimization algorithms applied to distribution network reconfiguration can be divided intwo groups, Heuristic and Stochastic Methods, both with particular advantages and disadvan-tages as reported in Table 2.1.

Some popular algorithms of the Heuristic family are Branch Exchange and Optimal Flowpattern [53]. Branch Exchange algorithm fundamentally takes as objective function the min-imization of active power losses and reconfiguration is performed sequentially, first forming

2.2. DISTRIBUTION NETWORK RECONFIGURATION FOR POWER LOSSMINIMIZATION AND VOLTAGE PROFILE IMPROVEMENT 13

loops by closing tie switches, eventually a tie switch within loops is opened until the systemis radial. Such method reduces the search, although it may not find optimal configuration andmay be prohibitive for large scale networks with high degree of flexibility (i.e. tie and sec-tionalizing switch number). Merlin and Back [54] were the first to propose an algorithm fordistribution system reconfiguration; their method was a based on branch exchange search withall tie lines initially closed thus creating a arbitrarily meshed system; subsequent switch open-ing was continued until radial configuration was achieved. Similarly, in [55] a method wasproposed where the network was initially meshed, switches were ranked based on current car-rying magnitude, the top-ranked switch was opened and the power flow calculation was carriedout. The process was repeated until system was radial. The branch exchange was performedwherever a loop had been identified. Configuration with lower power losses was kept. In [56],a generalized approach has been proposed in which the tie line with the highest voltage differ-ence is closed and the neighboring branch is opened in the loop formed, leading to reductionof power losses. Another heuristic method applied to DNR is the Optimal Flow Pattern. Here,similarly to branch exchange, all tie lines are closed thus forming a weakly meshed network.Loads at corresponding connection busses are converted into currents, and only the real partof the complex impedance of the branches is taken into consideration. Once Kirchhoff currentand voltage law is satisfied, the optimal load flow pattern is reached, at this point, the switchcarrying the lowest current value is opened at each corresponding loop, thus making the systemradial again. This method is typically fast and can be done in multi-feeder networks.

As found in the literature, algorithms of the meta-heuristics optimization family are someof the most applied to DNR because the problem is, in general, multi-model one. Thuan ThanhNguyen, Anh Viet Truong [16], perform DNR using cuckoo search algorithm on 33-, 69- and119-bus distribution systems and compare to methods presented in other works [57]–[61]. Thepower loss values presented for the 33- and 69- node systems are the same as those implement-ing continuous genetic algorithm (CGA) and particle swarm optimization (PSO); the lossesare smaller than those applying fireworks algorithm (FWA) [57], GA, refined GA (RGA), im-proved TS (ITS), harmony search algorithm (HSA) [59] and improved adaptive imperialistcompetitive algorithm (IAICA) [60]. For the 119 node system, CSA and CGA present thesame loss level, where FWA and HSA present the lowest. For larger distribution systems (withhundreds and thousands of buses), DNR can be computationally expensive and require un-acceptable amounts of computing time. [62] implemented DNR through node-depth encoding(NDE) which improves the performance of evolutionary algorithms and power flow algorithms;on average 27.64% in loss reduction was obtained on 30,880 buses, 5166 switch system. Al-though the majority of modern DNR solution methods exploit population-based algorithms,sequential stochastic methods such as simulated annealing [18] may also be potentially advan-tageous. In [19], a feasible approach to DNR under the presence of distributed generation (DG)has been discussed by using a hybrid optimization method involving simulated annealing andthe immune algorithm to avoid solution unfeasibility; DNR is implemented on a small distri-bution network consisting on 3 power nodes and 16 load nodes. In [63] authors approach DNRbased on convex relaxation of optimal power flow; the method proposed introduces a succes-sive branch reduction approach which is then simplified into a constant complexity algorithm.The algorithm is characterized by low computational complexity and it is shown to be globallyconvergent under very strict and impractical assumptions of only one switch to be opened. Aversion applicable for a general case is also provided which is essentially a heuristic algorithmbased on iterative application of the aforementioned scheme (without proving convergence),and demonstrated to provide good results for selected test cases.


Tabl

e2.

1:D

escr

iptiv

eta

ble

ofop

timiz

atio

nap

proa

ches

ford

istr

ibut

ion

reco

nfigu

ratio

nav

aila

ble

inth

elit

erat

ure

Alg

orith

mM

etho

dA

dvan

tage

sD

isad

vant

age

Avai

labl

eL

itera

ture

Gen

etic

Alg

orith

mM

eta-

Heu

rist

ics

Effi

cien

tsea

rch

capa

bilit

ies,

good

Prob

lem

spec

ific:

the

algo

rith

mm

ustb

ead

apte

dto

the

natu

reof

the

netw

ork.

[15]

,[16

],[4

8],[

60],

[64]

–[66

]

Sim

ulat

edA

nnea

ling

Met

a-H

euri

stic

sFa

stco

nver

genc

e,av

oids

getti

ngst

uck

inlo

calm

inim

aH

ighe

rco

mpu

tatio

nal

cost

rel-

ativ

eto

popu

latio

nba

sed

al-

gori

thm

s,ro

bust

ness

isre

duce

dw

ithne

twor

ksi

ze

[19]

,[67

]–[7

0]

Fire

wor

ksM

eta-

Heu

rist

ics

Effi

cien

tse

arch

capa

bilit

ies,

satis

-fa

ctor

ype

rfor

man

ceon

larg

esc

ale

netw

orks

Hig

hco

mpu

tatio

nal

cost

s,ro

-bu

stne

ssre

duce

dw

hen

cons

ider

-in

gm

ulti-

obje

ctiv

epr

oble

ms.

[12]

Cuc

koo

Met

a-H

euri

stic

sH

ighl

yro

bust

,sa

tisfa

ctor

ype

rfor

-m

ance

whe

nco

nsid

erin

gm

ulti-

obje

ctiv

eop

timiz

atio

n

Prob

lem

adap

tatio

nis

diffi

cult

[16]

Art

ifici

alN

eura

lN

etw

orks

Inte

llige

ntH

igh

robu

stne

ss,

nonl

inea

rfit

ting

abili

tyR

equi

res

exte

nsiv

eam

ount

sof

trai

ning

data

[17]

,[71

]

Bra

nch

Exc

hang

eH

euri

stic

sR

educ

esse

arch

spac

ePr

ohib

itive

inla

rge

scal

ene

t-w

orks

[54]

–[56

]

Opt

imal

Flow

Patte

rnH

euri

stic

sE

ffici

ent

sear

chan

dlo

wco

mpu

ta-

tiona

ltim

ePo

wer

flow

ism

odifi

ed,

hard

toim

plem

ents

witc

hing

[63]

2.3. DISTRIBUTION NETWORK RECONFIGURATION AND OPTIMALDISTRIBUTED GENERATION ALLOCATION 15

2.3 Distribution Network Reconfiguration and OptimalDistributed Generation Allocation

As reported in the literature, distribution network reconfiguration is proven to enhance the op-erational characteristics of distribution systems (i.e. power loss minimization, voltage profileimprovement, reliability increase, cost reduction, etc.). Other methods in addition to DNRhave been considered for further enhancement of system operational characteristics. One ofthe techniques is optimal allocation of DG sources. Power generation at the distribution levelfrom DG sources has recently increase significantly [20], [21], mainly attributed to the liber-alization of the energy market and the development of renewable energy technology. Some ofthe sources of DG generation are often, solar photovoltaics (PV) , wind, biomass, mini hydroand/or a combination of sources, typically under 10 MW capacity [72]. Although DG sourcesinterconnection to the distribution system may bring benefits for the system, the location andsize of DG sources is to be considered. Excess DG generation or misplacement may increasepower losses, reduce voltage profile and reduce reliability of the system. Thereupon, optimalsizing and location of DG sources should be considered to maintain a reliable system underoperational standards. Although several methods are reported in the literature applied to DNRand optimal allocation of DG (i.e. Artificial Intelligence and Heuristics), meta-heuristics areoften applied due to their capability of dealing with the multi-modal nature of the given prob-lem. Meta- heuristic algorithms implement learning strategies and innovative strategies for anefficient search, the former being of great advantage due the large search space accompaniedby large scale distribution networks. Table 4.4 reports various methods available in the liter-ature, their advantages and disadvantages. Several optimization methods have been proposedfor solving the optimal DG allocation problem. In [10], a Particle Swarm Optimization (PSO)algorithm has been applied to the optimum sitting and sizing of two DGs on a 13 bus network,however, the basis for selecting the optimum location of DG is not provided. In [8], the au-thors determine optimal location of DG based on the Power Loss Index (PLI) and sizing isperformed using the GA taking into account three objective functions, power loss reduction,voltage profile improvement and voltage stability index (VSI). In [11], the authors determinecandidate buses for DG allocation based on the loss sensitivity (LSF) and voltage sensitivityfactors (VSF); the optimum DG size is then determined with a hybrid PSOGSA optimizationalgorithm.

Although DNR and optimal allocation of DG can be performed individually, their benefitscan be further enhanced when performed collectively. In [12], DNR and optimum sitting andsizing of DG based on the fireworks algorithm (FWA) under three different loading scenarioshas been proposed. The buses with the lowest VSI are selected as candidate buses followed byoptimal DG sizing determined by the FWA; when both methods are executed simultaneously,the losses are reduced by up to 84 percent. In [13], the authors apply GA to a 16-bus networktaking as objective minimization of power costs; here, all buses are candidate for DG place-ment and DG sizes are generated randomly. Although all methods mentioned above solve thereconfiguration and optimal DG allocation problem, no statistical data is presented, which isessential for measuring quality of results obtained by the algorithm. Also, in majority of cases,very small networks are considered, which does not permit conclusive assessment of the pro-posed methodologies in terms of their prospective performance when applied to more realisticcases.


Tabl

e2.

2:C

ompa

rativ

eta

ble

ofm

etah

eris

ticop

timiz

atio

nal

gori

thm

sap

plie

dto

optim

alal

loca

tion

ofD

Gan

dD

NR

Alg

orith

mB

enefi

tsW

eakn

ess

Avai

labl

elit

erat

ure

FWA

Fast

orfe

lxib

le.

Abi

lity

tode

alw

ithan

ynu

mbe

rDG

sour

ces

Har

dto

find

optim

also

lutio

ns[1

2]

Evo

lutio

nary

/Gen

etic

Alg

o-ri

thm

(GA

)C

ompu

tatio

nally

effic

ient

,si

mpl

eim

ple-

men

tatio

n,ca

pabl

eof

serc

hing

larg

esp

aces

,ty

pica

llyfin

dsgl

obal

optim

a,ro

bust

For

larg

esc

ale

netw

orks

,co

mpu

ta-

tiona

ltim

em

ight

beco

nsid

erab

ledu

eto

larg

ernu

mbe

rofi

tera

tions

[11]

,[13

],[5

8],[

73]

HSA

Sim

ple

impl

emen

tatio

n,fa

st,s

hort

com

pu-

tatio

nalt

ime

Lik

ely

toge

tsto

ckin

loca

lopt

ima

[58]

,[74

],[7

5]

RG

AR

obus

t,co

mpu

tatio

nale

ffici

ency

Com

para

tivel

ym

ore

com

plex

impl

e-m

enta

tion

[58]

Sim

ulat

edA

nnea

ling

Lik

ely

tofin

dlo

calo

ptim

aan

dab

ility

toes

-ca

pelo

calo

ptim

aH

ighe

rco

mpu

tatio

nal

cost

com

pare

dto

gene

tical

gori

thm

.Q

ualit

yof

re-

sults

rela

tivel

ylo

wer

[47]

17

Chapter 3

Methodology

In this section, the proposed optimization algorithms are outlined. They are referred to asfeasibility-preserving evolutionary optimization (FPEO) and feasibility-preserving simulatedannealing (FPSA). FPEO, a population-based metaheuristic, is selected due to the ability forthis class of methods to perform global search and deal with mixed-integer problem types asDNR-DG allocation. On the other hand, FPSA involves efficient sequential (not population-based) stochastic optimization that adopts some mechanisms from simulated annealing [70],specifically those utilized to escape local optima. Both methods mentioned above are consid-ered appropriate for application to the DNR/DNR-DG allocation problems due to their intrinsicmulti-modal and mixed integer characteristics. Both, FPEO and PFSA algorithms are tailoredto the task at hand, in particular, radial network configuration is maintained throughout theoptimization run. The operation and performance of the methods is demonstrated in Chapterfour.

3.1 Power Distribution System Modeling

In this work, the open distribution system simulator (OpenDSS) is utilized to carry out powerflow simulations of the test distribution systems described in Section IV. For the radial circuits,as the ones presented in this work, OpenDSS exhibits convergence characteristics similar tothose of forward-backward sweep methods [46]. The test distribution systems are modeled aspower delivery elements (power lines) and power conversion elements (loads) in the OpenDSSenvironment. The OpenDSS script driven simulation engine has a Component Object Model(COM) which allows MATLAB to access OpenDSS features as illustrated in Fig 3.1. For moreinformation about the modeling of power distribution systems with OpenDSS, refer to [45].Switching operations require a switch control assigned to every branch in the network; givensectionalizing switches Nsec and tie switches Nts, the total number of switch controls Ns is:

Ns = Nsec +Nts · (P 2i +Q2

i

|V 2i |

) (3.1)

The switch status is expressed by a binary string x of sizeNs to represent the switch operationalstatus; zeros corresponding to an open switch and ones representing a closed switch.

18 CHAPTER 3. METHODOLOGY

Figure 3.1: MATLAB commands access the main simulation engine through the COM inter-face. Some of the commands include switch operations, power flow execution, results extrac-tion, etc.

3.2 Problem FormulationAlthough the power losses in a distribution network can occur in a variety of devices that makeup the system, the losses given in power line segments and transformers often take a majorshare on the overall losses. The power loss in the ith line segment shown in Fig 1.3 connectingbus i with bus i+ 1 is calculated [57] as:

Ploss(i, i+ 1) = Ri · (P 2i +Q2

i

|V 2i |

) (3.2)

Where Pi andQi are the active and reactive power flowing out of bus i respectively. Consid-ering a network comprised ofN line segments, the net power loss P net

loss given in the distributionsystem can be expresses as the summation of all losses given in theN power lines and the lossesgiven in other components (e.g. transformers, switches, fuses, capacitors):

P netloss =

Nbr∑i=1

Ri · (P 2i +Q2

i

|V 2i |

) (3.3)

where Nbr is the total number of line segments in the radial network:

Nbr = Nbus − 1 (3.4)

where: Nbus is the total number of buses in the system.Both DNR and DG interconnection have an influence in the operational characteristics

of the distribution network. An indicator of power loss change in the system after applying

3.2. PROBLEM FORMULATION 19

DNR/or in the case DG allocation is the loss reduction index, defined as the ratio of the powerloss of the base system and the net power loss after DNR/DG interconnection:

∆P ratioloss =

Prec/DGloss

P initloss

(3.5)

where P rec/DGloss and P init

loss are the net power losses after DNR/DG allocation and the initial netpower losses of the system, respectively.

As a result of topology changes and/or DG interconnection, loading of the line segmentsmay vary. Thereupon, the voltage of load buses may be altered. An indicator of the deviationin voltage levels is given as:

∆VD = maxi=1,...Nbus

(V1 − ViV1

)(3.6)

where V1 is the voltage level of the source bus and Vi is the load bus voltage level.By adopting the standard approach of [12], the reconfiguration and DG allocation process

attempts to minimize the sum of the two scalar values corresponding to voltage deviation andloss reduction index:

minF = P ratioloss + ∆VD (3.7)

s.t. PminDG ≤ PDG ≤ Pmax

DG (3.8)

V mini ≤ Vi ≤ V max

i (3.9)

where PDG and Vi are the DG real power supply and bus voltage respectively.Another constrain though out the DNR/DG allocation problems is to maintain the radial

configuration of the network. By adopting notation introduced in Section 3.2, the size of thesearch space can be calculated as C(Ns, Nts), where Ns is the number of switches, Nts is thenumber of tie lines, and C(n, k) is the number of k-combinations of an n-element set. Clearly,the number of combinations grows rapidly with both Nbr and Nts. Let us use a symbol G todenote a graph of the distribution network with buses being the graph vertices and connectionsbeing the edges. The number of radial configurations is equal to the number of the spanningtrees (G), which is considerably smaller thanC(Nbr, Nts). For example, for the 119-bus systemof [76] with 133 switches and 15 tielines, we have C(Ns, Nts) 2.4× 1019 and (G) 4× 1015 (thelatter estimated using the matrix-tree theorem, which establishes that the number of spanningtrees in a graph can be computed in terms of the determinant of a specific matrix derived fromconnected graph or also characterized as a generalization of Cayley’s formula which providesthe number of spanning trees in a complete graph [24]). Further detail about the evaluation ofradial systems is presented in Sections 3.3, 3.4 and, 3.5.


