The Price of Synchrony: Evaluating Transient Power Losses

The Price of Synchrony:

Evaluating Transient Power Losses in

Renewable Energy Integrated

Power Networks

EMMA SJODIN

Master’s Degree ProjectStockholm, Sweden August 2013

XR-EE-RT 2013:023

The Price of Synchrony:Evaluating Transient Power Losses in Renewable

Energy Integrated Power Networks

EMMA SJÖDIN

Master’s ThesisSupervisor: Dennice F. GaymeExaminer: Henrik Sandberg

XR-EE-RT 2013:023

iii

Abstract

This thesis investigates the resistive losses incurred in returning a powernetwork to a synchronous state following a transient stability event, or in main-taining this state in the presence of persistent stochastic disturbances. Wequantify these transient power losses, the so-called “Price of Synchrony”, usingthe squared H2 norm of a linear system of generator and load dynamics subjectto distributed disturbances. We first consider a large network of synchronousgenerators and use the classical machine model to form a system with cou-pled second order swing equations. We then extend this model to explicitlyinclude dynamics of loads and asynchronous generators, which represent solarand wind power plants. These elements are modeled as frequency-dependentpower injections (extractions), and the resulting system is one of coupled first-and second order dynamics. In both cases, the disturbance inputs representpower fluctuations due to transient stability events or the inherent variabilityof loads and intermittent energy sources.

The network structure is captured through a weighted graph Laplacian ofthe network admittance. In order to simplify the analysis for both models,we use the concept of grounded graph Laplacians to obtain an asymptoticallystable reduced system. We then evaluate the transient losses in the reducedsystem and show that this system’s H2 norm is in fact equivalent to the H2norm of the original system. Furthermore we show that although the transientbehaviours of the first order, second order or mixed dynamical systems are ingeneral fundamentally di�erent, for same-sized networks they may all have thesame H2 norm if the damping coe�cients are uniform.

The H2 norms for both system models are shown to be a function of trans-mission line and generator properties and to scale with the network size. Thesetransient losses do not, however, depend on the network connectivity. This isin contrast to related power system stability notions that predict better syn-chronous stability properties for highly connected networks. The equivalenceof the norms for di�erent order systems indicate that renewable energy sourceswill not increase transient power losses if their controllers can be adjusted tomatch the dampings of existing synchronous generators. However, since thelosses scale linearly with the number of generators, our results also demon-strate that increased amounts of distributed generation in low-voltage gridswill lead to larger transient losses, and that this e�ect cannot be alleviated byincreasing the network connectivity.

iv

Sammanfattning

I den här rapporten utvärderar vi de resistiva förluster som uppstår i ettelektriskt nätverk då det återgår till ett synkroniserat tillstånd efter en stör-ning. Dessa transientförluster, som vi benämner ”synkronismens pris”, utvär-deras med hjälp av H2 -normen för ett linjärt dynamiskt system. I ett förstasteg modellerar vi ett stort nätverk av synkrongeneratorer och erhåller ett sy-stem med kopplade svängningsekvationer av andra ordningen. Sedan utvidgasdenna modell för att även omfatta dynamiska laster och asynkrongenerato-rer, som ofta används tillsammans med sol- och vindkraft. Dessa modellerassom frekvensberoende kraftinjektioner och det slutgiltiga systemet beskriverett sammankopplat nätverk med både första och andra ordningens dynamik. Ibåda fallen utsätts systemet för spridda störningar, som kan representera bå-de nätverksfel och fluktuationer i elförsörjningen orsakade exempelvis av vind-eller solkraft.

För att utvädera transientförslusterna används först en typ av reducera-de, eller ”jordade”, laplacianer för att beskriva ett reducerat system som ärasymptotiskt stabilt. Vi visar sedan att H2 -normen för det ursprungliga sy-stemet inte påverkas av denna reduktion. Systemets H2 -norm visar sig beropå egenskaper hos generatorer och kraftlinor och växa linjärt med storleken pånätverket. I motsats till typiska resultat för stabilitet i elkraftsystem som visaratt starkt sammankopplade nätverk har bättre synkroniseringsegenskaper änsvagt sammankopplade, visar dock våra resultat att transientförlusterna inteberor på nätverkstopologin.

Vidare visar vi att, trots att transienter hos system med första ordning-ens, andra ordningens eller kombinerad dynamik skiljer sig kraftigt åt, så kanderas H2 -normer vara lika för nätverk av samma storlek med lika dämpnings-koe�cienter. Dessa resultat indikerar att nätanslutna förnybara energikällorinte kommer att öka transientförlusterna om deras regulatorer kan bli anpas-sade till dämpningen hos befintliga synkrongeneratorer. De visar dock ocksåatt en ökad utbredning av distribuerad generation, särskilt i mellan- och låg-spänningsnät, kommer att öka transientförlusterna eftersom de växer linjärtmed antalet generatorer, samt att denna e�ekt inte kan mildras genom att ökaantalet anslutningar.

v

Acknowledgements

I am most grateful to Prof. Dennice F. Gayme of the Department ofMechanical Engineering at the Johns Hopkins University (JHU) for herintelligent, supporting and friendly advising throughout this degree project.My sincere thanks also for making my visit at JHU possible.

Together, we are thankful to Bassam Bamieh of the University of Californiaat Santa Barbara for a fruitful collaboration. The support of NSF throughgrant ECCS-1230788 is also gratefully acknowledged.

Furthermore, I would like to express my gratitude to Henrik Sandberg ofKTH Royal Institute of Technology for his most insightful advice and severalrewarding discussions. I am thankful for his genuine interest in my work andfor taking the time to discuss it even while on travels.

I would also like to thank Prof. Louis L. Whitcomb and Prof. BenjaminF. Hobbs together with their research groups for a number of interestingdiscussions, which enriched both this thesis and my stay at JHU.

Emma SjödinStockholm, August 2013

Contents

Contents vi

1 Introduction 1

1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Preliminaries 7

2.1 Power System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 Classification of Power System Stability . . . . . . . . . . . . 82.1.2 The Swing Equation . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Network Descriptions and Graph Laplacians . . . . . . . . . . . . . . 102.2.1 The Admittance Matrix . . . . . . . . . . . . . . . . . . . . . 102.2.2 Consensus Dynamics and Graph Laplacians . . . . . . . . . . 112.2.3 Properties of Graph Laplacians . . . . . . . . . . . . . . . . . 12

2.3 The H2

Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Renewable Power Generation . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Synchronous vs. Asynchronous Generators . . . . . . . . . . 152.4.2 Wind Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.3 Other Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Resistive Losses in Synchronizing Power Networks 19

3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.1 System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.2 Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Evaluation of Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.1 System Reduction . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 H

2

Norm Calculation . . . . . . . . . . . . . . . . . . . . . . 233.2.3 Special Case: Equal Line Ratios . . . . . . . . . . . . . . . . 263.2.4 H

2

Norm Interpretations for Swing Dynamics . . . . . . . . . 273.3 Generalizations and Bounds . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.1 Network-Characteristic Bounds on Losses . . . . . . . . . . . 293.3.2 Generator Parameter Dependence . . . . . . . . . . . . . . . 30

3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

vi

CONTENTS vii

3.4.1 Line Ratio Variance . . . . . . . . . . . . . . . . . . . . . . . 313.4.2 Increased Network Size . . . . . . . . . . . . . . . . . . . . . 323.4.3 Marginal Losses for Added Lines . . . . . . . . . . . . . . . . 333.4.4 E�ects of Generator Placement . . . . . . . . . . . . . . . . . 35

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Losses in Renewable Energy Integrated Systems 39

4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1.1 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.2 Model of Asynchronous Machines . . . . . . . . . . . . . . . . 414.1.3 System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 434.1.4 System Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.5 Performance Metric . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Input-Output Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.2 H

2

Norm Calculations . . . . . . . . . . . . . . . . . . . . . . 474.2.3 Properties of the Augmented Network Laplacians . . . . . . . 484.2.4 Relation to Previous Results . . . . . . . . . . . . . . . . . . 49

4.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.1 Increased Synchronous Damping . . . . . . . . . . . . . . . . 514.3.2 E�ects of Generator Placement . . . . . . . . . . . . . . . . . 51

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Conclusions and Directions for Future Work 55

A Appendices to Chapter 3 57

A.1 Proof of Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 57A.2 Proof of Lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 59A.3 H

2

Norm With Simultaneously Diagonalizable Laplacians . . . . . . 59A.4 Proof of Corollary 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 61

B Appendices to Chapter 4 63

B.1 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

List of Figures 65

Bibliography 67

Chapter 1

Introduction

The electric power system is undergoing large and rapid changes, primarily due tothe growing interest in replacing fossil fuel-based power generation with renewableenergy sources. Factors driving this replacement are growing concerns about cli-mate change and global warming, diminishing supplies of fossil fuels causing priceincreases [7] and government mandates world-wide [31]. The Nordic countries, in-cluding Sweden, state some of the most ambitious goals for the energy sector andaim to have a carbon-neutral energy system by 2050 [32]. Although the transportsector and industry account for large portions of energy consumption, the powergrid will also need to become “greener” by substantial integration of renewable en-ergy. Figure 1.1 shows the projected total energy supply in the Nordic countries by2050 compared to 2010.

In the United States, Maryland’s Renewable Portfolio Standard (RPS) pre-scribes 20 % of the state’s electricity demand to be covered by renewables by2022 [38], and several similar initiatives exist in other states [54]. On a globallevel, the German Energiewende or Energy Transition initiative is also worth men-tioning. Its goal to phase-out all nuclear power by 2022 and subsequent policieshave led to remarkably large investments in residential solar panels and an over-all renewable penetration of 25 % in 2012, which is expected to rise to 40 % by2020 [41]. Furthermore, new types of decentralized power grids, often with highrenewable penetration, are becoming prevalent in the developing world, since theserequire smaller investments than conventional centralized power systems [61].

A high grid penetration of renewables, however, poses a number of challenges tothe power system. The inherent intermittency of wind and solar power generationcauses high levels of uncertainty [54,56], and their typically much smaller capacitiesthan conventional generators will make the future generation system much moredistributed than today’s [61]. The use of electricity in the tranport sector throughelectric vehicles and customer programs for demand response will also contributeto changing load patterns [45]. Many of these changes will, apart from posingoperational and market-related challenges, a�ect the dynamics and stability of thepower system. For example, the variability of wind and solar power will lead to

1

2 CHAPTER 1. INTRODUCTION

Figure 1.1: Primary energy mix in the Nordic countries (Sweden, Norway, Denmark,Finland and Iceland) in 2010 and 2050. Data source: [32]

more frequent and higher amplitude disturbances, that have the potential to a�ectthe rotor-angle or synchronous stability, which is the ability of the power systemto regain synchrony when subject to a disturbance [44]. Synchrony refers to thecondition when the frequency of all generators within a particular power networkare aligned, and there are no angular swings in the system [42,44]. Loss of synchronymay lead to black-outs [2] and a secure system operation therefore relies on stabilityof the power system. Renewable generators have di�erent dynamical propertiesthan conventional generators and as their penetration grows, this change has thepotential to a�ect the stability of the grid [23,52]. This thesis is part of an ongoingresearch trend to characterize the dynamics of renewable energy integrated powersystems.

The problem of synchronization in power networks is analogous to the problemof distributed control in complex networks, and we therefore review some recentwork on deriving stability conditions for such systems in Section 1.2. In this the-sis however, the concept of synchronization in renewable energy integrated powernetworks is studied in a di�erent context. We assume that the network will re-turn to a synchronized state after disturbances and instead focus on the controle�ort required to maintain this synchrony. Loss of synchronism leads to circulatingpower flows passing between generators whose angles are out of phase, which in turnleads to resistive losses over the power lines due to their non-zero line resistances.These transient losses are generally considered relatively small compared to the to-tal real power flow in a typical power network. It is, however, unclear whether theywill remain small in power grids of the future, which are expected to have highlydistributed generation, and consequently many more generators than today’s grid.The transient losses, i.e., the real power required to drive the system to a stable,synchronous operating condition is what we term the “Price of Synchrony”.

1.1. SCOPE 3

1.1 Scope

In this thesis, the transient resistive power loss – the price of synchrony – is eval-uated for large power networks, for which we formulate the dynamics as a lineartime-invariant (LTI) system of coupled generator swing equations. We considerscenarios in which the network encounters single distributed impulse disturbancesor is subjected to persistent stochastic noise, and show that the transient restistivelosses are, in both cases, given by the squared H

2

norm of this LTI system.We begin by considering a network of synchronous generators, which according

to the so-called classical machine model can be modeled by coupled second orderoscillator dynamics. The network structure is captured through a weighted graphLaplacian of the network admittance. In order to simplify the analysis, we usethe concept of grounded graph Laplacians to first evaluate the resistive losses for areduced, or grounded, system in which one of the generators is taken as a reference.We then show that the H

2

norm of the original system is equivalent to that of thereduced system. This squared H

2

norm is shown to be a function of the power lineand generator damping properties and to scale with the network size. However, incontrast to typical power systems stability notions, which predict highly connectednetworks to have better synchronous stability properties, our results show that thetransient losses are independent of the network connectivity. Therefore, if one wantsto minimize losses in a system where power flows are used to maintain synchrony,the size of the network is more important than its topology. The fact that the lossesgrow linearly with the number of generators is of increasing importance as powergeneration becomes more distributed, particularly in low-voltage distribution grids.

The aforementioned results remain valid in the second part of the thesis, wherethe model is extended to capture loads as well as renewable sources grid-connectedby asynchronous generators. This is done by coupling the previous second orderoscillators to nodes with first order dynamics, which are shown to capture the es-sential dynamical properties of asynchronous machines. The results here show thatalthough the transient behaviours of systems of first order, second order and mixedcoupled oscillators are in general fundamentally di�erent, for networks of equal sizethey may all have the same H

2

norm provided that their damping coe�cients areequal. This indicates that connecting renewable energy sources to a network will notincrease the system losses if their controllers can be adjusted to match the dampingcoe�cients of the existing synchronous machines.

The theoretical considerations and results outlined above are complementedby numerical examples and simulation studies. In particular, we study how, inheterogenous generator networks, the placement of generators a�ects the transientpower losses. These are found to be reduced if highly damped generators are alsoplaced at highly interconnected nodes in the network.

Since synchronization in power networks is a type of networked control problem,many results derived in this thesis are more widely applicable to e.g. robotic orbiological systems. What we term the price of synchrony can then be generalized toa type of energy measure and the results, particularly on topology and model orderindependence, may also have interesting consequences for these types of networks.

4 CHAPTER 1. INTRODUCTION

1.2 Related Work

A special case of the problem of rotor-angular or synchronous stability is the tran-sient stability problem, which is associated with large angular disturbances due toe.g. generator or line failures, or the intermittency of the power sources in a renew-able energy integrated system. There is a large body of transient stability literaturefrom the last decades, see [55] for an excellent survey. This work generally focuseson determining regions of attraction of synchronous states and finding Lyapunovlike energy functions to show stability in these regions, as in e.g. [43].

For general complex networks, such as biological or digital systems, the conceptof synchronization and formal stability criteria linked to network properties, havespurred interest across many fields, a good summary of such work is found in [51].Recently, connections between such distributed control problems and power systemsstability have been drawn. A particular set of works [10,11], which shows an equiva-lence between power system dynamics and a first order model of so-called Kuramotooscillators has gained much attention. That modeling framework provides su�cientanalytical conditions for frequency and phase synchronization [10], as well as a linkbetween structure preserving power system models [11], like the ones that will beused in Chapter 4 of this thesis, and reduced models such as those discussed inChapter 3. While the work in [10, 11] makes limiting assumptions on the networkproperties, the authors of [42] use a slightly di�erent approach to derive stabilitycriteria in heterogenous networks, but with uniform generators, considering a modelmuch like the ones employed in this thesis.

In this thesis, the damping properties of the generators, both synchronous andasynchronous, will prove to be important for the transient resistive power losses.In [37], a type of system-wide damping is studied, using a non-linear version ofthe coupled first- and second order oscillator dynamics similar to those which weintroduce in Chapter 4. In that work, principles are derived to improve this damp-ing, i.e., the rate of convergence in the system, by studying the connectivity of astate-dependent graph Laplacian. The model employed by the authors of [37], asin Chapter 4 of this thesis, is based on a network-preserving dynamical model firstintroduced in [5].

To our knowledge, this type of coupled first- and second order oscillator modelhas not previously been used in order to model dynamics of renewable integratedpower networks. Instead, much of the work on stability of such networks focuses onmodeling the dynamics of a particular subset of the system, such as the wind farm,as in [14, 21]. Alternatively, due to the complexity of the problem, such studiesare conducted purely by simulations as in [1, 33]. There is a hope that the controlsystems of modern wind farms with so-called doubly-fed induction generators (seeSection 2.4) can be employed to stabilize the power system, and there is a largeamount of ongoing work to explore this potential, see e.g. [16,17] or, for a survey, [58].

