kom.aau.dkkom.aau.dk/project/smartcool/public_repository/... · Contents Preface IX Abstract XI Synopsis XIII 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . .

Morten Juelsgaard

Utilizing Distributed Resources in Smart GridsA Coordination Approach

Utilizing Distributed Resources in Smart GridsPhD thesis

June 2014

Copyright 2011-2014 c© Morten Juelsgaard

Title:Utilizing Distributed Resources in Smart Grids - A Coordination Approach

Author:Morten Juelsgaard

Supervisors:Prof. Rafael Wisniewski, Assoc. Prof. Jan Dimon Bendtsen

List of published papers:Enclosed journal papers

• Morten Juelsgaard, Jan Bendtsen and Rafael Wisniewski, ”Utilization of Wind Turbines forUp-regulation of Power Grids”, IEEE Transactions on Industrial Electronics, pp. 2851–2863,July, 2013

• Morten Juelsgaard, Rafael Wisniewski and Jan Bendtsen, ”Fault Tolerant Distributed Port-folio Optimization in Smart Grids”, International Journal of Robust and Nonlinear Control,pp. 1317 - 1340, May-June, 2014

• Morten Juelsgaard, Palle Andersen and Rafael Wisniewski, ”Distribution Loss Reduction byHousehold Consumption Coordination in Smart Grids”, IEEE Transactions on Smart Grid,pp. 2133-2144, July, 2014

Enclosed conference papers

• Morten Juelsgaard, Palle Andersen and Rafael Wisniewski, ”Stability Concerns for IndirectConsumer Control in Smart Grids”, Proceedings of the European Control Conference, pp.2006 - 2013, July, 2013

• Morten Juelsgaard, Christoffer Sloth, Rafael Wisniewski and Jayakrishnan Pillai, ”LossMinimization and Voltage Control in Smart Distribution Grid”, Proceedings of IFAC World

Congress, August, 2014, To appear

• Morten Juelsgaard, André Teixeira, Mikael Johansson, Rafael Wisniewski and Jan Bendtsen,”Distributed Coordination of Household Electricity Consumption”, Proceedings of IEEE

Multi-Conference on Systems and Control, October, 2014, Submitted for publication

Additional conference papers, not enclosed

• Morten Juelsgaard, Jan Bendtsen and Rafael Wisniewski, ”Robust Utilization of Wind Tur-bine Flexibility for Grid Stabilization”, Proceedings of IFAC Symposium on Robust Control

Design, pp. 659-665, June, 2012

• Morten Juelsgaard, Palle Andersen and Rafael Wisniewski, ”Minimization of DistributionGrid Losses by Consumption Coordination”, Proceedings of IEEE Multi Conference on Sys-

tems and Control, pp. 501 - 508, August, 2013

• Morten Juelsgaard, Christoffer Sloth and Rafael Wisniewski, ”Low-Voltage ConsumptionCoordination for Loss Minimization and Voltage Control”, Proceedings of the 2nd Virtual

Control Conference, September, 2013

• Morten Juelsgaard, Luminita Totu, Ehsan Shafiei, Rafael Wisniewski and Jakob Stoustrup,”Control Structures for Smart Grid Balancing”, Proceedings of IEEE conference of Innova-

tive Smart Grid Technologies, pp. 1 - 5 , October, 2013

This thesis has been submitted for assessment in partial fulfillment of the PhD degree. The thesisis based on the submitted or published scientific papers which are listed above. Parts of the papersare used directly or indirectly in the extended summary of the thesis. As part of the assessment,co-author statements have been made available to the assessment committee and are also availableat the Faculty. The thesis is not in its present form acceptable for open publication but only inlimited and closed circulation as copyright may not be ensured.

Contents

Preface IX

Abstract XI

Synopsis XIII

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Smart Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 State of the Art and Background 7

2.1 Traditional Power Dispatch . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 General Portfolio Coordination . . . . . . . . . . . . . . . . . . . . . . . 82.3 Flexible Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Grid Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Wind Turbine Flexibility 27

3.1 Overproduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Reactive Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Demand Management 37

4.1 Consumer Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Indirect Loss Minimization Across Tie-Line . . . . . . . . . . . . . . . . 414.3 Direct Loss Minimization Across Tie-Line . . . . . . . . . . . . . . . . . 454.4 Spatially Distributed Portfolio Balancing . . . . . . . . . . . . . . . . . . 48

5 Grid Operation Through Coordination 53

5.1 Coordination Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Distributed Loss Minimization, Voltage Control

and Congestion Management . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Closing Remarks 69

Acronyms 73

Nomenclature 75

References 79

V

CONTENTS

Contributions 87

Paper A: Utilization of Wind Turbines for Up-regulation of Power Grids 89

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912 Background and Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 923 General Problem Description . . . . . . . . . . . . . . . . . . . . . . . . 974 Maximizing Overproduction Period . . . . . . . . . . . . . . . . . . . . 995 Robust Overproduction Strategy . . . . . . . . . . . . . . . . . . . . . . 1036 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 Conclusion and Further Work . . . . . . . . . . . . . . . . . . . . . . . . 111A Appendix: Proof of Feasible Solution . . . . . . . . . . . . . . . . . . . 111B Appendix: Lower Bound on Overproduction period . . . . . . . . . . . . 113References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Paper B: Stability Concerns for Indirect Consumer Control in Smart Grids 117

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203 Indirect control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132A Appendix: Eigenvalue of Greedy consumers . . . . . . . . . . . . . . . . 133References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Paper C: Distribution Loss Reduction by Household Consumption Coordina-

tion in Smart Grids 137

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454 Flexibility analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505 Distributed Consumption Coordination . . . . . . . . . . . . . . . . . . . 1526 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Paper D: Fault Tolerant Distributed Portfolio Optimization in Smart Grids 163

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1652 Portfolio Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1673 Distributed Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 1694 Fault tolerant implementation . . . . . . . . . . . . . . . . . . . . . . . . 1755 Early Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1786 Numerical Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 187A Appendix: Supporting Results . . . . . . . . . . . . . . . . . . . . . . . 187References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Paper E: Loss Minimization and Voltage Control in Smart Distribution Grid 195

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1972 Modeling and Problem Formulation . . . . . . . . . . . . . . . . . . . . 1993 Optimization and benchmark . . . . . . . . . . . . . . . . . . . . . . . . 203

VI

CONTENTS

4 Test-Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2065 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 2076 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Paper F: Distributed Coordination of Household Electricity Consumption 215

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2172 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2183 Coordination problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2244 Distributed Consumption Balancing . . . . . . . . . . . . . . . . . . . . 2255 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2286 Conclusion and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . 233References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

VII

Preface

This thesis is submitted as a collection of papers in partial fulfillment of a PhD studyat the Section of Automation and Control, Department of Electronic systems, AalborgUniversity, Denmark. The work has been conducted during the period August 2011 toJune 2014, and has been supported by the Southern Denmark Growth Forum and theEuropean Regional Development Fund, under the project ”Smart & Cool”.

The thesis consists of two parts. The first part encompasses Chapters 1 - 6, containingthe motivation, state of the art and extended summary of the contributions. The contri-butions are presented in detail, through the publications enclosed as Paper A - Paper F,comprising the second part of the thesis. The order of the presented papers and results,largely represents the progression of our work throughout the duration of the study.

Besides modifying the layout, all publications are enclosed in the same form as theyhave been published, meaning that no modifications has been made to phrasings, notation,etc. One exception is a few minor errata which has been found and corrected in some ofthe papers. Footnotes have been inserted to indicate where any changes have been madeto the original papers. As no other changes have been made, the notation varies some-what from one paper to the next, reflecting the fact that it has been updated and refinedthroughout the duration of our work. As Chapters 1 - 6 represent a distilled version of thecombined contributions, we have used a unified notation, which may thus in some casesdiverge from the one used in the individual papers covering the discussed material. Thenotation employed throughout Chapters 1 - 6 has been summarized in the nomenclaturepresented in the pages succeeding Chapter 6, along with a list of applied acronyms.

The work presented here has been completed under the supervision of professorRafael Wisniewski and associate professor Jan Bendtsen, to whom I owe great apprecia-tion for their excellent guidance. Besides my supervisors, I am much obliged to everyoneat the section of Automation and Control, in particular Christoffer Sloth and Palle Ander-sen, who have co-authored several of the enclosed papers, as well as John Leth, who hasbeen a great resource in numerous technical discussions. I am further very thankful forthe hospitality extended to my by professor Mikael Johansson, by allowing me to visit hisgroup at The Royal Institute of Technology in Stockholm, Sweden, in the fall of 2013,and for providing me with both supervision and insights during my stay. During thisvisit I further received invaluable inputs from doctoral students André Teixeira and Eu-hanna Ghadimi, through numerous technical discussions, for which I am equally grateful.

Morten JuelsgaardAalborg University, June 2014

IX

Abstract

The Danish electricity grid is expected to face numerous challenges in the future. Theseare the combined effect of several factors, such as an increased use of renewable re-sources for power production; an increased use of electricity for previously fossil-fueledconsumption; and an increase in local power generation at household level. These fac-tors pose challenges relating to how supply and demand should be balanced when supplyrelies on volatile resources; how to avoid grid congestion when demand increases; howto deal with increasing costs of transport losses; and how to maintain quality of powerif the flow reverses on account of residential production. The focus of this thesis is toshow that proper utilization and coordination of distributed resources in the grid can bean important tool to alleviate these challenges.

We initially elaborate on the expected challenges, where we focus specifically ongrid balancing, congestion management, minimization of transport losses, and controlof power quality. Subsequently, we discuss various classes of resources in the grid andintroduce the concept of a smart grid, within which we derive optimization strategies todemonstrate the resources applicability for alleviating the presented challenges.

Our strategies relies on coordination of power producers and consumers by utilizingthe flexibility provided by the ability of shifting either production or consumption, fromone time instance to another. We present models describing this flexibility as well as itslimitations. Enforcing coordination through temporal shifts of consumption and produc-tion requires the problems we consider to be solved across some predefined time-horizon.

Utilizing flexibility of consumers through coordination, is known as demand man-agement, and considers how consumers may be motivated to mobilize their flexibility toobtain a trade-off between private desires, and social expenses. We discuss two conceptu-ally different approaches for demand management, outlining their benefits and disadvan-tages. For this we formally present several problems concerning the central challenges,and demonstrate how demand management may solve these. We solve the problems us-ing several methods based on distributed optimization, catering to the distributed natureof the resources in the grid.

Our work adds to the current insight in the field of smart grids, by demonstratinghow the presented issues, concerns and challenges may be solved within a smart gridframework, by identifying, coordinating and utilizing the available flexibility of variousdistributed resources through optimization based techniques.

XI

Synopsis

Det forventes at det danske elnet vil stå overfor talrige udfordringer i fremtiden. Dis-se vil være den kombinerede effekt af flere faktorer, så som øget elektricitetsproduktionfra vedvarende energikilder; øget anvendelse af elektricitet til forbrug der tidligere varbrændselsdrevet; og øget installation af elektricitetsproduktion på husholdningsniveau.Disse faktorer giver udfordringer i forhold til blandt andet hvorledes forbrug og produk-tion balanceres når produktionen i stigende grad bliver uforudsigelig; hvordan undgåsoverbelastninger i nettet når forbruget stiger; hvordan håndteres de øgede omkostnin-ger til energitransport; og hvordan garanteres spændingskvaliteten hvis effektransportenomvendes på grund af øget elektricitetsproduktion fra husholdninger. Fokus for denneafhandling er at vise hvordan korrekt udnyttelse af distribuerede ressourcer kan være etvigtigt værktøj til at afhjælpe disse udfordringer.

Indledningsvist uddyber vi de forventede udfordringer, med specifikt fokus på net-balancering, håndtering af overbelastning, minimering af nettab, og regulering af spæn-dingskvalitet. Efterfølgende diskuteres forskellige typer af ressourcer og det intelligenteelnet introduceres som begreb, for hvilket vi udleder strategier til at demonstrere ressour-cernes anvendelighed i forhold at afhjælpe de præsenterede udfordringer.

De strategier vi udleder bygger på koordinering af energiproducenter og -forbrugereved at udnytte deres respektive fleksibilitet igennem muligheden for at flytte enten pro-duktion eller forbrug fra et tidspunkt til et andet. Vi præsenterer modeller der beskriverdenne fleksibilitet såvel som dens begrænsninger.

At udnytte forbugsfleksibilitet igennem koordinering, kendes som forbrugshåndteringog omhandler blandt andet hvorledes forbrugere kan motiveres til at mobilisere deres flek-sibilitet således at der opnås et kompromis mellem private ønsker og sociale omkostnin-ger. Vi diskuterer to konceptuelt forskellige metoder til at opnå dette, og beskriver deresrespektive fordele og ulemper. Til dette formål opstilles flere formelle problemer relaterettil de beskrevne udfordringer, og det påvises hvorledes forbrugshåndtering kan bruges tilat løse disse problemer. Problemerne løses igennem forskellige teoretiske tilgange, base-ret på distribueret optimering, således at den distribuerede beskaffenhed af ressourcerne inettet imødekommes.

Vores arbejde bidrager til den nuværende indsigt indenfor intelligente elnet, ved at på-vise hvorledes de præsenterede problemer og udfordringer kan afhjælpes i en intelligentnet- og kommunikationsstruktur, ved at identificere, koordinere og udnytte den tilgænge-lige fleksibilitet af distribuerede ressourcer igennem optimeringsbaserede teknikker.

XIII

1 Introduction

This chapter presents the motivation behind the thesis. We discuss future ex-

pected issues of the electrical grid and introduce the concepts of demand man-

agement and smart grid, as a framework for alleviating the challenges. This

equips the reader with an initial understanding of the motivation of our work,

to be substantiated in subsequent chapters.

1.1 Motivation

The Danish electrical grid, as conceptually illustrated in Fig. 1.1, is expected to undergosignificant changes during the coming decades with respect to both electricity productionand consumption [Danish Energy Association and Energinet.dk, 2010a].

The electricity production has traditionally been governed primarily by a combinationof large centralized power plants, and smaller distributed combined heat and power (CHP)plants, both of which are usually driven by fossil fuels. However, both environmental, sci-entific and political interests aim at removing the Danish dependency on fossil-fuels [Dan-ish Ministry of State, 2011], implying that the use of renewable resources, in particularwind and solar power, is expected to gradually replace fossil fuels. This transition fromfossil to renewable fuels does not only affect the conventional power producers. It alsochallenges the traditional distinction between power producer and power consumer, sincethe installation of solar power in Denmark has predominantly been in the low-voltagegrid [Constantin et al., 2012], typically as household solar panels. This implies thatconsumers may contribute to the power production during periods where their privateproduction exceeds their consumption.

Besides affecting the way power is produced, the desire to phase out fossil fuels alsochallenges the traditional way in which power is consumed. This is due to various in-centives and legislations motivating consumers to replace fossil fueled appliances, byelectric appliances. This creates the possibility that these may be driven by renewableenergy, thereby further reducing the need for fossil fuels [Danish Energy Associationand Energinet.dk, 2010a, Danish Energy Association and Energinet.dk, 2012, Interna-tional Energy Agency, 2011]. This could be by replacing oil-fired burners by electric heatpumps (EHPs), and traditional combustion based vehicles, by electric vehicles (EVs), etc.

These tendencies in electricity production and consumption, bring various challengesto the electrical grid which must be overcome for the penetration of renewable resourcesto reach the desired level. We focus on four challenges relating to grid balancing, conges-

1

Introduction

High Voltage

Medium Voltage

Low Voltage

InternationalCollaboration

Central plants

Offshore wind

Decentral CHPs

Consumers Wind

Central plants

InternationalCollaboration

Figure 1.1: Conceptual outline of the Danish electrical grid structure, where the voltagelevels are separated by transformers ( ). The high and medium voltage sections are inDenmark considered the transmission system, whereas the low voltage section composesthe distribution system. The figure is derived but modified from [Energinet.dk, 2014].

tion management, loss minimization and voltage control, which encapsulates the mostsignificant future challenges, as summarized by [Danish Energy Association and En-erginet.dk, 2010b].

Balancing and Congestion

Imbalances between the produced and consumed power are manifested as variations inthe grid frequency. Frequency variations may result in large currents in induction motorsand transformers, carrying a risk of damaging the equipment [Kundur, 1993]. Substantialfrequency variations may further carry a breakdown, either of parts of the system, or theentire system itself. It is thus important for balance to be maintained.

In the Nordic region, production is scheduled to meet consumption ahead of time.This is achieved by trading most of the electricity through a day-ahead market, at a com-mon energy exchange [Nord Pool Spot, 2013]. At the exchange, balance responsibleparties (BRPs) submit price bids for quotas of energy of production and consumption, foreach hour of the following day. After gate closure of the exchange, the transmission sys-tem operator (TSO) allocates production quantities of each hour of the following day, sothat the consumption and production is scheduled to balance ahead of time. As a result ofthe planning and market trading, each BRP obtains production or consumption referencesahead of time, which must be met during each hour of the day of operation.

Due to unforeseen faults, variations in consumption, weather conditions, etc. online

2

1.1 Motivation

updates to the references obtained from the market are required, in order to ensure balanceduring runtime. Since renewable resources are inherently volatile, increasing their cov-erage of the total power production, while taking traditional, controllable power plantsout of operation, complicates this balancing task and renders it more difficult to main-tain a stable and reliable electricity source, where supply meets demand. This challengecomprises one of the focal points of our work.

The balance must be maintained while obeying capacity limits of the cables and trans-formers used for transmission and distribution of power. Increasing electricity consump-tion increase the challenge of avoiding grid congestions, if many consumers activate theirappliances simultaneously [Masoum et al., 2011]. Congestion in the transmission sys-tem is avoided by accounting for limitations in transmission capacity, as part of the day-ahead planning at the energy exchange. However, these mechanisms does not considerbottlenecks in the low-voltage distribution grid [Danish Ministry of Climate, Energy andBuilding, 2011], and as argued by [Pillai et al., 2012], the current tendencies in electricityconsumption may soon challenge the capacity of distribution grids, thus revealing anotherimportant obstacle to be considered in the work presented in the following chapters.

Losses and Voltage Variations

Increasing installation of electric appliances carries an increase in power consumptionwhich in turn causes larger voltage drops throughout the grid. Conversely, massive instal-lations of solar panels may cause reversed power flow, if the locally produced power isnot correspondingly consumed locally. This in turn causes the voltage to rise throughoutthe grid. Variations in the voltage should be limited since a persistent increase or decreasemay affect the operation of connected appliances [Kundur, 1993]. The recommendationsof [Danish Energy Association, 2011a, Danish Energy Association, 2011b] provide volt-age limits of ±10 % in the Danish grid, but these limits will become increasingly difficultto comply with in the low-voltage grids, presenting the third of the main challenges weconsider throughout.

An increased power flow, in either direction of the grid, will additionally increasepower transport losses and the associated cost. The cost of losses is initially covered bythe grid operators [Energinet.dk, 2007,Energinet.dk, 2011], but following the recommen-dations of [Danish Energy Association, 2005], the cost of losses in the Danish system isincluded when calculating grid tariffs, and are thereby indirectly passed on to the con-sumer. In that sense, both grid operators and consumers have a financial incentive tominimize power losses, and cost thereof.

The challenges pertaining to voltage variations and power losses are interconnectedand closely related to the consumption of end users. This is exemplified in Fig. 1.2,representing a simplified low-voltage grid. The voltage source represents an abstractionof the medium voltage grid and step-down transformer, capable of maintaining a fixed,normalized supply voltage vs = 1 pu (per-unit). The grid supports a load with fixedaverage active and reactive consumption p, q ∈ R, fed through a single distribution linemodeled as an impedance z ∈ C. Let vl(p, q), ı(p, q) ∈ C denote the average loadvoltage and current, where ı(p, q) = (p − jq)/v∗l (p, q). Then the load voltage and theactive losses incurred in the distribution line are given by

vl(p, q) = vs − zı(p, q) = vs − zp− jqv∗l (p, q)

, l(p, q) = r|ı|2 =r

|vl|2(p2 + q2) (1.1)

3

Introduction

vsz

ı vl

p, q Load

Figure 1.2: Fixed p, q load drawing power through a single impedance.

where r = Re(z) ∈ R+ is the cable resistance. In (1.1) it is apparent how the active andreactive power affect the losses and load voltage. To illustrate this graphically, Fig. 1.3traces out the load voltage magnitude and active losses, parametrized by active and reac-tive power. In the left case, the reactive power consumption is fixed at q = q′, and loadflow analyses are used to calculate the transport loss and load voltage, when sweeping fora range of both positive and negative active powers, thus mimicking a load that can bothconsume and produce power. For future reference, the right case presents the conversescenario, where the active power is fixed p = p′, and load flow analyses are performedfor a range of both positive and negative reactive powers. This mimics a load that canboth absorb and produce reactive power.

p < 0

p > 0

l(p,q′ )

[pu]

|vl(p, q′)| [pu]

0.9 1 1.10

5

10

15

q < 0

q > 0

l(p′ ,q)

[pu]

|vl(p′, q)| [pu]

1.08 1.082 1.084 1.086 1.0886

8

10

12

Figure 1.3: Left: Trace of losses and load voltage magnitude parametrized, by p for qfixed at q′ = 0. The dashed line denotes p = 0 pu, and the arrows indicate the direc-tion of the black dot when p is reduced from positive to negative. Right: Similar plotparametrized by q while keeping p fixed at some positive value p′.

From Fig. 1.3 it is apparent that both load voltage and transport losses varies with theactive and reactive consumption, and thereby also how an increased consumption in thegrid may eventually cause unacceptable voltage variations and costs of transport losses.From Fig. 1.3 it is further indicated how minimization of losses and minimization of volt-age magnitude deviation from the source, may be conflicting objectives. However, as thelimits on voltage variations originates from operational ratings of connected appliances,these are often prioritized above loss minimization, in the sense that losses can only beminimized to an extent that still guarantees satisfactory voltages throughout the grid.

As evident from Fig. 1.3, the magnitude of the voltage may either increase or de-crease, depending on whether a load produces or consumes power. This may have severeimplications in the future for more elaborate grid structures with several loads, when

4

1.1 Motivation

each load represent a consumer that may additionally produce power on occasion. This isdemonstrated in Fig. 1.4, where Fig. 1.4(Top) illustrates 50 consumers ( ) connected ina line, and separated by cable sections ( ). The feeder is connected to the remaininggrid through a step-down transformer ( ), providing a fixed secondary side voltage.

Fig. 1.4(Bottom) present a conceptual voltage profile along the feeder, for a fixed timeinstance where the 18 consumers closest to the transformer (indexed 1 − 18) introducemassive consumption (e.g. from EV charging), and the 14 consumers furthest from thetransformer (indexed 36 − 50), present massive production (e.g. from abundant solarpower). The figure demonstrates how this situation may result in an unacceptable voltagedrop in the initial part of the feeder, due to the consumption, while the production resultsin a local voltage increase in the furthest part of the feeder, violating the maximum voltagelimit. Thus, there is both massive over- and under-voltages at the same time throughoutthe feeder.

|u(h

)|[p

u]

Distance to transformer (h) [-]0 10 20 30 40 50

0.9

1

1.1

1.2

Figure 1.4: Top: Consumers connected in a line topology, where each arrow indicates aconsumer. Blue arrows indicates power consumption, whereas red arrows indicate con-sumers that produce power above their own self consumption. Bottom: The resultingvoltage profile along the feeder (Green, dotted), and allowed voltage limits (Red, dashed).

The Danish low-voltage grids, containing the residential consumption, are generallynot equipped to relieve such voltage fluctuations since voltage regulating equipment isgenerally only installed in higher layers of the grid [Danish Ministry of Climate, En-ergy and Building, 2011]. This further underlines the challenge of the desired renewableconversion of consumption and production.

Demand Management

It has been argued that a lot of both the existing as well as the future electricity con-sumption is highly flexible, in the sense that it may be temporally shifted with respectto normal operation, with no practical implications for the consumer. For instance, ina household heated by an EHP, the temperature of the household would only changeslowly if the EHP was turned off, due to the large thermal mass of the house. In thisway, the power consumption of the household heating may be postponed or expedited,with little or no discomfort to the residents. Similar flexibility considerations exists forvarious other appliances, such as dish washers, washing machines, driers, EVs, etc., andin addition, flexibility also exists for some classes of production. By utilization of this

5

Introduction

flexibility through proper coordination of temporal shifts of production and consump-tion, it is expected that the impact of the issues discussed above, may be reduced [Strbac,2008, Ipakchi and Albuyeh, 2009, Rahimi and Ipakchi, 2010]. This could for instancebe by coordinating consumption to avoid large voltage fluctuations, or coordinating con-sumption against predicted variations of wind energy production to maintain balance.

Employing the flexibility through management of electricity consumption in responseof supply conditions, is known as demand management [Varaiya et al., 2011]. Demandmanagement is not limited to active power, and as indicated in the above examples, co-ordination and management of reactive power may also assist in mitigating the presentedchallenges.

In [Danish Energy Association and Energinet.dk, 2010a, Danish Energy Associationand Energinet.dk, 2012] it was concluded that the most financially sound approach foralleviating the expected grid issues outlined above, is to invest in strategies that introducedemand management capable of exploiting the flexibility of consumers. However, suit-able approaches for achieving this are still to be defined, thus comprising the main focusof our study:

How to mobilize the flexibility presented by various electricity

consumption as well as production units, in order to alleviate the

main expected future challenges of the electrical grid.

1.2 The Smart Grid

Mobilizing the flexibility posed by consumption and production units in the grid, requiresa transition from the current grid structure into a smart grid structure. This refers to astructure with advanced metering, communication and control infrastructure, enabling amore dynamic interplay between the grid and individual units [Danish Energy Associationand Energinet.dk, 2010a, Ipakchi and Albuyeh, 2009]. This is opposed to the currentsituation where, in particular, consumers are simply passive components with respect tothe operation of the grid.

An advanced metering infrastructure implies that measurement equipment is installedin a more extensive manner than is currently the case. In this work we employ the standingassumption that it is possible to obtain close to real-time measurement of quantities suchas voltage, current, power, etc. both in the connection point of each power producer orconsumer, as well as in various other locations throughout the grid. Although this isnot currently the case, many grid companies are working towards an advanced meteringinfrastructure by installing smart meters on consumer premises for demand management[Varaiya et al., 2011].

We assume additionally that some abstract advanced communication infrastructureis installed, enabling data transfer and communication between both production unitsand consumers as well as higher level entities such as power retailers, grid operators orbalance responsible parties (BRPs). Similarly, we require that some abstract frameworkhas been implemented to enable producers and consumers to interact as well as react andrespond on external control signals or incentives, allowing them to participate in gridoperation and demand management.

6

2 State of the Art and

Background

This chapter outlines the background, previous work as well as currently pro-

posed solutions concerning the challenges discussed in Chapter 1. We start

by discussing the balancing task and then proceed to discuss in more detail

the topic of demand management and flexible resources. We subsequently dis-

cuss how the available flexibility may be employed for general grid operation,

pertaining to the outlined challenges. Finally we discuss our conception of

the primary shortcomings of the presented state of the art, and use this as a

basis to outline the specific research objectives of our work. A condensed sum-

mary of our contributions to these objectives is presented, to be elaborated in

subsequent chapters.

2.1 Traditional Power Dispatch

Section 1.1 describes how the present initial step to obtain a balanced grid is by schedulingproduction and consumption through the electricity market. As a result of the planningand market trading, each balance responsible party (BRP) obtains production or consump-tion references ahead of time, composing a demand which must be met during each hourof the day of operation in order to maintain balance of the grid. If a BRP fails to followthe provided demand, the incurred imbalance will be covered by the TSO at the expenseof the responsible BRP, following the regulations of [Energinet.dk, 2014]. This wouldtypically result in increased expenses of the responsible BRP.

If a BRP or power producer operates several power plants the task is to subdivide anddistribute the demand obtained through the market, as individual production references toeach plant such that the demand is met. This should be done in a way that minimizes theincurred operating cost while satisfying any individual constraints of each plant. This isknown as the dispatch problem and can formally be stated as:

Problem 1 (Optimal Dispatch).Let the following data be provided:

• a market demand pdem(t), spanning a horizon t ∈ T = [0, T ]

• a portfolio N = 1, . . . , n, containing n ∈ N power plants

7

State of the Art and Background

• functions fi : R → R, describing the operating cost of each unit i ∈ N

• sets Pi ⊂ R, encompassing local operating constraints of each unit i ∈ N .

Find power production references pi : T → R, i ∈ N , solving:

minimizepi,i∈N

∫

t∈T

∑

i∈N

fi(pi(t))dt

subject to∑

i∈N

pi(t) = pdem(t)

pi(t) ∈ Pi

(2.1)

for all t ∈ T .

The process from market trading to reference dispatching is presented in Fig. 2.1. Thedispatch problem is in general an NP-hard problem, but simpler formulations have beenconsidered in the literature by e.g. [Wood and Wollenberg, 1984,Saadat, 2002,Wangsteen,2007], where static examples are presented to demonstrate the nature of the problem, andvarious approaches for solving the dispatch problem are discussed. Recent work on powerdispatch includes [Kim et al., 2002], which considered a more complex formulation ofthe problem, and introduced genetic optimization for solving it. A more modular and dy-namic approach was designed by [Edlund et al., 2011], where the problem was formulatedas a model predictive control (MPC) problem, and solved in a hierarchical way.

Even though the power references obtained through the market are supposed to bringbalance between supply and demand, volatility of renewable resources and unforeseenevents such as faults of generators, etc. introduce uncertainties which may cause devi-ations from the spot market schedules, resulting in imbalances to the grid. The resultsof [Edlund et al., 2011] were extended by [Standardi et al., 2012] by including these un-certainties in the dispatch, and similarly in [Varaiya et al., 2011] where a dispatch strategywas designed to limit the risk of imbalances between consumption and production.

Traditional control strategies for removing imbalances resulting from uncertaintiesare described in the literature, e.g. by [Kundur, 1993, Saadat, 2002]. The technical speci-fications of these contingency strategies are described by the regulations of the TSO, [En-erginet.dk, 2014], and are currently governed primarily by traditional thermal plants [Ed-lund et al., 2009]. However, as outlined in Section 1.1, it is theoretically possible tointroduce demand management for coordination of consumers thereby including them inthe balancing, as well as other grid related challenges. In the following, we elaborate ondemand management as a tool to mobilize the flexibility of consumers, such as to enablethem to participate in grid operation, both in the planning stage, as well during runtime incontingency events.

2.2 General Portfolio Coordination

Problem 1 considered specifically the task of maintaining the balance of a portfolio bymeeting an accumulated power demand pdem. However, as outlined earlier, other objec-tives may be equally important and our work considers in particular the problems of mini-mization of transport losses, avoiding grid congestion and control of voltage fluctuations,

8


MarketParticipation

t

pde

m(t)

DispatchStrategy

t

p2(t)

t

p1(t)

t

p3(t)

Individualreferences

Subdivideportfolio demand

Portfoliodemand

Electricityspot market

Figure 2.1: A portfolio demand is obtained from the market, and dispatched among theindividual plants, such that the accumulated plant references corresponds to the demandobtained through the market.

in addition to the balancing problem. Additionally, the dispatch treated by Problem 1includes only traditional power plants in the problem but in the following, we broaden theportfolio to include a mix of consumers as well as producers whereby e.g. the balancingproblem may be treated by coordination of either, rather than coordination of produc-tion exclusively. With these comments in mind, we formally cast the general portfoliocoordination problem as:

Problem 2 (General Portfolio Coordination).Let the following data be provided:

• a coordination horizon T = [0, T ]

• a portfolio N = 1, . . .N, containing n ∈ N production and consumption units

• sets Pi ⊂ R, encompassing local operating constraints of each unit i ∈ N

• extended value function F : Rn → R ∪ ∞, describing some high level coordi-nation objective.

9


Find power references pi : T → R, i ∈ N , solving:

minimizepi,i∈N

∫

t∈T

F (p1(t), . . . , pn(t))dt

subject to pi(t) ∈ Pi

(2.2)

for all t ∈ T .

The power references pi, i ∈ N above refer to both consumption and production,depending on each specific unit i. From this we remark that the notions of production andconsumption throughout this work, may be used interchangeably, by the understandingthe production is simply negative consumption, and vice versa.

Problem 2 is very general, and may encompass a large variety of high-level objectives,including the four main concerns outlined for this work. For the purpose of illustration,the optimal dispatch Problem 1 can be recovered from Problem 2, by

F (p1(t), . . . , pn(t)) =

∑

i∈N fi(pi(t)), if∑

i∈N pi(t) = pdem(t)

+∞, otherwise,

which implicitly includes the balancing constraint in the extended value definition of F .In the dispatch problem, the entire portfolio is governed by a single BRP, i.e. all

portfolio units can be considered collaborative. For other formulations of the portfoliocoordination, this need not always be the case, and in some of the problems we presentthroughout, the individual portfolio units may be competing. The objective of the port-folio coordination might for instance be to ensure satisfactory grid operation from theviewpoint of a grid-company, requiring the behavior of several entities in some grid sec-tion to be coordinated. Individual entities might be competitors unwilling to cooperate,however their participation in coordination may be required, e.g. to ensure safe and sta-ble grid operation. As discussed in a later section, this presents additional challenges onthe methods employed for coordination, with respect to the required communication andsharing of information.

The flexibility of consumers and producers, and their ability to perform temporalshifts of power in order to participate in portfolio coordination, enters through the privateconstraints Pi, as well as each units specific influence on the objective F . We remarkhere that where fi for the power plants in Problem 1 was a strictly monetary cost, theprivate objective of consumers may be of much more subjective nature. We return to thisdiscussion in Chapter 4.

Problem 2 provides the generic representation of the class of coordination problemstreated in this work. It may be considered as an off-line planning problem to be solvedahead of run-time, thereby providing a power reference for each portfolio unit. Alter-natively Problem 2 can be considered an online control problem, where the problem issolved during runtime, e.g. in a receding horizon fashion, providing real-time referencesto each unit. In our work, we generally employ the former interpretation, but we includediscussions as to how our results would generalize to the latter.

The presentation of Problem 2 is simplified, in that it considers only coordination ofactive power. However, as apparent from Section 1.1, coordination of reactive power maybe equally important depending on the formulation of the objective F . Although we do

10


include discussions concerning reactive power management throughout, our main focusin the first part of the thesis is on active power management, in order to make discussionsclearer and notation simpler throughout. We refer to the enclosed papers for results onreactive power flow, when relevant.

We remark that Problem 2 simply defines the high-level goal of portfolio coordinationwhen including consumers through demand management. We have yet to discuss how tomobilize the flexibility of consumers, that is how to introduce demand management, al-lowing a solution to be obtained for any particular instance of Problem 2. The conceptualbackground of this is presented in the following.

2.2.1 Demand Management

The concept of demand management was discussed in broad terms by [Gellings, 1985],which outlined the main services that temporal shifts in consumption might provide tothe grid, such as load shifting and peak shaving, as illustrated in Fig. 2.2. The figureillustrates how the diurnal peaks in consumption load may be reduced, by employingdemand management for utilizing consumer flexibility to shift consumption to low loadperiods. This peak shaving could be employed for instance to avoid grid congestion.

Loa

d

Time

Traditionalconsumption

Demandmanagement L

oad

Time

Managedconsumption

Figure 2.2: Demand management may be employed to shift load for intra-day peak shav-ing and similar.

More recent discussions on demand management was provided by [Strbac, 2008,Ipakchi and Albuyeh, 2009, Rahimi and Ipakchi, 2010] which, besides discussing con-gestion issues, further elaborated and outlined how demand management might alleviatebalancing issues pertaining to intermittent energy supply. The work by [Varaiya et al.,2011] presented a conceptual framework to include consumers in the dispatch problem,such that consumption and production could be planned jointly, to maintain the power bal-ance. They accounted for stochastic variation of e.g. consumption and renewable energyproduction, to devise a dispatch strategy, limiting the risk of imbalances. Another moretangible example of demand management was provided in [Fazeli et al., 2011], demon-strating with numerical results, the applicability of short term demand management forenergy balancing.

11


2.2.2 Direct and Indirect Control

A more formal analysis of demand management and how it might be utilized for coor-dination, was presented by [Petersen et al., 2013]. Here, consumers were modeled asfirst order dynamic systems, where the state of the system described the energy storedby the consumer. This could correspond to the thermal energy stored in the floors andwalls of a household, or the energy stored in the battery of an EV. By assigning variousconstraints to states and inputs, several archetypical classes of consumption were derived,mimicking different appliances. In addition to presenting a general framework to modelconsumer flexibility, [Petersen et al., 2013] also derived a hierarchical coordination strat-egy to employ the flexibility of a portfolio of consumers, by controlling their individualconsumption patterns in a way that caused the accumulated consumption to track a pro-vided reference. As reference tracking could be employed to counteract variations inintermittent energy production, these results demonstrate how demand management maybe used for grid balancing. A similar approach was presented by [Trangbaek et al., 2011].

The earlier work by [Pedersen et al., 2011] used consumer models similar to theframework in [Petersen et al., 2013], but took a different approach to demand manage-ment, by considering control of individual consumers, rather than an aggregation of sev-eral. In [Pedersen et al., 2011] the notion of discomfort was introduced, as a measureof deviation from some preferred consumption pattern of the consumer. An estimatedprice of electricity was similarly introduced and used to optimize electricity consumptionacross some time-horizon, with the purpose of minimizing accumulated price of energy,on the one hand, and incurred discomfort on the other. The price signal thus served as anincentive for the consumer, to shift the consumption from the preferred pattern. In otherwords, the results presented by [Pedersen et al., 2011] indicated how an external incentivemight motivate consumers to utilize their flexibility by shifting consumption.

The above works by [Petersen et al., 2013] and [Pedersen et al., 2011] illustrate anatural subdivision of demand management concepts into two classes: direct and indirectcontrol [Danish Energy Association and Energinet.dk, 2010a]. Indirect control refers to ascheme where consumers are presented with an incentive, e.g. a price signal, motivatinga shift of consumption from high price periods to low price periods. The price signal isprovided by some external third party. This scheme was employed by [Pedersen et al.,2011], and have since been conceptually elaborated by [Heussen et al., 2012] and itsapplicability have been investigated further by e.g. [Corradi et al., 2013] and [Sossanet al., 2013]. Indirect control is conceptually illustrated in Fig. 2.3.

External Party Incentive Consumer Consumption

Figure 2.3: In indirect control, an external party provides incentives to the consumer,motivating a certain consumption pattern.

Converse to indirect control, is direct control. This refers to a scheme where a con-sumer has signed off control rights to some external third party, for some part of the

12


consumption. The external party is then able to control consumption, following the spec-ifications of some predetermined agreement or contract. The agreed upon contract wouldalso describe the benefit of the consumer for participating in the demand management pro-cess, such that no other incentives should be provided. Simple examples of the consumerbenefit was discussed in [Danish Energy Association and Energinet.dk, 2012], but directcontrol has been the topic of several research works, e.g. [Kraning et al., 2011,Sundströmand Binding, 2012, Kraning et al., 2014], among others. It is conceptually outlined inFig. 2.4, indicating the structural difference to indirect control in Fig. 2.3.

Consumer Agreement External Party Consumption

Figure 2.4: In direct control, an external party and the consumer enters into an agreement,allowing the external party to control the consumption pattern, honoring the constraintsand obligations set forth by the agreement.

Both the direct and indirect scheme involve an external party either to perform the de-mand management directly, or to supply the incentive for demand management [Heussenet al., 2012]. The conceptual illustrations in Fig. 2.3 and Fig. 2.4 outline demand manage-ment for a single consumer, however, with the exception of large industrial facilities, mostelectricity consumers are small scale, and their individual consumption may therefore beinsignificant with respect to a larger grid section. On this basis, it is expected that theexternal parties perform direct or indirect control, respectively, for a portfolio containingnumerous consumers. Coordination used to shape their aggregated consumption or be-havior thereby has more significant impact when participating in the energy markets, or invarious ancillary services [Danish Energy Association and Energinet.dk, 2012, Gkatzikiset al., 2013]. The external parties are therefore referred to as aggregators.

Even though the underlying premises of direct and indirect control are different, theobjective is still the same: consumers should be coordinated to behave in a beneficial way,with the purpose of satisfying some high-level objective, as encapsulated by Problem 2.This could be aggregated reference tracking, [Petersen et al., 2013], but other objectivescould be loss minimization or avoidance of grid congestion, [Pillai et al., 2012, Biegelet al., 2012]. In direct control, the aggregator can directly coordinate consumers to presentthe desired behavior. For indirect control, the aggregator needs first to learn the consumersresponse to external incentives, and thereafter design incentives to promote the desiredbehavior [Heussen et al., 2012, Corradi et al., 2013]. In that sense, it may be arguedthat indirect control presents an additional layer of complexity. In fact the works by[Heussen et al., 2012, Sossan et al., 2013] discussed how the lack of controllability andunknown consumer response could easily affect the functionality and even stability ofindirect control frameworks. This has however not yet been fully explored.

13


2.3 Flexible Resources

Demand management as discussed above focuses solely on the flexibility of consumers.However, the renewable resources accounting for some of the challenges outlined in Sec-tion 1.1, also present some level of flexibility which can be utilized when included in theportfolio coordination, Problem 2. Our work focuses on the utilization of wind and solarpower, which is thereby also the focus in the following.

2.3.1 Flexibility of Wind Power

As previously discussed, intermittency of wind power may challenge the balancing of thegrid. However as we outline below, there are various approaches to use wind turbines formitigating this.

Grid balancing through down-regulation, i.e. lowering electricity production to main-tain grid balance, can be supported by wind turbines simply by lowering their poweroutput. The converse case of up-regulation can be aided by wind turbines provided theyoperate with a certain margin to the available power, such that the power output may beincreased by this margin if the need occurs. This is known as ∆-mode operation [Chang-Chien et al., 2011, Energinet.dk, 2010], and is illustrated in Fig. 2.5.

Time

Power

Margin Avl. Pwr.

Prod. Pwr.

Figure 2.5: In ∆-mode operation, a production margin is maintained as a spinning reservebetween the available power (Green) and the produced power (Blue).

The production margin can be considered as a spinning reserve that can be activatedif required, for instance during various contingency events. The work by [Chang-Chienet al., 2011] demonstrated how a controller for this operating mode could be designed ina way that ensured stable operating conditions of the turbine. The ability to maintain ∆-mode operation is a requirement to all wind turbines in Denmark, as per the regulationsof [Energinet.dk, 2010].

As an alternative to ∆-mode operation, recent works by [Anaya-Lara et al., 2006,Ul-lah et al., 2008, Tarnowski et al., 2009], have discussed up-regulating grid support by theinertial energy of variable speed wind turbines (VSWTs). This can be used even in caseswhere the wind turbine is producing a power corresponding to the available wind power,i.e. if no production margin is maintained. The idea is that a power output above availablepower is obtained by extracting mechanical energy of the turbine rotor, thereby slowing itdown. As the rotor speed must no pass a lower threshold, the overproduction is only tem-porary, whereafter the rotor must be accelerated back to normal operation. Acceleratingthe rotor requires the electrical power output to be lowered, rendering a period of under-

14

2.3 Flexible Resources

production, which must be covered by other resources. The idea is outlined in Fig. 2.6.

pop

t

Normal OP UP Normal

p(t)

pmek(t)

pavl(t)

Figure 2.6: The electrical (p), mechanical (pmek) and available power (pavl), over time (t). Thefigure illustrates operating periods; normal: p(t) = pavl(t), OP (overproduction): p(t) = pop,and UP (underproduction) p(t) < pavl(t), for some fixed overproduction reference pop, [Tarnowskiet al., 2009].

The work by [Anaya-Lara et al., 2006] designed a controller for a VSWT, with thepurpose of enabling overproduction capabilities. The functionality of the controller wasdocumented through simulations, but no analysis or examination of the extent of its sup-port was presented in terms of the amount of energy that could be delivered, or the lengthof the period of support. In [Ullah et al., 2008, Tarnowski et al., 2009] the work wasextended through numerical simulations, demonstrating the potential of grid support byoverproduction. These works took the converse approach and focused on investigatingthe extent of the support in terms of energy delivered and length of the overproductionperiod, but did not derive controllers to utilize the support. The influence of various fac-tors was investigated. Both works showed similar results, although these were limitedto simulation based inferences, using a high-end turbine model. No general analyticalresults were provided. In that sense, extending the results of these works to other scenar-ios, is only possible to the extent that these scenarios falls under the parameter variationsinvestigated by the works.

The work by [Hovgaard et al., 2013] similarly used turbine inertia to extract energyfrom the wind turbine rotor. However, as opposed to the works described above, theuse of rotational inertia was not employed for grid support in terms of overproduction.Rather, it was employed along with an energy storage, for smoothing the power output ofthe wind turbine by limiting the gradient of the produced power. Reducing the gradientof the produced power correspondingly reduces the variability and thus the uncertainty ofthe wind power. Similar works on power smoothing using various types of storages waspresented in e.g. [Cimuca et al., 2006, Cárdenas et al., 2006]. In [Cimuca et al., 2006]it was demonstrated how a local flywheel installed at a VSWT could assist in smoothingthe power output of a wind turbine, in order for it to provide ancillary services to thegrid. Additionally, [Cimuca et al., 2006] discussed controllers to store and extract powerfrom the flywheel, and further discussed the magnitude of the losses associated to this, asopposed to operating the wind turbine without storage. Wind turbine power smoothingusing local storages was similarly investigated in [Cárdenas et al., 2006].

15


The above discussion concerning flexibility of wind power relates solely to the flex-ibility of active power production, and we return to the implications of this in Chapter3. For some wind turbines, it is however also possible to control the reactive powerflow. As we outlined Section 1.1, this may help to mitigate the remaining challengespertaining to losses and voltages. Flexibility of reactive power flow of wind turbines hasbeen discussed by e.g. [Cimuca et al., 2006], which included the option of controllingreactive power flow in their controller. Reactive power control of modern wind turbineswas also discussed in [Margaris et al., 2010] for doubly-fed induction generator (DFIG)wind turbines. We have not considered reactive power control of wind turbines in thisthesis. Reactive power control as a tool against the presented challenges, has only beenconsidered for solar generated power as discussed in the following.

2.3.2 Flexibility of Solar Power

The active power output of solar panels is generally governed by weather conditions, suchas direct and indirect solar radiation, cloud coverage, ambient temperature, etc. [EuropeanPhotovoltaic Industry Association, 2012], and thus the active power production cannot beincreased above the available power defined by ambient conditions. Control of activepower output of solar panels is thereby limited to strategies lowering the active powerproduction, which may be necessary e.g. for grid balancing in terms of down-regulation,or in order to respect congestion limits or allowed voltage variations in low voltage grids.

Lowering the photo-voltaic (PV) power output could be done directly by curtailment,or by installing a local energy storage that may absorb excess power of the solar panel[Constantin et al., 2012,Gaudin et al., 2012,European Photovoltaic Industry Association,2012]. Charging a local energy storage, rather than employing curtailment, stores excessenergy, rather than wasting it. Discharging the storage may later be used to either increasethe power output, or assist in smoothing the power output of the solar panel [Marinelliet al., 2013]. For solar panels installed privately at residential households, the excessenergy may be stored by an increased self-consumption of the household, e.g. as thermalenergy, [Pedersen et al., 2011], rather than installing a dedicated storage.

Converse to the active power, the reactive power of the solar panel may be controlledmuch more freely, and is primarily limited by the constraint that the apparent power ofthe PV inverter needs to be below an upper limit, as illustrated in Fig. 2.7. This is dueto the power electronics of the inverter connecting the solar panel to the grid. This wasthe topic of [Constantin et al., 2012, Turitsyn et al., 2011] which demonstrated variousstrategies for control of reactive power flow from solar panels.

The purpose of these works was to reduce solar induced voltage variations, as well asto reduce active power losses. However, as pointed out by [Turitsyn et al., 2011], the twoobjectives of minimizing both losses and voltage variations, are to some extent conflictingso various trade-offs were implemented in the derived controllers. The results obtainedin [Constantin et al., 2012] demonstrated how reactive power control may increase thepossible installation of PVs, while satisfying grid constraints with respect to voltage vari-ations. The considered control structures of both [Constantin et al., 2012] and [Turitsynet al., 2011] are readily implementable in the grid, given the current inverter technologyand grid design. However, it was argued by [Constantin et al., 2012] that improved resultsmay be obtainable in a smart grid setting, where an elaborate communication infrastruc-ture allows for more sophisticated control schemes.

16

2.4 Grid Operation

q

p

smax

s2 = p2 + q2,

|q| ≤=√

s2max − p2,

Figure 2.7: The constraint on active power p and reactive power q of the PV inverter,as given by the apparent power s, and the inverter limit smax. The figure is modifiedfrom [Turitsyn et al., 2011].

The work by [Constantin et al., 2012, Turitsyn et al., 2011] was solely simulationbased, and did not provide any analytical assessment of their results. Similarly, no guar-antees towards effectiveness of the considered approaches were provided. Also, the extentto which their results would generalize to other networks than the ones investigated, wasnot discussed. Both works mention briefly the issue of inverter capacity: the apparentpower limit of the inverter reduces the potential impact of the reactive power control,since the high active power output in solar intensive periods restricts the allowed rangeof the reactive power, as indicated in Fig. 2.7. We explore this further in the end ofSection 5.2.

The reader should notice that the possibility of reactive power flow control describedabove, relates to the PV inverter, and not the PV panel itself. In that sense, the samereactive power control could be employed for other inverter based devices, e.g. EVs orsome classes of wind turbines. Our focus here does however reside on PV systems.

2.4 Grid Operation

Traditional approaches for mitigating the main challenges outlined in Section 1.1 whenoperating the grid, include grid reconfiguration ([Baran and Wu, 1989]), and control ofvoltage or reactive power flow through various equipment such as tap-changers, capacitorbanks, static var compensators, etc. ( [Vournas and Christoforidis, 2012]). Other works,such as [Hoff and Shugar, 1995, Guo et al., 2011], consider loss minimization as a prob-lem of optimal placement picking for installation of distributed generators (DGs). A morerecent approach for loss minimization was presented by [Šulc et al., 2013], utilizing dis-tributed control of the reactive power flow in selected nodes of a grid. Additionally, [Šulcet al., 2013] utilized the reactive power flow to control voltage fluctuations along feeders.

The flexibility and demand management discussed in Section 2.2 and Section 2.3,was not included for grid operation in most of the works cited above. This was howeverthe focus of [Pillai et al., 2012, Biegel et al., 2012, Kraning et al., 2014, Sossan et al.,2013, Turitsyn et al., 2010, Turitsyn et al., 2011], among others.

The work described in [Pillai et al., 2012] considered the installation of EVs in a low-

17


voltage (LV) grid, and examined how many vehicles could be installed without breakingthe voltage constraints or overloading the transformers in the grid. The results obtainedby [Pillai et al., 2012] indicated that the main obstacle for installing EVs is not transformercapacity, but rather the voltage constraint.

In [Biegel et al., 2012] a coordination method was presented to avoid grid congestionwhen several BRPs service consumers in the same grid. Here, no concerns were includedtowards consumer behavior, and besides the concern towards congestion, no grid relatedissues were included. The work by [Kraning et al., 2014] considered an extended coordi-nation problem for a portfolio of consumers, by also including the loss of power in eachcable, although no mechanism to minimize losses was implemented. Additionally, theissue of voltage stability was disregarded altogether. The work by [Turitsyn et al., 2011]did not consider control of active power consumption, but considered instead the reactiveflexibility of PV panels, and designed local controllers of each of the reactive power flowof each PV, in order to minimize voltage variation and incurred losses.

Demand management was included in all the above works for grid operation, in or-der to solve coordination problems of similar type as Problem 2. Regardless of whetherconsumers are included for grid operation in a direct or indirect fashion, the hierarchicalstructure of the control strategies designed by the above works, vary greatly. At the oneextreme, there is a completely flat and distributed structure, where each unit operates in-dependently, and simply uses local measurements to perform some control action. Theindividual controllers may be designed for some complicated high-level purpose, relatedto the interaction of several units, but after implementation, each controller runs com-pletely locally. At the other extreme there is a central coordinator that gathers data andmeasurements from all units, and dictates the control action to be taken by each unit.

The coordination strategies employed in our work range in between theses extremes:each unit makes local measurements and control decisions, i.e. there is no central decisionmaking, however, the local control decisions are coordinated against the behavior of otherunits in the rest of the portfolio. This thereby requires some level of interaction betweenunits. We elaborate on this structure in Chapters 4 and 5, where our contributions to theabove works on grid operations are discussed.

2.5 Optimization

The exposition outlined in Section 2.1 - Section 2.4 above, demonstrates the state of theart relating to both the identification and classification of flexibility of power producersand consumers, as well as how it may be employed to solve the challenges presentedinitially. In the following Chapters 3 - Chapter 6 we summarize our contributions tothis, and as we have previously mentioned, this will rely heavily on optimization basedstrategies for solving various specific instances of Problem 2. To ease the discussiononwards, the following provide an overall outline of the methods we employ.

2.5.1 Convex and Non-Convex Problems

Many of the problems we consider throughout have natural formulations as constrainedconvex problems ([Boyd et al., 2010]), for which known algorithms are available to glob-ally and reliably solve the problem. Some of the problems we consider does however

18

2.5 Optimization

contain non-convex components, for which various approaches and theory relates, de-pending on the problem structure. One approach we have found suitable for the class ofproblems considered in our work, is to convexify the non-convex components, or applyconvex heuristics to the non-convex problem. From this, a convex approximation to theoriginal non-convex problem may be obtained and solved.