3.3 Feasibility Preserving Evolutionary Optimization(FPEO) Applied to the Distribution NetworkReconfiguration

3.3.1 Algorithm Flow

The proposed algorithm follows the basic steps of the generational evolutionary algorithms.The flow is as follows (P stands for the population):

1. P = initialize();

2. F = evaluate_population(P );

3. Pbest = find_best(P, F );

4. Pnew = select_parents(P );

5. Pnew = crossover(Pnew);

6. Pnew = mutation(Pnew);

7. Pbest = find_best(Pnew, F );

8. P = Pnew; insert Pbest into P ;

9. S = population_diversity(P );

10. adjust_mutation_probability(S);

11. If ∼ termination_condition go to 2; else END.

The algorithm uses binary tournament selection [77] during the parent selection process.It also features elitism and adaptive adjustment of mutation probability based on populationdiversity. Diversity is measured as the average standard deviation of solution components. Thetwo critical (and novel) components of the algorithm are mutation and crossover operators,both designed to maintain radial structure of the network. They are briefly described in thefollowing two subsections.

3.3.2 Feasibility-Preserving Mutation Operator

The mutation operator is supposed to introduce a small random change in the network configu-ration. Here, it is implemented as follows. First, one of the open switches is randomly selectedand closed creating a loop in the network. Then, the loop created this way is identified. Fi-nally, one of the connections on the loop is randomly selected to be opened. As a result, thenetwork is transformed from one radial configuration to another. The mutation operator is alsoexplained in Fig.3.2.

3.3. FEASIBILITY PRESERVING EVOLUTIONARY OPTIMIZATION (FPEO) APPLIEDTO THE DISTRIBUTION NETWORK RECONFIGURATION 21

Figure 3.2: Operation of feasibility-preserving mutation operator: (a) part of the radial net-work with one of the switches randomly selected to be closed and create connection (markedthick dashed line), (b) a loop created by closing the switch (thick solid line), (c) one of the con-nections along the loop randomly selected to be opened, thus retaining the radial configurationof the network..

3.3.3 Feasibility-Preserving Crossover OperatorThe crossover operator, similarly as the mutation one, is designed to maintain radial configura-tion of the network. The first step is a conventional uniform crossover [78], where componentsof the offspring x are randomly selected from one of the two parent individuals. In the secondstep, a repair procedure is launched which works as follows (here, G(x) represents the networkgraph corresponding to the configuration x):

1. If G(x) is a connected graph go to 4; otherwise go to 2;

2. Modify x by closing one of randomly selected open switches;

3. If G(x) is not connected go to 2; otherwise go to 4;

4. If G(x) is a tree, go to 7;

5. Modify x by opening a randomly selected connection;

6. If G(x) is not connected, reverse modification made in Step 5; go to 4;

7. END

In the above algorithm, connectivity of the network graph is first checked. In case the graphis not connected, subsequent switches are closed until connectivity is achieved. At this step,only those switches that do not lead to creating loops can be closed. Otherwise, subsequentconnections are opened until the network graph becomes a tree which corresponds to a radialnetwork configuration. While modifying the individual, the switches that have already beentried out are stored in order to avoid unnecessary repetitions. The flow diagram of the crossoveroperator is shown in Fig.3.3. It should be emphasized that due to the repair procedure, thenetwork modification made by a crossover operation can be quite extensive, especially in theinitial stages of the optimization run when diversity of the population is large. Consequently, alow crossover probability is used (here, 0.2), which is more advantageous as indicated by theinitial experiments.


Figure 3.3: Flow diagram of feasibility-preserving crossover operator.

3.4 Feasibility Preserving Simulated Annealing Applied tothe Distribution Network Reconfiguration

3.4.1 Algorithm StructureThe operation of the proposed feasibility-preserving simulated annealing (FPSA) algorithmhas been explained below. The algorithm processes a single solution by subjecting it to bothdeterministic and stochastic modifications. The conditions for accepting the new candidatesolution are similar to those utilized in simulated annealing [70]. We use the following notation:x(0) is the initial solution; T and T0 are the current and the initial system temperature; r ∈ (0, 1)is a random number drawn with uniform probability distribution; imax is the maximum numberof algorithm iterations. The algorithm flow can be summarized as follows:

1. Set i = 0 (iteration counter);

2. Set Fbest = Fcurrent = F (x(0));

3. Set xbest = x(0);

4. Generate random candidate solution: xrand = random_radial_configuration();

5. If F (xrand) < Fcurrent OR exp((Fcurrent−F (xrand))/T ) > r then set x(i+1) = xrand andFcurrent = F (xrand) and go to 9;

3.4. FEASIBILITY PRESERVING SIMULATED ANNEALING APPLIED TO THEDISTRIBUTION NETWORK RECONFIGURATION 23

6. Perform (deterministic) local_search: xlocal = local_search(x(i));

7. F (xlocal) < Fcurrent then set x(i+1) = xlocal and Fcurrent = F (xlocal) and go to 9;

8. Generate (local) random modification: x(i+1) = random_change(x(i)); Fcurrent =F (x(i+1));

9. If Fcurrent < Fbest then set xbest = x(i+1);Fbest = Fcurrent;T = T0(1− i/imax);

10. Set i = i+ 1;

11. If i < imax then go to 4, else END (return xbest);

It should be noted that the algorithm stores the best solution found due to deterministic andrandom modifications (Fcurrent) as well as the best solution found so far (Fbest). The reasonis that acceptance of a random modification (Step 5) may lead to temporary degradation ofthe solution and further improvement steps (local search of Step 6) has to be carried out withrespect to this solution rather than the overall best solution, so that various regions of the designspace can be explored.

The FPSA algorithm features three major steps:The function random_radial_configuration() produces a random network configuration,which is then compared to the current solution x(i) and accepted under two conditions, (i)it exhibits a better value of the objective function than F (x(i)), or (ii) it is worse but the con-dition exp((Fcurrent − F (xrand))/T ) > r is satisfied for the random number 0 < r ≤ 1. Thisis the main mechanism borrowed from simulated annealing [70] which enables the algorithmto escape from possible local minima. It should be noticed that satisfaction of this condition ismore difficult for poor solutions (because of the term Fcurrent−F (xrand)) and at later stages ofthe optimization run (because the system temperature T is gradually reduced to zero). The mainsearch step is the local search which aims at finding a (local) modification of the current con-figuration x(i), resulting in maximum reduction of the objective function value (which can beconsidered as equivalent of the hill-climbing typically used in simulated annealing schemes).In case the local search is successful, the resulting network configuration is accepted, other-wise, a (local) random change is introduced (function random_change()). This is necessaryto avoid dead ends due to failure of the local search. The overall best solution is updated inStep 10. This step allows for preserving the best solution found so far (which could be lostotherwise due to random changes of Steps 4 and 8).

The flow diagram of the algorithm has been shown in Fig.3.4. The three functions outlinedabove are described in detail in Sections 3.4.2 through 3.4.3.

3.4.2 Function random_radial_configuration

A random radial configuration is generated by starting from an initial configuration correspond-ing to all switches being closed, and sequentially opening randomly selected switches until aradial configuration is obtained. After selecting and opening a switch, the network graph con-nectivity is checked and in case the graph is not connected, the switch is closed, and replacedby another (randomly selected) switch. This is sufficient because the configuration obtainedafter opening Nts switches is radial if and only if the corresponding graph is connected.


3.4.3 Function local_search

Local search is the major search step of the algorithm. It works as follows (F0 is the objectivefunction value corresponding to the current network configuration x(i)):

1. For all open switches sk, k = 1, . . . , Nts:• Close the switch sk;• Identify the loop created by closing the switch;• Find two connections on the loop, adjacent to sk, and the corresponding switches skj ,j = 1, 2;• For j = 1, 2, open the switch sk.j and calculate objective function Fk.j of the resultingradial network x(k.j) ;

2. If mink = 1, . . . , Nts; j = 1, 2 : Fk.j < F0 then return x(k.j) that realizes the minimum;else return x(i);

The local search sequentially modifies the network configuration by closing all open switches,one by one, and opening adjacent connections, looking for the modifications that is the mostbeneficial from the point of view of objective function value reduction. It should be noted thatthe local search preserves radial configuration of the network. Its computational cost is 2Ntsevaluations of the objective function.

3.4.4 Function random_change

Random modification of the network configuration is realized with preserving solution feasi-bility (i.e., radial configuration). Here, only one connection is being changed as follows.

1. Randomly select one of the open switches and close it;

2. Identify the loop created this way;

3. Randomly select one of the switches along the loop and open it. Operation of the proce-dure has been explained in Fig. 3.3.

Remarks

The algorithm presented here is a sequential stochastic procedure, which is quite different fromtypical state-of-the-art solution approaches adopted for solving distribution network reconfig-uration problems in the literature. Vast majority of these methods rely on population-basedmetaheuristics, which is due to multi-modal nature of the problem at hand. The proposedFPSA offers global search capability by using mechanisms borrowed from simulated annealingschemes, yet, the rest of the components are tailored to the DNR task. As indicated in Chapter4, it results in outperforming the benchmark algorithms, especially in terms of robustness ofthe optimization process.

3.4. FEASIBILITY PRESERVING SIMULATED ANNEALING APPLIED TO THEDISTRIBUTION NETWORK RECONFIGURATION 25

Figure 3.4: Flow diagram of the feasibility-preserving simulated annealing (FPSA) algorithm.


3.5 Feasibility-Preserving Evolutionary Search Applied toDNR and Optimal Allocation of Distributed Generation

The optimization approach presented in this section is a generalized evolutionary algorithmas the one applied to the DNR problem described in section 3.3. There are two similaritiesbetween the approach applied to the problem at hand and the DNR problem: (i) The algorithmflow is the same, (ii) both, mutation and crossover operators preserve radial configuration ofthe network. For the DNR/DG allocation problem, two additional variables are introduced: (i)Allocation of the DG sources and (ii) The power magnitude of the DG source.

The problem is constrained threefold: (i) to maintain the radial configuration of the network,and (ii) the powers of the DG sources are limited as in (3.8) and, (iii) meet voltage understandard operational values. Thus, the task is a mixed-integer one with both integer-based andcontinuous constraints.

3.5.1 Data RepresentationThe candidate solutions are represented by structures x containing the following three fields:

• x.L – a binary string with zeros corresponding to open switches (no connection) and onescorresponding to close switches (existing connections);

• x.G – a vector of integers from the set 1, 2, . . . ,Nbus representing allocation of distributedgenerators (the length of the string corresponds to the number of generators Ndg);

• x.P - a vector of floating point numbers of the length Ndg representing the distributedgenerator capacities.

3.5.2 Recombination Operators for Distributed Generator AllocationLocations of distributed generators are encoded using a vector of integers x.G. In this case,straightforward uniform crossover is utilized with a simple repair procedure to ensure that allcomponents of x.G are different. The mutation operator is slightly more complex. It is appliedindependently (with probability pmG) to each component gk of x.G and consists of the followingsteps:

• Find the distances (lengths of the paths) between the bus gk and all other buses;

• Identify a set of buses gn1, . . . , gnp for which the distances to gk are not larger thandmax (a user-defined parameter);

• Randomly select an integer j ∈ 1, . . . , p;

• Replace gk by gnj .

The above procedure describes a local mutation (in our numerical experiments dmax = 1 hasbeen used). A global version of the operator is also used, in which gk is replaced by a randomlyselected bus from the set 1, . . . , Nbus \ x.G. The global mutation is applied with a probabilitypmg/Mg (typically, Mg = 5).

3.5. FEASIBILITY-PRESERVING EVOLUTIONARY SEARCH APPLIED TO DNR ANDOPTIMAL ALLOCATION OF DISTRIBUTED GENERATION 27

3.5.3 Recombination operators for Distributed Generator capacitiesIn case of the vector x.P representing distributed generator capacities, standard floating-pointrecombination operators are utilized. In particular, we use arithmetic crossover of the form

z.P = r x.P + (1− r) y.P (3.10)

The mutation operator works as follows (applied separately to each component pk of thevector z.P ) [77]:

Where:

∆pk =

(pkmax − pk) · (2(r − 0.5))β if r > 0.5

(pkmin − pk) · (2(0.5− r))β otherwise(3.11)

in which r ∈ [0, 1] is a random number; β is larger than 1 in order to increase probabilityof smaller changes; p_k.min and where x and y are parent individuals, z is an offspring, and ris a random vector of numbers between 0 and 1, drawn with uniform probability distribution; denotes component-wise multiplication. pk.max are lower and upper bounds of the kth generatorcapacity.

RemarksIt should be noted that the proposed algorithm simultaneously handles and adjusts networkconfiguration, distributed generation allocation and sizes. Although complexity of the algo-rithm is slightly increased (as compared, to, e.g., sequential handling of particular aspects ofthe problem), the quality of the solution obtained is expected to be better, which has indeedbeen demonstrated in Chapter 4.

29

Chapter 4

Test Results

In order to demostrate the proposed methodology descrived in Chapter 3, the algorithms aretested on the 69-bus network of [79]. This section presents the results of distribution networkreconfiguration based on the FPEO. Section 4.1 reports the performance of the algorithm basedon different control parameters. Chapter 5 reports further results of FPEO, FPSA and FPESon multiple test networks. The results are presented as journal paper manuscript format. Theresults from the three algorithm tests are compared with state of the art methodologies reportedin the literature. The results from the three methods report an outperformance of those methodsreported in the literature.

4.1 Distribution Network Reconfiguration By Means OfFeasibility-Preserving Evolutionary Optimization forPower Loss Reduction

The 69-bus system at hand is treated as balanced, positive sequence component. Table I exhibitsthe initial system conditions obtained from the power flow analysis performed with OpenDSS.For power flow resolutions, the L-N base voltage level for the 69-bus systems is set to be 12.66kV; a tolerance threshold of 0.005 is predetermined.

Table 4.1: Initial condition of the test systems

Test Case Initial Loss(kW)

Min V (p.u) Initial open switches

69 Bus [12] 223.725 0.9094 69, 70, 71, 72, 73

In order to determine the optimum setup of control parameters of the FPSO algorithm, anumber of experiments have been executed using all combinations of the control parameters:

• Population size: Np 10, 20;

• Crossover probability pc 0.1, 0.2, 0.3, 0.4, 0.5;

• Initial mutation probability pm 0.1, 0.2;

• Number of mutation trials 0.1, 0.2;

30 CHAPTER 4. TEST RESULTS

There are 40 combinations altogether and 20 algorithms runs have been performed for eachcombinations and for all three benchmark problems. The number of cost function evaluationshas been set to 500 for the 69-bus system at hand. As it turns out, the best combination isconsistent for all test cases, and it is: Np = 10, pc = 0.1, pm = 0.2, rm = 0.2. Table 4.2indicates six best algorithm setups for all test problems ordered with respect to ascending orderof the standard deviation of the cost function. Note that for almost all best control parametercombinations we have Np = 10 (so that smaller population size is preferred and larger number ofiterations for a given maximum number of function calls), as well as small crossover probability(typically 0.1 or 0.2) and higher mutation rate (0.2).

The proposed algorithm has been executed using the following set setup: Np = 10, pc =0.1, pm = 0.2, rm = 0.2. In order to obtain statistical data, 20 algorithms run have been carriedout. Maximum number of function evaluations have been set to 500, 1,000, and 2,000

Table 4.2: Control parameter setup vs algorithm performance

TestNetwork

Np pc pm rm StandardDeviation

69-bus 10 0.1 0.2 0.2 0.002210 0.3 0.2 0.1 0.003910 0.2 0.2 0.1 0.004510 0.1 0.2 0.1 0.004610 0.5 0.2 0.1 0.005110 0.2 0.1 0.1 0.0067

Table 4.3: Statistics of the optimization results of the 69-bus test network

Test Case N#evals Best

(kW)Average

(kW)Worst(kW)

STD$ Fitness

FPEO 500 98.9299 98.9299 98.9299 0.0022 0.4492FPEO 1000 98.9299 98.9299 98.9299 0 0.4492FPEO 2000 98.9299 98.9299 98.9299 0 0.4492CSA [16] 3000 98.9418 N/A* N/A* N/A* 0.4993GA [60] 900 98.9 418 101.34 104.73 N/A* 0.4993AC [60] 900 99.1225 103.18 110.28 N/A* 0.5001IAICA[60] 900 98.9418 100.57 104.25 N/A* 0.4993# Number of cost function evaluations (= population size times number of iterations)$ Standard deviation of the fitness value over the performed number of algorithm runs∗ Relevant data has not been provided

4.1. DISTRIBUTION NETWORK RECONFIGURATION BY MEANS OFFEASIBILITY-PRESERVING EVOLUTIONARY OPTIMIZATION FOR POWER LOSSREDUCTION 31

Results attained for the 69-bus test network are compared to those presented in [16],and[60]; configurations presented by the specified references are simulated using OpenDSS aspower flow analysis tool. As exhibited by Table 4.3, the power losses are almost identical asthose reported in [16] and [60]. Evidently, as indicated in (3.7), i.e., the objective functionincorporates the power loss and bus voltage index. It is precise to emphasize that repeatabilityof the results is exceptional for the proposed FPEO: in this case the algorithm yields the globallyoptimum results for all runs. Consequently, reliability of FPEO by far exceeds reliability ofother techniques. Switches configuration exhibiting best fitness using the proposed methodare, 14, 55, 61, 69 and 70. Table 4.4 shows the distribution system characteristics of the bestconfiguration obtained by the proposed algorithm and those methods reported in the literature.The results include power losses, switch configuration, fitness value and minimum voltage.