There is also a body of related work on the theoretical concepts applied inthis thesis. Consensus dynamics in large-scale networks, such as vehicle formationproblems, result in models similar to the ones used in this thesis. The coherence

1.2. RELATED WORK 5

of such networks was explored in a recent well-cited study [4]. In that study theH

2

norm is used as a performance measure which quantifies the error variance.The authors then apply di�erent control strategies, and study how this norm scalesasymptotically with the network size. The authors of [49] use a similar notionof the H

2

norm in dynamical networks, and define a concept of “LQ

-energy” as arobustness measure. Bounds on this energy measure are presented and characterizedfor various graph types, and it is shown that the “L

Q

-energy” corresponds to the“Price of Synchrony”, which was first introduced in [3] and later studied in thisthesis.

Chapter 2

Preliminaries

In the remainder of the thesis, dynamical models of the power system will be derivedand evaluated. This chapter provides some theoretical background to the concept ofpower system dynamics, network descriptions as well as to some aspects of renewablepower generation. A brief review of the main means of evaluation applied in thisthesis, the H

2

system norm, will also be presented.

2.1 Power Systems Dynamics and Stability

An electricity consumer in an industrial country is used to a secure and reliablesupply of electricity in the wall socket, with correct voltage and frequency. Thissupply is ensured by a functioning grid infrastructure and power generators, whichat every instant inject to the grid an amount of power that exactly balances theaggregated demand. If this balance is fulfilled, and there is an equilibrium betweenthe rotating generators and the grid, we say that the power system operates at asteady state.

However, the system is constantly exposed to disturbances, and several dynamicphenomena occur on di�erent time scales. A prerequisite for a secure system op-eration is therefore that the power system is stable. Power system stability canbe defined as the ability of an electric power system to regain a state of operatingequilibrium after being subjected to a physical disturbance [44]. Lack of stabilitymay lead to blackouts, like the one in southern Sweden in 1983 when 2/3 of thecountry’s network was shut down [35], or the major Northeastern blackout of 2003which a�ected 50 million people in the United States and Canada [22].

In this section, we will review di�erent forms of power system stability beforeintroducing the swing equation, which is used to analyze the rotor angular, orsynchronous, dynamics and stability, which will be the focus of this thesis.

7

8 CHAPTER 2. PRELIMINARIES

Figure 2.1: The principle of a synchronous generator. In steady state, the mechanicalpower input P

m

and its torque Tm

balance the output electrical power Pe

to thegrid and its counter torque T

e

on the generator. The generator rotor’s frequency Êis then equal to the system frequency, but an imbalance will cause an accelerationor deceleration of the rotor.

2.1.1 Classification of Power System StabilityAlthough the issue of power system stability is essentially a single problem, it is use-ful to look at the di�erent forms of instabilities that may occur separately [44]. Onethen obtains three di�erent stability notions. Issues related to the global generation-load balance mentioned in the introduction to this section are frequency stabilityphenomena. Voltage stability refers to the ability of the system to maintain asteady and high voltage level by avoiding local imbalances in reactive power, oftendue to large loads. In this thesis however, phenomena connected to rotor angularor synchronous stability will be considered. This refers to the ability of the powersystem to regain synchrony after a disturbance and depends on the ability of thesynchronous machines to maintain or restore an equilibrium between their rotatingcomponents and the grid’s electromagnetic torque [44]. We will elaborate on thisin the following section.

Power system dynamics are inherently non-linear and whether or not the sys-tem will stabilize after a disturbance is therefore highly dependent on the initialoperating point and the size of the disturbance. However, a subset of the rotorangle stability issues concern small-signal (or small-disturbance) stability, which isthe ability of the system to maintain synchrony when subject to small disturbancesthat allow the system to be analyzed in terms of linearized equations. This thesiswill only model such small disturbances and the considered power system dynamicswill be linear.

2.1.2 The Swing EquationAccording to a model often referred to as the classical machine model [55], thepower system can be regarded as a network of oscillators. The electromechanicaloscillations that arise due to an imbalance are then described by the swing equationfor synchronous generators, which we will now derive.

Under steady state conditions, each generator i œ {1, . . . , N} is fed a mechanicalpower P

m,i

from the plant which is equal to the electrical power output to the gridP

e,i

. The generator rotor will then rotate with a constant frequency Êi

and a certain

2.1. POWER SYSTEM DYNAMICS 9

phase angle ◊i

(also called bus or rotor angle). If the system is perturbed, however,so that the equilibrium between the power input and output is lost, the rotor willaccelerate or decelerate according to

Mi

◊i

+ —i

◊i

= Pm,i

≠ Pe,i

, (2.1)

where Mi

is the generator’s inertia constant and —i

its damping coe�cient. Theresulting rotor angle deviations propagate to the other generator buses over thenetwork lines according to the power flow equation:

Pe,i

= gi

|Vi

|2 +ÿ

j≥i

gij

|Vi

| |Vj

| cos(◊i

≠ ◊j

) +ÿ

j≥i

bij

|Vi

| |Vj

| sin(◊i

≠ ◊j

), (2.2)

where |Vi

| is the voltage magnitude at bus i and j ≥ i denotes a line between buses iand j. The coe�cients b

ij

and gij

are respectively the conductance and susceptanceof that line and g

i

is the shunt conductance of bus i (see also Section 2.2.1).We now apply the standard DC power flow approximation to linearize equation

(2.2). This type of linearization, which is particularly applicable to power transmis-sion systems [34], assumes:

i. a flat voltage profile; Vi

= V0

, ’i = 1, ..., N ,

ii. that the line resistance is negligible compared to the reactance in all lines, and

iii. that the voltage angle di�erences (◊i

≠ ◊j

) are small between all nodes i, j.

Enforcing these assumptions and without loss of generality assuming V0

= 1 p.u.1,we obtain

Pe,i

¥ÿ

j≥i

bij

(◊i

≠ ◊j

). (2.3)

Substituting this into (2.1) leads to

Mi

◊i

+ —i

◊i

= ≠ÿ

j≥i

bij

[◊i

≠ ◊j

] + Pm,i

. (2.4)

This is the linear version of the swing equation in the classical machine model, whichcaptures the power system dynamics relevant to this thesis.

A mechanical analogy to these power system dynamics is shown in Figure 2.2,which depicts a network of three coupled oscillators. Each oscillator has a phaseangle ◊

i

and a speed Êi

= ◊. Any deviations from a steady state will propagateacross the network over the springs, whose sti�ness coe�cients are analogous to thesusceptances b

ij

in (2.4).1“p.u.” stands for “per unit” and indicates that the quantity is normalized with respect to a

system-wide base unit quantity, in this case a base voltage. The per unit system is widely usedwithin power systems analysis and power engineering to simplify calculations [47].


b12

b23

b13

◊1

Ê1

Ê2

Ê3

Figure 2.2: Mechanical analogy to the power system dynamics and the swing equa-tion (2.4). A deviation of one oscillator’s phase angle ◊

i

and/or its derivative Êi

willpropagate across the springs with sti�ness b

ij

to the other oscillators. This analogyis due to the authors of [13].

2.2 Network Descriptions, Graph Laplacians andConsensus Problems

In the previous section, we derived the power system dynamics as oscillations acrossa network. In this section, we will introduce the admittance matrix, which is usedto describe the topology and physical properties of the power network. This admit-tance matrix is a type of graph Laplacian or Laplacian matrix; a matrix represen-tation of a network or graph.

Graph Laplacians arise naturally in state space formulations of so-called consen-sus problems, in which a system of agents cooperate with a certain control objective.Since the coupled oscillator dynamics (2.4) are a type of such consensus dynamics,this type of problem will also briefly be reviewed at this stage, along with propertiesof graph Laplacians that will be made use of later on.

This section’s review is largely based on [6], [30] and [57], in which elaborationson the introduced concepts can be found. The literature on these subjects, however,is vast.

2.2.1 The Admittance MatrixThe admittance matrix (also called nodal, graph or bus admittance matrix) is amathematical abstraction of the electric network which describes the network’stopology and the physical properties of its lines.

Consider a network (graph) of the set N = {1, . . . , n} nodes (buses) and let thetwo nodes i, j œ N be connected by a line (edge) with the impedance z

ij

= rij

+jxij

,where r

ij

is the line’s resistance and xij

is its reactance. An example of such anetwork for N = 7 is found in Figure 3.1. The inverse of the impedance is calledthe admittance:

yij

= 1z

ij

= gij

≠ jbij

,

where gij

= rij

r

2ij+x

2ij

and bij

= xij

r

2ij+x

2ij

are respectively the conductance and suscep-

2.2. NETWORK DESCRIPTIONS AND GRAPH LAPLACIANS 11

tance of the line. Furthermore, each node i œ N may have a shunt conductance gi

which is the conductance of the node’s connection to ground.Now we can define the admittance matrix Y by:

Yij

:=

Y___]

___[

gi

+ÿ

k≥i

(gik

≠ jbik

), if i = j,

≠(gij

≠ jbij

), if i ”= j and j ≥ i,

0 otherwise.

(2.5)

where j ≥ i denotes a line between nodes i and j. The diagonal elements Yii

of theadmittance matrix is the self-admittance of node i and is equal to the sum of theadmittances of all lines incident (including the shunt) to that node.

Y can be partitioned into a real and an imaginary part and we continue to define

Y = (LG

+ G) ≠ jLB

, (2.6)

where LG

is called the conductance and LB

the susceptance matrix. G is a diagonalmatrix of the shunt conductances, which will be irrelevant for the remainder of thisthesis.

LG

and LB

are equivalent to weighted graph Laplacians, where the weights arerespectively the conductance and susceptance of each edge in the graph. In thefollowing sections, another context where such weighted Laplacians arise as well astheir properties will be discussed.

2.2.2 Consensus Dynamics and Graph LaplaciansConsider a system of n agents: x

i

= ui

, i = 1, . . . , n where the control objectiveis for all agents to eventually reach the same state x

1

(t) = x2

(t) = · · · = x(t), i.e.,consensus. If the control u

i

is decentralized and merely based on the relative errorsx

j

≠ xi

that agent i measures to its neighbors j œ Ni

, one control strategy is

ui

(t) =ÿ

jœNi

aij

(xj

≠ xi

).

In order to write this system on state space form, we define the weighted graphLaplacian L by

Lij

:=

Y___]

___[

ÿ

kœNi

aik

, if i = j,

≠aij

if i ”= j and j œ Ni

,

0 otherwise,

(2.7)

where aij

are positive weights of the graph which describes how the agents (nodes)are connected. The elements on the diagonal L

ii

, are called the degree of node iand is the sum of the weights of all edges incident to that node. In the special casewhere all edge weights a

ij

= 1, the degree is the number of incident edges.


Figure 2.3: A network of n robots, where the lines symbolize communication linkswith positive weights a

ij

.

Now, if we define the state vector x = (x1

, . . . , xn

)T , the consensus dynamicscan be written:

x = ≠Lx. (2.8)

If the graph is connected, i.e., if there is a path between any two agents in thenetwork, then the control objective, consensus, will be achieved (see e.g. [30] for aproof). The coupled oscillator dynamics derived in the coming chapters will be atype of second order consensus dynamics, but the principle is the same as in (2.8),and x represents the synchronized state.

2.2.3 Properties of Graph LaplaciansWe now consider a n-dimensional weighted graph Laplacian L defined as in (2.7)and list some of its properties:

i. Symmetry. For undirected graphs considered in this thesis, the edge from nodei to node j is identical to the edge from node j to node i. Therefore, L

ij

=L

ji

’i, j œ {1, . . . , n}, and L is symmetric.

ii. Zero row/column sums. Since Lii

= ≠ qj ”=i

Lij

, all rows and columns sum to0. That means that all graph Laplacians have as common eigenvector the vector1 with all components equal to 1, i.e.,

L1 = 0,

corresponding to the eigenvalue 0. Graph Laplacians are thus singular.

iii. Positive semidefiniteness. If the graph underlying the Laplacian is connected(i.e. any two nodes are connected by a path of edges), then, apart from thesimple zero eigenvalue, remaining n ≠ 1 eigenvalues are positive. If the graph isnot connected, the multiplicity of the zero eigenvalue will equal the number ofisolated graphs.

2.3. THE H2 NORM 13

iv. Diagonalizability by unitary matrix. Since L is symmetric, it can be diagonalizedby a unitary matrix U whose columns are orthonormal (i.e., UúU = I), suchthat L = Uú�U , where � = diag{⁄

1

, ⁄2

, . . . , ⁄n

} is a diagonal matrix of L’seigenvalues 0 = ⁄

1

Æ ⁄2

Æ . . . Æ ⁄n

.

2.3 The H2

NormIn this thesis, power system dynamics will be formulated as a linear time-invariant(LTI) system, representing swing dynamics as derived in Section 2.1.2 excited bydisturbance inputs. We will also define an output signal representing the resistivelosses in the network. A general such input-output system H can be written

x(t) = Ax(t) + Bw(t) (2.9)y(t) = Cx(t),

where x œ Rn, w œ Rm and y œ Rp. Its p ◊ m-dimensional transfer matrix is givenby G(s) = C(sI ≠ A)≠1B. If the system is asymptotically stable, we can define itsH

2

norm by||G||2H2 = 1

2fi

⁄ Œ

≠Œ||G(jÊ)||2

F

dÊ, (2.10)

where || · ||F

denotes the Frobenius norm.2 The H2

norm characterizes the system’sinput-output behaviour by, in a sense, quantifying the e�ect an input w has on theoutput y, alternatively the “size” or energy of the system. In control design, it isoften a control objective to keep the H

2

norm below a given limit, and the feedbackis chosen accordingly [26].

The integral in (2.10) is however rarely evaluated in the frequency domain usingG(jÊ), but can instead be evaluated conveniently in the time domain, directly fromthe state space representation H. This will be the only representation used in thisthesis. Through calculations omitted here it can then be found that

||H||2H2 = tr(BúXB), (2.11)

where X is the observability Gramian given by

AúX + XA = ≠CúC.3 (2.12)

The matrix equation (2.12) is referred to as the Lyapunov equation.In this thesis, we will use the H

2

norm to evaluate the resistive losses in sys-tems of oscillating generators during the synchronization transient. This usage issupported by some of the H

2

norm’s standard interpretations, of which three are2The Frobenius norm is defined as the sum of the absolute values of all entries in a matrix:

||A||2F =qn

i=1qm

j=1 |aij |2 = tr(AúA).3||H||H2 can also be calculated using the controllability Gramian XC ; ||H||2H2 = tr(CXCCú),

with AXC + XCAú = ≠BBú. This formulation will however not be used in this thesis.


presented below. The physical meaning of these interpretations for our particularsystem and the context in which they can be used to quantify the transient resistivelosses will be discussed Section 3.2.4.

The H2

norm of the LTI system (2.9) can be interpreted as follows (see e.g. [24]or [53]):

i. Response to white noise. Let the input w be “white noise”, i.e., a stochasticprocess such that the covariance E{w(·)wú(t)} = ”(t≠·)I, where I is the identitymatrix and ” the Dirac delta function. Then the (squared) H

2

norm representsthe sum of the steady-state variances of the output’s components:

||H||2H2 = limtæŒ E{yú(t)y(t)}.

This variance is the expected value of the sum of the squares of all the output’scomponents.

ii. Response to a random initial condition. The H2

norm can also be used torepresent a system response to a certain initial condition when there is no inputto the system, i.e. w(t) = 0 ’t. If the initial condition is a zero-mean randomvariable x

0

which has covariance E{x0

xú0

} = BBú, then the H2

norm (squared)is the time integral

||H||2H2 =⁄ Œ

0

E{yú(t)y(t)}dt

of the resulting transient response. This interpretation is closely related to inter-pretation (iii):

iii. Sum of impulse responses. If H were a single-input-single-output (SISO) system,the H

2

norm would be the signal energy of a simple impulse response at sometime t

0

: w(t) = ”(t ≠ t0

). Here, we are considering a system with multiple inputsand outputs (MIMO) and the H

2

norm then represents the sum of many suchimpulse responses; one over each channel.Let e

i

denote the ith unit vector in the m-dimensional input space and let therebe m “experiments” where the system is fed an impulse at the ith channel, i.e.,w

i

(t) = ei

”(t ≠ t0

). If the corresponding output signal is yi

(t), then the systemH

2

norm (squared) is the sum of the L2

norms of these outputs, i.e.:

||H||2H2 =mÿ

i=1

⁄ Œ

0

yúi

(t)yi

(t) dt.