One heuristic approach for non-convex optimization, is the method of sequential con-vex programming (SCP) [Dihn and Diehl, 2010]. The method relies on sequential stepsof convexifying and solving a non-convex problem, around an estimated solution, and af-terwards updating the estimate. Repeating this process in an iterative fashion, often leadsto local optima of the original problem. This approach has been applied with success forvarious applications in e.g. [Biegel et al., 2011, Hovgaard et al., 2013]. However, as SCPis a heuristic for solving non-convex problems, only few convergence results exists forthe method [Dihn and Diehl, 2010].

One formulation of SCP is provided by [Boyd, 2014]: Let a non-convex problem beformulated as

minimizeu

ψ0(u)

subject to φi(u) = 0, i = 1, . . . , rψj(u) < 0, j = 1, . . . , s

(2.3)

where ψi : Rn → R, i ∈ 0, . . . , s are (possibly non-convex) functions, and φ : Rn →R, j ∈ 1, . . . , r are (possibly non-affine) functions.

Let u denote an estimate of the solution to (2.3), and let U(u) denote a convex trustregion of this estimate. SCP can then be formulated as in Algorithm 2.5.1.

Algorithm 2.5.1 Sequential convex programming

Initialize estimate u(0)

for k=0,1, . . . do

u = u(k)

Form convex approx. ψi of ψi and affine approx. φj of φj over U(u)Obtain u(k+1) as solution to convex approximated problem:

minimizeu

ψ0(u)

subject to φj(u) = 0, ψi(u) < 0u ∈ U(u)

Next iterationend for

In Algorithm 2.5.1 the notation u(k) refers to the value of u at iteration k, i.e. the super-script (k) is an iteration index and not an argument of u. We use this superscript notationin general for iterative procedures.

Alternative formulations and methods for deriving the trust-region are covered in[Boyd, 2014]. A convergence measure of Algorithm 2.5.1 would be given by the residual

ζ(k) = u− u(k+1),

19


that is how much is the estimated solution updated between iterations. A related termi-nation criteria would be to terminate when ‖ζ(k)‖2 ≤ ǫabs, for some absolute toleranceǫabs > 0.

2.5.2 Distributed Coordination

For any specific instance of the portfolio coordination, Problem 2, it might be difficultfor any single entity to solve the problem, as this would require a central collection andstorage of the parameters, constraints etc. describing each unit in the considered portfolio.This would include e.g. power plant operating costs, comfort descriptions of consumersand constraints of all portfolio participants.

As we mentioned previously, it is not a matter of course that all units in the portfolioare collaborative. Thus the private performance criteria, constraints, model parameters,etc. of each participant is privileged information that must not or should not be disclosed.Additionally, even if all the required information could be collected, the problem mightstill be intractable to solve, given the size it may obtain when a large number of consumersand distributed resources are included in a problem formulation.

Various approaches have been employed for simplifying the main problem, in orderto mitigate these complications and make the problem more tractable to solve. Someapproaches involves simplifying the problem through various heuristics to apply a suitablecontrol structure, as was the case in [Trangbaek et al., 2011, Turitsyn et al., 2011]. Otherworks employ decomposition methods to divide the main problem into smaller problems,which may be distributed among each portfolio unit participating in the problem. Theunits may subsequently solve the problem collaboratively, without disclosing privilegedinformation, as in [Biegel et al., 2012, Šulc et al., 2013, Kraning et al., 2014]. This oftenhas the benefit that sensitive information can be kept private to each participant, and onlyhigh level information needs to be shared.

Depending on the nature of the coordination problem, numerous decomposition meth-ods exists with various prerequisites, benefits and drawbacks [Bertsekas and Tsiksiklis,1997]. Two decomposition methods which have received significant attention, are themethods of primal and dual decomposition [Palomar and Chiang, 2006], but other ap-proaches such as Dantzig Wolfe decomposition, [Edlund et al., 2011, Standardi et al.,2012], and Alternating Direction Method of Multipliers (ADMM), [Boyd et al., 2010,Šulcet al., 2013, Kraning et al., 2014] have also been employed for various problems relatingto power systems and demand management. Our work focus primarily on the two meth-ods of dual decomposition and ADMM, as the problems we consider often has a naturalformulation which fit seamlessly within the framework of these approaches.

The procedure of dual decomposition can be conceptualized by considering the fol-lowing problem

minimizeu1,...,un

ψ(u1, . . . , un) =

n∑

i=1

ψi(ui)

subject to ui ∈ Ui∑ni=1 ui = c,

(2.4)

for ui, c ∈ RN , i ∈ 1, . . . , n, where ψi : RN → R are strictly convex, and Ui ⊂ RN

are convex sets. The reader may notice the structural resemblance between (2.4) and the

20

2.5 Optimization

general dispatch problem on Page 7. The dual problem of (2.4) is

maximizeν

g(ν) =

n∑

i=1

infui∈Ui

(ψi(ui) + ν⊤ui)− ν⊤c (2.5)

where ν is the Lagrange multiplier of the equality constraint in (2.4). We denote thesolution of (2.5) by ν⋆. Let u⋆i (ν) = arg infui∈Ui

(ψi(ui) + ν⊤ui). The strict convexityof ψi, implies that g(ν) is differentiable ( [Bertsekas, 2008]) with gradient

∇g(ν) =n∑

i=1

u⋆i (ν)− c.

A distributed optimization method utilizing the differentiability of g can thus be formu-lated as Algorithm 2.5.2.

Algorithm 2.5.2 Dual decomposition

Initialize ν(0), α(0), α(1), , . . . for k=0,1, . . . do

Obtain: u⋆i (ν(k)) = arg infui∈Ui

(ψi(ui) + u⊤i ν(k)), i ∈ 1, . . . , n

Update: ν(k+1) = ν(k+1) + α(k)(∑n

i=1 u⋆i (ν

(k))− c)Next iteration

end for

For suitable choices of the step size α(k), we have limk→∞ ν(k) = ν⋆, and(ν⋆, u⋆1(ν

⋆), . . . , u⋆n(ν⋆)) comprises a dual-primal optimum. As Algorithm 2.5.2 is sim-

ply a gradient update scheme, a common termination criteria is

‖∇g(ν)‖2 = ‖n∑

i=1

u⋆i (ν) − c‖2 ≤ ǫabs,

for some absolute tolerance ǫabs > 0.As evident from Algorithm 2.5.2, dual decomposition yields a problem structure

where each agent in a portfolio solves a subproblem involving the externally provided La-grange multipliers. Afterwards, the solution of each subproblem is gathered, and the mul-tipliers are updated for the following iteration. This structure was employed by [Biegelet al., 2012] to coordinate a number of BRPs, in order to avoid grid congestion.

In some applications, it is intractable to require that all units are able to communicatewith one central entity for gathering and distributing the Lagrange multipliers. Instead,the portfolio may be considered as a graph, where each unit may only communicate witha few neighbors, keeping the communication structure localized, as indicated in Fig. 2.8.

In this case, it may be possible to combine the decomposition method with a graph-wide method for sharing information through local communication. One such method isdistributed averaging [Xiao and Boyd, 2004, Xiao et al., 2007]. A combination of dualdecomposition and distributed averaging was employed by [Johansson and Johansson,2005, Johansson et al., 2008], with an application towards distributed optimization ofcommunication networks. We elaborate this discussion in Chapter 4.

21


1

2

3

4

5

6

1

2

3

4

5

6

Figure 2.8: Left: Central gathering and distribution of data among units in portfolio.Right: Neighbor-based data sharing across portfolio.

For some problem formulations, a more intuitive neighbor-based communication strat-egy can be obtained through the method of ADMM. This can be employed for problemsof type

minimizeu,y

ψ(u) + φ(y)

subject to Au+By = c(2.6)

where ψ, φ are convex functions, possibly defined as extended value functions. TheADMM method can be summarized as in Algorithm 2.5.3, where ν is the Lagrange mul-tiplier of the equality constraint in (2.6).

Algorithm 2.5.3 Alternate Direction Method of Multipliers

Initialize u(0), y(0), ν(0), ρ > 0for k=0,1, . . . do

Obtain:u(k+1) = arg inf

u(ψ(u) + ρ

2‖Au+By(k) − c+ ν(k)‖22

y(k+1) = arg infy

(φ(y) + ρ2‖Au

(k+1) +By − c+ ν(k)‖22

ν(k+1) = ν(k) +Au(k+1) +By(k+1) − cNext iteration

end for

Assuming that

1. Equation (2.6) is feasible, with solution u⋆, y⋆,

2. Functions ψ, φ are convex, possibly defined as extended value functions on nonemptypolyhedral sets

3. Matrices A,B are full column-rank,

then it can be shown that Algorithm 2.5.3 converges in the sense that y(k) → y⋆ andu(k) → u⋆ for k → ∞, for any value of ρ > 0 [Bertsekas and Tsiksiklis, 1997, Motaet al., 2011]. We refer to u as the primary variable and y as the ADMM variable. Otherconvergence results exists for milder criteria than the ones listed above [Boyd et al., 2010].

22

2.6 Research Objectives

In [Boyd et al., 2010] is also described how the quantities

ζ(k) = Au(k+1) +By(k+1) − c, and, ξ(k) = ρA⊤B(y(k+1) − y(k)), (2.7)

can be interpreted as residuals, thus suggesting the termination criteria

‖ζ(k)‖2, ‖ξ(k)‖2 ≤ ǫabs,

for some absolute tolerance ǫabs > 0.Algorithm 2.5.3 updates the variables in three steps, 1: u-update, 2: y-update, 3:

ν-update. In order to calculate u(k+1), the values (y(k), ν(k)) needs to be available. Sim-ilarly, to calculate y(k+1) and ν(k+1) respectively, the values (u(k+1), ν(k)) and (u(k+1),y(k+1)) need to be available. Thus, even though u and y are coupled through the con-straints of (2.6) they can be iteratively updated separately. The required data passing isillustrated in Fig. 2.9, where the right-facing arrows indicate data to be used in the currentiteration, and the left-facing arrows indicate data to be used in the coming iteration.

u y ν

u(k+1)

u(k+1) y(k+1)

ν(k)

ν(k)y(k)

Figure 2.9: Data passing procedure for the three stage variable update in ADMM. Eachbox represent a update procedure, and each arrow represent the passing of an update result

It may not be obvious how wide a range of problems may be solved by Algorithm 2.5.3.An extensive review was provided by [Boyd et al., 2010], and we shall demonstrate itsapplicability to different smart grid related problems in Chapters 4 and 5, where we alsodemonstrate how the data passing structure outlined in Fig. 2.9 may allow for neighborbased communication.

2.6 Research Objectives

During the above outline of the state of the art we have highlighted various shortcomingsand weaknesses of the cited works. We summarize these below, thereby comprising thetopics to consider onwards by our work.

Flexible Resources:

• The cited works on flexibility of wind power through temporary overproduction, islimited to numerical inferences. That is, only simulation based results are presentedand no analytical assessment of the extent and applicability of overproduction isprovided.

23


Demand Management:

• The framework of indirect consumer control has yet not been unambiguously de-fined, and thus, the possible drawbacks of the lacking controllability of consumershas not been explored in depth. It has primarily been discussed on a high level, andonly few works present tangible examples.

• The potential of demand management, either direct or indirect, has been discussedby several. However, there is still a lacking understanding on how the formula-tion of constraints and operating criteria of various appliances and consumers, mayaffect their respective flexibility.

Grid Operation:

• The works on employing flexibility of consumers and producers, considers primar-ily only a subset of the challenges we have discussed, without assessing how theremaining challenges may be accounted for in the presented solutions.

• Utilizing flexibility for grid operation is additionally considered only for active andreactive power, separately. To the best of our knowledge, no work has yet presentedapproaches for joint management of active and reactive power.

Communication Strucure:

• The cited works on grid operation present frameworks for mitigating the discussedchallenges. The bulk of these frameworks are either fully distributed, relying onheuristics requiring no communication, or fully centralized, employing optimiza-tion theory with full information disclosure. Only few works present frameworksthat are both based in optimization theory, while also accounting for the possiblecompetitive nature of the participating entities, by not requiring key information tobe disclosed.

We approach these shortcomings by posing the following research objectives of our work,pertaining to the four main challenges of grid balancing, congestion avoidance, loss min-imization and voltage control, in relation to portfolio coordination, Problem 2:

Flexibility - Extent and Limitations

Formulate the flexibility of selected power producers and consumers in relation to

mitigation of the presented main challenges. Examine the extent and limitations of the

flexibility for different instances of the coordination problem, and explore how this varies

for alternative flexibility formulations. Further, investigate how the formulation of

flexibility may affect different coordination procedures.

Coordination and Communication

Derive optimization based coordination strategies to harvest the flexibility of power

production and consumption, for different instances of the coordination problem. The

strategies must be applicable to subsets, as well as the full set of outlined challenges. As

such, the strategies should enable joint coordination of both active and reactive power.

In addition, the strategies should be distributed and require only localized and limited

communication, without requiring disclosure of sensitive information.

24

2.7 Summary of Contributions

2.7 Summary of Contributions

As a response to the research objectives provided above, the following outlines the con-tributions of this work, by highlighting the key content of each of the enclosed papers.

• Extent of wind turbine grid support (Paper A)

We have expanded the cited results on grid support by temporary wind turbine over-production. We have done this by analyzing the limitations of the overproduction,and we have shown how the overproduction period may be maximized, in order tofully utilize the applicability for grid support. Our work additionally derive robustestimates of the overproduction period, under various parametric uncertainties. Itis further argued that the overproduction period will only increase slightly, evenwhen the increased load of the overproduction is shared between a larger numberof turbines. This work is published as [Juelsgaard et al., 2013c].

• Stability concerns of indirect demand management (Paper B)

We have outlined a framework for indirect control of a portfolio of consumers withthe purpose of obtaining a trade-off between the private concerns of a portfolioof consumers on the one hand, and the resulting cost of transport losses on theother. For this we introduce a simplified grid model, where all cable impedancesand power losses are aggregated into a common tie-line. To enforce control, theconsumers are provided an estimated price signal spanning a future horizon. Theprice signal is iteratively updated to incentivise consumers to plan consumption,such as to obtain the described trade-off. We demonstrate that even though thisframework functions as desired for one class of consumers, it may be unstable fora different class, thus demonstrating how the specific consumer behavior greatlyaffects the functionality of the indirect framework. Our work thereby demonstratesthe importance of ensuring that an indirect control framework will persist to be welldefined for all admissible consumer classes. This work is published as [Juelsgaardet al., 2013b].

• Loss reduction through direct coordination of consumption (Paper C)

We have extended the work presented in Paper B by deriving a direct rather thanindirect coordination strategy for a similar problem. It is discussed how the di-rect framework does not suffer from same stability issues as the indirect approachof Paper B. The direct coordination is used to obtain a trade-off between privateconcerns and shared cost of losses for a portfolio of consumers. The strategy di-rectly includes grid congestion as a concern, by including grid capacities in theproblem formulation. The presented coordination framework is distributed to thepoint where all consumers need to exchange information with a shared data cen-ter. Besides providing a direct and distributed coordination framework, this workadditionally demonstrates the flexibility and responsiveness of various classes ofconsumers, against shared concerns such as cost of losses. This work is publishedas [Juelsgaard et al., 2014a].

• Distributed portfolio balancing (Paper D)

We have derived a framework for maintaining power balance for a portfolio con-taining both power producers and consumers. Consumers and producers are coor-dinated in a distributed manner, where the distributed framework is more elaborate

25


than in Paper C, in the sense that coordination requires only neighbor-based com-munication, i.e., the central data storage is avoided. We show that the presentedframework converges to a sub-optimal, but arbitrarily good solution to the balanc-ing problem, even when faced with limited calculation time. Additionally, it isshown that the derived framework is robust against unit or communication linkfailures. This work is published as [Juelsgaard et al., 2014d].

• Voltage regulation by consumer coordination (Paper E, Paper F)

With vantage point in our work leading to the above results, we have expanded theframework to encompass a more elaborate grid model, also accounting for voltagedrops throughout the grid. For this, both centralized and distributed methods arepresented to coordinate consumers, such that limits on voltage variations are sat-isfied, while also including the private and shared objectives from our initial workon this subject. Similar to the work in Paper D, the derived distributed strategy re-quires only communication between neighbors in the grid. This work is publishedas [Juelsgaard et al., 2014b] and [Juelsgaard et al., 2014c].

The following Chapters 3 through 6, elaborates on the results outlined above, whereChapter 3 considers our work relating to the flexibility of production units, specificallyfocusing on wind turbines, and their flexibility to participate in grid balancing through up-regulation. Chapter 4 models the flexibility of consumers, and analyzes several problemswhere sub-components of the main challenges are treated individually. This serves topave the way for a more complex and complete coordination framework, presented inChapter 5, while Chapter 6 provides closing remarks.

Additional Publications

For the sake of brevity, not all publications completed throughout the study are enclosed.Rather, the extent has been limited to encompass only the most significant and thus, theworks described in [Juelsgaard et al., 2012, Juelsgaard et al., 2013a, Juelsgaard et al.,2013d, Juelsgaard et al., 2013e] contain intermediate work and results, leading up to themain publications included in the thesis, but are not explicitly included.

26

3 Wind Turbine Flexibility

In this chapter, our work relating to wind turbine flexibility is described, and

how it has contributed to the state of the art outlined in Section 2. As mentioned

in Section 2.3, the work on wind turbine flexibility is related to overproduction

of the active power output. This chapter first considers the task of maximizing

the overproduction period for a portfolio of wind turbines, and subsequently

considers the task of providing robust bounds on the overproduction, when

faced with parametric model uncertainties. This chapter focus on centralized

operation of the entire wind turbine portfolio. Distributed coordination meth-

ods are considered in the subsequent chapters.

3.1 Overproduction

As illustrated in Fig. 2.6 on Page 15, and discussed by [Anaya-Lara et al., 2006, Ul-lah et al., 2008, Tarnowski et al., 2009], it is possible for variable speed wind turbines(VSWTs) operating below rated power, to increase their power output above availablepower, by controlling the generator torque to extract the rotational energy of the rotor.This slows down the rotor. The overproduction is thus temporary and will be followedby a period of underproduction, where the turbine rotors are accelerated back the originaloperating conditions. Such overproduction can potentially be used for grid balancing insituations where a temporary, fast-acting boost of energy may aid in up-regulating thegrid, while slower resources are ramped up to relieve the overproduction of the turbines,and further cover the subsequent deficit originating from the underproduction.

From the above it follows that the material in this chapter primarily addresses the bal-ancing challenge, however, comments concerning the remaining challenges are broughtin Section 3.2.

The works cited above examine the extent of the overproduction ability quantitatively,through numerical experiments, using high-end turbine models. However, no qualitative,analytical assessment is provided. In Paper A, we present an analytical assessment of theoverproduction and how to maximize its duration, as summarized in the following.

We model a single turbine by a flywheel, as illustrated in Fig. 3.1. The turbine ischaracterized by an inertia J > 0, viscous friction B > 0, rotational velocity ω(t) ∈ R

and generator torque τg(t) ∈ R, at time t ∈ T = [0, T ] for some final time T > 0. Theturbine experiences a driving torque τw(t) ∈ R produced by the incoming wind field.

27

Wind Turbine Flexibility

J

τg(t) τw(t)Bω(t)

ω(t)

Figure 3.1: Flywheel model of a wind turbine.

The dynamics of the flywheel are given by

Jω(t) = -Bω(t)− τg(t) + τw(t)p(t) = ω(t)τg(t),

(3.1)

for all t ∈ T , with starting condition ω(0) = ω, where p(t) denotes the power producedby the turbine, and ω denotes a desired rotational velocity. In (3.1), τg(t) is a controllableinput, and p(t) is the corresponding output. Given the initial conditions, (3.1) thus definesa mapping τg(t) 7→ p(τg(t)) from generator torque to power output.

We define the available power for a turbine as the power generated in steady state, i.e.,

pavl(t) = τ g(t)ω, where τ g(t) = τw(t)−Bω. (3.2)

That is, τ g(t) is the generator torque that would result in ω(t) = 0, when operatingat ω(t) = ω. It is assumed that the wind has a constant power, denoted pw ∈ R+. Thepossibility of including a more elaborate model of the wind is discussed in Paper A, whereit is also argued that a constant wind power pw, translates to a constant wind torque, i.e.,τw(t) = τw. From (3.2), we see that a constant wind torque implies that the availablepower of the turbine will be constant.

We now extend the discussion to consider a portfolio of n ∈ N turbines, not necessar-ily located at the same geographical location, i.e. they could be distributed across a largerregion. We assign to each turbine an index i ∈ N = 1, . . . , n and use this to relate theentities introduced above, to each specific turbine, e.g. τg,1(t), τg,2(t), . . . . With a slightmisuse of notation we let

τg(t) = (τg,1(t), . . . , τg,n(t)) ∈ Rn,

and extend this soτg(T ) = τg(t)| t ∈ T ⊂ Rn,

i.e. τg(T ) is a torque trajectory across the entire horizon, for all turbines. We modela portfolio of turbines simply as an aggregation of individual turbines, disregarding anyinter-turbine effects such as changes in wind speed or turbulence throughout the portfolio.These effects can be included in the modeling of the wind torque, but is omitted in thiswork. The aggregation of individual turbines into a portfolio is illustrated in Fig. 3.2.We impose an aggregated power demand pdem(t) ∈ R+ to the portfolio, defined by

pdem(t) =

∑

i∈N

pavl,i, t < tc

(1 + γ)∑

i∈N

pavl,i, t ≥ tc, (3.3)

28

3.1 Overproduction

To grid

τw,1

τw,2

τw,n

p1(t)

p2(t)

pn(t)

Figure 3.2: Aggregation of turbines into a portfolio.

for some tc ∈ (0, T ), where γ > 0. For t < tc, the demand does not exceed the ac-cumulated available power and can thereby be tracked closely. For t ≥ tc, the portfoliois required to overproduce, and turbines will slow down. As described, the portfolio canonly overproduce in a limited time period. For any torque trajectory, we define this over-production period by

Top(τg(T )) = supt∈T

t− tc > 0

∣∣∣∣∣∀s∈[tc, t] pdem(s) ≤

∑

i∈N

pi(τg,i(s)),

, (3.4)

with pi defined as a mapping from generator torque into power output, as introduced in(3.1), and the demand is as defined in (3.3). That is, (3.4) provides the duration betweentc and the first following instance, where the farm production no longer meets or exceedsthe demand.

To increase the ability of the wind portfolio to participate in grid stabilization, the taskis to operate each turbine through control of generator torque, in a way that maximizesthe overproduction period. That is, given (3.4) we wish to solve

maximizeτg(T )

Top(τg(T ))

subject to ωi(t) =-BiJiωi(t) +

1

Ji(-τg,i(t) + τw,i)

p(τg,i(t)) = ωi(t)τg,i(t)ωi(t) ≥ ωmin, τg,i(t) ≥ 0,t ≥ tc,

(3.5)

with ωmin ∈ R denoting the lower limit of the rotational velocity. The problem presentedin (3.5) can be viewed as an abstract instance of Problem 2, for a portfolio of wind tur-bines, with the objective of meeting or exceeding a demand for as long as possible.

The physical relation p(τg,i(t)) = ωi(t)τg,i(t) is introduced as a constraint in (3.5).Given its non-linear nature, (3.5) is difficult to solve. A detailed discussion concerningthis is presented in Paper A. Here we shall simplify it by a linear approximation

p(τg,i(t)) ≈ ωiτg,i(t) + τ g,iωi(t)− ωiτ g,i (3.6)

29


where (ωi, τ g,i) is the operating point of the approximation. By writing the rotationalvelocity and generator torque as

ωi(t) = ωi + δωi(t), τg,i(t) = τ g,i + δτg,i(t),

where δωi(t) and δτg,i(t) describe the deviation from operating point, we see that theerror introduced by the linear approximation is δωi(t)δτg,i(t) which is negligible in thevicinity of the operating point.

By making the linear approximation (3.6), we can solve Problem (3.5) for a specificpower demand, by reformulating it as a feasibility problem. We do this by picking a fixedTop ∈ (0, T ), and afterwards, solve the feasibility problem

find τg(T )


1

Ji(-τg,i(t) + τw)

pdem(t) =∑

i∈N

(ωiτg,i(t) + τ g,iωi(t)− ωiτ g,i)

ωi(t) ≥ ωmin, τg,i(t) ≥ 0,

0 < t < tc + Top.

(3.7)

To solve (3.7), we first discretize it, whereafter the problem is convex for any Top, andcan be solved efficiently by known methods [Boyd and Vandenberghe, 2004].

Given the results presented in Paper A, (3.7) will be always feasible for sufficientlysmall values of Top. However, it will not be feasible for large values. By iterating betweenlarge and small values for Top e.g. in a bisection manner [Boyd and Vandenberghe, 2004],we can converge on a value where (3.7) is just feasible, thereby approximating a solutionto (3.5). This approach is outlined in Fig. 3.3, illustrating how large an overproductiona portfolio is able to maintain, if it is required to maintain it for T 1

op, T2op, T

3op and T 4

opseconds respectively.

T 1op

T 2op

T 3op

T 4op

pdem

pavl

ttc

∑n i=

1pi(t)

T ∗op

Figure 3.3: Illustration of the iterative solution approach to the feasibility problem, for 5values of Top (short dashed), the demand pdem (long dash), and the available power pavl

(solid).

For T 1op, which is a very small period, the portfolio is capable of producing a very large

power before the energy stored in the rotor is depleted. This power spike is far larger than

30

3.1 Overproduction

the demand. For T 2op > T 1

op, the portfolio can produce less power, but still more than the

required demand. Similarly for T 3op. However, T 4

op is too large, in that the portfolio is notable to obey the demand for the required overproduction period, before the stored energyis depleted. By iterating, we can eventually find the value T ∗

op, where T 3op < T ∗

op < T 4op,

such that the demand is exactly met throughout the period. The solution found in this wayonly approximately solves (3.5), on account on the simplification made in (3.6).

The bisection approach outlined above is completely general, in the sense that itmakes no assumptions on how large part any one turbine should play in the overall over-production, or when any one individual turbine should start to overproduce, with respectto tc. Instead it simply finds the torques that should be applied to each turbine over time,in order to reveal the longest overproduction period, given the demand. We will illustratethis with an example.

Numerical Example: Maximizing Overproduction Period

Consider a portfolio consisting of n identical turbines, with

B1 = · · · = Bn = B, J1 = · · · = Jn = J,

where B and J are well known. The available power defined by (3.2) is thereby constantand equal for all turbines, and the available power of the portfolio becomes Pavl = npavl.The portfolio is subjected to a demand

pdem(t) =

Pavl, t < tc,

Pavl +1

2nPavl, tc ≤ t < tc + Top,

Pavl, t ≥ tc + Top,

and we use the approach outlined above for maximizing Top.In the following, we consider a portfolio with n = 10 turbines, for which we instan-

tiate the overproduction at tc = 50 s. When using B = 1 kg · m2/s, J = 80 kg · m2,ω = 4π rad/s, τw = 201 Nm and ωmin = 0.7ω rad/s, we obtain the results presented inFig. 3.4 through 3.5, and an overproduction period of Top = 10.7 s.

Fig. 3.4(Left) presents the power produced by each of the 10 turbines. As it ap-pears, they all produce the same power. The production increases at t = 50 s where theoverproduction period starts. Following the overproduction, comes the underproductionperiod, where all turbines recover to their original operating conditions. By summingthe production from each turbine, the portfolio production is obtained. It is presentedin Fig. 3.4(Right) along with the demand. As seen from the figure, the demand is nottracked during the recovery period, leaving a production deficit that must be covered bysome other strategy. This strategy has not been derived in our work, but as mentionedinitially, one approach could be to use the fast-acting nature of the wind turbines, to makea fast-acting boost of energy to the grid, while ramping up slower resources to cover theimpending deficit.

The final figure presented in Fig. 3.5, shows the rotational velocities of all turbinesalong with the allowed lower limit. As seen from this figure, the overproduction is main-tained until all stored energy for the turbines has been depleted, whereafter the turbinesstart to recover.

31


pi(t)

[kW

]

t [s]20 40 60 80 100

500

1000

1500

2000

2500

3000

pi(t)

[kW

]

t [s]20 40 60 80 100

×104

0.5

1

1.5

2

2.5

3

Figure 3.4: Left: The power produced by each individual turbine. Right: The accumu-lated production of the portfolio (Solid), and the portfolio demand (Dashed).

ωi(t)

[rad

/s]

t [s]20 40 60 80 100

8

9

10

11

12

13

Figure 3.5: The rotational velocity of each turbine in the portfolio (Solid) and the allowedlower limit (Dashed).

We remark to the reader that the example presented here is meant to be illustrative,and the reader should not put too much emphasis on the actual numbers employed in theexample.

Robust Overproduction Bound

As described above, the general approach for maximizing the overproduction suffers fromthe disadvantage of requiring exact information about the parameter values for all turbinesin the wind farm model. In the following we present an alternative formulation that is lessgeneral, but provides a certain robustness against parametric uncertainties.

We employ a simple dispatch strategy to calculate power references pref,i(t) ∈ R+, i ∈N , for each turbine:

pref,i(t) = pavl,i(t) +pdem(t)−

∑ni=1 pavl,i(t)

n. (3.8)

Employing a dispatch strategy to find the overproduction period can obviously only besuboptimal compared to solving the general problem in (3.7). However, as the task isto provide a robust lower bound on the overproduction period, this approach is moresuitable. Further discussion concerning this is provided in Paper A.

32

3.1 Overproduction

Following (3.8), if pdem(t) ≤∑ni=1 pavl,i(t), the references dispatched by (3.8) will

all be less than the available power for the individual turbines, i.e., pref,i(t) ≤ pavl,i(t),thus by tracking the reference, it will be possible for the production to meet the demand.If, on the other hand pdem(t) >

∑ni=1 pavl,i(t), then pref,i(t) > pavl,i(t), and the turbine

will only be able to follow this reference for a limited time, as the rotor will slow down.If we employ (3.8), for the demand in (3.3), we see that the production reference to

all turbines can be expressed as

pref,i(t) =

pavl,i, t < tc

pavl,i + γ

∑nj=1 pavl,j

n, t ≥ tc

,

where we have omitted the time dependency on pavl(t), since the assumption of constantwind implies constant available power.

Since the demand is constant for t > tc, this entails that the references will be constantfor t > tc, i.e.,

pref,i = pavl,i + γ

∑nj=1 pavl,j

n, t > tc

for i = 1, . . . , n. When tracking this power reference, the model in (3.1), with the lin-earization in (3.6), is expressed by

Jiωi(t) = -Biωi(t)− τg,i(t) + τw

pref,i = ωiτg,i(t) + τ g,iωi(t)− ωiτ g,i,

for t > tc, and ωi(tc) = ωi.In Paper A the differential equations are solved, revealing closed form expressions for

the required τg,i(t) and resultingωi(t), when tracking the overproduction reference. Fromthese, an analytical expression can be found for the time intervalTop,i, where pi(τg,i(t)) =pref,i, before ωi(t) reaches the lower bound ωmin. We obtain

Top,i =ωiJi

τ g,i −Biωiln

(ωmin − aiωi − ai

)

. (3.9)

where

ai =Bipref,i +Biωiτ g,i − τg,iτw

Biτ g,i −B2i ωi

+τw

Bi.

Let the model parameters be unknown, except that they are known to lie in certain inter-vals, i.e,

Bi ∈ [Bmin; Bmax], Ji ∈ [Jmin; Jmax],

for i ∈ 1, . . . , n. It is shown in Paper A that if τw > 2Bω , and if the power referenceduring overproduction is expressed as pref,i = βpavl,i, β > 1, and the lower limit onrotational velocity as ωmin = αω , 0 < α < 1, then the lowest value for Top,i over allpossible parametrizations of Bi ∈ [Bmin; Bmax] and Ji ∈ [Jmin; Jmax], is obtained forBi = Bmin and Ji = Jmin.

This means that even when the specific parameter values for the parameters acrossour portfolio model are unknown, a lower bound on the over production period can stillbe obtained. The implications of this can be illustrated by the following simple example.

33


Numerical Example: Robust Overproduction

We consider a portfolio consisting of n identical turbines, meaning that

J1 = · · · = Jn = J, B1 = · · · = Bn = B,

which yields the same available power, pavl,i = pavl, i = 1, . . . , n. The power demandduring the overproduction period is given by

pdem =

(

n+1

2

)

pavl,

corresponding to γ = 1/(2n) in (3.3).From the dispatch strategy (3.8), the production from each turbine during overpro-

duction is

pref,i =

(

1 +1

2n

)

pavl, i = 1, . . . , n

The overproduction with respect to the available power for each individual turbine isthereby inversely proportional to the number of turbines in the portfolio.

Using (3.9), Fig. 3.6 illustrates the mapping between n and Top, for n = 1, . . . , 1000.The figure presents four graphs, corresponding to the four combinations of maximum andminimum B and J . This corresponds to examining the four vertices of the parametricuncertainty region.

In Fig. 3.6, we have used [Bmin; Bmax] = [1; 4] kg · m2/s, and [Jmin; Jmax] =[80; 120] kg · m2. We have further used ω = 4π rad/s, τw = 201 Nm and ωmin =0.7ω rad/s.

Top

[s]

n [-]0 200 400 600 800 1000

0

20

40

60

80

Figure 3.6: The overproduction period, as a function of portfolio size. All turbines have the sameinertia and friction, with the four combinations: Bmin; Jmin(Solid, Plain), Bmax; Jmin(Solid,Dots), Bmin; Jmax(Solid, Asterisk), Bmax; Jmax(Dashed).

Fig. 3.6 reveals several interesting points. First, the gain in Top, obtained by increasingthe number of turbines n in the portfolio, decreases as n increases. That is, even thoughthe required overproduction of the portfolio remains fixed, the increase in Top obtainedby increasing n, is only minor. Secondly, the specific value of the model parametershave significant impact on the overproduction period. However, as expected, the lowestvalue of Top is obtained for Bi = Bmin and Ji = Jmin, i = 1, . . . , n. This examplethus reveals how upper and lower bounds on the overproduction period may be obtained

34

3.2 Reactive Compensation

by the portfolio operator, in order for him to asses the applicability of the portfoliosparticipation for grid balancing in various situations. The work in Paper A additionallypresent a method for turbine operation, capable of realizing the obtained the lower boundon the overproduction period, when faced with parametric uncertainties.

3.2 Reactive Compensation

The above discussion focuses solely on flexibility of active power output as a possibletool for grid balancing. As mentioned in Section 1.1, the additional challenges pertainingto losses and voltage variations, may be accommodated by proper compensation of thereactive power flow. Several works have discussed how wind turbines may be used forreactive compensation (e.g. [Cimuca et al., 2006,Margaris et al., 2010]), however this hasnot been included in our work. Rather, reactive compensation has only been consideredfor demand management and household related power production from solar panels, tobe elaborated onwards.

This concludes the overview of our work on wind turbine flexibility. The followingchapter provides a similar overview on our work on consumer flexibility, and demandmanagement.

35

4 Demand Management

In this chapter we introduce the notation used to describe consumer flexibil-

ity for demand management. We derive consumer models, and use these to

demonstrate how demand management can be included in different formula-

tions of general portfolio coordination, Problem 2 introduced on Page 9. We

discuss several instances of the coordination problem, presented in a some-

what simplified, and high level manner. This serves to pave the way for a more

complex and complete coordination problem presented in the following Chap-

ter 5, to which we devote a more detailed exposition. This reflects the bottom

up nature of our work.

4.1 Consumer Flexibility

In the following, we consider the coordination horizon [0, T ] as discretized into N ∈ N

intervals of equal length Ts = T/N , representing the sample time. We reuse the symbolT = 1, . . . , N to denote the discretized coordination horizon. We consider a portfolioN = 1, . . . , n containing n consumers and introduce pi, pi, pi : T → R to expressthe average flexible, inflexible and total power consumption, respectively, for each periodt ∈ T and each unit i ∈ N . That is

pi(t) = pi(t) + pi(t), t ∈ T , i ∈ N . (4.1)

The inflexible component relates to the traditional consumption of light fixtures, cooking,television, etc, that is consumption that does not allow for temporal shifts. Conversely, theflexible consumption allows for temporal shifts, following a set of constraints, and couldencompass consumption of e.g. heating, transportation, washing, etc. We will in generalassume that the inflexible consumption is known either by estimation or from historicaldata, whereby (4.1) simply defines an offset between flexible and total consumption.

Notice that pi(t) is the average power during period t ∈ T , i.e. pi(t) should not beconfused as an instantaneous power. Similarly for the remaining quantities. As pi(t) isan average measure of power, it is equivalent to corresponding energy of Tspi(t).

We extend the notation in (4.1) such that

p(t) = (p1(t), . . . , pn(t)) ∈ Rn, and pi = (pi(1), . . . , pi(N)) ∈ RN , (4.2)

37

Demand Management

for i ∈ N , and thus

p =

p⊤1...p⊤n

=

| |p(1) . . . p(N)| |

∈ Rn×N . (4.3)

We employ corresponding notation for all similar maps and matrices. We refer to p as theconsumption profile for the consumers. The consumption profile of each consumer is gov-erned by a private objective function fi : RN → R, i ∈ N , over some set of constraintsPi ⊂ RN , in the sense that if no coordination is enforced, the preferred consumptionpattern of each consumer would be given as

pi = arg infy∈Pi

fi(y), ∀i ∈ N . (4.4)

The flexibility of a consumer is its ability or willingness to deviate from the preferredbehavior, and the task of coordination is to align the uncoordinated behavior in (4.4)to also account for high-level concerns. In the following we elaborate on the specificformulation of the private constraints and objective.

Objective Function

The objective function is a measure of displeasure of the consumer, associated to anyconsumption profile. The objective may consist of several components reflecting eitherincurred financial expenses or discomfort.

To define the objective formally, we initially consider a consumer as a dynamic sys-tem, where a linear approximation of the consumer dynamics is given by

xi(t+ 1) = aixi(t) + bipi(t) + εi(t), i ∈ N , t ∈ T , (4.5)

with some initial condition xi(1) = xs,i. For simplicity, we derive the following for firstorder models, but our work allows directly for higher order models. In (4.5), ai ∈ (0, 1),bi, ǫ(t) ∈ R are the private model parameters and external disturbance, respectively. Bothmodel parameters and disturbances are specifically related to the appliance installed bythe consumer.

The state xi(t) can be considered as the energy stored by the consumer. This could beas electric energy stored in the battery of an EV, or thermal energy stored in a household,etc. Letting xi = (xi(1), . . . , xi(N)) ∈ RN , εi = (εi(1), . . . , εi(N)), then, given xs,iand a known estimate of εi, the state trajectory of (4.5), may be written as an affine map:

xi : RN → RN , pi 7→ xi(pi) = Φixs,i +Ψipi + Γiεi, (4.6)

where

Φi =

aia2i...

aN+1i

, Γ =

1ai 1...

.... . .

aNi aN−1i · · · 1

,

and Ψi = biΓi.

38

4.1 Consumer Flexibility

One concern to include in the consumer objective, could be the discomfort related toa given consumption profile. To model discomfort we introduce a family of maps, suchthat for any γ ∈ N ∪ ∞ and any euclidean space E, we associate the map

dγ : E × E → R, (u, y) 7→ dγ(u, y) = ‖u− y‖γ

where ‖ · ‖γ refer to any standard norm. Then, provided psp ∈ RN as a desired set-pointconsumption profile, or xsp ∈ RN as a desired state trajectory, the discomfort could beexemplified as

d2(psp, pi), or d2(xsp, xi(pi)) (4.7)

respectively. As indicated, we commonly employ the 2-norm, or squared 2-norm, butother conventional norms such as 1-norm ( [Pedersen et al., 2011]), or ∞-norm may beequally justified. Alternative discomforts could be introduced, however, we limit ourattention to equivalents of (4.7).

Another common concern included when modeling consumers for demand manage-ment, is the desire to minimize the cost of energy [Pedersen et al., 2011, Corradi et al.,2013, Biegel et al., 2012]. Thus, provided an estimate of the electricity price, w(t) ∈R+, ∀t ∈ T , the objective of consumers would be to minimize the accumulated cost ofelectricity w⊤pi.

When combining the cost of energy with one or more measures of discomfort, the ob-jective of a consumer essentially becomes multi-valued. We scalarize the objective func-tion by introducing one or more trade-off parameters [Boyd and Vandenberghe, 2004], forexample for the case of dual objectives including w⊤pi and d2(xsp, xi(pi)), the objectivefunction becomes

fi(pi) = w⊤pi + λd2(xsp, xi(pi))

where λ ≥ 0 is the trade-off parameter.It is important to stress that the above only represents possible formulations of the

private objective of a consumer. Each individual consumer may employ different objec-tives, according to whatever appliances are installed in the household. In the followingas well as in Chapter 5, we present various alternative objective formulations, includingthe extreme case where fi = 0, i.e. all consumption profiles are equally acceptable to theconsumer, provided they satisfy the provided constraints.

Operating Constraints

The power consumption of a household is constrained either by appliance ratings, con-sumer comfort demands or similar. To describe the constraints included in our work, weassociate to any mapping h(p), and scalars h, h ∈ R a set Ch(h, h) ⊂ RN , defined as

Ch(h, h) = p | h ≤ h(p) ≤ h. (4.8)

In case h(p) is vector valued, the inequalities are to be understood entry-wise. Using(4.8), we can formulate constraints on e.g. power range, state range, set-point deviation,etc., as

pi ∈ CI(p, p), pi ∈ Cxi(x, x), pi ∈ Cxi

(x+ xsp, x+ xsp),

39

Demand Management

respectively, where I refers to the identity and p, p, x, x are provided limits. If h = h,then (4.8) reduces to

Ch(h, h) = p | h ≤ h(p) ≤ h = p | h(p) = h,

that is, the inequality constraint has been reduced to an equality constraint and we simplywrite Ch(h). When two or more of the above constraints are to be enforced, the jointconstraint set can be formed using intersections of the form

pi ∈ CI(p, p) ∩ Cxi(x, x) ∩ . . .

This represents constraints on the flexible consumption. To translate this into constraintson the total consumption we write

pi ∈ Pi = pi = pi + pi|pi ∈ CI(p, p) ∩ Cxi(x, x) ∩ . . . . (4.9)

As for the formulation of the objective function in Section 4.1, it is important tostress that the above only represents possible formulations of the private constraints ofa consumer. Each individual consumer may subject different constraints, according towhatever appliances are installed in the household.

Convexity

Many works on demand management include constraints formulated as in (4.8), wherethe map h is usually linear or affine, such as the state map (4.6), or an identity mapping[Pedersen et al., 2011,Petersen et al., 2013,Kraning et al., 2014]. We conform to the sametype of constraints in our work. When enforcing affine or linear mappings in (4.6), theconstraints of the consumer compose a convex set, which is the focus of this work.

Some appliances may present non-convex constraints, such as thermostatically con-trolled loads (TCLs) [Callaway, 2009, Totu and Wisniewski, 2014]. These are on-offcontrolled, whereby their consumption cannot be controlled continuously between an up-per and lower limit. Additionally, these appliances usually present minimum up or downtimes, which are difficult to formulate in a convex framework. However as these appli-ances are usually quite small, their individual consumption would be insignificant withrespect to a larger grid section. As discussed in Section 2.2 an aggregator would be re-quired to combine flexibility from a large number of units.

As presented in [Callaway, 2009,Totu and Wisniewski, 2014] a fleet of TCLs may ag-gregately be approximated accurately as a single continuous large-scale consumer, whichcan be described by convex constraints. That is, even though each individual applianceis non-convex, the aggregation may be approximately convex. Each convex consumerformulation in the portfolio considered in this work, may thus be interpreted as a high-level abstraction of a fleet of small scale appliances. The operation of the fleet may beapproximated as convex from the general point of a high-level aggregator, whereas thenon-convex operation of each individual appliance may be governed by specialized localcontrol loops, outside the scope of this work.

With the above outline of consumer flexibility, the following sections presents threeintroductory formulations of the general portfolio coordination, Problem 2, specificallyincluding consumers for demand management. We discuss the challenges outlined in

40

4.2 Indirect Loss Minimization Across Tie-Line

Section 1.1, as approached from a demand management perspective. Additionally, werevisit the discussion of direct and indirect control, and discuss potential drawbacks ofindirect control, related to the unknown response of consumers to external incentives.


In Section 2.2 we mentioned briefly how the uncertainty of consumers reaction to anindirect incentive makes such approaches less controllable compared to a direct approach.This is demonstrated in the following, where a simple and intuitive formulation of indirectcontrol is presented, and it is shown how the lack of controllability may affect the stabilityof the approach.

The lack of controllability of indirect control was also the focus of [Sossan et al.,2013], who showed that the unknown behaviors of consumers could easily cause conges-tions in the grid, and introduced a local price controller at each substation in the grid,to alleviate the congestion. As further discussed by [Heussen et al., 2012], uncertaintyof consumers reaction to varying incentives, in particular prices, may even challenge thestability of the system. The discussion concerning instability presented in [Heussen et al.,2012] was on a conceptual level, and no tangible demonstrations were provided. Also,both of the works cited above considers real-time operation of indirect control, and howconsumers react to real-time price updates. In the following we take a different approach,in that we examine indirect control with respect to a planning problem, e.g. for day-aheadscheduling and coordination of consumption of a portfolio of residential consumers.

The coordination is conducted with focus on the loss minimization challenge dis-cussed in Section 1.1, meaning that the following seeks to coordinate consumers to obtaina trade-off between optimizing private objectives, on the one hand, and minimizing costof transport losses on the other. As we argued in Section 1.1, consumers have a financialmotivation to minimize losses, since the cost of losses incurred by the distribution systemoperator (DSO) are passed onto the consumer through tariffs. To minimize losses, a pricesignal is introduced and iteratively updated with the purpose of incentivising consumersto coordinate their individual consumption paterns in a fashion that weighs private con-cerns against increasing prices du to losses. The following relies on the results presentedin Paper B.

We consider a portfolio of n ∈ N closely located households such as a single streetor small suburban town. We refer collectively to such a group of households, as a com-munity. The power feed to the community from the remaining grid, is introduced througha lossy tie-connection. These losses represent ohmic losses, transformer losses, etc. Weassume that the households of the community are located sufficiently close, so the losseswithin the community are minor compared to losses in the tie-connection, and may bedisregarded. The community can conceptually be illustrated as in Fig. 4.1. In the follow-ing analysis we assume that the grid is balanced, allowing us to model the losses for asingle phase equivalent system [Kundur, 1993].

To model the cost of losses, we introduce l : Rn×N → RN as

l(u)(t) =

(n∑

i=1

ui(t)

)2

, (4.10)

41

Demand Management

Grid

tie-connection

Figure 4.1: Conceptual schematic outline of the community.

where the indexing of u should be understood as introduced in (4.2). Paper B demon-strated, assuming that the community collectively has constant power factor, that the av-erage tie-line losses are expressed as βl(p), for some grid-related parameter β ∈ (0, 1),depending on the tie-line resistance. Given a price signal w ∈ RN+ , the cost of lossesbecomes βw⊤l(p), and the coordination problem is thus

minimizep

βw⊤l (p) +∑

i∈N fi(pi)

subject to pi ∈ Pi,(4.11)

where (4.11) represents a specific (discretized) instance of the portfolio coordination,Problem 2, where the community of households constitutes a portfolio of consumers. Weconsider (4.11) for two cases, each referring to a community consisting solely of one oftwo types of consumers

• Greedy consumers: The greedy consumer-type represents consumers who areonly concerned with the total price of energy across the horizon. This could beconsumers who only provide flexibility to the grid through, for example, an EV.The consumer is thereby only concerned that the EV is charged to some requiredlevel, at the end of the horizon, i.e., that the flexible consumption integrates to somefixed value pdem,i > 0, ∀i. If this is achieved, the specific charge pattern is of noconcern, or discomfort. The uncoordinated behavior is thus the solution of the pri-vate problem

minimizepi

fi(pi) = w⊤pi

subject to pi ∈ Pi = pi = pi + pi|pi ∈ C1⊤(pdem,i)

• Comfort consumers: As opposed to the greedy consumer, the comfort consumeris not only accounting for the cost of energy when optimizing a consumption pro-file, in that the comfort consumer also includes a cost of discomfort. This couldbe exemplified as a consumer with an installed EHP, where the discomfort is mea-sured as deviation from a desired set-point. The uncoordinated behavior of comfortconsumers thus follow

minimizepi

fi(pi) = w⊤pi + λid22(xi(pi), xsp,i)

subject to pi ∈ Pi = RN

42


where the consumer state is modeled as described in (4.6). The specific value ofthe parameters used in the model are not important for our current purpose.

Various relevant constraints have been omitted in this formulation, and the reader is re-ferred to Paper B for a discussion on including these. The above outlines the behaviorof two widely different classes of consumers, in some way representing two extremes offlexibility. Many combinations and extensions can be derived from these generic classes.

One way of incentivising consumers to take into account the cost of losses, is to dis-tribute cost of losses among the consumers based on their individual share factorspi(t)/(

∑

j∈N pj(t)) during each period t ∈ T . The cost of losses of an individual con-sumer then becomes

β∑

t∈T

w(t)l(p)(t)pi(t)

∑

j∈N pj(t)=∑

t∈T

βw(t)pi(t)

(∑

j∈N

pj(t)

)

= βp⊤i W

(n∑

j=1

pj

)

(4.12)

for i ∈ N , where W = diag(w). To simplify notation, we let

p!i =∑

j 6=i

pj,

such that (4.12) may be written

βp⊤i W

n∑

j=1

pj

= βp⊤i Wpi + βp⊤i Wp!i.

Including the cost of losses thus mean that each consumer has preferred consumption

p⋆i (p!i) = arg infpi∈Pi

(βp⊤i Wpi + βp⊤i Wp!i + fi(pi)),

for i ∈ N , when provided the value of p!i.For each individual household i ∈ N , the consumption of the remaining community,

p!i, is unknown. Our approach to coordination in this indirect control case, is thereforeto assume that an estimate of p!i is available to each household. Such estimates couldbe made available, for instance, by the distribution system operator (DSO). Based onthe estimate of p!i for i ∈ N , each consumer can optimize consumption locally, andthe estimate of p!i for all i, can subsequently be updated in an iterative fashion. This issummarized in Algorithm 4.2.1.

Algorithm 4.2.1 Indirect control of consumers

Initialize estimates p(0)!i , i = 1, . . . , nfor k=0,1, . . . do

Obtain local solutions:p⋆i (p

(k)!i ) = arg inf

pi∈Pi

((βp⊤i Wpi + βp⊤i Wp!i + fi(pi)), i = 1, . . . , n

Update estimates:p(k+1)!i =

∑

j 6=i p⋆j (p

(k)!j ), i = 1, . . . , n

end for

43

Demand Management

In Algorithm 4.2.1, we have used p(k)!i , to denote the estimated p!i, at iteration k. Conver-gence of Algorithm 4.2.1, in the sense that

limk→∞

(p⋆i (p(k+1)!i )− p⋆i (p

(k)!i )) = 0, ∀i, (4.13)

implies that there is no update of the consumption patterns for any consumer. The pro-cess has then reached a Nash equilibrium, in the sense that no consumer desires to altertheir current consumption pattern, provided that the remaining community refrains fromchanging theirs as well.

In the coordination approach outlined above, no consumer discloses any informa-tion concerning their private objective or constraints. Instead each individual consumerprivately plan and optimize their consumption pattern, accounting for private comfortconcerns, as well as the cost of energy and grid-losses. The coordination does howeverrequire some exchange service, or shared data center for collecting p⋆i (p

(k)!i ) and distribut-

ing p(k+1)!i for each iteration of the algorithm, for all i. This is illustrated in Fig. 4.2.

Shared Data Center

Consumers

p(k+1)!1p⋆1(p

(k)!1 ) p

(k+1)!np⋆n(p

(k)!n )

Figure 4.2: Accounting for grid losses, requires data exchange with a shared data center.

The communication structure outlined in Fig. 4.2 requires local computation devicesto be available to each consumer, as well as a way of passing information between localand high-level computations. This illustrates the smart grid framework introduced inSection 1.2, and is similarly employed onwards in the subsequent sections.

Granting consumers access to information in the fashion outlined by Fig. 4.2, is whatallows them to account for grid losses, by considering the action of the remaining mem-bers of the community. Notice, that no consumer receives direct information about anyother consumer. Instead, only information concerning the community as a whole is dis-tributed, so no privacy concerns are violated.

We shall not present numerical demonstrations here, and the reader is referred toPaper B. The work in Paper B proved that the derived coordination framework will bestable and converge to a Nash equilibrium for arbitrarily large communities consistingof comfort consumers, provided that the trade-off parameter is sufficiently large. It willhowever not be stable for communities consisting of greedy consumers, for n > 2, in thatthe volatility of the consumer behavior would never allow the iterative update to converge.

With this we have illustrated that an indirect strategy that is suitable for one class ofconsumers, may be completely inappropriate for another class, and this illustrates the im-portance of the behavior of each specific consumer. Since the behavior of each consumer

44

4.3 Direct Loss Minimization Across Tie-Line

cannot be controlled, and may even be unknown in the framework of indirect control,deriving a coordination framework that is unambiguously appropriate for all admissibleconsumer classes, may be a complicated task to complete.

For our simple case of two types of consumers, the framework may be remedied byenforcing a more elaborate data exchange strategy, e.g. by include some filtering processbefore distributing the estimates p!i to the households, in order to prevent divergence ofthe iterative process. It can be shown that even simple filtering schemes present stabi-lizing capabilities for both types of consumers outlined here. However, in that case it isnecessary to analyze the risk of cheating, i.e. the risk of some consumers deviating fromthe agreed data exchange, if this presents potential benefits.