Table 4.4: Distribution system characteristics after reconfiguration

Test Case Initial Loss(kW)

Open Switches Fitness Min voltage(p.u)

FPEO 98.9299 14,55,61,69,70 0.4492 0.943CSA [16] 98.9418 14,57,61,69,70 0.4993 0.9298GA [60] 98.9418 14,57,61,69,70 0.4993 0.951AC [60] 99.1225 12,55,61,69,70 0.5001 0.943

IAICA [60] 98.9418 14,57,61,69,70 0.4993 0.943

Figure 4.1: Flow diagram of the feasibility-preserving simulated annealing (FPSA) algorithm.

It can be observed in the optimization history presented Fig.4.1 that the evolution of theobjective function vs iteration is consistent for all algorithm runs thus confirming its robustnessof the implemented algorithm.

The statistical data for the proposed FPEO algorithm, reported in Table 4.3, also indicatesthat the number of function evaluations necessary to obtain perfect repeatability of results (zeroor close to zero standard deviation) is very low compared to other algorithms. For 69-bus athand, 1000 evaluations. This indicated the robustness of the method but also that it might be


Figure 4.2: Number of objective function evaluation required by FPEO to obtain perfect re-peatability (i.e., optimum solution at every algorithm run) versus feasible search space size.The solid line plot obtained for the test cases considered in the test cases reported in Section 5.The dashed line is a rough estimation for a potential larger problem of 300-400 buses

Figure 4.3: Network voltage profile before and after reconfiguration/optimization

suitable for handling larger systems (a few hundred buses and beyond). Fig. 4.2 shows a plotindicating dependence of the number of objective function evaluations necessary to ensure per-fect repeatability versus (feasible) search space size, obtained for the considere test problem,as well as an estimation for a potential larger system with the size of 300-400 buses and 20to 30 tie lines. Clearly, the required number of function calls without a requirement of perfectrepeatability would be much smaller. The impact on power losses and voltage profile as conse-quence of distribution network reconfiguration are of significant magnitude. Fig. 4.3 and Fig.4.4 report the voltage profile and power losses before and after reconfiguration, respectively.

4.1.1 DiscussionFPEO proves to be a feasible method for solving the DNR problem. For the most part, usinga different number of iterations present feasible results at a low computational cost comparedto the results of state of the art methods. Based on the formulation of (3.7), both, power

4.1. DISTRIBUTION NETWORK RECONFIGURATION BY MEANS OFFEASIBILITY-PRESERVING EVOLUTIONARY OPTIMIZATION FOR POWER LOSSREDUCTION 33

Figure 4.4: Network power losses before and after reconfiguration/optimization

losses as well as voltage levels are globally encompassed, consequently not only improvementsin power loss have been obtained but also an evident improvement in the voltage profile. Itis precise to emphasize on two aspects of FPSO. The first one is robustness. In contrast tomajority of benchmark approaches, FPEO features excellent repeatability of results, namely,globally optimum solutions obtained in the 69-bus systems at hand. Second, the computationalcost of the algorithm is intensely lower than that of the benchmark approaches. For example,the CSA, CGA, and PSO methods of [16] are set to 3,000 function evaluations. FPEO works1,000 maximum function evaluations for the 69-bus system at hand. In other words, FPEOoffers a much faster convergence rate. Comprehensive numerical studies demonstrate that ourtechnique is superior over the state-of-the-art methods reported in the literature both in terms ofrobustness (i.e., it exhibits excellent repeatability of solutions) and computational complexity.Additional experiments confirm consistency of the optimum setup of the control parameters ofthe algorithm for the entire set of considered benchmark problems, as well as good scalabilityproperties with respect to the distribution network size. The future work will be focused onmulti-objective DNR (for both power loss reduction and voltage provide control), constrainedoptimization, as well as handling other types of networks (multi-source, non-radial).

4.1.2 ConclusionIn this work, an evolutionary algorithm preserving radial configuration of the network has beenproposed to solve the distribution network reconfiguration problem in the context of power lossreduction and voltage profile improvement. The algorithm features custom mutation and re-combination operators which allow us to maintain radial structure of the network at all steps ofthe optimization process. This results in a dramatic reduction of the search space size as com-pared to the space of all network configurations. Comprehensive numerical studies demonstratethat our technique is superior over the state-of-the-art methods reported in the literature bothin terms of robustness (i.e., it exhibits excellent repeatability of solutions) and computationalcomplexity. Additional experiments confirm consistency of the optimum setup of the con-trol parameters of the algorithm for the entire set of considered benchmark problems, as wellas good scalability properties with respect to the distribution network size. The future work


will be focused on multi-objective DNR (for both power loss reduction and voltage providecontrol), constrained optimization, as well as handling other types of networks (multi-source,non-radial).

35

Chapter 5

Journal Manuscript Annexations

This chapter present a series of journal article versions that are currently under review 1.Thethree methods presented in Chapter 4 have been tested on three distribution networks. The firstjournal article presents the results obtained by FPEA applied to 33-, 69- and 119-bus networks[76], [79], [80]. The system data and images are reported in Appendix A. The second versionreports the results obtained by FPSA applied to the same distribution networks described above.Finally, the third version reports the results of FPES applied to the 33- and 69-bus test networksfor DNR and optimal allocation of DG.

1Manuscript journals reported and the content of this thesis shall not be reproduced for any purpose unlessnotified and authorized by the authors

XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE

Distribution Network Reconfiguration By Means Of

Feasibility-Preserving Evolutionary Optimization for

Power Loss Reduction

Alberto Landeros Rojas

School of Science and Engineering

Reykjavik University

Reykjavik, Iceland

[email protected]

Slawomir Koziel



Reykjavik, Iceland

[email protected]

Mohamed F.M. Abdelfattah



Reykjavik, Iceland

[email protected]

Abstract—Distribution network reconfiguration (DNR) can

significantly reduce power losses, improve the voltage profile, and

increase the power quality. DNR studies require implementation of

the power flow analysis and complex optimization procedures

capable of handling large combinatorial problems. The size of the

distribution network influences the type of the optimization

method to be applied. In particular, straightforward approaches

can be computationally expensive or even prohibitive whereas

heuristic or meta-heuristic approaches can yield acceptable results

with less computation cost. In this paper, a customized

evolutionary algorithm has been introduced and applied to power

distribution network reconfiguration. The recombination

operators of the algorithm are designed to preserve feasibility of

solutions (here, the radial structure of the network) thus

considerably reducing the size of the search space. Consequently,

an improved repeatability of results as well as lower overall

computational complexity of the optimization process have been

achieved. The figures of interest considered in the optimization

process are power losses and the system voltage profile aggregated

into a scalar cost function. The power flow analysis is carried out

with open distribution system simulator (OpenDSS), a simple and

efficient electrical system simulation tool for electric utility

distribution systems. Our approach is demonstrated using several

network of various sizes (from 33 to 119 nodes) and compared to

state-of-the-art methods from the literature.

Keywords—Distribution network reconfiguration Optimization,

power loss reduction, voltage profile improvement, genetic

algorithm, evolutionary programing.

I. INTRODUCTION

Power losses in distribution systems may be significant and

may negatively affect the finances of utilities [2].

Consequently, it is of interest to study the reduction of losses

through techniques such as distribution network

reconfiguration (DNR). The topology of power distribution

systems is typically radial, whereas transmission systems can

operate in loop or radial configurations [3]. The radial

distribution systems often feature sectionalizing switches and

tie switches mainly used for fault isolation, power supply

recovery and system reconfiguration. These switches allow for

reconfiguring the topology of the network, with the objectives

being reduction of power losses, load balancing, voltage

profile improvement and increasing reliability of the system

[1]. A large number of permutations resulting from all

possible switch configurations make the network

reconfiguration task a complex and non-linear combinatorial

problem, particularly for large systems. Due to the power flow

calculations involved [17], the incurred computational cost

can be considerable.

One of the most relevant objectives in reconfiguration of

power distribution systems is reduction of power losses.

Merlin and Back [4] were the first to propose a method for

distribution system reconfiguration, which was a branch and

bound search with all tie lines initially closed thus creating a

meshed system; subsequent switch opening was continued

until radial configuration was achieved. Similarly, in [6] a

method was proposed where the network was initially meshed,

switches were ranked based on current carrying magnitude,

the top-ranked switch was opened and the power flow

calculation was carried out. The process was repeated until

system was radial. The branch exchange was performed

wherever a loop had been identified. Configuration with lower

power losses was kept. In [5], a generalized approach has been

proposed in which the tie line with the highest voltage

difference is closed and the neighboring branch is opened in

the loop formed, leading to reduction of power losses.

Because DNR problem is, in general, multi-model one,

computational intelligence algorithms are normally more

appropriate [3], [7] - [11], even though they generally do not

converge to the global optima for larger size systems [3].

Thuan Thanh Nguyen, Anh Viet Truong [3], performs DNR

using cuckoo search algorithm on 33, 69 and 119 node

distribution systems and compare to methods presented in

other works [10]-[16]. The power loss values presented for the

33 and 69 node systems are the same as those implementing

continuous genetic algorithm (CGA) and particle swarm

optimization (PSO); the losses are smaller than those applying

fireworks algorithm (FWA) [11], GA, refined GA (RGA),

improved TS (ITS), harmony search algorithm (HSA) [14]

and improved adaptive imperialist competitive algorithm

(IAICA) [15]. For the 119 node system, CSA and CGA

present the same loss level, where FWA [11] and HSA[14]

36

present the lowest. For larger distribution systems (with

hundreds and thousands of buses), DNR can be

computationally expensive and require unacceptable amounts

of computing time. [7] implemented DNR through node-depth

encoding (NDE) which improves the performance of

evolutionary algorithms and power flow algorithms; on

average 27.64% in loss reduction was obtained on 30,880

buses, 5166 switch system.

Power flow analysis is an essential task of DNR for it

allows for evaluating the status of the distribution network

before and after reconfiguration, thus serving as a reference to

determine the effects of network reconfiguration on figures of

interest (e.g., line losses or voltage profile). Due to the multi-

combinatorial nature of DNR, a large number of power flow

analyses is required [18]. The usual radial topology of

distribution networks has driven power flow analysis to adopt

the techniques such as the forward-backward sweep ladder

methods [20] for traditional methods such as decoupled

version of Newton Raphson have shown convergence

problems leading to undesirable solutions [21]. Open DSS is

an efficient and fast simulation tool for distribution network

analysis; it can solve power flow analysis although it evolved

as a harmonic flow analysis tool [19]; it handles not only

radial networks but arbitrarily-meshed, multi-phase networks.

The power flow solving methods implemented by OpenDSS is

based on the “current injection mode” and the “Newton”

making OpenDSS a fast and robust power flow analysis tool

[22].

In this paper, a customized evolutionary algorithm has

been proposed for solving a DNR problem for radial networks.

By enforcing feasibility of solutions, in particular, maintaining

radial network structure at all stages of the process including

recombination operations, considerable reduction of the search

space size has been obtained. This leads to faster convergence

of the optimization process as well as improved repeatability

of the results as compared to the state-of-the-art benchmark

algorithms run against the 33-, 69-, and 119-bus problems.

II. PROBLEM FORMULATION

A. Objective Functions

Network reconfiguration is realized by changing the state

of the switches (open/close). Here, the objective of the change

in the system is to minimize the power losses as well as to

control the voltage index. In more rigorous terms, the

objective is to minimize the expression [18]:

ratio

loss DF P V (1)

In (1), one takes into account the change in losses after the

reconfiguration as well as the deviation of voltage in relation

to the base voltage and aims to minimize the sum of both

parameters. The constraint is to maintain the radial

architecture of the network. The notation used is as follows. ratio

lossP is the ratio of the total power loss given in the

branches before reconfiguration and the branch losses after

reconfiguration, whereas VD is the voltage variation index.

Moreover, we have

init

loss

rec

lossratio

lossP

PP

(2)

where rec

lossP is the branch power losses after reconfiguration

and init

lossP is the initial branch power losses.

Power losses are obtained by adding up the losses in all

active network branches:

Nbr

i

iiiloss

Vi

QPRP

1

22

(3)

where Nbr is the number of active branches; it is given by

1 busbr NN (4)

The voltage at each bus is often preferred to be as close as

possible to the source voltage, therefore ∆VD is calculated

with respect to the voltage level of the source:

1,...,1

max 1bus

iD

i N

VV

V

(5)

B. Power Flow

In this work, the open distribution system simulator

(OpenDSS) is utilized to carry out power flow simulations of

the test distribution systems described in Section IV. For the

radial circuits, as the ones presented in this work, OpenDSS

exhibits convergence characteristics similar to those of

forward-backward sweep methods [19].

The test distribution systems are modeled as power

delivery elements (power lines) and power conversion

elements (loads) in the OpenDSS environment. The OpenDSS

script-driven simulation engine has a Component Object

Model (COM) which allows MATLAB to access OpenDSS

features as illustrated in Fig. 1. Switching operations require a

switch control assigned to every branch in the network; given

sectionalizing switches Nsec and tie switches Nts, the total

number of switch controls Ns is:

tss NNN sec

(6)

The switch status is expressed by a binary string x with

zeros representing an open switch and ones representing a

closed switch.

Fig. 1. MATLAB commands access the main simulation engine through the

COM interface. Some of the commands include switch operations, power flow execution, results extraction, etc.

37

III. OPTIMIZATION METHODOLOGY

In this section, we outline the proposed optimization algorithm. It is referred to as feasibility-preserving evolutionary optimization (FPEO). Similarly as in many works reported in the literature, e.g., [3]-[11], a population-based metaheuristic is selected due to the ability of this class of methods to perform global search. This is necessary because DNR is intrinsically a multi-modal problem. The algorithm is tailored to the task at hand, in particular, a radial network configuration is maintained throughout the optimization run. The operation and performance of the method is demonstrated in Section IV.

A. Representation

The solutions are represented as binary strings with zeros corresponding to open switches (no connection) and ones corresponding to close switches (existing connections). Given Nbr and Nts, the number of possible network configurations is C(Nbr,Nts) which is large and grows quickly with both Nbr and Nts. At the same time, the number of radial configurations,

equal to the number of the spanning trees (G) of the network graph G, is much smaller. For example, for the 119-bus system

of [24] the numbers are 2.41019 and 41015 (the latter estimated using the matrix-tree theorem [26]). Consequently, it is beneficial to maintain feasibility of solutions throughout the entire optimization run. In the proposed algorithm, it is realized by appropriate definition of the recombination and mutation operators (cf. Sections III.C and III.D).

B. Algorithm Flow

The proposed algorithm follows the basic steps of the generational evolutionary algorithms. The flow is as follows (P stands for the population):

1. P = initialize(); 2. F = evaluate_population(P); 3. Pbest = find_best(P,F); 4. Pnew = select_parents(P); 5. Pnew = crossover(Pnew); 6. Pnew = mutation(Pnew); 7. Pbest = find_best(Pnew,F); 8. P = Pnew; insert Pbest into P; 9. S = population_diversity(P); 10. adjust_mutation_probability(S); 11. If ~termination_condition go to 2; else END.

The algorithm uses binary tournament selection [27]. It also features elitism and adaptive adjustment of mutation probability based on population diversity. Diversity is measured as the average standard deviation of solution components. The two critical (and novel) components of the algorithm are mutation and crossover operators, both designed to maintain radial structure of the network. They are briefly described in the following two subsections.

C. Feasibility-Preserving Mutation Operator

The mutation operator is supposed to introduce a small random change in the network configuration. Here, it is implemented as follows. First, one of the open switches is randomly selected and closed creating a loop in the network.

Then, the loop created this way is identified. Finally, one of the connections on the loop is randomly selected to be opened. As a result, the network is transformed from one radial configuration to another. The mutation operator has also been explained in Fig. 2.