2.4 Renewable Power GenerationA large scale introduction of renewable energy sources to the power grid is, as men-tioned in Chapter 1, apart from introducing high levels of disturbances, likely tochange the dynamic behaviour of the power system. This is due to a new kind ofgeneration; while a power system with mostly conventional generation is dominated

2.4. RENEWABLE POWER GENERATION 15

by few very large synchronous generators with large inertias, the renewable energyintegrated system has many generators, often asynchronous, with small or no iner-tia. To a certain extent, power injections by renewable sources can be modeled asnegative load, such that the resulting load is a type of net demand, but as integra-tion levels grow, more physically accurate models are required. We will propose asimple such model for a dynamical analysis of renewable energy integrated systemsin Chapter 4.

In this section, we will briefly review some basic properties of synchronous andasynchronous generators and discuss their usage with di�erent type of power sources.The reader should be aware that the term “asynchronous” is in this thesis somewhatabused, and used to denote all machines which are not synchronous, i.e., not onlyinduction machines for which the term is commonly used, but also e.g. powerconverters.

2.4.1 Synchronous vs. Asynchronous Generators

Traditionally, the power system is dominated by synchronous generators, or alterna-tors. As discussed in Section 2.1.2, the rotor of a synchronous generator rotates witha speed corresponding precisely to the grid frequency f

0

(provided a synchronousstate), according to

Ê0

= 2fif0

p,

where p is the number of magnetic poles in the rotor. An example where p = 4is shown in Figure 2.4. Very simplified, power is generated when the rotor angleleads the grid angle. When a synchronous generator is started, it needs to be runto synchronous speed o�-line, before being connected to the grid [39].

In an induction generator however, there is no obvious relationship between thefrequency and phase of the power output and the generator rotor position. Usually,the induction generator rotor spins about 2 ≠ 3% faster than synchronous speed,generating a certain slip s;

s = Ê0

≠ Ê

Ê0

.

The stator, which surrounds the rotor, is namely excited by the grid, and for apower to be induced, there needs to be a negative slip so that the rotor cuts themagnetic flux in the stator coils [39]. The same machine can also operate as a motor,if the rotor spins at a speed slower than synchronous speed. If s = 0, active powerwill neither be generated nor withdrawn from the grid, but the stator will remainexcited and therefore act as an impedance load drawing reactive power, which maybe disadvantageous from a grid perspective [39,60]. Note also that since the powerinput or output from an induction machine depends on the slip, it is also dependenton the grid frequency.


Figure 2.4: A 4-pole synchronous generator.

2.4.2 Wind PowerWind power generation stands for the largest portion of installed renewable energy(disregarding hydro power) [9] and during the last decades, the technology has beenrefined in order to increase the e�ciency of wind turbines. Apart from improvingthe blade design, di�erent types of turbines and generators have been developed,e.g.:

i. Fixed-speed wind turbines. Until now, the most common type of wind tur-bines is fixed- or constant-speed turbines, depicted in Figure 2.5a [28]. Theseare connected to the grid via a simple induction generator. A fixed-speedturbine is designed to spin at a certain speed and transfers the mechanicalenergy of that rotation via a shaft to the generator, which then operates at agiven slip. If the wind speed does not match the generator’s operating speed(within about 1%), the blades may be controlled to extract the correct amountof wind energy, or a gearbox may be used to alter the operating speed, butthe e�ciency of the generator drops.

ii. Doubly-fed generators. Modern wind farms are often connected to the gridvia doubly fed induction generators (DFIGs), which decouple the electrical andmechanical rotor frequencies, thus allowing the generator to operate e�cientlyat all wind speeds. The DFIG combines the classical induction generator witha controlled power electronic converter, such that the stator is excited by thegrid, but the rotor windings through the converter [15], see Figure 2.5b. Thisway, a desired slip can be obtained, and the output frequency matches the grid.However, since the rotating parts of the generator are entirely decoupled fromthe grid, a variable speed wind generator does not contribute with any inertia,i.e., stored energy, to the power system.

iii. Grid-coupled synchronous generators. Some wind turbines, usually in stand-alone systems, are connected to the grid via a synchronous generator. The

2.4. RENEWABLE POWER GENERATION 17

(a) Fixed-speed (b) DFIG

Figure 2.5: Principles of fixed-speed wind turbines with squirrel cage inductiongenerators (a) and doubly-fed induction generators (DFIGs). Since induction gen-erators consume reactive power, they are often combined with a so-called VARcompensator, consisting of capacitors, as seen in (a).

synchronous generator may be of a conventional type and use a gearbox totransfer the mechanical energy from the rotor blades to the generator, orit may have a converter as an interface towards the grid, which excites thegenerator stator and decouples the rotor frequency from the grid. The latteris preferrable and more common, since wind gusts may otherwise cause lossof synchronism [28].

2.4.3 Other Sources

While wind energy is the world-wide largest renewable energy source (apart fromhydro power), solar energy is expected to be the fastest growing in the comingyears [9]. The term solar power denotes both photovoltaics (PV) and the lesscommon so-called concentrated solar power (CSP) generation, which works like aconvetional thermal plant, but where the sun is used as the thermal source. PV cellshowever, convert the solar energy directly to electricity and generate a DC poweroutput. If the PV cell is grid-connected, this power needs to be converted to AC.The DC/AC converter (inverter) is controlled in such a way that the AC frequencymatches that of the grid, but since there are no rotating parts in a PV system, suchgeneration provides no inertia, i.e., stored energy, to the system [60].

The two next largest renewable energy sources for electricity generation world-wide are geothermal energy and biomass and biofuels. These di�er from wind- andsolar power in that they are dispatchable and therefore more similar to conventionalgeneration. Still, mainly for financial reasons, but also to enable a fast ramp-up,this type of energy sources are often combined with asynchronous generators [20].

The future power system with high renewable integration levels is thereforelikely to be much more heterogenous in terms of generation than today’s grid,regardless of dominating energy source. A continued stable and secure operation ofthe power system therefore relies upon an understanding of the altered dynamics due


to asynchronous generation as well as appropriate control of the power electronicsin the grid.

Chapter 3

Resistive Losses in Synchronizing Power

Networks

In this Chapter, we will use a coupled set of swing equations as derived in Sec-tion 2.1.2 to model the power system dynamics for a large network of synchronousgenerators. The network which determines the coupling of the swing equations isdescribed through the admittance matrix, which is a weighted graph Laplacian, asseen in Section 2.2.1. We consider several scenarios such as the power network en-countering isolated disturbance events, or being subjected to persistent stochasticdisturbances where the system is continuously correcting for errors. In both of thesescenarios, we quantify the total power lost during the synchronization transient dueto non-zero line resistances and show that this is given by the squared H

2

norm ofthe system of generator swing dynamics.

This H2

norm is evaluated by regarding a reduced, or grounded version of thesystem, in which one of the system nodes behaves as an infinite bus with fixedstates. The network is then described by so-called grounded Laplacians, as previ-ously studied by e.g. [25,40], in which the inherent singularity of graph Laplacians iseliminated. We show that, in the case of uniform generators, this grounded systemis equivalent to the original system in terms of the H

2

norm.Our main result shows that the transient resistive losses are a function of the

power line and generator damping properties and scale linearly with the networksize. The losses are however shown to have little or no dependence on networktopology, i.e., a loosely connected network will, in principle, incur the same lossesduring the transient as a highly connected network. Through numerical examplesand bounds, we illustrate this network topology independence for heterogenousnetworks and study the e�ect of altered generator dampings on the losses.

The remainder of this chapter is organized as follows. Section 3.1 derives thesystem dynamics through the classical machine model and defines the resistive powerlosses as the performance metrics. We then introduce the grounded system andderive algebraic expressions for the its H

2

norm in Section 3.2, where interpretationsof the norm along with operating scenarios in which it can be used to quantify the

19

20 CHAPTER 3. RESISTIVE LOSSES IN SYNCHRONIZING POWER NETWORKS

r12 + jx12 r23 + jx23

r

24+jx

24

r

34+jx3

4

r

45+jx4

5

r

46+jx

46

r56 + jx56 r67 + jx67

G1

{V1, ✓1}G2

{V2, ✓2}G3

{V3, ✓3}

G4

{V4, ✓4}

G5

{V5, ✓5}G6

{V6, ✓6}G7

{V7, ✓7}

Figure 3.1: An example of a network with N = 7 generator nodes. Each line has theimpedace z

ij

= rij

+ jxij

, where rij

is the line resistance and xij

the line reactance.For the coming examples, it is also worth noting that nodes 1 and 7 are the leastconnected nodes while node 4 is the most interconnected node.

transient resistive losses are also provided. In Section 3.3 we discuss bounds andgeneralizations of the norm and proceed to illustrate some of these in the numericalexamples of Section 3.4. We conclude this chapter and discuss the main findings inSection 3.5.

3.1 Problem FormulationIn this section, we model the power system as a linear time-invariant (LTI) systemof coupled swing equations with distributed disturbances. The output of this systemwill represent the dissipated power in the network, so that the squared input-outputH

2

norm of the system gives the total resistive losses during the synchronizationtransient.

For this purpose, we consider a simplified model of the power system, consistingof a network of N nodes (buses) and a set E of edges (lines), as depicted in Figure 3.1for N = 7. At every node i = 1, . . . , N there is a generator with inertia constant M

i

,damping coe�cient —

i

, voltage magnitude |Vi

| and voltage phase angle ◊i

. Each lineE

ij

œ E is characterized by its impedance zij

= rij

+ jxij

. Without loss of generality,this system can be assumed to also capture constant impedance loads lumped intothe lines.

3.1.1 System DynamicsWe use the classical machine model, see e.g. [55], and standard linear power flow as-sumptions, see e.g. [34], to represent the interactions between the generators through

3.1. PROBLEM FORMULATION 21

the network of impedances. The dynamics of each generator i œ {1, . . . , N} are thengiven by (2.4). By also making use of the susceptance matrix L

B

, defined by Equa-tions (2.5)-(2.6), we can write rewrite the di�erential equation (2.4) in state spaceform as:

ddt

C◊Ê

D

=C

0 I≠M≠1L

B

≠M≠1B

D C◊Ê

D

+C

0M≠1

D

w (3.1)

(3.2)

where M = diag{Mi

}, B = diag{—i

}. By a slight abuse of notation, we have letthe states above represent deviations from a steady-state operating point and froma synchronously rotating reference frame, and let the constant power input P

m,i

belumped into the disturbance w.Remark 3.1 The case where the input w is assumed to be pre-scaled by the gen-erator inertia M

i

so that B = [0 I]T is also meaningful. In that case one assumesthat a disturbance on a “heavy” large-inertia generator is inherently larger than adisturbance influencing a “lighter” generator. This is opposed to the current formu-lation (3.1), which allows a uniformly sized disturbance to have a larger influenceon small-inertia generators. Depending on the character of disturbances, both defi-nitions may be suitable. While events such as a generator failure or sudden changesin generator operation would be served better by the second choice of input defini-tion, small disturbances due to e.g. net demand fluctuations are more likely to bebetter captured by (3.1). A result for the second input definition is however alsopresented, see Corollary 3.5.

3.1.2 Performance MetricsIn order to evaluate the performance of the system (3.1) we choose to measure thecontrol actuation required to drive the system to a synchronous state after a faultevent (disturbance). Synchrony is achieved through circulating power flows thatarise due to the phase angle di�erences between the generator buses, and we willmeasure the control e�ort as the resistive power losses associated with these flowsdue to non-zero line resistances.

The real power flow over an edge Eij

is, according to Ohm’s law,

Pij

= gij

|Vi

≠ Vj

|2 .

Since we are regarding ◊i

as the deviation from the ith generator’s operating point,this power is equivalent to the resistive power loss over an edge during the transient.Using a small angle approximation and standard trigonometric identities this canbe approximated as

P loss

ij

= gij

|◊i

≠ ◊j

|2 . (3.3)The corresponding sum of resistive losses over all links in the network is then

P

loss

=ÿ

i≥j

gij

|◊i

≠ ◊j

|2 . (3.4)


We can now make use of the conductance matrix LG

defined by Equations (2.5)-(2.6) to rewrite (3.4) as the quadratic form P

loss

= ◊úLG

◊, where ◊ is the statevector introduced in (3.1). We therefore choose to define an output of (3.1) as

y = CÂ =:ËC

1

0È C

◊Ê

D

, (3.5)

where P

loss

= yúy. Since LG

is positive semidefinite, see Section 2.2.3, we can takeC

1

as the unique positive semidefinite matrix square root L1/2

G

, which is what weassume from now on.

Equations (3.1) and (3.5) can then be rewritten as

ddt

C◊Ê

D

=C

0 I≠M≠1L

B

≠M≠1B

D C◊Ê

D

+C

0M≠1

D

w (3.6a)

y =ËL

12G

0È C

◊Ê

D

(3.6b)

We denote the input-output mapping of (3.6) by H.The total real power losses incurred in returning this system to a synchronous

state after a disturbance can be quantified using the input-output H2

norm, whichhas several standard interpretations that were discussed in Section 2.3. In thefollowing section, we will calculate the H

2

norm from disturbance w to output yof the system (3.6) and then further discuss the physical implications on the norminterpretations for our system.Remark 3.2 Although the linearization of the dynamics which give (3.6a) involvesassuming negligible line resistances, the output (3.6b) captures the e�ect of non-zero line resistances in terms of transient power losses, given the system trajectoriesthat result from the linearized swing dynamics.Remark 3.3 In a more general context, the dynamics (3.6a) is a type of secondorder consensus dynamics, see Section 2.2.2. Considering the simpler first orderconsensus dynamics (2.8), we can let L

Q

define another weighted Laplacian for thesame graph. The quadratic form xúL

Q

x, which is analogous to (3.3), can thenbe thought of as an “L

Q

norm”; ||x||2LQ

, which is an energy measure with variousinterpretations and applications, see [49]. In [30], the quadratic form xúL

Q

x isalso proposed as a Lyapunov function, which will be non-increasing along all statetrajectories if the system is controllable and the graph connected.

For the multirobotic system depicted in Figure 2.3, LQ

could e.g. be definedthrough communication costs, and the conclusions regarding the H

2

norm and whatwe term the price of synchrony in power systems could be interpreted as the “costof consensus” in the robotic system.

3.2. EVALUATION OF LOSSES 23

3.2 Evaluation of Losses

In order to compute the input-output response of (3.6), we first define a reducedsystem H and derive an expression for its H

2

norm. We then show that this normis equal to that of the original system (3.6). Following that, we consider the specialcase when all lines have equal resistance to reactance ratios. Finally, we discussinterpretations of the H

2

norm and their implications for this particular system.Throughout this section we will assume identical generators, i.e., M = MI andB = —I.

3.2.1 System Reduction

As previously discussed, LG

and LB

are graph Laplacians, and as such each havea zero eigenvalue, see Section 2.2.3. This also leads to a singularity in the system(3.6), which is therefore not asymptotically stable. In order to properly definethe H

2

norm of (3.6) we will therefore instead regard a reduced system which isasymptotically stable.

Following the approach in [25], we derive the reduced system by first defininga reference state k œ {1, . . . , N}. We denote the reduced or grounded Laplaciansthat arise from deleting the kth rows and columns of L

G

and LB

respectively, by LG

and LB

. The states of the reduced system ◊ and Ê are then obtained by discardingthe kth elements of each state vector. This leads to a system that is equivalent toone in which ◊

k

= Êk

© 0 for some node k œ {1, 2, . . . , N}, and all other states aremeasured towards this reference. This has the physical meaning of connecting thekth node to ground, hence the terminology, and a mechanical analogy can be seenin Figure 3.2. We call the resulting reduced, or grounded, system H:

ddt

C◊Ê

D

=C

0 I

≠ 1

M

LB

≠ —

M

I

D C◊Ê

D

+C

01

M

I

D

w (3.7a)

=: A„ + Bw;

y =ËL

12G

0È C

◊Ê

D

=: C„. (3.7b)

By the assumption of a network where the underlying graph is connected, thegrounded Laplacians L

G

and LB

are positive definite Hermitian matrices (see e.g.[40]). All of the poles of system H are thus strictly in the left half plane and theinput-output transfer function from w to y has a finite H

2

norm.

3.2.2 H2 Norm Calculation

The (squared) H2

norm of the system H is given by Equations (2.11) - (2.12).We call the obsevability Gramian X and partition it into four submatrices. The


b12

b23

b13

◊2

Ê1

© 0Ê2

Ê3

Figure 3.2: Mechanical analogy to grounded power system dynamics. If the kth,here the 1st, oscillator is fixed, ◊

1

= Ê1

© 0, and the other angles are measuredwith respect to that reference. It is intuitively apparent that in this system, theoscillators will settle at their initial points after a disturbance, as opposed to thesystem depicted in Figure 2.2 where the final angles are likely to di�er from theinitial ones. This drift in the original (ungrounded) system, i.e., the change inoperating state, is information lost by reducing a system of consensus dynamics.Such an angle drift is however irrelevant in power systems, where only phase angledi�erences are relevant and where the zero phase angle can be defined entirelyarbitrarily.