The above discussion has spurred us to go in the direction of direct control. The fol-lowing therefore derives a direct coordination framework to solve a similar problem asdiscussed in Section 4.2. The derived approach is distributed and relies on a communi-cation strategy similar to the one used in Section 4.2. The main difference between theindirect framework and the direct approach outlined below, is solely that in the following,we assume that the consumer response to external signals may be decided based on anagreed contract with the consumer, as illustrated in Fig. 2.4. In that sense, we decide andthereby know, how consumers react to external signals. We shall demonstrate how thisknowledge allows us to derive a framework that is not affected by consumer behavior inthe same way as the previous indirect approach. The following is built upon the work andresults presented in Paper C.

We consider the same portfolio of consumers as in Section 4.2 arranged in a commu-nity as the one illustrated in Fig. 4.1. As before, we assume that an estimated price signalw ∈ RN+ is available, spanning the coordination horizon T = 1, . . . , N. Again, wemodel losses and cost of losses as

βl(p), and βw⊤l(p), (4.14)

where the loss map l was introduced in (4.10). We expand the problem of Section 4.2 bydividing the grid into m ∈ N radials as indicated in Fig. 4.1, and enforce capacity limitson each radial as well as the tie-line itself. For this we introduce θ0, θ1, . . . , θm, whereθ0 denotes the tie-line capacity and θj , j ≥ 1 is the capacity of the respective radial. Bydefining

h : 2N × Rn×N → RN , (N , p) 7→ h(N , p) =

∣∣∣∣∣∣

∑

i∈N

pi

∣∣∣∣∣∣

,

andNj = i ∈ N| household i is on radial j, j = 1, . . . ,m,

the capacity constraints are formulated as

h(N , p) ≤ θ0, h(Nj , p) ≤ θj , j = 1, . . . ,m. (4.15)

45

Demand Management

We model consumer behavior in a similar fashion as with the greedy and comfortcharacteristics in Section 4.2, but we combine the two such that each consumer occupiesboth characteristics. In that sense, each consumer has two appliances installed, exem-plified by one EHP and one EV. Each consumer is privately concerned with both thecomfort maintained by their EHP, the charging of their EV, and the overall cost of thetotal power consumption. Additionally, we add the shared objective of minimizing losses.

To model each consumer with two appliances, we temporarily modify the notationintroduced in Section 4.1, in the sense that the flexible consumption is now

pi : T → R2, that is, pi(t) =

[

pehpi (t)pevi (t)

]

reflecting the consumption of each flexible appliance. We further write

pehpi = (pehp

i (1), . . . , pehpi (N)) ∈ RN ,

and similarly for the EV. The total power consumption of a household is then

pi = pi + pehpi + pev

i

We also expand the dynamic model described in (4.5) such that

xi : RN × RN → RN × RN , that is xi(pehpi , pev

i ) = (xehpi (pehp

i ), xevi (p

evi )),

where we design xehpi , xev

i in the same fashion as in (4.6), for i ∈ N .Provided a desired set-point xsp,i ∈ RN of the EHP, and a desired final state of charge

xF,i ∈ R of the EV, we formulate the consumer objective function as

fi(pi) = w⊤pi︸︷︷︸

Cost

+λehpi d22(xsp,i, x

ehpi (pehp

i ))︸︷︷︸

EHP discomfort

+λevi d

22(xF,i, x

evi (p

evi )(N))

︸︷︷︸

EV discomfort

, (4.16)

for all i ∈ N , where λehpi , λev

i > 0, are the household trade-off parameters.For both the EV and EHP we enforce power and state limits, that is

pevi ∈ CI(p

evi, pevi ) ∩ Cxev

i(xevi , x

evi ), i ∈ N ,

where xevi , x

evi , p

evi, pevi are the constraint limits. We introduce similar constraints for the

EHPs, whereby the constraint for the complete power consumption becomes

pi ∈ Pi =

pi = pi +pehpi + pev

i | pji ∈ CI(pj

i, pji ) ∩ C

xji(xji , x

ji ), j ∈ ehp, ev

,

for i ∈ N . We can implicitly include these constraints, by formulating the objective ofeach consumer as an extended value function, such that

f+i (pi) =

fi(pi), for pi ∈ Pi

+∞, otherwise(4.17)

Similarly, we can define the grid losses as an extended value map

l+(p) =

l(p), for h(N , p) ≤ θ0, h(Nj , p) ≤ θj ,

+∞, otherwise.(4.18)

46


From (4.17) and (4.18), the coordination problem becomes

minimizep

βw⊤l+(p) +∑

i∈N f+i (pi) (4.19)

which essentially is the same problem as presented in (4.11), only expanded by the im-plicit capacity constraints.

In (4.19) consumers are coupled through the losses, and the constraints on powertransport capacity. However, introducing auxiliary variables yi, i ∈ N , as well as consis-tency constraints yi = pi, the problem can be brought to the equivalent form

minimizep,y

βw⊤l+(y) +∑

i∈N f+i (pi)

subject to p = y(4.20)

By the identifications

p ∼ u, A ∼ I, B ∼ −I, ψ ∼∑

i∈N

f+i , φ ∼ βw⊤l+,

the reader may verify that (4.20) is on the ADMM-form discussed in Section 2.5, as pre-sented in (2.6). As l+ and f+

i , i ∈ N represent convex functions of closed polyhedralsets, the ADMM Algorithm 2.5.3 can be applied to obtain the global optimum in an itera-tive fashion, enforcing sequential updates of p, y and ν which is the Lagrange multipliersfor the equality constraint in (4.20).

As elaborated in Paper C, the objective in (4.20) is separable in pi meaning that theiterative update of p in Algorithm 2.5.3 can be performed in parallel for each pi, i ∈ N .This ultimately implies that Algorithm 2.5.3 applied to (4.20) requires a communicationstructure as illustrated in Fig. 4.3, where (y(k)i , ν

(k)i ) are distributed to each consumer by

a central data storage, whereafter p(k+1)i are collected from each consumer.

Shared Data Center

Consumers

(y(k)1 , ν

(k)1 )p

(k+1)1 (y

(k)n , ν

(k)n )p

(k+1)n

Figure 4.3: Data passing for the direct coordination approach for minimization of tie-linelosses.

This communication structure resembles closely that illustrated in Fig. 4.2. The al-gorithm does not require the individual consumer to disclose their private objective orconstraints. Rather, it is only required that each consumer reacts in a specific fashion tothe quantity (y

(k)i , ν

(k)i ) provided by the data storage, as dictated by Algorithm 2.5.3.

47

Demand Management

The direct framework described above has in essence the same benefits as the indirectapproach in terms of limited communication requirements and information sharing. How-ever, the direct approach is guaranteed to converge to the global optimum for any con-sumer behavior that complies with the prerequisites for the applied algorithm. The spe-cific behavior of the consumers therefore cannot influence the stability of the approach,in the fashion that was discussed in Section 4.2. Requiring that consumers behave in aspecific way when presented an external signal, thus represent the gained controllabilityof the direct framework.

Numerical examples are presented in Paper C, where it is also demonstrated how theproblem size of (4.20) may be reduced such that the number of added auxiliary variablesscales with the number of radials, rather than the number of consumers. The numericalexamples in Paper C further demonstrate how the discomfort formulation of EVs andEHPs in (4.16) affect their flexibility and impact on (4.19). To recap from the paper: thesetpoint enforced for EHPs restricts their flexibility, as a significant temporal shift of theirconsumption, would incur an unacceptable discomfort. On the other hand, the lack ofset-point tracking for EVs makes them more flexible, since discomfort is only measuredin the end of the time frame, and temporal shifts thereby has less influence on the incurreddiscomfort.

The work described above focus on coordination of consumers for loss-minimizationand congestion management, but disregards the balancing and voltage control challengesoutlined in Section 1.1. Further, the communication framework outlined in Fig. 4.2 re-quires all consumers to communicate with a central data storage. The following presentsa different coordination framework, focusing on neighbor-based communication, thusavoiding the need for a central data storage. For this, we focus exclusively on the balanc-ing challenge, before a more complex coordination framework is presented in Chapter5, dealing with all the challenges discussed throughout, while incorporating all of thepresented benefits.

4.4 Spatially Distributed Portfolio Balancing

We consider again the discrete coordination horizon T = 1, . . . , N and portfolio N =1, . . . , n. As opposed to the preceding sections, the portfolio treated in the followingcontains both consumers as well as producers in form of traditional thermal power plants.

Below is presented an instance of the dispatch problem of Section 2.1, where theinclusion of consumers as well as producers in the portfolio allows a portfolio demand tobe tracked by adjusting either production or consumption. The following is based uponthe results presented in Paper D.

In the following p ∈ Rn×N as introduced in (4.3), denotes the power productionfrom each unit in the portfolio, where the term production is used even for consumers,in the sense that consumption can be considered negative production when maintaining abalanced portfolio. The portfolio is conceptually illustrated in Fig. 4.4, where the thermalpower plants are subdivided into two types, classified by their operating characteristics:

• Type 1:These have very large production capacity and a low cost of operation. However,

48


these units also have a very low gradient, such that ramping the production up anddown can only be performed slowly.

• Type 2:These have a smaller production capacity and a higher operating cost compared totype 1 generators. On the other hand, their power gradients are significantly larger.

Type 1 plants could be coal fired CHP plants whereas type 2 units could be oil- or gas-fired plants [Kragelund et al., 2012]. Fig. 4.4 also include external actors in the portfolioin order to cover the fact that any power deficit of the portfolio will be covered by otherentities, at the expense of the portfolio owner (PO), which was discussed in Section 2.1.

Type 1Type 2

Consumers

External actors

Grid

Σ

Figure 4.4: Conceptual sketch of how the production of the portfolio accumulates. Theaccumulated production should balance the grid by tracking a demand.

Each unit in the portfolio is associated with a objective function fi : RN → R, andconstraint set Pi ⊂ RN . For production units, we let fi denote operating cost, referringto Section 2.1. For consumers we use again the discomfort measures described in Sec-tion 4.1. The constraints represent operational limits of both consumers and producers.

Provided a power demand pdem ∈ RN , we wish to solve the (discretized) dispatchproblem described in (4.21), as recapped from Section 2.1.

minimizep1,...,pn

∑

i∈N fi(pi)

subject to pi ∈ Pi∑

i∈N pi = pdem

(4.21)

where we assume that a solution does in fact exist.Based on the arguments pertaining to geographical location, juridical affiliation, pri-

vate information sharing and problem size discussed in Section 2.5, we wish to solve thisproblem in a distributed rather than centralized fashion. That is, when considering theportfolio as a graph as illustrated in Fig. 4.5, each unit should communicate only withneighboring units, and not with any central service provider.

The reader should be able to verify, that with the identifications

p ∼ u, ψ ∼∑

i∈N

fi, Pi ∼ Ui, pdem ∼ c,

then (4.21) is similar to (2.4). Thus, with the two assumptions:

• Assumption 1: all fi’s are strictly convex

49

Demand Management

p1

p2

p3

p4

p5

p6

pn

Figure 4.5: Illustration on the portfolio and the distributed communication framework.Each unit in the portfolio is a node in the graph, and each link is a communication path.The gray lines conceptualize the underlying electrical grid.

• Assumption 2: all Pi’s are closed convex,

then (4.21) may be solved by the methods in Section 2.5. This is shown in Paper D whichadditionally demonstrates that the methods may be refined to achieve a robust, distributedand decentralized coordination scheme.

Assumption 1 above holds for consumers provided their local objective may be for-mulated as in (4.7) or similar. Additionally, it is common to model the cost of operationof thermal power plants as an increasing quadratic [Wood and Wollenberg, 1984, Saadat,2002, Kim et al., 2002, Wangsteen, 2007], which is similarly strictly convex.

Assumption 2 above is satisfied when all constraints are formulated as in Section 4.1,which is the case for all constraints included in Paper D.

The work presented in Paper D combines the dual decomposition discussed in Sec-tion 2.5, with distributed averaging, in a nested and iterative fashion. Defining

gi : RN → R, ν 7→ gi(ν) = infpi∈Pi

(fi(pi) + ν⊤pi)

and further letting p⋆i (ν) = arg infpi∈Pi(fi(pi) + ν⊤pi) the dual to the original problem

in (4.21) is given by

maximize g(ν) =∑

i∈N

gi(ν) − ν⊤pdem (4.22)

the gradient of which is given by ∇g(ν) =∑

i∈N p⋆i (ν) − ν⊤pdem, which can be usedfor a gradient ascend update on the Lagrange multiplier ν, in an iterative fashion:

ν(k+1) = ν(k) + α(k)∇g(ν(k)). (4.23)

Calculating the gradient for the update in (4.23) seemingly requires that the solutions ofeach portfolio unit be gathered, thereby requiring a central data storage similar to Fig. 4.3.However, an alternative formulation can be derived using the mean value

µ : Rn×N → RN , u 7→ µ(u) =1

n

n∑

i=1

ui,

50


then the gradient can be equivalently formulated as

∇g(ν) =∑

i∈N

p⋆i (ν)− ν⊤pdem = nµ(p⋆(ν)) − ν⊤pdem. (4.24)

Thus the ν−update can be performed using the average, rather than the specific value ofthe p⋆i ’s. This average can be found distributed across the graph, by weighted exchangeamong neighbors. To illustrate this, we first let Ni ⊂ N denote the units in the port-folio with a direct communication path to node i. Then, at each iteration k, we defineΞ(0)i = p⋆i (ν

(k)), ∀i ∈ N , and introduce an additional iterative procedure, indicated bythe counter h, which is given by

Ξ(h+1)i = σiiΞ

(h)i +

∑

j∈Ni

σijΞ(h)j

where σij are averaging weights. By proper choice of these weights, the above updateconverges in the sense that

limh→∞

Ξ(h)i = µ(Ξ(0)) = µ(p⋆(ν(k)). (4.25)

That is, by only transferring data to immediate neighbors, the average can be found acrossthe entire graph. Assuming each node knows the demand and portfolio size, this allowseach portfolio unit to calculate the gradient (4.24) locally, and subsequently make a localupdate of ν, following the rule presented in(4.23).

Even though the average in (4.25) is obtained only in the limit, the work presented inPaper D shows that the local update of ν can come arbitrarily close to the global optimumwhen running a finite number h′(k) of averaging steps, at each gradient step k, providedthat h′(k) is sufficiently large. A similar result was provided in [Johansson et al., 2008],who considered a problem with a shared constraint set, known to all agents, but did notdiscuss how this could be reformulated to encompass constraints private to each unit.

The weights σij , i, j ∈ N can be designed in different ways [Xiao and Boyd, 2004,Xiao et al., 2007]. The design employed in Paper D renders our approach fault tolerant inthe sense that units may leave or enter the portfolio, without requiring major revisions tothe algorithm. Only local updates are required, as indicated in Fig. 4.6.The figure illustrates the different faults that may occur in the portfolio. The node markedin red, along with appertaining links may fail, and the node marked in green and apper-taining links, may join the portfolio. These changes to the portfolio would only effect theimmediate neighbors as marked in teal, and not the remaining units of the portfolio.

For numerical examples, as well as the details of the above discussion, the reader isreferred to Paper D.

Notice the benefits and drawbacks of the two coordination approaches investigated inSection 4.3 and Section 4.4. The dual decomposition introduced in Section 4.4 allowsa coordination scheme requiring only neighbor based communication, whereas the ap-proach outlined in Section 4.3 requires a central data storage. On the other hand, the classof problems that may be handled by the method of Section 4.3 is much more general inthat we are not bound by the assumption on strict convexity enforced by the dual decom-position in Section 4.4. For example, the ADMM approach would allow consumers that

51

Demand Management

p1

p2

p3

p4

p5

p6

pn

p7

Figure 4.6: Illustration of possible faults occurring in the portfolio when either nodes orlinks break down or enters the portfolio.

are defined solely by constraints, i.e. with a zero objective function. We demonstrate thisin the following, where we present a more complete case of portfolio coordination, en-compassing all of the challenges treated in this chapter. Additionally it is illustrated howthe ADMM method can be formulated such as to enable almost the same neighbor-basedcommunication capabilities as the dual decomposition, and omit the need of a central datastorage.

52

5 Grid Operation Through

Coordination

In this chapter we extend the discussion from the previous chapter, and present

a more elaborate application of demand management for portfolio coordina-

tion, combining the benefits from the examples of the previous chapter, accom-

modating to a larger extent the challenges discussed in Section 1.1.

5.1 Coordination Outline

In Section 4.3 (Paper C) we presented a strategy to coordinate a portfolio of consumerssuch that their individual preferred consumption was aligned to also account for the jointobjective of minimizing transport losses through the grid. In Section 4.4 (Paper D) thefocus was on grid balancing and it was demonstrated how a portfolio containing bothproducers and consumers could be coordinated to obtain balance in an effective manner.In both examples the focus was on distributed methods, where Section 4.3 required centralunit to communicate with all consumers, whereas Section 4.4 derived a strategy requiringonly neighbor based communication.

In the following we combine the preceding sections to derive a more complete in-stance of portfolio coordination and demand management, involving all the challenges ofSection 1.1 while requiring only a neighbor-based communication structure. For this, weexpand the grid model of Section 4.3 to include losses and voltage variations across theentire grid, and not only for tie-lines. As in Section 4.3 we focus on a portfolio describinga community of residential households, that is the thermal plants included Section 4.4 aredisregarded and power production is only introduced in the portfolio through householdsolar panels.

5.2 Distributed Loss Minimization, Voltage Control

and Congestion Management

The following builds on the work and results presented in Paper E and Paper F, whichdiscusses extensively consumer coordination in tree-structured grids as the one illustratedin Fig. 5.1. The grid contains again cables ( ) and consumer connections ( ). Eachcable section is characterized by a lumped parameter impedance z ∈ C. The externalgrid is connected through a step-down transformer ( ).

53

Grid Operation Through Coordination

z1

z2

vs

z4

z5

z3

z6

z7

Figure 5.1: Tree structured grid.

Contrary to the work presented in Section 4.3 and Section 4.4, which only consideredactive power coordination, both Paper E and Paper F consider a combination of activeand reactive power flows of consumers for coordination. Including the reactive powerflows gives additional leverage for loss minimization and voltage control, as outlined inSection 1.1 and discussed further in Section 5.2.

In Paper E, the coordination is performed in a centralized fashion, where the coordina-tion problem is recognized to be non-convex. The coordination is subsequently conductedin an iterative manner, relying on sequential convexification as described in Section 2.5by Algorithm 2.5.1.

In Paper F, the coordination is conducted for a similar problem as in Paper E, althoughin a distributed rather than centralized fashion. The distributed approach requires onlyneighbor communication, as described in Section 2.5, and employs Algorithm 2.5.3. Inthe following, we focus on demonstrating the approach of Paper F with an example.Examples of the work presented in Paper E are not included here, and the reader is referredto the paper itself.

To ease notation and discussion in the following, the full alternate current (AC) caseconsidered by Paper F, is simplified by omitting reactive power flows. The followingexposition thus essentially equates to a direct current (DC) grid. This obviously impliesthat the reactive control capabilities cannot be used to manage voltage and losses, but asour purpose is only to outline and generalize our method of Paper F, we are not impededby the lack of reactive capabilities. An overall discussion relating to the reactive capabil-ities of consumers is brought on Page 66, although the reader is referred to the papers adetailed analysis of this.

Grid Structure and Cable Modeling

As Paper E and Paper F focused exclusively on tree-structured grids, the following willfor the sake of completeness demonstrate how our work generalizes to non-tree structuredgrids. In particular, we consider the low-voltage grid depicted in Fig. 5.2, where each bus-bar is indicated by , and connections to the remaining electrical grid are indicated by

. As was also the case in both Paper E and Paper F, we assume that the voltage in theconnections to the remaining grid, is maintained fixed at vs = 1 pu (per-unit) (0.4 kV).

54

5.2 Distributed Loss Minimization, Voltage Controland Congestion Management

The grid contains m, b, n ∈ N bus-bars, cables and consumers respectively. Let

M = 1, . . . ,m, B = 1, . . . , b, N = b+ 1, . . . , b+ n,

denote their respective index sets, and further, denote by F ⊂ M the bus-bars at theconnections to the remaining grid. We consider the grid as a graph, where each cablecomprises an edge, and bus-bars comprises the nodes.

r11

r3

r 4

r 5

r6

3

6

5

r24 2

r7

r9

r8

r10

7

9 810

2 13

r11

r13

r12

r14

11

13 1214

5 46

r15

r17

r16

r18

15

17 1618

8 79

r19

r21

r20

r22

19

21 2022

11 1012

Figure 5.2: Example grid for numerical demonstration. The cable sections are labeled inblue and bus-bars in red.

Each cable is considered as a two-terminal device, to which we assign a left and rightside. We introduce mappings

Pa : B ∪ N → M, Ch : B → M

to denote the parent and child node of each cable, and the parent node of each consumerin the following sense:

• Pa(j) = i if left side of cable j ∈ B is connected to node i ∈ M.

• Ch(j) = i if right side of cable j ∈ B is connected to node i ∈ M.

• Pa(j) = i if consumer j ∈ N is connected to node i ∈ M.

55


zj = rjvPa(j)(t) vCh(j)(t)

pl,j(t) pr,j(t)

ıj(t)

Figure 5.3: Isolated cable section in the grid, with all related quantities indicated.

Each cable section can be depicted as in Fig. 5.3, where we let each cable be purelyresistive zj = rj , for some resistance rj > 0, j ∈ B. We let r = (r1, . . . , rb) ∈ Rb+ andintroduce

pl,j : T → R, pr,j : T → R, j ∈ B

in the same fashion as in (4.2), to denote the average power flowing into the left and rightterminal respectively of the cables, for the discrete coordination horizon T . Additionally,we let ıj : T → R, j ∈ B denote the current through the cables, and vj : T → R, j ∈ Mthe voltage in all nodes of the grid. We define the current flow through each cable asgoing from left to right.

Voltage Drop

The current through a cable is

ıj(t) =pl,j(t)

vPa(j)(t)=

-pr,j(t)

vCh(j)(t), ∀j ∈ B, (5.1)

rendering a voltage difference

vPa(j)(t)− vCh(j)(t) = rj ıj(t) ⇔ vCh(j)(t) = vPa(j)(t)− rj ıj(t).

Inserting (5.1) gives

vCh(j)(t) = vPa(j)(t)− rjpl,j(t)

vPa(j)(t), j ∈ B. (5.2)

The voltage requirements discussed in Section 1.1 state that the voltage must be withinmaximum and minimum magnitudes v, v ∈ R i.e.,

vj ∈ CI(v, v), ∀j ∈ M. (5.3)

The Danish system uses v = .9vs, v = 1.1vs, i.e., an allowed deviation of 10 % from thesource voltage [Danish Energy Association, 2011a, Danish Energy Association, 2011b].

Since we consider a DC grid, the voltage is a real number, whereby the voltage con-straint (5.3) is convex. In the AC case considered in Paper E and Paper F the voltage iscomplex, and the equivalent to (5.3) extends to a non-convex magnitude constraint. Inthe iterative procedure enforced in Paper E, this is convexified by an affine approxima-tion, which is updated in each iteration. In Paper F a simpler approach was successfully

56


employed, by realizing that the constraint (5.3) entails that voltages will remain close tothe supply voltage and in that sense vj ≈ 1 pu. We employ the same approximation here,whereby (5.2) is simplified to

vCh(j)(t) = vPa(j)(t)− rjpl,j(t), ∀j ∈ B, t ∈ T . (5.4)

By the notation introduced in (4.2) and (4.3), we can collect all equations in (5.4) as

Csv = Cpv −Dpl,

where Cs, Cp ∈ 0, 1b×m are selector matrices:

[Cs]j,i =

1, i = Ch(j)

0, otherwise[Cp]j,i =

1, i = Pa(j)

0, otherwise,

and D = diag(r) ∈ Rb×b.

Cable Capacity

As in Section 4.3, the cables may represent capacities on allowed power transport. Fromthe illustration of a cable-section in Fig. 5.3, this constraint may be expressed as [Kraninget al., 2014]:

1

2|pl,j − pr,j | ≤ θj , (5.5)

for j ∈ B with the capacity limit θj .

Losses

From (5.1), the squared current magnitude is:

ıj(t)2 =

pl,j(t)2

vPa(j)(t)2=

pr,j(t)2

vCh(j)(t)2=

1

2

(pl,j(t)

2

vPa(j)(t)2+

pr,j(t)2

vCh(j)(t)2

)

.

However, as argued above, the voltage constraint entails that vCh(j)(t) ≈ vPa(j)(t) ≈1, j ∈ B, t ∈ T , i.e.,

ıj(t)2 ≈

1

2

(pl,j(t)

2 + pr,j(t)2). (5.6)

The losses are given by rj ıj(t)2, so by defining l : RN × RN → RN as

l(u, y)(t) =1

2

(u(t)2 + y(t)2

), t ∈ T (5.7)

for any u, y ∈ RN , the losses can be approximated as rj l(pl,j, pr,j), for each cable j ∈ B.As these losses represent power dissipated in each cable, the physical relation betweenleft and right terminal power flow, is given by

pl,j + pr,j = rj l(pl,j, pr,j), j ∈ J . (5.8)

57


Since (5.8) represents a quadratic equality, it is a non-convex constraint. In [Kraninget al., 2014] the convex relaxation

pl,j + pr,j ≥ rj l(pl,j , pr,j), j ∈ B, (5.9)

was suggested. Although this represents a simplification of the actual constraint, thedesire to minimize losses, would typically make the inequality tight. With this simplifica-tion, the task of minimizing losses thereby means to minimize 1⊤(pl,j+pr,j), constrainedby (5.5) and (5.9). To describe the operation of each cable, we define cl : B×RN×RN →R ∪ ∞

c+l (j, u, y) =

1⊤(u+ y), u+ y ≥ rjl(u, y), |u− y| ≤ 2θj

+∞, otherwise,(5.10)

whereby the losses in each cable can then be represented as c+l (j, pl,j, pr,j), j ∈ B. Theconstraints in (5.5) and (5.9) are implicitly included as an operating constraint.

Power Flow Consistency

Similar to pl and pr, we introduce pc,i : T → R, i ∈ N to denote the power of consumers.Power flow consistency throughout the grid require that

∑

pr,i(t)i∈B|j=Ch(i)

+∑

pl,i(t)i∈B|j=Pa(i)

+∑

pc,i(i)i∈N|j=Pa(i)

= 0, ∀j ∈ M\F , ∀t ∈ T (5.11)

that is, the power entering any node, must equal the power leaving the node. LettingHl, Hr ∈ 0, 1|M\F|×|B|, Hc ∈ 0, 1|M\F|×|N| be given as

[Hr]j,i =

1, j = Ch(i)

0, otherwise[Hl]j,i =

1, j = Pa(i)

0, otherwise[Hc]j,i =

1, j = Pa(i)

0, otherwise

The equations in (5.11) may be collected as

Hrpr +Hlpl +Hcpc = 0 ⇔ H

pl

pr

pc

= 0

for H = [Hl Hr Hc].

Consumer Coordination

With the models and notation introduced above, the portfolio coordination problem con-sidering private consumer objectives, grid losses, and grid capacity constraints, can nowbe condensed as

minimizepc,pl,pr,v

∑

i∈N

f+i (pc,i) +

∑

j∈B

c+l (j, pl,j , pr,j)

subject to Csv = Cpv −Dpl

Hlpl +Hrpr +Hcpc = 0

vj ∈ CI(v, v), ∀j ∈ M.

(5.12)

58


where the consumer behavior is modeled in the same framework as in Chapter 4. Thespecific formulation of consumer objectives f+

i (pi) is introduced in a numerical exampleto be presented.

Our strategy for solving (5.12) in a distributed fashion, relying on neighbor basedcommunication, is derived from the initial observation that the objective in (5.12) is com-pletely separable with respect to consumers, cables and voltages. The only coupling isintroduced from the consistency constraints. This coupling can be removed by realizingthat (5.12) can be written in ADMM form, whereby (5.12) can be solved by the methodoutlined in Section 4.3. For this we introduce auxiliary variables wl, wr ∈ Rb×N , wc ∈Rn×N and z ∈ Rm×N along with constraints

wl = pl, wr = pr, wc = pc,v = Csz, Dpl + v = Cpz.

(5.13)

We additionally introduce indicator functions

IC(u) =

0, u ∈ C

+∞, otherwise,

whereby the voltage and power conservation constraints, may be expressed implicitly, byexpanding the objective in (5.12) by the terms

ICI (v,v)(vj) and, ICH(0)([wl, wr, wc])

for j ∈ M. We may then form the following problem as an equivalent to (5.12):

minimizepc,pl,pr,v

∑

i∈N

f+i (pc,i) +

∑

j∈B

c+l (j, pl,j, pr,j)

+∑

j∈M

ICI(v,v)(vj) + ICH(0)([wl, wr, wc])

subject to wl = pl, wr = pr, wc = pc,

v = Csz, Dpl + v = Cpz

(5.14)

The reader may verify that this is indeed on the ADMM form (2.6), by the identifications

u ∼ (pl, pr, pc, v), y ∼ (wl, wr, wc, z)

ψ ∼∑

i∈N

f+i (pc,i) +

∑

j∈B

c+l (j, pl,j , pr,j) +∑

j∈M

ICI(v,v)(vj), φ ∼ ICH(0)([wl, wr, wc])

A =

II

II

D I

, B = −

II

ICs

Cp

.

The problem arranged in (5.14) can thus be solved using the method in Algorithm 2.5.3.To avoid confusion, we remark to the reader that the auxiliary variables and consis-

tency constraints introduced above, are included in a different fashion that in Paper F. Wefind that the methodology used above is simpler than the one originally employed in theenclosed paper.

59


Interpretation as Distributed Coordination

Running Algorithm 2.5.3 on (5.14) introduce alternate updates on

(pl, pr, pc, v), and, (wl, wr, wc, z),

respectively, as well as on the Lagrange multipliers associated to the constraints (5.13).To see that this alternate update on the problem formulation in (5.14) corresponds to aneighbor based communication scheme, we write the constraints in (5.13) explicitly

pl,i(t) = wl,i(t), pr,i(t) = wr,i(t), pc,j(t) = wc,j(t)

vi(t) = zCh(i)(t), ripl,i(t) + vi(t) = zPa(i)(t),

for i ∈ B, j ∈ N , t ∈ T . By the smart grid framework discussed in Section 1.2, welet each consumer, cable section and bus-bar be installed with a private computation andmetering device. In a practical setup, some of these private computation devices mayphysically be the same, however, we shall treat them individually in the following. Let

pl,j, pr,j , vj, ∀j ∈ B

be private variables, governed by the computation device of each specific cable, and sim-ilarly let pc,j be private to each consumer j ∈ N . Also, for each j ∈ M let

wr,i|i ∈ B, j = Ch(i), wl,i|i ∈ B, j = Pa(i),

wc,i|i ∈ N , j = Pa(i) and, zj ,

be private variables governed by the device of each bus-bar, along with the correspondingLagrange multipliers.

From the constraints we see that according to Algorithm 2.5.3, all cables may conductparallel updates of

p(k)l,j , p

(k)r,j , v

(k)j ,

when provided the current value of

w(k)l,j , w

(k)r,j , z

(k)Ch(j), z

(k)Pa(j),

as well as the associated Lagrange multipliers. This data is governed by the parent andchild node of each cable. In the same fashion, each household may update p(k)c,j , provided

the value w(k)c,j and associated Lagrange multiplier, which is governed by the parent node

of that consumer. Thus, the variable update of either cables or consumers require onlydata transferred from the buses they are directly connected to.

Subsequently, all bus-bars j ∈ M may in parallel update

w(k)r,i |i ∈ B, j = Ch(i), w

(k)l,i |i ∈ B, j = Pa(i),

w(k)c,i |i ∈ N , j = Pa(i) and, z

(k)j ,

when provided the values of

p(k+1)r,i |i ∈ B, j = Ch(i), p

(k+1)l,i |i ∈ B, j = Pa(i), p

(k+1)c,i |i ∈ N , j = Pa(i)

60


v(k+1)i |i ∈ B, j = Ch(i), v

(k+1)i |i ∈ B, j = Pa(i),

and the associated Lagrange multipliers. As was also the case before, this shows that theupdate of the ADMM variables requires only data to be forwarded from the cables andconsumers directly connected to each bus-bar. This data exchange can be visualized as inFig. 5.4, for a fictive grid section with B = 1, . . . , 4,M = 1, . . . , 5. The introducedcommunication structure for solving (5.12) distributed, thereby coincides with the layoutof the grid.

11

22

33

44

5

(w(k)r,1 , z

(k)5 )(w

(k)l,1 , z

(k)1 )

(w(k)r,2 , z

(k)5 )(w

(k)l,2 , z

(k)2 )

(w(k)l,3 , z

(k)5 ) (w

(k)r,3 , z

(k)3 )

(w(k)l,4 , z

(k)5 ) (w

(k)r,4 , z

(k)4 )

11

22

33

44

5

(p(k+1)r,1 , v

(k+1)1 )(p

(k+1)l,1 , v

(k+1)1 )

(p(k+1)r,2 , v

(k+1)2 )(p

(k+1)l,2 , v

(k+1)2 )

(p(k+1)l,3 , v

(k+1)3 ) (p

(k+1)r,3 , v

(k+1)3 )

(p(k+1)l,4 , v

(k+1)4 ) (p

(k+1)r,4 , v

(k+1)4 )

Figure 5.4: Top: Data passing prior to the update of primary variables: In order for eachcable section to update private variables, it needs the current ADMM and Lagrange vari-ables from the neighboring nodes in the grid. The Lagrange variables are not explicitlydrawn in the figure, since they are each associated to one of the existing arrows. Bottom:

Corresponding data passing prior to update of ADMM variables of each bus-bar.

Numerical Example

In this example we define consumers simply as constraints, conceptually equivalent to anEV user, with the sole concern that the battery is charged at the end of the horizon. Addingalso charge and energy constraints on the battery, we can formulate these consumers asfi(pi) = 0, and

pi ∈ CI(p, p) ∩ Cxi(x, x) ∩ C1T (pdem)

pi ∈ Pi = pi = pi + pi|pi ∈ CI(p, p) ∩ Cxi(x, x) ∩ C1T (pdem),

61


for limits p, p, x, x ∈ R and charge requirement pdem ∈ R. In this formulation, we havenot even included a concern towards cost of energy. This could thus be viewed as acontemporary consumer formulation, as in the current system, small consumers do notreceive time differentiated prices whereby electricity is equally expensive at all times.

As the reactive power is not included, the reactive control abilities cannot be em-ployed for voltage control, which was otherwise considered in both Paper E and Paper F.The only control action against over and under-voltages are then to coordinate consump-tion between consumers such that voltage variations remain acceptable. Even though theactual limit on voltage variations are 10 %, we have used a limit of 8 % in this example,to allow for deviations caused by the approximations introduced throughout.

We run the coordination for the grid layout presented in Fig. 5.2, that is with b = m =22 cables and bus-bars and n = 12 consumers. We introduce a horizon of T = 24 hours,discretized into Ts = 15 minute slots. The use a horizon of 24 hours, both here as well asin most of the enclosed papers, allows the simulation to capture any diurnal variations ininflexible consumption patters, solar power production, etc.

In the simulation, we have used an accumulated flexible demand of each consumerof pdem,i = 20 pu, with a 1 kW base-value. Each consumer is modeled as a loss-lessintegrator. All consumers are governed by the same operating limits

pi = −pi= 10 pu, xi = 0 pu, xi = 20 pu, ∀i ∈ N ,

and a starting condition of xs,i = 0, ∀i ∈ N . The baseline consumption used in theexample is presented in Fig. 5.5. The negative baseline for some consumers is on accountof installed solar panels, rendering a baseline production through solar intensive periods.This example has been run with 30 % solar penetration, i.e. 4 consumers have solar panelsinstalled, with a maximum power output of 6 pu. The solar production is superimposedon the regular base-line consumption of each household with solar arrays.

pc

[pu]

Time

12:45 17:45 22:45 03:45−6

−4

−2

0

Figure 5.5: Each curve show the baseline consumption of a consumer. Negative con-sumption corresponds to a local production from solar panels, exceeding the remaininginflexible consumption.

The grid has been dimensioned such that ri = rj , ∀i, j ∈ B, where the resistancehas been chosen such that the baseline consumption, not including solar power, ren-ders transport losses of 3.51 %. The power capacity of each cable has been set toθi = 5 pu, ∀i : Pa(i) ∈ F , that is the capacity limits are enforced on the tie-lines import-ing power from the remaining grid.

62


As we have previously argued, (5.12) and the equivalent (5.14) both represent con-vex problems. As the scale of this example is quite limited it is numerically possible tosolve the problem in a centralized way, to obtain the global minimum. We use this as abenchmark for comparison of the results obtained by the distributed coordination.

The distributed coordination is conducted by running Algorithm 2.5.3 on (5.14) withρ = 1, which seems to work well for this example. We employ a termination toleranceof ǫabs = 3 · 10−4 ·

√

3(n+ b)N , where the scaling is to account for problem size. Theconvergence in terms of residuals, is presented in Fig. 5.6.

‖ζ(k

)‖2,‖ξ(k

)‖2

k

100 200 300 400

10−2

100

102

Figure 5.6: Convergence of primal (blue) and dual residual (red) residuals when runningAlgorithm 2.5.3 on the presented problem instance.

The coordinated consumption patterns are presented in Fig. 5.7(Left), and the left-terminal power of each cable is presented in Fig. 5.7(Right). In both figures, the solutionobtained distributed are presented in red, and the centralized solution is presented in bluefor comparison. The two solutions are almost indistinguishable, meaning that the dis-tributed procedure has indeed converged to the solution of the problem.

DistributedCentral

pc

[pu]

Time

12:45 17:45 22:45 03:45−2

0

2

pl

[pu]

Time

12:45 17:45 22:45 03:45−2

0

2

4

6

Figure 5.7: Left: Consumption profiles found distributed (red) and centralized (blue).Right: Left terminal power flow of each cable, as found distributed (red) and centralized(blue).

In Fig. 5.8, the state of each consumer is presented, and how it reaches the requireddemand, without breaking the state limit. When compared to Fig. 5.5, it is clear that the

63


consumers installed with solar power charges exclusively during solar intensive periods,so that they can effectively avoid importing power. Even when fully charging their ve-hicle, the solar panels still produce excess power. This excess solar power is absorbedby the remaining consumers, which can be seen by the increased charge rate during thesolar intensive period, even though the total power import to the grid is reduced, as ev-ident from cable power transport in Fig. 5.7(Right). Absorbing the excess solar powerin this fashion reduces the incurred losses, whereby this is a result of the coordination ofconsumers.

x(p

c)[p

u]

Time

12:45 17:45 22:45 03:450

10

20

30

Figure 5.8: The accumulated demand (dashed), and consumer state (solid) across thecontrol horizon for consumers with (teal) and without (magenta) solar panels, after coor-dination.

The grid voltages are presented in Fig. 5.9, where the voltages obtained through dis-tributed coordination is presented in red. In Fig. 5.9(Left), the centralized solution ispresented in blue for comparison, and in Fig. 5.9(Right) the voltage profiles obtained byrunning a load flow analysis on the consumption of Fig. 5.7(Left) is shown in green. Inboth cases, the solutions are almost indistinguishable, and well within the limits.

v[p

u]

Time

12:45 17:45 22:45 03:450.9

0.95

1

DistributedLoad flow

v[p

u]

Time

12:45 17:45 22:45 03:450.9

0.95

1

Figure 5.9: Left: Coordinated voltage (red), voltage obtained from central solution (blue),and allowed lower voltage limit (dashed). Right: Coordinated voltage (red), voltageobtained from load flow analysis (green), and lower voltage limit (dashed).

From the load flow analysis used to obtain the green voltage profiles in Fig. 5.9(Right),we asses the impact of the approximation vj(t) ≈ 1, ∀j ∈ M, t ∈ T , introduced in (5.4)

64


and (5.6). Despite this approximations, there is only a maximum voltage deviation of0.35 % between the ADMM profiles in Fig. 5.9(Right), and those found through loadflow analysis.

The deviation with respect to losses, between the estimated losses found distributed,and the actual losses found through load flow analysis, does however accumulate to7.61 % across the entire horizon. In Paper F, the loss deviation was experienced simi-larly high, and it was argued that a better guess on the voltage throughout the grid, wouldreduce the deviation. To verify this, we can combine the two methods used in Paper E(SCP) and Paper F (ADMM), respectively. That is; first we conduct the ADMM coordina-tion under the assumption that vj(t) ≈ 1, ∀j ∈ M, t ∈ T . The voltage profiles resultingfrom distributed coordination and ADMM, would be as presented in red in Fig. 5.9. Welet these be denoted by vest, and use this as the guess on voltage used for a subsequentrepetition of the coordination, i.e. we rerun the ADMM procedure with the updated as-sumptions vj(t) ≈ vest,j(t). We can repeat the procedure several times, however by run-ning ADMM until convergence just one additional time, the maximum voltage deviationhas dropped to 0.15 % and the accumulated loss-error has dropped to 0.33 %.

Scalability

It can be seen from Fig. 5.6 that ADMM terminates this example in 420 iterations, whichis about a third of the 1500 iterations used for the example presented in Paper F, indi-cating that the problem presented in the paper is numerically more difficult to solve. Ithas been experienced that the number of iterations required for the ADMM algorithmto converge, to some extent depends on the number of constraints active in the solution.For instance for the example presented in Paper F, several of the voltage constraints areactive, whereas in the above example, all voltage constraints are satisfied with a margin.It has however also been experienced, that when given the structure of a specific instanceof the coordination problem, the required number of ADMM-iterations required to solvethe problem, scales fairly well with respect to problem size.

To demonstrate this, we rerun the above example, still for a period of T = 24 hours,but with different sampling times, thus giving a different number of discrete steps in thecoordination horizon, i.e. a different number of variables in the coordination problem.The resulting number of ADMM iterations before convergence is presented in Table 5.1.

Ts (min.) 60 30 20 15N 24 47 70 93

# itt. 423 411 456 420

Table 5.1: Iterations required to obtain convergence of ADMM for different sample timesTs, and thereby different horizon lengths N .

As evident from Table 5.1, the number of required iterations is fairly unaffected, whendecreasing the sample time from Ts = 1 hour to Ts = 15 minutes, i.e. when quadruplingthe number of variables in the problem.

65


Extensions

As has also previously been the case with the example we present, there is a vast numberof extensions we could still impose on the problem considered above, without requiringany modifications to the framework. A simple example could be to combine the portfoliobalancing problem discussed in Section 4.4 with the distributed coordination describedabove. Provided the demand pdem ∈ RN , and recalling that F ⊂ M denotes the bus-bars connecting the grid section to the remaining grid, this extension could be included in(5.12) by adding the constraint:

∑

pl,i(t)i∈B|∃j∈F :i=Ch(j)

= pref(t), t ∈ T , (5.15)

that is; the power imported through the tie-lines to the grid, should correspond to somereference. The employed reference could then be designed for balancing purposes. Animportant assumption here is of course, that (5.12) is still feasible when also introducing(5.15) as a constraint. We discuss a few additional extensions in the following chapter,where we will also provide a brief conclusion and outlook of our work, before the indi-vidual contributions are presented.

Reactive Compensation

Paper E and Paper F included reactive power control, which was however omitted above.As is clear from the papers, and discussed in Section 1.1, including reactive compensationin the demand management, provides an additional tool for loss minimization and voltagecontrol. We have however not discussed the limits of this ability, and thus, for the sake ofcompleteness, the following discussion should be seen as an extension to the influence ofreactive power on losses and voltage variations, introduced in Section 1.1. We limit ourfocus to reactive capabilities of photo-voltaic (PV) arrays.

As described in Section 2.3.2, the inverter of the PV array has a maximum apparentpower rating smax such that PV operation is constrained by |s(t)| ≤ smax, as indicatedin Fig. 2.7. An example of the practical implication of the PV limitations is presentedin Fig. 5.10, for a solar panel with a 6 pu inverter limit. In Fig. 5.10(Top) a conceptualexample of the active power production is plotted during the course of a day, as a Ts = 15min. average. In Fig. 5.10(Bottom), the blue curve shows the corresponding range forthe reactive power, in that at every time interval, the reactive power must be kept withinthe upper and lower limit, for the inverter capacity to be satisfied. As evident, when theactive power output is close to the inverter limit, there is very little room for varying thereactive power, and the corresponding potential for voltage control and loss minimizationwould be equally restricted.

The remaining curves in Fig. 5.10(Bottom) presents the reactive limits, when intro-ducing different degrees of overdimensioning of the inverter, i.e. they show the admissiblerange of q(t), when p(t) is maintained as in Fig. 5.10(Top), but the inverter capacity isincreased following:

pmax = maxt∈T

(p(t)), smax = (1 + γ)pmax, γ ∈ 0, 0.05, 0.1, 0.15.

To demonstrate the effect of such over dimensioning, we repeat the abstract exampleoutlined by Fig. 1.2 and Fig. 1.3 in Section 1.1. We let the load consist on an aggregation

66


p(t)

[pu]

0123456

15 %10 %5 %0 %

q(t)

[pu]

Time09:00 15:00 21:00 03:00 08:00

−10

−5

0

5

10

Figure 5.10: Top: Active power output of solar panel during a day. Bottom: Correspond-ing range of reactive power flow of PV inverter for different overdimensionings.

of n = 15 consumers, each with a self-consumption of 0.5 pu, and solar power productioncorresponding to Fig. 5.10(Top). We run a load flow analysis for the simplified grid inFig. 1.2, at the instance where the solar power production is 6 pu from each consumer,i.e. the aggregated load has an active power production of some fixed value p = p′.We sweep the admissible range of reactive power, for various sizes of the inverter, topresent the available trade-offs between loss and voltage magnitude, similar to Fig. 1.3.In Fig. 5.11, the resulting load voltage and loss magnitude is presented.

0 %5 %10 %15 %20 %

l(p′ ,q)

[pu]

|vl(p′, q)| [pu]

1.08 1.082 1.084 1.086 1.0886

7

8

9

10

11

Figure 5.11: Effect of reactive power compensation for different overdimensionings ofthe inverter, where each color represent traces of the obtainable operating points throughreactive power control, for a specific inverter size. For heavily overdimensioned inverters,the operating range increases, since the allowed reactive control range increases.

The figure demonstrates the limited voltage control ability of the reactive power flowof the PV inverter when the active power production is large, i.e. when the inverter satu-rates, and how a small over-dimensioning, increase the reactive capabilities significantly.

67

6 Closing Remarks

We summarize our work below and subsequently present overall outlooks and

perspectives. We touch on a few future extensions and how these could com-

plement our work.

Conclusion

The focus of this work has been to find ways for mobilizing the flexibility of consumersand producers in the electrical grid, with the purpose of mitigating expected challenges inthe future electrical grid, following the trends in electricity consumption and production.

We have initially elaborated on the expected challenges and stressed the necessityof finding suitable solutions. Subsequently, the notion of flexibility was discussed interms of both electricity production and consumption, and it was further discussed howthe identified flexibility may potentially alleviate challenges pertaining to grid balancing,congestion management, loss minimization and voltage control. In extension of this,we have introduced the smart grid framework, within which several approaches havebeen derived to iteratively coordinate consumption and production, with the purpose ofmobilizing the available flexibility, allowing the potential benefits to be harvested.

Using models of the presented flexibility for a portfolio of consumers and producers,we have formulated numerous problems encompassing different aspects of the discussedchallenges. By applying the presented coordination methods to the portfolio, it has beenshowed how the problems may be solved, and the appertaining grid related issues may bereduced or avoided altogether. We have initially formulated simple problems, focusingonly on different sub-problems of the overall challenges. Subsequently, a more complexproblem has been considered, encompassing the majority of the discussed challenges.

The presented work has focused on distributed methods in order to accommodateboth the distributed nature of the resources in the grid, but also to limit the informationsharing required by the entities involved in the underlying portfolio, when solving theposed problems. This serves to enable problems to be solved, even when the underlyingportfolio involves e.g. a large number of participants, competing entities, participantswith different juridical affiliations, sensitive information, etc.

Our work provides solutions to many of the issues and challenges we initially dis-cussed to motivate our work. In addition, the methods are very general and compre-hensive, in the sense that numerous extensions can be readily incorporated, the only re-quirement being that the conditions outlined for each presented approach is satisfied, e.g.

69

Closing Remarks

advanced metering possibilities, convex problem formulations, possibility for neighborcommunication, etc.

Outlook and Perspectives

As we have mentioned previously, the iterative coordination strategies we have derivedfor portfolio coordination, may be interpreted in different ways. That is, both as onlineor off-line strategies. In the work we have presented here, we have mainly adopted theoff-line point of view, where coordination is conducted ahead of time. This could be seenas a day-ahead process employed to introduce the available flexibility to the day-aheadelectricity market. In this case any prices, weather conditions, baseline consumption etc.,required to solve the problems we have presented, should be understood as estimatedsignals to be used for coordination. These estimates could be obtained, for example, fromhistorical data.

The result from the off-line coordination would be references that each unit in theportfolio should track during real-time operation. For this, local controllers could bedesigned for real-time reference tracking of each unit, as discussed in [Juelsgaard et al.,2013e] and illustrated in Fig. 6.1.

Portfolio demand,Portfolio size, etc.

Coordination(Central or Distributed)

Iterations

Units

Local Optimizer(Unit 1)

Local Optimizer(Unit n) O

ff-L

ine

Pla

nnin

g

Local Controller(Unit 1)

Process Dynamics(Unit 1)

Local control

Local Controller(Unit n)

Process Dynamics(Unit n)

Local control

Power Reference Power ReferenceR

eal-

Tim

eO

pera

tion

Figure 6.1: Control structure illustrating the combination of off-line coordination, andreal-time control, figure modified from [Juelsgaard et al., 2013e].

The figure demonstrates the hierarchical nature of this process with a high-level coordi-nation process, and low-level real-time controllers for reference tracking.

The converse view of our work as real-time control, can be extracted by repeatedcoordination, whereby a receding horizon strategy can be implemented, in the fashionof [Maciejowski, 2000]. In this way, the coordination problem is solved across the entire

70

considered horizon, but only the first sample of the resulting power schedules are appliedby the portfolio units, whereafter the coordination process is repeated. When employinga distributed scheme for coordination as discussed in e.g. Section 4.3, 4.4 or 5.2, sucha receding horizon strategy would result in nested iterative procedures, where an innerloop performs the coordination, and an outer loop governs the receding horizon strategyby progressing the control horizon. Such a strategy would carry an increased robustnessagainst stochastic variations of external disturbances, such as weather conditions, whichmay vary greatly in the day-ahead estimates.

The success in such a strategy would lie in the ability to satisfy real-time demands,which in turn would relate to the problem under consideration. For example, real-timepower balancing presents real-time demands in the seconds to minutes time scale, how-ever, slower requirements could also be valid, e.g. requirements to voltage magnitudesacross 10 minute intervals, or even intra hour energy balances.

It is difficult to determine which real-time guarantees can be made for the methods wehave presented, in that such guarantees would depend greatly on the number of iterationsrequired. This in turn depends greatly on the problem and structure of the the problem,as well as the tuning parameters of the specific methods. For example, the ρ parameterfor ADMM greatly affects the convergence speed, and ongoing research is still tryingto determine suitable strategies for determining an optimal value for this, in terms ofminimizing the number of iterations required for convergence.

Although the methods presented for distributed coordination accommodates numer-ous concerns, there are still aspects of the coordination we have omitted or only lightlydiscussed in our work, such as distributed synchronization and termination of the coordi-nation processes, and alternative approaches for dealing with non-convex problems. Forthe latter we have however demonstrated that the effect of the convex approximations in-troduced in this work, may be remedied by iterated updates of the operating point used forapproximation. Additionally, it does in fact appear as if the introduced approximationshave only limited influence on the solution.