D. Feasibility-Preserving Crossover Operator

The crossover operator, similarly as the mutation one, is designed to maintain radial configuration of the network. The first step is a conventional uniform crossover [28], where components of the offspring x are randomly selected from one of the two parent individuals. In the second step, a repair procedure is launched which works as follows (here, G(x) represents the network graph corresponding to the configuration x):

1. If G(x) is a connected graph go to 4; otherwise go to 2; 2. Modify x by closing one of randomly selected open

switches; 3. If G(x) is not connected go to 2; otherwise go to 4; 4. If G(x) is a tree, go to 7; 5. Modify x by opening a randomly selected connection; 6. If G(x) is not connected, reverse modification made in

Step 5; go to 4; 7. END

In the above algorithm, connectivity of the network graph is first checked. In case the graph is not connected, subsequent switches are closed until connectivity is achieved. At this step, only those switches that do not lead to creating loops can be closed. Otherwise, subsequent connections are opened until the network graph becomes a tree which corresponds to a radial network configuration. While modifying the individual, the switches that have already been tried out are stored in order to avoid unnecessary repetitions. The flow diagram of the crossover operator has been shown in Fig. 3.

It should be emphasized that due to the repair procedure, the network modification made by a crossover operation can be quite extensive, especially in the initial stages of the optimization run when diversity of the population is large. Consequently, a low crossover probability is used (here, 0.2), which is more advantageous as indicated by the initial experiments.

(a) (b) (c) Fig. 2. Operation of feasibility-preserving mutation operator: (a) part of the radial network with one of the switches randomly selected to be closed and create connection (marked thick dashed line), (b) a loop created by closing the switch (thick solid line), (c) one of the connections along the loop randomly selected to be opened, thus retaining the radial configuration of the network.

38

Fig. 3. Flow diagram of feasibility-preserving crossover operator (Section III.D).

IV. NUMERICAL RESULTS AND BENCHMARKING

In this section, the FPEO algorithm introduced in Section III has been comprehensively validated using the three standard test networks [22-24] consisting of 33, 69 and 119 buses. The network topologies are briefly discussed in Section IV.A. Numerical results are presented in Section IV.B along with comparison to state-of-the-art methods from the literature.

A. Test Cases

Here, the relevant details of the three verification cases are

given. The systems at hand are treated as balanced, limited to

positive sequence component. Initial conditions of the systems

described in Table I obtained from the power flow method as

described in Section II.B. For power flow purposes, the L-N

base voltage level for the 33 and 69 bus systems is set to be

12.66 kV, while for the 119 bus system, the base voltage level

set is 11kV; a tolerance threshold of 0.005 is set. The initial

system voltage profiles are shown in Fig. 10.

The initial topologies of the considered systems have been

shown in Figs. 4 through 6. Each bus is indexed for easy

identification, the discontinuous lines represent the tie lines

and an each is labeled with the switch number the switch that

operates the tie line.

Based on the fact that the voltage source operates as a

Thevenin equivalent source, the complex impedance set

during the modeling in the OpenDSS environment is the

impedance of the first line connected to the voltage source

bus.

TABLE I INITIAL PARAMETERS OF TEST DISTRIBUTION NETWORKS

Fig. 4. 33-bus distribution system [22].

Fig. 5. 69-bus distribution system [23].

B. Results and Benchmarking

The proposed algorithm has been executed using the

following setup: population size 10, crossover probability 0.2,

mutation probability 0.2. Maximum number of function

evaluations was set to 500, 1000, and 5000 for 33-, 69- and

119-bus systems, respectively, which corresponds to 50, 100,

and 500 algorithm iterations. This setup has been established

based on the initial experiments performed for all three test

cases considered in this work as well as several other

networks.

FPEO was executed 20 times in order to obtain meaningful

statistical data. The results obtained for the 33 bus test

network are compared to those presented in [3], [11] and [14];

configurations presented by the specified references are

Test Case Initial Loss

(kW) Min V (p.u)

Initial open switches

33 Bus [22] 200.745 0.9107 33, 34, 35,36, 37

69 Bus [23] 223.725 0.9094 69, 70, 71, 72, 73

119 Bus [24] 1298.1 0.8667 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133

39

simulated using the power flow method described in Section

II.B. As indicated in Table I, the power losses are slightly

higher than those presented in [3], RGA[14] and GA[14],

however, the fitness function value corresponding to the

proposed optimization method exhibits the lowest value. The

reason is objective aggregation as indicated in (1), i.e., the

objective function value is a composition of the power lossess

and the bus voltage index. It should be emphasized that

repeatability of the results is excellent for the proposed FPEO:

in this case the algorithm returns the globally optimum results

in all runs (i.e., zero standard deviation). This is not the case

for the benchmark methods as shown in Table I. Thus,

reliability of FPEO by fat exceeds reliability of other

techniques.

For the 69-bus system, the results are consistent with those

obtained for the 33-bus network, i.e., globally optimum results

have been obtained in all algorithm runs (cf. Table III).

The result for the 119-bus test network are presented in

Table IV. The configuration obtained is the same as that of

CSA[3], FWA[11] and CGA[3]. As indicated in Table IV, the

statistical data reflects a clear indication that feasibility

preserving evolutionary optimization obtains global optima for

vast majority of runs (19 out 29; the standard deviation is very

low).

Figures 7 through 9 show the optimization history for the

three considered test cases. It can be observed that the

evolution of the objective function (versus iteration index) is

consistent for all algorithm runs thus confirming its

robustness. Figure 10 shows the voltage profiles before and

after optimization.

Fig. 6. 119 Bus test distribution system [24].

TABLE II STATISTICS OF THE OPTIMIZATION RESULTS OF THE 33 BUS TEST SYSTEM

* Relevant data has not been provided

TABLE III STATISTICS OF THE OPTIMIZATION RESULTS OF THE 69 BUS TEST SYSTEM


Method Best (kW) Average

(kW) Worst(kW)

Standard

Deviation

Open Switches Fitness

FPEO 140.3350 140.3350 140.3350 0.0 7,9,14,28,32 0.7607

CSA [3] 139.8476 N/A* N/A* N/A* 7, 9, 14, 32, 37 0.7618

FWA [11] 140.3350 147.02 157.243 5.39 7,9,14,28,32 0.7607

HSA [14] 142.8780 153.82 197.01 11.28 7,10,14,36,37 0.7810

RGA [14] 139.8476 166.51 200.34 13.34 7,9,14,32,37 0.7618

ITS [14] 142.8780 165.1 198.22 12.11 7,9,14,36,37 0.7810

GA [14] 139.8476 167.82 204.68 14.54 7,9,14,32,37 0.7618

Method Best (kW) Average

(kW) Worst(kW)

Standard

Deviation

Open Switches Fitness

FPEO 98.9299 98.9299 98.9299 0.0 14,55,61,69,70 0.4492

CSA [3] 98.9418 N/A* N/A* N/A* 14,57,61,69,70 0.4993

GA [ 25] 98.9418 101.34 104.73 N/A* 14,57,61,69,70 0.4993

AC [ 25] 99.1225 103.18 110.28 N/A* 12,55,61,69,70 0.5001

IAICA [25] 98.9418 100.57 104.25 N/A* 14,57,61,69,70 0.4993

40

TABLE IV STATISTICS OF THE OPTIMIZATION RESULTS OF THE 119 BUS TEST SYSTEM


Fig. 7. Optimization history for 33-bus system. Gray lines indicate objective

function value versus iteration index for 20 algorithm runs; the black line is an

average value. As indicated in Table II, the optimum value is obtained for all but one algorithm runs with the algorithm being virtually converged after

around 35-40 iterations (which corresponds to 350-400 system simulations).



average value. The optimum value has been obtained for all algorithm runs (cf. Table III) with the algorithm being virtually converged after around 50-60

iterations (which corresponds to 500-600 system simulations).



average value. The optimum value has been obtained for most of the algorithm runs (cf. Table IV).

C. Discussion

Feasibility-preserving evolutionary optimization has been demonstrated to be a viable method for solving power distribution system reconfiguration based on the radial nature of the distribution systems. The globally optimum configurations have been found for all considered test cases. The improvement in power losses is significant and although the voltage profile was not directly controlled (cf. (1)), the optimized configurations exhibit significant improvement with this respect. OpenDSS is used in this work to solve power flow, and although power flow method is different than that utilized by the benchmark methods, the numerical results are very close.

There are two features of FPEO that have to be emphasized. The first one is robustness. As opposed to majority of benchmark approaches, FPEO features excellent repeatability of results (globally optimum solutions obtained in all algorithm runs for 33- and 69-bus systems and all but one for 119-bus network). Second, the computational cost of the algorithm is dramatically lower than that of the benchmark approaches. For example, the CSA, CGA, and PSO methods of

Method Best (kW) Average (kW) Worst(kW) Standard Deviation Open Switches Fitness

FPEO 856.8 861.19 865.585

1.9 24, 26, 35, 40, 43, 51, 59, 72, 75,

96, 98, 110, 122, 130, 131 0.73

CSA [3] 856.8 N/A* N/A*

N/A*

24, 26, 35, 40, 43, 51, 59, 72, 75,

96, 98, 110, 122, 130, 131 0.73

FWA [11] 856.8 887.53 942.63

29.58 24, 26, 35, 40, 43, 51, 59, 72, 75,

96, 98, 110, 122, 130, 131 0.73

CGA [3] 856.8 N/A* N/A*

N/A*

24, 26, 35, 40, 43, 51, 59, 72, 75,

96, 98, 110, 122, 130, 131 0.73

PSO [3] 898.6068 N/A* N/A*

N/A* 9, 23, 35, 43, 52, 60, 71, 74, 82,

96, 99, 110, 120, 122, 131 0.7626

MTS [] 884.9 N/A* N/A*

N/A* 23,27,33,40,43,49,52,62,72,74,77,

83,110,126,131 0.7493

41

[3] are set to 3,000 function evaluations for 33- and 69-bus system, and 15,000 evaluations for 119-bus system. FPEO works with 500, 1,000, and 5,000 maximum function evaluations for the three systems, respectively. In other words, FPEO offers a much faster convergence rate.

(a)

(b)

(c)

Fig. 10. Network voltage profile before and after reconfiguration: (a) 33-bus

system, (b) 69-bus system, (c) 119-bus system.

V. CONCLUSION

In the paper, a customized evolutionary algorithm for solving distribution network reconfiguration problem has been presented. It permits reduction of the power losses within the network improvement of the voltage profile. The proposed algorithm features feasibility preserving mutation and recombination operators which allow us to maintain radial structure of the network at all steps of the optimization process. This results in a dramatic reduction of the search space size. As demonstrated through comprehensive numerical validation, our technique is superior over majority of the state-of-the-art methods reported in the literature. In particular, the computational cost of the optimization process is much lower than for most of competitive approaches, and, even more importantly, repeatability of results is excellent, allowing for obtaining a globally optimum solution in almost each and every algorithm run. The future work will be focused on extending the range of applications of the method, among others, multi-objective DNR (for both power loss reduction and voltage provide control), constrained optimization, as well as other types of networks (multi-source, non-radial).

ACKNOWLEDGMENT

This work was supported in part by the National Science Centre of Poland Grant 2014/15/B/ST8/02315 and Mexico´s National Council for Science and Technology-Sustentabilidad Energetica SENER-CONACYT (2016).

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[26] J.M. Harris, J.L. Hirst, and M.J. Mossinghoff, Combinatorics and Graph Theory, 2nd Ed., Springer, New York, 2008.

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Alberto Landeros (S’2016) received the B.Sc.

degree in energy engineering from Polytechnic University of Aguascalientes, Aguascalientes,

Mexico, in 2015. He is currently working toward a

M.Sc. degree in Sustainable Energy Engineering-Power Systems at Reykjavik University, Reykjavik,

Iceland. He has done research in applied

photovoltaics, solar power concentration, wind power, thermal power cycles, fluid dynamics, energy

systems, high voltage conductors, underground

conductors, corona effect and power systems. His current research interests include, engineering optimization, numerical modeling, power systems, power

system stability and control, transient simulations and machine learning. His

employment experience include, Optics Research Center, Enersol Energy Solutions and Green Energy Geothermal.

Slawomir Koziel received the M.Sc. and Ph.D.

degrees in electronic engineering from Gdansk

University of Technology, Poland, in 1995 and 2000,

respectively. He also received the M.Sc. degrees in

theoretical physics and in mathematics, in 2000 and 2002, respectively, as well as the PhD in mathematics

in 2003, from the University of Gdansk, Poland. He is

currently a Professor with the School of Science and Engineering, Reykjavik University, Iceland. His

research interests include CAD and modeling of

microwave and antenna structures, simulation-driven design, surrogate-based optimization, space mapping, circuit theory, analog signal processing,

evolutionary computation and numerical analysis.

Mohamed F. Abdel-Fattah was born in Qualiobia,

Egypt on June 11, 1972. He received his PhD degree in 2006 from Zagazig University, Egypt, after

completing his BSc and MSc degrees, and then

appointed as a PhD lecturer with the department of electrical power and machines engineering. He

joined Aalto University, Finland, as a post-doctoral

researcher and lecturer, with the power systems group at the department of electrical engineering

(2007-2009), and after that he joined the “smart grids

and energy markets” project (2009-2014), for the development of the self-healing capability of the smart distribution networks.

Currently, he is an assistant professor at Reykjavik University, Iceland, with

the school of science and engineering, since 2014. His research activities are mainly related to electrical power systems protection including earth-fault

detection, diagnosis and location in medium and high-voltage networks, self-

extinction property of arcing earth faults, development of the transient-based protection systems, and self-healing for smart-grids.

43

1

Abstract—In this paper, a novel algorithm for power loss

reduction in distribution networks through network

reconfiguration is presented. Our approach is based on sequential

stochastic optimization that utilizes mechanisms adopted from

simulated annealing (to avoid getting stuck in local minima), and

customized network modification procedures that aim at improving

the cost function while maintaining the radial architecture of the

distribution system. The proposed technique, referred to as

feasibility-preserving simulated annealing (FPSA), is

comprehensively validated using three IEEE test cases, 33-, 69- and

119-bus systems. Comparison with the state-of-the-art methods

reported in the literature reveals superiority of FPSA both with

respect to the robustness of the optimization process (excellent

repeatability of solutions), and computational complexity.

Index Terms-- Distribution network reconfiguration, simulated

annealing, feasibility preserving, power loss reduction, radial

networks, voltage profile.

I. INTRODUCTION

HE electric power distribution network is an elemental

component of the power distribution system. It deserves

particular attention for the most significant challenges, both

design- and operational-wise, emerge at this level [1]. One

particular area of interest for power engineers working on

distribution systems is finding feasible solutions for power loss

reduction; power losses in distribution systems can account for up

to 70 percent of the total power losses [2], consequently increasing

the operational costs of the system [3]. Introduction of the smart

grid (SG) concept that aims at supporting the transition into a safe,

efficient and sustainable power system [4], requires the use of

computational intelligence methods to meet the aforementioned

objectives. Distribution network reconfiguration (DNR) has been

shown to be a feasible approach, often involving computational

intelligence algorithms to optimize power delivery by reducing

power losses, balancing loads, increasing power quality and

improve reliability [5]. Conceptually, the DNR process lays upon

This work was supported in part by Mexico´s National Council for Science and Technology-Sustentabilidad Energetica SENER-CONACYT (2016) and

National Science Centre of Poland Grant 2014/15/B/ST8/02315.

S. Koziel is with the School of Science and Engineering, Reykjavik University, 101 Reykjavik, Iceland, and with the Faculty of Electronics,

Telecommunications and Informatics, Gdansk University of Technology, Poland (e-mail: [email protected]). A. Landeros is with the School of Science and Engineering, Reykjavik University, 101 Reykjavik, Iceland, on a scholarship provided by Mexico´s National

Council for Science and Technology (e-mail: [email protected]).

M. F. Abdel-Fattah is with the School of Science and Engineering, Reykjavik University, 101 Reykjavik, Iceland (e-mail: [email protected]). S. Moskwa is with Department of Electrical and Power Engineering, AGH University of Science and Technology, Krakow, Poland (e-mail:

[email protected]).

modifying the topological structure of the distribution network to

achieve desired operational characteristics (e.g. minimum system

power loss, voltage profile improvement, load balance, reliability

levels, etc.). DNR involves two fundamental steps: (i) varying

network topology by changing the status of normally open/closed

switches of the system, and (ii) subsequent execution of power

flow analysis to determine the operational characteristics of the

modified system (e.g. power loss levels, node and line voltage, load

levels in power lines, etc.).

The network topology adopted for power distribution

networks is often radial and its analysis may be a complex task

by itself. Due to a large number of possible network

architectures (increasing exponentially with the size of the

system), as well as multi-modal nature of the problem (i.e., the

presence of many local optima), DNR is a highly complex

combinatorial problem. Its practical solution requires

implementation of efficient optimization algorithms with global

search capability.