Lyapunov equation (2.12) expanded for our system (3.7) is thenC0 ≠ 1

M

LB

I ≠ —

M

I

D CX

1

X0

Xú0

X2

D

+C

X1

X0

Xú0

X2

D C0 I

≠ 1

M

LB

≠ —

M

I

D

= ≠CL

G

00 0

D

.

From this, we obtain

X0

≠ —

MX

2

+ Xú0

≠ X2

—

M= 0 ∆ —

Mtr(X

2

) = tr(Re{X0

}),

where Re{·} extracts the real part of a complex matrix. Moreover,

≠ 1M

LB

Xú0

≠ X0

1M

LB

= ≠LG

,

which, since LB

is nonsingular, gives

LB

Xú0

L≠1

B

+ X0

= MLG

L≠1

B

.

Since tr(LB

Xú0

L≠1

B

) = tr(L≠1

B

LB

Xú0

) = tr(Xú0

) we have that

tr(Re{X0

}) = M

2 tr(L≠1

B

LG

). (3.8)

Finally, noting that tr(BúXB) = 1

M

2 tr(X2

), these equations give the result statedin the following lemma.


Lemma 3.1 The H2

norm (squared) of the input-output mapping (3.7) is

||H||2H2 = 12—

tr(L≠1

B

LG

), (3.9)

where LB

and LG

are the grounded Laplacians obtained by deleting row and columnk from the susceptance and conductance matrices L

B

and LG

and where — is agenerator’s self damping.

Lemma 3.1 is derived using the reduced, or grounded system H. However, itturns out that the choice of grounded node k has no influence on the resulting H

2

norm. We illustrate this point through the following lemmas, which are used derivethe main result of Theorem 3.4.

Lemma 3.2 Let H be the input-output mapping (3.6) with M = MI and B = —Iand H denote the corresponding reduced system (3.7). Then, the norm ÎHÎ2

H2 existsand is equal to ÎHÎ2

H2 for any grounded node k.

Proof: See Appendix A.

Lemma 3.3 Let LG

and LB

be the reduced, or grounded, Laplacians obtained bydeleting the kth rows and columns from L

G

and LB

respectively. Then:

tr(L≠1

B

LG

) = tr(L†B

LG

), (3.10)

where † denotes the Moore-Penrose pseudo inverse.


The result can now be stated in the following theorem.

Theorem 3.4 Given a system of N generators with equal damping and inertiacoe�cients —

i

= — and Mi

= M, ’i œ {1, . . . , N} whose input-output response isgiven by (3.6). The squared H

2

norm of the system is given by

ÎHÎ2

H2 = 12—

tr1L†

B

LG

2. (3.11)

Thus, the total transient resistive losses of the system are a function of what weterm the generalized Laplacian ratio of L

G

to LB

.

Proof: Follows directly from Lemmas 3.1 - 3.3.


Remark 3.4 In the above derivation, we have assumed that the input w is definedrelative to the mechanical input P

m,i

to each generator i. If instead, one choosesto scale the input by the generator’s inertia, i.e., let wÕ represent P

m,i

/M and setBÕ = [0 I]T , one obtains the following result:

Corollary 3.5 Consider the modified input-output mapping H Õ:

ddt

C◊Ê

D

=C

0 I

≠ 1

M

LB

≠ —

M

I

D C◊Ê

D

+C0I

D

wÕ (3.12)

y =ËL

12G

0È C

◊Ê

D

.

The H2

norm (squared) of this system is

ÎH Õ||2H2 = M2

2—tr(L†

B

LG

).

Proof: The result is easily obtained in analogy to the derivation of Lemma 3.1, not-ing that in this case, tr(BÕúXBÕ) = tr(X

2

). The rest then follows from Lemmas 3.2and 3.3.

3.2.3 Special Case: Equal Line RatiosThe result in Theorem 3.4 states that the transient resistive losses in a synchronizingnetwork are linearly dependent on what can be thought of as a generalized ratio be-tween the conductance and susceptance matrices. We now consider the assumptionthat this generalized ratio is a scalar matrix, which is the case when all lines of thesystem have equal ratios between their conductance and susceptance, equivalentlytheir resistance to reactance ratios, i.e., we assume that for all edges E

ij

œ Eg

ij

bij

= rij

xij

= –,

which gives LG

= –LB

. While still assuming identical generators, by Lemmas 3.1and 3.2, we have that

||H||2H2 = 12—

tr(L≠1

B

–LB

) = –

2—(N ≠ 1), (3.13)

which is the result presented in [3]. This result is remarkable in that it says that theloss growth depends only on the network size and is independent of the topology.Remark 3.5 The constant – can be defined as a weighted mean of the ratios –

ij

=gij

bijof all lines E

ij

in the system. The authors of [49] propose such a mean whichmakes the result (3.13) exact.


A choice of – = –max

Ø gij

bijfor all edges E

ij

œ E , will make (3.13) conservative, andvice versa if – = –

min

Æ gij

bij. The H

2

norm of the system can thus be bounded as:

–min

2—(N ≠ 1) Æ ||H||2H2 Æ –

max

2—(N ≠ 1). (3.14)

These bounds are independent of the network topology, but increase unboundedlywith the number of generators. The accuracy of these bounds in comparison to suchthat reflect network characteristics will be discussed in the coming sections.

While the ratio of power lines’ resistances to reactances of, in particular, trans-mission systems is generally small and, as in (2.3), often neglected in power flowcalculations [34], the result (3.13) shows that increasing the number of generatorswill increase resistive losses, regardless of the network topology. This is a fact thatwill become increasingly important as generation becomes more distributed.

In particular, the envisioned future smart grid is likely to involve a large numberof generators connected to low voltage distribution grids, which have higher r/xratios than transmission systems (typically, this ratio is 1/16 in 400 kV lines but2/3 in 11 kV systems) [23]. The result (3.13) thus indicates that a large number ofgenerator nodes in this type of network, which in any case will be subject to highlevels of disturbances due to intermittency in the generation, will lead to higherlosses due to grid synchronization.

The equal line ratio assumption is not unreasonable for power systems, as thereare a select number of materials and line configurations used for transmission sys-tems and for all of these the ratio of resistances to reactance ratio tend to lie withina small interval. In order to quantify this notion we examined four IEEE trans-mission system benchmark cases, representative of parts of the American powertransmission system, and found that a high percentage of the lines fell within anarrow range, see Figure 3.3. For example in the 118 bus system, 90% of the lineshad a ratio below 0.34, and 72% lay in the interval 0.20 ≠ 0.30. According to theclassical machine model considered in this chapter, the network is reduced so thatthe lines also represent impedance loads, a case not captured by the IEEE testsystems. A recent study [42], however, suggests that uniformity in line propertiesalso applies to such Kron reduced networks, by quantifying the homogenity in nodedegrees of several reduced actual power networks.Remark 3.6 The result (3.13) for when L

G

= –LB

is also a special case of a resultwhich applies when L

G

and LB

are simultaneously diagonalizable. If LB

and LG

aresimultaneously diagonalizable, the H

2

norm can be expressed directly in terms ofthe Laplacian eigenvalues. A derivation of this more general result and a discussionof cases when it applies is found in Appendix A.

3.2.4 H2 Norm Interpretations for Swing DynamicsBy the formulation in Section 3.1.2, the square of the Euclidean norm yúy of theoutput vector is defined to equal the dissipated real power in the network lines


0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2

1.4

Line

Rat

io _

k

Line index k

14 bus30 bus57 bus118 bus

Figure 3.3: Resistance to reactance ratios r/x for the lines Eij

= Ek

œ E for the IEEE14, 30, 57 and 118 bus benchmark cases. Note that the “lines” of zero resistancecorrespond to transformers, which are not part of the model considered in thischapter.

during the synchronization of the system after a disturbance. We choose to evaluatethis lost power by calculating the H

2

norm of the input-output system (3.6). Theconcept of the H

2

norm and its interpretation were reviewed in Section 2.3. We willnow discuss, in relation to these interpretations, physical scenarios which permitthe H

2

norm in 3.11 to quantify the resistive losses of the system (3.6).

i. Response to persistent stochastic disturbance. The H2

norm (squared) canbe interpreted as the steady-state total variance, when the input signal is whitenoise. For the system considered in this chapter, white noise can be thought ofas a persistent stochastic forcing at each generator. These disturbances, whichwould be uncorrelated across the system’s generators may be due to e.g. localvariations in gereration and load. The H

2

norm would then exactly correspondto the expected total power losses.

ii. Response to a random initial condition. If the system is not subject to anydisturbance, but is driven from an initial condition „

0

which is a random variable

with covariance „0

„ú0

= BBú =C0 00 M≠2

D

then the H2

norm (squared) will be

the total expected resistive losses due to the system’s returning to a synchronizedstate. This random initial condition „

0

corresponds to each generator having arandom initial velocity perturbation that is uncorrelated across the generators(since BBú is diagonal), and zero initial phase perturbation.

iii. Sum of impulse force responses. If each generator is subject to an impulseforce disturbance, then ||H||2H2 is the total power loss over all time and over alllines. Such an impulse disturbance could occur e.g. due to planned changedoperation of the generator, a sudden lost load at the bus or a fault event.

3.3. GENERALIZATIONS AND BOUNDS 29

3.3 Generalizations and BoundsIn this section, we will present further bounds on the expression (3.11) and discusstheir implications for resistive losses in syncronizing power grids. We will alsoaddress the more general case of non-identical generators as well as applicationsoutside the field of power systems.

3.3.1 Network-Characteristic Bounds on LossesAs previously mentioned, the term tr(L†

B

LG

) in Theorem 3.4 can be interpretedas a generalized ratio between the power network’s conductance matrix L

G

and itssusceptance matrix L

B

, i.e., the real and imaginary part of the admittance matrixY . We denote the respective eigenvalues of L

G

and LB

as ⁄G

N

Ø ... Ø ⁄G

2

> 0 and⁄B

N

Ø ... Ø ⁄B

2

> 0. The generalized ratio of these two Laplacians can then be lowerbounded in terms of their eigenvalues as

tr(L†B

LG

) ØNÿ

i=2

⁄G

i

⁄B

i

. (3.15)

(See e.g. [59] for a proof.) In the case of identical line ratios, equality holds, andeach eigenvalue ratio is equal to –. The literature on Laplacian eigenvalues and theirrelationships to the underlying graphs is vast, [6] and [8] for good general overviews.But L

G

and LB

describe the same topology, which makes it is reasonable to assumethat when the graph undergoes changes, their degrees and eigenvalues both changein the same fashion. Also, when the network grows, the number of eigenvaluesincrease. The sum of the Laplacians’ N ≠ 1 non-zero eigenvalue ratios will thusboth be topology independent and grow unboundedly with N . Therefore we canconclude that the bound (3.15) leads to resistive losses that scale unboundedlywith the network size and are independent of network connectivity, similar to theconclusions drawn in Section 3.2.3. We illustrate this in the example of Section 3.4.2.

The resistive losses can, as derived in Section 3.2.3, also be lower and upperbounded by (3.14), which allows for a simple and convenient analysis of the network.These bounds will increase unboundedly with N , but become loose if the system isheterogenous in terms of the line resistance to reactance ratios. This may be thecase if a combined transmission and distrubution network is considered, or if theimpedance loads, that are lumped into the lines in the reduced generator networkconsidered in this paper, are very di�erent. In some cases, it is then better to boundthe losses in terms of graph-theoretical quantities. This can be done as:

⁄G

2

tr(L†B

) Æ tr(L†B

LG

) Æ tr(LG

)⁄B

2

, (3.16)

(again, see [59] for a proof). ⁄G

2

and ⁄B

2

are the smallest non-zero eigenvalues of LG

and LB

, or the algebraic connectivities of the graphs weighted by line conductancesand susceptances respectively. It holds that ⁄G

2

Æ N

N≠1

gii,min

and ⁄B

2

Æ N

N≠1

bii,min

,


the gii

, bii

being respectively the self conductances and susceptances (degrees) of thenodes. Furthermore, the quantity tr(L†

B

) is proportional to what we can interpretas the total e�ective reactance of the network, in analogy with the concept of graphtotal e�ective resistance, as recently discussed in e.g. [19] and [25].

By Rayleigh’s monotinicity law (see [12]), the total e�ective reactance can de-crease unboundedly by adding lines and increasing line susceptances. However,the algebraic connectivites are very small for weakly connected networks and de-crease with network size, so while the bounds (3.16) are accurate for small, andwell-interconnected networks, they become loose for the type of networks that mostoften characterize a power grid.

In a more general context, Theorem 3.4 however applies to all networks withsecond order consensus dynamics, and the H

2

norm can be interpreted as an energymeasure, see Remark 3.3. Such dynamics may describe several types of mechanicalor biological systems [13], which may be of a di�erent character in terms of edgeratios and connectivities than power systems. The di�erent considerations andbounds discussed here, e.g. those for highly connected networks, can then be ofrelevance, especially when the network topology is subject to design, in order toreduce this general energy measure.

3.3.2 Generator Parameter DependenceFrom Theorem 3.4 we can deduce that

12—

max

tr(L†B

LG

) Æ ÎHÎ2

H2 Æ 12—

min

tr(L†B

LG

),

where —min

= miniœ{1,...,N} —

i

and —max

= maxiœ{1,...,N} —

i

. The losses are thusbounded by the properties of the and most strongly and lighly damped generatorsrespectively. However, envisioning an increased penetration of renewable generationto the grid, more insight to the e�ects of varying generator properties is desired.The results derived by considering a grid with identical generators suggest that thelosses scale with the network size. However, the marginal losses for an added lineconnecting a “light” (low-inertia) and highly damped generator to the system willnot be as large as if the new line is attached to a “heavy” generator. Consider thefollowing corollary to Lemma 3.1:

Corollary 3.6 Consider a network of N generators, and let its resistive losses berepresented by ||H

0

||2H2 for some choice of grounded node k œ {1, ..., N}.Connect anadditional generator with damping —

N+1

and inertia MN+1

to node k by the lineE

k,N+1

.The new system’s losses will be given by:

||H1

||2H2 = ||H0

||2H2 + 12—

N+1

–k,N+1

,

where –k,N+1

= rk,N+1xk,N+1

is the line ratio of Ek,N+1

.

If the system is described by (3.12) the additive term is instead M

2N+1

2—N+1–

k,N+1

.

3.4. NUMERICAL EXAMPLES 31


Remark 3.7 Note that Corollary 3.6 allows for the N generators represented byH

0

to be non-uniform in terms of damping and inertia.

In the face of increased penetration of renewable generation, this result impliesthat while the synchronization losses do scale with the network size, the impact oftypically low inertia renewable generators will be relatively low, compared to thatof conventional generators.

An analysis of the resistive losses in networks of non-uniform generators withdi�erent dynamics is given in Chapter 4, which will provide more insight to thee�ects of renewable power integration on grid synchronization.

Apart from Corollary 3.6, a general result for the resistive losses in generatornetworks with non-uniform generator dampings and inertias is not derived in thisChapter. However, as may be intuitively apparent, the generator parameters do in-teract with the network topology to influence coherence and synchronization losses.As we illustrate by an example in Section 3.4.4, losses are reduced if generators thatdominate the system, i.e., that have large dampings, are placed at highly intercon-nected nodes, i.e., nodes with high self conductances and susceptances (degrees),and vice versa. If the formulation (3.12) is chosen, the generator inertias also play arole for the losses, and the same argument applies; losses are reduced if high-inertiagenerators are also highly interconnected.

3.4 Numerical ExamplesThe results derived and discussed in the previous sections indicate that the resistivelosses in a network of generators depend on the number of generators in the system,the system’s resistance to reactance ratios and the generator properties. In thissection, we illustrate these results by numerical examples and simulations.

3.4.1 Line Ratio VarianceIn the Section 3.2, we showed that the H

2

norm depends on what we term the gen-eralized ratio between the conductance and susceptance matrices. In Section 3.2.3,we discussed how this ratio can be bounded by actual conductance to susceptanceratios in the system, which tend to lie in a small interval. In Section 3.3.1 we alsoshowed that the generalized ratio is essentially topology-independent and lies closeto the actual line ratios, even for heterogenous networks. In the example presentedin this subsection, we will illustrate numerically that the assumption of equal lineratios is a good approximation, by considering the IEEE benchmark topologies withincreasingly heterogenous line properties.