71

Acronyms

AC alternate current

ADMM Alternating Direction Method of Multipliers

BRP balance responsible party

CHP combined heat and power

DC direct current

DFIG doubly-fed induction generator

DG distributed generator

DSO distribution system operator

EHP electric heat pump

EV electric vehicle

LV low-voltage

MPC model predictive control

pu per-unit

PV photo-voltaic

SCP sequential convex programming

TCL thermostatically controlled load

TSO transmission system operator

VSWT variable speed wind turbine

73

Nomenclature

General notation

2N The power set of set N

1 Vector of ones: (1, . . . , 1)

C Set of complex numbers

E General euclidean space

Im(x) Imaginary part of x

j Imaginary unit

N Set of natural numbers

Re(x) Real part of x

R Set of real numbers

R+ Set of positive real numbers

|x| Absolute value/Complex magnitude of x

x Derivative of x w.r.t. time

‖x‖p p-norm of x

x∗ Complex conjugate of x

x⊤ Transposed of x

x(k) Value of x at kth iteration

Additional sets

Ch Constraint set induced by map h

N Portfolio index set

P Private constraint set

T Coordination horizon

75

Closing Remarks

Wind turbine notation

α Minimum velocity scaling w.r.t. steady state

γ Overproduction factor

β Reference scaling w.r.t. available power

B Wind turbine viscous friction

Bmax Upper uncertainty limit of friction

Bmin Lower uncertainty limit of friction

δω Small signal velocity deviation

δτg Small signal torque deviation

J Wind turbine inertia

Jmax Upper uncertainty limit of inertia

Jmin Lower uncertainty limit of inertia

n Portfolio size

p Wind turbine power production

Pavl Portfolio available power

pavl Wind turbine available power

pref Wind turbine power reference

pw Power of wind field

τ g Wind turbine steady state generator torque

τg Wind turbine generator torque

τw Wind generated torque

Top Overproduction period

T Final control time

tc Overproduction initiation time

ω Wind turbine rotational acceleration

ω Wind turbine rotational velocity

ωmin Lower rotational velocity

ω Wind turbine steady state velocity

76

Demand management notation

α Gradient update step size

ǫabs Distributed convergence tolerance

Γ Expanded consumer model parameter

γ Norm indicator

λ Trade-off parameter

µ Mean value map

ν Lagrange multiplier

Φ Expanded consumer model parameter

Ψ Expanded consumer model parameter

σ Averaging weights

θ Cable capacity limits

ε Consumer model disturbance

ξ Dual residual

ζ Primal residual

A ADMM matrix

a Consumer model parameter

β Grid loss parameter

B ADMM matrix

b Consumer model parameter

Cp Parent indicator matrix

Cs Child indicator matrix

Ch Child map

D Diagonal loss matrix

d Distance map

F High level portfolio objective

f Private objective function

H Complete adjacency matrix

Hc Consumer adjacency matrix

77

Closing Remarks

Hl Left terminal adjacency matrix

Hr Right terminal adjacency matrix

IC Indicator function of set C

ı Phasor current

I Identity map

i Indexing variable

j Indexing variable

k Iteration counter for iterative methods

l Active power loss map

N Disicrete coordination horizon steps

n Portfolio size

p Inflexible power consumption

p Flexible power consumption

p Active power

psp Desired consumption trajectory

xsp Desired state trajectory

Pa Parent map

q Reactive power

r Resistance

s Complex power

smax Apparent power inverter limit

Ts Discretization sample time

v Phasor voltage

vs Fixed supply voltage

w Price signal

x Consumer state

xs Initial consumer state

z Impedance

78

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85

Contributions

Paper A: Utilization of Wind Turbines for Up-regulation of Power Grids

Paper B: Stability Concerns for Indirect Consumer Control in Smart Grids

Paper C: Distribution Loss Reduction by Household Consumption Coordina-

tion in Smart Grids

Paper D: Fault Tolerant Distributed Portfolio Optimization in Smart Grids

Paper E: Loss Minimization and Voltage Control in Smart Distribution Grid

Paper F: Distributed Coordination of Household Electricity Consumption

Paper A

Utilization of Wind Turbines for Up-regulation of Power Grids

Morten Juelsgaard, Jan Bendtsen and Rafael Wisniewski

This work is published in:IEEE Transactions on Industrial Electronics, July, 2013

Copyright c© IEEEThe layout has been revised

1 Introduction

Abstract

This work considers the use of wind turbines for aiding up-regulation of an elec-trical grid, by employing temporary overproduction with respect to available power.We present a simple model describing a turbine, and show how the possible periodof overproduction can be maximized by solving a series of convex problems, wherethe load is distributed among several turbines in a farm. Thereafter, we present anoptimization scheme that guarantees a lower limit for the overproduction period andsubsequently propose an adaptive implementation that is robust against parameteruncertainties.

1 Introduction

For a number of years, the Danish use of wind turbines for electrical power generation hasincreased. It is further expected to increase in the future, while it is desired to reduce theuse of fossil fired thermal plants, which currently holds the majority of power generation.

Wind is inherently volatile and hard to predict. Thus, increasing the penetration ofwind power, while at the same time reducing the use of controllable resources such asthermal plants, incurs a lower degree of production reliability in the overall system. Re-duced production reliability imposes significant problems in terms of maintaining gridstability and ensuring security of supply. For this reason, the issue of energy manage-ment and how to maintain the stability of the grid while incorporating a large percentageof volatile energy sources, has become an area of significant research interest in recentyears. See [1–6], among others.

One measure of grid stability is the grid frequency. In a network, considerable dropsin frequency could cause large currents in motors and transformers. The frequency de-viation of the network is related to the difference between the mechanical and electricalload of the generators, such that an insufficient electrical load would cause an increasedfrequency, and vice versa [7]. Therefore, if the grid frequency decreases below the nom-inal value, more active power needs to be produced and similarly, less power should beproduced when the frequency increases.

Accommodating frequency deviations by an increase or decrease of production iscurrently handled by thermal plants [8], [9], but recent research has also revolved aroundutilizing wind generated power for up- or down-regulation in response to frequency de-viations [2], [10], [11]. Down-regulation of the grid, i.e., lowering production, can beaided by wind turbines, simply by lowering their production. Up-regulation can be aidedby combining several strategies, such as power production set-points, droop control and∆-mode operation [4], [12]. In ∆-mode operation, production set-points are employedby turbines rather than a maximum production strategy, in order to maintain a productionmargin with respect to the available power. In this way, wind turbines and -farms main-tain a spinning reserve that can be activated during grid faults, for instance as a traditionaldroop control corresponding to frequency deviations. This would be applicable in situ-ations where the spinning reserve is more valuable than maximizing the power output,e.g. when part of the grid operates as an island or similar. Set-point tracking and ∆-modecontrol is discussed further by [4]. However, below rated power, wind turbines usuallyemploy a power maximizing strategy, i.e. they produce whatever power is available in the

91

Paper A

wind. In this case, the question is whether wind turbines are also capable of participatingin up-regulating the grid, i.e., by increasing production above available power.

It has been illustrated by for instance [11] that turbines are capable of temporarilyincreasing production above the available power, by extracting kinetic energy from therotating mass in the rotor plane. Consequently, the rotor will slow down, so the overpro-duction can only be maintained for a limited period of time.

In this paper, we investigate the potential and limitations of utilizing this overproduc-tion for up-regulating the grid. We introduce the optimization problem of exploiting therotational energy stored in a wind farm, such that the overproduction period is maximized.This is desirable since an increased period of overproduction would entail that turbinescould be used for grid stabilization on a larger scale, as an alternative to thermal plants.We show that, assuming perfect knowledge about model parameters, this problem can besolved as a series of convex problems. Our results illustrate that, contrary to our intuition,the overproduction period only increases slightly when increasing the farm size. This isthe case, even when the required overproduction remains fixed.

Afterwards, we present an alternative approach, robust against uncertainties in modelparameters. This alternative approach initially provides a lower bound for the overproduc-tion period, given known bounds on the uncertainties. Subsequently, we apply an adaptiveoptimization scheme for achieving this period while subject to the same uncertainties.

We will initially elaborate on the background for the flexibility introduced by em-ploying overproduction from wind turbines. This is presented in Section 2, along withthe wind turbine model. Section 3 provides a formal statement of the problem of max-imizing the overproduction period, whereafter Section 4 illustrates how maximizing theoverproduction period can be solved as a number of convex problems. Section 5 consid-ers parametric uncertainties for the purpose of formulating a robust lower bound for theoverproduction period, and also presents our adaptive implementation. This is followedby numerical examples in Section 6. Section 7 presents the conclusions and suggestionsfor further work.

2 Background and Modeling

In this paper, we extend the work presented in [11], which explores the potential of over-production from a VSWT-DFIG (variable speed wind turbine, doubly fed induction gen-erator). The basic idea is that when the turbine operates at normal conditions, below ratedpower, and produces power corresponding to the available power of the wind, the rotatingmass of the rotor can be seen as an energy storage. Extracting energy from this storage en-tails that the power production can be increased above available power by increasing thegenerator torque. However, this slows down the rotor. In this way, the turbine is regardedas having a built in flywheel for storing energy, that can be used for power smoothing.The use of flywheels as dedicated energy storages, and how power imbalances can besmoothed by such, has been further investigated by [13] and [14].

The turbine rotor must not be slowed below some lower limit, so at this point, the gen-erator torque must be reduced, allowing the rotor to accelerate back to normal operatingconditions. This is illustrated in Fig. 1 [11].

Fig. 1 illustrates the power production from a turbine, divided into the three dif-ferent periods denoted ’Normal’, ’Overproduction (OP)’ and ’Underproduction (UP)’ -

92


pop

p(t),p

mek(t),p

avl(t)

t

Normal OP UP Normal

p(t)pmek(t)pavl(t)

tc

Figure 1: The electrical and mechanical power of the turbine, during normal, OP (overproduction)and UP (underproduction) periods, [11].

operation. In normal operation, the power p(t) ∈ R+, produced by the turbine, equalsthe available power in the wind pavl(t) ∈ R+, i.e., p(t) = pavl(t). Here t ∈ [0; ∞) is thetime and R+ refers to the non-negative real numbers. The power p(t) is limited above bythe rated power, pmax, of the turbine, but we will disregard this concern in this work.

At some time tc, the turbine power reference is increased, for instance as a result ofa frequency imbalance in the grid. The turbine is now required to produce p(t) = pop >pavl(t), which is an overproduction. The electrical power can increase, but as illustrated,the mechanical power in the rotor will consequently decrease, because the rotor slowsdown. When the lower limit is reached, the electrical power will have to be reducedin order to return the mechanical power to the operating point by accelerating the rotorplane. This causes a period of underproduction with respect to the available power.

By use of state-of-the-art simulation models, [11] explores how the period of over-production, is affected by a number of parameters, e.g. the magnitude of overproduc-tion, acceleration during underproduction, available power etc. However, the results arepurely simulation based, and no general mathematical results are presented. Our workemploys a different strategy. We use a simple, dynamic model for a VSWT, that mimicsthe results of [11], while allowing us to investigate the potential and limitations posed byoverproduction. We do this by using the model for deriving mathematical statements andguarantees for the period of overproduction.

2.1 Turbine modeling

We can model the described behavior by approximating the turbine as a flywheel, asillustrated in Fig. 2. The flywheel is characterized by an inertia J , a rotational velocity

J

τg(t) τw(t)Bω(t)

ω(t)

Figure 2: Flywheel model of a wind turbine.

93

Paper A

of ω(t) ∈ R+ and a viscous friction B, where J and B are positive constants specificto the given turbine. The rotating motion is driven by a torque τw(t) ∈ R+ produced bythe wind. The turbine generator affects the rotation by a torque τg(t) ∈ R+, opposite thedirection of rotation.

The power p(t), is given by p(t) = τg(t)ω(t), so a model describing the turbine, canbe arranged as

Jω(t) = -Bω(t)− τg(t) + τw(t)p(t) = ω(t)τg(t),

(1)

where ω(t) = dω(t)/dt, with initial conditions ω(tc) = ω , where ω is some desiredsteady state rotational velocity of the turbine. This model has one controllable inputτg(t). The use of the generator torque counteracts the incoming wind torque τw(t), andaffects the rotational velocity ω(t) as well as the output p(t). Given the initial conditions,(1) defines the mapping τg(t) 7→ p(t), i.e, from generator torque to power output.

In Fig. 3, we have illustrated how this model contains the dynamics illustrated byFig. 1, in that applying the torque presented in Fig. 3 (Top), we obtain the power output,and rotational velocity depicted in Fig. 3 (Middle) and (Bottom) respectively.

The available power for the turbine model (1) is defined as the power generated insteady state operation, i.e.,

pavl(t) = τgω, (2)

whereτ g = (τw(t)−Bω). (3)

That is, τg is the generator torque that would result in ω(t) = 0, when operating atω(t) = ω . We assume that τg > Bω , meaning that in the operating point, more power isproduced in the generator than lost to friction.

2.2 Farm modeling

We expand the model of a single turbine to cover an entire farm, simply by aggregation.Any inter-turbine effects, such as changes in wind speed or turbulence throughout thefarm, can be included in the modeling of the wind torque τw,i(t), affecting the ith turbine.The aggregation of n individual turbines into a farm is illustrated in Fig. 4. This workfocuses solely on the active power output of the wind farm, and does not consider theremaining electrical grid with respect to dynamic behavior, transmission losses, etc.

When aggregating turbines into a farm, the scalar signals from each turbine will com-prise entries in vector signals, which we present with a bold-font notation, such that

τg(t) = (τg,1(t), . . . , τg,n(t)) ∈ Rn+

andτw(t) = (τw,1(t), . . . , τw,n(t)) ∈ R

n+.

Similar notation is employed for the remaining signals. The accumulated available powerof the farm is denoted Pavl ∈ R+, and is expressed as

Pavl(t) =n∑

i=1

pavl,i(t). (4)

In the following, we will assume that the wind has constant power denoted, pw ∈ R+.Even though this assumption to some extend reduces the accuracy of our results, we

94


τg(t)

[Nm

]

20 40 60 80100

150

200

250

300

p(t)

[kW

]

20 40 60 80500

1000

1500

2000

2500ω(t)

[rad

/s]

t [s]20 40 60 80

8

9

10

11

12

13

Figure 3: Top: An example of the torque τg(t) input to the model. Middle: The result-ing power output p(t) (Solid), where we have also illustrated the available power pavl(t)(Dashed, Asterisk), and the overproduction pop (Dashed, plain). Bottom: The resultingrotational velocity ω(t) (Solid), along with the allowed lower limit ωmin (Dashed).

To grid

τw,1(t)

τw,2(t)

τw,n(t)

p1(t)

p2(t)

pn(t)

Figure 4: Aggregation of turbines into a farm.

maintain this assumption throughout this work in order to easier illustrate the implicationsof our results. We discuss this further in Section 7.

The fraction of wind power applied to the turbine rotor, is related to the power coeffi-cient by

pr(t) = Cp(λ(t), θ(t))pw, (5)

where pr(t) ∈ R+ is the power applied to the rotor by the wind, Cp : R × R → R is the

95

Paper A

power coefficient, where λ(t) is the so-called tip-speed ratio, and θ(t) is the pitch angle ofturbine blades [15]. The tip-speed ratio is defined by λ(t) = Rω(t)/V , where the scalarsR and V are the turbine blade radius and incoming wind speed, respectively. The windspeed is in this work assumed constant.

An illustration of a power-coefficient as a function of λ is presented in Fig. 5, for fixedpitch θ = θ. The basic shape of the curve is similar for all pitch angles.

Cp(λ

,θ)

λ0 5 10 15 20

0

0.1

0.2

0.3

0.4

0.5

Figure 5: Power coefficient as a function of tip-speed ratio, for fixed pitch. This exampleis for the NREL CART3 turbine [16], found using WT_perf.

The right-most vertical dashed line in Fig. 5 indicates the steady state tip-speed ratioλmax, where maximum available power is extracted from the wind. When overproductionis initiated, the rotor slows down, i.e. the tip-speed ratio decreases, where the left-mostdashed line in Fig. 5 indicates the lower limit. We will employ a linear approximation ofthe Cp curve in the range between the dashed lines. The error introduced by this approxi-mation depends on the lower limit as presented in the table below, for the curve in Fig. 5:

Lower limit.9 .8 .7 .6 .5 .4

(Fraction of λmax)Error (%) 4.7 8.6 11.9 12.2 31.8 81.8

Employing the linear approximation entails

Cp(λ) ≈ c1λ(t) = c1R

Vω(t), (6)

where c1 is the best fit coefficient obtained for the approximation. The power applied tothe rotor by the wind can be described by

pr(t) = τw(t)ω(t) ⇔ (7)

τw(t) =pr(t)

ω(t)≈c1Rpwω(t)

V ω(t)=c1RpwV

. (8)

So, employing the linear approximation of Cp in the desired range of operation, entailsthat a constant wind power can be translated to a constant wind torque, as seen from thefly-wheel model. We employ this subsequently.

96

3 General Problem Description

3 General Problem Description

In the following, we present the general problem of maximizing the overproduction periodfor a farm, when subjected to a demand exceeding the accumulated available power. Westart by providing a formal definition of the overproduction period.

Consider a wind farm consisting of n turbines. The farm is subjected to a demandpdem(t) ∈ R+, given by

pdem(t) =

n∑

i=1

pavl,i(t), t < tc

(1 + γ)n∑

i=1

pavl,i(t), t ≥ tc

, (9)

where γ > 0.For t < tc, the demand does not exceed the accumulated available power and can

thereby be tracked closely. For t ≥ tc, the farm is required to overproduce. As described,the farm can only overproduce in a limited time period. We define this overproductionperiod by1

Top(τg) = supt

t− tc

∣∣∣∣∣t > tc, ∀s∈[tc,t] pdem(s) ≤

n∑

i=1

pi(τg,i(s))

,

with pi(t) defined as in (1), and the demand as defined in (9).The overproduction period Top is only defined for τ g(t) ∈ Tpdem, where2

Tpdem =

τg ∈ Ln∞([0, ∞))

∣∣∣∣∣∃s > tc such that ∀t∈[tc;s] pdem(t) ≤

n∑

i=1

pi(τg,i(t))

As evident, Tpdem depends on the demand pdem.As the demand can be met for all t < tc, Top is the time between tc and the first

following time instance, where the farm production decreases below the demand. Aspi(t) depends on τg,i(t), Top depends on τ g(t).

Given the model (1), the task is to choose τ g(t) such as to maximize the overproduc-tion period. This can be formulated as

maximizeτg

Top(τg)


1

Ji(-τg,i(t) + τw(t))

pi(t) = ωi(t)τg,i(t)ωi(t) ≥ ωmin, τg,i(t) ≥ 0,t ≥ tc,

(10)

for i ∈ 1, . . . , n, with initial conditions ωi(tc) = ωi. The first two constraints describethe dynamics of the model. The inequalities express the previously mentioned practicalconstraints, where ωmin is the allowed lower limit for the rotational velocity. It shouldfurther be stressed that the cost function in Problem (10) implies the constraint

pdem(t) ≤n∑

i=1

pi(t), tc ≤ t < tc + Top, (11)

1Errata corrected in equation compared to original publication2Errata corrected in equation compared to original publication

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Paper A

due to the definition of Top(τ g). Note that Problem (10) is non-convex, on account ofnon-convex constraints. We will analyze this problem in the following.

The wind farm as a whole is required to produce a power demand pdem(t), such thatn∑

i=1

ωi(t)τg,i(t) = pdem(t), (12)

where each turbine is subject to a set of dynamic equations as in (1). In the following wewill assume a constant demand, that is pdem(t) = pdem. We will further assume that eachturbine produces a constant power, following a constant reference pref,i, such that

n∑

i=1

pref,i = pdem. (13)

Focusing on a single turbine, this entails

ω(t)τg(t) = pref, (14)

where we omit the subscript i. By differentiation, we get

ω(t)τg(t) + ω(t)τg(t) = 0. (15)

Inserting ω(t) from (1), yields(−B

Jω(t)−

1

Jτg(t) +

τw(t)

J

)

τg(t) + ω(t)τg(t) = 0, (16)

which reduces to

−B

Jω(t)τg(t)−

1

Jτg(t)

2 +τw(t)

Jτg(t) + ω(t)τg(t) = 0. (17)

Now, using (14), this is reformulated to

τg(t) =1

Jpref,iτg(t)

3 −τw

Jpref,iτg(t)

2 +B

Jτg(t), (18)

where we have assumed a constant torque from the wind, τw(t) = τw on account of thediscussion in Section 2.2.

Equation (18), is a nonlinear, first order differential equation, known as an Abel differ-ential equation of first kind [17]. Given that all coefficients in (18) are constants, we canin principle solve it by the method described in [18], using the results collected in [19].However, an explicit expression for τg(t) cannot be derived. We therefore choose toemploy a different strategy, and introduce a linear approximation of pi(t) such that

pi(t) ≈ ωiτg,i(t) + τ g,iωi(t)− ωiτ g,i (19)

where (ωi, τ g,i) is the operating point of the approximation. By writing the rotationalvelocity and generator torque as

ωi(t) = ωi + δωi(t), τg,i(t) = τ g,i + δτg,i(t), (20)

where δωi(t) and δτg,i(t) describe the deviation from operating point, we see that theerror ei(t) introduced by the linear approximation, is

ei(t) = δωi(t)δτg,i(t), (21)

which is negligible in the vicinity of the operating point.

98

4 Maximizing Overproduction Period


By making the linear approximation (19) of pi(t), we can solve Problem (10) for a specificpower demand, by reformulating it as a convex feasibility problem. We do this by defining

Top = Top(τg), (22)

and afterwards, solve the feasibility problem

find τg(t) ∈ Tpdem


1

Ji(-τg,i(t) + τw)

pdem(t) =n∑

i=1

(ωiτg,i(t) + τg,iωi(t)− ωiτg,i)

ωi(t) ≥ ωmin, τg,i(t) ≥ 0,

0 < t < tc + Top.

(23)

By iteratively increasing Top until (23) becomes infeasible, we find an approximate solu-tion to (10). This approach is illustrated in Fig. 6.

T 1op

T 2op

T 3op

T 4op

pdem

pavl

ttc

∑n i=

1pi(t)

T ∗op

Figure 6: Illustration of the iterative solution approach to the feasibility problem, for 5values of Top (short dashed), the demand pdem (long dash), and the available power pavl

(solid).

Fig. 6 illustrates how large an overproduction a farm is able to maintain, if it is re-quired to maintain it for T 1

op, T2op, T

3op and T 4

op seconds respectively. For T 1op, which is

a very small period, the farm is capable of producing a very large power before the en-ergy stored in the rotor is depleted. This power spike is far larger than the demand. ForT 2

op > T 1op, the farm can produce less power, but still more than the required demand.

Similarly for T 3op. However, T 4

op is too large, in that the farm is not able to obey the de-mand for the required overproduction period, before the stored energy is depleted. Byiterating, we can eventually find the value T ∗

op, where T 3op < T ∗

op < T 4op, such that the

demand is exactly met throughout the period.The solution found in this way, by (23), only approximately solves (10), on account

on the simplification made in (19).Ignoring mechanical limitations, Proposition 1 in Appendix A shows that we can

always find a sufficiently small Top, such that (23) has a solution. For any fixed value ofTop, (23) is convex, and can thereby be solved efficiently [20].

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Paper A

The approach outlined above is completely general, in the sense that it makes noassumptions on how large part any one turbine should play in the overall overproduction,or when any one individual turbine should start to overproduce, with respect to tc. Insteadit simply finds the torques that should be applied to each turbine over time, in order toreveal the longest overproduction period, given the demand. We will illustrate this with afew examples.

4.1 Example - Identical Turbines

First consider a farm consisting of n identical turbines, with

B1 = · · · = Bn = B, J1 = · · · = Jn = J, (24)

where B and J are well known. The available power defined by (2) is thereby constantand equal for all turbines, and we further have Pavl = npavl. The farm is subjected to ademand

pdem(t) =

Pavl, t < tc,

Pavl +1

2nPavl, tc ≤ t < tc + Top,

Pavl, t ≥ tc + Top,

(25)

and we use the approach outlined above for maximizing Top.In the following, we consider a farm with n = 10 turbines, for which we instantiate

the overproduction at tc = 50 s. When using B = 1 kg · m2/s, J = 80 kg · m2,ω = 4π rad/s, τw = 201 Nm and ωmin = 0.7ω rad/s, we obtain the results presented inFig. 7 through 9. We obtain an overproduction period of Top = 10.7 s.

Fig. 7 presents the power produced by all 10 turbines. As it appears, they all pro-duce the same power. The production increases at t = 50 s where the overproductionperiod starts. Following the overproduction, comes the underproduction period, where allturbines recover to their original operating conditions.

pi(t)

[kW

]

t [s]20 40 60 80 100

500

1000

1500

2000

2500

3000

Figure 7: The power produced by each individual turbine.

By summing the production from each turbine, the farm production is obtained. It ispresented in Fig. 8 along with the demand.

Finally, the rotational velocities of all turbines are presented in Fig. 9, along with theallowed lower limit. As seen from this figure, the overproduction is maintained until allstored energy for the turbines has been depleted, whereafter the turbines start to recover.

100


pi(t)

[kW

]t [s]

20 40 60 80 100

×104

0.5

1

1.5

2

2.5

3

Figure 8: The accumulated production of the farm (Solid), and the farm demand (Dashed).

ωi(t)

[rad

/s]

t [s]20 40 60 80 100

8

9

10

11

12

13

Figure 9: The rotational velocity of each turbine in the farm (Solid) and the allowed lowerlimit (Dashed).

4.2 Example - Non-identical Turbines

Next, we consider n non-identical turbines, i.e.,

B1 6= · · · 6= Bn, J1 6= · · · 6= Jn. (26)

In this case, the model parameters are uniformly distributed over known intervals,

Bi ∈ [1; 4], Ji ∈ [80; 120], i = 1, . . . , n. (27)

We still assume that we know all the parameter values. Using the same simulation valuesas before, we obtain the results in in Fig. 10 through 12. In this example, we obtain anoverproduction period of Top = 34.4 s.

The power produced by each turbine is presented in Fig. 10. Unlike the examplewith identical turbines, the individual turbines now produce different powers. They alloverproduce, but with different levels, and the individual overproduction is commencedat different time instances. However, in the same way as before, all turbines produce aconstant power during overproduction.

Fig. 11 presents the accumulated production across the farm, as well as the powerdemand. Notice that compared to the previous example, the demand is now lower. Thisis because the demand is calculated based on the available power across the farm, and theavailable power depends on the friction, as evident from (2).

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Paper A

pi(t)

[kW

]t [s]

20 40 60 80 1000

2000

4000

6000

8000

10000

Figure 10: The power produced by each turbine in the farm. Each color corresponds toan individual turbine.

pi(t)

[kW

]

t [s]20 40 60 80 100

×104

1

1.5

2

2.5

Figure 11: The accumulated production across the farm (Solid), and the farm powerdemand (Dashed).

ωi(t)

[rad

/s]

t [s]20 40 60 80 100

8

9

10

11

12

13

Figure 12: The rotational velocity of each turbine (Solid), and the allowed lower limit(Dashed). Each color corresponds to an individual turbine.

The rotational velocities of all turbines are shown in Fig. 12. Similar to Fig. 9, theoverproduction is maintained as long as there is stored energy in any turbine.

The two examples presented above suggest that the optimal strategy, with respectto the power production in overproduction periods, is that all turbines should produce aconstant power, although not necessarily the same constant power, and not necessarily atthe same time. Nonetheless, we will use this general strategy in the following.

102

5 Robust Overproduction Strategy


The general approach for maximizing overproduction period described in Section 4 suf-fers from the disadvantage of requiring information about the parameter values for all tur-bines in our wind farm model. In the following, we will present an alternative formulationthat is less general, but provides a certain robustness against parametric uncertainties.

We shall employ a two step approach:

1. Find a lower bound on the overproduction period, assuming worst-case parameter val-ues.

2. Find the generator torques, for achieving this lower bound, given parametric uncer-tainties.

In Step 1, we find a guaranteed overproduction period, even under parametric un-certainties. In a practical setting, this worst case period would give a farm operator aguaranteed period in which a given farm can participate in up-regulating the grid, even ifthe operator is uncertain about specific parameter values.

In Step 2, we adaptively compute the torques that should be applied each turbine, inorder for the farm to overproduce for a time corresponding to the lower bound on theoverproduction period.

5.1 Overproduction Period Bound

Dispatch Strategy

Instead of solving the general problem outlined in (23), we can decide on the productionof each turbine based on a predetermined dispatch strategy. A dispatch strategy calculatespower references pref,i(t) ∈ R+, i = 1, . . . , n, for each turbine. We assume that the elec-trical dynamics of the turbine generator, are much faster than the mechanical dynamicsof the rotor, such that pi(t) = pref,i(t) if pref,i(t) < pavl,i(t).

In the following, we use the dispatch strategy

pref,i(t) = pavl,i(t) +pdem(t)−

∑ni=1 pavl,i(t)

n. (28)

If pdem(t) ≤∑ni=1 pavl,i(t), the references dispatched by (28) will all be less than the

available power for the individual turbines, i.e., pref,i(t) ≤ pavl,i(t), and the productionwill meet the demand. If however pdem(t) >

∑ni=1 pavl,i(t), then pref,i(t) > pavl,i(t), and

the turbine will only be able to follow this reference for a limited time, as the rotor willslow down.

If we employ (28), for the demand in (9), we see that the production reference to allturbines can be expressed as

pref,i(t) =

pavl,i, t < tc

pavl,i + γ

∑nj=1 pavl,j

n, t ≥ tc

, (29)

where we have omitted the time dependency on pavl(t), since the assumption of constantwind, entails constant available power.

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Paper A

Since the demand is constant for t > tc, this entails that the references will be constantfor t > tc, i.e.,

pref,i = pavl,i + γ

∑nj=1 pavl,j

n, t > tc (30)

for i = 1, . . . , n. When tracking this power reference, the model in (1), with the lineariza-tion in (19), is expressed by

Jiωi(t) = -Biωi(t)− τg,i(t) + τwpref,i = ωiτg,i(t) + τg,iωi(t)− ωiτg,i,

(31)

for t > tc, and ωi(tc) = ωi.Solving the differential equations reveals

τg,i(t) = di + cieτg,i−Biωi

Jiωi(t−tc), ωi(t) = ai + bie

τg,i−BiωiJiωi

(t−tc), (32)

for t > tc, where

ai =Bipref,i +Biωiτg,i − τg,iτw

Biτg,i −B2i ωi

+τwBi

bi =τwτg,iωi −Bipref,iωi −Biω

2i τg,i

τ2g,i −Biωiτ g,i−pref,i

τ g,i

ci =pref,i

ωi−τ g,iτw −Bipref,i −Biωiτ g,i

τg,i −Biωi

di =τg,iτw −Bipref,i −Biωiτg,i

τ g,i −Biωi

are all constants, with ai > 0, bi < 0, ci > 0, di > 0, and

τg,i −BiωiJiωi

> 0. (33)

From this, we can derive an expression for the time interval Top,i, where pi(t) = pref,i,before ωi(t) reaches the lower bound ωmin. We obtain

Top,i =ωiJi

τg,i −Biωiln


)

. (34)

Employing the dispatch strategy (28) to obtain (34) can obviously only be suboptimal,compared to solving the general problem in (23). However, as the task is to provide arobust lower bound on the overproduction period, this approach is more suitable, as weshall illustrate shortly. We have further remarks concerning this in Section 7.

Equation (34) represents a relation between the possible overproduction period, andthe number of turbines in the farm. We will illustrate this in a later example.

Parametric Uncertainty

Let the model parameters be unknown, except that they are known to lie in certain inter-vals, i.e,

Bi ∈ B = [Bmin; Bmax], Ji ∈ J = [Jmin; Jmax], (35)

for i ∈ 1, . . . , n. Proposition 2 in Appendix B , shows that if τw > 2Bω , and if thepower reference during overproduction is expressed as pref,i = βpavl,i, β > 1, and the

104


lower limit on rotational velocity as ωmin = αω , 0 < α < 1, then the lowest value forTop,i over all possible parametrizations of Bi ∈ B and Ji ∈ J , is obtained for Bi = Bmin

and Ji = Jmin.This implies that even when the specific parameter values for the parameters across

our farm model are unknown, a lower bound on the over production period can still beobtained. We can illustrate the implications of this by the following simple example.

Example

We again consider a farm consisting of n identical turbines, meaning that

J1 = · · · = Jn = J, B1 = · · · = Bn = B, (36)

which yields the same available power, pavl,i = pavl, i = 1, . . . , n. The power demandduring the overproduction period is given by

pdem =

(

n+1

2

)

pavl, (37)

corresponding to γ = 1/(2n) in (9).From the dispatch strategy (28), the production from each turbine during overproduc-

tion is

pref,i =

(

1 +1

2n

)

pavl, i = 1, . . . , n. (38)

The overproduction with respect to the available power for each individual turbine isthereby inversely proportional to the number of turbines in the farm.

Using (34), Fig. 13 illustrates the mapping between n and Top, for n = 1, . . . , 1000.The figure presents four graphs, corresponding to the four combinations of maximum andminimum B and J . This corresponds to examining the four vertices of the parametricuncertainty region.

In Fig. 13, we have used B = [1; 4] kg · m2/s, and J = [80; 120] kg · m2. We havefurther used ω = 4π rad/s, τw = 201 Nm and ωmin = 0.7ω rad/s.

Top

[s]

n [-]0 200 400 600 800 1000

0

20

40

60

80

Figure 13: The overproduction period, as a function of farm size. All turbines have the sameinertia and friction, with the four combinations: Bmin; Jmin(Solid, Plain), Bmax; Jmin(Solid,Dots), Bmin; Jmax(Solid, Asterisk), Bmax; Jmax(Dashed).

Fig. 13 reveals several interesting points. First, as mentioned in Section 1, the gain inTop, obtained by increasing the number of turbines n in the farm, decreases as n increases.

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Paper A

That is, even though the required overproduction of the farm remains fixed, the increasein Top obtained by increasing n, is only minor. This is due to the exponentials in (32).

Secondly, the specific value of the model parameters have significant impact on theoverproduction period. However, as expected, the lowest value of Top is obtained forBi = Bmin and Ji = Jmin, i = 1, . . . , n.

5.2 Robust Strategy

We now propose a robust overproduction strategy capable of obtaining the lower bound,without knowing the parameters of the individual turbines. That is, we only know B andJ as given by (35). Further, we assume to know the available power for the farm, albeitnot necessarily for the individual turbines. The demand is expressed by (9) as before.

Our approach can be expressed by the following procedure, which we elaborate below,

1. Assume that all turbine parameters are equal, and Bi = Bmin and Ji = Jmin, ∀ i.

2. Estimate available power for all turbines.

3. Find optimal control input to the estimated farm model.

4. Apply control signal, obtain measurements.

5. Reestimate parameters for all turbine models.

6. Continue from 2.

The first step in the procedure above is to assume that all turbines in the farm are locatedin the same corner of the uncertainty set. This gives us an initial estimate, which we canupdate during runtime.

The available power is estimated as

pestavl,i = τ g,iωi, (39)

where τ g,i can be calculated as

τg,i = τw −Besti ωi, (40)

Besti being the estimated friction coefficient, and ωi is the desired stationary rotational

velocity of the rotor. Besti is initially estimated as Bmin, as given by Step 1. Afterwards, it

is estimated based on measurements as described below.Given our current estimate of the model parameters Best

i and J esti , we want to find

the optimal control input for the turbines to follow their references as close as possible.The references are distributed using the dispatch strategy (28) and the estimated availablepower values.

We solve the optimization in a discretized fashion, employing a model predictivestrategy, looking H steps ahead [21]. At discrete time k the problem can be expressed as

minimizeτg,i

H−1∑

j=0

n∑

i=1

((pref,i(k + j) − pi(k + j))2 + µτg,i(k + j)2

)

subject to ωi(k + 1) = φesti (k)ωi(k) + ψest

i (k)(τw − τg,i(k))pi(k) = ωiτg,i(k) + τg,iωi(k)− ωiτg,iωi(k) > ωmin

τg,i(k) > 0,

(41)

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6 Numerical Example

for i = 1, . . . , n and j = 0, . . . , H − 1, where µ ∈ R+ is a trade-off parameter, andφesti (k) and ψest

i (k) are the discretized, current estimate of the coefficients in the dynamicequations:

φesti = 1−

Besti

J esti

Ts, ψesti =

1

J esti

Ts. (42)

Here we have used zero-order-hold for discretization, with a sample time of Ts.We use a receding horizon strategy, and only apply the first sample of τg,i(k). From

this we obtain a measurement of ωi(k + 1). We can use this to reestimate the modelparameters via the least squares problem

minimizex

‖F (k)x(k)− g(k)‖2

subject to φesti (k) ∈

[(

1−Bmax

JminTs

)

;

(

1−Bmin

JmaxTs

)]

ψesti (k) ∈

[TsJmax

;TsJmin

]

,

(43)

with variable x(k) = [ψesti (k) φest

i (k)], and

F (k) =

ωi(k) τg,i(k)− τwωi(k − 1) τg,i(k − 1)− τw

......

ωi(1) τg,i(1)− τw

, g(k) =

ωi(k + 1)ωi(k)

...ωi(2)

. (44)

After updating the model, we can again solve (41), apply first control sample, obtainmeasurement, reestimate model etc.

It should be noted that, during runtime, we alter the optimization horizon H . In nor-mal operation where pref,i(k) = pavl,i(k), we use a quite large value for H . However, inan overproduction situation where pref,i(k) > pavl,i(k), we set H = 1, i.e., we effectivelyremove the prediction, in our optimization. If we had continued with a long horizon, ouroptimization algorithm would have completely omitted to track the increased demand inorder to avoid the underproduction period illustrated in Fig. 1. Our interest is instead totrack the overproduction for as long a period as possible.

It can seem redundant to make this online estimation of the model parameters, andafterwards introducing an MPC control scheme for calculating power references. In-stead it would seem more straightforward to just solve Problem (23) after estimating themodel parameters. This is not possible however, since solving Problem (23) is an of-fline approach for obtaining the references, and we need an online implementation forincorporating measurements, and making updates and corrections during runtime. On afinal note, the MPC problem and receding horizon strategy described above could not beexchanged with a LQR or LQG controller, since the constraints imposed in (41) becomeactive when the stored energy is depleted, and the overproduction period is terminated.

6 Numerical Example

The following presents a numerical example of both obtaining the lower bound of theoverproduction period, as well as arranging the robust strategy. Consider a farm of n = 10

107

Paper A

turbines, whose parameter values are distributed uniformly between a lower and uppervalue. We assume that we do not know their specific parameter values, only their ranges:

Bi ∈ [1; 4], Ji ∈ [80; 120], i = 1, . . . , n. (45)

We initially want to find a lower bound on the overproduction period, and afterwards useour adaptive optimization scheme to achieve it.

6.1 Overproduction Bound

As explained in Section 5.1, the worst case overproduction period, for a demand in theform (9) is obtained by assuming

Bi = Bmin, Ji = Jmin, i = 1, . . . , n. (46)

Using (34), we can depict the lower bound on the overproduction for our farm as a func-tion of γ; see Fig. 14

Top

[s]

γ [-]0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

Figure 14: The lower bound on overproduction period as a function of overproduction.

6.2 Adaptive Optimization

We implement our adaptive optimization for a case where we know the available powerof the farm. Denote this available power for the farm, Pavl. The power demand for thefarm is given by

pdem(k) =

Pavl, k < kc

Pavl +1

2nPavl, kc ≤ k < kc + Top

Pavl, k ≥ kc + Top

(47)

meaning that γ = 1/(2n) = 0.05. From Fig. 14, this gives a lower bound of Top =11 s on the overproduction period. We therefore apply a power demand, with an 11 soverproduction period, and expect that this can be obeyed. We initiate the overproductionat time kc = 50. After the overproduction period, we want all turbines to recover to theirinitial conditions, yielding a period of underproduction.

Following the procedure in Section 5.2, we obtain the results presented in Fig. 15through Fig. 19. In this example, we have used ω = 4π rad/s, τw = 201 Nm and ωmin =0.7ω rad/s, similar to the example in Section 5.1.

108

6 Numerical Example

To make the situation more realistic, we add noise to all measurements of the rota-tional velocity, such that at time k, we obtain

ωi(k + 1) = φesti ωi(k) + ψest

i (τw − τg,i(k)) + νi(k), (48)

where νi(k) is zero-mean, normally distributed, random noise, with standard deviation σ.In the following we have chosen σ to be 1 % of the allowed range for ω(k).

In Fig. 15, the references and production from each turbine are shown, with the ref-erences calculated by the dispatch strategy (28). Similarly, the demand tracking for theentire farm is presented in Fig. 16.

pi(k)

andp

ref,i(k)

[kW

]

k [s]0 20 40 60 80 100

1000

1500

2000

2500

Figure 15: The reference to each turbine (Dashed), and the actual production (Solid).

∑n i=

1pi(k)

andp

dem

[kW

]

k [s]0 20 40 60 80 100

×104

1.4

1.6

1.8

2

2.2

2.4

Figure 16: The power demand (Dashed), and the farm production (Solid).

We see that the demand tracking is quite accurate in the overproduction period, despitethe addition of noise. This suggests that the noise have only minor effect on the modelestimation, and rather effects the expected outcome throughout the prediction. We recallthat in the overproduction period, we reduce the optimization horizon to H = 1, and bythis, we see in the figure that a small horizon limits the effect of the noise.

The rotational velocity is plotted in Fig. 17. The figure shows that after the 11 soverproduction period, only 1 turbine is close to the lower limit ωmin = 8.78 rad/s. Allthe remaining turbines could in principle have continued to overproduce, which illustratesthat the result from Section 6.1, is conservative.

The applied torques are presented in Fig. 18. We see that there is an initial process offinding a steady operating point, which is on account of estimating the correct model.

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Paper A

ωi(k)

[rad

/s]

k [s]0 20 40 60 80 100

8

9

10

11

12

13

14

Figure 17: The obtained rotational velocity of the turbines in the farm (Solid), and the allowedlower limit (Dashed). Only one turbine has depleted its energy storage after the overproductionperiod.

τg,i(k)

[Nm

]

k [s]0 20 40 60 80 100

100

150

200

250

300

Figure 18: The torque applied to each turbine in the farm.

In Fig. 19, we have plotted both the online, linear estimation of the produced power, aswell as the true nonlinear production. We see that by continuously updating the operatingpoint, the two are practically indistinguishable.

pi(k)

[kW

]

k [s]0 20 40 60 80 100

1000

1500

2000

2500

Figure 19: Comparison between the linear approximation of the turbine power (Solid), and theactual produced power (Dashed).

110

7 Conclusion and Further Work

7 Conclusion and Further Work

In this work, we have presented an optimization-based strategy for employing the energystored in the rotor of a wind turbine for up-regulation of an electrical grid, as a period ofoverproduction. We have illustrated how the time an overproduction can be achieved isrelated to the size of a wind farm, and further how the maximum overproduction periodcan be found by solving a sequence of convex feasibility problems. Further we havedemonstrated how the stored energy can be utilized via a strategy robust to parametricuncertainties. Our results have illustrated that farm size has only minor influence onthe overproduction period, even when the required overproduction remains fixed, whileincreasing the number of turbines in the farm.

The approach described in Section 4 for maximizing the overproduction period, isvery basic, and numerous extensions can be implemented. For instance, a constraint onthe maximum production of each individual turbine could be included in order to accom-modate their respective rated power limitations. Another simple extension could be toinclude a constraint on maximum underproduction, following the overproduction, i.e.,how large a dip in the farm production we would accept. These are just a few possibleextensions to basic feasibility approach from problem (23). The point is that these ex-tensions are convex [20], so including them as additional constraints in (23), would notchange convexity, and we would still be able to solve the series of feasibility problemsefficiently.

Throughout the design of the robust utilization of the stored energy in the turbinerotors, we have employed the dispatching strategy presented in (28). Even though numer-ical results suggest it, we have at no point argued the optimality of this strategy. Rather,we have illustrated how one might come about parametric uncertainties, in a way thatguarantees a conservative performance.

This work assumed constant wind fields across the farm, which will not be the casein practical situations. We have made this assumptions in order to better illustrate ourapproach and results. The remedy would be to introduce more accurate estimates of thewind fields in Equations (10), (23), and (41). This would be a natural extension of thework presented here.

In the examples throughout the paper, we have addressed overproduction for turbinesin the same farm, and it could therefore be argued that parametric uncertainty might notbe relevant, since turbines in the same farm are usually of the same type and thereforehave roughly the same parameter specifications. However, it is well known that param-eters tend to change over time during actual operation; for instance, friction is known todepend on temperature as well as general wear and tear. Furthermore, considering dif-ferent parameter values allows greater flexibility in terms of defining which turbines toinclude in a ’farm’: The overproduction scheme could be planned for a larger aggregationof individual turbines, or perhaps aggregation of entire farms, distributed across differentgeographical locations, where it is not certain that all parameters are equal or known.

A Appendix: Proof of Feasible Solution

In the following, we will illustrate that it is possible to find a sufficiently small Top, forwhich the problem (23) has a solution, for any γ > 0.

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Paper A

Proposition 1. Let n be the number of wind turbines in a farm, each with power pro-duction pi(t), i = 1, . . . , n, given by the mapping (1). For any γ > 0, there exists asufficiently small Top > 0, such that the demand

pdem(t) =

n∑

i=1

pavl,i(t), t < tc

(1 + γ)n∑

i=1

pavl,i(t), t ≥ tc

, (49)

is obtainable in the period tc ≤ t ≤ tc + Top, i.e,

pdem =n∑

i=1

pi(t), tc ≤ t ≤ tc + Top. (50)

Proof: When attempting to find the largest demand a farm can obey, without regardsfor the duration, the only sensible utilization of the stored energy is to have all turbinesproduce maximum power, immediately. In the following we disregard the constraintposed by the rated power on each turbine.

For a turbine delivering all stored energy, over a small time interval Top, with initialconditions ωi(tc) = ωi, the average change in rotational velocity is described by

ωi(t) =ωmin − ωi

Top. (51)

Inserting this into the model (1), we get

ωmin − ωi

Top

=-BiJiωi −

1

Jiτg,i(t) +

1

Jτw

pi(t) = ωiτg,i(t),(52)

where we have inserted ωi(t) = ωi, on account of the initial conditions. Rearranging Eq.(52) gives

τg,i(t) =

(Ji

Top−Bi

)

ωi −Jiδtωmin + τw, (53)

and further

pi(t) = ωi

((Ji

Top

−Bi

)

ωi −Ji

Top

ωmin + τw

)

. (54)

The maximum power a farm can deliver during the period Top, is then obtained by sum-ming (54) for i = 1, . . . , n.

The available power is defined as

pavl,i(t) = ωi(τw −Biωi), (55)

and so, the demand presented in (9) can, during overproduction, be expressed as

pdem(t) = (1 + γ)

n∑

i=1

ωi(τw −Biωi), t ≥ tc (56)

112

B Appendix: Lower Bound on Overproduction period

In order for the power demand to be feasible, there exists a sufficiently small Top > 0,such that

pdem(t) =n∑

i=1

pi(t), tc ≤ t ≤ tc + Top, (57)

and the demand must staisfy

(1 + γ)

n∑

i=1

ωi(τw −Biωi) ≤

n∑

i=1

ωi

((Ji

Top

−Bi

)

ωi −Ji

Top

ωmin + τw

)

,

(58)

which reduces to

γ ≤

∑ni=1 ωiJi(ωi − ωmin)

Top∑ni=1 ωi(τw −Biωi)

. (59)

Thus for any γ > 0, there is a sufficiently small Top, such that the demand is feasible,which proves Proposition 1. 2

This is of course a theoretical rather than practical result, in that in practical situa-tions, the demand would also be limited by the rated power of the farm, which we havedisregarded here.


Proposition 2. Define constants

Bi, Ji, τ g,i, ωi, τw > 0, (60)

such that τg,i > Biωi, Bi ∈ [Bmin, Bmax] and Ji ∈ [Jmin, Jmax]. Further, define

ai =Bipref,i +Biωiτg,i − τg,iτw

Biτg,i −B2i ωi

+τwBi

(61)

wherepref,i = βpavl,i, pavl,i = τ g,iωi,τg,i = τw −Biωi, ωmin = αωi,

(62)

for β > 1 and 0 < α < 1.Consider the overproduction period given by

Top,i(Bi, Ji) =ωiJi

τ g,i −Biωiln


)

, (63)

The minimum value for Top,i is obtained for Bi = Bmin, and Ji = Jmin.

Proof: We omit index i for ease of notation. Using (62), we can rewrite (63) to

Top(B, J)

=ωJ

τw − 2Bωln

((α− β)τw + (β + 1− 2α)Bω

(1− β)(τw −Bω)

)

,

= f(B, J) ln(g(B)) (64)

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Paper A

wheref(B, J) =

ωJ

τw − 2Bω, g(B) =

r + qB

u+ vB, (65)

andr = (α− β)τw, q = (β + 1− 2α)ω,u = (1− β)τw, v = (β − 1)ω.,

(66)

Given our assumption τg,i > Biωi, we can show that f(B, J) > 0 for B, J > 0, andg(B) > 1 for B > 0, so ln(g(B)) > 0.

Calculating the gradient of Top(B, J), gives

∂Top(B, J)

∂J=

ω

τw − 2Bωln

(r + qB

u+ vB

)

, (67)

which is always positive, given the arguments above, hence Top(B, J) decreases when Jdecreases. Similarly, the sign of ∂Top(B, J)/∂B can be determined by first noticing that

∂

∂BTop(B, J) =

∂

∂Bf(B, J) ln(g(B)) + f(B, J)

g′(B)

g(B), (68)

where

g′(B) =dg(B)

dB, (69)

and∂

∂Bf(B, J) =

2ω2J

(τw − 2Bω)2=

2

Jf2(B, J). (70)

Furthermore

g′(B) =q(u+ vB)− v(r + qB)

(u+ vB)2

= q(u+ vB)(u+ vB)−2 − v(r + qB)(u+ vB)−2

= q(u+ vB)−1 − vg(B)(u+ vB)−1

=q − v g(B)

u+ vB=

1− α

1− β

τwω

(τw −Bω)2. (71)

We will show that∂

∂BTop(B, J) > 0, (72)

which, by (68), is true if and only if

2

Jf2(B, J) ln(g(B)) + f(B, J)

g′(B)

g(B)> 0. (73)

We can reduce this tog′(B) > -h(B) ln(g(B))g(B), (74)

where we have made the substitution

h(B) =2

Jf(B, J). (75)

Let

P (B) = −k(B − a)

(B − b)(B − c)(76)

114


withk =

(1− α)τw(β + 1− 2α)ω

, a =τw2ω

,

b =τwω, c =

(β − α)τw(β + 1− 2α)ω

.(77)

and further letz(B) = P (B)− ln(g(B)). (78)

Then, by some manipulation, it can be seen that (74) is equivalent to

z(B) < 0. (79)

We notice that k, a, b, c > 0 and a < c < b. Recall that by assumption τw > 2Bω , thuswe have 0 ≤ B < τw/(2ω) = a. Inserting B = a yields g(a) = 1, whereby z(a) = 0.

Calculating the derivative of z(B) yields

z′(B) = P ′(B)−g′(B)

g(B)=k

2

(B − a)(B − (b+ c)/2)

((B − b)(B − c))2. (80)

Recall that 0 < α < 1 < β, hence we have a < (b + c)/2. From (80) we see thatz′(B) > 0 for all B ∈ [0, a), so z(B) is a monotonically increasing function of B andmust therefore be strictly negative for all B < τw/(2ω). Hence (79) holds, and thereby(72) holds. 2

Acknowledgment

This work is supported by the Southern Denmark Growth Forum and the European Re-gional Development Fund, under the project ”Smart & Cool”.

The authors would like to thank John Leth and Christoffer Sloth, Aalborg University,for constructive discussions and suggestions.

References

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[2] G. Tarnowski, P. Kjær, P. Sørensen, and J. Østergaard, “Study on variable speedwind turbines capability for frequency response,” transactions of EWEC2009, Mar.2009.

[3] I. Margaris, A. Hansen, P. Sørensen, and N. Hatziargyriou, “Illustration of modernwind turbine ancillary services,” Energies, vol. 3, pp. 1290–1302, Jun. 2010.

[4] L. Chang-Chien, W. Lin, and Y. Yin, “Enhancing frequency response control byDFIGs in the high wind penetrated power systems,” IEEE Transactions on Power

Systems, vol. 26, no. 2, pp. 710–718, May 2011.

[5] T. Zhou and B. Francois, “Energy management and power control of a hybrid activewind generator for distributed power generation and grid integration,” IEEE Trans.

on Ind. Electron., vol. 58, no. 1, pp. 95 – 104, 2011.

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[6] H. Kanchev, D. Lu, F. Colas, V. Lazarov, and B. Francois, “Energy management andoperational planning of a microgrid with a pv-based active generator for smart gridapplications,” IEEE Trans. on Ind. Electron., vol. 58, no. 10, pp. 4583 – 4592, 2011.

[7] P. Kundur, Power system stability and control. McGraw-Hill, 1993.

[8] K. Edlund, “Dynamic load balancing of a power system portfolio,” Doctoral thesis,2010.

[9] G. Wood and W. Hung, “Generating plant frequency control services,” pp. 1–5, Feb.1995.

[10] N. Ullah, T. Thiringer, and D. Karlsson, “Temporary primary frequency control sup-port by variable speed wind turbines - potential and applications,” Power Systems,

IEEE Transactions on, vol. 23, no. 2, pp. 601 –612, May 2008.

[11] G. Tarnowski, P. Kjær, P. Sørensen, and J. Østergaard, “Variable speed wind turbinescapability for temporary over-production,” IEEE PES General Meeting, Jul. 2009.

[12] Energinet.dk, “Technical Regulations 3.2.5 for wind power plants with a power out-

put greater than 11 kW,” www.energinet.dk/, Nov. 2010.

[13] R. Cárdenas, R. Pena, M. Pérez, J. Clare, G. Asher, and P. Wheeler, “Power smooth-ing using a flywheel driven by a switched reluctance machine,” IEEE Transactions

on Industrial Electronics, vol. 53, no. 4, pp. 1086–1093, Aug. 2006.

[14] G. Cimuca, C. Saudemont, B. Robyns, and M. Radulescu, “Control and perfor-mance evaluation of a flywheel energy-storage system associated to a variable-speedwind generator,” IEEE Transactions on Industrial Electronics, vol. 53, no. 4,pp. 1074–1085, Aug. 2006.

[15] F. Bianchi, H. . Battista, and R. Mantz, Wind turbine control systems. Springer,2007.

[16] E. Bossanyi, A. Wright, and P. Fleming, “Controller field tests on the NREL CART3

turbine,” www.upwind.eu, 2006, upWind project report.

[17] E. Cheb-Terrab and A. Roche, “An abel ordinary differential equation class general-izing known integrable classes,” European Journal of Applied Mathematics, vol. 14,pp. 217–229, 2003.

[18] A. Polyanin and V. Zaitsev, Handbook of exact solutions for ordinary differential

equations, 2nd ed. Chapmann & Hall, 2003.

[19] M. R. Spiegel and J. Liu, Schaums Outlines - Mathematical Handbook of Formulas

and Tables, 2nd ed. McGraw-Hill, 1999.

[20] S. Boyd and L. Vandenberghe, Convex optimization, 1st ed. Cambridge UniversityPress, 2004.

[21] J. Maciejowski, Predictive control with constraints. Prentice Hall, 2000.