There have been various methods for solving a DNR problem

reported in the literature [6]. Heuristic approaches such as trial

and error and branch exchange [7-8] typically exhibit low

computational complexity; however, they can be trapped in

local minima. On the other hand, population-based meta-

heuristic algorithms (e.g., [9]-[14]) have proven to be feasible

when applied to DNR since in most cases these methods are

able to find optimal or near optimal solutions. Their efficiency

can be attributed to various learning strategies built into the

optimization procedures as well as integration of concepts that

enable global search capability, especially when working with

large-scale systems. In [9], DNR is performed on three test

distribution networks using a cuckoo search algorithm,

considering a dimensionless objective function which combines

power loss ratio and voltage deviation index [FWA]. The

comparative study exhibits certain advantages over competitive

methods in terms of lower power losses than those obtained

using a harmony search algorithm (HSA) [11], a fireworks

algorithm (FWA) [12], GA, refined GA (RGA), improved TS

Sequential Stochastic Optimization Algorithm

for Power Loss Reduction in Distribution

Networks

S. Koziel, Senior Member, IEEE, A. Landeros, Student Member, IEEE, M. F. Abdel-Fattah,

and S. Moskwa

T

44

2

(ITS), and an improved adaptive imperialist competitive

algorithm (IAICA) [13]. For the larger scale network, CSA and

CGA achieve identical loss levels, whereas FWA [12] and HAS

[11] exhibit the lowest losses. In [15] authors approach DNR

based on convex relaxation of optimal power flow; the method

proposed introduces a successive branch reduction approach

which is then simplified into a constant complexity algorithm.

The algorithm is characterized by low computational

complexity and it is shown to be globally convergent under very

strict and impractical assumptions of only one switch to be

opened. A version applicable for a general case is also provided

which is essentially a heuristic algorithm based on iterative

application of the aforementioned scheme (without proving

convergence), and demonstrated to provide good results for

selected test cases.

Although the majority of modern DNR solution methods

exploit population-based algorithms, sequential stochastic

methods such as simulated annealing [14] may also be

potentially advantageous. In [16], a feasible approach to DNR

under the presence of distributed generation (DG) has been

discussed by using a hybrid optimization method involving

simulated annealing and the immune algorithm to avoid

solution unfeasibility; DNR is implemented on a small

distribution network consisting on 3 power nodes and 16 load

nodes.

Feasible handling of DNR require utilization of efficient

power flow methods, particularly when applied to large scale

systems and optimization procedures which entail the analysis

of a large number of network configurations. A robust power

flow method is necessary to deal with the unique characteristics

of power distribution networks [17] (e.g. radiality, high R / X

ratios, multiphase unbalanced operation, unbalanced loads and

distributed generation). Available power flow methods include

forward/backward sweep methods [18], modified Newton

methods [19], and others [20]. The Open distribution system

simulator (OpenDSS) is a robust simulation tool for the

operational simulation of distribution networks capable of

performing fast power flow analysis [21].

In this paper, a novel algorithm for distribution network

reconfiguration for power loss reduction and voltage profile

improvement has been proposed. Our approach involves

sequential (not population-based) stochastic optimization that

adopts some mechanisms from simulated annealing [22] as a

way to avoid being stuck in local optima. Furthermore, it

implements the procedures for maintaining the radial network

topology throughout the optimization process. OpenDSS is

used to perform power flow analysis. Comprehensive

numerical studies concerning 33-, 69- and 119- bus systems

reveal superiority of our approach over state-of-the-art

metaheuristic and artificial intelligence algorithms [9, 11-13,

23] in terms of reliability, solution repeatability, as well as

computational complexity.

II. FORMULATION OF DNR PROBLEM FOR POWER LOSS

REDUCTION

A. Power Losses and Voltage Deviation

Although the power losses in a distribution network can

occur in a variety of devices that make up the system, the losses

given in power line segments and transformers may take a

major share of the overall losses. In this work, only the power

losses in the line segments are considered. The power loss in

the ith line segment shown in Fig.1 connecting bus i with bus

i+1 is calculated as in (1), [23]:

2

22

)1,(

i

iiiloss

V

QPRiiP (1)

where Pi and Qi represent the active and reactive power flow

out of bus i, respectively, Ri is the resistance of line segment i,

and Vi is the voltage at the ith bus. The net power losses of the

distribution system, obtained by summing up the losses in all

actively operating lines, is given by:

Nbr

ii

iiiloss

V

QPRP

12

22

(2)

where Nbr is the number of power line segments:

1 busbr NN (3)

where Nbus is the total number of buses in the system.

An indicator of the DNR performance is the loss reduction

index which is defined as the ratio of power loss before and

after reconfiguration:

init

loss

rec

lossratio

lossP

PP (4)

where rec

lossP and init

lossP are the power losses after reconfiguration

and the initial power losses of the system, respectively.

As a consequence of the topology changes, loading of the

line segments may vary. Thereupon, the voltage of load buses

may be altered. An indicator of the deviation in voltage levels

after reconfiguration is given as:

1

1

,..1max

V

VVΔV i

NiD

bus

(5)

where V1 is the voltage level of the source bus and Vi is the ith

bus voltage level.

By adopting the standard approach of [9, 12], the

reconfiguration process attempts to minimize only the loss

reduction index, this, due to the strong correlation between

losses and voltage deviation:

D

rec

loss ΔVΔPF (6)

B. Power Flow Analysis

Due to its advantageous characteristics, OpenDSS is used to

perform power flow analysis on the testing platforms presented

in Section IV. In this work, all line segments in the system are

assigned a sectionalizing switch. Let Nsec, be the number of

sectionalizing switches and Nts the number of tie switches, the

total switch controls Ns can be expressed as:

tss NNN sec (7)

45

3

A binary string x of size Ns is used to represent the switch

operational status; zero corresponding to an open switch and

one corresponding to a closed switch.

Fig. 1. General representation of a line segment connecting two adjoining buses.

III. FEASIBILITY-PRESERVING SIMULATED ANNEALING

The purpose of this section is to provide description of the

proposed optimization framework. It is referred to as

feasibility-preserving simulated annealing (FPSA) because

some of the mechanisms have been adopted from SA [22],

specifically those utilized to enable escaping from local optima.

The algorithm is based on processing of a single agent, unlike

majority of modern methods utilized to solve the DNR problem,

which are typically variations of population-based

metaheuristics. Also, network modifications implemented in

FPSA preserve radial architecture of the system at hand.

A. Solution Representation

As mentioned in Section II.B, the network configuration is

represented using binary strings (denoted as a vector x), in

which zeros correspond to open switches (no connection) and

ones correspond to close switches (connection). Adopting

notation introduced in Section II, the size of the search space

can be calculated as C(Nbr,Nts), where Nbr is the number of

active branches, Nts is the number of tie lines, and C(n,k) is the

number of k-combinations of an n-element set. Clearly, the

number of combinations grows rapidly with both Nbr and Nts.

On the other hand, in this work, only radial network

architectures are of interest. Let us use a symbol G to denote a

graph of the distribution network with buses being the graph

vertices and connections being the edges. The number of radial

configurations is equal to the number of the spanning trees (G),

which is considerably smaller than C(Nbr,Nts). For example, for

the 119-bus system of [24] with 15 tie lines, we have C(Nbr,Nts)

2.41019 and (G) 41015 (the latter estimated using the

matrix-tree theorem [26]).

One of the fundamental features of the optimization

algorithm introduced in this work is that all of its components

(search steps and network modifications) are designed to ensure

radial configurations so as to operate in the reduced space

mentioned in the previous paragraph.

B. Algorithm Structure

The operation of the proposed feasibility-preserving

simulated annealing (FPSA) algorithm has been explained

below. Some of its important components are described in

detail in Section III.C through III.E. The algorithm processes a

single solution by subjecting it to both deterministic and

stochastic modifications. The conditions for accepting the new

candidate solution are similar to those utilized in simulated

annealing [22]. We use the following notation: x(0) is the initial

solution; T and T0 are the current and the initial system

temperature; r (0,1) is a random number drawn with uniform

probability distribution; imax is the maximum number of

algorithm iterations. The algorithm flow can be summarized as

follows:

1. Set i = 0 (iteration counter);

2. Set Fbest = Fcurrent = F(x(0));

3. Set xbest = x(0);

4. Generate random candidate solution:

xrand = random_radial_configuration();

5. If F(xrand) < Fcurrent OR exp((Fcurrent – F(xrand))/T) > r

then set x(i+1) = xrand and Fcurrent = F(xrand) and go to 9;

6. Perform (deterministic) local search:

xlocal = local_search(x(i));

7. If F(xlocal) < Fcurrent then set x(i+1) = xlocal and Fcurrent =

F(xlocal) and go to 9;

8. Generate (local) random modification:

x(i+1) = random_change(x(i)); Fcurrent = F(x(i+1));

9. If Fcurrent < Fbest then set xbest = x(i+1); Fbest = Fcurrent;

10. T = T0(1 – i/imax);

11. Set i = i + 1;

12. If i < imax then go to 4, else END (return xbest);

It should be noted that the algorithm stores the best solution found due to deterministic and random modifications (Fcurrent) as well as the best solution found so far (Fbest). The reason is that acceptance of a random modification (Step 5) may lead to temporary degradation of the solution and further improvement steps (local search of Step 6) has to be carried out with respect to this solution rather than the overall best solution, so that various regions of the design space can be explored.

The FPSA algorithm features three major steps. The function random_radial_configuration() produces a random network configuration, which is then compared to the current solution x(i) and accepted under two conditions: (i) it exhibits a better value of the objective function than F(x(i)), or (ii) it is worse but the condition exp((Fcurrent – F(xrand))/T) > r is satisfied for the random number 0 < r ≤ 1. This is the main mechanism borrowed from simulated annealing [22] which enables the algorithm to escape from possible local minima. It should be noticed that satisfaction of this condition is more difficult for poor solutions (because of the term Fcurrent – F(xrand)) and at later stages of the optimization run (because the system temperature T is gradually reduced to zero). The main search step is the local search which aims at finding a (local) modification of the current configuration x(i), resulting in maximum reduction of the objective function value (which can be considered as equivalent of the hill-climbing typically used in simulated annealing schemes). In case the local search is successful, the resulting network configuration is accepted, otherwise, a (local) random change is introduced (function random_change()). This is necessary to avoid dead ends due to failure of the local search. The overall best solution is updated in Step 9. This step allows for preserving the best solution found so far (which could be lost otherwise due to random changes of Steps 4 and 8).

The flow diagram of the algorithm has been shown in Fig. 2. The three functions outlined above are described in detail in Sections III.C through III.E.

C. Function random_radial_configuration()

A random radial configuration is generated by starting from

an initial configuration corresponding to all switches being

closed, and sequentially opening randomly selected switches

until a radial configuration is obtained. After selecting and

opening a switch, the network graph connectivity is checked

ith

i i+1

46

4

and in case the graph is not connected, the switch is closed, and

replaced by another (randomly selected) switch. This is

sufficient because the configuration obtained after opening Nts

switches is radial if and only if the corresponding graph is

connected.

D. Function local_search()

Local search is the major search step of the algorithm. It

works as follows (F0 is the objective function value

corresponding to the current network configuration x(i)):

1. For all open switches sk, k = 1, …, Nts:

• Close the switch sk;

• Identify the loop created by closing the switch;

• Find two connections on the loop, adjacent to sk, and

the corresponding switches sk.j, j = 1,2;

• For j = 1, 2, open the switch sk.j and calculate objective

function Fk.j of the resulting radial network x(k.j) ;

2. If mink = 1, …, Nts; j = 1, 2 : Fk.j < F0 then return x(k.j)

that realizes the minimum; else return x(i);

The local search sequentially modifies the network

configuration by closing all open switches, one by one, and

opening adjacent connections, looking for the modifications

that is the most beneficial from the point of view of objective

function value reduction. It should be noted that the local search

preserves radial configuration of the network. Its computational

cost is 2Nts evaluations of the objective function.

E. Function random_change()

Random modification of the network configuration is realized

with preserving solution feasibility (i.e., radial configuration).

Here, only one connection is being changed as follows.

1. Randomly select one of the open switches and close it;

2. Identify the loop created this way;

3. Randomly select one of the switches along the loop and

open it.

Operation of the procedure has been explained in Fig. 3.

F. Remarks

The algorithm presented here is a sequential stochastic

procedure, which is quite different from typical state-of-the-art

solution approaches adopted for solving distribution network

reconfiguration problems in the literature. Vast majority of

these methods rely on population-based metaheuristics (e.g.,

[9]-[13]), which is due to multi-modal nature of the problem at

hand. The proposed FPSA offers global search capability by

using mechanisms borrowed from simulated annealing

schemes, yet, the rest of the components are tailored to the DNR

task. As indicated in Section IV, it results in outperforming the

benchmark algorithms, especially in terms of robustness of the

optimization process.

IV. RESULTS AND DISCUSSION

The proposed FPSA algorithm has been applied, for the sake

of validation and performance analysis, to a range of test

networks of various sizes. The testing platform contains three

distribution networks consisting of 33-, 69- and 119-buses [23-

25]; further network characteristics are presented in

Section IV.A. Network modeling is carried out in the OpenDSS

environment and the power flow analysis is executed via the

COM interface; Section IV.B presents numerical results along

with a comparison to state of art methods reported in the

available literature.

A. Testing Platform

Here, relevant characteristics of the three test networks are

presented. The initial power loss, minimum bus voltage and open

switches corresponding to the 33-, 69- and 119-bus systems are

presented in Table I; the voltage profile corresponding to the

three systems is shown in Fig. 6. The L-N base voltage for the

33-and 69-bus systems is set to 12.66 kV, while for the 119-bus

system it is set to 11kV. The three systems at hand are limited to

the positive sequence component; the above characteristics are

necessary for the modeling of the test networks in the OpenDSS

environment. For power flow analysis purposes, the tolerance

threshold is set to 0.005.

The initial topology configuration for the three test

networks has been shown in Fig. 4. The network buses are

indexed, facilitating identification. Tie lines (operated by

normally open switches) are represented by the dashed lines and

each is labeled with the switch number respectively.

Fig. 2. Flow diagram of the feasibility-preserving simulated annealing (FPSA)

algorithm (Section III.B).

47

5

(a) (b) (c) Fig. 3. Feasibility-preserving random configuration modification: (a) part of the

radial network with one of the switches randomly selected to be closed and

create connection (marked thick dashed line), (b) a loop created by closing the switch (thick solid line), (c) one of the connections along the loop randomly

selected to be opened, thus retaining the radial configuration of the network. OpenDSS treats the power source bus as a Thevenin equivalent

source, thereupon, requiring the input of a complex impedance for

the modeling the source, often treated as the complex impedance

of the power lines incident with the source bus.


The FPSA algorithm has been executed on the systems described in Section IV.A. In order to yield meaningful statistical data and for a comprehensive analysis of the FPSA algorithm performance, twenty algorithm runs have been carried out. The experiments have been repeated for various maximum numbers of objective function evaluations as follows:

• 200, 500, 1,000 and 2,000 for the 33-bus network;

• 500, 1,000, 2,000 and 5,000 for the 69-bus network;

• 1,000, 2,000, 5,000 and 10,000 for the 119-bus network.

Figure 5 shows the optimization history corresponding to 33- 69- and 119-bus networks with maximum number of function evaluations of 500, 2,000 and 5,000, respectively. Tables II through IV exhibit the results obtained using FPSA and compared to those reported in the literature [7,11,12,13,22]. The FPSA algorithm run on the 33-bus network consistently yields the global optimum for all 20 algorithm runs (zero standard deviation), which is significantly better than for all the benchmark methods. For the 69-bus network, the results are consistent with those obtained for the 33-bus system; the standard deviation is not zero, nonetheless it is very small. Furthermore, the worst values obtained by FPSA are better than the average results of the competitive algorithms. Finally, for the 119-bus system, the results obtained by FPSA are again better both with respect to the best/average objective function value and the standard deviation than majority of the benchmark methods. It should also be noted that FPSA is competitive even for a relatively small number of evaluations of 2,000.

C. Discussion

Feasibility-preserving simulated annealing optimization has been proven a viable method for solving distribution system reconfiguration applied to networks of radial topology. Optimization of the network configuration results not only in significant reduction of the net power losses but also substantial improvement of the voltage profile index as shown in Fig. 6.

Regarding the algorithm performance, not only FPSA

applied to the 33-bus network yields global optima for all

algorithm runs even if the maximum number of objective

function evaluations is limited to 500. Also, as indicated in Fig.

5(a), it exhibits very fast convergence with the global optimum

reached in about 170 iterations. For the 69-bus network, both

the fitness function and best power loss values obtained are

lower than for the state of the art methods in the comparison

pool. In relation to the 69- and 119-bus networks, global optima

are obtained for all algorithm setups, yet, not for all algorithm

runs. Nonetheless, the average power loss values is only slightly

higher than that of the global optima meaning that the global

optima were found in vast majority of runs.