Figure 3.4 shows the resistive losses according to Theorem 3.4 for a hypo-thetical set of identical generators (— = 1) connected by the topologies of the


0 0.02 0.04 0.06 0.08 0.1 0.120

2

4

6

8

10

||H|| H 22

_ij Standard Deviation

Figure 3.4: Resistive losses in the modified IEEE 14 (lower points) and 30 bus (upperpoints) benchmark networks with lines of increasingly varied ratios –

ij

= rij

xij. The

figure shows the mean of 100 randomly generated systems. The bars illustrate thebounds in (3.14).

IEEE 14 bus and 30 bus benchmark systems respectively. The line ratios –ij

=rij

xij= gij

bijfor the lines E

ij

œ E are randomly uniformly distributed on the intervals0.4, 0.4 ± 0.025, 0.4 ± 0.05, ..., 0.4 ± 0.2, and the horizontal axis represents the re-sulting standard deviation of the line ratios. We take the values for x

ij

from thebenchmark systems and let r

ij

= –ij

xij

. The bars in the figure represent the upperand lower bounds of the inequality (3.14).

We note that while increased standard deviation of the line ratios leads to aless precise bounding by (3.14), the resistive losses of the system themselves varyvery little when the average line ratio remains constant. They are instead highlydependent on the network size (here 14 or 30 nodes), as deducible from equation(3.13) and discussed previously in [3]. The merely small changes in the norm subjectto the increased variance can also be understood by considering Theorem 3.5 andwriting the conductance matrix as L

G

= –LB

+ LG

. Then

||H||2H2 = –

2—(N ≠ 1) + 1

2—tr(L†

B

LG

).

The entries of LG

in general take on both positive and negative values, and the lessdistributed the line ratios are around –, the smaller their absolute values are. Fora meaningful choice of –, like the average value over the network, tr(L†

B

LG

) willbe small. In Figure 3.4, this quantity is represented by the small deviations of thepoints from the horizontal lines respresenting equal line ratios.

3.4.2 Increased Network SizeAccording to our results, the resistive losses in a network of synchronizing gener-ators will be largely independent of the network topology, but instead depend onthe ratio between susceptances and reactances and increase unboundedly with thenetwork size. In this example, we will compare the H

2

norm of two increasinglylarge hypothetical power networks, one whose underlying graph is radial, where all


(a) Radial (b) Complete

Figure 3.5: Example of a 4 node (a) radial and (b) complete graph. The example ofSection 3.4.2 calculates the H

2

norm for these types of networks when the numberN of nodes grows large.

except two nodes have two neighbors, and one where it is a complete graph, inwhich every node is connected to every other node. Examples of these graphs canbe seen in Figure 3.5.

For simplicity, we regard the case where — = 1, and assign random line param-eters to the increasingly large networks, letting each line’s reactance x

ij

and lineratio –

ij

both be drawn from a normal distribution with mean 0.2 and standarddeviation 0.1 (replacing any negative values with the mean). As shown in the pre-vious example, one can then expect the norm for each network to mainly dependon the mean ratio – = 0.2 and the number of nodes N .

Figure 3.6 shows how the norm increases for the radial and the complete graphsas the network size increases from a 5 node to a 50 node system. The boundspresented in Section 3.3 are also displayed. The eigenvalue ratio bound (3.15)provides the tightest bound and grows with N in the same fashion as the norm. Wealso note that the network-parameter dependent bounds (3.16) are more accuratethan the line ratio bounds (3.14) for the complete graph, but are not very accuratefor the radial network.

3.4.3 Marginal Losses for Added LinesAs a first step of characterizing a case with non-uniform generators, we will nowstudy the situation in Corollary 3.6 and simulate the 7 bus network depicted inFigure 3.1. We let all lines E

ij

œ E have the impedances zij

= z0

= 0.04 + j0.2,node 1 be the grounded node, and let all generators i = 2, ..., N = 7 have theparameters [48]: M

i

= 20

2fif

and —i

= 10

2fif

= —0

with a frequency f = 60 Hz. Let thisoriginal system be denoted H

0

.To node 1, three di�erent additional generators will be connected, with —

N+1

=—

8

œ {0.1—0

, —0

, 10—0

}. The connecting line has the impedande z1,8

= z0

. Figure 3.7then shows the system trajectories of the three resulting reduced systems H

1

, whenthe system is subject to a random initial angular velocity disturbance, correspondingto the H

2

norm interpretation (ii) in Section 2.3.The expected power losses during the transient response for these respective

systems are given by Corollary 3.6 and will be ||H1

||2H2 = ||H0

||2H2 + –1,82—8

= 60.3, 26.4


5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

N

H2 norm_min − _max bounds

Eigenvalue ratio bound

(a) Radial Graph

5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

N

H2 norm_min − _max bounds

Eigenvalue ratio boundh2−bound

(b) Complete Graph

Figure 3.6: H2

norms for (a) a radial network and (b) a complete graph, each withN nodes, with some of the bounds presented in Section 3.3. Despite some variationdue to the randomness in the line parameters, the H

2

norm scales linearly withthe network size and is largely the same for the complete graph as for the radialnetwork. The bound related to the Laplacian eigenvalue ratios (3.15) is the mostaccurate bound, and for the complete graph, the inequality (3.16) linked to thealgebraic connectivities ⁄

2

, also provides accurate bounds (for the radial graph, thelatter have been omitted since they are o� by orders of magnitude).


0 5 10 15 20

−2

0

2

e

0 5 10 15 20−40

−20

0

20

40

t

t

(a) —N+1 = 10—0

0 5 10 15 20

−2

0

2

e

0 5 10 15 20−50

0

50

t

t

(b) —N+1 = —0

0 5 10 15 20

−2

0

2

e

0 5 10 15 20−50

0

50

t

t

(c) —N+1 = 0.1—0

Figure 3.7: Simulation of the grounded 7 bus network of Figure 3.1 with identicalgenerators in the main network and one additional generator of (a) 10 times, (b)equal, and (c) a tenth of the damping of the other generators connected to thegrounded node (k = 1). The system is subject to random velocity and zero phaseinitial conditions, so that the expected power losses correspond to the H

2

norm. Thelosses are the largest in system (c), where the lightly damped generator maintainsits oscillation for a very long time, as predicted by Corollary 3.6.

and 23.0 respectively. For the particular example in Figure 3.7, the losses arerepectively 110, 32.2 and 23.7. The poorly damped generator will experience strongoscillations and incur large transient losses before it stabilizes to the same statesas the grounded node. The highly damped generator, on the other hand, incursless oscillations and losses than in the case where an equally damped generator isconnected.

3.4.4 E�ects of Generator PlacementNow consider again the network depicted in Figure 3.1. We assign the impedancesr

ij

+jxij

= 0.1+j0.6 to all lines of the system, which results in nodes 1 and 7 havingthe smallest degree, node 4 having four times, nodes 2 and 6 having three timesand nodes 3 and 5 having twice that degree. We will now compare the behaviourof this system when a set of generators is distributed across the network so that (a)the strongly damped generators are placed at highly interconnected nodes (matcheddampings and degrees) to when (b) strongly damped generators are placed at theleast connected nodes (mismatched dampings and degrees).

For the simulations, we use the following parameters: M = 20

2fif

and — œ1

2fif

{2, 8, 14, 20}, with a frequency f = 60 Hz. Figure 3.8a and 3.8b respectivelyshow the state trajectories of a reduced version of the system (node 1 grounded)where in Figure 3.8a the node degrees have then been matched to the size of thedamping coe�cients — as in situation (a) above, and in Figure 3.8b they have beenmismatched as in situation (b). Call these systems H

match

and Hmismatch

respec-


0 5 10 15 20 25−4

−2

0

2

4

e

0 5 10 15 20 25−40

−20

0

20

40

t

Time

(a) Damping size matched to node degrees

0 5 10 15 20 25−4

−2

0

2

4

e

0 5 10 15 20 25−40

−20

0

20

40

t

Time

(b) Damping size mismatched to node degrees

Figure 3.8: Simulation of the reduced 7 bus system in Figure 3.1 with node 1grounded and with (a) the strongly damped generators placed at the highly inter-connected nodes and vice versa, and (b) the strongly damped generators placed atweakly interconnected nodes and vice versa. The system in (b) is less coherent andexperiences larger resistive losses during the transient response.

tively.In order for the simulations to correspond to the H

2

norm interpretation (ii) inSection 2.3, the initial conditions Ê

i

(0) were drawn from a uniform distribution on[≠

Ô12

2M

,Ô

12

2M

] and ◊(0) was set to zero.The expected power losses during the transient response for these respective

systems are ||Hmatch

||2H2 = 18.9 and ||Hmismatch

||2H2 = 20.7. For the particular casesdepicted in Figures 3.8a and 3.8b, the losses are 13.2 and 27.9 respectively. Fromthe figures, it is seen that the transient behaviour of the system H

mismatch

is less“coherent” than that of H

match

, most clearly by regarding the time at which thesmall oscillations in the respective phase angles ◊ have died out.

3.5 DiscussionIn this Chapter, we have formulated power system dynamics as a linear system ofcoupled swing equations. The system’s output was defined such that the system’ssquared input-output H

2

norm corresponds to the transient resistive losses incurredafter the system is subjected to either impulse disturbances or persistent stochasticforcings. This H

2

norm is calculated by making use of the concept of grounded,Laplacians, which eliminate the inherent singularity of consensus problems (seeSection 2.2) by choosing one node in the network as a fixed reference. By showingthat the H

2

norm is the same for the original system as for the grounded system,we demonstrate the usefulness and legitimity of such a system reduction. A relatedresult is discussed in [25] where the e�ective graph resistance (or Kirchho� index)is calculated from a grounded conductance matrix and is shown to be indi�erent to

3.5. DISCUSSION 37

the choice of grounded node.Our main result, which considers a general network in the case of identical

generators, shows that the resistive power losses depend on what can be thoughtof as a generalized ratio between the conductance and susceptance matrices. Thesetransient losses are shown to increase with the network size, and in the limit ofall lines in the network having identical ratio of line conductance to susceptance,they are entirely independent of the topology and scale directly with the numberof nodes. In this limit it is directly deducible that highly connected and looselyconnected networks incur the same resistive power losses in recovering synchrony.By considering bounds on the generalized Laplacian ratio, these same conclusionscan be shown to hold also for more heterogenous networks, as is demonstrated inthe numerical examples of Section 3.4.

These results showing that the transient power losses are independent of networktopology are in contrast to typical power system stability notions and performancemetrics such as network coherence. For example, the topology of the system playsan important role in determining whether a system can synchronize [4, 10, 11] andthe rate of convergence or damping of a power system is directly related to thenetwork connectivity [37]. One intuitive explanation for the price of synchrony be-ing independent of network topology is as follows. A highly connected network isexpected to have more phase coherence than a loosely connected network. Conse-quently, the power flows per line in a highly connected network are relatively small,but there are many more links than in the loosely connected network, and in theaggregate, the total power losses of the two networks are the same. One should keepin mind however, that in the less coherent network, disturbances are more likely tocause instabilities. The issues of stability and the cost of synchrony are two di�erentconcerns.

When considering a network of non-uniform generators, we also show that themarginal losses incurred by adding a well-damped low-inertia generator to the sys-tem are small compared to adding a poorly damped high-inertia generator. Thisgenerator parameter dependence is relevant in the face of increasingly distributedrenewable generation which may quickly increase the number of nodes in low volt-age grids with typically high resistances, and thus, according to our results, leadto large transient power losses. This fact, combined with the more intense distur-bances that can be expected from the source variability in a renewable integratedgrid, indicate that the significance of the transient resistive losses will increase,and di�erent strategies to reduce them may become relevant. Our numerical re-sults related to generator placement, which show that losses are decreased if highlydamped generators are also highly interconnected, may provide more insight to suchstrategies.

In this chapter, however, we have not considered the changes in system dynamicsintroduced by renewable generation sources. Also, the classical machine model’sreduction of power network topologies to a set of only generator nodes may limitthe legitimity of the linear DC power flow approximation (see Section 2.1.2) andgives a model which does not reflect the grid topology. In Chapter 4 we will extend


this model so that both renewable generation and loads, as well as topology, can bemodeled more accurately.

Chapter 4

Power Loss Scaling of Transients in

Renewable Energy Integrated Power

Systems

In this chapter, the power system analysis described in Chapter 3 will be extendedto capture renewable power sources as well as loads. As mentioned in Chapter 1,the grid integration of substantial amounts of renewable generation will significantlychange the dynamical properties of the power system, such as the manner in whichthe system returns to a synchronous state after a disturbance [23,52]. As in Chapter3, we will not directly address questions about regions of synchronous stability orhow disturbances propagate in the face of intermittent energy, but instead assumethe disturbances to be su�ciently small to ensure an asymptotically stable linearsystem. We thus presume that the system will recover synchrony after a disturbanceevent and evaluate the control e�ort in terms of the real power required to achievethis synchrony. This real power loss during the transient is again quantified as thesquare of the system’s H

2

norm.The new system will be modeled by adding power injections with first order dy-

namics to the existing oscillating system in order to create a coupled system, whichwe will, by some abuse of terminology, refer to as one of coupled first and secondorder oscillators. These types of complex oscillator networks have been studied fora number of applications, see [13] for a survey along with applications in mechanicaland biological systems. In the context of power systems stability, they were firstanalyzed by Bergen and Hill [5], who proposed a model which preserves networkstructure and explicitly includes loads. This model is in contrast to the commonlystudied classical machine model introduced in Chapter 3 which incorporates loadbuses into the lines, thus altering the topology. The loads of the Bergen-Hill modelwere assumed to be frequency-dependent and thus described by first order dynam-ics. In this chapter, we use this model to represent renewable generation, such asthat from wind or solar power plants. We show that this usage to capture thedynamics of buses dominated by asynchronous machines is well supported by the

39

40 CHAPTER 4. LOSSES IN RENEWABLE ENERGY INTEGRATED SYSTEMS

physics of the problem. While the literature o�ers many more complicated or highorder dynamical descriptions of asynchronous machines in general and wind powergenerators in particular [14,18,46], the appealingly simple coupled first and secondorder oscillator model has so far, to our knowledge, not been used to represent theaddition of renewable generation.

Throughout this chapter, we will again make use of the reduced, or grounded,version of the network, which will simplify the notation significantly compared to [5].The use of grounded Laplacians will eliminate the singularity of the system and alsosimplify the stability analysis of the linear system. We will also present a resulton the generalized ratio between graph Laplacians similar to that of (3.10), andshow that a certain type of network augmentation used to combine di�erent orderdynamics at the same node has no e�ect on this ratio.

Our results show that although the transient behaviour of systems of first order,second order and mixed coupled oscillators is in general fundamentally di�erent, fornetworks of equal number of nodes they may all have the same H

2

norm providedthat their damping coe�cients are equal. This indicates that adding additionalrenewable energy sources to a network will not increase the system losses if theircontrollers can be adjusted to match the damping coe�cients of the existing syn-chronous machines.

The remainder of this chapter is organized as follows. Section 4.1 introducesthe new problem formulation derives the linear system with extended dynamics.In Section 4.2 we show that this system is stable and derive an expression forits H

2

norm, which is then compared to the results in Chapter 3. We presentnumerical examples in Section 4.3 and summarize the chapter through a discussionin Section 4.4.

4.1 Problem Formulation

In this section, the problem formulation of Section 3.1, will be extended to incorpo-rate loads and asynchronous machines such as wind generators, using the machineryintroduced in [5]. We now consider a network of n

0

nodes with m Æ n0

synchronousgenerators of the kind modeled in Chapter 3. We will assume that the n

0

≠ mremaining nodes are dominated by frequency-dependent first order dynamics, rep-resenting loads, asynchronous generation or controlled power electronics. We willlet the system at steady state be subject to a disturbance input and define theresistive losses that arise in the system as it returns to synchrony as the output ofthe system.

The remainder of this section is organized as follows. First, a means of augment-ing the network to allow for di�erent dynamics at the same bus will be introduced.Then, the first order oscillator model of asynchronous machines, i.e. loads or renew-able generation, will be presented and justified. We will then derive the state-spacedescription of the system, briefly discuss the disturbance input and finally define anoutput of the system that is analogous to the definition in Section 3.1.


4.1.1 Network ModelConsider a network of n

0

nodes (buses) connected by the set E0

of edges (lines) andlet m of these nodes each be attached to a synchronous generator. Now, we augmentthe system by introducing m ficticious buses associated with these generators andconnect them by purely reactive, i.e. lossless, lines, representing the generator’sinternal reactance, the so-called transient reactance. Call the set of these m losslesslines used to augment the system E

aug

. Figure 4.1 shows an example of how a systemis augmented in this manner.