116

Paper B

Stability Concerns for Indirect Consumer Control in Smart Grids

Morten Juelsgaard, Palle Andersen and Rafael Wisniewski

This work is published in:Proceedings of the European Control Conference, July, 2013

Copyright c© EUCAThe layout has been revised

1 Introduction

Abstract

Demand side management will be an important tool for maintaining a balancedelectrical grid in the future, when the penetration of volatile resources, such as windand solar energy increases. Recent research focuses on two different managementapproaches, namely direct consumer control by an external third party, and indirectconsumer control through incentives and price signals. In this work we present asimple formulation of indirect control, where the behavior of each consumer, is gov-erned by local optimization of energy consumption. The local optimization accountsfor both cost of energy and distribution losses, as well as any discomfort incurredby consumers from any shift in energy consumption. Our work will illustrate that inthe simplest formulation of indirect control, the stability is greatly affected of boththe behavior of consumers, and the number of consumers to include. We will showhow instability is related to the local optimization problem of the consumer, and theinformation made available to him.

1 Introduction

Current scientific and political interests are directed towards increasing the use of renew-able energy and reduce power production from fossil fuels. A major concern related tothis, is how to maintain a stable and balanced electrical grid, when a large part of thepower comes from volatile resources, such as wind and sun. A suggested approach tothis problem, is load shifting by use of different energy storages [1], for instance activelycontrolled consumption. The possibility of adjusting consumption is based on prognosesthat a large part of the future consumption will be for transportation, i.e, electric vehi-cles (EVs), and heating systems in form of electric heat pumps (EHPs). Load shiftingby active control of such consumption, is feasible for instance if a private house is to bemaintained at a certain temperature, then shutting off the EHP would not be noticeablefor some period of time. Similar considerations can be made for EVs, whereby these andsimilar consumption types, are usually called flexible consumption. It is expected thatgrid balancing in the future could be handled to some extend by adjusting the flexibleconsumption, rather than adjusting production. On this basis, recent research has focusedon how to employ different types of consumers for grid balancing [2, 3].

Two different approaches are investigated for including consumers: direct and indi-rect control. By direct control is understood that consumers sign off control rights forpart of their consumption to some third party, e.g., a power company, power retailer orsimilar. The third party would then be able to control for instance the EHP of a number ofhouseholds, honoring some prior agreement on for instance temperature bounds or otherdiscomfort constraints. By aggregating the flexible consumption from a large number ofconsumers, a large energy storage can be maintained. Direct control is considered in [2,3]among others.

The other approach is indirect control, where each consumer is in full control of hisown consumption, however, a third party would present an incentive to act in a certainway. This could for instance be as a price signal indicating a high cost of energy whenvolatile resources are scarce, and low cost of energy when resources are ample. Based onsuch a price signal, consumers can plan their energy consumption as a trade-off on costof energy, and the discomfort experienced for instance by lowering the indoor tempera-ture. Indirect control and concepts relating to demand side management in general, have

119

Paper B

been outlined by [4–6], among others. However, the stability properties of indirect con-trol schemes have not yet been fully analyzed. As the stability is affected by the specificbehavior of the consumers, and further as the behavior of consumers cannot be decideddirectly, but only affected by incentives, it is important to consider how different types ofconsumers affect stability. Therefore the focus of this work is to suggest a basic mathe-matical formulation of indirect control, and show how the stability of this is affected bydifferent consumer strategies.

In this work, the price of energy will not only be affected by the available amountof renewable energy. In our model, the cost of electricity will also encompass the costof grid-losses. Grid-losses are included for two reasons. Firstly, in the Danish system,the cost of grid losses are covered by the distribution system operator [7, 8], and passedon to consumers through tariffs [9]. On this basis, consumers will have an incentive tominimize losses, not individually but rather as a society of consumers. Further, froma socio-economic point of view, as energy generation becomes more volatile by the in-creased penetration of renewable resources, it is beneficial to reduce losses and increasethe efficiency of energy utilization.

This paper is organized as follows: Section 2 outlines the modeling of consumers, gridlosses and energy prices, whereafter Section 3 outlines our indirect control framework.In Section 4, we analyze the stability of the presented framework, followed by numericalexamples in Section 5. Concluding remarks are presented in Section 6. A final Appendixelaborates part of the stability analysis.

2 Modeling

This section first presents the modeling of both the cost of energy and cost of losses foreach consumer. Subsequently, the consumer behavior is modeled.

2.1 Cost of energy

We consider a time horizon of lengthm, starting from the current time tc. Without loss ofgenerality, we set tc = 1. We consider hourly measurements from consumers, meaningthat the interval [1, m] is divided into m discrete samples at hourly instances,

T ≡ 1, 2, . . . ,m.

Consider n households, and let xi = (xi(1), . . . , xi(m)) ∈ Rm, i ∈ 1, . . . , n be theenergy consumption in units of Watt-hours (Wh), for each household, during each hourt ∈ T . Further, let w(t) ∈ R+, t ∈ T be a known price signal, providing the price ofenergy during each hour t, in units of ¤/Wh, where ¤ represents an arbitrary currency.As indicated, we assume that w(t) > 0, ∀t. Strictly speaking, at times it is possible thatw(t) < 0, especially when introducing a significant amount of wind and solar power. Weshall however, leave the analysis of these cases for future study. We let

W = diag(w(1), . . . , w(m) ∈ Rm×m+ ,

be the diagonal, fixed, price matrix with the main diagonal consisting of the price ofenergy. Given the price signal, the cost of energy ce,i : Rm → R for each household, isexpressed by

ce,i(xi) = 1TWxi, i = 1, . . . , n, (1)

120

2 Modeling

in units of currency (¤), where 1 = (1, . . . , 1) ∈ Rm.

2.2 Cost of losses

Let all households be located closely together, for instance as a single street or smallsuburban town. We refer collectively to such a group of households, as a community.The power feed to the community from the remaining grid, is introduced through a lossytie-connection. These losses represents ohmic losses, transformer losses, etc.

The community can conceptually be illustrated as in Fig. 1, as a number of radialsconnected to the remaining grid.

Grid

tie-connection

ztie

Figure 1: Conceptual schematic outline of the community.

We assume that the households of the community are located sufficiently close, so thelosses within the community are minor compared to losses in the tie-connection, and maybe disregarded. The loss of energy in the tie-line is modeled in the following, where weassume that the grid is balanced, allowing us to conduct the analysis for a single phaseequivalent system [10].

Remember that xi(t) in units of kWh, is the total energy consumption through thehour t ∈ T . Let pi(t) ∈ R in units of kW, denote the corresponding average powerconsumption through the hour t, i.e.;

xi(t) = Tspi(t) ∀i, t,

where Ts = 1 hrs is the length of the interval t ∈ T . Conservation of power entails thatthe accumulated average power consumption of the community p(t) ∈ R, is expressed asp(t) =

∑ni=1 pi(t). It is important to notice that p(t) is not an instantaneous power, but

instead, the average power during the hour t.

Let γ ∈ [0, 1] denote the power factor of the community, when aggregating consump-tion from all individual households. We assume γ constant, i.e., γ(t) = γ, ∀t ∈ T . As aconsequence, the apparent power of the community, during hour t, is then s(t) = p(t)/γ[11]. Since we have disregarded losses within the community, all households will con-sume power at the same voltage v, so the magnitude of the current is given by

|i(t)| =|s(t)|

v=

|p(t)|

γv.

121

Paper B

The average tie-line losses, xtie(t) during hour t, depends on the squared current magni-tude, and is thus represented as

xtie(t) =p(t)2

(γv)2rT = β

(n∑

i=1

xi(t)

)2

∀t ∈ T , (2)

where rT > 0 is the tie-line resistance, and β = rT /(Tsγv)2 is a loss parameter.

The cost of losses are expressed as cl(t) = w(t)xtie(t), The total cost of losses isdistributed among the individual consumers, by use of their individual share factor, i.e,their share of the total consumption. That is

cl,i(t) = cl(t)xi(t)

∑nj=1 xj(t)

= βw(t)xi(t)

(n∑

j=1

xj(t)

)

,

for i ∈ 1, . . . , n. The above is in units of currency (¤). With a slight misuse of notationwe write the total cost of losses over the horizon, of a single household as

cl,i(x1, . . . ,xn) =∑

t∈T

cl,i(t) = βxTi W

n∑

j=1

xj

. (3)

Adding (3) to the cost of energy in (1), the total cost of energy and losses across thehorizon for a single household, becomes

ci(x1, . . . ,xn) = ce,i(xi) + cl,i(x1, . . . ,xn)

= 1TWxi + βxTi W

(n∑

j=1

xj

)

, (4)

from which it is clear that the cost incurred by single household, depends of the consump-tion of the entire community, due to the added cost of losses.

2.3 Consumer flexibility

As mentioned in Section 1, the electricity consumption will involve some level of flexi-ble consumption. It will also contain amounts of inflexible consumption, which cannotbe temporally shifted, i.e., lights, television, etc. We will therefore model the energyconsumption as

xi(t) = xi(t) + xi(t), ∀t ∈ T , (5)

where xi(t) denotes the inflexible consumption and xi(t) denotes flexible consumption.It is reasonable to consider xi(t) as the known, traditional consumption, i.e., a baselineconsumption. The losses presented in (2) are typically in the order of 5 % of this baseline[12].

2.4 Consumer behavior

As also outlined in Section 1, the flexible consumption of each individual consumer, mayhave different characteristics, which may also effect the way energy is consumed. Inthe following we outline two different types of consumers. The purpose of this is to

122

2 Modeling

illustrate, that when relying on indirect control, there is no way of deciding, or perhapseven knowing how any individual consumer will behave and react to a price signal. Inother words, the local objective of any consumer remains unknown. Below we presentthe behavior of two types or classes of consumers which are fundamentally different, butboth likely to be present in the grid. In Section 4, we present a stability analysis of theindirect control, in the extreme cases where a community consists solely of consumers ofeither type. We refer to the two consumer types as greedy and comfort consumers.

Greedy consumer

The greedy consumer-type represents consumers who are only concerned with the totalprice of energy across the horizon. This could be consumers who only provide flexibilityto the grid through, for example, an EV. The consumer is thereby only concerned thatthe EV is charged to some required level, at the end of the horizon, i.e., that the flexibleconsumption integrates to some fixed value. If this is achieved, the specific charge patternis of no concern, or discomfort.

Given the relation between flexible consumption and total consumption (5), the localoptimization problem of the greedy consumer is then

minimizexi

ci(x1, . . . ,xn)

subject to 1T xi = αi,(6)

for i = 1, . . . , n, where we have extended the bold notation introduced previously, i.e.,xi = (xi(1), · · · , xi(m)) ∈ Rm, for all i. Above, αi > 0, in units of Wh, is the requiredaccumulated consumption of flexible energy during the horizon. The constraint ensuresthat the EV is charged to the required level.

As we have argued previously, the cost of any single consumer is affected by theconsumption of all consumers, through the cost of losses. We have indicated this in thenotation of (6), as the cost function depends on x1, . . . ,xn, but the consumer can onlycontrol his local consumption xi.

Comfort consumer

The second type of consumer, is the comfort consumer. As opposed to the greedy con-sumer, the comfort consumer is not only accounting for the cost of energy and losseswhen optimizing a consumption profile, in that the comfort consumer also includes a costof discomfort. In this work, we shall model this as a consumer with an installed EHP,where the discomfort is measured as deviation from a desired set-point temperature inunits of C:

Tsp,i = (Tsp,i(1), . . . , Tsp,i(m)) ∈ Rm,

for i = 1, . . . , n. Let the indoor temperature of household i be denoted Ti ∈ R, i =1, . . . , n, also in units of C. Then, given some initial value Ti,0, the household tempera-ture is modeled as

Ti(t+ 1) = aiTi(t) + bixi(t) + eiTa(t), t ∈ T , (7)

for all i, where Ta(t) ∈ R is the ambient temperature, and

0 < ai < 1, bi, ei > 0,

123

Paper B

are known, household specific parameters, accounting for heat dissipation, EHP effi-ciency and outside conditions. The model could be expanded to also include elementssuch as direct and indirect solar radiation, but we shall save this for future work. Formore thorough discussion of the thermal modeling, consult [2].

With the same bold-font notation as previously, we let

Ti(xi) = (Ti(1), . . . , Ti(m)) ∈ Rm,

where we have written Ti(xi) since the temperature Ti(τ) for any τ ∈ T , depends onxi(t) for t = 1, . . . , τ − 1. Given the temperature model and the desired set-point, thediscomfort of the consumer is modeled by di : Rn → R as

di(xi) = (Tsp,i −Ti(xi))T (Tsp,i −Ti(xi)), (8)

that is; the discomfort is quadratic in temperature deviation. Including both the total costof energy from (4), and the cost of discomfort in (8), results in the following optimizationproblem for the comfort consumer

minimizexi

ci(x1, . . . ,xn) + λidi(xi) (9)

for i = 1, . . . , n, where λi > 0 is a unitless local trade-off parameter.We could include a number of constraints, such as discomfort limits or actuator limits

as bounds on xi. For the time being, we shall omit to do so, and the reader is referred to,e.g., [2].

3 Indirect control

In this section we outline the indirect control framework. As evident, the local cost func-tion of both Problem (6) and Problem (9), is affected by the consumption pattern acrossthe entire community, which is a consequence of the cost of losses in (3). To simplifynotation, we let

qi =∑

j 6=i

xj .

With this notation, we have

ci(xi,qi) = xTi W1+ βxTi W(xi + qi)

= (1+ βqi)TWxi + βxTi Wxi (10)

and we further writex⋆i (qi) = arg inf

xi|1T xi=αi

(ci(xi,qi)),

andx⋆i (qi) = arg inf

xi

(ci(xi,qi) + λidi(xi)),

to denote the solution of the optimization for greedy and comfort consumers respectively,given qi.

For each individual household, the consumption of the remaining community, qi, isunknown. Our approach to indirect control, is therefore to assume that an approximationof qi is available to each household. Such approximations could be made available,

124

3 Indirect control

for instance, by the distribution system operator (DSO). Based on the approximation ofqi for i ∈ 1, . . . , n, each consumer can optimize consumption locally, and we cansubsequently improve the approximation of all qi, through iterative updates. This issummarized in Algorithm 3.1.

Algorithm 3.1 Indirect control of consumers

Initialize estimates q(0)i , i = 1, . . . , n

for k=0,1, . . . do

Obtain local solutions:x⋆i (q

(k)i ), i = 1, . . . , n

Update estimates:q(k+1)i =

∑

j 6=i x⋆j (q

(k)j ), i = 1, . . . , n

end for

In Algorithm 3.1, we have used q(k)i , to denote the estimated qi, at iteration k. Conver-

gence of Algorithm 3.1, in the sense that

limk→∞

(x(k+1)i − x

(k)i ) = 0, ∀i, (11)

would entail that there is no update of the consumption patterns for any consumer. Theprocess has then reached a Nash equilibrium, in the sense that no consumer desires toalter their current consumption pattern, provided that the remaining community refrainsfrom changing theirs as well.

The approach outlined above, allows each individual consumer to privately plan andoptimize the optimization pattern, accounting for private comfort concerns, as well as thecost of energy and cost of grid-losses. However, it requires some exchange service, orshared data center for collecting xi(q

(k)i ) and distributing q

(k+1)i for each iteration of the

algorithm, for all i. This is illustrated in Fig. 2.

Shared Data Center

Consumers

q(k+1)1x

⋆1(q

(k)1 ) q

(k+1)nx

⋆n(q

(k)n )

Figure 2: Accounting for grid losses, requires data exchange with a shared data center.

Granting consumers access to information in this fashion, is what allows them toaccount for grid losses, by considering the action of the remaining members of the com-munity. This poses a benefit for both the consumer, and for society as a whole, since theconsumer desires to consider losses as an expense, and society desires to reduce the wasteof energy.

Notice, that no consumer receives direct information about any other consumer. In-stead, only information concerning the community as a whole is distributed, so no privacyconcerns are violated.

125

Paper B

With this outline of the indirect control framework, the following section examinesthe stability for the two types of consumers described in Section 2.4, in the sense ofconvergence of Algorithm 3.1, as given by (11).

4 Stability Analysis

The following analysis is performed for the two extremal cases, where the communityconsists solely of greedy or comfort consumers. We shall leave mixed populations forfuture work. Following the stability analysis, Section 5 gives numerical examples.

4.1 Stability of Greedy Consumers

For greedy consumers, with ci(xi,qi) replaced by the expression (10), the Lagrangian ofProblem (6) becomes

Li(xi, µi) = (1+ βqi)TW(xi + xi) + β(xi + xi)

TW(xi + xi) + µi(1

Txi − αi),

where we remind the reader that xi = xi + xi, and the optimization variable is only xi.The baseline consumption xi is assumed fixed and known. Similarly, at each iteration ofthe previously described algorithm, qi represents a fixed and known parameter. Above,µi ∈ R is the Lagrange multiplier for the equality constraint. The Karush-Kuhn-Tucker(KKT) conditions are then

1. ∇xiL(xi, µ) = W(1+ β(qi + 2xi)) + 2βWxi + 1µi = 0.

2. 1T xi = αi

For fixed xi and qi, Problem (6) is convex, and the KKT conditions are both necessaryand sufficient [13]. We let I denote the identity, and introduce the matrix

I =

[I

0 . . . 0

]

∈ Rn+1×n.

The conditions above may then be formulated as[2βW 1

1T 0

]

︸︷︷︸

M

[xiµ

]

︸︷︷︸

yi

=

[−W(1+ 2βxi)

αi

]

︸︷︷︸

hi

−βIWqi.

From this we have

x⋆i (qi) = ITy⋆i (qi) = ITM−1hi − βITM−1IWqi. (12)

Introducing (12) in the iterative process of Algorithm 3.1 entails that

x⋆i (q(k)i ) = x⋆i (q

(k)i ) + xi = pi +Φq

(k)i ,

wherepi = ITM−1hi + xi, Φ = −βITM−1IW.

Let p = (p1, . . . ,pn), q = (q1, . . . ,qn), x = (x1, . . . ,xn), and further

H =

Φ. . .

Φ

, J =

0 I · · · I

I 0 · · · I

.... . .

...I I · · · 0

,

126

4 Stability Analysis

thenx⋆(q(k)) = p+Hq(k), (13)

andq(k+1) = Jx⋆(q(k)) = Jp+ JHq(k). (14)

Let νi(JH), i = 1, . . . , n denote the eigenvalues of JH, and let V(JH) = νi(JH), i =1, 2, . . . denote the spectrum. Then the iterative update in (14), converges only in thesense of (11), provided that

maxi

|νi(JH)| < 1, (15)

as this is the requirement for (14) to be a contraction [14].The block-diagonal structure of H, entails that ( [15])

ν ∈ V(Φ) ⇒ ν ∈ V(H).

Further, it is easily shown that for n ≥ 2

ν ∈ V(H) ⇒ (n− 1)ν ∈ V(JH).

Finally, as elaborated in the Appendix, −1/2 ∈ V(Φi), ∀i, independent of β. With thearguments above, this entails that for n > 1

1− n

2∈ V(JH),

whereby the requirement in (15) is only obeyed for n ≤ 2. Therefore, if a communityconsists solely of greedy consumers, and contains more than 2 households, convergenceand thereby stability of Algorithm 3.1, is not achieved. This shows that there is a riskpertaining to the indirect control, since a certain behavior of consumers could lead toinstability. To ensure stability against greedy consumers would require a revision of theframework. For instance, by including some filtering scheme for the estimates of qi.We shall illustrate the behavior of greedy consumers with examples in Section 5, andcomment further on this in Section 6.

4.2 Stability of Comfort Consumers

We perform a similar analysis of the stability of comfort consumers. From a logicalperspective, it would make sense that indirect control in this case, would be stable, at leastwhen supplying a sufficiently large trade-off coefficient λi. This is so, since it would notmake sense for the consumer to enforce a very large positive or negative consumption,since this would incur a significant cost of discomfort. We shall show that contrary to thegreedy consumers, stability of comfort consumers can be guaranteed for arbitrary largecommunity, n, provided that some lower limit on the trade-off parameter λi is guaranteed,i.e, comfort is sufficiently important for all consumers, compared to the cost of energy.To this end, we introduce the notation

Ai =

aia2i...ami

, Gi =

1 0 0 · · · 0 0ai 1 0 · · · 0 0a2i ai 1 · · · 0 0...

......

......

am−1i am−2

i am−3i · · · ai 1

,

127

Paper B

Di = b2iGTi Gi, ζi = biG

Ti (Ti,sp −AiTi,0 − eiGiTa)

where Di is symmetric and positive definite. When employing the cost of energy (10),the KKT conditions of Problem (9) reduces to the objective function having zero-gradient,i.e.

∇xici(xi,qi) + λ∇xi

di(xi) = 0.

Given the dynamics in (7) for the thermal process, and the notation introduced above, thezero-gradient requirement is expressed by

W(1+ β(qi + 2xi)) + 2βWxi + 2λiDixi − 2λiζi = 0, (16)

from which it is clear that

x⋆i (qi) = Φ−1i (λζ −

1

2W1− βWxi))

︸︷︷︸

hi

−1

2βΦ−1

i Wqi,

where we introduceΦi = βW+λiDi, with the matrix Φi being full rank if ai, bi, λi 6= 0,and therefore, it is nonsingular. The reader will notice that we have reintroduced andredefined some of the notation from Section 4.1. We do this partly to limit the extend ofour notation, and partly in order to emphasize the similarities between the two consumertypes, even though the basic behavior between them, is different.

Similar to the greedy case, we can collect the expression for each household intomatrix form;

x⋆(q(k)) = p+Hq(k), (17)

andq(k+1) = Jx⋆(q(k)) = Jp+ JHq(k). (18)

with p = (p1, . . . ,pn), pi = Φ−1i hi + xi and

H =−β

2

Φ−11 W

. . .Φ−1n W

,

and J is similar to previous. As before, convergence for comfort consumers requires

maxi

|νi(JH)| < 1, (19)

In the following we show that for any n > 0, there exists a sufficiently large λi, i =1, . . . , n, such that (19), is obeyed. Notice that

Φ−1i W = (βW + λiDi)

−1W =1

λi

(β

λiW +Di

)−1

W.

Hence, increasing λi entails that the row sum of any row of Φ−1i W can get arbitrarily

close to zero. So, increasing λi for all i, entails that any row sum of JH can comearbitrarily close zero, and so by the Geršgorin disc theorem [15], the eigenvalues of JHcan come arbitrarily close to zero. Therefore, if all consumers increase λi sufficiently, theconvergence criteria (19) is guaranteed.

128

5 Examples

This concludes our stability analysis for the two consumer types. The following sec-tion illustrates our results with a few numerical examples. As mentioned initially, con-vergence of the indirect control, in the case of arbitrary many consumers, can only beguaranteed for comfort consumers, and this only if a sufficiently large trade-off parame-ter is employed by the local optimization.

5 Examples

In the following, we present numerical examples to illustrate the main results from theprevious sections. In all examples we consider a time-horizon for the local optimiza-tion of m = 24, corresponding to for instance 24 hours. The energy price and ambienttemperature during this period is depicted in Fig. 3(Top) and (Middle). The curve inFig. 3(Bottom) is representative for the baseline consumption; however, the baseline con-sumption for each individual consumer will contain some stochastic perturbation of thisgeneric curve. In the examples, these perturbations are known in advance.

x(t)

[kW

h]

t

Ta(t)

[C

]w(t)

[¤/k

Wh]

04:00 09:00 14:00 19:00

0.81.01.2

121620

0

0.3

0.6

Figure 3: Top: Price of energy through the period of optimization. Middle: Ambienttemperature. Bottom: Baseline consumption.

The following examples illustrate our results for the following scenarios:

1. Community of greedy consumers with convergence

2. Community of greedy consumers where conver-gence is not obtained

3. Community of comfort consumers where conver-gence is obtained

129

Paper B

Greedy consumers, convergence obtained

A community of greedy consumers in a setting where convergence is obtained, impliesas we have shown, that n ≤ 2. Below we illustrate the case for n = 2. In the example,we assume that the greedy consumers represent owners of EVs, that needs to be chargedduring the simulation horizon. Since no constraints towards maximum EV battery chargecapacity where included in the convergence analysis, the converged solution cannot beexpected to obey any considerations towards such constraints. Hence, to make the exam-ple more realistic, we include a maximum battery capacity constraint in the optimizationfor each of the two consumers. This does not affect the convergence.

After a number of iterations of Algorithm 3.1, Fig. 4 shows the converged flexibleconsumption pattern for each of the two consumers. By analyzing the figure, it is seenthat energy is bought whenever there is a local minimum in the price, and subsequentlysold, i.e. the battery is discharged, when prices are subsequently high.

xi(t)

[kW

h]

t

04:00 09:00 14:00 19:00−10

0

10

Figure 4: Optimized, flexible consumption pattern, for the two households.

The accumulated flexible consumption is shown in Fig. 5, along with the consumptionconstraint and charge limit. Even though the battery is both charged and discharged sev-eral times throughout the horizon, the final charge level meets the constraint.

∑t r=1xi(r)

[kW

h]

t

04:00 09:00 14:00 19:000

5

10

Figure 5: Accumulated flexible consumption (Solid), and consumption constraint(Dashed), for both households.

130

5 Examples

We remark that the reader should focus on the characteristics of the consumption be-havior, rather than the actual numbers presented in the example. That is, since greedyconsumers acquires no discomfort by their flexible consumption, there can be introducedfluctuating consumption from buying and selling power, whenever the price signal indi-cates this to be beneficial.

Greedy consumers, no convergence

For n = 20, the indirect control framework diverges for greedy consumers. This isillustrated by Fig. 6, where the maximum entry of q(k)

i is plotted for k = 1, 2, 3, for eachof the 20 households. As evident, the algorithm does not converge, and the communityestimates q(k) diverges.

maxj((q(k

)) j)

k

1 2 3

×106

0

5

10

Figure 6: Divergence of the community consumption estimates q(k)i , for the unstable case

of n = 20.

Adding constraints on battery charge limits as in the previous example, will not rem-edy this situation. Introducing such constraints would simply entail that rather than di-verging towards infinity, as in Fig. 6, the local optimization of each household will simplyalternate between two solutions, between each iteration of Algorithm 3.1, but it will notconverge.

Comfort consumers, convergence obtained

Below we present an example including n = 100 comfort consumers, characterized bythe parameters

ai ∈ [0.936; 0.961], bi ∈ [0.9; 0.95], ei ∈ [0.039; 0.064]

Ti,0 = 21 C, Ti,sp(t) = 21 C, ∀t,

for all i. The parameters above entail that the thermal dynamics of each householdhas a time-constant of 15-25 hours. The baseline consumption is again representedby Fig. 3(Bottom), and the grid losses corresponds to about 5 % of this consumption.The trade-off parameter has been set equal for all households, such that λi = 0.5, i ∈1, . . . , 100, which is large enough to ensure stability.

131

Paper B

T(t)

[C

]

t

xi(t)

[kW

h]

04:00 09:00 14:00 19:0020.8

21

21.2

0

0.3

0.6

Figure 7: Top: Consumption pattern in the implementation of 100 comfort consumers.Bottom: The temperature setpoint (Dashed), and the household temperatures, resultingfrom the consumption patterns above (Solid).

After convergence of Algorithm 3.1, the flexible consumption pattern of each house-hold appears as in Fig. 7(Top).

Initially, an energy storage is built when the price is low. The storage is then depletedlater when prices are higher in the period 09.00-19.00. This is visible in Fig. 7(Bottom)where the temperature is increased beyond the set-point when prices are low. The tem-perature then drops gradually when prices are higher.

This shows that even for larger communities, the coordination among consumers asdescribed through Algorithm 3.1, can be stable, provided that the trade-off parameter λibetween cost and discomfort, is sufficiently large, for each consumer. In other words;each consumer needs to put a sufficiently large weight on comfort as a trade-off with costof energy.

We have omitted to present an example illustrating an unstable implementation of acommunity of comfort consumers, since this would simply result in similar results asillustrated in Fig. 6.

In the examples presented here, we have adjusted the loss parameter β according tocommunity size, such that the losses are roughly 5 % of the baseline consumption. Sincethe baseline increases when more consumers are added to the community, this entails thatβ is reduced accordingly. This would correspond to a distribution network being designedwith wider cables, ensuring lower resistance for larger communities.

6 Conclusion

In this work we have outlined a simple and intuitive framework for the concept of indirectcontrol. We have described two types of consumers that are likely to exist in a smart grid,and their appertaining optimization problem for planning consumption across a time hori-zon, when faced with a price signal. We have included a cost of energy losses throughoutthe grid, which ties together all consumers. In order to account for cost of losses, we haveintroduced an iterative approach for sharing information between consumers. Despitethe intuitive formulation, we have shown that the iterative approach for local optimiza-

132

A Appendix: Eigenvalue of Greedy consumers

tion of consumption, in general fails to converge for communities consisting of greedyconsumer. However, convergence can be guaranteed for arbitrary large communities ofcomfort consumers, provided the comfort trade-off parameter is sufficiently large.

With this we have illustrated that since local consumer behavior cannot be controlled,and perhaps is unknown in the framework of indirect control, several stability issues couldpotentially complicate the balance of the grid, and the risk of instability is related to thespecific behavior and strategy of consumers. Therefore, deeper analysis of main consumertypes and behavior, and the consequences for stability should be considered.

The results presented here also illustrates that a more sophisticated incentive schemeshould be formulated, or a more elaborate data exchange must be devised, compared tothe intuitive and naive approach outlined here. A first attempt could be to include somefiltering process before distributing the estimates qi to the households, in order to pre-vent divergence. It can be shown that even simple filtering schemes present stabilizingcapabilities for both types of consumers outlined here. In that case however, it becomesnecessary to analyze the risk of cheating, i.e. the risk of some consumers deviating fromthe agreed data exchange, if this presents potential benefits. However, such game the-oretic considerations have been deemed outside the scope of this study. Further, whenfiltering processes, or similar attempts to affect consumers, are included, the frameworkshifts to some extend from indirect control, to direct control, where consumers are con-trolled directly, rather than through incentives. In our framework, we have not affectedconsumers in any other way, than to present them with incentives through price signals,and information about the total community consumption.

Acknowledgment

The authors would like to thank John Leth, AAU, for inputs during technical discussions.The authors further appreciate the received feedback from the anonymous reviewers.

A Appendix: Eigenvalue of Greedy consumers

In the section relating to greedy consumers, we have introduced the matrix

Φ = −βITM−1IW, with M =

[2βW 1

1T 0

]

and W = diag(w(1), . . . , w(m) ∈ Rm×m+ , where M ∈ Rm+1×m+1, and Φ ∈ Rm×m.

Let wi ≡ w(i), and ρ =∑m

i=1 w−1i , then by straight calculation, we see that

Φ =−1

2ρ

∑

j 6=1 w−1j −1/w1 · · · −1/w1

−1/w2

∑

j 6=2 w−1j · · · −1/w2

.... . .

...−1/wm −1/wm · · ·

∑

j 6=m w−1j

Let u = (u1, · · · , um) ∈ Rm+1 be any vector that satisfies

m∑

i=1

ui = 0 ⇒∑

j 6=i

uj = −ui, ∀i, (20)

133

Paper B

and observe that

Φu =−1

2ρ

(

(∑

j 6=i

w−1j )ui − w−1

i

∑

j 6=i

uj

)

=−1

2ρ

(

(∑

j 6=i

w−1j )ui + w−1

i ui

)

=−1

2ρρui =

−1

2ui,

for all i ∈ 1, . . . , n. This entails that u is an eigenvector for Φ, with eigenvalue ν =−1/2.

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135

Paper C

Distribution Loss Reduction by Household ConsumptionCoordination in Smart Grids

Morten Juelsgaard, Palle Andersen and Rafael Wisniewski

This work is published in:IEEE Transactions on Smart Grid, July, 2014


1 Introduction

Abstract

In this work, we address the problem of optimizing the electrical consumptionpatterns for a community of closely located households, with a large degree of flex-ible consumption, and further some degree of local electricity production from solarpanels. We describe optimization methods for coordinating consumption of electricalenergy within the community, with the purpose of reducing grid loading and activepower losses. For this we present a simplified model of the electrical grid, includingsystem losses and capacity constraints.

Coordination is performed in a distributed fashion, where each consumer opti-mizes his or her own consumption pattern, taking into account both private objectives,specific to each individual consumer, as well as objectives common to all consumers.In our work, the common objective is to minimize active losses in the grid, and ensurethat grid capacity limits are obeyed. These objectives are enforced by coordinatingconsumers through a nonlinear penalty on power consumption. We present simula-tion test-cases, illustrating that significant reduction of active losses, can be obtainedby such coordination. The distributed optimization algorithm employs the alternatingdirections method of multipliers.

1 Introduction

In Denmark, the future domestic use of fossil fuels for heating and transportation is ex-pected to decrease. This could be realized, for instance, by replacing oil-fired boilers withelectric heat pumps (EHPs), and combustion based cars with electric vehicles (EVs) [1].Further, current trends tends towards an increased level of local power production, athousehold level, for instance by small scale wind turbines and solar panels [2].

The growing use of electricity increases grid loading, power losses, and the risk ofcongestion. However, employing electricity for heating and transportation, also introducea significant level of flexibility to the traditional consumption pattern. For instance, whenshutting off an EHP, the indoor temperature of a household, will only drop slowly, and thetemperature drop would be unnoticeable for some period. The energy used for heating,may therefore be temporally shifted, with little or no discomfort to the user. Similarly,when charging EVs; the specific time the battery is charged would usually be unim-portant, provided that a certain charge level was guaranteed by the end of a given timeframe. By utilizing this flexibility, the risks and drawbacks of the increased consump-tion, can be reduced, by planning and coordinating the consumption profile for severalhouseholds. Specifically, this work focus on how consumers may be coordinated suchthat active power losses in the distribution grid, are minimized.

The important issue of reducing power losses, has been considered by many, e.g.[3–8]. A common approach to loss reduction, is to consider grid modifications. For in-stance, [3] considers loss reduction by reconfiguring the radials in a low-voltage distribu-tion grid, and [4–6] consider the introduction of distributed generation, and the effect ondistribution- and transformer losses. The work presented in [7] illustrates how consump-tion balanced locally, by power produced by photo-voltaic (PV) systems, may reducecurrents and thereby losses, in the distribution grid. A different study on loss reductionby PV systems was provided by [8], who illustrates the potential of loss reduction by con-trol of reactive power flow in the inverter of PV systems. Since reactive power control isalso a common approach for maintaining stable voltages, the issues of loss reduction and

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voltage control is commonly considered jointly. However, we shall limit our attention tothe issue of loss reduction, and leave voltage control for future work, but comment furtheron this aspect in Section 6.

Common for all the works described above is that they do not consider utilizing theflexibility available in the consumption pattern, that is, they either consider grid modifica-tions, or reactive power control, but not control and temporal shifts of active power. Also,most of the provided analyses considers only constant consumption, and do not accountfor intraday variations of consumption. We take a different approach. We employ theflexibility of consumers, and coordinate them through temporal shifts of consumption,such that distribution losses are minimized.

Employing demand side management for consumption coordination, has been ex-plored in several previous works. In [9], it was assumed that an estimate of a real-timeprice signal was available, so that power consumption could be scheduled to minimize ac-cumulated cost of energy, as a trade-off with incurred discomfort. A similar problem wasinvestigated by [10], who illustrated how consumption could be coordinated to obey gridlimitations, i.e., avoid overloading. However, the approach employed in these works, didnot include any concerns towards grid-losses, which is one of the added concerns in ourwork. In [11], a strategy for charging a large number of EVs was derived when consider-ing both grid capacity limitations and the cost of energy. However, this work also omittedlosses as a concern, and further the derived charge strategy was calculated centralized,meaning that one single entity, i.e., an aggregator, power retailer or similar, controls anumber of vehicles collectively. However, as different consumers may have agreementswith different power retailers, it is uncertain whether one entity will be able to control allunits. Therefore, in our work, we employ a decentralized strategy, where each unit actsautonomously. The task is to coordinate their autonomous behavior, which is achievedthrough iterative updates of nonlinear penalties relating to the private consumption pat-tern. The behavior of consumers is further planned on basis of a price estimate, similarto [9], such that consumers have an incentive to minimize the cost of their consumption.This price signal could be provided by, e.g., the transmission system operator (TSO) [12],or the power retailer of the individual consumer. In our work, we include a cost of lossesin the price of energy, and illustrate that including this concern specifically when planningconsumption patterns, will not only minimize losses in the distribution grid, but also havethe added effect of reducing the overall load of the grid.

We limit our focus to losses incurred in the low-voltage distribution grid. These lossesare financially covered by the distribution system operators (DSOs), managing the differ-ent grid sections. Specifically in Denmark, the recommendations of [13], suggest thatthe expenses of the Danish DSOs related to losses, should be included when calculatingthe distribution grid tariffs, passed on to the consumer. This implies that even thoughconsumers are not directly charged for grid losses, they will still cover the cost of losses,through tariffs. From this point of view, the consumers have a financial incentive to reducelosses. Currently, the consumption of residential households is only metered a few timesevery year, however current trends, tends towards more consumers being installed withsmart meters, capable of making online measurements of the consumption for individualhouseholds, which we employ in this work.

In the following Section 2, we outline our modeling approach concerning individualhouseholds, the grid including losses and capacity limitations, as well as the price signal.Section 3 formulates the problem of optimizing consumption patterns for each household,

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2 Modeling

while minimizing losses and obeying grid constraints. We show that the overall problemis in general convex, and can be solved efficiently in a centralized fashion. Section 4elaborates further on the how consumer behavior affects the grid-loading, and under whatcircumstances, loss minimization will have significant effect on the loading. Thereafter,Section 5 outlines our approach for distributed consumer coordination. The decentral-ized approach guarantees that the consumption pattern for each household converges tothe global optimum. Section 5 further contain numerical examples. Finally, Section 6summarizes our work, and discuss further perspectives.

2 Modeling

The following describes the modeling of individual households, the electrical grid, as wellas consumer behavior.

2.1 Household modeling

We consider a time horizon of length L, starting from the current time tc. Without loss ofgenerality, we set tc = 1. We consider hourly measurements from consumers, meaningthat the interval [1, L] is divided into L discrete samples at hourly instances, i.e., the setof sample times are

T ≡ 1, 2, . . . , L.

In the following, we consider n households, where each household i has a state vectorxi : T → R2, i ∈ 1, . . . , n, where

xi,1 : is the indoor temperature [C],xi,2 : is the energy in the EV [kWh].

For all i ∈ 1, . . . , n, we introduce the bold-font notation; xi,1 = (xi,1(1), . . . , xi,1(L)) ∈

RL, and associate to this a temperature set-point xi,sp ∈ RL, to which xi,1 preferablyshould remain close. This is a matter of consumer comfort.

Each household is characterized by a controllable flexible energy consumption ui :T → R2, where

ui,1 : is the energy consumed by the EHP [kWh],ui,2 : is the energy consumed by the EV [kWh].

Since t ∈ T , ui(t) refer to the total energy consumed by the EHP and EV respectively,through the hour t.

We approximate the dynamics of each household, by a linear model:

xi(t+ 1) = Aixi(t) +Biui(t) + EiTa(t) (1)

where Ta : T → R denotes the ambient temperature, and

Ai =

[ai,1 00 ai,2

]

, Bi =

[bi,1 00 bi,2

]

, Ei =

[ei0

]

,

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for some set of known scalar coefficients ai,1, ai,2, bi,1, bi,2, ei. Further, let

Xi = xi(t)| xi ≤ xi(t) ≤ xi, t ∈ T

Ui = ui(t)| ui ≤ ui(t) ≤ ui, t ∈ T ,

where xi, xi, ui, ui ∈ R2 are allowed upper and lower bounds on states and energy con-sumptions respectively. A feasible flexible consumption pattern is then defined by satis-fying

ui(t) ∈ Ui and xi(t) ∈ Xi. (2)

Besides the flexible consumption ui(t), each household is also characterized by aninflexible consumption ui : T → R+. The inflexible consumption cannot be controlled.It is an accumulation of different types of consumption like lights, television, cooking,etc. which does not allow for temporal shifts. The inflexible consumption ui(t) refers tothe energy consumed during the hour t ∈ T .

In this work we assume some households are installed with solar panels, providingsome level of local energy production. Let ui,s(t), t ∈ T , i ∈ 1, . . . , n be the energyproduction from the solar panels of each household, where ui,s(t) ≡ 0, if no solar panelsare installed. The total energy consumption (or production) ui(t), for each household, isthen the difference between the consumption from flexible and inflexible appliances, andthe production from solar panels. This can be expressed by

ui(t) = ui(t) + ui,1(t) + ui,2(t)− ui,s(t), t ∈ T , (3)

with ui(t) in units of kWh. Here, ui(t) > 0 indicates consumption and ui(t) < 0 in-dicates production, in situations where solar production exceeds consumption. Onwards,we shall employ the same bold-font nation as introduced previously, and let

ui = (ui(1), . . . , ui(L)) ∈ RL. (4)

2.2 Grid modeling

Let all households be located closely together, for instance as a single street or smallsuburban town. We refer collectively to such a group of households, as a community. Thecommunity can conceptually be illustrated as in Fig. 1, divided into m radials.

r1

r2

rm

PCC

Grid

tie-connection

Figure 1: Conceptual schematic outline of the community.

Let rj ⊂ 1, . . . , n, be the set of households in radial j, where j ∈ 1, . . . ,m. Fur-ther, card(rj) = nj , where card(·) denotes cardinality. All households are connected

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2 Modeling

to one and only one radial, hence

ri ∩ rj = ∅, ∀i 6= j andm⋃

j=1

rj = 1, . . . , n.

The radials are joined at a single point of common connection (PCC), where the powerfeed from the remaining grid is introduced through a lossy tie-connection. These lossesrepresents ohmic losses, transformer losses, etc.

Both the tie-connection and each individual radial has capacity constraints, such thatthe total energy import or export, from each individual radial, as well as the tie-connectionitself, must be below some limit, i.e.

∣∣∣∣∣∣

∑

i∈rj

ui(t)

∣∣∣∣∣∣

≤ νj , j ∈ 1, . . . ,m, and

∣∣∣∣∣

n∑

i=1

ui(t)

∣∣∣∣∣≤ νm+1 (5)

for t ∈ T , where ν1, . . . , νm are the radial capacities and νm+1 is the tie line capacity.We introduce the notation

fj(u1(t), . . . , un(t)) =

∣∣∣∣∣∣

∑

i∈rj

ui(t)

∣∣∣∣∣∣

, j ∈ 1, . . . ,m,

fm+1(u1(t), . . . , un(t)) =

∣∣∣∣∣

n∑

i=1

ui(t)

∣∣∣∣∣.

The grid constraints (5) can then be expressed as

fj(u1(t), . . . , un(t)) ≤ νj , j ∈ 1, . . . ,m+ 1. (6)

The households of the community are located sufficiently close, so that the losses withinthe community are minor compared to losses in the tie-connection, and may be disre-garded. The loss of energy in the tie-line are modeled in the following, where we assumethat the grid is balanced, allowing our analysis to be performed for a single phase [14].

Recall that ui(t), in units of kWh, is the total energy consumption through the hour t ∈T . Let pi(t) ∈ R, in units of kW, denote the corresponding average power consumptionthrough the hour t, i.e.;

ui(t) = Tspi(t),

where Ts, in units of hrs, is the length of the interval t ∈ T . The accumulated averagepower consumption of the community, p(t) ∈ R, is expressed as p(t) =

∑ni=1 pi(t),

where it is important to notice that p(t) is not an instantaneous power, but instead, theaverage power during the hour t.

Let α ∈ [0, 1] denote the power factor of the community, when aggregating consump-tion from individual households. The apparent power of the community, during hour t, isthen s(t) = p(t)/α ∈ R [15]. The magnitude of the current drawn by the community isgiven by

|i(t)| =s(t)

v,

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where v is the voltage magnitude at the PCC. We assume v to be fixed. The averagetie-line losses, l(t) during hour t is given by

l(t) = |i(t)|2rT =p(t)2

(αv)2rT = β

(n∑

i=1

ui(t)

)2

∀t ∈ T , (7)

where rT > 0 is the tie-line resistance, and β = rT /(Tsαv)2.

2.3 Price modeling

Let w(t) ∈ R+, t ∈ T denote the cost of energy, for each period t ∈ T . As indicated,we assume that w(t) > 0, ∀t. Strictly speaking, at times it is possible that w(t) < 0,however we shall save the analysis of these cases for future work. On a similar note, weassume that all households have the same estimated price signal. Our results generalizeto the case with individual price signals for each consumer, but to ease the discussionthroughout the paper, we shall not include this concern.

For ease of notation, we let

W = diag(w(1), . . . , w(L) ∈ RL×L+ ,

with diag(· · · ) denoting a diagonal matrix. The cost of energy ci,e ∈ R, acquired fromeach household can then be described by

ci,e(ui) = 1TWui, (8)

where 1 = (1, . . . , 1) ∈ RL, and superscript T denotes transpose. As mentioned inSection 1, the DSO is billed for grid losses in the same fashion as consumers are billed forconsumption, so the cost experienced by the DSO on account of losses can be expressedas

cl(u1, . . . ,un) =L∑

t=1

w(t)l(t)

= β(u1 + · · ·+ un)TW(u1 + · · ·+ un).

(9)

Equation (9) entails that if all consumption is balanced by local production from solarpanels and vice versa, then there will be no net-import/export of energy, and thereby, thelosses would vanish.

2.4 Consumer behavior

The consumer behavior is affected both by the cost incurred by the use of energy, as wellas an incurred discomfort as described in the following.

Recall that xi,1(ui) is the temperature of household i across the horizon T . Weexplicitly include ui in the notation, to indicate that the state trajectory xi,1 is the resultof a specific consumption profile ui, given some initial conditions for (1). We furtherremind the reader, that xi,sp ∈ RL denotes a desired temperature set-point. Similarly, we

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3 Problem description

also introduce xi,F ∈ R2 as a target final state to each household, representing a finalindoor temperature and EV charge level, desired by the inhabitants. Given xi,sp and xi,F ,we shall model discomfort of household inhabitants by

di(ui) = (xi,sp − xi,1(ui))T (xi,sp − xi,1(ui))

+(xi(L)− xi,F )T (xi(L)− xi,F ), (10)

that is the discomfort is quadratic in temperature deviation, and deviation from desiredfinal state.

The discomfort could be modeled in many other ways, and the choice of a quadraticis based on an assessment that the average deviation from the set-point, should be min-imized, as opposed to for instance maximum deviation. Therefore, in this work we willconform to a standing hypothesis that (10) is a valid measure of discomfort. However, theresults presented throughout, are valid for any proper convex formulation of the discom-fort.

Any consumer desires to minimize both the cost of energy (8), as well as the incurreddiscomfort (10). In order to balance these two objectives into a scalar expression, weintroduce trade-off parameters λi > 0, i ∈ 1, . . . , n, as illustrated in the following.

2.5 Stochastic estimation

From the preceding sections, it is apparent that our models require knowledge about sev-eral uncontrollable and uncertain quantities, such as ambient temperature Ta(t), solarpower production us(t) and energy prices w(t). In order to include these signals in thecoordination process, suitable prediction models must be designed to provide estimatevalues of these uncertain quantities, spanning the horizon of the coordination. The task ofdesigning these estimation mechanisms is beyond the scope of this paper. The remainderof this paper therefore employ the standing assumption that such estimation proceduresexists, and provides deterministic predictions of the uncertain quantities. In subsequentexamples, future values of e.g. price signals, should thus be interpreted as a best estimateat the current point in time, provided by a higher level estimation procedure. We discussthis further in Section 6.


When planning consumption patterns for each household, it is desirable to minimize bothexpenses pertaining to the use of energy (8), the overall cost of losses (9), as well as thediscomfort incurred (10). This minimization should be performed while obeying both thelocal, household specific constraints (2), as well as the overall grid constraints (6).

3.1 Centralized Consumption Coordination

Consider a centralized scenario, where all information is globally available, and one sin-gle external governor, for instance a power company or retailer, has obtained control rightsfor the consumption of all households in the community. The optimization problem ofplanning the consumption pattern of each household, can be formulated as the followingProblem 1:

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Problem 1. From the relations (1), (3) (8), (9), (10), and provided

• known values of parameters:Ai, Bi, Ei,Xi,Ui,xi,sp, xi,F, i ∈ 1, . . . , n, as well as β and νj ,j ∈ 1, . . . ,m+ 1

• known values of:Ta(t), ui(t), ui,s(t), w(t), i ∈ 1, . . . , n, t ∈ T

• known, fixed trade-off parameters λi > 0, i ∈ 1, . . . , n

minimizeu1,...,un

n∑

i=1

(ce,i(ui) + λidi(ui))

+cl(u1, . . . ,un)

subject to xi(t) ∈ Xi, ui(t) ∈ Ui, ∀ifj(u1(t), . . . un(t)) ≤ νj ,t ∈ T , j ∈ 1, . . . ,m+ 1.

(11)

.As mentioned, the parameters λi > 0 are introduced in order to balance the two objectivesfor each household, i.e., cost and discomfort minimization, into a scalar cost function.This is known as regularization [16].

All the constraints of (11) are linear, and the cost function is further convex quadratic.Hence, (11) is a convex problem which can readily be solved, in a centralized man-ner [16]. We shall denote the minimal value of (11) by ζ⋆, and the corresponding con-sumption profiles we denote u⋆i , i ∈ 1, . . . , n.

As initially discussed, the nature of the problem is distributed. This is on account ofseveral conditions: Solving (11) centralized, corresponds to all consumers having signedoff control rights to one and the same entity, e.g. their power company, or similar. How-ever, as consumers can freely choose their energy provider, it is more likely that therewill be several companies, each allowed to control some subset of the consumers in thecommunity. As such companies are competitors, it is not certain that they are willing tocooperate and share information, in order to solve (11). Further, a large majority of theinformation required to solve (11) is household specific, and as such it is to some extentsensitive information that should not be distributed. Therefore, Section 5 discusses howthe problem might be solved in a decentralized fashion.

3.2 Numerical example

Before discussing how Problem 1 may be solved distributed, the following present anumerical example illustrating how including the cost of losses directly when optimizingconsumption patterns, will also have the added effect of reducing the overall loading ofthe grid. In the following, we solve Problem 1 centralized, and compare the results tothose obtained from solving the following Problem 2.

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Problem 2. From the relations (1), (3) (8), (10), and provided the same parameters anddata, as in Problem 1,

minimizeu1,...,un

n∑

i=1

ce,i(ui) + λidi(ui)

subject to xi(t) ∈ Xi, ui(t) ∈ Ui,fj(u1(t), . . . un(t)) ≤ νj ,t ∈ T , j ∈ 1, . . . ,m+ 1.

(12)

.The difference between Problem 1 and Problem 2 is solely that Problem 2 omits the costof losses cl from the objective, i.e., (12) only considers the cost of energy and incurreddiscomfort of each household.

We present an example, for a horizon of 24 hours, starting at 8 AM. We consider acommunity size of n = 30 households, divided into m = 3 radials. We use the estimatesfor electricity prices, inflexible consumptions and production from solar panels, presentedin Fig. 2, Fig. 3 and Fig. 4, respectively. The prices are in units of ¤/kWh, where ¤ refersto a generalized currency. In the example, 11 out of the 30 households have installedsolar panels, randomly divided as 5, 2 and 4 households in each of the three radials. Wehave adjusted the loss coefficient β such that the inflexible consumption results in around5 % losses. The individual households are modeled such that all EVs behave as loss-lessintegrators, and the thermal dynamic of each household has a time constant between 15and 25 hours.

w(t)

[¤/k

Wh]

t

13:00 18:00 23:00 04:00

0.32

0.34

0.36

0.38

0.4

0.42

Figure 2: The fixed energy prices across a 24-hour period, where ¤ refers to a genericcurrency.

ui(t)

[kW

h]

t

13:00 18:00 23:00 04:000

0.2

0.4

0.6

0.8

Figure 3: The inflexible consumption for all 30 households.

In Fig. 5 and Fig. 6 the consumption pattern resulting from solving Problem 1 respec-tively Problem 2, are shown.

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replacements

ui,s(t)

[kW

h]

t

13:00 18:00 23:00 04:000

2

4

Figure 4: The solar power generated by 11 households in the community.

ui(t)

[kW

h]

t

13:00 18:00 23:00 04:00

−2

0

2

Figure 5: The energy consumption of all households during each hour of the day, whenlosses are considered. The figure illustrates houses without (plain), and with solar panelsinstalled (dots)

ui(t)

[kW

h]

t

13:00 18:00 23:00 04:00−4

−2

0

2

4

Figure 6: The energy consumption of all households during each hour of the day, whenlosses are disregarded. The figure illustrates houses without solar panels (plain), and withsolar panels installed (dots).

As evident from Fig. 5, when losses are considered, the consumption increases forall households without solar panels, during the periods where solar power is produced byhouseholds with solar panels installed. This is because the joint cost of losses are reducedby consuming the solar power locally, rather than incurring losses both when transportingexcess power back to the grid, as well as importing it again later on.

This tendency is not present in the case when losses are not considered, shown inFig. 6. Here we see a large energy production from households with solar panels duringsolar intensive period, but the remaining households do not increase consumption accord-

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ingly, to absorb this production. This is because the price signal gives no incentive forthe remaining community to increase consumption. Instead, consumers without panelsimport a large amount of energy at a later time, around 19.00, entailing that the tie-linepower import/export fluctuates heavily in the period from 13.00 to 19.00.

∑

i∈riui(t)

[kW

h]

t

13:00 18:00 23:00 04:00

−50

0

50

Figure 7: The grid capacity (dashed, cyan), radial capacity (dashed red) and utilization ofgrid (solid, cyan) and radials (solid,blue/green/red), when losses are considered.

∑

i∈riui(t)

[kW

h]

t

13:00 18:00 23:00 04:00

−50

0

50

Figure 8: The grid capacity (dashed, cyan), radial capacity (dashed red) and utilization ofgrid (solid, cyan) and radials (solid,blue/green/red), when losses are disregarded.