In this work, OpenDSS has been used as a power flow analysis

tool. Although the results are slightly different—in terms of the

final network configuration—than those obtained with the

benchmark methods, numerical results are very similar. For the

33-bus network, FPSA obtains the same configuration topology

as that obtained in [9] and [11] and, although the power losses are

slightly higher, the fitness function is smaller due to the

aggregation of indices in (1) and the numerical results obtained

with the power flow methodology of choice.

TABLE I

INITIAL PARAMETERS OF TEST DISTRIBUTION NETWORKS

TABLE II STATISTICS OF THE OPTIMIZATION RESULTS: 33 BUS SYSTEM

Method Nevals# Best (kW)

Average

(kW) Worst(kW) STD$

Fitness

Value

FPSA 200 139.8861 139.8976 140.1017 0.0011 0.6968

FPSA 500 139.8861 139.8861 139.8861 0.0000 0.6968

FPSA 1000 139.8861 139.8861 139.8861 0.0000 0.6968

FPSA 2000 139.8861 139.8861 139.8861 0.0000 0.6968

CSA [9] 3000 139.8476 N/A* N/A* N/A* 0.6968

FWA [12] 1000 139.9800 145.6300 155.7500 5.49 0.6968

HSA [11] 2500 142.8780 153.82 197.01 11.28 0.7119

RGA [11] N/A* 139.8861 166.51 200.34 13.34 0.6968

ITS [11] 600 142.8780 165.1 198.22 12.11 0.7119

GA [11] 21000 139.8476 167.82 204.68 14.54 0.6968


TABLE III

STATISTICS OF THE OPTIMIZATION RESULTS: 69 BUS SYSTEM

Method Nevals Best (kW) Average

(kW) Worst(kW) STD$

Fitness

Value

FPSA 500 98.9299 100.1362 108.2560 0.0109 0.4421

FPSA 1000 98.9299 99.6760 102.2336 0.0059 0.4421

FPSA 2000 98.9299 99.8777 99.9276 0.0013 0.4421

FPSA 5000 98.9299 98.9299 98.9299 0.0000 0.4421

CSA [9] 3000 98.9418 N/A* N/A* N/A* 0.4421

GA [ 13] 900 98.9418 101.34 104.73 N/A* 0.4421

AC [ 13] 900 99.1225 103.18 110.28 N/A* 0.5001

IAICA

[13] 900 98.9418 100.57 104.25 N/A* 0.4424


Test Case Initial Loss

(kW) Min V (p.u) Initial open switches

33 Bus [23] 200.745 0.9107 33, 34, 35,36, 37

69 Bus [24] 223.725 0.9094 69, 70, 71, 72, 73

119 Bus [25] 1298.1 0.8667

119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133

48

6

TABLE V

STATISTICS OF THE OPTIMIZATION RESULTS: 119 BUS SYSTEM

Method Nevals Best (kW) Average

(kW) Worst(kW) STD$

Fitness

Value

FPSA 1000 856.8 887.3888 1017.7 0.0362 0.6595

FPSA 2000 856.8 859.2656 872.5508 0.0044 0.6595

FPSA 5000 856.8 857.6013 872.5508 0.0040 0.6595

FPSA 10000 856.8 858.4318 872.5508 0.0047 0.6595

CSA[9] 15000 856.8 N/A* N/A* N/A* 0.6595

FWA [11] 3000 856.8 887.53 942.63 N/A* 0.6595

CGA[9] 15000 856.8 N/A* N/A* N/A* 0.6595

PSO[9] 15000 898.6068 N/A* N/A* N/A* 0.6922

MTS[27] N/A* 884.9 N/A* N/A* N/A* 0.6816

* Relevant data has not been provided.

(a)

(b)

(c)

Fig. 4. The testing platform consists of three test networks: (a) 33-bus, (b) 69-

bus, and (c) 119-bus.

(a)

(b)

(c)

Fig. 5. Optimization history for considered test cases (gray lines indicate

objective function value versus iteration index for 20 algorithm runs; the black line is an average value): (a) 33-, (b) 69-, (c) 119-bus system.

V. CONCLUSION

Feasibility-preserving simulated annealing (FPSA)

algorithm has been developed and applied to solve a

distribution system reconfiguration problem with the objective

being power loss reduction and voltage profile improvement.

FPSA utilizes the mechanisms adopted from simulated

annealing in order to enable global search. Also, it implements

problem-tailored operators for local configuration changes as

well as random alterations that ensure maintenance of radial

configuration at every step of the optimization process. The

distribution system reconfiguration by means of FPSA reduces

net power losses significantly and improves the voltage profile

for the 33-, 69-, and 119-bus distribution networks. The best

results obtained by FPSA are comparable to those obtained by

the state-of-the-art benchmark methods. On the other hand,

FPSA exhibits a much better repeatability of results with global

optima obtained in the majority of algorithm runs.

0 1

2 4 5 6 7 8 9

3

10 11 18 19 20 21 22 23 24 25 26 27

29 30 31 32 33 34 35 36 49 50 51 52 53 54 55 56

66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

116

12 13 14 15 16 17

40 41 42 43 44 45 46 47 48

37 38

58 59 60 61 62 63 64 65

100 101 102 103

81 82 83 84 85 86 87 88

89 90 91

105 106 107 108 109 111 112 114 115 117 118

113119 120 121 122 123

S120

S121

S133S125

S119

S122

S123

S124

S126

S12993 94 95 96 97 98 99

S127

S128

S130S131

S132

110

49

7

(a)

(b)

(c)

Fig. 6. Network voltage profile before and after reconfiguration: (a) 33-bus

system, (b) 69-bus system, (c) 119-bus system.

The results are consistent for all test problems. Furthermore,

FPSA features fast convergence. For example, for a 33-bus

system, globally optimum solutions are obtained for the

maximum number of objective function evaluations of 500,

which is considerably faster than for all benchmark methods

considered. The future work will aim at applying FPSA to

larger scale networks, multi-source networks, constrained

optimization, as well as non-radial networks. An extension of

FPSA for multi-objective DNR (w.r.t. power loss, reliability

and operational costs) will also be developed.

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VII. BIOGRAPHIES

Slawomir Koziel received the M.Sc. and Ph.D.

degrees in electronic engineering from Gdansk

University of Technology, Poland, in 1995 and 2000, respectively. He also received the M.Sc. degrees in

theoretical physics and in mathematics, in 2000 and

2002, respectively, as well as the PhD in mathematics in 2003, from the University of Gdansk, Poland. He is

currently a Professor with the School of Science and

Engineering, Reykjavik University, Iceland. His research interests include CAD and modeling of

microwave and antenna structures, simulation-driven design, surrogate-based

optimization, space mapping, circuit theory, analog signal processing, evolutionary computation and numerical analysis.

Alberto Landeros (S’2016) received the B.Sc. degree

in energy engineering from Polytechnic University of Aguascalientes, Aguascalientes, Mexico, in 2015. He

is currently working toward a M.Sc. degree in

Sustainable Energy Engineering-Power Systems at

Reykjavik University, Reykjavik, Iceland. He has

done research in applied photovoltaics, solar power

concentration, wind power, thermal power cycles, fluid dynamics, energy systems, high voltage

conductors, underground conductors, corona effect

and power systems. His current research interests include, engineering optimization, numerical modeling, power systems, power system stability and

control, transient simulations and machine learning. His employment

experience include, Optics Research Center, Enersol Energy Solutions and Green Energy Geothermal.

Mohamed F. Abdel-Fattah was born in Qualiobia,

Egypt on June 11, 1972. He received his PhD degree in 2006 from Zagazig University, Egypt, after

completing his BSc and MSc degrees, and then

appointed as a PhD lecturer with the department of electrical power and machines engineering. He joined

Aalto University, Finland, as a post-doctoral

researcher and lecturer, with the power systems group

at the department of electrical engineering (2007-

2009), and after that he joined the “smart grids and

energy markets” project (2009-2014), for the development of the self-healing capability of the smart distribution networks.

Currently, he is an assistant professor at Reykjavik University, Iceland, with the school of science and engineering, since 2014. His research activities are mainly

related to electrical power systems protection including earth-fault detection,

diagnosis and location in medium and high-voltage networks, self-extinction property of arcing earth faults, development of the transient-based protection

systems, and self-healing for smart-grids.

Szczepan Moskwa received the M.Sc. the Ph.D. degrees in electrical engineering from AGH University of

Science and Technology, Kraków, Poland, in 2000 and

2007, respectively. Currently, he is Assistant Professor at

Department of Electrical and Power Engineering in UST-

AGH. His areas of interest are power reliability,

renewable power sources and distributed generation as well as smart grids technology.

51

Power Loss Reduction By Means Of Network

Reconfiguration and Distributed Generation Using

Feasibility-Preserving Evolutionary Search

Slawomir Koziel, Alberto Landeros Rojas,

Mohamed F. Abdel-Fattah


Engineering Optimization & Modeling Center

Reykjavik, Iceland

e-mail: koziel, alberto16, [email protected]

Szczepan Moskwa

AGH University of Science and Technology

Department of Electrical and Power Engineering

Kraków, Poland

e-mail: [email protected]

Abstract—The paper proposes a novel algorithm for power loss

reduction through distribution network reconfiguration (DNR)

and optimization-based allocation of distributed generation (DG)

sources. In this work, DNR is solved simultaneously with DG

allocation. The problem at hand is a complex mixed-integer task. A

customized evolutionary algorithm has been developed with

recombination operators preserving a radial structure of the

network, integer-based operators for DG placement, and floating-

point operators for handling their power output capacities.

Comprehensive numerical validation performed on standard IEEE

33- and 69-bus systems indicates that our methodology

outperforms state-of-the-art algorithms available in the literature

in terms of the obtained power loss reduction. Furthermore, it

features good repeatability of results as demonstrated through

statistical analysis of multiple algorithm runs.

Keywords—Distribution network reconfiguration, distributed

generation, evolutionary programming, feasibility preserving,

power loss reduction, voltage profile, radial networks, smart grid.

I. INTRODUCTION

Reduction of power losses and improvement of

voltage profile in power distribution networks has been often

approached by methodologies such as distribution network

reconfiguration (DNR), capacitor bank placement and

Volt/Var control [1]-[2]. Recently, optimal allocation of

distributed generation (DG) has been taken into consideration

as a method for grid support due to its attributions towards

frequency drop stabilization [3], power loss reduction and

voltage profile improvement [4]-[5]. Combination of multiple

methodologies such as distribution reconfiguration and

optimal allocation of DGs have been proposed to increase the

benefits [6]-[11]; however, the practical complexity of

implementing such combination of methodologies increases

significantly.

Radial distribution networks often feature

sectionalizing switches and tie switches mainly used for fault

isolation, power supply recovery and system reconfiguration.

These switches allow for reconfiguring the topology of the

network, with the objectives being reduction of power losses,

load balancing, voltage profile improvement and increasing

reliability of the system. Reference [12] presents a

comprehensive overview of reconfiguration benefits and some

optimization algorithms applied to the DNR problem. A large

search space resulting from all possible switch configurations

makes the network reconfiguration task a complex non-linear

combinatorial problem, with the complexity dramatically

increasing with the number of degrees of freedom, i.e., both

the network size and the number of switches. Due to the

nature of the DNR problem, artificial intelligence and meta-

heuristic optimization techniques are often applied. In [13],

the authors implemented a reconfiguration algorithm for a

realistic distribution network, based on a genetic algorithm

(GA), and taking as objective power loss minimization and

load balancing index. In [14], an optimization algorithm based

on the parasitic behavior of the Cuckoo bird is proposed and

applied to 33-, 69- and 119-bus test networks takings as

objective the active power loss minimization and voltage

profile improvement; the results are then compared with other

methods applied to the same networks. In [15], the authors

propose a method for solving the DNR problem based on the

Hopfield neural network, having as objective power loss

reduction and reliability improvement.

Incentivized by factors such as liberalization of the

energy market, environmental concerns, and technology

development, incorporation of DG to the power systems has

recently been increasing [16]-[17], particularly at the

distribution level [6]. DG interconnections may bring multiple

technical benefits, i.e. active/reactive power loss reduction,

voltage profile improvement, reliability enhancement as well as

overall power delivery efficiency. However, if misallocated, DG

can actually lead to degradation of the network performance (in

particular, an increase of the power losses, and increase of the

voltage levels above the operational standards) and contribute to

instability [7]. Consequently, it is important to consider optimal

sitting and sizing of DG sources. Several optimization methods

have been proposed for solving the optimal DG allocation

problem. In [8], a Particle Swarm Optimization (PSO)

algorithm has been applied to the optimum sitting and sizing of

two DGs on a 13 bus network, however, the basis for selecting

the optimum location of DG is not provided. In [6], the authors

52

determine optimal location of DG based on the Power Loss

Index (PLI) and sizing is performed using the GA taking into

account three objective functions, power loss reduction, voltage

profile improvement and voltage stability index (VSI). In [9],

the authors determine candidate buses for DG allocation based

on the loss sensitivity (LSF) and voltage sensitivity factors

(VSF); the optimum DG size is then determined with a hybrid

PSOGSA optimization algorithm.

Although DNR and optimal allocation of DG can be

performed individually, their benefits can be further enhanced

when performed collectively. In [10], DNR and optimum

sitting and sizing of DG based on the fireworks algorithm

(FWA) under three different loading scenarios has been

proposed. The buses with the lowest VSI are selected as

candidate buses followed by optimal DG sizing determined by

the FWA; when both methods are executed simultaneously,

the losses are reduced by up to 84 percent. In [11], the authors

apply GA to a 16-bus network taking as objective

minimization of power costs; here, all buses are candidate for

DG placement and DG sizes are generated randomly.

Although all methods mentioned above solve the

reconfiguration and optimal DG allocation problem, no

statistical data is presented, which is essential for measuring

quality of results obtained by the algorithm. Also, in majority

of cases, very small networks are considered, which does not

permit conclusive assessment of the proposed methodologies

in terms of their prospective performance when applied to

more realistic cases.

In this paper, a customized evolutionary algorithm has

been proposed for solving the DNR and optimal allocation of

DG, limited to radial networks. The major differences between

conventional evolutionary algorithms and the proposed one

include dedicated data representation and recombination

operators embedding the problem specific knowledge. By

enforcing feasibility of solutions, in particular, maintaining

radial network structure at all stages of the process,

considerable reduction of the search space size has been

obtained. Furthermore, all relevant problem variables

(network configuration, DG allocation and sizes) are

processed simultaneously. This leads to a faster convergence

of the optimization process as well as better quality and

improved repeatability of the results, as demonstrated through

numerical studies carried out for 33- and 69-bus test networks.

Comprehensive benchmarking is also provided indicating

superiority of the proposed technique over the state-of-the-art

algorithms reported in the literature.

II. PROBLEM FORMULATION

This section provides formulation of the distribution

network reconfiguration problem involving allocation of

distributed generation sources (both allocation and sizes).

Formulation of the algorithm proposed to solve this problem

has been provided in Section III.

A. Power Losses and Voltage Deviation

Although power losses in a distribution network can occur

in a variety of devices that make up the system, the losses

given in power line segments and transformers may take a

major share of the overall losses. In this work, only the power

losses in the line segments are considered. The power loss in

the ith line segment shown in Fig.1 connecting bus i with bus

i+1 [23] is calculated as:

2

22

)1,(i

iiiloss

V

QPRiiP (1)

where Pi and Qi represent the active and reactive power flow

out of bus i into i + 1, respectively, Ri is the resistance of line

segment i, and Vi is the voltage at the ith bus. The net power

losses of the distribution system, obtained by summing up the

losses in all actively operating lines is given by:

Nbr

i

lossloss iiPP1

)1,( (2)

where Nbr is the number of power line segments in a radial

network:

1 busbr NN (3)

where Nbus is the total number of buses in the system.

An indicator of DNR and DG interconnection influence is

the loss reduction index, defined as the ratio of the base

system net power loss and the net power loss after DNR and

DG interconnection:

init

loss

DGrec

lossratio

lossP

PP

/

(4)

where DGrec

lossP / and init

lossP are the net power losses after

reconfiguration and DG and the initial net power losses of the

system, respectively.

As a result of topology changes and DG interconnection,

loading of the line segments may vary. Thereupon, the voltage

of load buses may be altered. An indicator of the deviation in

voltage levels is given as:

1

1

,..1max

V

VVΔV i

NiD

bus

(5)

where V1 is the voltage level of the source bus and Vi is the

load bus voltage level.

By adopting the standard approach of [10], the

reconfiguration and DG allocation process attempts to

minimize the sum of the two scalar values corresponding to

voltage deviation and loss reduction index:

D

rec

loss ΔVΔPF min (6)

s.t. maxmin

DGDGDG PPP (7)

maxmin

iii VVV (8)

where PDG and Vi are the DG power real power supply and bus

voltage respectively.