We now have n = n0

+m buses in the system. Let each of these have the voltagemagnitude |V

i

| and voltage phase angle ◊i

. Without loss of generality, the buses willbe numbered such that buses 1, ..., n

0

≠m are without synchronous generation, nodesn

0

≠ m + 1, ..., n0

are the the original generator buses, connected in rising order tonodes n

0

+ 1, ..., n0

+ m = n, which are the fictitious generator buses. See alsoTable 4.1.

In all of the derivations that follow, and unless otherwise stated, the fictitiousgenerator bus n will be the grounded node. This means that we let ◊

n

© 0 and allother phase angles and frequency deviations in the system will be measured withrespect to this reference. Node n thus behaves as a so-called infinite bus.Remark 4.1 With the system augmentation method outlined above, some non-zeroload must be co-located with every synchronous generator in the pre-augmented sys-tem, which is what we will assume without loss of generality. To model a bus withonly a synchronous generator, the corresponding node does not require augmenta-tion and its transient reactance can be absorbed into surrounding lines.

Table 4.1: Node numbering for the augmented network.

Indices Set Node type

1, . . . , n0

≠ m Nwind

Buses without synchronous generation. Assumedto be dominated by renewable generation.

n0

≠ m + 1, ..., n0

Nload

Generator/load buses of pre-augmented system,load buses in augmented network.

n0

+1, ..., n0

+m = n, Naug

Fictitious generator buses, connected to nodesn

0

≠ m + 1, ..., n0

in rising order.n - Grounded node (infinite bus)

4.1.2 Model of Asynchronous MachinesWe will consider a system where the dynamics at the n

0

≠ m buses without syn-chronous generation are dominated by asynchronous machines, such as wind powerinduction generators, which inject a frequency-dependent power to the grid. Wemodel these as:

Pwind,i

= P 0

wind,i

≠ Di

Êi

(4.1)


1

2

3 4

5

G G

G

r1

2 +jx Õ

1

2 r 2

3

+j

xÕ

2

3

r34

+ jxÕ34

r25

+ jxÕ25

r 4

5

+j

xÕ

4

5

(a) Original network

1

2

3 4

5

6 7

8

G G

G

r1

2 +jx

1

2 r 2

3

+j

x 2

3

r34

+ jx34

r25

+ jx25

r 4

5

+j

x 4

5

jx36

jx47

jx58

(b) Augmented network

Figure 4.1: Example of a 5 bus network with 3 generators (a), and the correspondingaugmented network (b) in which every generator/load bus is replaced by a load busand a fictitious generator bus. The lines in E

aug

= {E36

, E47

, E58

} are the purelyreactive lines that model the transient reactances of the generators, which in (a) areabsorbed into xÕ

ij

. The arrows directed towards buses 1 and 2 symbolize a powerinjection by an asynchronous generation source.


’i œ Nwind

where P 0

wind,i

is the constant steady-state input, Di

> 0 is the frequencydependence parameter of the generation and Ê is the frequency. The index “wind”will be used to represent all types of asynchronous generation.

This model was originally proposed in [5] as a dynamical representation of realpower loads and is one of the models that has been used since to represent inductionmotors drawing power from the grid [36]. Although induction machines are inmost cases modeled as a higher order system using voltage, current, power andfrequency as state variables, it is not generally clear how these models can be used toanalyze the transient stability of the overall power system. The model (4.1) capturesthe relationship between the active power and frequency in induction machines bymodelling it as a linear function of the slip (see Section 2.4). Since this model is interms of the same states as the classical machine model, it is also useful for studiesrelated to synchronous stability. An asynchronous induction generator works in thesame way as an induction motor, see Section 2.4, and an induction motor modelsuch as (4.1) can therefore be used to represent fixed speed wind turbines, that usecage type induction generators.

For the increasingly installed variable speed wind farms, that use more advancedcontrol systems, or other power sources such as photovoltaics, a DC/AC powerconverter (inverter) is used as an interface towards the grid. A means of controllingthis converter’s frequency to emulate a synchronous machine is by the droop controllaw

Êi

= Êú ≠ ki

(Pe,i

≠ P úi

), (4.2)where P

e,i

is the active power demand at bus i and P úi

is the constant power injectedto the grid when operated at the rated frequency Êú. The perameter k

i

is the so-called droop coe�cient and is subject to design. In [50] it is shown that there is anexact correspondence between the droop control law and our proposed model (4.1)1.This further supports that (4.1) is well-suited to capture the physics of several typesof renewable generation in power systems.

At the buses in Nload

, which are the generator buses of the pre-augmentedsystem, we assume that there is a certain power load drawn from the bus, with thesame dynamics as in (4.1):

Pload,i

= P 0

load,i

+ Di

Êi

. (4.3)

These will di�er from the asynchronous generators in the size of the parameter Dand will enter the bus power balance with a negative sign to signify that they aredrawing power from the system.

4.1.3 System DynamicsThe oscillator dynamics at the synchronous generator buses are still given by thesecond order swing equation (2.1). This together with (4.1) and (4.3) give that the

1Multiply (4.2) by 1ki

and rearrange to obtain Pe,i = P úi + 1

kiÊú ≠ 1

kiÊi. Since P ú

i and Êú

are rated, constant quantities, we can set P úi + 1

kiÊú = P 0

i , and by identfying 1ki

= Di we haveretrieved (4.1).


dynamics at each bus i œ {1, . . . n} can be described by

Mi

◊i

+ —i

◊i

+ Di

◊i

= P 0

m,i

+ P 0

wind,i

≠ P 0

load,i

≠ Pe,i

,

where all parameters are non-negative and Di

= 0, Mi

, —i

, P 0

m,i

> 0 only fori œ N

aug

, P 0

wind,i

> 0 only for i œ Nwind

and P 0

load,i

> 0 only for i œ Nload

.We continue to enforce standard linear power flow assumptions, so that the real

power injected at every node i œ {1, . . . n} is given by (2.3), and let the network’sn-dimensional susceptance and conductance matrices L

B

and LG

be given by (2.5)-(2.6). As in Section 3.1, we then define the reduced, or grounded, Laplacians L

B

and LG

, by the matrices that arise by deleting the row and column of LB

and LG

corresponding to the nth grounded node.Now, let

M = diag{Mn0+1,...,Mn≠1

}, D = diag{D1

, ..., Dn0}, B = diag{—

n0+1

, ..., —n≠1

},

where the nth parameters are omitted since they correspond to the grounded node.We then write the dynamics of the synchronouns generator nodes and the asyn-chronous machine nodes separately as

DÊD

= ≠ËI

n0 0È

(LB

◊ ≠ P 0) (4.4)

MÊG

+ BÊG

= ≠Ë0 I

m≠1

È(L

B

◊ ≠ P 0), (4.5)

where P 0 = diag{P 0

i

} with P 0

i

= P 0

m,i

+ P 0

wind,i

≠ P 0

load,i

(note that for each i,only one of these quantities is non-zero). The state vectors are defined such that◊ =

Ë◊

1

· · · ◊n≠1

ÈT

and

◊ =CI

n00

D

ÊD

+C

0I

m≠1

D

ÊG

= T1

ÊD

+ T2

ÊG

. (4.6)

Now, from (4.4), we get:

T1

ÊD

= ≠T1

D≠1T T

1

(LB

◊ ≠ P 0)

which allows the state ÊD

to be eliminated. To simplify the notation, we can defineT

D

:= T1

D≠1T T

1

and finally write the state equations for ◊ and ÊG

as:

ddt

C◊

ÊG

D

=C

≠TD

LB

T2

≠M≠1T T

2

LB

≠M≠1B

D C◊

ÊG

D

+C

TD

M≠1T T

2

D

w, (4.7)

where we have lumped the constant power P 0 into the disturbance input w. Byslight abuse of notation, we will in the following let the formulation (4.7) representdeviations from a steady-state operating point.


4.1.4 System InputsAs in Chapter 3, we have here let the system states represent deviations from asteady state operating point and in doing so lumped the constant power input oroutput P 0

i

at each bus i into the disturbance input wi

. Even though P 0 originallyhas both positive and negative quantities in its entries:

P 0 =

S

WWWWWWWWWWWWWWWWWU

P 0

wind,1

...P 0

wind,n0≠m

≠Pload,n0≠m+1

...≠P

load,n0P 0

m,n0+1

...P 0

m,n0+m≠1

T

XXXXXXXXXXXXXXXXXV

,

the disturbance w is a zero-mean stochastic variable which represents deviationsfrom P 0. The original signs of P 0’s entries are thus insignificant.

In analogy to the Remark 3.1, the input w could be pre-scaled by the coe�cientsD

i

for i = 1, . . . , n0

and Mi

for i = n0

+ 1, . . . , n ≠ 1 respectively. However, for thesake of brevity, that case is not considered in this chapter.

4.1.5 Performance MetricTo evaluate the performance of the system (4.7), we will as in Chapter 3 measurethe control e�ort needed to drive the system to a synchronous state2. This controle�ort is defined as the real power flowing between the nodes due to the phase angledi�erences arising from disturbances and is the same as the resistive power lossesin the system during the transient phase.

Under the assumptions that phase angle di�erences are small and that the sys-tem has a flat voltage profile, the resistive power loss over an edge E

ij

œ E = E0

fiEaug

is P loss

ij

= gij

|◊i

≠ ◊j

|2 and thus the total losses in the network are:

P

loss

=ÿ

i≥j

gij

|◊i

≠ ◊j

|2. (4.8)

This is equivalent to the quadratic form P

loss

= ◊úLG

◊. We can therefore define the(n ≠ 1)-dimensional output y of the system (4.7) as:

y =ËL

1/2

G

0È C

◊Ê

G

D

, (4.9)

2Since the system is reduced and all states are measured against node n, the synchronous stateis given by ◊ © 0.


by which P

loss

= yúy. Since LG

is positive semidefinite, we let L1/2

G

be the uniquepositive semidefinite square-root of L

G

. Note that over the purely reactive linesthat connect the fictitious generator buses, no real power will flow and y

i

© 0 fori œ N

aug

.Remark 4.2 The positive semidefiniteness of L

G

is not immediately obvious, sincethe graph underlying L

G

is not connected (recall the lines connecting the fictitiousnodes being purely reactive). By this construction, the m last rows and columns ofL

G

and the m ≠ 1 last rows and columns of LG

will be zero. However, the (n0

◊ n0

)submatrix with the non-zero entries of L

G

and LG

will still be a graph Laplaciandescribing the connected network between the n

0

nodes in the pre-augmented net-work. For L0

G

, the properties listed in Section 2.2.3 all hold. LG

will therefore havem zero eigenvalues and n

0

≠ 1 positive ones, and remains positive semidefinite.

We can now define the system (4.7) and (4.9) as the input-output mapping Hfrom w to y:

ddt

C◊

ÊG

D

=C

≠TD

LB

T2

≠M≠1T T

2

LB

≠M≠1B

D C◊

ÊG

D

+C

TD

M≠1T T

2

D

w

=: AÂ + Bw (4.10)

y =ËL

1/2

G

0È C

◊Ê

G

D

=: CÂ

4.2 Input-Output AnalysisIn this section, we will first show that the linear system (4.10) is asymptoticallystable in order to ensure that its H

2

norm exists. We will then derive an expressionfor this norm and finally discuss upper and lower bounds on its value, as well as limitswhere it reduces to norms of lower order dynamics. In order to evaluate the results,it is useful to be able to compare non-augmented systems as in Figure 4.1a withaugmented systems such as those of Figure 4.1b. A theorem relating these systemsis presented, which describes the cases in which the augmentation procedure doesnot a�ect the H

2

norm and the associated power losses.

4.2.1 StabilityWe will follow the arguments in [5] to show that the system (4.10) is asymptoticallystable. Consider the Lyapunov function candidate

V (◊, ÊG

) = 12◊T L

B

◊ + 12ÊT

G

MÊG

.

4.2. INPUT-OUTPUT ANALYSIS 47

Clearly, V (0, 0) = 0. LB

is positive definite, since it is the reduced Laplacian of acomplete graph (see e.g. [40]), and M > 0, thus V (◊, Ê

G

) > 0, ’◊, ÊG

”= (0, 0).The derivative of V evaluated along the state trajectories, will after some algebraicoperations be given by

V (◊, ÊG

) = ≠ÊT

G

BÊG

≠ ◊T LB

T T

D

LB

◊

which is non-positive for all ◊, ÊG

, since B > 0 and LB

T T

D

LB

Ø 0. For globalasymptotic stability, we also require V (◊, Ê

G

) © 0 … (◊, ÊG

) © (0, 0). Clearly, V ©0 requires Ê

G

© 0 and T T

D

LB

◊ © 0. The latter is equivalent to T T

1

LB

◊ © 0. By (4.7),however, Ê

G

© 0 implies ≠M≠1T T

2

LB

◊ © 0. Now, if T =ËT

1

T2

È= I

n0+m≠1

, thetwo give T L

B

◊ © 0. Since T is the identity matrix and LB

is positive definite, ◊ © 0.We therefore conclude that the last criterion holds and all Lyapunov’s conditionsfor global asymptotic stability are fulfilled.

4.2.2 H2 Norm Calculations

Since the system H in (4.10) is asymptotically stable, its H2

norm exists. It canbe calculated by evaluating (2.11), where the observability Gramian X is given bythe Lyapunov equation (2.12). If we partition X œ C(n+m≠2)◊(n+m≠2) into foursubmatrices as

X =CX11 X0

X0ú X22

D

,

where X11 œ C(n≠1)◊(n≠1), X0 œ C(n≠1)◊(m≠1) and X22 œ C(m≠1)◊(m≠1), the Lya-punov equation translates into the following three linearly independent equations:

LB

TD

X11 + LB

T2

M≠1X0ú + X11TD

LB

+ X0M≠1T T

2

LB

= LG

(4.11a)L

B

TD

X0 ≠ LB

T2

M≠1X22 + X11T2

≠ X0M≠1B = 0 (4.11b)T T

2

X0 ≠ M≠1BX22 + X0úT2

≠ X22M≠1B = 0. (4.11c)

Evaluating (2.11) gives

||H||2H2 = tr(T 2

D

X11) + tr(M≠2X22). (4.12)

Assuming uniform dampings and inertia, i.e. M = MIm≠1

, D = DIn0 and B =

—Im≠1

, (4.12) can be simplified to:

||H||2H2 = 12D

tr(L≠1

B

LG

) + (1 ≠ —

D) 1M2

tr(X22), (4.13)

where X22 can be evaluated using equations (4.11a)-(4.11c).


4.2.3 Properties of the Augmented Network LaplaciansBefore we proceed to discuss the H

2

norm of the system (4.10) and how the result(4.13) relates to the results of Chapter 3, we will present some observations regardingwhat we term the generalized graph Laplacian ratio tr(L≠1

B

LG

) in the case wherethe underlying graph is that of an augmented network.

The graph underlying LB

is connected and LB

is thus non-singular (otherwiseL≠1

B

would not be defined). The graph underlying LG

is, as discussed in Remark 4.2,not connected and L

G

is therefore singular. Despite this fact, Lemma 3.3 still holds,since its proof does not put any requirements on L

G

other than its being a graphLaplacian. This observation will be used to prove the main result of this section.

The notion that tr(L≠1

B

LG

) is a generalized ratio between the conductance andsusceptance matrices is further strengthened by the following result, which showsthat, for our augmented network, the purely reactive edges (with a zero conductanceto susceptance ratio) do not contribute to the generalized Laplacian ratio. Thus, thesusceptances of the lines connecting the augmented, fictitious nodes are irrelevantand only the main network, connecting nodes 1, . . . , n

0

, influence tr(L≠1

B

LG

). Thisidea is formalized through the following theorem:

Theorem 4.1 Consider a network of n0

nodes connected by the set E0

= {Eij

} ofedges with weights b

ij

> 0 and gij

> 0. Denote the Laplacians associated withthe suseptance and conductance matrices for these weighted graphs as L0

B

and L0

G

respectively.Consider m Æ n

0

of these nodes, which can without loss of generality be renum-bered n

0

≠ m + 1, . . . , n0

, and divide each of these nodes into two nodes to create maugmented nodes. Call the set of edges connecting these augmented nodes E

aug

, andlet the weights be b

ij

> 0 and gij

= 0, for Eij

œ Eaug

. Let the full graph Laplacians ofthe augmented network with n

0

+ m = n nodes and the set E = E0

fi Eaug

of edges bedenoted as L

B

and LG

respectively and their grounded versions, with the nth rowsand columns deleted be called L

B

and LG

. Then

tr(L≠1

B

LG

) = tr(L†B

LG

) = tr(L0†B

L0

G

).