Some of these tendencies are more clearly visible from Fig. 7 and Fig. 8, where gridloading is shown in the two cases. In Fig. 7 we see that since households without solarpanels increase consumption to balance out production from the those who have panels,the overall loading of each radial and the tie-line itself, is fairly even throughout thesolar intensive periods, whereas the loading varies greatly for the other case, illustrated inFig. 8.

The tie-line losses are 5.3 % from the inflexible consumption. When optimizing theflexible consumption with respect to both price, losses and discomfort, the added flexibleconsumption increases the losses to 19.8 %, and the tie-line is fully loaded 16.7 % ofthe considered period. However, when losses are not included in the optimization, lossesincreases to 24.0 %, and the tie-connection is fully loaded for 45.8 % of the period.

The reader should not put too much emphasis on the magnitude of these loss percent-ages, since this example is meant to be illustrative, rather than present practical results.The reader should, however, notice the significant loss and load reduction, obtained fromthe consumption coordination proposed through Problem 1.

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4 Flexibility analysis

From the numerical example presented in the previous section, it is apparent that there isgreat potential for reducing the overall loading of the grid, by including a concern towardslosses in the optimization. The potential load reduction does however, to some extent, de-pend on the specific consumer behavior, i.e., how the individual consumers evaluatesdiscomfort, and what kind of flexibility is provided. We show this through numericalexperiments. By solving Problem 1 and Problem 2, we illustrate the following:

H1. Including loss-minimization when planning consumption patterns of EHPs, doesnot introduce significant load shift, on account of the setpoint deviation included inthe discomfort measure presented in (10).

H2. Including loss-minimization when planning consumption of EVs, introduces sig-nificant load shift, since the discomfort in (10), is only measured at the final sampleof the horizon.

H3. Load shifting of EVs introduces a temporal smoothing of the import/export ofpower from the community.

To support H1 and H2, we execute again the example from Section 3.2, with two mi-nor changes; 1: The parameters of the thermal dynamics are set equal for each household,2: We disregard the grid limitations in the optimization. The resulting grid-load across thehorizon is presented in Fig. 9 for the two cases, where losses are considered (Problem 1)and where they are disregarded (Problem 2), in the optimization.

From the figure, it is seen that there is a severe grid-violation at night, when lossesare not considered; whereas this violation is significantly reduced if losses are includedin the planning. To explain this, the corresponding energy consumption of flexible units,i.e., EHPs and EVs, is presented in Fig. 10. Here, case 1 is when losses are disregarded,and case 2 is when losses are considered.

Comparing Fig. 9 and Fig. 10, it is apparent that when losses are disregarded, the peakgrid loading is completely dictated by the EVs. This is because the discomfort of EVs isonly related to the final state of charge, and therefore, the battery is completely chargedwhenever the price is lowest, in this case at night. This, however, has the consequence ofsignificantly overloading the grid. The energy consumption of EHPs is quite similar inboth cases, but EV charging is far more distributed across the time horizon, when lossesare considered. This is to avoid the high cost of losses, incurred by a significant peak inpower import, similar to the one appearing when losses are disregarded in Fig. 9. Thissupports our claims:

H1: The load profiles of EHPs are almost unchanged whether losses are accounted foror not. Since the use of EHPs is already distributed over time, they introduce noextraordinary losses. Including concern towards losses for this consumption type,therefore have only little effect. The flexibility of EHPs is restricted, in the sensethat consumption is closely related to set-point control. A significant temporal shift

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4 Flexibility analysis

of this consumption, would therefore incur an unacceptable discomfort, so loadshift from these consumption units must be short-term.

H2: On the other hand, for EVs, the grid loading is significantly reduced. The lackof set-point tracking for EVs makes them more flexible, since discomfort is onlymeasured in the end of the time frame, and therefore extraordinary losses are intro-duced, if no concern is included to reduce losses.

This illustrates a major difference between the EHPs and EVs as flexible consumptionunits. A more formal classification of flexible consumption units are presented by [17];however, the example above shows that a naive approach for load shifting, by simplyintroducing a price-signal to consumers, might be a risky approach, since a significantpresence of EVs and similar battery type storages, might introduce congestion problemsin low-price periods.

To support H3, we repeat the previous example. However, this time we test the casewhere all households have solar panels. Fig. 11 shows the grid loading in this case, bothwhen losses are considered and disregarded.

Loss consideredLoss disregarded

∑

iui(t)

[kW

h]

t13:00 18:00 23:00 04:00

−50

0

50

100

150

200

250

Figure 9: Grid loading in both cases where losses are considered and disregarded. Asbefore, the dashed line indicates the capacity of the tie-line, which is however, disregardedin the optimization.

2. EV2. EHP1. EV1. EHP

u(t)

[kW

h]

t13:00 18:00 23:00 04:00

0

1

2

3

4

5

Figure 10: The consumption of all EHPs and EVs in case 1 where losses are disregardedand case 2 where losses are considered.

The results presented in Fig. 11, supports the final hypothesis:

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ConsideredDisregarded

∑

iui(t)

[kW

h]

t

13:00 18:00 23:00 04:00−100

0

100

200

Figure 11: The grid utilization, when all households have solar panels installed, bothwhen losses are disregarded, and considered in the planning.

H3: When losses are not considered, the massive installation of solar panels entails thatthere will be a large power transfer out of the community, since there is no incentiveto consume it locally. Rather, a corresponding power import is introduced later on,introducing the risk of grid congestion in both directions of the grid. When lossesare considered, the large production from solar panels, is consumed locally by theEVs. This removes the need to import power later on, whereby there is only littlevariation in the total consumption of the community.

This concludes the flexibility analysis of the consumers. The following describes ourapproach for solving Problem 1, distributed.

5 Distributed Consumption Coordination

The example and results presented in the previous section, was obtained by solving (11) ina centralized fashion, and served to illustrate the potential benefit from including lossesas a concern when planning consumption patterns. However, as we have mentioned,the nature of the problem requires an approach that employs decentralized, rather thancentralized optimization.

Had we disregarded losses in our problem formulation, (11) would have been simpli-fied to the problem treated in [10], who used dual decomposition for finding a distributedsolution. Dual decomposition has been greatly employed in several network related ap-plications, e.g., [10,18,19]. However, due to the cost of losses, cl(u1, . . . ,un), all house-holds are tied together, which complicates the process of arranging a distributed algo-rithm. Below we outline the approach employed in our work, relying on the AlternatingDirection Method of Multipliers (ADMM). Similar to the method of dual decomposition,this is an iterative approach for finding a distributed solution to the problem.

Elaborating all details of ADMM is beyond the scope of this paper. We will only out-line the approach, and discuss the benefits with respect to our application. For extensivetreatments of the method, refer to, e.g, [20, 21].

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5.1 Distributed consumption coordination through ADMM

First, we define hi : RL → R ∪ ∞, i ∈ 1, . . . , n:

hi(ui) =

ci(ui) + λid(ui), if ui ∈ Ui, and xi(ui) ∈ Xi

∞, otherwise

and g : RL × · · · × RL → R ∪ ∞:g(u1, . . . ,un) =

cl(u1, . . . ,un), if fj(u1(t), . . . un(t)) ≤ νj

∀j ∈ 1, . . . ,m+ 1

∞, otherwise,

i.e., hi is an indicator function of whether the local constraints of each household areobeyed, and g is an indicator function of whether the tie-line and radial capacity limitsare obeyed. It is then apparent that (11) is equivalent to the problem

minimizeu1,...,un

n∑

i=1

hi(ui) + g(u1, . . . ,un). (13)

Introducing auxiliary variables z1, . . . , zn ∈ RL, as well as consistency constraints ui −zi = 0, ∀i ∈ 1, . . . , n, (13) can be written in ADMM-form, as

minimizeui,zi

n∑

i=1

hi(ui) + g(z1, . . . , zn)

subject to ui − zi = 0, ∀i.

(14)

In order to solve (14) through ADMM, both g and all hi’s are required to be closed andproper convex functions [20]. However, as we argued in Section 2, this is indeed the casein our problem formulation. We further assume that a feasible solution to Problem (14)does indeed exist.

The ADMM approach for solving (14) distributed and iteratively, can then be formu-lated as [20]:

u(k+1)i = argmin

ui

(

hi(ui) + ρ/2‖ui − z(k)i + y

(k)i ‖22

)

z(k+1) = argmin

z

(

g(z) + ρ/2n∑

i=1

‖z(k)i − u(k+1)i − y

(k)i ‖22

)

y(k+1)i = y

(k)i + u

(k+1)i − z

(k+1)i

(15)

for i = 1, . . . , n, with z = (z1, . . . , zn) and ρ > 0 is a fixed parameter, k ∈ N is theiteration index, and yi ∈ RL are the Lagrange multipliers for the equality constraints in(14). The z-update above involves an optimization problem in nL variables, and thereforescales equally bad with the number of households, as the original Problem (11). However,in a similar fashion as illustrated in [20], we can reformulate this to involve only mLvariables, i.e., the complexity increases with number of radials m, rather than the numberof households n. Since the number of radials should be expected to be quite significantlylower than the number of households, this is indeed a significant improvement.

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5.2 Problem reduction

We introduce the notation

zj =1

nj

∑

i∈rj

zi ∈ RL, z = (z1, . . . , zm) ∈ RmL, (16)

with j ∈ 1, . . . ,m. We further define the set Z ⊂ RmL

Z =

z |

∣∣∣∣∣∣

m∑

j=1

njzj

∣∣∣∣∣∣

≤ νm+1, |njzj | ≤ νj ,

j ∈ 1, . . . ,m

,

where νj are the radial and tie-line limitations introduced earlier, and nj is the number ofhouseholds in each radial. All the inequalities are to be understood entry-wise.

We define the extended value functiong : RmL → R ∪∞ as

g(z) =

β(∑m

j=1 njzj

)T

W(∑m

j=1 njzj

)

, if z ∈ Z

∞, otherwise.(17)

By the definitions (16) and (17), we have g(z) = g(z), for any z. The z-update in (15)can thereby be equivalently formulated as the optimizer of the problem

minimizez

g(z) + ρ/2

n∑

i=1

‖zi − u(k+1)i − y

(k)i ‖22

Subject to zj =1

nj

∑

i∈rj

zi, j ∈ 1, . . . ,m.(18)

For any fixed value of z = (z1, . . . , zm), the Lagrangian of (18) is expressed by

L(z, µ) = g(z) + ρ/2

n∑

i=1

‖zi − u(k+1)i − y

(k)i ‖22

+m∑

j=1

µTj (zj −1

nj

∑

i∈rj

zi),

where µ1, . . . , µm ∈ RL are the Lagrange multipliers, associated to the equality con-straints in (18). As (18) is a convex problem, the necessary and sufficient conditions foroptimality, is given as

∇zL(z, µ) = 0 and, zj =1

nj

∑

i∈rj

zi, j ∈ 1, . . . ,m. (19)

For fixed z, g(z) is just a scalar, and the condition ∇zL(z, µ) = 0 translates to;

ρ(zi − u(k+1)i − y

(k)i )−

1

njµj = 0, ∀i ∈ rj , (20)

154


and j ∈ 1, . . . ,m. After some manipulation, (19) and (20) collectively implies

zi = u(k+1)i + y

(k)i − u

(k+1)j − y

(k)j + zj , ∀i ∈ rj (21)

where we have used the same overline-notation for ui and yi, as introduced in (16):

uj =1

nj

∑

i∈rj

ui, yj =1

nj

∑

i∈rj

yi, j ∈ 1, . . . ,m.

By employing (21) in (18), the z-update introduced in (15), can be equivalently formu-lated as

z(k+1) = argmin

z

(

g(z) + ρ/2

m∑

j=1

nj‖zj − u(k+1)j − y

(k)j ‖22

)

,

with z = (z1, . . . , zm). This now involves a problem of only mL variables. Inserting(21) in the update to y(k+1) gives

y(k+1)i = u

(k+1)j + y

(k)j − z

(k+1)j , ∀i ∈ rj , j ∈ 1, . . . ,m (22)

from which we see that the Lagrange multipliers are equal for all households in the sameradial. We thereby let

y(k)i ≡ y

(k)j , ∀k, i ∈ rj ,

for j ∈ 1, . . . ,m. The update (22) then simplifies to

y(k+1)j = u

(k+1)j + y

(k)j − z

(k+1)j , j ∈ 1, . . . ,m

With these simplifications, the ADMM algorithm is formulated by Algorithm 5.1.

Algorithm 5.1 Distributed optimization

Initialize z(0)j ,u(0)i ,y

(0)i , j = 1, . . . ,m, i = 1, . . . , n,

for k=0,1, . . . do

Local optimization at each household:u(k+1)i = argmin

ui

(

hi(ui) + ρ/2‖ui − u(k)i + u

(k)j − z

(k)j + y

(k)j ‖22

)

,

i ∈ rj , j ∈ 1, . . . ,m

Central coordination:z(k+1) = argmin

z

(

g(z) + ρ/2∑mj=1 nj‖z

(k)j − u

(k+1)j − y

(k)j ‖22

)

y(k+1)j = u

(k+1)j + y

(k)j − z

(k+1)j , j = 1, . . . ,m

end for

Below, we give a practical interpretation of the iterations in Algorithm 5.1.

5.3 Practical Interpretation

The ui-update in Algorithm 5.1 can be conducted locally at each household. Subse-quently, all results from the local problems are then collected in order to make the up-dates:

z(k+1) = (z(k+1)1 , . . . , z(k+1)

m )

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y(k+1) = (y(k+1)1 , . . . ,y(k+1)

m ),

i.e., these updates are performed centralized, and then distributed to each household forthe following iteration, along with the average consumption within each radial. Thiscommunication process is illustrated in Fig. 12.

Distribution System

Consumers

uk+11 + y

k+11u

k+11

−zk+11

uk+1m + y

k+1mu

k+1n

−zk+1m

Operator

r1 rmRadials

Figure 12: Iterative data exchange between the local optimization of each individualhousehold, and the centralized coordination from the DSO.

With this communication process, Algorithm 5.1 consists of both localized and central-ized computations, in the sense that each household solves a local optimization problem,the results of which are collected by, e.g., the DSO, in order to solve a centralized prob-lem.

This can be interpreted as a clearing process, similar to the market clearing describedby [10]: At each iteration k; uk+1

i is the local, desired consumption pattern for each

household i ∈ rj , based on the current values of z(k)j , y(k)j and u

(k)j , for all j. Con-

sidering the desired consumption patterns, the DSO updates zj and yj for the followingiteration, with the intention of minimizing active losses, and ensures that grid constraintsare obeyed. The iterations continue until some certificate is obtained, that constraints areobeyed, and losses are minimized within some tolerance. The iterations then stop, i.e.,the clearing process is completed.

When updating the u’s, the private optimization of each consumer computes a trade-off between the local cost hi(ui) and the added term ρ/2‖ui−u

(k)i +u

(k)j −z

(k)j +y

(k)j ‖22.

The added norm thus represent a penalty enforced by the DSO to incentivise consumersto shift their consumption from their individual preferred consumption pattern, towardsthe global optimum accounting also for coupling through losses. By iteratively updatingthis term, the DSO can coordinate consumers to minimize the losses as well as ensuringthat grid constraints are satisfied. As mentioned in Section 1, the consumers are currentlybilled for losses, indirectly through fixed tariffs, set by the DSO, i.e. there is no benefitto be obtained by temporally shifting consumption. In the coordination scheme derivedhere, the penalty introduced to each consumer, allows the consumer to obtain a benefitby shifting consumption to low-load periods, since this reduces his cost of losses. Thisstrategy may thus be considered as a more direct billing strategy than the current fixedtariff, reflecting the fact that the cost of losses changes as the grid loading does.

In the modeling, Section 2, we have described the local objectives, manifested by hiin the algorithm above, to have the same format for all households. However, the con-

156


vergence properties presented in Section 5.4, requires only that hi is convex and strictlyproper, and so does not require the objective of each household to be similar.

On a final note, we remark to the reader, that the size of each subproblem in Algo-rithm 5.1, does not increase when the number of participating households increase, i.e.,the complexity is unaffected by community size. Similarly, as we showed above, the sizeof the central problem scales with the number of radials, rather than households, so thecomplexity of this problem is largely unaffected by the specific number of households inthe community.

5.4 Convergence

For the consistency of our presentation, we convey the main convergence results on for theADMM algorithm. Convergence of ADMM follows from convergence proofs presentedby [20, 21], among others. Following these, in our case, ADMM satisfies

1a) strategy convergence: u(k)i → u⋆i as k → ∞, ∀i

1b) objective convergence: ζ(k) → ζ⋆ as k → ∞

2) residual convergence: q(k) = u(k)j − z

(k)j → 0

as k → ∞ for j ∈ 1, . . . ,m

3) auxiliary convergence: s(k) = z(k+1)j − z

(k)j → 0

as k → ∞ for j ∈ 1, . . . ,m

Item 1 above entails that when solving Problem (11) distributed via ADMM, the localconsumption strategies of each household converges to the global optimum, implyingthat the total cost converges to the global optimum.

Items 2 and 3 above can be used for termination criteria of the iterations of Algo-rithm 5.1, [20]. The algorithm can be stopped when

q(k), s(k) < ǫ,

for some termination tolerance ǫ > 0.In the following, we illustrate the distributed algorithm with an example.

5.5 Numerical example

We present a full example, considerably larger, than was the case in Section 3.2. Weinclude n = 342 households, whereof 144 have solar panels installed. The householdsare divided into m = 19 radials, resulting in 18 households pr. radial. We consideragain a horizon of 24 hours, divided into 1-hour samples. The size of this problem corre-sponds to a small town, specifically, the number of households and radials included here,corresponds to the Danish town of Aasted as described in detail by [22].

We solve this problem using the same estimated price signal, and general pattern ofthe inflexible consumption and generated solar power, as illustrated in Fig. 2, Fig. 3 andFig. 4, respectively. As was also the case previously, the electric vehicles are modeledas integrators, and the thermal dynamics of each household has a time-constant of 15-25hours. The grid coefficient β is adjusted so that the inflexible consumption corresponds

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to 4.25 % losses. For comparison, we again solve the problem centralized, this givesus the global optimal value ζ⋆. In our example, running ADMM gives the convergenceillustrated in Fig. 13, showing both residual, auxiliary and objective convergence. In theexample, we have used the termination criteria, ‖q(k)‖, ‖s(k)‖ < 5E-4. The resulting

Dev

iatio

n

k20 40 60 80 100 120 140

10−4

10−2

100

Figure 13: Convergence of the distributed average approach as a function of iterationnumber k. The figure shows ζ(k) − ζ⋆ (asterisk), ‖q(k)‖ (solid) and ‖s(k)‖ (dashed).

consumption profiles are illustrated in Fig. 14, for 20 randomly selected consumers. Thefigure shows the consumption patterns found from both the centralized and decentralizedsolutions, and also which consumers have solar panels installed. As is apparent from thefigure, the consumption profiles found decentralized, are virtually indistinguishable fromthose found centralized. As a consequence hereof, the overall consumption pattern of the

ui(t)

[kW

h]

t

13:00 18:00 23:00 04:00−2

0

2

Figure 14: The centralized solution (solid), for consumers without (plain, light blue),and with solar panels (dotted, dark blue). Similarly, the distributed solution (dashed), forconsumers without (plain, orange), and with solar panels (dotted, red). The figure showsthe consumption pattern for 20 selected consumers.

entire community, is the same as that found centralized. This is illustrated in Fig. 15. TheADMM method was proven by [21] to converge for any ρ > 0. The convergence speedmay, however, be greatly affected by the specific value chosen. A suitable choice of thevalue of ρ greatly depends on the problem structure. Suitable strategies are for pickingρ are still an open-ended question, although some results are provided in [23], for someclasses of problems.

With this example, we have illustrated that even in the case where (11), cannot besolved centralized, either on account of problem size, or conflicting interest, e.g, thepresence of multiple power retailers, it is still possible for the DSO to coordinate each

158

6 Conclusion

∑

i

ui(t)

[kW

h]

t

13:00 18:00 23:00 04:00200

400

600

800

Figure 15: Community consumption found centralized (solid), and distributed (asterisk).Also plotted is the tie-line capacity limit (dashed).

consumer, towards a global optimum, where losses are minimized as a trade-off withconsumer cost and discomfort, and further obey grid limitations.

5.6 Extensions

A wide range of alternative problems would fit into the framework we outline in this work.Alternative formulations could for instance be consumers that are without any discomfort,and are solely defined by constraints; an example could be obtained by formulating theEV charge limit as a hard constraint, rather than allowing any deviation which we do inthis paper.

There might also be consumers with EHPs installed that would only allow for veryminor deviations from the temperature set-point, in which case the introduced constraintset X would be smaller, in other words, the flexibility is reduced, and the consumerresembles to a larger degree one who only has inflexible consumption.

In the same way, a more elaborate price scheme could be seamlessly employed for thecost of energy, introducing e.g. different prices for consumption and production of en-ergy, the requirement simply being that cost (revenue) is convex in the local consumption(production).

In light of this discussion, the consumption and production classification outlined inthis paper, can be regarded as archetypical consumer classes, which may be combined orextended to other types of consumers, while still obeying the presented framework. Thisillustrates the versatility of the described approach.

6 Conclusion

In this work we have illustrated how consumption may be coordinated in a community ofclosely located households, such that losses in the electric distribution grid are minimizeddirectly, when optimizing the consumption profile of each household against an estimatedreal-time price signal. Through examples, we have illustrated that including cost of lossesdirectly when optimizing consumption, results in a general reduction of the grid loading,and further that large fluctuations in energy import/export from the community, can bedampened.

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In this paper we have further shown how the centralized problem can be solved in adistributed fashion, catering to the distributed nature of the households in the community,and the lack of affiliation between each household.

We have assumed that all losses can be translated to the tie-connection between thecommunity and the remaining electrical grid. However, as this is a simplification, a wayof including losses distributed throughout the community should be considered. Theframework outlined in this work, directly generalizes to the case where losses are lumpedwithin each radial, but distributed losses along the radials, are not covered here, and issaved for future work.

Extending the analysis of this work, to include distributed losses within the commu-nity, would also entail that the illustrated consumption coordination concept, could beused for voltage control and stabilization. For instance, it is easy to imagine that coor-dination between solar power from one household, and the charge pattern of the EV ofanother, could alleviate issues concerning either local over-voltages caused by unused PVproduction, and under-voltages caused by heavy consumption for EV charging.

In this work we assume that methods are available for estimating the stochastic andtime varying components of the problem, e.g. energy price, ambient temperature, so-lar radiation, etc. so that by employing these estimates, we complete the coordination.To increase robustness against uncertainties, we anticipate to expand the off-line coordi-nation described in this work, to an online coordination employing a receding horizonstrategy. This means that at each time-step, consumers update their local estimates ofsolar-production, ambient temperature, etc., and participates in the coordination proce-dure as described in this paper. In short; we repeatedly coordinate consumptions patternsfor the entire horizon, but we only make use of the first sample before we repeat the co-ordination. This receding horizon strategy entails a lot of computational overhead, buthas the benefit of allowing to incorporate measurements of the stochastic components assoon as they are available, thereby improving robustness against uncertainties. It shouldbe remarked, that the iterative process derived in this paper will convergence regardlessof uncertainties and disturbances, although the global optimum, and thus the point ofconvergence, will be affected to some degree.

References

[1] Danish Energy Association and Energinet.dk, “Smart Grid i Danmark,”www.danskenergi.dk, 2010.

[2] The Danish Energy Agency, “Energistatistik 2010,” www.ens.dk/, 2010, publicationonly in Danish.

[3] M. Baran and F. Wu, “Network Reconfiguration in Distribution Systems for LossReduction and Load Balancing,” IEEE Transactions on Power Delivery, vol. 4,no. 2, 1989.

[4] F. Viawan and A. Sannino, “Voltage Control with Distributed Generation and ItsImpact on Losses in LV Distribution Systems,” IEEE Russia Power Tech, 2005.

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6 Conclusion

[5] M. Masoum, P. Moses, and K. Smedley, “Distribution transformer losses and perfor-mance in smart grids with residential Plug-In Electric Vehicles,” Innovative Smart

Grid Technologies (ISGT), pp. 1–7, 2011.

[6] Y. Guo, Y. Lin, and M. Sun, “The impact of integrating distributed generations onthe losses in the smart grid,” IEEE Power and Energy Society General Meeting,2011.

[7] T. Hoff and D. Shugar, “The value of grid-support photovoltaics in reducing distri-bution system losses,” IEEE Transactions on Energy Conversion, vol. 10, no. 3, pp.569–576, 1995.

[8] K. Turitsyn, P. Sulc, S. Backhaus, and M. Chertkov, “Options for Control of Re-active Power by Distributed Photovoltaic Generators,” Proceedings of the IEEE,vol. 99, no. 6, pp. 1063–1073, Jun. 2011.


Proceedings, 2011.

[10] B. Biegel, P. Andersen, J. Stoustrup, and J. Bendtsen, “Congestion management ina smart grid via shadow prices,” Sep. 2012.

[11] O. Sundström and C. Binding, “Flexible charging for electric vehicles consideringdistribution grid constraints,” IEEE Transactions on Smart Grid, vol. 3, no. 1, pp.26–37, Mar. 2012.

[12] Nord Pool Spot, www.nordpoolspot.com/, 2013, common Nordic Power Exchange.

[13] Danish Energy Association, “Vejledning til tarifberegningsmodel,”www.danskenergi.dk, 2005, guidelines for grid tarif calculations.


[15] J. Irwin and R. Nelms, Basic Engineering Circuit Analysis. Wiley, 2005.


[17] M. Petersen, K. Edlund, L. Hansen, J. Bendtsen, and J. Stoustrup, “A taxonomy formodeling flexibility and a computationally efficient algorithm for dispatch in smartgrids,” Proceedings of IEEE American Control Conference, 2013.

[18] D. Palomar and M. Chiang, “A tutorial on decomposition methods for network util-ity maximization,” IEEE Journal on Selected Areas in Communications, vol. 24,no. 8, pp. 1439–1451, Aug. 2006.

[19] H. Terelius, U. Topcu, and R. Murray, “Decentralized multi-agent optimization viadual decomposition,” Proceedings of the 18th IFAC World Congress, Aug. 2011.

[20] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimizationand statistical learning via the alternating direction method of multiplier,” Founda-

tions and Trends in Machine Learning, vol. 3, no. 1, 2010.

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[22] J. Pillai, P. Thøgersen, J. Møller, and B. Bak, “Integration of electric vehicles in lowvoltage Danish distribution grids,” Power and Energy, 2012.

[23] E. Ghadimi, A. Teixeira, I. Shames, and M. Johansson, “Optimal parameter selec-tion for the alternating direction method of multipliers (admm): quadratic prob-lems,” Jun. 2013, available from Arxiv: www.arxiv.org/abs/1306.2454.

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Paper D

Fault Tolerant Distributed Portfolio Optimization in Smart Grids

Morten Juelsgaard, Rafael Wisniewski and Jan Bendtsen

This work is published in:International Journal of Robust and Nonlinear Control, May-June, 2014

Copyright c© John Wiley and SonsThe layout has been revised

1 Introduction

Abstract

This work considers a portfolio of units for electrical power production and theproblem of utilizing it to maintain power balance in the electrical grid. We treatthe portfolio as a graph in which the nodes are distributed generators and the linksare communication paths. We present a distributed optimization scheme for powerbalancing, where communication is allowed only between units that are linked in thegraph. We include consumers with controllable consumption as an active part of theportfolio. We show that a suboptimal, but arbitrarily good power balancing can beobtained in an uncoordinated, distributed optimization framework, and argue that thescheme will work even if the computation time is limited. We further show that ourapproach can tolerate changes in the portfolio, in the sense that increasing or reducingthe number of units in the portfolio requires only local updates. This ensures that unitsexperiencing faults or need for maintenance can be removed from the graph withoutaffecting the overall performance or convergence of the optimization. The results areillustrated by numerical case studies.

1 Introduction

Current scientific and political interests are directed towards increasing the use of renew-able energy and reducing power production from fossil fuels. A major concern relatedto this is how to maintain a stable and balanced electrical grid when a large part of thepower comes from volatile resources such as wind and sun. A suggested approach tothis problem is load shifting by use of different energy storages [1], for instance activelycontrolled consumption. The idea of actively controlling consumption is motivated by hy-pothesis that a significant part of the future electricity consumption will be flexible, andmay therefore be temporally shifted without noticeable discomfort for the consumer. Forinstance, this could be through an increased use of electric heat pumps (EHPs) for heating,or electric vehicles (EVs) for transportation. Consider for instance a private householdmaintained at a certain temperature by an EHP: Shutting off the EHP would not causea noticeable temperature drop for some period of time due to the thermal mass of thebuilding. The electric load caused by the EHP may thus be temporally shifted without in-curring any significant discomfort to the inhabitants. Similar considerations can be madefor EVs. By utilizing such flexibility, it is expected that grid balancing in the future couldbe handled to some extent by the shifting consumption, rather than adjusting production.The recent research in [2–4], has focused on how to include different types of consumersin different grid balancing scenarios.

The balance between consumption and production of electrical power is maintainedby national transmission system operators (TSOs). Specifically, in the Scandinaviancountries, most power consumption and production is traded through a common energyexchange, ensuring that balance between consumption and production is maintained atall times [5, 6]. This entails that if a power producer (PP) agrees to supply a quantityof power through the energy exchange, but fails to generate the agreed quantity, the PPwill have caused a power imbalance. In that case, several different balancing mechanismswill ensure that the power mismatch is compensated by other actors. The PP causing themismatch will subsequently be billed [7].

A PP typically operates a number of different production units, with different opera-tional costs and characteristics. We refer to them collectively as a production portfolio.

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Paper D

During operation, the units in the portfolio might experience faults caused by for instanceoverload or gradual deterioration, forcing it to shut-down unexpectedly, whereby the PPwill experience a deficit with respect to the promised production. Similar deficits mightoccur in portfolios containing significant levels of wind energy, if the predicted poweroutput from wind turbines is far from the true output, or if wind conditions force turbinesto shut-down, e.g, if the wind speed is too high for safe operation of the turbines. Onaccount of the incurred power deficit, the PP will either face the financial expenses ofcausing a power imbalance, or he will have to cover the deficit locally, by distributing itamong the remaining portfolio units. This work presents an optimization strategy, whichsolves the problem of optimally distributing such a deficit among the remaining units inthe portfolio. Besides traditional power plants, we include controllable consumers in theportfolio, such that a power deficit can be covered either by increasing production fromgenerators, or decreasing consumption from consumers.

The number of units in the portfolio such as thermal, wind, solar etc. combined withthe number of controllable consumers can cause the portfolio to become very large. Asa consequence, the task of collecting and distributing data across the portfolio, as well assolving the optimization in a centralized manner, may grow computationally intractable.To relieve this problem, we propose a distributed optimization scheme, where we dividethe main problem into smaller subproblems. Such approaches usually require some formof central coordination among subproblems [8, 9]. However, rather than introducing co-ordination, we employ a distributed consensus approach, that does not require centralhierarchical layering.

We consider the portfolio as a connected graph, where the nodes represent individualportfolio units, and the links represents communication paths. One of our main contri-butions is to derive an optimization strategy for solving the balancing problem with nocentral data collection or coordination, and further to show that a sub-optimal, but ar-bitrarily good solution is obtained when faced with limited calculation time. A similarresult was presented in [10] and [11]; however, these works omitted private constraints ateach node, which is one of the complicating factors included here.

Another contribution of this work is to show that our approach is robust againstchanges in the portfolio, in the sense that convergence towards optimal portfolio oper-ation is not affected by online modifications to the graph, due to for instance shut-downof units. Furthermore, our approach requires no network-wide changes in case of suchmodifications. This introduces a degree of fault tolerance on the network level, similar tothe strategy presented in [12]. The fault tolerant design of our approach thereby gives ita ’plug-and-play’-like nature, in the sense that individual units are allowed to leave andreenter the portfolio during operation.

In the following, Section 2 presents in greater detail the production portfolio and ac-companying optimization problem. Section 3 presents a short outline of the theory relatedto the distributed optimization employed in this work, along with an outline of the graphtopology of the portfolio. The main contributions are presented in Section 4, where thefault tolerant plug-and-play nature of our approach is investigated, and Section 5 wherewe analyze and discuss convergence of the optimization. Thereafter Section 6 presents anumerical case study, illustrating our contributions, and Section 7 summarizes the work.In Appendix A we provide technical lemmas and corollaries, which are necessary forderiving the statements we present throughout, but which are not strictly necessary forthe reader to understand the overall procedure.

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2 Portfolio Description

2 Portfolio Description

Let the portfolio consist of n power generating units each producing power xi(t) ∈ R, fori ∈ S = 1, . . . , n, at discrete time instance t ∈ T = t0, . . . t0+N − 1, withN ∈ Zbeing the length of the time-frame under consideration. Each unit is characterized by amaximum and minimum production capacity xmin,i, xmax,i ∈ R and a maximum powergradient dxmax,i ∈ R, i ∈ S, such that

xmin,i ≤ xi(t) ≤ xmax,i, |xi(t)− xi(t− 1)| ≤ dxmax,i, ∀i ∈ S . (1)

We let xi = (xi(t0), . . . , xi(t0+N − 1)) ∈ RN , and associate to each unit an operatingcost function fi : RN → R. The following elaborates and classifies several differentclasses of portfolio units: Thermal plants, consumers and external actors.

2.1 Classification of thermal plants

We divide the thermal plants into two classes:

• Type 1:These have very large production capacity, and a low cost of operation. However,these units also have a very low gradient, such that ramping the production up anddown can only be performed slowly.

• Type 2:These have a smaller production capacity and a higher operating cost compared totype 1 generators. On the other hand, their power gradients are significantly larger.

Type 1 plants could be coal fired combined heat and power (CHP) plants. Type 2 unitscould be oil- or gas-fired plants [13].

Let S1 ⊂ S be the set of indices corresponding to all type 1 units in the portfolio.Similarly, let S2 ⊂ S be all indices corresponding to type 2 units. Let P1, P2 ∈ R+

denote the cost of operating these plants over a horizon of N time instances from thecurrent time t = t0, where

P1 =∑

i∈S1

fi(xi), P2 =∑

i∈S2

fi(xi), (2)

In this work, we will assume that the lower capacity limit for these units are xmin,i = 0.We use the cost functions

fi(xi) = xTi Gixi, ∀i ∈ S1 ∪ S2, (3)

where the weight matrices Gi ∈ RN×N are positive definite. However, the results pre-sented in this work hold in general for strictly convex and C1 cost functions. The cost(3) resembles that of [14] and [15] who use 2nd order polynomials to describe simplifiedoperating costs of power plants.

2.2 Classification of consumers

As described in Section 1, consumers with controllable consumption can be employed inseveral ways for grid balancing. We focus on consumers in the form of electric heating

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of private households, in a similar fashion as described by [2]. We model the temperatureT of a household as

Ti(t+ 1) = aiTi(t) + bixi(t), i ∈ S3 (4)

where S3 ⊂ S are the indexes corresponding to all households in the portfolio, andai, bi ∈ R are household specific model parameters. The capacity limits for house-holds are xmin,i < 0 and xmax,i = 0, since these units represents consumption ratherthan production. The consumers further introduce constraints on allowed discomfort, i.e.,maximum temperature deviation from a set-point Tsp,i(t) ∈ R:

|Tsp,i(t)− Ti(t)| ≤ δTmax, i ∈ S3 (5)

where δTmax ∈ R is the maximum allowed temperature deviation. The cost of employingconsumers for balancing is not financial, as was the case with the thermal plants. Rather,we let the incurred cost be related to discomfort of inhabitants, measured as the deviationfrom the set-point:

fi(xi) = (Tsp,i − Ti(xi))TGi(Tsp,i − Ti(xi)), ∀i ∈ S3 (6)

where Gi is positive definite and

Tsp,i = (Tsp,i(t0), . . . , Tsp,i(t0+N−1)) ∈ RN , Ti(xi) = (Ti(t0), . . . , Ti(t0+N−1)) ∈ R

N ,(7)

that is, the cost of discomfort relates to the squared two-norm of the temperature devia-tion, similar to the work of [2]. We express the total cost P3 ∈ R+ of using consumersfor balancing as

P3 =∑

i∈S3

fi(xi). (8)

2.3 Classification of external actors

As mentioned in Section 1, the PP faces financial expenses for any imbalance it introducesby deviating from a predetermined production schedule. In this work, we introduce thisexpense through external actors, meaning that if a power imbalance is created, it will becompensated by some external actors, at some cost. These actors could either representthe balancing mechanisms of the TSO, or it could be bilateral contracts made betweena PP and some third party. In this work, we employ the latter approach, and modelthe operating cost of external actors similar to (3), since they represent thermal plantsoperated by some third party. It is reasonable to assume that the cost of using externalactors is higher than using the thermal plants operated the PP itself.

We consider the external actors collectively as a single unit in the portfolio. That is,let S4 ⊂ S be the singleton denoting external actors, where the operating cost of thesebecomes

P4 = fi(xi), i = S4, (9)

with fi modeled as in (3). Since external actors in essence could represent all remainingpower plants in the grid, we will assume that they have unlimited slew rate capabilities,and very large capacity limits.

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3 Distributed Optimization

2.4 Collected Portfolio

Fig. 1 illustrates how all units in the portfolio contributes with power to the grid. Aswe elaborate through the following sections, we are not concerned with any particulartopology of the geographical location of each portfolio unit with respect to the electricalgrid. Since we do not assume that portfolio units are located in the same grid section,we refrain from considering issues such as grid congestion, and simply regard the gridas an ideal bus. For more detailed work on grid related issues such as power losses andcongestion, the reader is referred to studies such as [9, 16, 17].

Type 1Type 2

Consumers

External actors

Grid

Σ

Figure 1: Conceptual sketch of how the production portfolio collectively balances thegrid.

The optimal use of the power portfolio minimizes the operating costs when coveringpower demand c(t) ∈ R. This problem can be formulated as

minimizex1,...,xn

P1 + P2 + P3 + P4

subject to∑

i∈S

xi = c

xi ∈ Xi, ∀i ∈ S

(10)

where c = (c(t0), . . . , c(t0 + N − 1)) ∈ RN , and Xi ∀i ∈ S are compact, convexand encapsulate all local constraints for each unit in the portfolio, i.e., any capacity, slewrate or temperature constraints. Notice that S =

⋃4i=1 Si, and Si ∩ Sj = ∅, ∀i, j ∈

1, . . . , 4, i 6= j.We have omitted renewable resources such as wind in the portfolio. Wedo this in order to illustrate a future scenario, where faulty wind predictions, combinedwith a significant penetration of wind energy, may contribute to significant imbalances tobe handled by the remaining non-renewable units in the portfolio. We elaborate on thisthrough the case studies in Section 6.


In this section we briefly outline two generic optimization strategies, and how we employand combine them to solve the overall problem presented in Section 2.

Every unit i in the portfolio, is assumed to have a strictly convex individual costfunction fi : RN → R+. The objective is to minimize the accumulated cost of all units,

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subject to local constraints as well as an overall resource constraint. We formulate this as

minimizex1,...,xn

n∑

i=1

fi(xi)

subject to xi ∈ Xi, ∀i ∈ 1, . . . , nn∑

i=1

xi = c,

(11)

where xi ∈ RN is the local production and Xi are compact, convex sets expressing thelocal constraints for unit i. Further, c ∈ RN is the required resource. The resourceconstraint couples all units together and without it, Problem (11) would be completelyseparable. We assume (11) is strictly feasible.

3.1 Problem Decomposition

Let qi(ν) = infxi∈Xi

(fi(xi) + νTxi

). The dual problem of (11) is then formulated as

maximizeν

q(ν) =n∑

i=1

qi(ν)− νT c (12)

where ν ∈ RN are Lagrange multipliers for the rescource constraint. The duality gapbetween (11) and (12) is zero, as given by Slaters condition [18]. Let

x⋆i (ν) = arg infxi∈Xi

(fi(xi) + νTxi

), (13)

then∑n

i=1 x⋆i (ν) − c ∈ ∂q(ν), where ∂q(ν) denote the subdifferential of q(ν) [19]. In

fact, as the fi’s are strictly convex, and the Xi’s are convex, q(ν) can be shown to bedifferentiable [20], i.e. ∂q(ν) = ∇q(ν); hence

∇q(ν) =∑

i∈S

x⋆i (ν)− c. (14)

In this work we employ dual decomposition [19] for solving (11), such that each individ-ual unit solves the local subproblem

minimizexi

fi(xi) + νTxi

subject to xi ∈ Xi,(15)

for i = 1, . . . , n. At a higher level, the subproblems are coordinated by iteratively updat-ing and distributing the dual variables

ν(k+1) = ν(k) + α(k)∇q(ν(k)), for ∇q(ν(k)) =n∑

i=1

x⋆i (ν(k))− c, (16)

where α(k) ∈ R+ is a step length and k is the iteration index. This corresponds to agradient-ascend algorithm for (12), so for specific choices of α(k), we have ν(k) → ν⋆

for k → ∞, where ν⋆ = arg supν(q(ν)).Problem (12) is unconstrained, however as the following lemma shows, we can design

a compact convex set V such that ν⋆ ∈ Int(V), whereby the optimization can be limitedto ν ∈ V . We use this in Section 5 when analyzing convergence.

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Lemma 1. Let q(ν) as defined in (12), be the dual of the feasible problem (11) where fiand Xi are convex functions and sets respectively, for all i. Let the local constraint setsXi contain maximum and minimum bounds on the entries of xi, for all i.

Arrange S such that S4 = n, i.e., let the singleton representing external actors bedenoted by index n. Further, let

xmin,n < min(c)−∑

i/∈S4

xmax,i, xmax,n > max(c)−∑

i/∈S4

xmin,i, (17)

Letν⋆ = arg sup

ν∈RN

(q(ν)), x⋆n(ν⋆) = arg inf

xn∈Xn

(fn(xn) + xTnν⋆). (18)

Then x⋆n(ν⋆) ∈ Int(Xn) and

ν⋆ ∈ Int(V), V = ν ∈ RN | -max

xn∈Xn

dfndxn

− ǫ ≤ ν ≤ -minxn∈Xn

dfndxn

+ ǫ, (19)

for any ǫ > 0, where the inequalities are read elementwise.

Proof:

Firstly, x⋆n(ν⋆) ∈ Int(Xn) since if xn = xmin,n or xn = xmax,n in any entry, then

∑ni=1 xi 6= c, which would be a constraint violation. Hence xmin,n < xn < xmax,n

entrywise, i.e., xn ∈ Int(Xn).Since x⋆n(ν

⋆) ∈ Int(Xn), we have by Fermat’s rule

dfndxn

(x⋆n(ν⋆)) + ν⋆ = 0, (20)

so

ν⋆ = −dfndxn

(x⋆n(ν⋆)). (21)

We do not know x⋆n(ν⋆), but we know the range, so we can make the bound

− maxxn∈Xn

dfndxn

≤ ν⋆ ≤ − minxn∈Xn

dfndxn

. (22)

This proves the Lemma. 2

Lemma 1 implies that Problem (12) can be constrained to the set V , without changingν⋆, which we employ in later sections.

The decomposition approach presented in this section involves a scatter/gather step ateach iteration k, where x⋆i (ν

(k)) is gathered from all subproblems in order to calculate theupdate (16), and the updated ν(k+1) is scattered for the following iteration, thus requiringa central coordination among subproblems. In [21] central coordination was avoided bypropagating the result from each subproblem through the network, whereby each unit canmake correct local updates of the Lagrange multipliers. In the following, we propose asimilar strategy of locally updating the Lagrange multipliers, but rather than propagatingvalues through the network, we suggest an alternative approach relying on averaging,inspired by the material in [8].

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3.2 Distributed Averaging

Eliminating central coordination across the portfolio essentially entails that even thoughall units are connected to the same power line as illustrated in Fig. 1, they can not nec-essarily communicate with all other units in the portfolio. In order for two units to com-municate and exchange data, there has to be a communication link between them, asillustrated in Fig. 2. Here, the location of each node may be translated to a geograph-ical location, and the limited communication paths can be translated to each unit onlycommunicating with other units that are within a reasonably distance.

x1

x2

x5

x3

x4

x6xn

Figure 2: Illustration of the portfolio. The dark arrows indicate communication paths,and the lighter buses indicates electricity paths.

The graph structure of the portfolio as illustrated by the communication links in Fig. 2will be used as a general framework for the remainder of this work. We therefore considerthe portfolio as a simple, undirected and connected graph. The graph consists of n ∈ N

nodes representing the individual units of the portfolio, and m ∈ N links representingcommunication paths. The communication paths should not be confused with the powerlines illustrated in Fig. 1.

Let A = [aij ] ∈ 0, 1n×n be the graph adjacency matrix where

aij =

1, nodes i and j are linked

0, otherwise,(23)

meaning that nodes i and j may exchange information only if aij = 1. Further, definethe neighbors of the node i as

Ni = j 6= i|aij = 1. (24)

Let x(ν(k)) ∈ RN denote the average over all the results from the subproblems atiteration k of the dual decomposition. It is given by

x(ν(k)) =1

n

n∑

i=1

x⋆i (ν(k)). (25)

By this, we see that (16) can be formulated with

∇q(ν(k)) = nx(ν(k))− c, (26)

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implying that each node does not need the specific values of x⋆i (ν(k)) for i = 1, . . . , n; it

suffices to know the average which can be obtained in a distributed fashion. The updateof the Lagrange multipliers can then be performed locally at each node, rather than at acentral coordinator. To obtain the average locally, we define the sequence x(k,h)i ∈ RN ,

where x(k,1)i = x⋆i (ν(k)), and

x(k,h+1)i = wiix

(k,h)i +

∑

j∈Ni

wij x(k,h)j , i = 1, . . . , n, (27)

where we want x(k,h)i → x(ν(k)) as h → ∞ for i = 1, . . . , n, such that each node canlocally calculate the gradient as in (26). The update (27) is a linear combination of the subproblem solution of node i and the nodes j ∈ Ni, entailing that only local information isused by any node.

In order to formally analyze convergence of (27) towards the average, we assignwij =0, if j /∈ Ni and rewrite (27) for all i, as a matrix-matrix multiplication

X(k,h+1) =WX(k,h), i = 1, . . . , n, (28)

where W ∈ Rn×n, W = [wij ], is a weight matrix reflecting the structure of the graph. Itwill be constructed explicitly in the following. The matrix X ∈ Rn×N is expressed as

X(k,h) =[

x(k,h)1 , . . . , x

(k,h)n

]T

. (29)

The necessary and sufficient conditions for

X(k,h) →[

x(ν(k)), . . . , x(ν(k))]T, for h→ ∞, (30)

are

W1n = 1n, 1TnW = 1

Tn , lim

h→∞W h = J and J =

1n1Tn

n, (31)

where 1n = [1, . . . , 1]T ∈ Rn, which is shown by [8]. There it is further shown that (31)is equivalent to ρ (W − J) < 1 i.e., the spectral radius of W −J should be less than one.We design the weight matrix from the Laplacian matrix of the grid L ∈ Rn×n, defined by

L = D − A. (32)

Here, D ∈ Rn×n is a diagonal matrix where the diagonal entries dii equals the degreeof node i, i.e., the number of links connected to it, and A is the adjacency matrix definedin (23). The Laplacian matrix is symmetric, and positive semi-definite. The smallesteigenvalue of L is zero, and the multiplicity of this will be larger than one, only if thegraph is disconnected, which is not the case by our prior assumptions. Inspired by [8],we design W as

W = I − ξL, (33)

where ξ ∈ R and I ∈ Rn×n is the identity. This design entails that

wij =

1− diiξ, i = j,

ξ, j ∈ Ni,

0, otherwise,

(34)

which yields W =WT , andΛ(W ) = 1− ξΛ(L), (35)

where Λ(·) denotes spectrum. The following further links the eigenvalues of W to thoseof W − J .

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Lemma 2. Take W ∈ Rn×n symmetric, obeying (31), and take J = 1n1Tn/n. Then

Λ(W − J) = Λ(W )\1 ∪ 0, (36)

where Λ(·) denotes the set of unordered eigenvalues.

Proof:First see that 1n is an eigenvector of (W − J), with an eigenvalue of 0, simply by

(W − J)1n =W1n − J1n = 1n − 1n = 0n. (37)

As (W − J) is symmetric, any remaining eigenvector v must be orthogonal to 1n, i.e.,v ∈ 1⊥

n . Notice that the range R of the linear operator J is a one dimensional subspacespanned by 1n, i.e.,

R (J) = span1n. (38)

Hence, for any remaining eigenvector v ∈ span1n⊥, we have

(W − J) v =Wv = λwv, (39)

where λw is an eigenvalue for W , and thus, given the limit in (31)

Λ (W − J) = Λ(W )\1 ∪ 0. (40)

2

As bothW and J are symmetric, ρ(W −J) = ‖W −J‖ [22], so Lemma 2 shows thatthe requirement ρ(W − J) < 1, is equivalent to max(|Λ(W )|\1) < 1. Let λ1(L) ≥λ2(L) ≥ · · · ≥ λn(L) = 0 be the eigenvalues of L, and recall that Λ(W ) = 1− ξΛ(L).Then the design of W in (33) entails that convergence of (28) is guaranteed for

0 < ξ <2

λ1(L), (41)

which are also the limits described by [8].

3.3 Applying Distributed Averaging

Designing the weight matrix as in (33) implies that each node can perform an individualupdate of (16) which can be written on the form

ν(k+1)i = [ν

(k)i + α(k)∇q(ν

(k)i )]V , (42)

where ν(k)i is the local copy of the Lagrange multipliers, and ∇q(ν(k)i ) is a local copy of

the gradient at node i. The notation [·]V denotes the projection

[ν]V = argminν∈V

‖ν − ν‖, (43)

where V is the set defined by Lemma 1. Obviously, when the distributed averaging pro-vides the precise value for the average, then

ν(k)i = ν

(k)j = ν(k), and ∇q(ν(k)i ) = ∇q(ν(k)j ) = ∇q(ν(k)), ∀i, j, k > 0, (44)

where ν(k) and ∇q(ν(k)) refer to the Lagrange multipliers and gradient, had there been acentral coordinator for the portfolio.

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4 Fault tolerant implementation

Note that we have introduced two different iterative processes. In the outer iterations,with index k, the subproblems are solved and the Lagrange multipliers are updated inorder to reach a global optimum. The gradient for updating the Lagrange multipliers isfound in the inner iterations, indexed by h, where the distributed averaging is performed.We can clarify this approach as follows:

Outer loop:

1. Initialize ν(1)i , i = 1, . . . , n and k = 1;

2. Solve subproblem x⋆i (ν(k)) = argminxi∈Xi

(fi(xi) + xTi ν(k)), for all i;

3. Continue with Inner loop;

Inner loop:

a Initialize h = 1, x(k,h)i = x⋆i (ν(k)) for all i;

b Perform update across neighborhood as shown in (27), for all i;

c If averaging process has not converged, increase h, and repeatStep 3b;

d Otherwise, continue with outer loop;

4. Perform local Lagrange update as in (42), where the gradient is given by (26);

5. Increase k and repeat from Step 2;

As the outline with our approach has now been set up, we will continue to present ourmain contributions in the following sections.


As mentioned in Section 1, it is possible that units temporarily may leave the portfolioon account of overload, repairs, maintenance, upgrades etc. Similarly, a unit may rejointhe portfolio at a later time. It is also possible that individual communication links mayfail, or additional links may be established during runtime. This illustrates that it mustbe possible to introduce and remove portfolio units during runtime, without affecting theconvergence or introducing the need for major network wide changes of the optimizationalgorithm previously derived. In the following, we analyze the modifications of the port-folio caused by these faults. We analyze the effect for both parts of our approach, i.e., thedistributed averaging, and the dual decomposition, and further show how modificationscan be accommodated. We remark that the following focuses solely on faults and errorsthat require units to leave the portfolio (and possibly subsequently rejoin), i.e., we do notconsider gradual degradation or faults resulting in partial performance of individual units.Such faults could be included in our framework, for instance by updating the constraintset or cost function at affected units as faults emerge. However, this is outside the scopeof this paper.

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4.1 Modifications to decomposition

Modifications where units leave or rejoin the portfolio implies that the number of unitschanges from n to n. This affects the gradient for updating the Lagrange multipliers. Thegradient changes accordingly:

∇q(ν(k)) = nx(ν(k))− c. (45)

where q denotes the dual (cf., (12)), after the portfolio modifications. Hence, changingthe number of units in the portfolio, does not alter the decomposition algorithm. It simplychanges the gradient, since the number of generators, and thereby the number of subprob-lems, has changed. However as indicated each production unit requires knowledge aboutn, so the number of units in the portfolio needs to be network wide available informationat all times. We elaborate on this in Section 4.4 below.

4.2 Modifications to distributed averaging

Modifying the portfolio translates to modifying the graph by either adding or removingnodes and links. This entails that the Laplacian is modified. We describe any modificationby the mapping H : Rn×n → Rn×n, where n and n are the number of units in the port-folio before and after the modifications. Using this mapping, L = H(L) is the Laplacianof the modified graph. In order to define H , the following describes the conventions wefollow when modifying the graph.