53

The variables of the problem at hand are the

following:

• Positions (open/close) of sectionalizing switches of

the network;

• Allocation of the DG sources;

• Power magnitude of the DG sources.

The problem is constrained twofold: (i) to maintain

the radial configuration of the network, and (ii) the powers of

the DG sources are limited as in (7). Thus, the task is a mixed-

integer one with both integer-based and continuous

constraints.

B. Power Flow Analysis

Here, OpenDSS is used to perform power flow analysis on

the testing platforms presented in Section IV. In this work, all

line segments in the system are assigned a sectionalizing

switch. Let Nsec be the number of sectionalizing switches and

Nts the number of tie switches, the total switch controls Ns can

be expressed as:

tss NNN sec (9)

A binary string x of size Ns is used to represent the switch

operational status; zero corresponding to an open switch and

one corresponding to a closed switch.

III. OPTIMIZATION ALGORITHM

This section provides a detailed description of the

proposed optimization algorithm, referred to as feasibility-

preserving evolutionary search (FPES). Because the problem

at hand is a multi-modal one and of mixed-integer type,

population-based metaheuristics are selected for the core

optimization engine. At the same time, the algorithm

incorporates the problem-specific knowledge, which, in this

case, is utilized to maintain a radial network configuration.

Comprehensive numerical validation of our optimization

framework has been discussed in Section IV.

A. Data Representation

The candidate solutions are represented by structures x

containing the following three fields:

• x.L – a binary string with zeros corresponding to open

switches (no connection) and ones corresponding to

close switches (existing connections);

• x.G – a vector of integers from the set 1, 2, …, Nbus

representing allocation of distributed generators (the

length of the string corresponds to the number of

generators Ndg);

• x.P - a vector of floating point numbers of the length Ndg

representing the distributed generator capacities.

ith

i i+1

Fig. 1. General representation of a line segment connecting two adjoining

buses.

It should be mentioned that the search space according to

the representation selected for network configurations is

redundant in the sense that even if the number of zeros in x.L

is kept equal to Nts (the number of tie lines), majority of the

configurations will not be radial. In particular, the number of

possible network configurations is C(Nbr,Nts) which is large

and grows quickly with both Nbr and Nts. At the same time, the

number of radial configurations, equal to the number of the

spanning trees (G) of the network graph G, is much smaller.

For example, for the 119-bus system of [18] the numbers are

2.41019 and 41015 (the latter estimated using the matrix-tree

theorem [19]). In other words, from the point of view of the

computational complexity of the optimization process as well

as its reliability, it is advantageous to maintain feasibility of

solutions throughout the entire optimization run. In the

proposed algorithm, it is realized by appropriate definition of

the recombination and mutation operators (cf. Section III.C).

B. Algorithm Flow

The core of the proposed optimization approach is a

generational evolutionary algorithm [20]. The algorithm flow

can be summarized as follows (P stands for the population):

1. P = initialize();

2. F = evaluate_population(P);

3. Pbest = find_best(P,F);

4. Pnew = select_parents(P);

5. Pnew = crossover(Pnew);

6. Pnew = mutation(Pnew);

7. Pbest = find_best(Pnew,F);

8. P = Pnew; insert Pbest into P;

9. S = population_diversity(P);

10. adjust_mutation_probability(S);

11. If ~termination_condition go to 2; else END.

In the above algorithm, we use random initialization as

well as binary tournament selection [20]. The algorithm also

implements elitism and adaptive adjustment of mutation

probability based on population diversity. Diversity is measured

as the average standard deviation of solution components.

In the following sections, recombination operators

pertinent to the three components of the solution (L, G, and P)

are described. In case of binary strings encoding network

configuration, the mutation and crossover operators have been

designed to maintain radial structure of the network, which is

critical for the algorithm performance.

C. Recombination Operators for Network Configuration

Both mutation and crossover operators are supposed to

preserve radial architecture of the network which, as argued in

Section III.A, allows for considerable reduction of the search

space.

The mutation operator is introducing small random

changes in the network configuration. It works by executing

the following steps:

• Randomly select one of the open switches and close it;

• Identify the loop created by closing the switch;

• Randomly select one of the connections along the loop

and open a sectionalizing switch on it.

54

The above procedure (see also Fig. 2) transforms one

radial network configuration to another because closing an

arbitrary open switch creates exactly one loop, whereas

opening any switch along this loop necessarily creates a tree

(i.e., radial configuration). The mutation operator is applied

with the initial probability pm, which is then adaptively

adjusted as explained in Section III.B.

The crossover operator is also designed to maintain

radial configuration of the network. The first step is a

conventional uniform crossover [21], where components of

the offspring x.L are randomly selected from one of the two

parent individuals. In the second step, a repair procedure is

launched which works as follows (here, G(x.L) represents the

network graph corresponding to the configuration x.L):

(a) (b) (c)

Fig. 2. Operation of feasibility-preserving mutation operator: (a) part of the

radial network with one of the switches randomly selected to be closed and create connection (marked thick dashed line), (b) a loop created by closing the

switch (thick solid line), (c) one of the connections along the loop randomly

selected to be opened, thus retaining the radial configuration of the network.

Fig. 3. Flow diagram of feasibility-preserving crossover operator (Section III.D).

1. If G(x.L) is a connected graph go to 4; otherwise go to 2;

2. Modify x.L by closing one of randomly selected open

switches;

3. If G(x.L) is not connected go to 2; otherwise go to 4;

4. If G(x.L) is a tree, go to 7;

5. Modify x.L by opening a randomly selected connection;

6. If G(x.L) is not connected, reverse modification made in

Step 5; go to 4;

7. END

The above algorithm features two paths executed

depending on whether a graph of the network obtained upon

uniform crossover is connected or not. In case the graph is not

connected, subsequent switches are closed until connectivity is

achieved. At this step, only those switches that do not lead to

creating loops can be closed. Otherwise, subsequent

connections are opened until the network graph becomes a tree

which corresponds to a radial network configuration. While

modifying an individual, the switches that have already been

tried out are stored and cannot be reused. This is to avoid

unnecessary repetitions. The flow diagram of the crossover

operator is shown in Fig. 3.

An interesting observation is that due to the repair

procedure the crossover operator may be quite disruptive,

especially at the initial stages of the optimization run when

diversity of the population is large. Therefore, it is

recommended to use low crossover probability pc. The initial

experiments indicate that values around 0.2 are provide the

best results for majority of considered test cases.

D. Recombination Operators for Distributed Generator

Sitting

Locations of distributed generators are encoded using a

vector of integers x.G (cf. Section 3.1). In this case,

straightforward uniform crossover is utilized with a simple

repair procedure to ensure that all components of x.G are

different. The mutation operator is slightly more complex. It is

applied independently (with probability pmG) to each

component gk of x.G and consists of the following steps:

• Find the distances (lengths of the paths) between the bus

gk and all other buses;

• Identify a set of buses gn1,…,gnp for which the

distances to gk are not larger than dmax (a user-defined

parameter);

• Randomly select an integer j 1, …, p;

• Replace gk by gnj.

The above procedure describes a local mutation (in our

numerical experiments dmax = 1 has been used). A global

version of the operator is also used, in which gk is replaced by

a randomly selected bus from the set 1, …, Nbus \ x.G (here,

the symbol ‘\‘ denotes set substraction). The global mutation

is applied with a probability pmg/Mg (typically, Mg = 5).

E. Recombination Operators for Distributed Generator

Capacities

In case of the vector x.P representing distributed

generator capacities, standard floating-point recombination

55

operators are utilized. In particular, we use arithmetic

crossover of the form:

. . (1 ) .P P P z r x r y (10)

where x and y are parent individuals, z is an offspring, and r is

a random vector of numbers between 0 and 1, drawn with

uniform probability distribution; denotes component-wise

multiplication.

The mutation operator works as follows (applied

separately to each component pk of the vector z.P) [20]:

.max

.min

2( 0.5) if 0.5

2(0.5 ) otherwise

k k

k

k k

p p r rp

p p r

(12)

where:

k k kp p p (11)

in which r [0,1] is a random number; is larger than 1 in

order to increase probability of smaller changes; pk.min and pk.max

are lower and upper bounds of the kth generator capacity.

F. Remarks

It should be noted that the proposed algorithm

simultaneously handles and adjusts network configuration,

distributed generation allocation and sizes. Although

complexity of the algorithm is slightly increased (as

compared, to, e.g., sequential handling of particular aspects of

the problem), the quality of the solution obtained is expected

to be better, which has indeed been demonstrated in

Section III.

IV. RESULTS AND DISCUSSION

For the sake of validation and performance analysis,

the proposed FPES algorithm has been applied to three

loading levels, 0.5 the nominal level, nominal and 1.6 the

nominal level. The data of the systems is taken from [22] and

[23]. Three DG sources are considered here for comparative

studies with [10] and [24], although the analysis can be

performed with any number of sources. The statistical data

obtained for multiple optimization runs is also provided and

utilized to measure the algorithm robustness.

A. Test Networks and DG Characteristics

Here, relevant characteristics of the two test networks

are presented. The initial power loss, minimum bus voltage

and open switches corresponding to the 33-,69-bus systems

are presented in Table II and Table IV respectively; the

voltage profile corresponding to the two systems is shown in

Fig. 4 and Fig. 5. The L-N base voltage for the 33-, 69-bus

systems is set to 12.66 kV. The two systems at hand are

assumed to be balanced, therefore, single phase representation

is sufficient. Power flow analysis is carried out with

OpenDSS. The analysis tolerance threshold is set to 0.005. DG

is assumed to be solar photovoltaic power sources supplying at

a power factor of 1, consequently, voltage constrain is not

applied in this study. Specifying these characteristics is

necessary for optimization purposes and modeling of the test

networks and DG sources.

OpenDSS treats the power source bus as a Thevenin

equivalent source, thereupon, requiring the input of a complex

impedance for source modeling, often treated as the complex

impedance of the power lines incident with the source bus.


The proposed algorithm has been executed using the

following setup: population size 10, crossover probability 0.2,

and mutation probabilities 0.2, 0.1, and 0.2 for x.L, x.G, and

x.P components of the solution structure (cf. Section III.A). It

should be noted that these values are different from typical

one utilized in evolutionary algorithms (say, 0.5 to 0.9 for

crossover). As already explained in Section 3.4, the dedicated

crossover operator designed for the proposed algorithm is

rather disruptive so that lower crossover probability has to be

used. Suitable values of this and other parameters (population

size, mutation probability) have been obtained through initial

experiments.

The algorithm has been executed on the systems

described in Section IV.A. To yield meaningful statistical data

and for a comprehensive analysis of the algorithm

performance, twenty algorithm runs for 5000 function

evaluations each have been executed.

Tables II and IV gather the results obtained for the

33- and 69-bus test systems under the three different loading

levels. The results are compared to those obtained by recent

state-of-the-art methods reported in the literature [10], [24]. It

can be observed that when the systems are under nominal

loading, FPES yields lower power losses than the benchmark

methods. Also, in most cases, it ensures higher minimum

voltage values, which is mainly due to aggregation of the

power loss index and the voltage deviation into the objective

function, cf. (1). On the other hand, for both networks under

nominal loading, the DG size is significantly higher,

consequently increasing the penetration of the DG sources.

Both the 33- and 69-bus networks under light and heavy

loading exhibit consistent results with those obtained under

nominal loading, except for the 69-bus network under heavy

loading. Here, the proposed methodology yields power losses

slightly higher than those obtained by the benchmark

techniques.

TABLE I. STATISTICAL ANALYSIS OF THE FPES ALGORITHM

PERFORMANCE ON THE 33-BUS NETWORK

Scenario Min

(kW)*

Av

(kW)*

Worst

(kW)*

STD

(kW)*

Light

(0.5) 12.467 13.272 13.966 0.3936

Nominal

(1.0) 51.759 55.209 57.179 1.7414

Heavy (1.6) 138.55 147.60 154.71 5.805

56

* The values obtained for 20 independent algorithm runs.

TABLE II. COMPARATIVE ANALYSIS OF ALGORITHM PERFORMANCE ON THE 33-BUS NETWORK

Scenario Characteristic FPES FWA[10] GA[24] HSA[24] RGA[24]

Base Case

Switches Open

Ploss (KW)

Vmin in p.u (bus no.)

33,34,35,36,37

202.2

0.910 (18)

33,34,35,36,37

202.67

0.9131 (18)

33,34,35,36,37

202.67

0.9131(N/A*)

33,34,35,36,37

202.67

0.9131 (N/A*)

33,34,35,36,37

202.67

0.9131 (N/A*)

Only reconfiguration

Switches Open

Ploss (KW)

%Loss reduction


7,9,14,28,32

140.3350

30.76

0.9383 (32)

7,9,14,28,32

139.98

30.93

0.9413 (32)

9,28,33,34,36

141.6

30.15

0.931(N/A*)

7,9,14,32,37

138.06

31.88

0.9342 (N/A*)

7,14,9,32,37

139.46

31.2

0.9315 (N/A*)

Simultaneous reconfiguration and DG

(0.5)

Switches Open

DGtotal size in KW (bus no.)

Ploss (KW)

%Loss reduction


10,28,31,33,34

629.8 (24)

471.6 (5)

387.1 (17)

12.467

73.3

0.9875 (14)

7,10,14,28,32

258.6 (32)

321.8 (29)

280.3 (18)

16.22

65.53

0.9862 (14)

N/A*

7,11,14,27,32

195.4 (32)

419.5 (31)

274.9 (33)

17.78

62.22

0.9859 (N/A*)

N/A*


(Nominal)

Switches Open


Ploss (KW)

%Loss reduction


11,28,31,33,34

919.7 (25)

765.4 (17)

1314 (24)

51.75

74.3

0.9763(14)

7,11,14,28,32

536.7 (32)

615.8 (29)

531.5 (18)

67.11

66.89

0.9713 (14)

7,10,28,32,34

1963.3 (N/A*)

N/A

N/A

75.13

62.92

0.9766

7,14,10,32,28

525.8 (32)

558.6 (31)

584.0 (33)

73.05

63.95

0.97

7,9,12,27,32

1774 (N/A*)

74.32

63.33

0.9691


(1.6)

Switches Open


Ploss (KW)

%Loss reduction


11,28,31,33,34

1392.9(16)

1907.9 (28)

1629.2 (6)

138.55

76.6

0.9661(32)

7,11,14,28,32

959.0 (32)

1190.1 (29)

1020.6 (18)

172.97

69.93

0.9554 (14)

N/A*

7,10,14,28,32

572.44 (32)

1254.8 (31)

925.7 (33)

194.22

66.23

0.9516 (N/A*)

N/A*


TABLE III. STATISTICAL ANALYSIS OF THE FPES

ALGORITHM PERFORMANCE ON THE 69-BUS NETWORK

* The values obtained for 20 independent algorithm runs.

Tables I and III provide the statistical analysis results

corresponding to the 33- and 69-bus networks, respectively.

Good repeatability of results is reflected by low values of

standard deviation (STD) across various algorithm runs.

Fig. 4 (d)-(f) and Fig. 5 (d)-(f) show the power loss profile

corresponding to both, the 33- and 69-bus networks.

Simultaneous execution of DNR and optimal allocation of DG

sources clearly permits improvement of the power loss levels

beyond those exhibited by the base case and reconfiguration

scenarios. As reported in Fig. 4 and Fig. 5, the voltage profile

is also improved significantly relative to the two scenarios

mentioned above.

C. Discussion

Feasibility preserving evolutionary search has been

proven a viable tool for solving the DNR and optimal

allocation of DG sources. OpenDSS is used in this work to

solve power flow, and although power flow method is

different than that utilized by the benchmark methods, the

numerical results are very close, making the comparative

study meaningful and conclusive. The results obtained in [10]

and [24] were re-simulated manually for the purpose of

benchmarking and the results have been corroborated.

Scenario Min

(kW)*

Av

(kW)*

Worst

(kW)*

STD

(kW)*

Light

(0.5) 8.621 9.100 9.659 0.3649

Nominal

(1.0) 35.53 36.312 39.648 1.087

Heavy

(1.6) 107.16 110.655 115.1458 2.075

57

TABLE IV. COMPARATIVE ANALYSIS OF ALGORITHM PERFORMANCE ON THE 69-BUS NETWORK


There are three features of FPES that should be be

highlighted. The first one is robustness confirmed by excellent

repeatability of results for all loading scenarios considered.

The second feature is high performance. In all but one of the

loading scenarios, FPES outperforms state-of-the-art methods

reported in the literature. The last feature is flexibility. The

FPES is capable to perform DNR and optimal allocation of

any number of DG sources. The particular setup of three

sources was selected in order to allow benchmarking of the

proposed method.