Proof: See Appendix B.

In the case where all conductance to susceptance ratios (equivalently resistanceto reactance ratios) of all lines (edges) are equal, the generalized Laplacian ratioscales directly with the number of nodes in the non-augmented system:

Corollary 4.2 Consider the network described in Theorem 4.1. Let the edges Eij

œE

0

be equal in terms of their resistance to reactance-ratio, i.e.:

–ij

= rij

xij

= gij

bij

= –.

4.2. INPUT-OUTPUT ANALYSIS 49

Thentr(L≠1

B

LG

) = –(n0

≠ 1). (4.14)

Proof: We have tr(L≠1

B

LG

) = tr(L0†B

L0

G

). In this case, L0

G

= –L0

G

, and byLemma 3.3: tr(L0†

B

L0

G

) = tr((L0

B

)≠1L0

G

) = tr((L0

B

)≠1–L0

B

) = tr(–In0≠1

) = –(n0

≠1).

Remark 4.3 Note that these Theorem 4.1 and Corollary 4.2 allow for the edgesE

ij

œ Eaug

to have any non-zero reactance xij

.Remark 4.4 For a general network, with the structure described in Theorem 4.1,but where g

ij

”= 0 for Eij

œ Eaug

, the generalized Laplacian ratio would be:

tr(L≠1

B

LG

) = tr(L0†B

L0

G

) +ÿ

EijœEaug

–ij

,

where –ij

= gij

bij. This can easily be shown in analogy to the proof of Theorem 4.1,

but such networks will not be considered in the remainder of this chapter.

4.2.4 Relation to Previous ResultsFrom (4.12) we see that in the case — = D, the H

2

norm of the system H reducesto

||H||2H2 = 12D

tr(L≠1

B

LG

). (4.15)

The frequency dependence coe�cient D in (4.1) and (4.3) characterizing the asyn-chronous generators and power loads can be interpreted as a type of damping,analogous to the dampings — of the synchronous generators. However, since windgenerators and loads are generally made out of smaller machines with less inertiathan traditional generators it is more physically meaningful to assume — Ø D. Wecan then upper bound (4.12) by

||H||2H2 Æ 12D

tr(L≠1

B

LG

). (4.16)

Even for non-uniform asynchronous dampings, a choice of D = mini=1,...,n0 D

i

=D

min

makes the above conservative.An interesting special case occurs if m = 0 (in this case, we can no longer ground

a synchronous generator node and may instead choose a node k œ {1, ..., n0

} as thereference). This would correspond to a network of only first order dynamics, i.e.,a system without synchronous generation in our modeling framework. Althoughsuch a system may not be realistic because a power system requires inertia, theresulting model: a linear version of the Kuramoto oscillator model, is relevant.For instance, the authors of [50] show that the (nonlinear) Kuramoto model is an


exact representation of droop-controlled inverters in a microgrid, e.g. a system withdistributed renewable generation in a low-voltage network.

To look further into this simpler model, let L0

B

and L0

G

be the graph Laplaciansfor this network of n

0

nodes and L0

B

and L0

G

the corresponding grounded Laplacians,with the kth rows and columns removed. Let all oscillators have the same dampingcoe�cient D. We can then write the input-output mapping H

1

st with first orderdynamics as:

ddt

◊ = ≠ 1D

L0

B

◊ + 1D

Iw (4.17)

y = (L0

G

)1/2◊,

where ◊ is the state vector of the n0

≠ 1 bus angles of the reduced system. It isthen easy to show that ||H

1

st ||2H2 = 1

2

¯

D

tr((L0

B

)≠1L0

G

), which by Lemma 3.3 andTheorem 4.1 is equivalent to (4.15).

By considering D to be a surrogate for the damping coe�cient —, and providedthe network remains unchanged, (4.15) is also the norm of a system with n

0

secondorder oscillators as derived in Chapter 3 (the system (4.10) reduces to the secondorder model if n

0

= m and the network is not augmented, see Remark 4.1). And,according to the main result of this section, this H

2

norm is also that of a networkof n

0

first order oscillators with dampings D, augmented with m second orderoscillators with dampings — = D. The remarkable result that the m additionaloscillators, while a�ecting the dynamics, do not change the norm again follows.

We can conclude that if the same uniform parameters are used, the H2

norm andtherefore the resistive losses will be the same in a same-sized network, regardless ofwhether the first order (4.17), the second order (3.6) or the combined model (4.10)is used. Any di�erences between the models in terms of the resistive losses will thusessentially depend on the damping parameters of the di�erent types of generators.These di�erences, as well as any loss-related synergies arising in the combined modeldue to coupling of the parameters, will be explored through numerical examples inSection 4.3.

We should point out that these results do not claim the models themselves tobe equivalent. The transient responses of the respective systems are substantiallydi�erent and for a stability analysis, the model order and parameters should bechosen with care. What we term the price of synchrony and the network’s abilityto synchronize are two di�erent issues, and it is only in terms of the H

2

norm thatthe di�erent order models are equivalent.Remark 4.5 In analogy to Lemma 3.2, it can be shown that for H

1

st in (4.17),||H

1

st ||2H2 = ||H1

st ||2H2 , where H1

st would be the non-grounded system with thefull Laplacians. The same holds for H in (4.10); if —

i

= Di

= D ’i = 1, ..., n anon-grounded system’s norm would also be given by (4.15).Remark 4.6 The case m = 1 also results in a network of only first order oscillatorsif the nth node is grounded. It must then however be noted that the norm of thegrounded system is not in general equivalent to that of the non-grounded one.

4.3. CASE STUDIES 51

4.3 Case StudiesThe result (4.13) indicates that the resistive losses in the network of coupled firstand second order oscillators depends on the sub-network of first order oscillatorswith damping coe�cients D, and increasingly on the second order oscillators, thesynchronous generators, as the damping coe�cient — increases. In this section, wepresent a brief case study to illustrate how the second term of (4.13) a�ects the H

2

norm. We will also, as in Section 3.4.4, discuss how the location of synchronous andasynchronous generators of di�erent dampings a�ect the transient resistive losses,in the case of non-uniform asynchronous dampings.

4.3.1 Increased Synchronous DampingWe now present an example of how the resistive losses in the network depicted inFigure 4.1b scale with the synchronous damping — and the transient reactances inE

aug

.To simplify the analysis, we assign to all edges E

ij

œ E0

the same impedancesz

ij

= 0.05+ j0.25 and to all asynchronous machines the same frequency dependencecoe�cient D = 5

2fif0, where f

0

= 60 Hz. We let the three synchronous generatorshave inertia M = 20

2fif0and vary their damping — as multiples of D. Figure 4.2 then

shows how this increase in synchronous damping a�ects the resistive losses in thenetwork for three cases with the transient reactances x

ij

œ {0.01, 0.05, 0.1}, for allE

ij

œ Eaug

.We note that while larger synchronous dampings reduce the synchronization

losses of the system, the marginal e�ect of the synchronous damping is decreasing.The example indicates that, in accordance with the bound (4.16), it is the size ofthe asynchronous dampings that drives the size of the losses. The losses also dependon the transient reactances, where a smaller reactance implies smaller losses. Anintuitive explanation for this is that a larger susceptance allows for more power toflow, which lets the larger damping of the synchronous generator attenuate more ofthe remaining system’s oscillations. Note that the transient reactances are howeverirrelevant when — = D. In that case, the losses are given by (4.14): ||H||2H2 =–

2

¯

D

(n0

≠ 1) = 0.2

2· 52fi60

· 4 = 30.16.

4.3.2 E�ects of Generator PlacementWe will now consider a similar situation as in the example of Section 3.4.4 andexamine how the placement of generator/load buses and asynchronous generationbuses at highly and weakly interconnected nodes respectively a�ect the losses. Sucha study is of course highly parameter-dependent, but we will assume that the fre-quency dependence coe�cient of the renewable generation sources D

wind

are equalto the synchronous dampings — (this may be achieved in a real system by appro-priately designing the droop-control, see Section 4.1.2), and that the loads have avery small frequency dependence D

load

.


1 2 3 4 5 6 7 818

20

22

24

26

28

30

32

`/D

||H|| 22

xaug = 0.01

xaug = 0.05

xaug = 0.1

Figure 4.2: The (squared) H2

norm as given by (4.13) for the system depicted inFigure 4.1b with increasingly large synchronous dampings and for di�erent transientreactances of the synchronous generators.

For this purpose, consider the network topology of Figure 3.1 and let node 1be the grounded node. Then assume that three of the six remaining nodes aresynchronous generator/load buses with — = 10

2fif0, M = 20

2fif0and D

load

= 1

2fif0.

The transient reactances are assumed to be xaug

= 0.05. The remaining nodesare dominated by renewable generation with D

wind

= 10

2fif

. All lines in E0

haveequal impedances: z

ij

= 0.1 + j0.6. We then compare the cases where (a) thegenerator/load buses are placed at the nodes with the highest degrees (nodes 2, 6and 4) and the renewable buses are placed at the nodes with the lowest degrees(nodes 7, 3 and 5) to (b) the generator/load buses are placed at the loads with thelowest degrees (nodes 7, 3 and 5) and the renewable buses are placed at the nodeswith the highest degrees (nodes 2, 6 and 4).

The low dampings of the load buses do not have a significant contribution tothe attenuation of oscillations in the synchronization transient, and the co-locatedsynchronous generator dampings only have a limited e�ect on the losses (as seenin Section 4.3.1). Therefore, allowing the generator/load buses to be most inter-connected as in (a) increases the losses compared to the alternative (b) in whichthe better damped renewable generation buses are placed at those nodes. The H

2

norms for these two cases are ||H(a)

||2H2 = 39.7 and ||H(b)

||2H2 = 33.8 respectively.Figure 4.3 displays the transient behaviour of these two systems. Note that

these impulse responses are significantly di�erent from the examples in Section 3.4,in particular in the way the phase angles oscillate. The almost non-existent oscilla-tions in the system 4.10 are due to the fact that what we have termed “first orderoscillators” are in reality not oscillators and will simply asymptotically dampen outany initial disturbance. This may have practical consequences in real systems; anincreased integration of renewable energy with dynamics captured by (4.1) maypotentially reduce oscillations and thus wear-and-tear in synchronous generators.

4.4. DISCUSSION 53

0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

θ

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

ωG

Time

(a) Configuration (a)

0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

θ

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

ωG

Time

(b) Configuration (b)

Figure 4.3: Simulation of the network in Figure 3.1 with (a) three generator/loadbuses located at the three nodes with the largest degrees and three renewable gen-erator buses located at the three nodes with the smallest degrees and (b) vice versa.The configuration (b) invokes smaller losses than (a), although this fact is di�cultto ascertain from the figure.

4.4 DiscussionIn this chapter, we have extended the model of Chapter 3 to include both loads andasynchronous generation, representative of renewable energy sources, by addingso-called first order oscillator dynamics to the existing model. The usage of thefirst order dynamical model to capture power injections from renewable generationsources is justified both by its common use as a model for induction machines andits equivalence with the droop control law used to control the frequency of powerconverters. This new combined first and second order model is deemed better suitedto characterize oscillations in power networks with highly heterogenous generationthan the classical machine model.

The resistive losses incurred in returning this system of coupled first and sec-ond order oscillators to a synchronous state was evaluated using the system’s H

2

norm. The main result of this chapter is that this H2

norm is indi�erent to the newdynamics. That is, provided damping coe�cients are uniform, the H

2

norm of anetwork with N second order oscillators derived in Chapter 3 is the same as that ofa network of N first order oscillators and also as that of N first order osccillatorsaugmented with m Æ N second order oscillators. All these models are thus equiv-alent in terms of the H

2

norm, and any di�erences can be attributed to di�erentdamping parameters. This result indicates that adding renewable energy sourcesto the network, either at existing load buses or as a replacement of synchronousgeneration, will not increase the transient resistive losses, provided their controllerscan be adjusted to match the dampings of the original synchronous machines.

However, there is a large and very significant di�erence in the transient be-haviours of the systems with di�erent order models. These behaviours are illus-


trated by simulations found in Sections 3.4 and 4.3. Even for uniform dampingparameters, the models are by no means equivalent, and both model order andparameters should be chosen with care if studying synchronous stability. What weterm the price of synchrony and the ability to synchronize are di�erent issues, andit is only in terms of the former that we have found he models to be equivalent.

An important part of the result that the H2

norm and thus the transient resistivelosses are the same for same-sized networks of first- and/or second order oscillatorswith uniform dampings is the fact that the purely reactive lines used to augmentthe network, in order to combine di�erent dynamics at the same node, do not a�ectthe generalized Laplacian ratio tr(L†

B

LG

). It may not seem remarkable that zeroresistance lines do not contribute to the resistive losses, but the generators at theselines are included in the dynamics and could therefore be expected to influencethe input-output behaviour of the system. In conclusion, for systems with uniformdampings, it is not the number of oscillators or their order, but the number of nodesand introduced disturbances, that a�ect the transient resistive losses.

If the damping parameters are not uniform, we can conclude through boundson the H

2

norm and numerical examples, that large synchronous dampings mayreduce transient resistive losses, but that this e�ect is bounded, and decreases withincreasing transient reactances. Instead, the asynchronous dampings (or frequencydependence parameters) have a large influence on the transient losses. Since thesemay be subject to design through the droop coe�cient, used to control the outputfrequency of e.g. modern wind farms or photovoltaics, the losses can be reduced byappropriate adjustment of the controllers. In these cases, our examples show that itis particularly advantageous to increase the damping at highly interconnected nodesin the network.

Chapter 5

Conclusions and Directions for Future

Work

In this thesis, we have formulated two linear dynamical power systems models; aclassical machine model of coupled swing equations describing the dynamics of thesynchronous generators that dominate current power systems, as well as a networkpreserving model for a heterogenous renewable energy integrated system. We haveapplied distributed disturbances to each of these networks and quantified the resis-tive power losses that they incur in regaining or maintaining a synchronous state.These transient losses, which we have termed the “Price of Synchrony”, have beenevaluated through the H

2

norm of the linear dynamical systems with the distur-bance input and the resistive losses as the output.

By considering networks of identical synchronous generators, we have shown thatthe transient losses depend on the generator dampings and a quantity that can beinterpreted as a generalized ratio between the conductance and susceptance matricesof the power network. They also grow linearly with the network size, i.e., the numberof nodes. However, one of our main result shows that these transient power lossesare independent of the network connectivity. This conclusion is in contrast to typicalpower systems stability notions, which imply that well interconnected networks havebetter syncronous stability properties than loosely connected networks. Therefore,given two networks of equal size, their ability to synchronize may di�er, but thecontrol e�ort required to reach synchrony will be the same.

A similar conclusion holds for the heterogenous networks in which mixed firstand second order dynamics describe the oscillations. Our results therefore indicatethat although the transient behaviours of the two considered models are funda-mentally di�erent, they may incur the same transient power losses, if the dampingcoe�cients are uniform.

For an envisioned future power system with high levels of renewable energysources, these results have several consequences. In general, renewables cause largerand more frequent disturbances that will induce synchronization transients more fre-quently. In addition, the grid of the future is expected to feature highly distributed

55

56 CHAPTER 5. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK

generation with potentially orders of magnitude more generator nodes than today’spower networks. Since the transient power losses scale linearly with the networksize, and are larger in the low-voltage distribution grids where much of this inte-gration is expected to take place, the magnitude and importance of these losses canbe expected to increase. Our results also imply that e�orts to improve the syn-chronous stability properties by increasing the grid connectivity will not alleviatethis e�ect. On the other hand, the losses also depend solely on the dampings ofthe generators and synchronous machines. This indicates that although renewable,asynchronous generators will alter the power system dynamics, they will not nec-essarily increase the transient power losses, if their dampings can be adjusted tomatch the existing synchronous generators. Since controlled power inverters are re-quired for grid-integration of many renewable sources, the damping is often a designparameter. Therefore, these asynchronous machines can even be used to improvethe system’s damping and decrease transient power losses. Such a strategy is shownto be particularly advantageous at highly interconnected nodes.

There are, however, some limitations to the models considered in this thesis.While the linearization allows for a theoretical analysis that yields significant in-sight to the synchronization transients of power systems, the results are only validfor small-signal disturbances. The applicability of the linear power flow approxi-mation in low-voltage grids is also disputed. A simulation study of the non-linearpower system dynamics is therefore an important direction for future work. Furtherdirections of future study also include comparing simulations of the simple renew-able energy integrated power system dynamics proposed here to higher order modelsfound in literature. There are also remaining open questions related to the linearsystems that have been discussed in this thesis. For example, it was shown that theH

2

norm is indi�erent to whether first order, second order or mixed dynamics aremodeled, but the HŒ norm was not studied. While the physical meaning of thisnorm would no longer be the total transient power losses, its evaluation may giveadditional insights regarding the transients of the di�erent systems.