All nodes in the graph are numbered 1, . . . , n. Let oi(L) be the node labeled i in thegraph defined by L. If node i is removed, then all nodes will be relabeled such that

oj(H(L)) =

oj(L), j < i

oj+1(L), j ≥ i(46)

for j ∈ 1, . . . , n− 1.Conversely, when adding nodes to the graph, i.e. n > n, the added nodes are num-

bered consecutively from n + 1, . . . , n. In order to add nodes, or add and remove links,we define C = [cij ] ∈ Rn×n as

cij =

1, create link between node i and j

−1, remove link between node i and j

0, otherwise

(47)

where the numbering of nodes, refer to H(L), i.e, the graph after modifications. Ourconvention further assumes that you cannot create links that already exist, or removelinks that do not exist. We can now define

H(L) = V TLV + P, (48)

where V = [vij ] ∈ Rn×n and P = [pij ] ∈ Rn×n are defined by

vij =

1, oi(L) = oj(H(L)),

0, otherwise,pij =

−cij , i 6= j,n∑

j=1

cij , otherwise.(49)

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4.3 Fault Tolerant Design

By designing the weight matrix after the graph modifications, similar to earlier

W = I − ξH(L), for 0 < ξ <2

λ1(H(L)), (50)

where λ1(H(L)) denotes the maximum eigenvalue after the modification, then the dis-tributed averaging converges, as it obeys requirement (41). However if ξ 6= ξ, the entireweight matrix will have changed. This is undesirable since the entire network will be ef-fected by the modifications, as we would need to update all the averaging weights acrossthe network, in order to ensure that our decentralized optimization would converge. Wetherefore need to ensure that ξ = ξ, for all allowed modifications, and that this wouldstill cause our algorithm to converge. To this end we define dmax as the upper limit ofthe number of links any node in our graph may support, such that all nodes must have adegree strictly less than dmax. The work presented in [23] then states

λ1(H(L)) < 2dmax. (51)

So, by letting

ξ = ξ =1

dmax, (52)

the distributed averaging is guaranteed to converge for any modifications to the originalgraph, as long as the modified graph is still simple, connected and further obeys themaximum degree limit. Further, any modifications to the graph will only affect the nodesthat are directly affected by the modification, and not the rest of the graph. This is becausethe weights of the averaging is now only dependent on each node’s local degree, whichwe shall illustrate through the examples in Section 6.

4.4 Fault detection latency

As explained above, the outlined strategy is robust against changes in the portfolio, pro-vided that the local weights used in the averaging algorithm are updated accordingly, andfurther that the updated number of portfolio units is distributed across the graph. Thisrequires an appropriate fault detection algorithm, which can determine the new graphtopology and, importantly, the new number of nodes. Such algorithms are typically exe-cuted intermittently, giving rise to a certain detection latency, [24, 25].

To examine the effect of detection latency on the algorithm outlined previously, weconsider a situation where a portfolio of n units, at some point experiences node failures,such that the number of nodes is reduced to n < n. Without loss of generality, thefollowing discussion is conducted with n = n− 1, i.e., loss of a single unit, and adjacentcommunication lines. The neighbors to the failing unit may soon realize that no dataexchange takes place, and may thus perform the required updates of their local weightswith respect to the averaging process, according to the rules described in Section 4.3.However, at this point it is uncertain whether it is simply a communication line or in factan entire node that is failing, so no other changes are made.

Let W be the revised weight matrix after the local changes performed on account ofthe new communication structure. Similarly, let x(ν(k)) be the average of the remainingn nodes. Since the neighbors of the failing node updates their weights employed for

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averaging, the averaging process does in fact converge to correct value x(ν(k)), eventhough the failure has not yet been detected.

After the failure occurs, but before it is detected, the local gradient used to update theLagrange multipliers is thus

∇q(ν(k)) = nx(ν(k))− c = nx(ν(k))− c+ x(ν(k))︸︷︷︸

error

, (53)

where the error term is on account of the local gradient being calculated with a wrongnumber of portfolio participants.

From the constraints described in Section 2, it is clear that the error x(ν(k)) is boundedfor all k. As we shall elaborate in the following section, updating the local Lagrangemultipliers with a gradient subject to a bounded error, still leads to convergence, althoughto a sub-optimum of the dual problem (12). Whenever the failure is detected, i.e. thecorrect number of units is employed in the gradient, convergence towards the optimum isagain achieved. We shall illustrate this further with a numerical example in Section 6.3.

5 Early Termination

In Section 3, two iteration loops were discussed: The inner used for the computationof the average of solutions from the individual subproblems, and the outer for updatingthe Lagrange variables locally. As convergence of the distributed averaging in the innerloop is achieved only at infinity, we are forced to terminate the averaging process at somepoint, causing inaccuracies to the local update of the Lagrange multipliers. There are twosources of inaccuracies: Averaging errors, and errors caused by ǫ−gradient updates. Inthe following we first elaborate on the origin of these errors, and analyze how they affectthe convergence of the distributed optimization.

On account of the inaccuracies it follows that in general ν(k)i 6= ν(k)j for i 6= j. The

inaccuracies cause the local Lagrange variables to be updated using approximate ratherthan accurate gradients, denoted g(h,k)i , such that (42) becomes

ν(k+1)i = [ν

(k)i + α(k)g

(k,h(k))i ]V , (54)

where the approximate gradient g(k,h(k))i = g(k,h(k))i + r

(k,h(k))i consists of an added

error term r(k,h(k))i and an ǫ-gradient g(k,h(k))i ∈ ∇ǫ(k,h(k))q(ν(k,h)). As indicated in the

notation, both the ǫ-gradient and the error term are affected by the number of performedaveraging steps h(k), at each iteration k. We elaborate on this in the following, whichrelies on some of the material of Appendix A .

5.1 Averaging error

The first source of error comes directly from the early termination of the averaging pro-cess. Let V (k) be the set of all the local Lagrange variables at outer iteration k, i.e.,

V (k) = ν(k)1 , . . . , ν(k)n . (55)

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5 Early Termination

The solution of each subproblem x⋆i (ν(k)i ) = argminxi∈Xi

(fi(xi)+xTi ν

(k)i ) depends on

local Lagrange variables and thus, the average value of all subproblem solutions, dependson the whole set V (k). We denote this by

x(V (k)) =1

n

n∑

i=1

x⋆i (ν(k)i ). (56)

When stopping the averaging process in Eq. (27) after h(k) iterations, the approximateaverage at any node i, can be described by

x(k,h(k))i = x(V (k)) + γ

(k,h(k))i , i ∈ 1, . . . , n, (57)

where γi ∈ RN is the inaccuracy between the local estimate and the true average at thecurrent iteration. This inaccuracy is caused by terminating the averaging process after afinite number of iterations, whereby it is not converged to the true average. The update ofthe local Lagrange variables at each node then follows

ν(k+1)i = [ν

(k)i + α(k)g

(k,h(k))i ]V (58)

with

g(k,h(k))i = nx

(k,h(k))i −c = n(x(V (k))+γ

(k,h(k))i )−c = nx(V (k))−c+r

(k,h(k))i , (59)

with the error r(k,h(k))i = nγ(k,h(k))i .

5.2 Updates with ǫ-gradients

The second source of error comes from the fact that ν(k)i 6= ν(k)j for i 6= j. As the update

in (54) for any node i depends on the whole set V (k) and not just ν(k)i , we have in generalthat,

g(k,h(k))i = nx(V (k))− c 6= ∇q(ν

(k)i ), (60)

which would have been the case if ν(k)i = ν(k)j for all i, j, k. By Lemma 7 and Lemma 8,

g(h,k)i is, however, an ǫ−gradient, such that

g(k,h(k))i ∈ ∇

ǫ(k,h(k))i

q(ν(k)i ), (61)

forǫ(k,h(k))i =

∑

j 6=i

(

(ν(k)i − ν

(k)j )T ν

(k)j + qj(ν

(k)j )− qj(ν

(k)i ))

. (62)

for all i and j.This leads us to the following proposition, which comprises the main result of this

section. We will employ the notation

Vψ = ν ∈ V|q(ν) ≥ ψ, V⋆ = ν ∈ V|q(ν) = q⋆, and dist(ν,V⋆) = minν⋆∈V⋆

‖ν − ν⋆‖,

(63)

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where V is defined by Lemma 1, whereby it is convex and compact. Given compactnessof Xi, i ∈ S, x(V (k)) is bounded, and further, given compactness of V , we can define twoconstants c > 0 and d > 0 such that

‖g(k,h(k))i ‖ ≤ c and dist(ν(k)i ,V⋆) ≤ d, ∀i, k. (64)

Proposition 3 (Convergence of Distributed Portfolio Optimization). For the problem de-fined by (11) through (16), let the local Lagrange multipliers be updated as

ν(k+1)i = [ν

(k)i + α(k)g

(k,h(k))i ]V , with g

(k,h(k))i = g

(k,h(k))i + r

(k,h(k))i , i = 1, . . . , n,

(65)

Where r(k,h(k))i ∈ RN is an error as described by (59), g(k,h(k))i ∈ RN is an ǫ-gradient asdescribed by (61) and V is compact, convex. Further, let α(k) > 0 satisfy

limk→∞

α(k) = 0, and∞∑

k=1

α(k) = ∞. (66)

(a)

There exist an upper bound r on the error, i.e., ‖r(k,h(k))i ‖ ≤ r for all i, h, k. Further, forany fixed pair r, ǫ > 0, there exists η, δ ∈ (0, 1), such that if

h(k) > h′(k) =log(

ηδk

2α(k)r

)

log(‖W − J‖), ∀k (67)

then‖r

(k,h(k))i ‖ ≤ r and ǫ

(k,h(k))i ≤ ǫ ∀i, k. (68)

(b)

These bounds guarantee convergence of the distributed portfolio optimization, in thesense that for any ψ < q⋆ − ǫ− rd

dist(ν(k)i ,Vψ) → 0 for k → ∞. (69)

Proof:

(a)

Existence of r such that ‖r(k,h)i ‖ ≤ r, for all i, h, k, is implied by Theorem 5. Further, byTheorem 5, for all r > 0 and for

h > h′(r) =log(r/r)

log(‖W − J‖), (70)

we have ‖r(k,h)i ‖ ≤ r for all i and k.Corollary 9 implies that for all ǫ > 0, there exists σ > 0 such that

‖ν(k)i − ν

(k)j ‖ < σ ⇒ ǫ

(h,k)i < ǫ, ∀i, k, (71)

and Proposition 6 entails that letting σ = η/(1 − δ) for some η, δ ∈ (0, 1) the bound

‖ν(k)i − ν

(k)j ‖ < σ is obtained when ‖r

(h,k)i ‖ ≤ ηδk/(2α(k)) for all i and k. Combining

this with (70), we obtain (67) as desired.

180

5 Early Termination

(b)Following similar calculations as presented in [26], (65) can be shown to imply

dist(ν(k+1)i ,V⋆)2 ≤ dist(ν(k)i ,V⋆)2 − 2α(k)(q⋆ − q(ν

(k)i )− ǫ− rd) + (α(k))2(c+ r)2. (72)

Provided that

0 < α(k) <2(q⋆ − q(ν

(k)i )− ǫ− rd)

(c+ r)2, ∀i, k > 0 (73)

Eq. (72) yields

dist(ν(k+1)i ,V⋆)2 < dist(ν(k)i ,V⋆)2, ∀i, k. (74)

Since the step size is diminishing, there exists k′ such that for all ψ < q⋆ − ǫ− rd

0 < α(k) <2(q⋆ − ψ − ǫ− rd)

(c+ r)2, ∀k > k′, (75)

which implies that (73) is satisfied for all ν(k)i /∈ Vψ, k > k′ and for all i. However, for

ν(k)i ∈ Vψ, we cannot guarantee that (73) is satisfied, so there might be some k where

dist(ν(k)i ,Vψ) = 0, and dist(ν(k+1)i ,Vψ) > 0. (76)

In that case we have from (72)

dist(ν(k+1)i ,V⋆)2 − dist(ν(k)i ,V⋆)2 ≤ −2α(k)(q⋆ − q(ν

(k)i )− ǫ− rd) + (α(k))2(c+ r)2

which implies that if dist(ν(k)i ,Vψ) = 0, then

dist(ν(k+1)i ,Vψ)

2 ≤ 2α(k)(q(ν(k)i ) + ǫ+ rd − q⋆) + (α(k))2(c+ r)2

≤ 2α(k)(ǫ+ rd) + (α(k))2(c+ r)2. ∀i (77)

Theorem 4 presented in Appendix A implies that for any ψ < q⋆ − ǫ − rd, and anyν(k)i /∈ Vψ, there exists ki > k, i ∈ 1, . . . , n, where dist(ν(ki)i ,Vψ) = 0, for all i. Letki be the first instance where

dist(ν(ki)i ,Vψ) = 0, (78)

for all i. Then for k ≥ ki, dist(ν(k+1)i ,Vψ) is bounded by (77). As α(k) → 0 for k → ∞,

we have dist(ν(k+1)i ,Vψ) → 0 for all i. 2

The above illustrates that global optimality of the distributed solution cannot be ob-tained. This is due to numerical drift between the iterative processes, running locallyat each portfolio unit. However, the algorithm converges to an arbitrarily good sub-optimum, by running a sufficiently large number of averaging steps, before letting eachunit update its local Lagrange multipliers.

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6 Numerical Case Study

In this section we present a few numerical examples to illustrate the method presentedthroughout the paper. We will illustrate both how our approach to distributed portfoliooptimization converges, and further that convergence is maintained if the portfolio is mod-ified, without modifying the optimization strategy. This shows that the approach outlinedin this paper, is robust against changes in the portfolio topology.

Consider a portfolio consisting of n = 16 units, arranged as the graph presentedin Fig. 3 with m = 24 links. The meshed structure of the graph is chosen purely forpresentational reasons. We could have used any layout that was simple, connected andundirected, which are the structual prerequisites for the averaging process to converge.Each node is numbered 1−16, as indicated in the figure. The averaging scheme presentedin (27) entails that each node assigns a weight to itself, and a weight to each of the linksconnected to it. We refer to these as node weights, and link weights. We define themaximum node degree dmax = 7, meaning that we allow any modification to the graph inFig. 3, provided that it remains simple, connected, and all nodes have strictly less than 7links connected.

Employing the weight design from (33) with ξ = 1/dmax, we obey the robust designlimits of (52). This entails that all node weights in the weight matrix becomes wii =1 − di/dmax, as indicated next to each node in Fig. 3. Further, all link weights becomeswij = 1/dmax.

16: 0.7115: 0.5714: 0.5713: 0.71

12: 0.5711: 0.4310: 0.439: 0.57

8: 0.577: 0.436: 0.435: 0.57

4: 0.713: 0.572: 0.571: 0.71

Figure 3: Graph presentation of our portfolio of 16 nodes (Dots) and 24 links. Thenumbering of the portfolio units, is indicated next to each node, along with the nodeweight. Link weights are not illustrated, as they are all equal: wij = 1/dmax = 0.14.

In the examples presented below, we have used S1 = 1, S2 = 2, 3, 4, 5, S3 =7, . . . , 16, S4 = 6, i.e. we have included the external actors as a node, since they areregarded as part of the portfolio.

The following presents the optimization results for a period of N = 30 time steps,with t0 = 1. We have introduced the power demand

c(t) =

50, t < 15,

130, t ≥ 15,(79)

182


where we enforce a known step in the demand at t = 15, which for instance could be onaccount of a wind farm disconnecting from the grid, on account of wind conditions. Thestep in demand would then be the power output initially expected from the wind farm,which now must be covered elsewhere. We have introduced the capacity and slew ratelimits

xmin,i =

0, i ∈ S1 ∪ S2

−5, i ∈ S3

−160 i ∈ S4

, xmax,i =

70, i ∈ S1

35, i ∈ S2

0, i ∈ S3

180 i ∈ S4

, dxmax,i =

3, i ∈ S1

10, i ∈ S2

,

(80)where we have omitted slew rate constraints on consumers and external actors. For allconsumers, we have introduced the discomfort constraints

|Tsp(t)− Ti(t)| ≤ 2, i ∈ S3 with Tsp(t) = 20, ∀t. (81)

The cost functions for each portfolio unit was described in Section 2. Let s = (1, 3, 5, 10),the weight matrices employed in the case studies are

Gi = sjI, ∀ i ∈ Sj , j ∈ 1, 2, 3, 4, (82)

where I ∈ RN×N is the identity matrix. The diagonal entries in Gi represents themarginal operating cost for each portfolio unit, at each time instance. We have usedconstant marginal costs for all generators, but distinct marginal costs for each generatortype. Time-varying costs could be introduced by inserting different values on the diagonalof the weight matrices, with the restriction that Gi must be positive definite.

6.1 Distributed Portfolio Optimization

By running the optimization strategy described in Section 3 and 5 we obtain the resultspresented in Fig. 4 and Fig. 5, where Fig. 4 illustrates the optimized utilization of the port-folio with respect to production and consumption, and Fig. 5 shows the demand trackingof the portfolio, and temperature development for the households. Fig. 4 presents both theoptimal portfolio utilization as found when solving the problem centralized, as well as theportfolio utilization obtained using the distributed approach described in this paper. Asevident, the two cases are indistinguishable, illustrating that the distributed optimizationhas indeed converged close to the global optimum. Fig. 4(Left) illustrates how the powerfrom type 1, type 2 and external actors contributes to the overall portfolio production. Weremark that the entire portfolio is plotted in Fig. 4, however as we have used the samevalues for characterizing units of the same type, they all behave in the same way, andplots appear as if there is only a single unit of each type.

In Fig. 4(Left) we see that since the step change of the demand is known in advance,the cheap but slow type 1 plant starts ramping up production early. In order to maintainthe power balance while the type 1 plant ramps up, consumption is increased by theconsumers, as seen in Fig. 4(Right). The temperature is thereby increased and a thermalstorage is created, cf. Fig. 5(Right). In this way, when the demand increase occurs, allconsumers can decrease their consumption almost instantly, thereby reducing the effectiveincrease of the demand.

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Paper D

xi(t)

t

5 10 15 20 25 30

0

20

40

60

xi(t)

t

5 10 15 20 25 30

0.5

1

1.5

2

2.5

Figure 4: Left: The power produced by each unit in the portfolio in the centralized(Dashed), and decentralized case (Solid). The four generator classes are represented:Type 1 (Dots), type 2 (Asterisk) and external actors (Diamond). Right: Controlledconsumption found centralized (Solid) and decentralized (Plus). The horizontal dashedline indicates the baseline consumption required at temperature 20C, in order to maintainthis temperature.

The capacity of the portfolio allows it to cover the power demand without using theexternal actors, who have a high marginal cost. However, on account of the nonlinear costfunction of all portfolio participants, it is still beneficial for the power producer to use acombination of its own generators and external actors.

The overall demand tracking is presented in Fig. 5(Left), where the dashed line in-dicates the power demand, which is satisfied by the portfolio. However as the portfolioadjusts both production and consumption, we further present the actual power productionas illustrated by the solid line in Fig. 5(Left). The discrepancies between the two curves iscaused by the consumption being either increased or decreased with respect to the base-line consumption. This discrepancy illustrates the flexibility obtained when includingcontrolled consumers for balancing. Adjusting consumption entails that we can increaseconsumption at one time, thereby building up (charging) a storage, and then lower con-sumption at a later time, when employing (discharging) the storage. Charging and dis-charging the thermal storage explains the periods where the dashed line is above/belowthe solid line in Fig. 5(Left), respectively. The process of charging and discharging the

∑n i=

1xi(t)

t

5 10 15 20 25 30

40

60

80

100

T(t)

t

5 10 15 20 25 30

18

20

22

Figure 5: Left: The power demand (Solid), and the reduced demand obtained by uti-lizing flexible consumption (Dotted Dashed). Right: The temperature (Solid) of theconsumers, and the allowed variation (Dashed).

184


thermal storage is also seen in Fig. 5(Right), illustrating household temperature. Beforethe demand increases, the household temperature is increased. As we saw in Fig. 5 thiswas in order to allow the slow type 1 plant to ramp up production ahead of the demandincrease, while still maintaining balance. At the instant when the demand increases, theconsumers effectively shut off their consumption, in order to even out the demand in-crease, resulting in a dropping temperature.

6.2 Fault Tolerant Portfolio Modifications

We now present a numerical example, to show the plug-and-play functionality of the opti-mization approach outlined in this paper. First observe the modified graph of Fig. 6(Left).Compared to the original graph in Fig. 3 we have added 4 nodes, each with 3 links, whilewe have removed a single node, along with its links. This leaves the graph with 19 nodesand 33 links.

19: 0.5718: 0.57

17: 0.57

16: 0.57

15: 0.5714: 0.2913: 0.2912: 0.43

11: 0.5710: 0.439: 0.438: 0.29

7: 0.716: 0.435: 0.434: 0.29

3: 0.712: 0.571: 0.57

maxt |∑

i∈S xi(t) − c(t)|‖x⋆i (ν

(k))− x⋆‖

Con

verg

ence

Iterations k

100 102

100

101

102

Figure 6: Left: Graph of the modified portfolio. Link weights are not illustrated, as theyare all equal: wij = 1/dmax = 0.14. Right: Convergence of the distributed optimization,after the graph is modified.

We observe in Fig. 6(Left) that only nodes directly affected by the portfolio modifica-tion have different node weights compared to Fig. 3. To summarize, wii = 1 − di/dmax,therefore the node weights only change if di does. The link weights are still equal, andthey still have the value wij = 1/dmax. The changes introduced by the portfolio modifi-cation are thus all local, as desired.

Running our proposed algorithm in the same fashion as previously, only modifyingthe indicated weights in the graph (and the number of portfolio participants), we obtainthe convergence presented in Fig. 6(Right). The solid line indicates the distance betweenthe decentralized solution and the global optimum. We can find this distance in our ex-ample, because the problem is small enough to solve in a centralized fashion, revealingthe global optimum. In practice, this is not an obtainable measure of convergence, andit is only included for illustrative purposes. On the other hand, the dashed line indicatesthe power deficit, measured as the difference between the power demand, and the output

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Paper D

of the portfolio for each iteration of the distributed optimization. This is a more practicalconvergence criteria.

As evident from the figure, when modifying the portfolio, our distributed optimizationconverges in a similar manner as for the original portfolio, even though only local changesare made to the algorithm on account of the portfolio modifications.

6.3 Fault detection latency

The final example presented below supports our discussion from Section 4.4 concerningthe effect of fault detection latency. We return to the graph presented in Fig. 3, and re-peat the example described in Section 6.1. However, during the optimization process,we remove a node and its appertaining links from the graph, resulting in the portfolio ofFig. 7(Left). Initially, the node is removed without updating the number of units in theportfolio employed by the local update of Lagrange multipliers for each unit. The neigh-boring units just assume that a link has been lost and update their respective averagingweights accordingly. At a later time, we assume that a fault detection procedure has cer-tified that a node has indeed failed, and that an updated number of portfolio units havebeen propagated to all units. Each remaining unit can thus again perform local updatesusing the correct number of units.

In Fig. 7(Right) the numeric results are presented for the case when the failure isintroduced after k = 75 iterations, but it is not detected until k = 150 iterations. Thefigure presents convergence towards the centralized optimal Lagrange multipliers for arandomly chosen unit. Since the simulation is conducted with a diminishing step size, wehave presented both the case where the diminishing step-size is maintained throughoutthe simulation, as well as the case where the step-size sequence is reset upon detectingthe fault.

15: 0.7114: 0.5713: 0.5712: 0.71

11: 0.5710: 0.439: 0.438: 0.57

7: 0.716: 0.435: 0.434: 0.57

3: 0.712: 0.571: 0.71

Step is resetStep not reset

‖ν(k

)−

ν⋆‖

Itterations k

0 100 200 30010−1

100

101

102

103

Figure 7: Left: Graph after a failure occurs at a single node. Right: Convergence of thedistributed optimization in the three cases: 1. No fault yet occurred, 2. Undetected faultoccurred, 3. Fault detected and corrected. The dashed vertical lines indicates transitionbetween the three cases.

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7 Conclusions and Future Work

As evident from the figure, the initial convergence is abruptly stopped when the failureoccurs. The sudden change in distance to optimum is because the number of units havechanged, so the optimum has changed accordingly. During the period where a faultynumber of units is used for updating, the distance to the optimum reduces only slightly.This was expected, since our discussion from Section 4.4 suggests that convergence isobtained towards a sub-optimum. However, what should be noticed is that the processdoes not diverge even though a wrong number of units is used for calculating the gradientused in updates. This was also argued in Section 4.4.

After the fault is detected, the convergence improves; however, as evident, conver-gence is considerably better when the diminishing step-size sequence is reset upon faultdetection.

7 Conclusions and Future Work

This work has presented an approach to optimal balancing of an electrical power defi-ciency across a portfolio of generators. We have presented the balancing in an uncoordi-nated, distributed fashion, where we have shown that a solution can be found arbitrarilyclose to the global optimum, when generators are allowed to communicate only with alimited number of neighbors. Throughout this work, we have shown that the proposedoptimization strategy can be designed to be robust against changes in the topology of thedistributed portfolio e.g., introducing or removing generators. This means that our ap-proach allows plug-and-play-like modifications of the portfolio structure without alteringthe optimization strategy, or convergence properties.

The presented numerical examples have illustrated how the strategy described herecan be employed in order to utilize the flexible consumption from households, such asto shape the power demand. We employed a time-invariant cost of employing consumerflexibility, however time-varying cost could easily be introduced. This could reflect thatthe incurred consumer discomfort is more costly during specific periods of the day. Sim-ilarly, the price of producing power from any other portfolio participant could be madetime variant.

Two important tasks have been disregarded in this work. The first is the task of syn-chronizing units in the portfolio, such that the iterative processes are executed in a syn-chronous fashion across the graph, in order for the updates and exchange of data to beperformed properly. This work assumed that an underlying synchronization strategy wasavailable. Secondly, an in-depth analysis of a distributed termination criterion should beinvestigated, that is, how to reach consensus on terminating the iterative optimization pro-cesses. Additional extensions of our work could include concerns towards grid capacityand congestion management.

A Appendix: Supporting Results

The following outlines the theorems, lemmas and corollaries referred throughout the pa-per.

Theorem 4 (Nedic and Bertsekas [26]). Let V ⊂ RN be nonempty, compact and convex.Let q : V → R be a concave function and denote q⋆ = maxν∈V q(ν) and V⋆ = ν ∈

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Paper D

V|q(ν) = q⋆. Compactness of V ensures that it is bounded, hence there is d > 0 suchthat dist(ν,V⋆) = min

ν⋆∈V⋆‖ν − ν⋆‖ ≤ d, ∀ν ∈ V .

Letν(k+1) = [ν(k) + α(k)g(k)]V , (83)

where α(k) > 0 obeys

limk→∞

α(k) = 0, and∞∑

k=1

α(k) = ∞, (84)

and the vector g(k) ∈ RN is an approximate subgradient:

g(k) = g(k) + r(k) (85)

where r(k) ∈ RN is an error, and g(k) ∈ ∂ǫ(k)q(ν(k)), such that for some ǫ(k)

q(w) ≤ q(ν(k)) + g(k)T (w − ν(k)) + ǫ(k), ∀w. (86)

Assume also that there exists r and ǫ such that

‖r(k)‖ ≤ r, ∀k ≥ 0, and lim supk→∞

ǫ(k) = ǫ. (87)

Thenlim supk→∞

q(ν(k)) ≥ q⋆ − ǫ− rd. (88)

Proof:

See [26]. 2

Theorem 5. Let there be given xmin,i, xmax,i ∈ R, i ∈ S = 1, . . . , n and xi ∈ RN

satisfyingxmin,i ≤ xi ≤ xmax,i, ∀i ∈ S, (89)

as well as W ∈ RN×N satisfying W1n = 1n, 1TnW = 1Tn , limh→∞Wh = J where

J = 1n1Tn/n.

Define

x =1

n

∑

i∈S

xi, x(1)i = xi, X(1) =

[

x(1)1 , . . . , x

(1)n

]T

, X(h+1) =WX(h), h ≥ 1,

(90)where X(h) = [x

(h)1 , . . . , x

(h)n ]T .

There exists γ > 0 such that for all γ > 0, and for

h > h′(γ) =log(γ/γ)

log(‖W − J‖), then ‖x

(h)i − x‖ ≤ γ, ∀i (91)

Proof:

From the bounds on xi, we have the element wise bound

mini(xmin,i) ≤ x ≤ max

i(xmax,i). (92)

188


This gives the bounds

|x(1)i − x| ≤ max

i(xmax,i)−min

i(xmin,i), ∀i, (93)

which are also to be read element wise. By defining,

γ =

√

N(

maxi

(xmax,i)−mini(xmin,i)

)2

, (94)

we are guaranteed that ‖x(1)i − x‖2 ≤ γ for all i. In general, we have [22]

‖x(h+1)i − x‖2 ≤ ‖W − J‖‖x(h)

i − x‖2 ⇒ ‖x(h)i − x‖2 ≤ (‖W − J‖)hγ, (95)

for all i ∈ S, where the conditions on W entail that ‖W − J‖ < 1. Hence, for any fixed

γ > 0, ‖x(h(γ))i − x‖2 ≤ γ is achieved if

γ < (‖W − J‖)h(γ)γ ⇒ h(γ) >log(γ/γ)

log(‖W − J‖)= h′(γ), (96)

as was initially argued. 2

Proposition 6. Let there be given strictly convex functions fi : RN → R, compactconvex sets Xi for i ∈ S = 1, . . . , n, and vectors ν(1)1 = · · · = ν

(1)n ∈ RN , as

well as scalars α(k) > 0 and matrix W ∈ RN×N satisfying W1n = 1n, 1TnW =

1Tn , limh→∞Wh = J, J = (1n1Tn )/n. Define

x(k)i = arg inf

xi∈Xi

(fi(xi) + xTi ν(k)i ), x

(k,1)i = x

(k)i , i ∈ S , x(k) =

1

n

∑

i∈S

x(k)i , (97)

X(k,1) =[

x(k,1)1 , . . . , x

(k,1)n

]T

, X(k,h+1) =WX(k,h), h ≥ 1, (98)

where X(k,h) = [x(k,h)1 , . . . , x

(k,h)n ]T . For any fixed k, h, let γ(k,h)i = x

(k,h)i − xk and

ν(k+1)i = ν

(k)i +nα(k)x

(k,h)i = ν

(k)i +nα(k)(x(k)+γ

(k,h)i ) = ν

(k)i +nα(k)x(k)+α(k)r

(k,h)i ,(99)

for each i ∈ S, with r(k,h)i = nγ(h,k)i . Then for any σ > 0, there exist r ∈ R+ and

η, δ ∈ (0, 1) such that σ = η/(1− δ) and

h(k) > h′(k) =log(ηδk

α(k)r

)

log(‖W − J‖)⇒ ‖ν

(k)i − ν

(k)j ‖ ≤ σ, ∀i, j, k. (100)

Proof:

Starting from ν(1)i , we can write ν(k)i as

ν(k)i = ν

(1)i + n

k−1∑

l=1

α(l)x(l) +

k−1∑

l=1

α(l)r(l,h(l))i , (101)

For any i, j ∈ S, the accumulated difference is

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ν(k+1)i − ν

(k+1)j =

k∑

l=1

α(l)(r(l,h(l))i − r

(l,h(l))j ) ⇒

‖ν(k+1)i − ν

(k+1)j ‖ =

∣∣∣∣∣

∣∣∣∣∣

k∑

l=1


(l,h(l))j )

∣∣∣∣∣

∣∣∣∣∣.

Furthermore,∣∣∣∣∣

∣∣∣∣∣

k∑

l=1


(l,h(l))j )

∣∣∣∣∣

∣∣∣∣∣≤

k∑

l=1

α(l)‖r(l,h(l))i − r

(l,h(l))j ‖. (102)

Theorem 5 implies that there exist r > 0, such that for any fixed r/2 > 0 and

h > log(r/(2r))/ log(‖W − J‖) then ‖r(k,h)i ‖ ≤ r/2. From this follows that for any

r/2 > 0 we can achieve

‖r(k,h(k))j − r

(k,h(k))i ‖ ≤ ‖r

(k,h(k))j ‖+ ‖r

(k,h(k))i ‖ ≤ r ∀i, j, k, (103)

by picking h(k) sufficiently large. This implies that we can bound each term in thesummation above. For any σ > 0, pick η, δ ∈ (0, 1) such that σ = η/(1− δ), and pick

h(k) ≥ h′(k) =log(

ηδk

2α(k)r

)

log(‖W − J‖)⇒ ‖r

(k,h(k))j − r

(k,h(k))i ‖ ≤

ηδk

α(k). (104)

Therefore if h(k) > h′(k)

‖ν(k+1)j − ν

(k+1)i ‖ ≤

k∑

l=1

α(k)‖r(k,h(k))j − r

(k,h(k))i ‖ ≤

k∑

l=1

ηδk ≤η

1− δ= σ ∀i, j, k,

(105)as initially stated. 2

Lemma 7. Let qi : RN → R be concave for i ∈ 1, . . . , n, and define q(ν) =∑n

i=1 qi(ν). If gi ∈ ∂ǫiqi(ν), then

n∑

i=1

gi ∈ ∂ǫq(ν), for ǫ ≥n∑

i=1

ǫi. (106)

Proof:

Each gi obeys the ǫ-subdifferential inequality qi(ω) ≤ qi(ν) + gTi (ω − ν) + ǫi ∀ω, andby summing together for i = 1, . . . , n we obtain

n∑

i=1

qi(w) ≤n∑

i=1

(qi(ν) + gTi (w − ν) + ǫi), (107)

which is exactly the implication of (106), which therefore must hold. 2

190


Lemma 8. Let q : RN → R be a continuous, concave function, and let g ∈ ∂q(ν), where∂q(ν) denotes the sub differential such that

q(ω) ≤ q(ν) + gT (ω − ν), ∀ω, (108)

Then for any ν ∈ RN and ǫ ≥ ǫ(ν, ν) = q(ν) − q(ν) + gT (ν − ν), we have g ∈ ∂ǫq(ν),where ∂ǫq(ν) denotes ǫ-subdifferential.

Proof:

Fix ν ∈ RN . Then by adding and subtracting q(ν) from the right side of (108), we obtainfor all ω

q(ω) ≤ q(ν) + gT (ω − ν) + q(ν)− q(ν)

= q(ν) + gT (ω − ν) + gT (ν − ν) + q(ν)− q(ν)

= q(ν) + gT (ω − ν) + ǫ(ν, ν). (109)

Given (108) we see that ǫ(ν, ν) = q(ν) − q(ν) + gT (ν − ν) ≥ 0, and so, (109) entailsthat g ∈ ∂ǫq(ν) for ǫ ≥ ǫ(ν, ν). 2

Corollary 9. Consider q, g and ǫ as defined in Lemma 8. Then for any ǫ > 0, there existsσ > 0 such that

‖ν − ν‖ ≤ σ ⇒ ǫ(ν, ν) ≤ ǫ. (110)

Proof:

Continuity of q(ν) implies that for all µ > 0, there exists σ > 0 such that

‖ν − ν‖ ≤ σ ⇒ |q(ν) − q(ν)| ≤ µ. (111)

For any ǫ > 0, it is thus possible to pick µ, σ > 0 satisfying (111), such that µ+‖g‖σ < ǫ.We then have

ǫ(ν, ν) = q(ν)−q(ν)+gT (ν−ν) ≤ |q(ν)−q(ν)|+‖g‖‖ν−ν‖ ≤ µ+‖g‖σ < ǫ. (112)

2

Acknowledgment

This work is supported by the Southern Denmark Growth Forum and the European Re-gional Development Fund under the project ’Smart & Cool’.

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193

Paper E

Loss Minimization and Voltage Control in Smart Distribution Grid

Morten Juelsgaard, Christoffer Sloth, Rafael Wisniewski and Jayakrishnan Pillai

This work is to appear in:Proceedings of IFAC World Congress, August, 2014

Copyright c© IFACThe layout has been revised

1 Introduction

Abstract

This work presents a strategy for increasing the installation of electric vehiclesand solar panels in low-voltage grids, while obeying voltage variation constraints.Our approach employs minimization of active power losses for coordinating con-sumption and generation of power, as well as reactive power control to maintainsatisfactory grid operation. Numerical case studies illustrate how our approach cansignificantly increase installation of both electric vehicles and solar panels, whileavoiding unsatisfactory over- and under-voltages throughout the grid.

1 Introduction

The low-voltage distribution grid faces increasingly significant challenges, compared tothe traditional operation of the grid. These challenges emerge from an increasing load inthe grid, as well as increasing levels of power production at household level.

The increased load is caused by an enhanced use of electricity for instance for trans-portation or heating, i.e., energy consumption that has previously been accommodated byfossil fuels, as forecasted by the [Danish Energy Association and Energinet.dk, 2010],as well as the [International Energy Agency, 2011]. The challenges emerging from thislies in the risk of overloading the distribution grid, as well as increased distribution gridlosses, and increased risk of unsatisfactory power quality. Specifically, the current lackof charging rules or guidelines for electric vehicles (EVs), entails that the low-voltagedistribution grid is currently not suited for large scale implementation of these, due to therisk of grid-overload and unacceptable voltage drops, as shown by [Pillai et al., 2012].

The increased penetration of household power production stems from installations ofsolar panels, household wind turbines, etc. Introducing significant levels of local powerproduction, may challenge the unidirectional power flow paradigm, under which the gridhas been designed. If the power produced locally is not also consumed locally, the powerwill not flow exclusively from the grid towards the consumers. Rather, power will alsoflow in the converse direction. This carries the risk of over-voltages occurring throughoutthe grid.

The combined effect of increased load, and local production, can be illustrated withthe following abstract example: Imagine a low-voltage distribution grid, connected tothe medium voltage grid through a step-down transformer. The grid is designed with aline topology, that is a single feeder, without branches. The grid contains 50 consumers,where the 18 consumers closest to the transformer (indexed 1 − 18), each have an EVinstalled, and the 14 consumers furthest from the transformer (indexed 36 − 50) eachhave a photo-voltaic (PV) array installed.

Fig. 1 shows the voltage profile of the feeder at one fixed time instance where allEVs are charging, while all solar panels produce power. As illustrated in Fig. 1, thecharging of vehicles causes an unacceptable voltage drop in the initial part of the radial,where the EVs are connected. Meanwhile, all PV owners produce excess solar power,which is not consumed locally, resulting in a local voltage increase in the furthest partof the feeder, violating the maximum voltage limit. Thus, there is both massive over-and under-voltages at the same time throughout the feeder. In this work, we present animplementation strategy on how these issues may be reduced through optimization andcoordination.

197

Paper E

|u(h

)|[p

u]

Distance to transformer (h) [-]

0 10 20 30 40 50

0.9

1

1.1

1.2

Figure 1: Voltage in the connection point h, of each consumer throughout the feeder(dots), and allowed voltage variation (dashed).

Traditional measures for maintaining stable voltages in low-voltage (LV) grids, arebased on an assumption of uni-directional power flow, such that the voltage will dropalong the feeder. As bi-directional power flows become increasingly common, this willnot pertain to be the case, requiring revisions of the traditional control strategies as dis-cussed by [Ipakchi and Albuyeh, 2009]. Specifically, in the example presented above,standard voltage control strategies, such as transformer tap-changers for voltage control,might very well raise the voltage to an acceptable level in the initial part of the feeder;however, it would conversely worsen the over-voltage issue at the end of the feeder.

This paper is a continuation of the work by [Juelsgaard et al., 2013], where it wasillustrated how control of EV power consumption could be employed for coordination,in order to minimize the incurred power loss, and reduce the overall grid loading. Inthe work at hand, we expand on this idea, and show how loss minimization can be usedfor coordinating consumption by EVs against production from solar panels, in order toincrease the possible installation of both, without unacceptable voltage variations. Wefurther include reactive power control of solar panel inverters, and illustrate both thepotential and limits of its effect on voltage stabilization.

Active control of consumers, with the purpose of avoiding voltage variations andgrid overload has been considered by e.g., [Pillai et al., 2012], who presented a heuristiccontrol strategy for charging a fleet of EVs. Consumer control was similarly investigatedby [Turitsyn et al., 2011], who discussed the design of local control of reactive power fromsolar panel inverters, with the purpose of reducing voltage variation and power losses.Compared to these works, we consider the installation of EVs and PVs collectively in thegrid, rather than separately, and illustrate how they may be coordinated to alleviate theabove issues. Also, we are not concerned with voltage variations specifically, rather weare concerned with obeying variational limits, while minimizing losses.

Loss minimization was also the main focus of [Baran and Wu, 1989], who consideredgrid reconfiguration for loss reduction, and [Hoff and Shugar, 1995, Guo et al., 2011],who investigated where to locate distributed generation, such as PVs, in the LV grid, inorder to reduce losses. Compared to these works, this paper does not attempt to modifythe grid or pick beneficial PV installation locations. Rather, we introduce a coordinationstrategy which can be employed for loss reduction and voltage control, in a predefinedgrid layout and PV installation.

The remainder of this paper is organized as follows: Section 2 outlines our mod-

198

2 Modeling and Problem Formulation

eling approach, and presents the formal problem description. Section 3 addresses ourapproach towards solving the coordination problem, and describes a benchmark strategyfor result comparison. A practical test-case, used for numerical experiments, is presentedin Section 4, followed by examples in Section 5. Concluding remarks are presented inSection 6.


In this section, we model the consumers in the LV grid, as well as the distribution linesfeeding them with electricity. Subsequently, we describe the optimization problem ofminimizing active power losses by coordinating consumers.

2.1 Consumer modeling

A household is modeled as a potential prosumer (combined producer and consumer), withactive and reactive power given by

ph(t) = ph(t) + ph(t), and qh(t) = qh(t) + qh(t),

for each household h ∈ H = 1, . . . , n and time step t ∈ T = 1, . . . , T . Above, phand qh refer to the part of the power that is inflexible, that is, it cannot be controlled ortemporally shifted. Contrary, ph and qh represent power that allows for a certain degreeof flexibility. In this work, the flexibility originates from the charging of EVs and reactivepower control of PVs. Throughout, we consider only average active and reactive powerover some time period; not instantaneous power. This entails that the time dependencyincluded in the notation, refers to the average power consumption throughout period t ∈T with fixed time period Ts.

We denote by Hev ⊆ H the households with EVs. The power used for charging eachvehicle h ∈ Hev represents flexible consumption at a constant power factor ψh. For eachvehicle, tev,h denotes the time the electric vehicle is plugged in. The energy in the EVbattery, i.e., the state of charge (SOC), then follows the charge pattern1

Eev,h(t) = Eev,h(tev,h) +

t∑

τ=tev,h

Tspev,h(τ), ∀h ∈ Hev,

where Eev,h(tev,h) denotes the charge when the vehicle is plugged in and pev,h(τ) is theaverage active power consumption of the vehicle during period τ ∈ T .

The power consumption, pev,h(t), of each EV is controllable, but is limited by thefollowing constraints

Eev,h(T ) = Edem,h, Emin,h ≤ Eev,h(t) ≤ Emax,h,

pmin,h ≤ pev,h(t) ≤ pmax,h,

qev,h(t) = pev,h(t) tan(acos(ψh)),

1Errata corrected in equation compared to original publication

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Paper E

for all t ∈ T and h ∈ Hev. Above, qev,h(t) is the reactive power consumed by the EV intime period t, Emin,h, Emax,h and Edem,h denote minimum, maximum and required finalSOC, respectively. Similarly, pmin,h and pmax,h are minimum and maximum EV powerlimits.

In this work, we introduce residential power production through photovoltaic solarpanels. We denote by Hpv ⊆ H the households with PV installed. These produce activepower ppv,h(t) ≥ 0, and reactive power qpv,h(t), h ∈ Hpv. The active power is uncontrol-lable and determined from weather conditions, i.e., direct/indirect radiation, clouds, etc.On the other hand, the reactive power is controllable, with the constraint

|qpv,h(t)| ≤√

s2max,h − p2pv,h(t), ∀t ∈ T (1)

where smax,h > 0 is a fixed upper limit of apparent power for the solar panel inverter. Thisconstraint is similar to the work by [Turitsyn et al., 2011], and is essentially a constrainton the maximum magnitude of the inverter current.

The inverter technology allowing reactive power control of PVs exists also in the EVinverters, whereby reactive power control of the EVs could equally well be introduced inthis work. We shall, however, limit our investigation to PV reactive power control, andleave the similar study of EVs for later work.

Given the modeling above, the total active and reactive power consumption of a con-sumer is then

ph(t) = ph(t) + pev,h(t)− ppv,h(t),

qh(t) = qh(t) + qev,h(t)− qpv,h(t).

We remind the reader that consumers at any time may produce rather than consumeractive and reactive power. This entails that the above is to be understood as consumptionif ph(t) > 0 and production when ph(t) < 0 and similarly for qh(t).

Throughout the remainder of the paper, we let it be implied that ppv,h, qpv,h ≡ 0 forh /∈ Hpv, and pev,h, qev,h ≡ 0 for h /∈ Hev.

2.2 Grid modeling

The low-voltage (LV) (0.4 kV) grid consists of the distribution cables feeding power toeach consumer.

The LV grid is connected to the medium voltage grid, through a transformer. Forsimplicity, the medium voltage grid and transformer station are abstracted by an idealvoltage source, i.e., the secondary side voltage of the transformer us ∈ R, has zero phaseand constant magnitude and frequency. In addition, we assume that the grid is balanced,allowing the analysis to be performed for an equivalent single phase system, see [Kundur,1993].

The distribution lines are modeled as approximate π-circuits, where the shunt capac-itances are neglected, since the cables in the considered grid are short. Thus, distributionlines are modeled as RL-series impedances. Let m ∈ N be the number of cable sec-tions in the considered grid. The value of the impedance for each grid section is denotedzk = rk + jωLk ∈ C for all k ∈ 1, . . . ,m, where rk, Lk and ω denote resistance andinductance of each cable, and grid frequency, respectively.

Given the notational convention introduced so far, the grid and associated residentsare illustrated conceptually in Fig. 2.

200


us

z1

i1

ppv,1qpv,1

p1,q1

pev,1

un

zm−1

in

ppv,nqpv,n

pn,qn

pev,n

zm

u1

z2

qev,1 qev,n

Figure 2: Conceptual outline of a radial of the low-voltage grid, illustrating the con-sumers, with active and reactive solar production (ppv, qpv), EV consumption (pev, qev),and inflexible consumption (p, q).

2.3 Grid losses

The root-mean-square (RMS) phasor-voltage in the connection point of consumer h, isdenoted uh(t) ∈ C. The corresponding RMS phasor-current ih(t) ∈ C, drawn by theconsumer is

ih(t) = f(ph(t), qh(t), uh(t)) =

(ph(t) + jqh(t)

uh(t)

)†

, ∀t ∈ T , (2)

where the functional f is introduced to ease notation onwards, and † denotes conjugatetranspose. From the current of each consumer, we can derive an expression for the powerlosses throughout the grid. We remark that the distribution grid can be visualized as agraph, where each cable represents an edge. Following the convention of [Desoer andKuh, 2010], the graph can be decomposed as a spanning tree, whose edges are calledbranches, and the remaining set of edges are called links. In practice, the branches wouldbe the cables composing each feeder of the grid, whereas links would be closed switches,creating paths between each feeder as illustrated in Fig. 3(Left).

For the sake of clarity, this work focuses on distribution grids without links, i.e., theirunderlying topology is a tree, similar to the grid illustrated in Fig. 3(Right). Our workgeneralizes to any connected grid topology, but the matrices to be introduced shortly arethen highly dependent on the topology of the specific network.

Conforming to the taxonomy of [Desoer and Kuh, 2010], we let the transformer sub-station denote the root node of the tree, and the consumers denote leaf nodes. For eachh ∈ H, let Zh ⊆ 1, . . . ,m denote the set of impedance indices of the unique simplepath between the transformer and consumer h. We define J = [Jx,y] ∈ Cn×n as

Jx,y =∑

h∈Zx∩Zy

zh,

and further define Jr = Re(J) ∈ Rn×n, as the entrywise real part of each entry of J .The matrix J is known as the loop impedance matrix, [Desoer and Kuh, 2010].

By letting i(t) = (i1(t), . . . , in(t)) ∈ Cn, u(t) = (u1(t), . . . , un(t)) ∈ Cn, for allt ∈ T , it can be shown, that the total active power loss in the feeder is

i(t)†Jri(t) > 0. (3)

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Paper E

z1

z2

us

z4

z5

z3

z6

z7

z1

z2

us

z4

z5

z3

z6

z7

Figure 3: Left: Graph structure of a small grid example. The arrows represent con-sumers. Each consumer and bus-bar in the grid is a node in the graph, and each cable-impedance is an edge. The dashed line illustrates a link, representing a closed switchbetween two feeders. Right: A similar grid, without links.

To see this notice that the losses through each cable, may be written as

rj(∑

k|j∈Zkik(t))

†(∑

k|j∈Zkik(t)), j = 1, . . . ,m.

Summing for all cables, gives (3).Additionally, the voltage at each household is

u(t) = us − Ji(t), ∀t ∈ T .

2.4 Voltage quality

The grid must be managed such that voltage variations throughout the radial are lim-ited, i.e., umin ≤ |u(t)| ≤ umax, where | · | denotes entry-wise complex magnitude, andumin, umax ∈ R are lower and upper bound on voltage magnitudes, respectively. This isequivalent to

u2min ≤ |u(t)|2 ≤ u2max, ∀t ∈ T . (4)

The inequalities above are to be read element-wise.

2.5 Problem formulation

Given the model and constraints of both consumers and the grid, as well as the expressionfor grid losses presented above, we state the following main problem:

Problem 1. For given data:

• sets: H,Hev,Hpv,

• profiles: ph(t), ppv,h(t), qh(t) for each h ∈ H, t ∈ T ,

• values: ψh, tev,h for each h ∈ Hev,

202

3 Optimization and benchmark

• matrices: J, Jr

solve

minimizepev,h(t), qpv,h(t)

T∑

t=1

i(t)†Jri(t)

subject to u2min ≤ |u(t)|2 ≤ u2

max,Eev,h(T ) = Edem,h,Emin,h ≤ Eev,h(t) ≤ Emax,h,pmin,h ≤ pev,h(t) ≤ pmax,h

qpv,k(t) ≤√

s2max,k − p2pv,k(t),

ij(t) = f(pj(t), qj(t), uj(t)),

(5)

for all t ∈ T , j ∈ H, h ∈ Hev and k ∈ Hpv.

The practical interpretation of Problem 1 is to coordinate consumers such that lossesare minimized, whilst obeying both grid constraints with respect to voltage variations, aswell as consumer constraints, with respect to EV charging.

We remind the reader that the problem is stated in discrete time; hence in (5), pev,h(t),qpv,h(t) each represent a discrete variable for each t ∈ T , and should not be confusedwith general time-domain functions.

In the next section, we elaborate on our approach for solving Problem 1, and formulatea benchmark strategy, which we use for comparison during numerical experiments.


Our strategy is to identify the non-convex elements of Problem 1, in order to make convexapproximations, and arrange a simplified problem, which we can solve globally, withknown methods.

3.1 Optimization

Large parts of Problem 1 are convex, and requires no simplifications. For instance, thecost function can be shown to be convex in the real and imaginary parts of i(t), respec-tively. The same applies for all EV and PV constraints. The only elements of (5) thatare not convex, are the relation between power and current (2), and the voltage variationlimits (4).

The voltage constraint (4) can be visualized as the annulus in Fig. 4, where the maxi-mum allowed amplitude is in fact convex in the real and imaginary part, respectively. Thelower limit is not a convex constraint. We approximate this by an affine constraint arounda fixed operating point u(t) ∈ Cn, for each t ∈ T :

u2min ≤ |u(t)|2 + 2Re(u(t))(Re(u(t))− Re(u(t)))

+2Im(u(t))(Im(u(t))− Im(u(t))).

This simplification is illustrated graphically in Fig. 4.The consistency constraint (2), is non-convex on account of the division by u†h(t). We

also replace this constraint by an affine approximation, around operating points p, q ∈ Rn

and u ∈ Cn. We denote this affine approximation by fp,q,u.

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Paper E

Im(u)

Re(u)

umin

umax

u

Figure 4: Constraint for the voltage amplitude, where the annulus visualizes the allowedrange of u. The dashed line illustrates a linear approximation for the lower limit, and theshaded region illustrates the resulting convexified constraint.

With these convexifications, Problem 1 can be approximated as:

Problem 2. Given:

• same information as in Problem 1,

• operating points uh(t), ph(t), qh(t) for each h ∈ H and t ∈ T

solve:

minimizepev,h(t), qpv,h(t)

T∑

t=1

i(t)†Jri(t)

subject to Re(u(t))2 + Im(u(t))2 ≤ u2max,

u2min ≤ |u(t)|2 + 2Re(u(t))(Re(u(t))− Re(u(t)))

+2Im(u(t))(Im(u(t))− Im(u(t)))Eev,h(T ) = Edem,h,Emin,h ≤ Eev,h(t) ≤ Emax,h,pmin,h ≤ pev,h(t) ≤ pmax,h

qpv,k(t) ≤√

s2max,k − p2pv,k(t),

ij(t) = fp,q,u(pj(t), qj(t), uj(t)),

(6)

for all t ∈ T , j ∈ H, h ∈ Hev and k ∈ Hpv.

.Problem 2 is convex and can be solved by known methods. Let the solution be denoted

i(t)⋆, q(t)⋆, p(t)⋆, with

p(t)⋆ = (p1(t)⋆, · · · , pn(t)

⋆), q(t)⋆ = (q1(t)⋆, · · · , qn(t)

⋆).

Since (6) was solved with an estimated voltage u(t), the true voltages resulting from thecurrent i(t)⋆, may now be found through the post calculation

utrue(t) = us − Ji⋆(t), ∀t ∈ T .