V. CONCLUSION

In the paper, feasibility preserving evolutionary search

(FPES) has been developed and applied to simultaneous

solving of a DNR problem and optimal allocation of DG

sources. The objective is minimization of power losses and

improvement of the voltage profile. The methodology has

been validated using two mainstream test networks consisting

of 33 and 69 buses. The proposed algorithm features a unique

combination of dedicated recombination and mutation

operators which, among others, allows to maintain a radial

structure of the network at all steps of the optimization process

and efficiently sit and size the DG sources. As demonstrated

through comprehensive numerical validation, under all but one

loading scenario (corresponding to the 69-bus network), the

proposed technique is superior over state-of-the-art methods

reported in the literature. Furthermore, a statistical analysis

carried out for twenty independent algorithm runs has been

presented. It confirms excellent repeatability of results

(implying algorithm robustness) for all three loading scenarios

considered. The future work will be focused on extending the

range of applications of the method, among others, solving

multi-objective tasks (e.g. switching cost reduction, optimal

sitting of automated switches) and validation for other types of

networks (multi-source, meshed/weakly meshed).

Scenario Characteristic FPES FWA[10] GA[24] HAS[24] RGA[24]

Base Case

Switches Open

Ploss (KW)


69,70,71,72,73

223.8

0.9094 (65)

69,70,71,72,73

224.96

0.9092 (65)

69,70,71,72,73

225

0.9092 (N/A*)

69,70,71,72,73

225

0.9092 (N/A*)

69,70,71,72,73

225

0.9092 (N/A*)

Only reconfiguration

Switches Open

Ploss (KW)

%Loss reduction


14,55,61,69,70

98.92

56

0.9430 (61)

14,56,61,69,70

98.59

56.17

0.9413 (61)

13,18,56,61,69

99.35

55.8

0.9428 (N/A*)

13,18,56,61,69

99.35

55.8

0.9428 (N/A*)

13,18,56,61,69

99.35

55.8

0.9428 (N/A*)

Simultaneous reconfiguration and

DG (0.5)

Switches Open


Ploss (KW)

%Loss reduction


14,55,63,69,70

286.6 (24)

761.9 (61)

256.6 (11)

8.62

83.1

0.9569 (65)

13,56,63,69,70

571.5 (65)

155.5 (62)

121.9 (64)

9.58

81.43

0.9905 (61)

N/A*

10,14,16,56,62

314.3 (62)

348.1 (61)

339.7 (64)

11.07

78.55

0.9860 (N/A)

N/A*


DG

(Nominal)

Switches Open


Ploss (KW)

%Loss reduction


14,55,61,69,70

487.8 (64)

537.3 (11)

1540.2 (61)

35.53

84.1

0.9774 (61)

13,55,63,69,70

1127.2 (61)

275.0 (62)

415.9 (65)

39.25

82.55

0.9796 (61)

10,15,45,55,62

2029.2 (N/A*)

46.5

73.38

0.9727 (N/A*)

13,17,58,61,69

1871.8

40.3

82.08

0.9736 (N/A*)

10,14,1655,62

2065.4 (N/A*)

44.23

80.32

0.9742 (N/A*)


DG (1.6)

Switches Open


Ploss (KW)

%Loss reduction


14,55,61,69,70

490.3 (64)

1662.7 (61)

538.0 (11)

107.1659

83.42

0.9437 (61)

13,57,63,69,70

1817.6 (61)

509.5 (62)

634.2 (65)

102.97

84.21

0.9685 (61)

N/A*

10,13,18,58,61

1593.5 (61)

821.9 (60)

967.4 (62)

104.67

83.96

0.9736

N/A*

58

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4. Voltage and power loss profile corresponding to the 33-bus network

under the three loading scenarios. (a), (b) and, (c) report the voltage profile of light, nominal and heavy loading respectively. (d), (e) and, (f) report the line

loss profile of light, nominal and heavy loading scenarios respectively.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 5. Voltage and power loss profile corresponding to the 69-bus network

under the three loading scenarios. (a), (b) and, (c) report the voltage profile of

light, nominal and heavy loading respectively. (d), (e) and, (f) report the line

loss profile of light, nominal and heavy loading scenarios respectively.

59

ACKNOWLEDGMENT

This work was supported in part by the National Science Centre of Poland Grant 2014/15/B/ST8/02315, and Mexico´s National Council for Science and Technology-Sustentabilidad Energetica SENER-CONACYT (2016).

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69

Appendix A

Distribution Test Networks Information

Figure A.1: 33-bus test distribution network

Table A.1: 33-bus distribution test network [80].

Lines No From bus To bus Resistance (Ω) Impedance (Ω) Load (kW) Load (kvar)1 0 1 0.0922 0.047 100 602 1 2 0.493 0.2511 90 403 2 3 0.366 0.1864 120 804 3 4 0.3811 0.1941 60 305 4 5 0.819 0.707 60 206 5 6 0.1872 0.6188 200 1007 6 7 0.7114 0.2351 200 1008 7 8 1.03 0.74 60 209 8 9 1.044 0.74 60 20

10 9 10 0.1966 0.065 45 3011 10 11 0.3744 0.1238 60 3512 11 12 1.468 1.155 60 3513 12 13 0.5416 0.7129 120 8014 13 14 0.591 0.526 60 1015 14 15 0.7463 0.545 60 2016 15 16 1.289 1.721 60 2017 16 17 0.732 0.574 90 4018 1 18 0.164 0.1565 90 4019 18 19 1.5042 1.3554 90 40

70

20 19 20 0.4095 0.4784 90 4021 20 21 0.7089 0.9373 90 4022 2 22 0.4512 0.3083 90 5023 22 23 0.898 0.7091 420 20024 23 24 0.896 0.7011 420 20025 5 25 0.203 0.1034 60 2526 25 26 0.2842 0.1447 60 2527 26 27 1.059 0.9337 60 2028 27 28 0.8042 0.7006 120 7029 28 29 0.5075 0.2585 200 60030 29 30 0.9744 0.963 150 7031 30 31 0.3105 0.3619 210 10032 31 32 0.341 0.5302 60 40

TIE LINES33 7 20 2 234 8 14 2 235 11 21 2 236 17 32 0.5 0.537 24 28 0.5 0.5

71


Table A.2: 69-bus distribution test network[79]

Lines No From bus To bus Resistance(Ω) Impedance (Ω) Load kW Load kvar1 1 2 0.0005 0.0012 0 02 2 3 0.0005 0.0012 0 03 3 4 0.0015 0.0036 0 04 4 5 0.0251 0.0294 0 05 5 6 0.366 0.1864 2.634 2.166 6 7 0.3811 0.1941 40.365 29.9467 7 8 0.0922 0.047 74.661 53.438 8 9 0.0493 0.0251 30 21.6249 9 10 0.819 0.2707 27.999 19.998

10 10 11 0.1872 0.0619 145.5 103.82711 11 12 0.7114 0.2351 145.5 103.82712 12 13 1.03 0.34 8.13 5.46313 13 14 1.044 0.345 8.13 5.46314 14 15 1.058 0.3496 0 015 15 16 0.1966 0.065 45.528 30.59416 16 17 0.3744 0.1238 49.5 35.32517 17 18 0.0047 0.0016 49.5 35.32518 18 19 0.3276 0.1083 0 019 19 20 0.2106 0.0696 0.948 0.63620 20 21 0.3416 0.1129 113.949 81.321 21 22 0.014 0.0046 5.286 3.55222 22 23 0.1591 0.0526 0 023 23 24 0.3463 0.1145 28.17 20.0124 24 25 0.7488 0.2475 0 025 25 26 0.3089 0.1021 14.001 9.9926 26 27 0.1732 0.0572 14.001 9.9927 3 28 0.0044 0.0108 26.001 18.43528 28 29 0.64 0.1565 26.001 18.55529 29 30 0.3978 0.1315 0 030 30 31 0.0702 0.0232 0 031 31 32 0.351 0.116 0 0

72

32 32 33 0.839 0.2816 13.746 9.7833 33 34 1.708 0.5646 19.503 13.64734 34 35 1.474 0.4873 5.76 3.8735 3 36 0.0044 0.0108 26.001 18.55536 36 37 0.064 0.1565 26.001 18.55537 37 38 0.1053 0.123 0 038 38 39 0.0304 0.0355 24 17.12739 39 40 0.0018 0.0021 24 17.12740 40 41 0.7283 0.8509 1.176 0.97541 41 42 0.31 0.3623 0 042 42 43 0.041 0.0478 6 4.28143 43 44 0.0092 0.0116 0 044 44 45 0.1089 0.1373 9.228 26.36145 45 46 0.0009 0.0012 9.228 26.36146 4 47 0.0034 0.0084 0 047 47 48 0.0851 0.2083 79.05 56.448 48 49 0.2898 0.7091 84.678 274.47649 49 50 0.0822 0.2011 384.678 274.47650 8 51 0.0928 0.0473 40.536 28.32651 51 52 0.3319 0.114 3.606 2.68252 9 53 0.174 0.0886 4.347 3.48653 53 54 0.203 0.1034 26.361 18.96654 54 55 0.2842 0.1447 24 17.12455 55 56 0.2813 0.1433 0 056 56 57 1.59 0.5337 0 057 57 58 0.7837 0.263 0 058 58 59 0.3042 0.1006 2.001 72.07559 59 60 0.3861 0.1172 0 060 60 61 0.5075 0.2585 1244.001 887.7361 61 62 0.0974 0.0496 32.001 22.83662 62 63 0.145 0.0738 0 063 63 64 0.7105 0.3619 227.01 161.61964 64 65 1.041 0.5302 59.01 41.73611 11 66 0.2012 0.0611 18 12.84666 66 67 0.0047 0.0014 18 12.84612 12 68 0.7394 0.2444 27.999 19.9868 68 69 0.0047 0.0016 27.999 19.98

TIE LINES69 11 43 0.5 0.570 13 21 0.5 0.571 15 46 1 172 50 59 2 273 65 27 1 1

73


Table A.3: 119-bus distribution test network [76]

Lines No From bus To bus Resistance(Ω) Impedance (Ω) Load kW Load kvar1 0 1 0 0 0 02 1 2 0.036 0.01296 133.84 101.143 2 3 0.033 0.01188 16.214 11.2924 2 4 0.045 0.0162 34.315 21.8455 4 5 0.015 0.054 73.016 63.6026 5 6 0.015 0.054 144.2 68.6047 6 7 0.015 0.0125 104.47 61.7258 7 8 0.018 0.014 28.547 11.5039 8 9 0.021 0.063 87.56 51.073

10 2 10 0.166 0.1344 198.2 106.7711 10 11 0.112 0.0789 146.8 75.99512 11 12 0.187 313 26.04 18.68713 12 13 0.142 0.1512 52.1 23.2214 13 14 0.18 0.118 141.9 117.515 14 15 0.15 0.045 21.87 28.7916 15 16 0.16 0.18 33.37 26.4517 16 17 0.157 0.171 32.43 25.2318 11 18 0.218 0.285 20.234 11.906

74

19 18 19 0.118 0.185 156.94 78.52320 19 20 0.16 0.196 546.29 351.421 20 21 0.12 0.189 180.31 164.222 21 22 0.12 0.0789 93.167 54.59423 22 23 1.41 0.723 85.18 39.6524 23 24 0.293 0.1348 168.1 95.17825 24 25 0.133 0.104 125.11 150.2226 25 26 0.178 0.134 16.03 24.6227 26 27 0.178 0.134 26.03 24.6228 4 29 0.015 0.0296 594.56 522.6229 29 30 0.012 0.0276 120.62 59.11730 30 31 0.12 0.2766 102.38 99.55431 31 32 0.21 0.243 513.4 318.532 32 33 0.12 0.054 475.25 456.1433 33 34 0.178 0.234 151.43 136.7934 34 35 0.178 0.234 205.38 83.30235 35 36 0.154 0.162 131.6 93.08236 31 37 0.187 0.261 448.4 369.7937 37 38 0.133 0.099 440.52 321.6438 30 40 0.33 0.194 112.54 55.13439 40 41 0.31 0.194 53.963 38.99840 41 42 0.13 0.194 393.05 342.641 42 43 0.28 0.15 326.74 278.5642 43 44 1.18 0.85 536.26 240.2443 44 45 0.42 0.2436 76.247 66.56244 45 46 0.27 0.0972 53.52 39.7645 46 47 0.339 0.1221 40.328 31.96446 47 48 0.27 0.1779 39.653 20.75847 36 49 0.21 0.1383 66.195 42.36148 49 50 0.12 0.0789 73.904 51.65349 50 51 0.15 0.0987 114.77 57.96550 51 52 0.15 0.0987 918.37 1205.151 52 53 0.24 0.1581 210.3 146.6652 53 54 0.12 0.0789 66.68 56.60853 54 55 0.405 0.1458 42.207 40.18454 54 56 0.405 0.1458 433.74 283.4155 30 58 0.391 0.141 62.1 26.8656 58 59 0.406 0.1461 92.46 88.3857 59 60 0.406 0.1461 85.188 55.43658 60 61 0.706 0.5461 345.3 332.459 61 62 0.338 0.1218 22.5 16.8360 62 63 0.338 0.1218 80.551 49.15661 63 64 0.207 0.0747 95.86 90.75862 64 65 0.247 0.8922 62.92 47.763 1 66 0.028 0.0418 478.8 463.7464 66 67 0.117 0.2016 120.94 52.00665 67 68 0.255 0.0918 139.11 100.3466 68 69 0.21 0.0759 391.78 193.5

75

67 69 70 0.383 0.138 27.741 26.71368 70 71 0.504 0.3303 52.814 25.25769 71 72 0.406 0.1461 66.89 38.71370 72 73 0.962 0.761 467.5 395.1471 73 74 0.165 0.06 594.85 239.7472 74 75 0.303 0.1092 132.5 84.36373 75 76 0.303 0.1092 52.699 22.48274 76 77 0.206 0.144 869.79 614.77575 77 78 0.233 0.084 31.349 29.81776 78 79 0.591 0.1773 192.39 122.4377 79 80 0.126 0.0453 65.75 45.3778 67 81 0.559 0.3687 238.15 223.2279 81 82 0.186 0.1227 294.55 162.4780 82 83 0.186 0.1227 485.57 437.9281 83 84 0.26 0.139 243.53 183.0382 84 85 0.154 0.148 243.53 183.0383 85 86 0.23 0.128 134.25 119.2984 86 87 0.251 0.106 22.71 27.9685 87 88 0.18 0.148 49.513 26.51586 82 89 0.16 0.182 383.78 257.1687 89 90 0.2 0.23 49.64 20.688 90 91 0.16 0.393 22.473 11.80689 68 93 0.669 0.2412 62.93 42.9690 93 94 0.266 0.1227 30.67 34.9391 94 95 0.266 0.1227 62.33 66.7992 95 96 0.266 0.1227 114.57 81.74893 96 97 0.266 0.1777 81.292 66.32694 97 98 0.233 0.115 31.733 15.9695 98 99 0.004 0.138 33.32 60.4896 95 100 0.001 0.18 531.28 224.8597 100 101 0.001 0.18 507.03 367.4298 101 102 0.1866 0.122 26.39 11.799 102 103 0.0746 0.318 45.99 30.392

100 1 105 0.0625 0.0265 100.66 47.372101 105 106 0.1501 0.234 456.48 350.3102 106 107 0.1347 0.0888 522.56 449.29103 107 108 0.2307 0.1203 408.43 168.46104 108 109 0.447 0.1608 141.48 134.25105 109 110 0.1632 0.0588 104.43 66.024106 110 111 0.33 0.099 96.793 83.647107 111 112 0.156 0.0561 493.92 419.34108 112 113 0.3819 0.1374 225.38 135.88109 113 114 0.1626 0.0585 509.21 387.21110 114 115 0.3819 0.1374 188.5 173.46111 115 116 0.2445 0.0879 918.03 898.55112 115 117 0.2088 0.0753 305.08 215.37113 117 118 2301 0.0828 54.38 40.97114 105 119 0.6102 0.2196 211.14 192.9

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115 119 120 0.1866 0.127 67.009 53.336116 120 121 0.3732 0.246 162.07 90.321117 121 122 0.405 0.367 48.785 29.156118 122 123 0.489 0.438 33.9 18.98

TIE LINES119 48 27 0.5258 0.2925120 17 27 0.5258 0.2916121 8 24 0.4272 0.1539122 56 45 0.48 0.1728123 65 56 0.36 0.1296124 38 65 0.57 0.572125 9 42 0.53 0.3348126 61 100 0.3957 0.1425127 76 95 0.68 0.648128 91 78 0.4062 0.1464129 103 80 0.4626 0.1674130 113 86 0.651 0.234131 110 89 0.8125 0.2925132 115 123 0.7089 0.2553

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