In a more general context, the coupled oscillators considered in this thesis modela type of consensus dynamics. Similar formulations arise e.g. in multirobotic or bi-ological systems and in vehicle formulation problems, where the outputs can bedefined so that the input-output H

2

norm is a meaningful measure of coherenceor robustness. For example in a standard first order consensus problem subject todisturbances, where the output is defined as an unweighted Laplacian, the H

2

normrepresents the variance of the steady-state local error and is again given by a gen-eralized Laplacian ratio. In this work, we have explored this generalized Laplacianratio analytically by proving its invariance with respect to network grounding andaugmentation. We have also evaluated bounds on its value and studied the specialcase where the Laplacians are simultaneously diagonalizable so that the generalizedratio reduces to a ratio of eigenvalues. Further work will also include formalizingsome of these concepts for general networked control problems and investigating theextent to which the conclusions drawn here transfer to such systems.

Appendix A

Appendices to Chapter 3

A.1 Proof of Lemma 3.2Consider the following state transformation of the system (3.6):

C◊Ê

D

=:CU 00 U

D C◊Õ

ÊÕ

D

,

where U is the unitary matrix which diagonalizes LB

, i.e., UúLB

U = �B

=diag{0, ⁄

2

, ..., ⁄N

}, where 0 = ⁄B

1

Æ ⁄B

2

Æ ...⁄B

N

are the eigenvalues of LB

. Wehave assumed, without loss of generality, that U = [ 1Ô

N

1 u2

... uN

], where ui

,i = 2, ..., N are the eigenvectors corresponding to the aforementioned eigenvalues.

Since the H2

norm is unitarily invariant, we can also define wÕ = Uúw andyÕ = Uúy to obtain an, in terms of the norm, equivalent system

d

dt

C◊Õ

ÊÕ

D

=C

0 I

≠ 1

M

�B

≠ —

M

I

D C◊Õ

ÊÕ

D

+C

01

M

I

D

wÕ

yÕ =ËUúL

12G

U 0È C

◊Õ

ÊÕ

D . (A.1)

Now, observe that

UúLG

U =

S

WU0 · · · 0... L

G

0

T

XV . (A.2)

so that the first rows and columns of both UúLG

U and �B

are zero. We thus havethat the states ◊Õ

1

= 1ÔN

qN

i=1

◊i

and ÊÕ1

= 1ÔN

qN

i=1

Êi

satisfy the dynamics:

◊Õ1

= ÊÕ1

ÊÕ1

= ≠ —

MÊÕ

1

+ 1M

wÕ1

yÕ1

= 0,

57

58 APPENDIX A. APPENDICES TO CHAPTER 3

which reveals that this mode, which we call H Õ1

and that corresponds to the singlezero eigenvalue of L

B

, is unobservable from the output. The remaining eigenvaluesof the system (A.1) lie strictly in the left half of the complex plane due to L

B

’spositive semidefiniteness and it follows that the input-output transfer function fromwÕ to yÕ is stable and has finite H

2

norm.By the equivalence of the system (A.1) and H, we have thus established the

existence of the H2

norm for the system H.We can now partition the system into the subsystems H Õ

1

and H. We take LG

asthe Hermitian positive definite submatrix in (A.2) and define �

B

= diag{⁄B

2

, ⁄B

3

, ..., ⁄B

N

}and write the input-output mapping H as:

d

dt

C◊Ê

D

=C

0 I

≠ 1

M

�B

≠ —

M

I

D C◊Ê

D

+C

01

M

I

D

w

y =ËL

12G

0È C

◊Ê

D (A.3)

or … d

dt

„ = A„ + Bw; y = C„.

Note that the systems H1

and H are completely decoupled and we thereforehave that ||H||2H2 = ||H Õ

1

||2H2 + ||H||2H2 = ||H||2H2 .The H

2

norm can then be calculated in perfect analogy to the derivations inSection 3.2.2 and we obtain that

ÎHÎ2

H2 = 12—

tr(�≠1

B

LG

). (A.4)

Now, we show that the result of Lemma 3.1 can be written in terms of the statetransformed matrices �

B

and LG

. Define the N ◊ (N ≠ 1) and the (N ≠ 1) ◊ Nmatrices R and P by:

R =

S

WU0 · · · 0

IN≠1

T

XV , P =

S

WUI

k≠1

0≠1

0 IN≠k

T

XV ,

where k is the index of the grounded node and ≠1 is the (N ≠ 1) ◊ 1 vector withall entries equal to ≠1. By this design, �

B

= Rú�B

R, LG

= RúUúLG

UR andL

B/G

= P úLB/G

P . Further, to simplify notation, we define the (N ≠ 1) ◊ (N ≠ 1)non-singular matrix V = PUR. Then we can write

tr(L≠1

B

LG

) = tr(V V ≠1L≠1

B

(V ú)≠1V úLG

),

since V V ≠1 = (V ú)≠1V ú = I. By the cyclic properties of the trace:

tr(V V ≠1L≠1

B

(V ú)≠1V úLG

) = tr(V ≠1L≠1

B

(V ú)≠1V úLG

V )= tr((V úL

B

V )≠1V úLG

V ).

But V úLB

V = RúUúP úLB

PUR = �B

and V úLG

V = RúUúP úLG

PUR = LG

.Hence,

tr(L≠1

B

LG

) = tr(�≠1

B

LG

).

A.2. PROOF OF LEMMA 3.3 59

In conclusion,

ÎHÎ2

H2 = 12—

tr(�≠1

B

LG

) = 12—

tr(L≠1

B

LG

) = ÎHÎ2

H2 ,

which proves the Lemma.

A.2 Proof of Lemma 3.3By the proof of Lemma 3.2, we have that tr(L≠1

B

LG

) = tr(�≠1

B

LG

). Now,

tr(�≠1

B

LG

) = trAC

0 00 �≠1

B

LG

DB

= trAC

0 00 �≠1

B

D

UúLG

U

B

.

By definition, see e.g. [27], UúL†B

U = diag{0, 1

⁄

B2

, ..., 1

⁄

BN

}, which makes the above

equivalent to: tr(UúL†B

UUúLG

U) = tr(UúL†B

LG

U). But since the trace is unitarilyinvariant, it follows that

tr(�≠1

B

LG

) = tr(L†B

LG

),

which concludes the proof.

A.3 H2

Norm With Simultaneously DiagonalizableLaplacians

In cases where the Laplacians LG

and LB

are simultaneously diagonalizable, theH

2

norm can be expressed directly in terms of the Laplacian eigenvalues. In thesimplest and most relevant of these cases, the two Laplacians are multiples of eachother, and the system can be analyzed in terms of the system lines’ resistance toreactance ratio, as is done in Section 3.2.3.

By definition, see e.g. [29], two matrices A and B are simultaneously diagonaliz-able if there exists a single nonsingular matrix S such that both S≠1AS and S≠1BSare diagonal. This is the case if and only if A and B commute, i.e. if AB = BA.

For our purposes, assume there is a unitary matrix U which diagonalizes bothL

B

and LG

, i.e., UúLB

U = �B

= diag{0, ⁄B

2

, ..., ⁄B

N

} and UúLG

U = �G

=diag{0, ⁄G

2

, ..., ⁄G

N

}. Without loss of generality, we have here assumed that U =Ë1/

ÔN1 u

2

· · · uN

È, where the u

i

are the eigenvectors corresponding to theeigenvalues ⁄B

i

, ⁄G

i

respectively. Since we are considering Hermitian matrices, theeigenvectors u

i

are orthogonal, which is why U is unitary. Then, one definition of theMoore-Penrose pseudo inverse reveals that UúL†

B

U = diag{0, 1/⁄B

2

, ..., 1/⁄B

N

} [27].


Now consider the result from Theorem 3.4. Since the trace is unitarily invariant,the squared H

2

norm for the special case of simultaneously diagonalizable Laplaciansis:

ÎHÎ2

H2 = 12—

tr(L†B

LG

) = 12—

tr(UúL†B

UUúLG

U)

= 12—

tr(

S

WWWWU

01/⁄B

2

. . .1/⁄B

N

T

XXXXV

S

WWWWU

0⁄G

2

. . .⁄G

N

T

XXXXV) (A.5)

= 12—

Nÿ

i=2

⁄G

i

⁄B

i

.

Case: Equal Line RatiosThe result considered in Section 3.2.3 is derived from the above as follows. IfL

G

= –LB

, then trivially LB

LG

= LB

–LB

= –LB

LB

= LG

LB

, i.e., the Laplacianscommute. Using (A.5), we then obtain:

ÎHÎ2

H2 = 12—

Nÿ

i=2

–⁄B

i

⁄B

i

= –

2—(N ≠ 1), (A.6)

The General CaseDespite a lack of direct relevance for power systems applications, general cases ofsimultaneously diagonalizable Laplacians are discussed here for the sake of com-pleteness.

The matrices X which commute with the Laplacian LG

and which are thussimultaneously diagonalizable with L

G

are solutions to the matrix equation

LG

X ≠ XLG

= 0, (A.7)

which is a Lyapunov equation AX ≠ XAú = ≠Q, but with the right hand side ≠Qbeing a zero matrix instead of what is normally assumed to be a negative definiteone (note that L

G

= LúG

).Given a certain conductance matrix L

G

, any solutions X of (A.7) for which italso holds:

i. X Hermitian and

ii. X has the same nonzero elements as LG

,

would be a candidate for the susceptance matrix LB

such that LG

and LB

aresimultaneously diagonalizable (assumption ii. can be relaxed if we need not assumethe system to be connected both in terms of conductance and susceptance). This

A.4. PROOF OF COROLLARY 3.6 61

problem has proven hard to solve, and a full exploration and characterization ofthe existence of candidates X for general graph Laplacians lies outside the focus ofthis thesis. It is however an interesting algebraic problem, and we can present oneresult in this context:

Lemma A.1 Consider a complete graph with n nodes and let L1

be a Laplacian inwhich every edge E

ij

carries the same weight “. Let L2

be a weighted Laplacian withweights w

ij

= wji

for every edge Eij

in the same n-node complete graph. Under thisconfiguration, L

1

and L2

commute and are therefore simultaneously diagonalizable.

Proof: L1

and L2

commute since:

L1

L2

= “(nIn

≠ Jn

)L2

= “(nL2

≠ L2

Jn

) = L2

“(nIn

≠ Jn

) = L2

L1

.

(In

is the n-dimensional identity matrix and Jn

is an n ◊ n matrix with all entriesequal to 1.)

A.4 Proof of Corollary 3.6Without loss of generality, let the grounded node k be node N and let M =diag{M

1

, ..., MN≠1

} and B = diag{—1

, ..., —N≠1

}. The reduced system H1

can thenbe written as

ddt

S

WWWU

◊◊

N+1

ÊÊ

N+1

T

XXXV =

S

WWWU

0 0 IN

0

0 0 0 1≠M≠1L

B

0 ≠M≠1B 0

0 ≠ bN,N+1MN+1

0 ≠ —N+1MN+1

T

XXXV

S

WWWU

0 0

0 0M≠1

0

0

1

MN+1

T

XXXV

Cw

wN+1

D

Cy

yN+1

D

=CL

1/2

G

0 0 0

0

Ôg

N,N+1

0 0

DS

WWWU

◊◊

N+1

ÊÊ

N+1

T

XXXV . (A.8)

Let the input-otput mapping HN+1

be the SISO subsystem of (A.8):

ddt

C◊

N+1

ÊN+1

D

=C

0 1≠ bN,N+1

MN+1≠ —N+1

MN+1

D C◊

N+1

ÊN+1

D

+C

01

MN+1

D

wN+1

yN+1

=ËÔ

gN,N+1

0È C

◊N+1

ÊN+1

D

From (A.8), it is clear that the systems H0

and HN+1

are entirely decoupled andsince we can write H

1

= diag{H0

, H1

},

||H1

||2H2 = ||H0

||2H2 + ||HN+1

||2H2 .


Now, the H2

norm of HN+1

can be calculated in scalar analogy to the derivation inSection 3.2.2 and is found to be:

||HN+1

||2H2 = 12—

N+1

gN,N+1

bN,N+1

= –N,N+1

2—N+1

,

which concludes the proof.

Appendix B

Appendices to Chapter 4

B.1 Proof of Theorem 4.1By Lemma 3.3, tr(L≠1

B

LG

) = tr(L†B

LG

). The Lemma is applicable since it requiresno other properties of L

G

than those of a graph Laplacian.We now apply Lemma 3.3 once again and regard the Laplacians reduced at

node n0

. Call these reduced Laplacians LB,n0 and L

G,n0 . It then holds:

tr(L†B

LG

) = tr(L≠1

B,n0LG,n0).

Without loss of generality, we assume that the numbering of the nodes is such thatnodes n

0

≠ m + 1, ..., n0

are each connected in rising order to one of the nodesn

0

+ 1, ..., n0

+ m, like in the example in Figure 4.1b. We can then consider thetransformation matrix

V =

S

WUI

n0≠m

0 00 I

m

00 I

m

Im

T

XV ,

for which it holds thatV úL

B

V =CL0

B

00 B

D

.

where B = diag{bn0≠m+1,n0+1

, ..., bn0,n0+m

}. We also observe that

V úLG

V =CL0

G

00 0

D

= LG

.

Continue to define the deleted transformation matrix V by the matrix that ariseswhen deleting row and column n

0

from V . Then

V úLB,n0 V =

CL0

B

00 B

D

, V úLG,n0 V =

CL0

G

00 0

D

,

where L0

B

and L0

G

are the reduced versions of the Laplacians L0

B

and L0

G

, with noden

0

grounded.

63

64 APPENDIX B. APPENDICES TO CHAPTER 4

It is evident from its structure that V is non-singular and its deleted version,V , has the same property. We can therefore write:

tr(L≠1

B,n0LG,n0) = tr(V V ≠1L≠1

B,n0(V ú)≠1V úLG,n0),

since V V ≠1 = (V ú)≠1V ú = In0+m≠1

. By the cyclic properties of the trace,

tr(V V ≠1L≠1

B,n0(V ú)≠1V úLG,n0) = tr(V ≠1L≠1

B,n0(V ú)≠1V úLG,n0 V )

= tr((V úLB,n0 V )≠1V úL

G,n0 V ).

It holds that1

(V úLB,n0 V )≠1 =

CL0

B

≠1

00 B≠1

D

.

Then

tr(L≠1

B,n0LG,n0) = tr(

CL0

B

≠1

00 B≠1

D CL0

G

00 0

D

) = tr(L0

B

≠1

L0

G

) + tr(B≠10)

= tr(L0

B

≠1

L0

G

).

Now, again by Lemma 3.3:

tr(L0

B

≠1

L0

G

) = tr(L0†B

L0

G

)

with concludes the proof.

1 This can be shown by:5

L0B

≠1 00 B≠1

6 5L0

B 00 B

6=

5L0

B

≠1L0

B 00 B≠1B

6=

5In0≠1 0

0 Im

6= In0+m≠1.

List of Figures

1.1 Energy mix in the Nordic countries in 2010 and 2050. . . . . . . . . . . 2

2.1 Principle of the electromechanical equilibrium in a synchronous genera-tor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Mechanical analogy to the power system dynamics. . . . . . . . . . . . 102.3 A network of n robots. Source: http://users.ece.gatech.edu/magnus/

projects.html . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 A 4-pole synchronous generator. Source: http://www.user.tu-berlin.de/

h.gevrek/ordner/ilse/wind/wind5e.html . . . . . . . . . . . . . . . . . . 162.5 Principles of fixed-speed and doubly fed wind generators. Source: http://

lejpt.academicdirect.org/A12/083_094.htm . . . . . . . . . . . . . . . . 17

3.1 Example of a network with N = 7 nodes. . . . . . . . . . . . . . . . . . 203.2 Mechanical analogy to grounded power system dynamics. . . . . . . . . 243.3 Resistance to reactance ratios r/x for the IEEE 14, 30, 57 and 118 bus

benchmark cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 H

2

norms for IEEE benchmark systems with increasingly varying lineratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Example of a 4 node radial and complete graph . . . . . . . . . . . . . . 333.6 H

2

norms for growing radial and complete networks. . . . . . . . . . . . 343.7 Simulation of 7 bus system with “light” and “heavy” generator added. 353.8 Simulation of a system with non-uniform dampings subject to di�erent

generator placements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 Example of network augmentation. . . . . . . . . . . . . . . . . . . . . . 424.2 H

2

norms for system with nonuniform synchronous and asynchronousdampings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Simulation of the renewable energy integrated system in two configurations. 53

65

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