Let u(t) = (u1(t), · · · , un(t)). If

‖utrue(t)− u(t)‖ > ǫ, ∀t ∈ T ,

204


for some tolerance ǫ > 0, then the true voltage is far from the approximated voltageemployed in the optimization. Our approach is then to update the voltage estimate, andre-solve Problem 2. This iterative approach can be formulated as:

Algorithm 1: Loss minimization procedure

1. Initialize γ > ǫ > 0 and u(t), p(t), q(t), for all t ∈ T ,

2. While γ > ǫ:

• Solve Problem 2 to obtain i⋆(t), q⋆(t), p⋆(t),for all t ∈ T

• set p(t) = p⋆(t) and q(t) = q⋆(t) for all t ∈ T

• Calculate true voltage:utrue(t) = us − Jzi

⋆(t), ∀t ∈ T

• Set γ = ‖utrue(t)− u(t)‖,

• Set u(t) = utrue(t), for all t ∈ T

3. Done

If the iterative procedure converges such that ‖utrue(t) − u(t)‖ < ǫ, then i⋆(t), q⋆(t),p⋆(t) are used as approximate solutions to the initial Problem 1.

This approach for finding an approximated solution to a non-convex problem, bysolving a series of approximated, convex problems, is commonly known as sequentialconvex programming. Convergence properties for the algorithm above, were discussedby [Dihn and Diehl, 2010], for certain classes of non-convex problems. However, themethod is in essence a heuristic, and general convergence and optimality guarantees aredifficult to provide. The method has however been applied with great success withinvarious fields, refer to e.g. [Hovgaard et al., 2013] and [Biegel et al., 2011].

3.2 Benchmark strategy

To illustrate the benefits of shifting the charge cycle of EVs, and utilizing reactive powercontrol of the PVs, we present a benchmark strategy that does not utilize this flexibility.That is, the benchmark strategy charges each EV, when it is plugged in and does notutilize the PV capability to absorb or produce reactive power.

The benchmark strategy thereby enforces

qpv,h(t) = 0, ∀h ∈ Hpv, t ∈ T ,

and for all h ∈ Hev:

pev,h(t) =

pmax,h, if t ≥ tev,h and Eev,h(t) < Edem,h

0, otherwise.

From the topology of the grid, and impedances of each cable, the radial admittancematrix Y can be arranged [Kundur, 1993]. Given Y , as well as ph(t), qh(t) for eachh ∈ H, t ∈ T , known methods exists for calculating the current and voltage of eachconsumer, e.g., Gauss-Seidel and Newton-Raphson.

The following section describes in detail a test-case used as a foundation for numericalexperiments in Section 5.

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4 Test-Case

We consider a low-voltage distribution grid, of a residential neighborhood, located inNorthern Jutland, Denmark. The entire low-voltage grid consists of three 10/0.4 kV trans-former substations, with a total of 19 feeders and 316 residential consumers. We limit ourattention to one of these feeders, servicing 34 residential consumers. The tree topologyof the feeder, is illustrated in Fig. 5. Each consumer is modeled as described in Section 2.

z1

z6

z7

z11

z15

z21

z20

z19

z32

z33

z41

us

z2

z5

z14

z12

z10

z8

z18

z16

z22

z23

z31

z29

z28

z26

z25

z24

z38

z34

z40

z39

z45

z42

14

57810

1113

1415

1618 1921 2223

24282930

3134

Figure 5: Outline of the feeder employed for numerical experiments. Boxes areimpedances and arrows indicate connection points of individual consumers. The numbersnext to each arrow indicate the consumer numbering convention, to be used for classifyingH,Hev,Hpv in subsequent examples.

The resistive and reactive parameters of impedances, are presented in the followingtable. There are four different types of cables, as elaborated by [Pillai et al., 2012]:

Impedance nr. Res. [Ω/m] Reac. [Ω/m]

1, 7, 11 0.21 0.07215, 19 0.32 0.0756, 20, 21, 32, 33, 41 0.64 0.0792-5, 8-10, 12-14, 16-18,

1.81 0.09422-31, 34-40, 42-45

Provided the length of each cable in the feeder, combined with the data above, theimpedances can be calculated. In the numerical experiments to follow, we consider atime-period of 21 hours, sampled in 10 minute intervals, starting at 14:00. The inflexibleconsumption of each consumer is modeled as known curves, presented in Fig. 6. The data

206

5 Numerical Experiments

in Fig. 6 is retrieved from [Nord Pool Spot, 2013], and is representative of the daily con-sumption pattern of residential houses. With the curves in Fig. 6, the average daily energyconsumption is 7.9 kWh. The inflexible consumption of all households are modeled witha constant power factor of 0.95.

ph(t)

[pu]

t13:56 18:58 24:00 04:52 09:54

0.2

0.3

0.4

0.5

Figure 6: The inflexible consumption of all households.


In the following numerical experiments, we explore several different scenarios, related tothe installation of EVs and PVs in the test-case. We examine the following scenarios:

1. Resilience of benchmark strategy against implementation of EVs and PVs separately.

2. Resilience of suggested coordination based strategy against implementation of EVsand PVs separately.

3. Benefit of coordination strategy over benchmark, with combined installations of EVsand PVs.

Scenario 1 substantiates the introductory example, and shows that by separately introduc-ing EVs and PVs, unacceptable over- and under-voltages occur, if the inherent flexibilityis not utilized. Scenario 2 shows that the voltage issues of Scenario 1 caused by EVconsumption and PV production, can be alleviated by altering the EV charge profile, andutilizing PV reactive power. Finally, Scenario 3 shows that by combining the installa-tion of EVs and PVs, all voltage issues can be alleviated by the proposed optimizationstrategy.

We employ pr-unit notation. The base voltage is set to 0.4 kV, and the base power isset to 1 kVA. All other quantities are transformed accordingly. The transformer phasorvoltage is us = 1∠0 pu. The Danish grid code requires that voltage variations are limitedto ±10%, however, following the example of [Pillai et al., 2012], we shall employ±6% asthe constraint in our coordination, such as to leave sufficient margin for medium voltagegrid variations. We refer to the 6% variation as a safety limit, and the 10% variation asthe strict limit.

Fig. 7 presents the known generation of active power from households with solarpanels. The active solar power output for all households is limited to 8 pu.

For all h ∈ Hev, we let pmin = −11 pu, pmax = 11 pu, Edem,h = 25 pu, and

tev,h ∈ U(14 : 00, 17 : 00), Eev,h(tev,h) ∈ U(0, 1),

where U(a, b) denotes a uniform distribution of [a, b].

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Paper E

ppv(t)

[pu]

t13:56 18:58 24:00 04:52 09:540

2

4

6

8

Figure 7: The solar power, ppv(t), based on data retrieved from the Danish MetrologicalInstitute.

5.1 Resilience of benchmark strategy

We evaluate the effect of installing EVs and PVs in the grid, when the benchmark strat-egy is utilized. The voltage profile at the connection point for each consumer in Fig. 5,is obtained by Gauss-Seidel iterations. Scenario 1 and 2 above are simulated for thebenchmark strategy, through the two configurations:

A. Hpv = ∅ and Hev = 30, . . . , 34;

B. Hpv = 1-7, 25-34 and Hev = ∅.

The resulting voltage profiles from both configurations throughout the horizon, are pre-sented in Fig. 8, with Configuration A illustrated in Fig. 8(top). Each curve shows thevoltage over time in the connection point of a consumer. The step-like nature of the curvesillustrates the time instances where each of the five EVs either start or finish charging,causing a voltage drop or increase, respectively.

uh(t)

[pu]

t

uh(t)

[pu]

13:56 18:58 24:00 04:52 09:54

0.9

1

1.1

0.9

1

1.1

Figure 8: Top: Voltage profile in the connection point of each consumer, resultingfrom benchmark strategy Configuration A (solid), and voltage variation limits (dashed).Bottom: Similarly, voltage profiles by Configuration B.

The voltage profiles violates the safety limit, and further, comes quite close to violat-ing the strict limit, even though only five EVs are connected.

The experiment for Configuration B, results in the voltage profiles presented in Fig. 8(Bottom).Here, we see similar to Configuration A, that PVs introduce local over-voltages in the

208


sense that the safety limit is violated in both the beginning and end of the simulation.Comparing these results, to the solar power presented in Fig. 7, it is clear that the timeof the over-voltages coincide with the period of highest solar intensity. Since the solarpower is not consumed locally, it is transported back to the medium voltage grid, whichcauses the over-voltages. For reference onwards, the maximum voltage experienced inFig. 8(Bottom), is 1.081 pu.

5.2 Resilience of coordination based strategy

Employing the coordination strategy described previously, we perform again two numer-ical experiments for Scenario 1 and 2, with configurations

A. Hpv = ∅ and Hev = H,

B. Hpv = 1-7, 25-34 and Hev = ∅.

The reader should notice that now all households are installed with EVs, and not only thefinal five. The voltage profile obtained in Configuration A is illustrated in Fig. 9(Top).As evident, the coordination performed by the algorithm previously introduced, is ableto support an EV for every household, without introducing under-voltages. The specific

uh(t)

[pu]

t

uh(t)

[pu]

13:56 18:58 24:00 04:52 09:54

0.9

1

1.1

0.9

1

1.1

Figure 9: Top: Voltage profile in the connection point of each consumer, resultingfrom optimization strategy Configuration A (solid), and allowed voltage range (dashed).Bottom: Similarly, voltage profiles by Configuration B.

charge pattern for the EVs are shown in Fig. 10. Here, it is observed that all EVs arecharged, roughly, with a constant power, in a way that averages out the load on the grid.Remember that the optimization minimizes losses, while obeying voltage constraints.Since losses are quadratic, averaging out the load of the grid reduces losses, comparedto a faster charge schedule of the vehicles. From further numerical studies, this resultappears to be reasonably consistent, also for other configurations of Hev.

In this work we have not included any price of electricity. Introducing for instance,a time-varying price signal would entail that both the cost of losses, as well as the costof energy would vary across the horizon. Thereby, it is likely that less intuitive chargeschedules would be prefferable. We shall leave this concern for future extensions.

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Paper E

Eev,h(t)

[pu]

t13:56 18:58 24:00 04:52 09:540

5

10

15

20

25

Figure 10: EV charge (solid) and desired final charge (dashed), when employing opti-mization based approach.

For Configuration B above, the optimization based approach renders the voltage pro-files presented in Fig. 9 (Bottom). We see that over-voltages are still present, however,the maximum voltage is now 1.073 pu, which is a reduction compared to the benchmarkstrategy, obtained solely through control of the reactive power for solar panels. There aretwo important comments related to the results in Fig. 9(Bottom):

• To avoid feasibility issues with the voltage profiles in the examples, we have refor-mulated the tight voltage constraints presented in Problem 1 and Problem 2, by softconstraints employing slack variables.

• We have over-dimensioned the solar panel inverters by 5 %, by letting smax =1.05ppv,max, where ppv,max is the maximum active power output. This entails thatwhen ppv(t) = ppv,max, the constraint in (1) still allows the PV inverter to ei-ther produce or consumer some amount of reactive power. Introducing this over-dimensioning is what allows the optimization to improve the voltage. If no over-dimensioning was introduced, the optimization would not be able to improve volt-age in periods where the active power output from the solar panels where saturated.

The table below illustrates the maximum voltage occurring in the grid for ConfigurationB above, for different levels of over-dimensioning of the solar panel inverters:

over-0 5 10 20 30

dimension [%]

voltage [pu] 1.081 1.073 1.070 1.064 1.060

Table 11.1: Reduction of over-voltages from various degrees of PV inverter over-dimensioning.

As illustrated by the table above, a sizeable overdimensioning of the PV inverter isrequired to fully remove the overvoltages. However, the following section illustrates howcoordination between flexible consumption from EVs and local production from PVs canimprove the voltage further. The benefit of over-dimensioning PV inverters was alsodiscussed by [Turitsyn et al., 2011].

210

6 Concluding remarks

5.3 Benefit of optimization strategy over benchmark

In this final experiment, we introduce EVs and PVs randomly throughout the feeder, suchthat the penetration of both PV and EV is around 50%. Employing both the benchmarkand optimization based strategy, yields the results in Fig. 11, where Fig. 11(Top) and(Middle) presents the total power consumption, and the power consumption solely fromEVs, respectively.

From the definition of the benchmark strategy, all EVs charge as soon as they areplugged in. This entails that there is a large peak in consumption in the beginning of thetime-span of the simulation. Similarly, in the end of the simulation, when the solar powerincrease, there is a large negative consumption. This entails that the voltage profiles cor-responding to the benchmark strategy, in Fig. 11(Bottom), initially show under-voltageswhen charging EVs, and later, over-voltages because the PV generated power is not ab-sorbed.

Conversely, when using the optimization based strategy, the EV charging is post-poned, and coordinated with the PV generation, such that consumption by EVs coun-teracts the production from PVs. Contrary to the benchmark strategy, this coordinationentails that both over- and under-voltages are avoided.

In Fig. 11(Bottom), we see over-voltages in the very beginning of the simulation ofboth the benchmark and the optimized strategy. This is because at this time, there issome PV production, however, no EVs are available for charging. These over-voltagescan thereby only be reduced by reactive power control of the solar panels. Pertainingto the previous discussion concerning over-dimensioning of converters, this example isconducted with 5 % inverter over-dimensioning. Even though the reactive power controlreduces over-voltages, it is evident from the figure, that they are not completely removed.This illustrates an important point; that to avoid the potential over-voltages caused by PVgeneration, may require some flexible consumption in order to absorb the power of solarpanels.

6 Concluding remarks

This work has addressed future challenges of the electric distribution grid, emerging fromexpected changes in the nature of consumers. We have described how a future increasein the use of EVs and PVs may cause the grid to be overloaded, and unacceptable voltagevariations to occur.

An optimization based strategy for employing flexibility of active and reactive powerfor EVs and PVs respectively, has been arranged. Using this strategy we have shown howcoordination among individual EVs and PV facilities may be employed to reduce powerlosses in the grid, as well as alleviate issues pertaining to voltage variations, compared toa benchmark control strategy which does not utilize flexibility.

Numerical experiments, based on a true distribution grid located in Northern Jutland,Denmark, has illustrated how the posed optimization problem can assist in maintaininggrid limitations, even when increasing the penetration of EVs and PVs far beyond thelevels currently present in the Danish electric grid.

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|uh(t)|

[pu]

t

pev,h(t)

[kW

]Bench.Opt.

ph(t)

[kW

]

13:56 18:58 24:00 04:52 09:54

0.9

1

1.1

0

5

10

−10

0

10

Figure 11: Top: The total consumption in the optimized case (blue, solid), and the bench-mark case (red, dashed). Middle: Corresponding EV charge schedule. Bottom: Resultingvoltage profiles, as well as voltage limits (dashed, magenta).

Acknowledgement

This work is supported in part by the Southern Danish Growth Forum and the EuropeanRegional Development Fund, under the project ”Smart & Cool”, and in part by the theDanish Council for Strategic Research (contract no 11-116843) within the ’ProgrammeSustainable Energy and Environment’ under the project ”EDGE” (Efficient Distributionof Green Energy).

References

[Baran and Wu, 1989] Baran, M. and Wu, F. (1989). Network Reconfiguration in Distri-bution Systems for Loss Reduction and Load Balancing. IEEE Transactions on Power

Delivery, 4(2).

[Biegel et al., 2011] Biegel, B., Juelsgaard, M., Kraning, M., Boyd, S., and Stoustrup, J.(2011). Wind turbine pitch optimization. IEEE Conference on Control Applications,

Proceedings, pages 1327–1334.

[Danish Energy Association and Energinet.dk, 2010] Danish Energy Association andEnerginet.dk (2010). Smart Grid i Danmark. www.danskenergi.dk.

[Desoer and Kuh, 2010] Desoer, C. and Kuh, E. (2010). Basic Circuit Theory. McGraw-Hill.

212

[Dihn and Diehl, 2010] Dihn, Q. and Diehl, M. (2010). Local convergence of sequentialconvex programming for nonconvex optimization. Recent Advances in Optimization

and its Applications in Engineering, pages 93–102.

[Guo et al., 2011] Guo, Y., Lin, Y., and Sun, M. (2011). The impact of integrating dis-tributed generations on the losses in the smart grid. IEEE Power and Energy Society

General Meeting.

[Hoff and Shugar, 1995] Hoff, T. and Shugar, D. (1995). The value of grid-support pho-tovoltaics in reducing distribution system losses. IEEE Transactions on Energy Con-

version, 10(3):569–576.

[Hovgaard et al., 2013] Hovgaard, T., Larsen, L., Boyd, S., and Jørgensen, J. (2013).MPC for wind power gradients - utilizing forecasts, rotor inertia, and central energystorage. Proceedings of European Control Conference.

[International Energy Agency, 2011] International Energy Agency (2011). Technology

Roadmap: Electric and plug-in hybrid electric vehicles. www.iea.org/.

[Ipakchi and Albuyeh, 2009] Ipakchi, A. and Albuyeh, F. (2009). Grid of the future.IEEE Power and Energy Magazine, (april).

[Juelsgaard et al., 2013] Juelsgaard, M., Andersen, P., and Wisniewski, R. (2013). Mini-mization of distribution grid losses by consumption coordination. Proceedings of IEEE

Multi-Conference on Systems and Control, pages 501–508.

[Kundur, 1993] Kundur, P. (1993). Power system stability and control. McGraw-Hill.

[Nord Pool Spot, 2013] Nord Pool Spot (2013). www.nordpoolspot.com/. CommonNordic Power Exchange.

[Pillai et al., 2012] Pillai, J., Thøgersen, P., Møller, J., and Bak, B. (2012). Integration ofelectric vehicles in low voltage Danish distribution grids. Power and Energy.

[Turitsyn et al., 2011] Turitsyn, K., Sulc, P., Backhaus, S., and Chertkov, M. (2011). Op-tions for Control of Reactive Power by Distributed Photovoltaic Generators. Proceed-

ings of the IEEE, 99(6):1063–1073.

213

Paper F

Distributed Coordination of Household Electricity Consumption

Morten Juelsgaard, André Teixeira, Mikael Johansson,Rafael Wisniewski and Jan Bendtsen

This work is submitted for:IEEE Multi-Conference on Systems and Control, October, 2014


1 Introduction

Abstract

This work presents a distributed framework for coordination of flexible electricityconsumption for a number of households in the distribution grid. Coordination isconducted with the purpose of minimizing a trade-off between individual concernsabout discomfort and electricity cost, on the one hand, and joint concerns about gridlosses and voltage variations on the other. Our contribution is to demonstrate howdistributed coordination of both active and reactive consumption may be conducted,when consumers are jointly coupled by grid losses and voltage variations. We furtherillustrate the benefit of including consumption coordination for grid operation, andhow different types of consumption present different benefits.

1 Introduction

Previous works have shown how several types of electrical power consumption is highlyflexible in the sense that it may be temporally shifted with little or no discomfort to theconsumer [1–4]. Proper utilization of this flexibility, through coordination of consump-tion, may be used to minimize grid losses, control voltage, avoid grid congestion, etc,thus comprising the focus of this work.

The information required to conduct coordination may encompass sensible informa-tion about each consumer. Such information should be kept private, thus requiring a co-ordination framework allowing this information to remain private and distributed amongeach individual consumer, rather than requiring the information to be collected and storedcentrally.

Previous works on distributed consumption coordination includes [3], which consid-ered a simplified grid model to include cost of losses in consumer coordination but disre-garded voltage variations; [5], who considered a more general and detailed grid structure,but also omitted voltage drops; [6], who managed voltage drops and power losses by op-timizing reactive power flow in simple grids with line topology, but disregarded activepower management and omitted any concerns towards consumer behavior.

This work extends the results of the references above, by presenting a framework forjoint coordination of active and reactive power consumption, such as to account for bothprivate consumer concerns, cost of transport losses and voltage quality throughout thegrid. To our knowledge, this is the first paper to consider joint coordination of active andreactive power while considering both loss minimization and voltage control.

Our work follows a framework similar to [5], but extends it to include voltage varia-tions as well as transport losses in the grid. We use the flexibility models of [3] and [4]to provide measures of consumer comfort, and demonstrate how the coordination maybe conducted distributed such that participants of the coordination are only required tocommunicate with a few neighboring units, with no need for centralized coordination.For this, we consider a future electrical grid with advanced metering and control infras-tructure that allows each consumer, cable, and grid junction, to play an active part in thecoordination.

The remainder is organized such that Section 2 presents the models required to for-mally state the coordination problem in Section 3. A framework for solving the coordi-nation in a distributed fashion is outlined in Section 4, for which a numerical example ispresented in Section 5. Final remarks and perspectives are provided in Section 6.

217

Paper F

2 Modeling

The following is divided into an outline of the grid structure, a derivation of the gridmodel and a derivation of the consumer models.

2.1 Grid structure

Our focus is low-voltage distribution grids, as illustrated in Fig. 1.

vs

Figure 1: Low voltage grid and the associated grid components.

The grid consists physically of a transformer substation ( ), cable sections ( ),and consumer connection points ( ). The links in between cables, as well as betweencables and consumers, represent grid junctions, also referred to as bus-bars. The bus-barat the secondary side of the transformer, indicated with a vertical line, is considered aslack-bus [7], with fixed normalized voltage magnitude vs = 1 pu.

We represent the network layout as a connected, undirected graph where cable sec-tions, represented by impedances, compose the edges and bus-bars compose the nodes.Consumers are connected to the network through a single private cable section. Thisgives a natural sub-division of cables into two categories; leaves, comprising the privatecable of each consumer, and branches, comprising shared cables. Let the network containn ∈ N consumers and b ∈ N branch cables. This means that the network contains n+ bbus-bars, excluding the secondary transformer side. We define sets

I = 1, . . . , n, J = 1, . . . , n+ b,

and assign an ordering to the network components as follows: leaves and consumers arenumbered by i ∈ I, such that leaves are assigned the same number as the consumer itconnects. Branches are numbered j ∈ n+ 1, . . . , n+ b.

To make the exposition clearer throughout, we limit our attention to networks whosegraphs compose a tree, rooted at the transformer, although our results can be generalizedto any connected, undirected graphs. We define mappings

Pa(j) ∈ J , Ch(j) ⊂ J , j ∈ J

denoting the unique parent edge, and the set of children edges of each cable, in a graph-theoretical sense.

2.2 Grid modeling

Each cable section is modeled as an RL-series connection with impedance zj = rj+jxj ∈C, with j ∈ J and rj , xj > 0 being the resistance and reactance of each cable section,respectively [7]. Consider an isolated cable j ∈ J illustrated in Fig. 2.

218

2 Modeling

zj = rj + jxj ,vl(t, j) vr(t, j)

sl(t, j) sr(t, j)

i(t, j)

Figure 2: Illustration of an isolated grid section, including the power flow through the twoterminals.

In the terminology of [5], each cable section is considered a two-terminal device, and weintroduce the map sl : T × J → C to denote the left-terminal complex power, whereT = 1, 2, . . . , N is a discrete coordination horizon of N steps. Similar to sl(t, j) wedefine sr(t, j), t ∈ T , j ∈ J as the right-terminal power. We extend the notation suchthat

sl(t, j) = sl,j(t) = sl,t(j) ∈ C, ∀j ∈ J , t ∈ T .

Additionally, we generalize this notation such that

sl,j = (sl(1, j), . . . , sl(N, j)) ∈ C|T |, j ∈ J ,

sl,t = (sl(t, 1), . . . , sl(t, n+ b)) ∈ C|J |, t ∈ T .

We employ this notation for all similar mappings.The voltage on either side of the cable section and the current through it are denoted

vl(t, j), vr(t, j), i(t, j) ∈ C, respectively.

Voltage drop

The current through the cable is [8]:

ij(t) =

(sl,j(t)

vl,j(t)

)∗

=

(sr,j(t)

vr,j(t)

)∗

, (1)

where (·)∗ denotes complex conjugate. The voltage difference across the section is

vl,j(t)− vr,j(t) = zjij(t) ⇔ vr,j(t) = vl,j(t)− zjij(t).

Inserting (1) and the impedance expression gives

vr,j(t) = vl,j(t)− (rj + jxj)

(sl,j(t)

vl,j(t)

)∗

. (2)

Power quality requirements state that the voltage must be within maximum and minimummagnitudes vmin, vmax ∈ R at all times, i.e.,

vmin ≤ |vr(t, j)| ≤ vmax, ∀j ∈ J , t ∈ T (3)

as illustrated in Fig. 3. The specific limits enforced in various countries may vary, how-ever, they would typically be around±10% around the transformer voltage. It is commonto operate the electrical grid with a small phase-shift of the voltage [5], effectively tight-ening the constraint in (3) to the hatched region of Fig. 3, denoted V . For this tighterconstraint, where vr,j(t), vl,j(t) ∈ V , we approximate v∗l,j(t) ≈ 1, whereby (2) reduces to

vr,j(t) = vl,j(t)− (rj + jxj)s∗l,j(t) (4)

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Paper F

Re(v j(t))

Im(vj(t))

vmin vmax

vmin

vmax

vs

Figure 3: Shaded: The annulus representing the feasible voltage region ∀j ∈ J , ∀t ∈ T .Hatched: A common operating subset (V) of the voltage.

The reader should notice from Fig. 1, that vl,j(t) = vr,Pa(j)(t), and thus

vr,j(t) = vr,Pa(j)(t)− (rj + jxj)s∗l,j(t), ∀j ∈ J . (5)

We may write (5) more compactly by defining P ∈ 0, 1b×b, and Dp, Dq ∈ Cb×b as

[P ]i,j =

1, Pa(j) = i

0, otherwise, [Dp]i,j =

rj + jxj , i = j

0, otherwise,

and Dq = −jDp, whereby (5) may be written

vr,t = Pvr,t −DpRe(sl,t)−DqIm(sl,t), t ∈ T .

Losses and cost

From (1), the squared current magnitude is:

|ij(t)|2 =

|sl,j(t)|2

|vl,j(t)|2=

|sr,j(t)|2

|vr,j(t)|2=

1

2

(|sl,j(t)|

2

|vl,j(t)|2+

|sr,j(t)|2

|vr,j(t)|2

)

.

However, as argued above, the voltage constraint entails that |vr,j(t)| ≈ |vl,j(t)| ≈ 1, j ∈J , t ∈ T , i.e.,

|ij(t)|2 ≈

1

2

(|sl,j(t)|

2 + |sr,j(t)|2) . (6)

The active losses are given by Re(ij(t)zjij(t)∗) = rj |ij(t)|2, and the reactive losses arecorrespondingly Im(ij(t)zjij(t)

∗) = xj |ij(t)|2. By defining l : CN ×CN → RN as

l(u, y)(t) =1

2

(|u(t)|2 + |y(t)|2

), t ∈ T (7)

for any u, y ∈ CN , the combined active and reactive losses may be expressed aszj l(sl,j , sr,j), for each cable j ∈ J . As these losses represents power dissipated in eachcable, the physical relation between left and right terminal power flow, is given by

sl,j = sr,j + zj l(sl,j , sr,j), j ∈ J . (8)

220

2 Modeling

In the sequel, we shall seek to implement the coordination procedure using convex op-timization tools. However, since (8) represents a quadratic equality, it is a non-convexconstraint. In [5] the convex relaxation

sl,j(t)− sr,j(t) ≥ zjl(sl,j , sr,j), j ∈ J , (9)

was suggested, and it was argued that in tree-networks, the relaxation would be tight.To define a cost of losses, we introduce a fixed, known estimate of the electricity price

w ∈ RN , and define cl : J ×CN ×CN → R ∪ ∞

cl(j, u, y) =

rj〈w,Re(u− y)〉, u ≥ y + zjl(u, y)

+∞, otherwise,(10)

where 〈·, ·〉 denotes inner product. The estimated cost of the losses in each cable canthen be represented as cl(j, sl,j , sr,j), where (9) are implicitly included as an operatingconstraint.

Bus-bar power conservation

Power conservation throughout the grid entails that the net in- and outflow of power ateach node, must coincide. Since we focus on tree structured networks, this is written as

sr,j =∑

h∈Ch(j)

sl,h, j ∈ J . (11)

2.3 Household consumption

Each consumer i ∈ I draws a complex power sc,i(t) = pc,i(t) + jqc,i(t) ∈ C, wherepc,i(t), qc,i(t) ∈ R represent the average active and reactive consumption during t ∈ T .Let each period t ∈ T be of length Ts, where the average power pc,i(t) is equivalent to anenergy Tspc,i(t).

Recall that consumers are connected to the grid through a private leaf cable. From thegrid ordering defined in Section 2.1, this entails

sr,i = sc,i, i ∈ I.

Not all consumption is flexible, so we let

pc,i(t) = pc,i(t) + pc,i(t), and qc,i(t) = qc,i(t) + qc,i(t),

sc,i(t) = pc,i(t) + jqc,i(t), and sc,i(t) = pc,i(t) + jqc,i(t),

such that sc,i(t) = sc,i(t) + sc,i(t), where pc,i(t), qc,i(t) ∈ R represents the estimatedinflexible consumption, which cannot be shifted. We refer to this as the baseline con-sumption. Conversely, pc,i(t), qc,i(t) ∈ R represents the flexible consumption, allowingfor some degree of temporal shifts.

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2.4 Discomfort and Appliance Constraints

The flexibility of a consumer, depends on the installed appliances and any discomfortand constraints associated to their use. We focus on the flexibility introduced by threegeneric appliances: electric heat pumps (EHPs), electric vehicles (EVs) and photo-voltaic(PV) arrays. For simplicity, we do not include consumers with more than one appliance,although our approach directly allows for this.

Electric heat pump installed

Let Iehp ⊂ I be consumers with an EHP installed. For i ∈ Iehp we introduce a statexi(t) ∈ R, representing the temperature of the household. A simple thermal model canbe approximated as in [1], by a linear first order model:

xi(t+ 1) = aixi(t) + bipc,i(t) + δi(t), i ∈ Iehp (12)

where ai ∈ (0, 1), bi ∈ R+ are estimated model parameters and δi(t) ∈ R is an estimateof the disturbance from ambient conditions. We let

xi = (xi(1), . . . , xi(N)) ∈ RN ,

and consider xi as a mapping: xi : CN → RN , taking the power sc to the temperaturexi(sc), for i ∈ Iehp.

To model comfort of a consumer employing an EHP, we introduce a known set-pointxsp,i ∈ RN , to which the indoor temperature preferably should remain close, and devia-tions are translated as a discomfort of the consumer. From this, we define the discomfortdi : C

N → R, asdi(sc,i) = ‖xi(sc,i)− xsp,i‖

22, i ∈ Iehp. (13)

The operation of the heat pump is bounded by upper and lower limits of the consump-tion, such that

pehp,i

≤ pc,i(t) ≤ pehp,i, t ∈ T , (14)

where pehp,i

, pehp,i ∈ R are known operating limits. We further enforce limits on temper-

ature xehp,i, xehp,i ∈ R, such that

xehp,i ≤ xi(sc,i) ≤ xehp,i, (15)

where the inequalities above are to be read entry-wise.Finally, in this work the flexibility of the EHP is solely related to the consumption of

active power, i.e., there is no flexibility for reactive consumption, so qc,i(t) = 0, t ∈ T .Provided known parameters in the model of (12), we collect the constraints in the set

Si = s = p+ jq | pehp,i

≤ p(t) ≤ pehp,i, q = 0

xehp,i ≤ xi(s) ≤ xehp,i ⊂ C|T |,

for i ∈ Iehp. For brevity of notation, we shall include these private constraints implicitlyin the discomfort of the consumer, by defining the extended value discomfort as

di(s) =

di(s), s ∈ Si

+∞, otherwise.(16)

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2 Modeling

for any s ∈ C|T | and i ∈ Iehp. We use a similar notation onwards for the remainingappliances.

Electric vehicle installed

Let Iev ⊂ I denote households with EVs installed. The EV charging is not subject toa setpoint or preferred charge schedule. Instead the vehicle is required to be chargedcompletely during the horizon, i.e.

∑

t∈T

Tspc,i(t) = edem,i, (17)

for some demand edem,i > 0. Additionally, the vehicle must, similarly to the EHP, obeylimits on charge and storage capacity

pev,i

≤ pc,i(t) ≤ pev,i, eev,i ≤τ∑

t=1

Tspc,i(t) ≤ eev,i, ∀τ ∈ T (18)

with limits pev,i, pev,i, eev,i, eev,i ∈ R. The vehicle will typically be away from the charg-

ing station for some period every day, where it cannot be charged, i.e,

pc,i(t) = 0, ∀t ∈ τ |τ ∈ T , τ ≤ tev,i. (19)

where tev,i ∈ T denote an estimate of the time-of-plug-in.It has been argued that the inverter based consumption such as EVs are capable of

both supplying and consuming reactive power [9]. The constraint is here that the capacityof the inverter must not be exceeded:

qc,i(t)2 + pc,i(t)

2 ≤ s2ev,i, t ∈ T , (20)

where sev,i > 0 is an upper limit of the apparent power of the inverter.Collecting the constraints in (17), (18), (19), (20), the feasible operating set of an EV

is

Si = s = p+jq |∑

τ∈T Tsp(τ ) = edem,i, pev,i≤ p(t) ≤ pev,i,

eev,i ≤∑tτ=1 Tspi(τ ) ≤ eev,i, t ∈ T

p(τ ) = 0, τ ≤ tev,i, q(t)2 + p(t)2 ≤ s2ev,i ⊂ C

|T |.

As any charge schedule fulfilling these constraints are equally acceptable, the discomfortis di(s) = 0, i ∈ Iev.

Photo-voltaics installed

Let Ipv ⊂ I denote consumers with solar panels installed. The active consumption ofa PV array is governed by weather conditions, and cannot be controlled, i.e. pc,i(t) =ppv,i(t), ∀i ∈ Ipv, t ∈ T , where ppv,i(t) ∈ R+ denotes some estimate of the solar produc-tion.

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Since PVs are inverter based, the same capabilities applies for the control of reactivepower, as was the case for EVs [10]:

qc,i(t)2 + ppv,i(t)

2 ≤ s2pv,i,

where spv,i > 0 is an upper limit of the apparent power of the inverter. The constraintscan be collected as

Si = s = p+ jq | p = ppv,i, q(t)2 + ppv,i(t)

2 ≤ s2pv,i ⊂ C|T |.

Similar to the case of the EVs, there is no discomfort related to employing the flexi-bility of PVs, so di(s) = 0.

2.5 Household objective

The objective of each household is to minimize the discomfort, as a trade-off with mini-mizing the cost of buying electricity. Additionally, recall that each consumer is connectedto the grid through a private leaf-cable and that sr,i = sc,i, i ∈ I. The power transportedthrough the leaf is thus defined solely by the consumer, and so are the losses introduced inthe leaf. We therefore assign the cost of losses in each leaf, specifically to the individualconsumer connected through it.

Given the price estimate introduced earlier, the estimated cost of buying electricity,the cost of leaf losses and the discomfort for each consumer is

ce(i, sl,i, sc,i) = 〈w,Re(sc,i)〉+ cl(i, sl,i, sc,i) + λidi(sc,i),

for i ∈ I, where λi > 0 is a trade-off parameter private to each consumer. The costof energy is only related to the active consumption, i.e. no monetary cost is directlyintroduced from the reactive power consumption.

3 Coordination problem

The primary task of the coordination is to ensure that the constraints of the grid and eachindividual consumer are satisfied.

The secondary task of coordination is to achieve a trade-off between the consumerscost of energy and discomfort, as well as the cost of losses incurred in the grid. Given themodels discussed in Section 2, this problem is stated as

Problem 1 (Centralized problem).Provided:

• mappings ce, and cl

• grid structure Pa(j),Ch(j), j ∈ J

• matrices P,Dp, Dq

• set V

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4 Distributed Consumption Balancing

Solve: minimizesc,i, sl,j , sr,j , ur,j

i ∈ I, j ∈ J

∑

i∈I

ce(i, sl,i, sc,i) +∑

j∈J\I

cl(j, sl,j , sr,j)

subject to vr,t = Pvr,t −DpRe(sl,t)−DqIm(sl,t)vr,t ∈ V, sr,j =

∑

h∈Ch(j) sl,h,

(21)

for t ∈ T , with the implicit constraint sc,i = sr,i, i ∈ I. The optimal cost of (21) isdenoted φ⋆ ∈ R.

Given the approximations introduced in Section 2, (21) is a convex problem. Tosee this, notice that each component of the objective function is convex in the real andimaginary parts of the variables, separately. The same holds for all constraints in (21).

We assume that Problem 1 is strictly feasible, i.e., that there exists consumption pro-files for each consumer that would be strictly within their individual private constraints,as well as strictly satisfy the grid constraints.

3.1 Benchmark strategy

Before deriving the distributed approach for solving Problem 1, we shall initially derivea benchmark strategy to be used for comparison in the numerical example in Section 5.In this strategy, each consumer only considers private objectives and constraints, and dis-regards any joint objectives and constraints. The strategy can be formulated through thefollowing problem:

Problem 2 (Benchmark).Provided:

• mappings di and trade-off parameters λi > 0, i ∈ I

• estimated price w ∈ R|T |+ ,

Solve: minimizesc,i, i ∈ I,

∑

i∈I

(〈w,Re(sc,i)〉+ λidi(sc,i)) . (22)

The benchmark could be considered a more contemporary strategy, where individualconsumers considers only private objectives.


The framework for distributed coordination builds on the approaches derived in [5, 6]. Itrelies on Alternating Direction Method of Multipliers (ADMM), [11, 12].

First, we introduce auxiliary variables zj(t), wl,j(t), wr,j(t) ∈ C, for j ∈ J , t ∈ T ,and define extended value function

g(wr,j , wl,h|h ∈ Ch(j), zj) =

0, zj ∈ V ∧ wr,j =∑

h∈Ch(j)

wl,h

∞, otherwise.

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Using the above mappings, and by adding consistency constraints, (21) may be equiva-lently formulated

minimizesc,i, sl,j , sr,j , ur,jwl,j , wr,j , zji ∈ I, j ∈ J

∑

i∈I


j∈J\I

cl(j, sl,j , sr,j)

+∑

j∈J

g(wr,j , wl,h|h ∈ Ch(j), zj)

subject to sl,t = wl,t, sr,t = wr,t,P vr,t = zt,vr,t = zt −DpRe(sl,t)−DqIm(sl,t).

(23)

All variables are complex, however, each constraint can be decomposed into separateconstraints of the real and imaginary part e.g.

sl,j(t) = wl,j(t) ⇔

Re(sl,j(t)) = Re(wl,j(t))

Im(sl,j(t)) = Im(wl,j(t))

By defining

F =

II

P

G1 = [Re(Dp) 0 I ]

G2 = [Im(Dp) 0 0]

H1 = [Re(Dq) 0 0]

H2 = [Im(Dq) 0 I ]

and I0 = [0 0 I], I+ = diag(I, I, I), the constraints in (23) are equivalent to

FF

G1 H1

G2 H2

︸︷︷︸

A

Re(sl(t))Re(sr(t))Re(vr(t))Im(sl(t))Im(sr(t))Im(vr(t))

︸︷︷︸

ζ(t)

+

-I+-I+

-I0-I0

︸︷︷︸

B

Re(wl(t))Re(wr(t))Re(z(t))

Im(wl(t))Im(wr(t))Im(z(t))

︸︷︷︸

η(t)

= 0

for t ∈ T , where ζ(t), η(t) are introduced simply to condense the notation in the follow-ing. Let

ζ = (ζ(1), . . . , ζ(N)), η = (η(1), . . . , η(N)),

then, by the definition of ζ(t) and η(t) above, we let

c(ζ) =∑

i∈I


j∈J\I

cl(j, sl,j, sr,j)

g(η) =∑

j∈J

g(wr,j, wl,h|h ∈ Ch(j), zj)

whereby (23) is equivalently expressed as

minimizeζ,η

c(ζ) + g(η)

subject to Aζ(t) +Bη(t) = 0, t ∈ T .

226


This equivalent expression of (23) is now on the standard ADMM form [11], and may besolved iteratively by the following sequential updates of each set of variables:

ζk+1 = argminζ

c(ζ) +∑

t∈T

ρ

2‖Aζ(t) +Bηk(t) + µk(t)‖22

(24)

ηk+1 = argminη

g(η) +∑

t∈T

ρ

2‖γk+1(t) +Bη(t) + µk(t)‖22

(25)

µk+1(t) = µk(t) + γk+1(t) +Bηk+1(t) t ∈ T , (26)

where γk+1(t) = αAζk+1(t)−(1−α)Bηk(t). Above, k is an iteration index, and shouldnot be read as an exponent. The quantity µ(t) is the Lagrange multipliers for the equalityconstraints scaled by ρ. The iterates start from some initial guess ζ0, η0 and µ0. Theparameters ρ > 0 and α ∈ [1, 2) are design parameters of the algorithm and are known asthe ADMM parameter and over-relaxation parameter [13]. These affect the convergencespeed of the algorithm. A method for picking suitable values for these parameters isstill an open-ended question, although [13] presents some results for specific classes ofproblems.

Termination of the algorithm is based on the residuals

ξk+1(t) = Aζk+1(t) +Bηk+1(t)ψk+1(t) = ρATB(ηk+1(t)− ηk(t)),

(27)

known as the primal and dual residuals [11]. Choosing some absolute tolerance ǫabs, thealgorithm is stopped when

max(‖ξk+1‖2, ‖ψk+1‖2) ≤

√

6(n+ b)Tǫabs, (28)

where the scaling by√

6(n+ b)T is simply to account for problem size.The reader is referred to [11,12], for proofs of convergence for the ADMM algorithm.

For our purposes, it suffices to mention that the algorithm converges both in cost andfeasibility, i.e.

c(ζk) + g(ηk) → φ⋆, ‖ξk‖2 → 0, as k → ∞.

The following describes how the updates (24)-(26) renders a coordination strategy relyingonly on distributed information sharing.

Interpretation as neighbor based communication

Observe that all elements of the cost function in (23) are separable, i.e, there are no sharedvariables. The complicating factors only appear due to constraints. Notice also that theconstraints Aζ(t) +Bη(t) = 0 can be explicitly formulated

vr,j + (rj + jxj)Re(sl,j) + (xj − jrj)Im(sl,j) = zj (29)

sl,j = wl,j, sr,j = wr,j, (30)

vr,j = zh, h ∈ Ch(j), j ∈ J . (31)

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As discussed in Section 1, the coordination procedure of this work is designed for atfuture grid, where an advanced metering and control infrastructure is available. For thisreason, we assume that each consumer in the grid has a local dedicated computationdevice, as does each cable section and grid junction. In a practical setup, some of theseprivate computation devices may physically be the same, however, we shall treat themindividually in the following. We let

(sl,i, sc,i, vr,i), i ∈ I,

be private variables of each consumer, and appertaining leaf. Correspondingly, we let(sl,j , sr,j, vr,j), j ∈ J \I, be private variables of each branch. The private variables aregoverned by the dedicated computation device of each consumer or cable. Similarly, weassign private variables

(wr,j , wl,h|h ∈ Ch(j), zj), j ∈ J ,

to each bus-bar. We further assign the lagrange multiplier associated to the each con-straint, as a private variable of the corresponding busbar.

Notice in (29)-(31) that we have written all variables private to node j on the left ofthe equalities, and all variables external to node j on the right. From this it can be seenthat in order to make ζ-update (24), each branch and leaf j ∈ J needs to be providedvalues

(wkr,j(t), wkl,j(t), z

kj (t), z

kh(t)|h ∈ Ch(j)),

along with the current value of the associated lagrange multipliers. That is, the currentADMM variables and lagrange multipliers must be forwarded, but only from the imme-diate bus-bar, and the parent and children bus-bars, as illustrated in Fig. 4(left). In thefigure, each arrow represents a set of variables being communicated from one point inthe grid to another. The arrow base indicates where the variables are stored, whereas thearrow head indicates where they are communicated to.

Similarly, in order to conduct the η-update of the auxiliary variables in (25), andsubsequently the lagrange update in (26), each bus-bar j ∈ J needs only the values

(sk+1r,j , sk+1

l,h |h ∈ Ch(j), vk+1r,j , vk+1

r,Pa(j)),

which again needs only local data to be passed around as illustrated in Fig. 4(right). Fromthis it is apparent that the approach outlined here requires only local data to be passedaround, whereby the need for a central governor or controller is avoided.

5 Numerical example

The following numerical example demonstrates the coordination approach. The examplespans a 24 hour horizon starting at 8 AM, divided into 1 hour samples. We coordinate anetwork containing n = 34 consumers and b = 11 branches, corresponding in size to thebenchmark network examined in [4]. The topology of the network is presented in Fig. 5.The estimated price-signal, solar generated power production and baseline consumption,used in the coordination, are presented in Fig. 6, in per-unit (pu) measures with basevalues of 1 kVA and 400 V for power and voltage respectively.

228

5 Numerical example

i

wkl,i(t)

wkr,i(t) zki (t)

zkh(t)h∈Ch(i)

i

vk+1r,Pa(i)(t)

sk+1r,i (t) vk+1

r,i (t)

sk+1l,h (t)h∈Ch(i)

Figure 4: Left: Data passing prior to ζ-update: In order for the ith cable section to updateprivate variables, it needs the current ADMM and lagrange variables from the neighboringnodes in the grid. The lagrange variables are not explicitly drawn in the figure, since theyare each associated to one of the existing arrows. Right: Corresponding data passingprior to η-update of the ith bus-bar.

The price signal is provided in a generalized currency (¤/pu). Solar power is pre-sented as a average curve, from which each individual consumer will exhibit some ran-domly generated deviations. The reactive baseline consumption is derived from the active,by use of a constant power factor of 0.9 lagging, for all consumers. Flexible appliancesare distributed at random between consumers, such that each consumer has a 90 % chanceof having an appliance, with equal probability of the appliance being either, an EHP, EVor PV. From this assignment procedure, the following example includes 7 EHPs, 10 EVs,and 10 PVs.

The voltage variation constraint has been set to 0.08 pu. The actual limit in the Danish

3

1

8

6

8

1

2

0

1

4

0

11

10

9

8

7

6

5

4

3

2

1

Figure 5: Tree structure of the electrical grid. The horizontal lines represents bus-bars andvertical or sloping lines, represents branch cables. Consumers and leaves are not drawnexplicitly. The number to the left of each bus-bar represents the number of consumersconnected to that point in the grid, whereas the right number refer to the ordering of thebus-bars.

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Paper F

w(t)

[¤/p

u]

0.3

0.35

0.4

ppv(t)

[pu]

0

2

4

6p

c[p

u]

Time

12:00 17:00 22:00 03:00

0.3

0.4

0.5

Figure 6: Top: Estimated electricity price. Middle: Estimated solar power production.Bottom: Estimated Baseline consumption.

system is 0.1 pu [2]. However, as we employ an approximate model, a tighter boundallows for some deviation. The grid impedances resemble those employed in [2], scaledto give a baseline loss of 3.26 %.

We conduct the coordination employing the distributed strategy derived in the preced-ing sections. As this example is fairly small, a centralized solution can also be obtained,giving the global optimum for comparison.

In the distributed coordination scheme, we employ parameter values ρ = 0.1 andα = 1.9 which appears to work well for this problem. We set the absolute residualtermination tolerance as ǫabs = 5E-4. It has been experienced that the quadratic, relaxed,loss constraint in (9) is very computationally demanding in simulations. To improvecomputation speed we have in this example approximated the quadratic map (7) as apiecewise affine function:

zjl(u, y)(t) ≈ lapprj (u, y)(t) = max

(

Hj

[u(t)y(t)

]

+ gj

)

, j ∈ J (32)

where Hj ∈ RK×2, gj ∈ RK are the coefficients of the approximation, and K ∈ N is thenumber of affine functions used in the approximation. This approximation is included asthe implicit constraint in (10).

The convergence of ADMM for this example is presented in Fig. 7, showing that ter-mination accuracy is obtained after roughly 1500 iterations. The primal residuals decreasein a fairly even way, whereas the dual residuals exhibit substantially more variation. Thesevariations have not been explored in depth, however we speculate that this behavior re-lates to the inertia of the ADMM method, inherited from the way past iterations influencefuture updates. This is due to the resemblance to the variations investigated in [14].

Upon termination of the distributed algorithm, the coordinated power consumptionappears as in Fig. 8, where the centrally found global optimum is also presented. Asevident, the distributed solution is almost indistinguishable from the global optimum, andthe cost of the decentralized solution deviates from that of the global optimum by only2.3E-3 %.

230

5 Numerical example

s(k),r(k)

[-]

k [-]200 400 600 800 1000 1200 1400

10−2

10−1

100

Figure 7: Convergence of the primal (top), and dual residuals (bottom).

The voltage magnitude is presented in Fig. 9, showing the correspondence betweenthe voltage profiles found by the distributed algorithm, and the actual true voltage profilesfound by Gauss-Seidel load flow analysis, when employing the coordinated consumptionprofiles in Fig. 8. Despite the approximations introduced in Equation (5), the maximumvoltage error is only of 0.28 %.

The total loss error when comparing the losses found by the dirstributed algorithm,and those found by load flow analysis, accumulates to 10.7 %. This is partly due to thecrudeness of the affine approximation introduced in (32), but is mainly caused by thevoltage approximation vr,j ≈ vl,j ≈ 1 introduced in (4) and (6). It has been experiencedthat the loss error can be greatly reduced by employing a more educated guess of thevoltage, e.g. based in historical measurements.

The flexible consumption profiles obtained through coordination are shown in Fig. 10,along with those obtained by the benchmark strategy. As evident from Fig. 10(top), thereis not much difference between the consumption of EHPs in the benchmark and coordi-nated case. This is because the set-point tracking embedded in the discomfort measure in

pc(t)

[pu]

Time

12:00 17:00 22:00 03:00−10

0

10

Figure 8: The coordinated consumption pattern found centralized (Red), and distributed(Blue).

vr(t)

[pu]

Time

12:00 17:00 22:00 03:00

0.9

1

1.1

Figure 9: The approximated voltage profiles found during distributed coordination (Blue),and the true voltage profiles calculated by load flow analysis (Green).

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Paper F

pc,i(t)

[pu]

0.5

1

1.5

2

pc,i(t)

[pu]

−10−505

10q

c(t)

[pu]

Time

12:00 17:00 22:00 03:00

−1

−0.5

0

Figure 10: Top: The active power consumption pattern for i ∈ Iehp in the coordinatedcase (Red) and the benchmark case (Blue). Middle: Similar to the above, for i ∈ Iev.Bottom: Similar as above, for reactive consumption of all consumers.

(13) naturally distributes the EHP consumption over time. This implicitly decreases thelosses, rendering no benefit to be obtained by introducing temporal shifts. In that regard,the distributed coordination strategy derived here, arrives at similar results as disucssedby [4].

For EVs on the other hand, there is a large difference between the benchmark andcoordinated case, Fig. 10(Middle). In the benchmark case, the optimal charge scheduleis fairly obvious, since the vehicle should be fully charged during low-price periods, andfully discharge during high-price periods, in a fashion that leaves the vehicle fully chargedat the end of the horizon. In this way it is possible for the EV owner to make money byselling energy back to the grid. This would however cause significant over and undervoltages, and incur massive losses. In the coordinated case, the charging of vehicles ismuch more distributed across the horizon, in order to accommodate the cost of losses,and to satisfy voltage constraints. This is more clearly visible in Fig. 11, where theaccumulated EV consumption is plotted. Here it is clear that the benchmark strategy givesa bang-bang charge and discharging of vehicles, whereas the charging is smoothed outwhen it is coordinated. The main charge period in the coordinated case is in the beginningof the horizon, which is in fact to absorb the locally produced solar power visible inFig. 6(middle), rather than introducing losses by first exporting the solar power, and laterimporting power for charging. This is not a concern in the benchmark case. Finallyin Fig. 10(bottom), the reactive consumption is presented for all consumers, where weremark that negative consumption corresponds to production of reactive power. From thefigure it is clear that consumers with reactive capabilities either balance their own reactivebaseline consumption such that their local reactive power flow is zero, or an amountof reactive power is produced, in order to accommodate the consumption of consumerswithout reactive capabilities.

232

6 Conclusion and Perspectives

∑

i∈I

evp

c,i(t)

[pu]

Time

12:00 17:00 22:00 03:00−100

0

100

Figure 11: Accumulated consumption of EVs at each time step, using the benchmarkstrategy (Blue), and the distributed coordination (Red).

6 Conclusion and Perspectives

In this paper we have extended results from previous works, and illustrated how voltagecontrol can be included in coordination framework of the of flexible energy consumptionof residential consumers. The voltage control has been included in a way that requiresno central control unit, and allows for a completely distributed optimization, where com-munication with neighbors is the only requirement. Our framework includes a detailedmodel of the electrical grid, and includes concerns towards losses as an objective in thecoordination, along with private objectives for each consumer in the grid. Numerical re-sults have shown how our distributed framework converges towards the global optimumof the posed problem, and we have demonstrated how various flexible appliances maycontribute differently to the coordination.

Although not implemented, the framework presented for tree-structured graphs, doesallow for distributed termination of the coordination: Each node in the network is able toevaluate their local residuals (27), and locally estimate if the termination criterion (28) issatisfied. If any node has received ’satisfied’ notifications from all its children, and if thenode itself also estimates that the termination criterion is satisfied, it may send a ’satisfied’notification to its own parent as well. In this way, local satisfaction can propagate fromthe leafs towards the root, which can ultimately decide to terminate the algorithm.

For the sake of brevity, various relevant concerns have been disregarded in this work,but may readily be included with little or no changes to the framework. This includeslocal capacity constraints on power transport of each cable, preferred charge schedules ofEVs, penalties for charge variations, etc.

Acknowledgement

This work is supported by the Southern Denmark Growth Forum and the European Re-gional Development Fund, under the project ”Smart & Cool”. The authors greatly appre-ciate the inputs and comments of Euhanna Ghadimi, KTH, and Christoffer Sloth, AAU,during technical discussions.

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kom.aau.dkkom.aau.dk/project/smartcool/public_repository/... · Contents Preface IX Abstract XI Synopsis XIII 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . .