The Price of Synchrony:
Evaluating Transient Power Losses in
Renewable Energy Integrated
Power Networks
EMMA SJÖDIN
Master’s Degree Project
Stockholm, 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 Thesis
Supervisor: Dennice F. Gayme
Examiner: Henrik Sandberg
XR-EE-RT 2013:023
iii
Abstract
This thesis investigates the resistive losses incurred in returning a power
network to a synchronous state following a transient stability event, or in maintaining this state in the presence of persistent stochastic disturbances. We
quantify these transient power losses, the so-called “Price of Synchrony”, using
the squared H2 norm of a linear system of generator and load dynamics subject
to distributed disturbances. We first consider a large network of synchronous
generators and use the classical machine model to form a system with coupled second order swing equations. We then extend this model to explicitly
include dynamics of loads and asynchronous generators, which represent solar
and wind power plants. These elements are modeled as frequency-dependent
power injections (extractions), and the resulting system is one of coupled firstand second order dynamics. In both cases, the disturbance inputs represent
power fluctuations due to transient stability events or the inherent variability
of loads and intermittent energy sources.
The network structure is captured through a weighted graph Laplacian of
the network admittance. In order to simplify the analysis for both models,
we use the concept of grounded graph Laplacians to obtain an asymptotically
stable reduced system. We then evaluate the transient losses in the reduced
system and show that this system’s H2 norm is in fact equivalent to the H2
norm of the original system. Furthermore we show that although the transient
behaviours of the first order, second order or mixed dynamical systems are in
general fundamentally different, for same-sized networks they may all have the
same H2 norm if the damping coefficients are uniform.
The H2 norms for both system models are shown to be a function of transmission line and generator properties and to scale with the network size. These
transient losses do not, however, depend on the network connectivity. This is
in contrast to related power system stability notions that predict better synchronous stability properties for highly connected networks. The equivalence
of the norms for different order systems indicate that renewable energy sources
will not increase transient power losses if their controllers can be adjusted to
match the dampings of existing synchronous generators. However, since the
losses scale linearly with the number of generators, our results also demonstrate that increased amounts of distributed generation in low-voltage grids
will lead to larger transient losses, and that this effect cannot be alleviated by
increasing the network connectivity.
iv
Sammanfattning
I den här rapporten utvärderar vi de resistiva förluster som uppstår i ett
elektriskt nätverk då det återgår till ett synkroniserat tillstånd efter en störning. Dessa transientförluster, som vi benämner ”synkronismens pris”, utvärderas med hjälp av H2 -normen för ett linjärt dynamiskt system. I ett första
steg modellerar vi ett stort nätverk av synkrongeneratorer och erhåller ett system med kopplade svängningsekvationer av andra ordningen. Sedan utvidgas
denna modell för att även omfatta dynamiska laster och asynkrongeneratorer, som ofta används tillsammans med sol- och vindkraft. Dessa modelleras
som frekvensberoende kraftinjektioner och det slutgiltiga systemet beskriver
ett sammankopplat nätverk med både första och andra ordningens dynamik. I
båda fallen utsätts systemet för spridda störningar, som kan representera både nätverksfel och fluktuationer i elförsörjningen orsakade exempelvis av vindeller solkraft.
För att utvädera transientförslusterna används först en typ av reducerade, eller ”jordade”, laplacianer för att beskriva ett reducerat system som är
asymptotiskt stabilt. Vi visar sedan att H2 -normen för det ursprungliga systemet inte påverkas av denna reduktion. Systemets H2 -norm visar sig bero
på egenskaper hos generatorer och kraftlinor och växa linjärt med storleken på
nätverket. I motsats till typiska resultat för stabilitet i elkraftsystem som visar
att starkt sammankopplade nätverk har bättre synkroniseringsegenskaper än
svagt sammankopplade, visar dock våra resultat att transientförlusterna inte
beror på nätverkstopologin.
Vidare visar vi att, trots att transienter hos system med första ordningens, andra ordningens eller kombinerad dynamik skiljer sig kraftigt åt, så kan
deras H2 -normer vara lika för nätverk av samma storlek med lika dämpningskoefficienter. Dessa resultat indikerar att nätanslutna förnybara energikällor
inte kommer att öka transientförlusterna om deras regulatorer kan bli anpassade till dämpningen hos befintliga synkrongeneratorer. De visar dock också
att en ökad utbredning av distribuerad generation, särskilt i mellan- och lågspänningsnät, kommer att öka transientförlusterna eftersom de växer linjärt
med antalet generatorer, samt att denna effekt inte kan mildras genom att öka
antalet anslutningar.
v
Acknowledgements
I am most grateful to Prof. Dennice F. Gayme of the Department of
Mechanical Engineering at the Johns Hopkins University (JHU) for her
intelligent, 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 California
at Santa Barbara for a fruitful collaboration. The support of NSF through
grant ECCS-1230788 is also gratefully acknowledged.
Furthermore, I would like to express my gratitude to Henrik Sandberg of
KTH Royal Institute of Technology for his most insightful advice and several
rewarding discussions. I am thankful for his genuine interest in my work and
for taking the time to discuss it even while on travels.
I would also like to thank Prof. Louis L. Whitcomb and Prof. Benjamin
F. Hobbs together with their research groups for a number of interesting
discussions, which enriched both this thesis and my stay at JHU.
Emma Sjödin
Stockholm, August 2013
Contents
Contents
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1 Introduction
1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Preliminaries
2.1 Power System Dynamics . . . . . . . . . . . . . . .
2.1.1 Classification of Power System Stability . .
2.1.2 The Swing Equation . . . . . . . . . . . . .
2.2 Network Descriptions and Graph Laplacians . . . .
2.2.1 The Admittance Matrix . . . . . . . . . . .
2.2.2 Consensus Dynamics and Graph Laplacians
2.2.3 Properties of Graph Laplacians . . . . . . .
2.3 The H2 Norm . . . . . . . . . . . . . . . . . . . . .
2.4 Renewable Power Generation . . . . . . . . . . . .
2.4.1 Synchronous vs. Asynchronous Generators
2.4.2 Wind Power . . . . . . . . . . . . . . . . . .
2.4.3 Other Sources . . . . . . . . . . . . . . . . .
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3 Resistive Losses in Synchronizing Power Networks
3.1 Problem Formulation . . . . . . . . . . . . . . . . . .
3.1.1 System Dynamics . . . . . . . . . . . . . . . .
3.1.2 Performance Metrics . . . . . . . . . . . . . .
3.2 Evaluation of Losses . . . . . . . . . . . . . . . . . .
3.2.1 System Reduction . . . . . . . . . . . . . . .
3.2.2 H2 Norm Calculation . . . . . . . . . . . . .
3.2.3 Special Case: Equal Line Ratios . . . . . . .
3.2.4 H2 Norm Interpretations for Swing Dynamics
3.3 Generalizations and Bounds . . . . . . . . . . . . . .
3.3.1 Network-Characteristic Bounds on Losses . .
3.3.2 Generator Parameter Dependence . . . . . .
3.4 Numerical Examples . . . . . . . . . . . . . . . . . .
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CONTENTS
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4 Losses in Renewable Energy Integrated Systems
4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . .
4.1.1 Network Model . . . . . . . . . . . . . . . . . . .
4.1.2 Model of Asynchronous Machines . . . . . . . . .
4.1.3 System Dynamics . . . . . . . . . . . . . . . . . .
4.1.4 System Inputs . . . . . . . . . . . . . . . . . . .
4.1.5 Performance Metric . . . . . . . . . . . . . . . .
4.2 Input-Output Analysis . . . . . . . . . . . . . . . . . . .
4.2.1 Stability . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 H2 Norm Calculations . . . . . . . . . . . . . . .
4.2.3 Properties of the Augmented Network Laplacians
4.2.4 Relation to Previous Results . . . . . . . . . . .
4.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Increased Synchronous Damping . . . . . . . . .
4.3.2 Effects of Generator Placement . . . . . . . . . .
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.5
3.4.1 Line Ratio Variance . . . . . . .
3.4.2 Increased Network Size . . . . .
3.4.3 Marginal Losses for Added Lines
3.4.4 Effects of Generator Placement .
Discussion . . . . . . . . . . . . . . . . .
vii
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5 Conclusions and Directions for Future Work
A Appendices to Chapter 3
A.1 Proof of Lemma 3.2 . . . . . . . . . . . . . . . . . . . . .
A.2 Proof of Lemma 3.3 . . . . . . . . . . . . . . . . . . . . .
A.3 H2 Norm With Simultaneously Diagonalizable Laplacians
A.4 Proof of Corollary 3.6 . . . . . . . . . . . . . . . . . . . .
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B.1 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
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List of Figures
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Bibliography
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Chapter 1
Introduction
The electric power system is undergoing large and rapid changes, primarily due to
the growing interest in replacing fossil fuel-based power generation with renewable
energy sources. Factors driving this replacement are growing concerns about climate change and global warming, diminishing supplies of fossil fuels causing price
increases [7] and government mandates world-wide [31]. The Nordic countries, including Sweden, state some of the most ambitious goals for the energy sector and
aim to have a carbon-neutral energy system by 2050 [32]. Although the transport
sector and industry account for large portions of energy consumption, the power
grid will also need to become “greener” by substantial integration of renewable energy. Figure 1.1 shows the projected total energy supply in the Nordic countries by
2050 compared to 2010.
In the United States, Maryland’s Renewable Portfolio Standard (RPS) prescribes 20 % of the state’s electricity demand to be covered by renewables by
2022 [38], and several similar initiatives exist in other states [54]. On a global
level, the German Energiewende or Energy Transition initiative is also worth mentioning. Its goal to phase-out all nuclear power by 2022 and subsequent policies
have led to remarkably large investments in residential solar panels and an overall renewable penetration of 25 % in 2012, which is expected to rise to 40 % by
2020 [41]. Furthermore, new types of decentralized power grids, often with high
renewable penetration, are becoming prevalent in the developing world, since these
require smaller investments than conventional centralized power systems [61].
A high grid penetration of renewables, however, poses a number of challenges to
the power system. The inherent intermittency of wind and solar power generation
causes high levels of uncertainty [54,56], and their typically much smaller capacities
than conventional generators will make the future generation system much more
distributed than today’s [61]. The use of electricity in the tranport sector through
electric vehicles and customer programs for demand response will also contribute
to changing load patterns [45]. Many of these changes will, apart from posing
operational and market-related challenges, affect the dynamics and stability of the
power 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 affect
the rotor-angle or synchronous stability, which is the ability of the power system
to regain synchrony when subject to a disturbance [44]. Synchrony refers to the
condition when the frequency of all generators within a particular power network
are aligned, and there are no angular swings in the system [42,44]. Loss of synchrony
may lead to black-outs [2] and a secure system operation therefore relies on stability
of the power system. Renewable generators have different dynamical properties
than conventional generators and as their penetration grows, this change has the
potential to affect the stability of the grid [23, 52]. This thesis is part of an ongoing
research trend to characterize the dynamics of renewable energy integrated power
systems.
The problem of synchronization in power networks is analogous to the problem
of distributed control in complex networks, and we therefore review some recent
work on deriving stability conditions for such systems in Section 1.2. In this thesis however, the concept of synchronization in renewable energy integrated power
networks is studied in a different context. We assume that the network will return to a synchronized state after disturbances and instead focus on the control
effort required to maintain this synchrony. Loss of synchronism leads to circulating
power flows passing between generators whose angles are out of phase, which in turn
leads 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 total real power flow in a typical power network. It is, however, unclear whether they
will remain small in power grids of the future, which are expected to have highly
distributed 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
1.1
3
Scope
In this thesis, the transient resistive power loss – the price of synchrony – is evaluated for large power networks, for which we formulate the dynamics as a linear
time-invariant (LTI) system of coupled generator swing equations. We consider
scenarios in which the network encounters single distributed impulse disturbances
or is subjected to persistent stochastic noise, and show that the transient restistive
losses are, in both cases, given by the squared H2 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 order
oscillator dynamics. The network structure is captured through a weighted graph
Laplacian of the network admittance. In order to simplify the analysis, we use
the concept of grounded graph Laplacians to first evaluate the resistive losses for a
reduced, or grounded, system in which one of the generators is taken as a reference.
We then show that the H2 norm of the original system is equivalent to that of the
reduced system. This squared H2 norm is shown to be a function of the power line
and generator damping properties and to scale with the network size. However, in
contrast to typical power systems stability notions, which predict highly connected
networks to have better synchronous stability properties, our results show that the
transient losses are independent of the network connectivity. Therefore, if one wants
to 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 losses
grow linearly with the number of generators is of increasing importance as power
generation becomes more distributed, particularly in low-voltage distribution grids.
The aforementioned results remain valid in the second part of the thesis, where
the model is extended to capture loads as well as renewable sources grid-connected
by asynchronous generators. This is done by coupling the previous second order
oscillators to nodes with first order dynamics, which are shown to capture the essential dynamical properties of asynchronous machines. The results here show that
although the transient behaviours of systems of first order, second order and mixed
coupled oscillators are in general fundamentally different, for networks of equal size
they may all have the same H2 norm provided that their damping coefficients are
equal. This indicates that connecting renewable energy sources to a network will not
increase the system losses if their controllers can be adjusted to match the damping
coefficients of the existing synchronous machines.
The theoretical considerations and results outlined above are complemented
by numerical examples and simulation studies. In particular, we study how, in
heterogenous generator networks, the placement of generators affects the transient
power losses. These are found to be reduced if highly damped generators are also
placed 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 or
biological systems. What we term the price of synchrony can then be generalized to
a type of energy measure and the results, particularly on topology and model order
independence, may also have interesting consequences for these types of networks.
4
1.2
CHAPTER 1. INTRODUCTION
Related Work
A special case of the problem of rotor-angular or synchronous stability is the transient stability problem, which is associated with large angular disturbances due to
e.g. generator or line failures, or the intermittency of the power sources in a renewable energy integrated system. There is a large body of transient stability literature
from the last decades, see [55] for an excellent survey. This work generally focuses
on determining regions of attraction of synchronous states and finding Lyapunov
like energy functions to show stability in these regions, as in e.g. [43].
For general complex networks, such as biological or digital systems, the concept
of synchronization and formal stability criteria linked to network properties, have
spurred interest across many fields, a good summary of such work is found in [51].
Recently, connections between such distributed control problems and power systems
stability have been drawn. A particular set of works [10,11], which shows an equivalence between power system dynamics and a first order model of so-called Kuramoto
oscillators has gained much attention. That modeling framework provides sufficient
analytical conditions for frequency and phase synchronization [10], as well as a link
between structure preserving power system models [11], like the ones that will be
used in Chapter 4 of this thesis, and reduced models such as those discussed in
Chapter 3. While the work in [10, 11] makes limiting assumptions on the network
properties, the authors of [42] use a slightly different approach to derive stability
criteria in heterogenous networks, but with uniform generators, considering a model
much like the ones employed in this thesis.
In this thesis, the damping properties of the generators, both synchronous and
asynchronous, 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 of
the coupled first- and second order oscillator dynamics similar to those which we
introduce in Chapter 4. In that work, principles are derived to improve this damping, i.e., the rate of convergence in the system, by studying the connectivity of a
state-dependent graph Laplacian. The model employed by the authors of [37], as
in Chapter 4 of this thesis, is based on a network-preserving dynamical model first
introduced in [5].
To our knowledge, this type of coupled first- and second order oscillator model
has not previously been used in order to model dynamics of renewable integrated
power networks. Instead, much of the work on stability of such networks focuses on
modeling 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 studies
are conducted purely by simulations as in [1, 33]. There is a hope that the control
systems of modern wind farms with so-called doubly-fed induction generators (see
Section 2.4) can be employed to stabilize the power system, and there is a large
amount 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 in
this thesis. Consensus dynamics in large-scale networks, such as vehicle formation
problems, 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 the
H2 norm is used as a performance measure which quantifies the error variance.
The authors then apply different control strategies, and study how this norm scales
asymptotically with the network size. The authors of [49] use a similar notion
of the H2 norm in dynamical networks, and define a concept of “LQ -energy” as a
robustness measure. Bounds on this energy measure are presented and characterized
for various graph types, and it is shown that the “LQ -energy” corresponds to the
“Price of Synchrony”, which was first introduced in [3] and later studied in this
thesis.
Chapter 2
Preliminaries
In the remainder of the thesis, dynamical models of the power system will be derived
and evaluated. This chapter provides some theoretical background to the concept of
power system dynamics, network descriptions as well as to some aspects of renewable
power generation. A brief review of the main means of evaluation applied in this
thesis, the H2 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 reliable
supply of electricity in the wall socket, with correct voltage and frequency. This
supply is ensured by a functioning grid infrastructure and power generators, which
at every instant inject to the grid an amount of power that exactly balances the
aggregated demand. If this balance is fulfilled, and there is an equilibrium between
the rotating generators and the grid, we say that the power system operates at a
steady state.
However, the system is constantly exposed to disturbances, and several dynamic
phenomena occur on different time scales. A prerequisite for a secure system operation is therefore that the power system is stable. Power system stability can
be defined as the ability of an electric power system to regain a state of operating
equilibrium after being subjected to a physical disturbance [44]. Lack of stability
may lead to blackouts, like the one in southern Sweden in 1983 when 2/3 of the
country’s network was shut down [35], or the major Northeastern blackout of 2003
which affected 50 million people in the United States and Canada [22].
In this section, we will review different forms of power system stability before
introducing the swing equation, which is used to analyze the rotor angular, or
synchronous, 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 mechanical
power input Pm and its torque Tm balance the output electrical power Pe to the
grid and its counter torque Te on the generator. The generator rotor’s frequency Ê
is then equal to the system frequency, but an imbalance will cause an acceleration
or deceleration of the rotor.
2.1.1
Classification of Power System Stability
Although the issue of power system stability is essentially a single problem, it is useful to look at the different forms of instabilities that may occur separately [44]. One
then obtains three different stability notions. Issues related to the global generationload balance mentioned in the introduction to this section are frequency stability
phenomena. Voltage stability refers to the ability of the system to maintain a
steady and high voltage level by avoiding local imbalances in reactive power, often
due to large loads. In this thesis however, phenomena connected to rotor angular
or synchronous stability will be considered. This refers to the ability of the power
system to regain synchrony after a disturbance and depends on the ability of the
synchronous machines to maintain or restore an equilibrium between their rotating
components and the grid’s electromagnetic torque [44]. We will elaborate on this
in the following section.
Power system dynamics are inherently non-linear and whether or not the system will stabilize after a disturbance is therefore highly dependent on the initial
operating point and the size of the disturbance. However, a subset of the rotor
angle stability issues concern small-signal (or small-disturbance) stability, which is
the ability of the system to maintain synchrony when subject to small disturbances
that allow the system to be analyzed in terms of linearized equations. This thesis
will only model such small disturbances and the considered power system dynamics
will be linear.
2.1.2
The Swing Equation
According to a model often referred to as the classical machine model [55], the
power system can be regarded as a network of oscillators. The electromechanical
oscillations that arise due to an imbalance are then described by the swing equation
for synchronous generators, which we will now derive.
Under steady state conditions, each generator i œ {1, . . . , N } is fed a mechanical
power Pm,i from the plant which is equal to the electrical power output to the grid
Pe,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 will
accelerate 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 coefficient. The
resulting rotor angle deviations propagate to the other generator buses over the
network 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 i
and j. The coefficients bij and gij are respectively the conductance and susceptance
of that line and gi 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 transmission 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 differences (◊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
ÿ
Pe,i ¥
bij (◊i ≠ ◊j ).
(2.3)
j≥i
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, which
captures 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 phase
˙ Any deviations from a steady state will propagate
angle ◊i and a speed Êi = ◊.
across the network over the springs, whose stiffness coefficients are analogous to the
susceptances bij 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 used
within power systems analysis and power engineering to simplify calculations [47].
10
CHAPTER 2. PRELIMINARIES
Ê2
b12
b23
Ê1
◊1
b13
Ê3
Figure 2.2: Mechanical analogy to the power system dynamics and the swing equation (2.4). A deviation of one oscillator’s phase angle ◊i and/or its derivative Êi will
propagate across the springs with stiffness bij to the other oscillators. This analogy
is due to the authors of [13].
2.2
Network Descriptions, Graph Laplacians and
Consensus Problems
In the previous section, we derived the power system dynamics as oscillations across
a network. In this section, we will introduce the admittance matrix, which is used
to describe the topology and physical properties of the power network. This admittance matrix is a type of graph Laplacian or Laplacian matrix; a matrix representation of a network or graph.
Graph Laplacians arise naturally in state space formulations of so-called consensus 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 properties
of graph Laplacians that will be made use of later on.
This section’s review is largely based on [6], [30] and [57], in which elaborations
on the introduced concepts can be found. The literature on these subjects, however,
is vast.
2.2.1
The Admittance Matrix
The admittance matrix (also called nodal, graph or bus admittance matrix) is a
mathematical abstraction of the electric network which describes the network’s
topology and the physical properties of its lines.
Consider a network (graph) of the set N = {1, . . . , n} nodes (buses) and let the
two nodes i, j œ N be connected by a line (edge) with the impedance zij = rij +jxij ,
where rij is the line’s resistance and xij is its reactance. An example of such a
network for N = 7 is found in Figure 3.1. The inverse of the impedance is called
the admittance:
1
yij =
= gij ≠ jbij ,
zij
where gij =
rij
2 +x2
rij
ij
and bij =
xij
2 +x2
rij
ij
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 :=
ÿ
Y
_
ḡ
+
(gik ≠ jbik ),
i
_
_
]
k≥i
≠(gij ≠ jbij ),
_
_
_
[
0
if i = j,
if i ”= j and j ≥ i,
otherwise.
(2.5)
where j ≥ i denotes a line between nodes i and j. The diagonal elements Yii of the
admittance matrix is the self-admittance of node i and is equal to the sum of the
admittances 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 + Ḡ) ≠ jLB ,
(2.6)
where LG is called the conductance and LB the susceptance matrix. Ḡ is a diagonal
matrix of the shunt conductances, which will be irrelevant for the remainder of this
thesis.
LG and LB are equivalent to weighted graph Laplacians, where the weights are
respectively the conductance and susceptance of each edge in the graph. In the
following sections, another context where such weighted Laplacians arise as well as
their properties will be discussed.
2.2.2
Consensus Dynamics and Graph Laplacians
Consider a system of n agents: ẋi = ui , i = 1, . . . , n where the control objective
is for all agents to eventually reach the same state x1 (t) = x2 (t) = · · · = x̄(t), i.e.,
consensus. If the control ui is decentralized and merely based on the relative errors
xj ≠ 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 graph
Laplacian L by
Yÿ
_
aik , if i = j,
_
Lij :=
_
]kœNi
≠aij
_
_
_
[
0
if i ”= j and j œ Ni ,
otherwise,
(2.7)
where aij are positive weights of the graph which describes how the agents (nodes)
are connected. The elements on the diagonal Lii , are called the degree of node i
and is the sum of the weights of all edges incident to that node. In the special case
where all edge weights aij = 1, the degree is the number of incident edges.
12
CHAPTER 2. PRELIMINARIES
Figure 2.3: A network of n robots, where the lines symbolize communication links
with positive weights aij .
Now, if we define the state vector x = (x1 , . . . , xn )T , the consensus dynamics
can be written:
ẋ = ≠Lx.
(2.8)
If the graph is connected, i.e., if there is a path between any two agents in the
network, then the control objective, consensus, will be achieved (see e.g. [30] for a
proof). The coupled oscillator dynamics derived in the coming chapters will be a
type of second order consensus dynamics, but the principle is the same as in (2.8),
and x̄ represents the synchronized state.
2.2.3
Properties of Graph Laplacians
We 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 node
i to node j is identical to the edge from node j to node i. Therefore, Lij =
Lji ’i, j œ {1, . . . , n}, and L is symmetric.
q
ii. Zero row/column sums. Since Lii = ≠ j”=i Lij , all rows and columns sum to
0. That means that all graph Laplacians have as common eigenvector the vector
1 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 the
simple zero eigenvalue, remaining n ≠ 1 eigenvalues are positive. If the graph is
not connected, the multiplicity of the zero eigenvalue will equal the number of
isolated graphs.
2.3. THE H2 NORM
13
iv. Diagonalizability by unitary matrix. Since L is symmetric, it can be diagonalized
by a unitary matrix U whose columns are orthonormal (i.e., U ú U = I), such
that L = U ú U , where
= diag{⁄1 , ⁄2 , . . . , ⁄n } is a diagonal matrix of L’s
eigenvalues 0 = ⁄1 Æ ⁄2 Æ . . . Æ ⁄n .
2.3
The H2 Norm
In 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 by
disturbance inputs. We will also define an output signal representing the resistive
losses in the network. A general such input-output system H can be written
ẋ(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 given
by G(s) = C(sI ≠ A)≠1 B. If the system is asymptotically stable, we can define its
H2 norm by
⁄
1 Œ
||G||2H2 =
||G(jÊ)||2F dÊ,
(2.10)
2fi ≠Œ
where || · ||F denotes the Frobenius norm.2 The H2 norm characterizes the system’s
input-output behaviour by, in a sense, quantifying the effect an input w has on the
output y, alternatively the “size” or energy of the system. In control design, it is
often a control objective to keep the H2 norm below a given limit, and the feedback
is chosen accordingly [26].
The integral in (2.10) is however rarely evaluated in the frequency domain using
G(jÊ), but can instead be evaluated conveniently in the time domain, directly from
the state space representation H. This will be the only representation used in this
thesis. Through calculations omitted here it can then be found that
||H||2H2 = tr(B ú XB),
(2.11)
where X is the observability Gramian given by
Aú X + XA = ≠C ú C.3
The matrix equation (2.12) is
In this thesis, we will use
tems of oscillating generators
supported by some of the H2
2
(2.12)
referred to as the Lyapunov equation.
the H2 norm to evaluate the resistive losses in sysduring the synchronization transient. This usage is
norm’s standard interpretations, of which three are
Theq
Frobenius
qm norm is defined as the sum of the absolute values of all entries in a matrix:
n
||A||2F = i=1 j=1 |aij |2 = tr(Aú A).
3
||H||H2 can also be calculated using the controllability Gramian XC ; ||H||2H2 = tr(CXC C ú ),
with AXC + XC Aú = ≠BB ú . This formulation will however not be used in this thesis.
14
CHAPTER 2. PRELIMINARIES
presented below. The physical meaning of these interpretations for our particular
system and the context in which they can be used to quantify the transient resistive
losses 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 stochastic
process such that the covariance E{w(· )wú (t)} = ”(t≠· )I, where I is the identity
matrix and ” the Dirac delta function. Then the (squared) H2 norm represents
the sum of the steady-state variances of the output’s components:
||H||2H2 = lim E{y ú (t)y(t)}.
tæŒ
This variance is the expected value of the sum of the squares of all the output’s
components.
ii. Response to a random initial condition. The H2 norm can also be used to
represent a system response to a certain initial condition when there is no input
to the system, i.e. w(t) = 0 ’t. If the initial condition is a zero-mean random
variable x0 which has covariance E{x0 xú0 } = BB ú , then the H2 norm (squared)
is the time integral
⁄
||H||2H2 =
Œ
0
E{y ú (t)y(t)}dt
of the resulting transient response. This interpretation is closely related to interpretation (iii):
iii. Sum of impulse responses. If H were a single-input-single-output (SISO) system,
the H2 norm would be the signal energy of a simple impulse response at some
time t0 : w(t) = ”(t ≠ t0 ). Here, we are considering a system with multiple inputs
and outputs (MIMO) and the H2 norm then represents the sum of many such
impulse responses; one over each channel.
Let ei denote the ith unit vector in the m-dimensional input space and let there
be m “experiments” where the system is fed an impulse at the ith channel, i.e.,
wi (t) = ei ”(t ≠ t0 ). If the corresponding output signal is yi (t), then the system
H2 norm (squared) is the sum of the L2 norms of these outputs, i.e.:
||H||2H2
2.4
=
m ⁄ Œ
ÿ
i=1 0
yiú (t)yi (t) dt.
Renewable Power Generation
A large scale introduction of renewable energy sources to the power grid is, as mentioned in Chapter 1, apart from introducing high levels of disturbances, likely to
change the dynamic behaviour of the power system. This is due to a new kind of
generation; 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 energy
integrated system has many generators, often asynchronous, with small or no inertia. To a certain extent, power injections by renewable sources can be modeled as
negative load, such that the resulting load is a type of net demand, but as integration levels grow, more physically accurate models are required. We will propose a
simple such model for a dynamical analysis of renewable energy integrated systems
in Chapter 4.
In this section, we will briefly review some basic properties of synchronous and
asynchronous generators and discuss their usage with different type of power sources.
The reader should be aware that the term “asynchronous” is in this thesis somewhat
abused, and used to denote all machines which are not synchronous, i.e., not only
induction machines for which the term is commonly used, but also e.g. power
converters.
2.4.1
Synchronous vs. Asynchronous Generators
Traditionally, the power system is dominated by synchronous generators, or alternators. As discussed in Section 2.1.2, the rotor of a synchronous generator rotates with
a speed corresponding precisely to the grid frequency f0 (provided a synchronous
state), according to
Ê0 =
2fif0
,
p
where p is the number of magnetic poles in the rotor. An example where p = 4
is shown in Figure 2.4. Very simplified, power is generated when the rotor angle
leads the grid angle. When a synchronous generator is started, it needs to be run
to synchronous speed off-line, before being connected to the grid [39].
In an induction generator however, there is no obvious relationship between the
frequency 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;
Ê0 ≠ Ê
s=
.
Ê0
The stator, which surrounds the rotor, is namely excited by the grid, and for a
power to be induced, there needs to be a negative slip so that the rotor cuts the
magnetic 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 power
will neither be generated nor withdrawn from the grid, but the stator will remain
excited and therefore act as an impedance load drawing reactive power, which may
be disadvantageous from a grid perspective [39, 60]. Note also that since the power
input or output from an induction machine depends on the slip, it is also dependent
on the grid frequency.
16
CHAPTER 2. PRELIMINARIES
Figure 2.4: A 4-pole synchronous generator.
2.4.2
Wind Power
Wind power generation stands for the largest portion of installed renewable energy
(disregarding hydro power) [9] and during the last decades, the technology has been
refined in order to increase the efficiency of wind turbines. Apart from improving
the blade design, different types of turbines and generators have been developed,
e.g.:
i. Fixed-speed wind turbines. Until now, the most common type of wind turbines is fixed- or constant-speed turbines, depicted in Figure 2.5a [28]. These
are connected to the grid via a simple induction generator. A fixed-speed
turbine is designed to spin at a certain speed and transfers the mechanical
energy of that rotation via a shaft to the generator, which then operates at a
given 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 amount
of wind energy, or a gearbox may be used to alter the operating speed, but
the efficiency of the generator drops.
ii. Doubly-fed generators. Modern wind farms are often connected to the grid
via doubly fed induction generators (DFIGs), which decouple the electrical and
mechanical rotor frequencies, thus allowing the generator to operate efficiently
at all wind speeds. The DFIG combines the classical induction generator with
a controlled power electronic converter, such that the stator is excited by the
grid, but the rotor windings through the converter [15], see Figure 2.5b. This
way, a desired slip can be obtained, and the output frequency matches the grid.
However, since the rotating parts of the generator are entirely decoupled from
the 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 standalone systems, are connected to the grid via a synchronous generator. The
2.4. RENEWABLE POWER GENERATION
(a) Fixed-speed
17
(b) DFIG
Figure 2.5: Principles of fixed-speed wind turbines with squirrel cage induction
generators (a) and doubly-fed induction generators (DFIGs). Since induction generators consume reactive power, they are often combined with a so-called VAR
compensator, consisting of capacitors, as seen in (a).
synchronous generator may be of a conventional type and use a gearbox to
transfer the mechanical energy from the rotor blades to the generator, or
it may have a converter as an interface towards the grid, which excites the
generator stator and decouples the rotor frequency from the grid. The latter
is preferrable and more common, since wind gusts may otherwise cause loss
of synchronism [28].
2.4.3
Other Sources
While wind energy is the world-wide largest renewable energy source (apart from
hydro power), solar energy is expected to be the fastest growing in the coming
years [9]. The term solar power denotes both photovoltaics (PV) and the less
common so-called concentrated solar power (CSP) generation, which works like a
convetional thermal plant, but where the sun is used as the thermal source. PV cells
however, convert the solar energy directly to electricity and generate a DC power
output. 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 frequency
matches that of the grid, but since there are no rotating parts in a PV system, such
generation provides no inertia, i.e., stored energy, to the system [60].
The two next largest renewable energy sources for electricity generation worldwide are geothermal energy and biomass and biofuels. These differ from wind- and
solar power in that they are dispatchable and therefore more similar to conventional
generation. 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 therefore
likely to be much more heterogenous in terms of generation than today’s grid,
regardless of dominating energy source. A continued stable and secure operation of
the power system therefore relies upon an understanding of the altered dynamics due
18
CHAPTER 2. PRELIMINARIES
to asynchronous generation as well as appropriate control of the power electronics
in 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 Section 2.1.2 to model the power system dynamics for a large network of synchronous
generators. The network which determines the coupling of the swing equations is
described through the admittance matrix, which is a weighted graph Laplacian, as
seen in Section 2.2.1. We consider several scenarios such as the power network encountering isolated disturbance events, or being subjected to persistent stochastic
disturbances where the system is continuously correcting for errors. In both of these
scenarios, we quantify the total power lost during the synchronization transient due
to non-zero line resistances and show that this is given by the squared H2 norm of
the system of generator swing dynamics.
This H2 norm is evaluated by regarding a reduced, or grounded version of the
system, in which one of the system nodes behaves as an infinite bus with fixed
states. The network is then described by so-called grounded Laplacians, as previously studied by e.g. [25,40], in which the inherent singularity of graph Laplacians is
eliminated. We show that, in the case of uniform generators, this grounded system
is equivalent to the original system in terms of the H2 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 network
size. The losses are however shown to have little or no dependence on network
topology, i.e., a loosely connected network will, in principle, incur the same losses
during the transient as a highly connected network. Through numerical examples
and bounds, we illustrate this network topology independence for heterogenous
networks and study the effect of altered generator dampings on the losses.
The remainder of this chapter is organized as follows. Section 3.1 derives the
system dynamics through the classical machine model and defines the resistive power
losses as the performance metrics. We then introduce the grounded system and
derive algebraic expressions for the its H2 norm in Section 3.2, where interpretations
of 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
G1
{V1 , ✓1 }
G2
{V2 , ✓2 }
r23 + jx23
r 45
+
jx
45
r56 + jx56
G5
{V5 , ✓5 }
r46 + jx46
r24 + jx24
r12 + jx12
G3
{V3 , ✓3 }
G6
{V6 , ✓6 }
r
34
+
jx 3
4
G4
{V4 , ✓4 }
r67 + jx67
G7
{V7 , ✓7 }
Figure 3.1: An example of a network with N = 7 generator nodes. Each line has the
impedace zij = 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 least
connected nodes while node 4 is the most interconnected node.
transient resistive losses are also provided. In Section 3.3 we discuss bounds and
generalizations of the norm and proceed to illustrate some of these in the numerical
examples of Section 3.4. We conclude this chapter and discuss the main findings in
Section 3.5.
3.1
Problem Formulation
In this section, we model the power system as a linear time-invariant (LTI) system
of coupled swing equations with distributed disturbances. The output of this system
will represent the dissipated power in the network, so that the squared input-output
H2 norm of the system gives the total resistive losses during the synchronization
transient.
For this purpose, we consider a simplified model of the power system, consisting
of a network of N nodes (buses) and a set E of edges (lines), as depicted in Figure 3.1
for N = 7. At every node i = 1, . . . , N there is a generator with inertia constant Mi ,
damping coefficient —i , voltage magnitude |Vi | and voltage phase angle ◊i . Each line
Eij 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 into
the lines.
3.1.1
System Dynamics
We use the classical machine model, see e.g. [55], and standard linear power flow assumptions, 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 then
given by (2.4). By also making use of the susceptance matrix LB , defined by Equations (2.5)-(2.6), we can write rewrite the differential equation (2.4) in state space
form as:
C D
d ◊
dt Ê
=
C
0
I
≠1
≠M LB ≠M≠1 B
DC D
C
D
◊
0
+
w
Ê
M≠1
(3.1)
(3.2)
where M = diag{Mi }, B = diag{—i }. By a slight abuse of notation, we have let
the states above represent deviations from a steady-state operating point and from
a synchronously rotating reference frame, and let the constant power input Pm,i be
lumped into the disturbance w.
Remark 3.1 The case where the input w is assumed to be pre-scaled by the generator inertia Mi so that B = [0 I]T is also meaningful. In that case one assumes
that a disturbance on a “heavy” large-inertia generator is inherently larger than a
disturbance influencing a “lighter” generator. This is opposed to the current formulation (3.1), which allows a uniformly sized disturbance to have a larger influence
on small-inertia generators. Depending on the character of disturbances, both definitions may be suitable. While events such as a generator failure or sudden changes
in generator operation would be served better by the second choice of input definition, small disturbances due to e.g. net demand fluctuations are more likely to be
better captured by (3.1). A result for the second input definition is however also
presented, see Corollary 3.5.
3.1.2
Performance Metrics
In order to evaluate the performance of the system (3.1) we choose to measure the
control actuation required to drive the system to a synchronous state after a fault
event (disturbance). Synchrony is achieved through circulating power flows that
arise due to the phase angle differences between the generator buses, and we will
measure the control effort as the resistive power losses associated with these flows
due 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 can
be approximated as
Pijloss = gij |◊i ≠ ◊j |2 .
(3.3)
The corresponding sum of resistive losses over all links in the network is then
Ploss =
ÿ
i≥j
gij |◊i ≠ ◊j |2 .
(3.4)
22
CHAPTER 3. RESISTIVE LOSSES IN SYNCHRONIZING POWER NETWORKS
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 Ploss = ◊ú LG ◊, where ◊ is the state
vector introduced in (3.1). We therefore choose to define an output of (3.1) as
y = CÂ =:
Ë
C1 0
C D
È ◊
Ê
(3.5)
,
where Ploss = y ú y. Since LG is positive semidefinite, see Section 2.2.3, we can take
1/2
C1 as the unique positive semidefinite matrix square root LG , which is what we
assume from now on.
Equations (3.1) and (3.5) can then be rewritten as
C D
C
d ◊
0
I
=
≠1
≠M LB ≠M≠1 B
dt Ê
Ë
1
2
y= L
G 0
C D
È ◊
Ê
DC D
C
D
◊
0
+
w
Ê
M≠1
(3.6a)
(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, which
has several standard interpretations that were discussed in Section 2.3. In the
following section, we will calculate the H2 norm from disturbance w to output y
of the system (3.6) and then further discuss the physical implications on the norm
interpretations for our system.
Remark 3.2 Although the linearization of the dynamics which give (3.6a) involves
assuming negligible line resistances, the output (3.6b) captures the effect of nonzero line resistances in terms of transient power losses, given the system trajectories
that result from the linearized swing dynamics.
Remark 3.3 In a more general context, the dynamics (3.6a) is a type of second
order consensus dynamics, see Section 2.2.2. Considering the simpler first order
consensus dynamics (2.8), we can let LQ define another weighted Laplacian for the
same graph. The quadratic form xú LQ x, which is analogous to (3.3), can then
be thought of as an “LQ norm”; ||x||2LQ , which is an energy measure with various
interpretations and applications, see [49]. In [30], the quadratic form xú LQ x is
also proposed as a Lyapunov function, which will be non-increasing along all state
trajectories if the system is controllable and the graph connected.
For the multirobotic system depicted in Figure 2.3, LQ could e.g. be defined
through communication costs, and the conclusions regarding the H2 norm and what
we term the price of synchrony in power systems could be interpreted as the “cost
of consensus” in the robotic system.
3.2. EVALUATION OF LOSSES
3.2
23
Evaluation of Losses
In order to compute the input-output response of (3.6), we first define a reduced
system H̃ and derive an expression for its H2 norm. We then show that this norm
is equal to that of the original system (3.6). Following that, we consider the special
case when all lines have equal resistance to reactance ratios. Finally, we discuss
interpretations of the H2 norm and their implications for this particular system.
Throughout this section we will assume identical generators, i.e., M = M I and
B = —I.
3.2.1
System Reduction
As previously discussed, LG and LB are graph Laplacians, and as such each have
a 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 define
the H2 norm of (3.6) we will therefore instead regard a reduced system which is
asymptotically stable.
Following the approach in [25], we derive the reduced system by first defining
a reference state k œ {1, . . . , N }. We denote the reduced or grounded Laplacians
that arise from deleting the k th rows and columns of LG and LB respectively, by L̃G
and L̃B . The states of the reduced system ◊˜ and Ễ are then obtained by discarding
the k th elements of each state vector. This leads to a system that is equivalent to
one in which ◊k = Êk © 0 for some node k œ {1, 2, . . . , N }, and all other states are
measured towards this reference. This has the physical meaning of connecting the
k th node to ground, hence the terminology, and a mechanical analogy can be seen
in Figure 3.2. We call the resulting reduced, or grounded, system H̃:
C D
C
d ◊˜
0
I
=
—
1
≠ M L̃B ≠ M
I
dt Ễ
=: A„˜ + B w̃;
Ë
1
2
ỹ = L̃
G 0
C D
È ◊˜
Ễ
DC D
◊˜
+
Ễ
˜
=: C „.
C
0
1
MI
D
w̃
(3.7a)
(3.7b)
By the assumption of a network where the underlying graph is connected, the
grounded Laplacians L̃G and L̃B are positive definite Hermitian matrices (see e.g.
[40]). All of the poles of system H̃ are thus strictly in the left half plane and the
input-output transfer function from w̃ to ỹ has a finite H2 norm.
3.2.2
H2 Norm Calculation
The (squared) H2 norm of the system H̃ is given by Equations (2.11) - (2.12).
We call the obsevability Gramian X̃ and partition it into four submatrices. The
24
CHAPTER 3. RESISTIVE LOSSES IN SYNCHRONIZING POWER NETWORKS
Ê2
Ê1 © 0
b12
◊2
b23
b13
Ê3
Figure 3.2: Mechanical analogy to grounded power system dynamics. If the k th ,
here the 1st , oscillator is fixed, ◊1 = Ê1 © 0, and the other angles are measured
with respect to that reference. It is intuitively apparent that in this system, the
oscillators will settle at their initial points after a disturbance, as opposed to the
system depicted in Figure 2.2 where the final angles are likely to differ from the
initial ones. This drift in the original (ungrounded) system, i.e., the change in
operating state, is information lost by reducing a system of consensus dynamics.
Such an angle drift is however irrelevant in power systems, where only phase angle
differences are relevant and where the zero phase angle can be defined entirely
arbitrarily.
Lyapunov equation (2.12) expanded for our system (3.7) is then
C
1
0 ≠M
L̃B
—
I ≠M
I
DC
D
C
X̃1 X̃0
X̃1 X̃0
+
X̃0ú X̃2
X̃0ú X̃2
DC
D
C
D
0
I
L̃
0
=≠ G
.
—
1
0 0
≠M
L̃B ≠ M
I
From this, we obtain
X̃0 ≠
—
—
—
X̃2 + X̃0ú ≠ X̃2
=0 ∆
tr(X̃2 ) = tr(Re{X̃0 }),
M
M
M
where Re{·} extracts the real part of a complex matrix. Moreover,
≠
1
1
L̃B X̃0ú ≠ X̃0 L̃B = ≠L̃G ,
M
M
which, since L̃B is nonsingular, gives
≠1
L̃B X̃0ú L̃≠1
B + X̃0 = M L̃G L̃B .
≠1
ú
ú
Since tr(L̃B X̃0ú L̃≠1
B ) = tr(L̃B L̃B X̃0 ) = tr(X̃0 ) we have that
tr(Re{X̃0 }) =
Finally, noting that tr(B ú X̃B) =
in the following lemma.
M
tr(L̃≠1
B L̃G ).
2
1
tr(X̃2 ),
M2
(3.8)
these equations give the result stated
3.2. EVALUATION OF LOSSES
25
Lemma 3.1 The H2 norm (squared) of the input-output mapping (3.7) is
||H̃||2H2 =
1
tr(L̃≠1
B L̃G ),
2—
(3.9)
where L̃B and L̃G are the grounded Laplacians obtained by deleting row and column
k from the susceptance and conductance matrices LB and LG and where — is a
generator’s self damping.
Lemma 3.1 is derived using the reduced, or grounded system H̃. However, it
turns out that the choice of grounded node k has no influence on the resulting H2
norm. We illustrate this point through the following lemmas, which are used derive
the main result of Theorem 3.4.
Lemma 3.2 Let H be the input-output mapping (3.6) with M = M I and B = —I
and H̃ denote the corresponding reduced system (3.7). Then, the norm ÎHÎ2H2 exists
and is equal to ÎH̃Î2H2 for any grounded node k.
Proof: See Appendix A.
Lemma 3.3 Let L̃G and L̃B be the reduced, or grounded, Laplacians obtained by
deleting the kth rows and columns from LG and LB respectively. Then:
†
tr(L̃≠1
B L̃G ) = tr(LB LG ),
where
†
(3.10)
denotes the Moore-Penrose pseudo inverse.
Proof: See Appendix A.
The result can now be stated in the following theorem.
Theorem 3.4 Given a system of N generators with equal damping and inertia
coefficients —i = — and Mi = M, ’i œ {1, . . . , N } whose input-output response is
given by (3.6). The squared H2 norm of the system is given by
ÎHÎ2H2 =
2
1 1 †
tr LB LG .
2—
(3.11)
Thus, the total transient resistive losses of the system are a function of what we
term the generalized Laplacian ratio of LG to LB .
Proof: Follows directly from Lemmas 3.1 - 3.3.
26
CHAPTER 3. RESISTIVE LOSSES IN SYNCHRONIZING POWER NETWORKS
Remark 3.4 In the above derivation, we have assumed that the input w is defined
relative to the mechanical input Pm,i to each generator i. If instead, one chooses
to scale the input by the generator’s inertia, i.e., let wÕ represent Pm,i /M and set
B Õ = [0 I]T , one obtains the following result:
Corollary 3.5 Consider the modified input-output mapping H Õ :
C D
d ◊
dt Ê
=
C
y =
Ë
0
I
—
1
≠ M LB ≠ M
I
1
2
LG 0
C D
È ◊
Ê
DC D
C D
◊
0 Õ
+
w
Ê
I
(3.12)
.
The H2 norm (squared) of this system is
ÎH Õ ||2H2 =
M2
tr(L†B LG ).
2—
Proof: The result is easily obtained in analogy to the derivation of Lemma 3.1, noting that in this case, tr(B Õú X̃B Õ ) = tr(X̃2 ). The rest then follows from Lemmas 3.2
and 3.3.
3.2.3
Special Case: Equal Line Ratios
The result in Theorem 3.4 states that the transient resistive losses in a synchronizing
network are linearly dependent on what can be thought of as a generalized ratio between the conductance and susceptance matrices. We now consider the assumption
that this generalized ratio is a scalar matrix, which is the case when all lines of the
system have equal ratios between their conductance and susceptance, equivalently
their resistance to reactance ratios, i.e., we assume that for all edges Eij œ E
gij
rij
=
= –,
bij
xij
which gives LG = –LB . While still assuming identical generators, by Lemmas 3.1
and 3.2, we have that
||H||2H2 =
1
–
tr(L̃≠1
(N ≠ 1),
B –L̃B ) =
2—
2—
(3.13)
which is the result presented in [3]. This result is remarkable in that it says that the
loss 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
bij of all lines Eij in the system. The authors of [49] propose such a mean which
makes the result (3.13) exact.
3.2. EVALUATION OF LOSSES
27
ij
A choice of – = –max Ø bij
for all edges Eij œ E, will make (3.13) conservative, and
gij
vice versa if – = –min Æ bij . The H2 norm of the system can thus be bounded as:
g
–min
–max
(N ≠ 1) Æ ||H||2H2 Æ
(N ≠ 1).
2—
2—
(3.14)
These bounds are independent of the network topology, but increase unboundedly
with the number of generators. The accuracy of these bounds in comparison to such
that reflect network characteristics will be discussed in the coming sections.
While the ratio of power lines’ resistances to reactances of, in particular, transmission systems is generally small and, as in (2.3), often neglected in power flow
calculations [34], the result (3.13) shows that increasing the number of generators
will increase resistive losses, regardless of the network topology. This is a fact that
will become increasingly important as generation becomes more distributed.
In particular, the envisioned future smart grid is likely to involve a large number
of generators connected to low voltage distribution grids, which have higher r/x
ratios than transmission systems (typically, this ratio is 1/16 in 400 kV lines but
2/3 in 11 kV systems) [23]. The result (3.13) thus indicates that a large number of
generator nodes in this type of network, which in any case will be subject to high
levels of disturbances due to intermittency in the generation, will lead to higher
losses due to grid synchronization.
The equal line ratio assumption is not unreasonable for power systems, as there
are a select number of materials and line configurations used for transmission systems and for all of these the ratio of resistances to reactance ratio tend to lie within
a small interval. In order to quantify this notion we examined four IEEE transmission system benchmark cases, representative of parts of the American power
transmission system, and found that a high percentage of the lines fell within a
narrow range, see Figure 3.3. For example in the 118 bus system, 90% of the lines
had a ratio below 0.34, and 72% lay in the interval 0.20 ≠ 0.30. According to the
classical machine model considered in this chapter, the network is reduced so that
the lines also represent impedance loads, a case not captured by the IEEE test
systems. A recent study [42], however, suggests that uniformity in line properties
also applies to such Kron reduced networks, by quantifying the homogenity in node
degrees of several reduced actual power networks.
Remark 3.6 The result (3.13) for when LG = –LB is also a special case of a result
which applies when LG and LB are simultaneously diagonalizable. If LB and LG are
simultaneously diagonalizable, the H2 norm can be expressed directly in terms of
the Laplacian eigenvalues. A derivation of this more general result and a discussion
of cases when it applies is found in Appendix A.
3.2.4
H2 Norm Interpretations for Swing Dynamics
By the formulation in Section 3.1.2, the square of the Euclidean norm y ú y of the
output vector is defined to equal the dissipated real power in the network lines
28
CHAPTER 3. RESISTIVE LOSSES IN SYNCHRONIZING POWER NETWORKS
1.4
14 bus
30 bus
57 bus
118 bus
Line Ratio
k
1.2
1
0.8
0.6
0.4
0.2
0
0
50
100
150
200
250
300
Line index k
Figure 3.3: Resistance to reactance ratios r/x for the lines Eij = Ek œ E for the IEEE
14, 30, 57 and 118 bus benchmark cases. Note that the “lines” of zero resistance
correspond to transformers, which are not part of the model considered in this
chapter.
during the synchronization of the system after a disturbance. We choose to evaluate
this lost power by calculating the H2 norm of the input-output system (3.6). The
concept of the H2 norm and its interpretation were reviewed in Section 2.3. We will
now discuss, in relation to these interpretations, physical scenarios which permit
the H2 norm in 3.11 to quantify the resistive losses of the system (3.6).
i. Response to persistent stochastic disturbance. The H2 norm (squared) can
be interpreted as the steady-state total variance, when the input signal is white
noise. For the system considered in this chapter, white noise can be thought of
as a persistent stochastic forcing at each generator. These disturbances, which
would be uncorrelated across the system’s generators may be due to e.g. local
variations in gereration and load. The H2 norm would then exactly correspond
to the expected total power losses.
ii. Response to a random initial condition. If the system is not subject to any
disturbance, but is driven from an
„0 which is a random variable
C initial condition
D
0
0
with covariance „0 „ú0 = BB ú =
then the H2 norm (squared) will be
0 M≠2
the total expected resistive losses due to the system’s returning to a synchronized
state. This random initial condition „0 corresponds to each generator having a
random initial velocity perturbation that is uncorrelated across the generators
(since BB ú is diagonal), and zero initial phase perturbation.
iii. Sum of impulse force responses. If each generator is subject to an impulse
force disturbance, then ||H||2H2 is the total power loss over all time and over all
lines. Such an impulse disturbance could occur e.g. due to planned changed
operation of the generator, a sudden lost load at the bus or a fault event.
3.3. GENERALIZATIONS AND BOUNDS
3.3
29
Generalizations and Bounds
In this section, we will present further bounds on the expression (3.11) and discuss
their implications for resistive losses in syncronizing power grids. We will also
address the more general case of non-identical generators as well as applications
outside the field of power systems.
3.3.1
Network-Characteristic Bounds on Losses
As previously mentioned, the term tr(L†B LG ) in Theorem 3.4 can be interpreted
as a generalized ratio between the power network’s conductance matrix LG and its
susceptance matrix LB , i.e., the real and imaginary part of the admittance matrix
G
Y . We denote the respective eigenvalues of LG and LB as ⁄G
N Ø ... Ø ⁄2 > 0 and
B
⁄B
N Ø ... Ø ⁄2 > 0. The generalized ratio of these two Laplacians can then be lower
bounded in terms of their eigenvalues as
tr(L†B LG ) Ø
N
ÿ
⁄G
i
.
B
i=2
⁄i
(3.15)
(See e.g. [59] for a proof.) In the case of identical line ratios, equality holds, and
each eigenvalue ratio is equal to –. The literature on Laplacian eigenvalues and their
relationships to the underlying graphs is vast, [6] and [8] for good general overviews.
But LG and LB describe the same topology, which makes it is reasonable to assume
that when the graph undergoes changes, their degrees and eigenvalues both change
in the same fashion. Also, when the network grows, the number of eigenvalues
increase. The sum of the Laplacians’ N ≠ 1 non-zero eigenvalue ratios will thus
both be topology independent and grow unboundedly with N . Therefore we can
conclude that the bound (3.15) leads to resistive losses that scale unboundedly
with the network size and are independent of network connectivity, similar to the
conclusions 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 upper
bounded 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 is
heterogenous in terms of the line resistance to reactance ratios. This may be the
case if a combined transmission and distrubution network is considered, or if the
impedance loads, that are lumped into the lines in the reduced generator network
considered in this paper, are very different. In some cases, it is then better to bound
the losses in terms of graph-theoretical quantities. This can be done as:
†
†
⁄G
2 tr(LB ) Æ tr(LB LG ) Æ
tr(LG )
,
⁄B
2
(3.16)
B
(again, see [59] for a proof). ⁄G
2 and ⁄2 are the smallest non-zero eigenvalues of LG
and LB , or the algebraic connectivities of the graphs weighted by line conductances
N
N
B
and susceptances respectively. It holds that ⁄G
2 Æ N ≠1 gii,min and ⁄2 Æ N ≠1 bii,min ,
30
CHAPTER 3. RESISTIVE LOSSES IN SYNCHRONIZING POWER NETWORKS
the gii , bii being respectively the self conductances and susceptances (degrees) of the
nodes. Furthermore, the quantity tr(L†B ) is proportional to what we can interpret
as the total effective reactance of the network, in analogy with the concept of graph
total effective resistance, as recently discussed in e.g. [19] and [25].
By Rayleigh’s monotinicity law (see [12]), the total effective reactance can decrease unboundedly by adding lines and increasing line susceptances. However,
the algebraic connectivites are very small for weakly connected networks and decrease with network size, so while the bounds (3.16) are accurate for small, and
well-interconnected networks, they become loose for the type of networks that most
often characterize a power grid.
In a more general context, Theorem 3.4 however applies to all networks with
second order consensus dynamics, and the H2 norm can be interpreted as an energy
measure, see Remark 3.3. Such dynamics may describe several types of mechanical
or biological systems [13], which may be of a different character in terms of edge
ratios and connectivities than power systems. The different considerations and
bounds discussed here, e.g. those for highly connected networks, can then be of
relevance, especially when the network topology is subject to design, in order to
reduce this general energy measure.
3.3.2
Generator Parameter Dependence
From Theorem 3.4 we can deduce that
1
1
tr(L†B LG ) Æ ÎHÎ2H2 Æ
tr(L†B LG ),
2—max
2—min
where —min = miniœ{1,...,N } —i and —max = maxiœ{1,...,N } —i . The losses are thus
bounded by the properties of the and most strongly and lighly damped generators
respectively. However, envisioning an increased penetration of renewable generation
to the grid, more insight to the effects of varying generator properties is desired.
The results derived by considering a grid with identical generators suggest that the
losses scale with the network size. However, the marginal losses for an added line
connecting a “light” (low-inertia) and highly damped generator to the system will
not be as large as if the new line is attached to a “heavy” generator. Consider the
following corollary to Lemma 3.1:
Corollary 3.6 Consider a network of N generators, and let its resistive losses be
represented by ||H̃0 ||2H2 for some choice of grounded node k œ {1, ..., N }.Connect an
additional generator with damping —N +1 and inertia MN +1 to node k by the line
Ek,N +1 .
The new system’s losses will be given by:
||H̃1 ||2H2 = ||H̃0 ||2H2 +
where –k,N +1 =
rk,N +1
xk,N +1
1
2—N +1
is the line ratio of Ek,N +1 .
–k,N +1 ,
If the system is described by (3.12) the additive term is instead
2
MN
+1
2—N +1 –k,N +1 .
3.4. NUMERICAL EXAMPLES
31
Proof: See Appendix A.
Remark 3.7 Note that Corollary 3.6 allows for the N generators represented by
H0 to be non-uniform in terms of damping and inertia.
In the face of increased penetration of renewable generation, this result implies
that while the synchronization losses do scale with the network size, the impact of
typically low inertia renewable generators will be relatively low, compared to that
of conventional generators.
An analysis of the resistive losses in networks of non-uniform generators with
different dynamics is given in Chapter 4, which will provide more insight to the
effects of renewable power integration on grid synchronization.
Apart from Corollary 3.6, a general result for the resistive losses in generator
networks with non-uniform generator dampings and inertias is not derived in this
Chapter. However, as may be intuitively apparent, the generator parameters do interact 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 that
dominate the system, i.e., that have large dampings, are placed at highly interconnected 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 a
role for the losses, and the same argument applies; losses are reduced if high-inertia
generators are also highly interconnected.
3.4
Numerical Examples
The results derived and discussed in the previous sections indicate that the resistive
losses 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 this
section, we illustrate these results by numerical examples and simulations.
3.4.1
Line Ratio Variance
In the Section 3.2, we showed that the H2 norm depends on what we term the generalized ratio between the conductance and susceptance matrices. In Section 3.2.3,
we discussed how this ratio can be bounded by actual conductance to susceptance
ratios in the system, which tend to lie in a small interval. In Section 3.3.1 we also
showed that the generalized ratio is essentially topology-independent and lies close
to the actual line ratios, even for heterogenous networks. In the example presented
in this subsection, we will illustrate numerically that the assumption of equal line
ratios is a good approximation, by considering the IEEE benchmark topologies with
increasingly heterogenous line properties.
Figure 3.4 shows the resistive losses according to Theorem 3.4 for a hypothetical set of identical generators (— = 1) connected by the topologies of the
32
CHAPTER 3. RESISTIVE LOSSES IN SYNCHRONIZING POWER NETWORKS
10
||H||2H
2
8
6
4
2
0
0
0.02
0.04
ij
0.06
0.08
Standard Deviation
0.1
0.12
Figure 3.4: Resistive losses in the modified IEEE 14 (lower points) and 30 bus (upper
r
points) benchmark networks with lines of increasingly varied ratios –ij = xijij . The
figure shows the mean of 100 randomly generated systems. The bars illustrate the
bounds in (3.14).
IEEE 14 bus and 30 bus benchmark systems respectively. The line ratios –ij =
rij
gij
xij = bij for the lines Eij œ E are randomly uniformly distributed on the intervals
0.4, 0.4 ± 0.025, 0.4 ± 0.05, ..., 0.4 ± 0.2, and the horizontal axis represents the resulting standard deviation of the line ratios. We take the values for xij from the
benchmark systems and let rij = –ij xij . The bars in the figure represent the upper
and lower bounds of the inequality (3.14).
We note that while increased standard deviation of the line ratios leads to a
less precise bounding by (3.14), the resistive losses of the system themselves vary
very little when the average line ratio remains constant. They are instead highly
dependent 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 subject
to the increased variance can also be understood by considering Theorem 3.5 and
writing the conductance matrix as LG = –LB + L̄G . Then
||H||2H2 =
–
1
(N ≠ 1) +
tr(L†B L̄G ).
2—
2—
The entries of L̄G in general take on both positive and negative values, and the less
distributed the line ratios are around –, the smaller their absolute values are. For
a meaningful choice of –, like the average value over the network, tr(L†B L̄G ) will
be small. In Figure 3.4, this quantity is represented by the small deviations of the
points from the horizontal lines respresenting equal line ratios.
3.4.2
Increased Network Size
According to our results, the resistive losses in a network of synchronizing generators will be largely independent of the network topology, but instead depend on
the ratio between susceptances and reactances and increase unboundedly with the
network size. In this example, we will compare the H2 norm of two increasingly
large hypothetical power networks, one whose underlying graph is radial, where all
3.4. NUMERICAL EXAMPLES
(a) Radial
33
(b) Complete
Figure 3.5: Example of a 4 node (a) radial and (b) complete graph. The example of
Section 3.4.2 calculates the H2 norm for these types of networks when the number
N of nodes grows large.
except two nodes have two neighbors, and one where it is a complete graph, in
which every node is connected to every other node. Examples of these graphs can
be seen in Figure 3.5.
For simplicity, we regard the case where — = 1, and assign random line parameters to the increasingly large networks, letting each line’s reactance xij and line
ratio –ij both be drawn from a normal distribution with mean 0.2 and standard
deviation 0.1 (replacing any negative values with the mean). As shown in the previous example, one can then expect the norm for each network to mainly depend
on 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 graphs
as the network size increases from a 5 node to a 50 node system. The bounds
presented 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. We
also note that the network-parameter dependent bounds (3.16) are more accurate
than the line ratio bounds (3.14) for the complete graph, but are not very accurate
for the radial network.
3.4.3
Marginal Losses for Added Lines
As a first step of characterizing a case with non-uniform generators, we will now
study the situation in Corollary 3.6 and simulate the 7 bus network depicted in
Figure 3.1. We let all lines Eij 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 the
20
10
parameters [48]: Mi = 2fif
and —i = 2fif
= —0 with a frequency f = 60 Hz. Let this
original system be denoted H̃0 .
To node 1, three different 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 3.7
then shows the system trajectories of the three resulting reduced systems H̃1 , when
the system is subject to a random initial angular velocity disturbance, corresponding
to the H2 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 ||H̃1 ||2H2 = ||H̃0 ||2H2 + 2—1,88 = 60.3, 26.4
34
CHAPTER 3. RESISTIVE LOSSES IN SYNCHRONIZING POWER NETWORKS
12
H norm
2
10
8
min
−
max
bounds
Eigenvalue ratio bound
6
4
2
0
5
10
15
20
25
30
35
40
45
50
35
40
45
50
N
(a) Radial Graph
12
10
8
H2 norm
min
−
max
bounds
Eigenvalue ratio bound
−bound
2
6
4
2
0
5
10
15
20
25
30
N
(b) Complete Graph
Figure 3.6: H2 norms for (a) a radial network and (b) a complete graph, each with
N nodes, with some of the bounds presented in Section 3.3. Despite some variation
due to the randomness in the line parameters, the H2 norm scales linearly with
the network size and is largely the same for the complete graph as for the radial
network. The bound related to the Laplacian eigenvalue ratios (3.15) is the most
accurate bound, and for the complete graph, the inequality (3.16) linked to the
algebraic connectivities ⁄2 , also provides accurate bounds (for the radial graph, the
latter have been omitted since they are off by orders of magnitude).
3.4. NUMERICAL EXAMPLES
35
2
2
0
0
0
−2
−2
−2
0
5
10
15
20
40
0
2
5
10
15
20
0
50
50
0
0
5
10
15
20
5
10
t
15
20
20
0
−20
−40
0
5
10
15
t
(a) —N +1 = 10—0
20
−50
0
5
10
t
15
(b) —N +1 = —0
20
−50
0
(c) —N +1 = 0.1—0
Figure 3.7: Simulation of the grounded 7 bus network of Figure 3.1 with identical
generators 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 the
grounded node (k = 1). The system is subject to random velocity and zero phase
initial conditions, so that the expected power losses correspond to the H2 norm. The
losses are the largest in system (c), where the lightly damped generator maintains
its 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 are
repectively 110, 32.2 and 23.7. The poorly damped generator will experience strong
oscillations and incur large transient losses before it stabilizes to the same states
as the grounded node. The highly damped generator, on the other hand, incurs
less oscillations and losses than in the case where an equally damped generator is
connected.
3.4.4
Effects of Generator Placement
Now consider again the network depicted in Figure 3.1. We assign the impedances
rij +jxij = 0.1+j0.6 to all lines of the system, which results in nodes 1 and 7 having
the smallest degree, node 4 having four times, nodes 2 and 6 having three times
and nodes 3 and 5 having twice that degree. We will now compare the behaviour
of 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 (matched
dampings and degrees) to when (b) strongly damped generators are placed at the
least connected nodes (mismatched dampings and degrees).
20
For the simulations, we use the following parameters: M = 2fif
and — œ
1
2fif {2, 8, 14, 20}, with a frequency f = 60 Hz. Figure 3.8a and 3.8b respectively
show 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 the
damping coefficients — as in situation (a) above, and in Figure 3.8b they have been
mismatched as in situation (b). Call these systems H̃match and H̃mismatch respec-
36
CHAPTER 3. RESISTIVE LOSSES IN SYNCHRONIZING POWER NETWORKS
4
4
2
2
0
0
−2
−2
−4
0
40
5
10
15
20
25
−4
0
40
10
5
10
15
20
25
15
20
25
20
20
0
0
−20
−20
−40
0
5
5
10
15
20
25
Time
(a) Damping size matched to node degrees
−40
0
Time
(b) Damping size mismatched to node degrees
Figure 3.8: Simulation of the reduced 7 bus system in Figure 3.1 with node 1
grounded and with (a) the strongly damped generators placed at the highly interconnected nodes and vice versa, and (b) the strongly damped generators placed at
weakly interconnected nodes and vice versa. The system in (b) is less coherent and
experiences larger resistive losses during the transient response.
tively.
In order for the simulations to correspond to the H2 norm interpretation (ii) in
Section
2.3, the initial conditions Ễi (0) were drawn from a uniform distribution on
Ô
Ô
12
12
˜
[≠ 2M , 2M
] and ◊(0)
was set to zero.
The expected power losses during the transient response for these respective
systems are ||H̃match ||2H2 = 18.9 and ||H̃mismatch ||2H2 = 20.7. For the particular cases
depicted in Figures 3.8a and 3.8b, the losses are 13.2 and 27.9 respectively. From
the figures, it is seen that the transient behaviour of the system H̃mismatch is less
“coherent” than that of H̃match , most clearly by regarding the time at which the
small oscillations in the respective phase angles ◊ have died out.
3.5
Discussion
In this Chapter, we have formulated power system dynamics as a linear system of
coupled swing equations. The system’s output was defined such that the system’s
squared input-output H2 norm corresponds to the transient resistive losses incurred
after the system is subjected to either impulse disturbances or persistent stochastic
forcings. This H2 norm is calculated by making use of the concept of grounded,
Laplacians, which eliminate the inherent singularity of consensus problems (see
Section 2.2) by choosing one node in the network as a fixed reference. By showing
that the H2 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 related
result is discussed in [25] where the effective graph resistance (or Kirchhoff index)
is calculated from a grounded conductance matrix and is shown to be indifferent 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 thought
of as a generalized ratio between the conductance and susceptance matrices. These
transient losses are shown to increase with the network size, and in the limit of
all lines in the network having identical ratio of line conductance to susceptance,
they are entirely independent of the topology and scale directly with the number
of nodes. In this limit it is directly deducible that highly connected and loosely
connected networks incur the same resistive power losses in recovering synchrony.
By considering bounds on the generalized Laplacian ratio, these same conclusions
can be shown to hold also for more heterogenous networks, as is demonstrated in
the numerical examples of Section 3.4.
These results showing that the transient power losses are independent of network
topology are in contrast to typical power system stability notions and performance
metrics such as network coherence. For example, the topology of the system plays
an important role in determining whether a system can synchronize [4, 10, 11] and
the rate of convergence or damping of a power system is directly related to the
network connectivity [37]. One intuitive explanation for the price of synchrony being independent of network topology is as follows. A highly connected network is
expected to have more phase coherence than a loosely connected network. Consequently, 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 the
aggregate, the total power losses of the two networks are the same. One should keep
in mind however, that in the less coherent network, disturbances are more likely to
cause instabilities. The issues of stability and the cost of synchrony are two different
concerns.
When considering a network of non-uniform generators, we also show that the
marginal losses incurred by adding a well-damped low-inertia generator to the system are small compared to adding a poorly damped high-inertia generator. This
generator parameter dependence is relevant in the face of increasingly distributed
renewable generation which may quickly increase the number of nodes in low voltage grids with typically high resistances, and thus, according to our results, lead
to large transient power losses. This fact, combined with the more intense disturbances that can be expected from the source variability in a renewable integrated
grid, indicate that the significance of the transient resistive losses will increase,
and different strategies to reduce them may become relevant. Our numerical results related to generator placement, which show that losses are decreased if highly
damped generators are also highly interconnected, may provide more insight to such
strategies.
In this chapter, however, we have not considered the changes in system dynamics
introduced by renewable generation sources. Also, the classical machine model’s
reduction of power network topologies to a set of only generator nodes may limit
the legitimity of the linear DC power flow approximation (see Section 2.1.2) and
gives a model which does not reflect the grid topology. In Chapter 4 we will extend
38
CHAPTER 3. RESISTIVE LOSSES IN SYNCHRONIZING POWER NETWORKS
this model so that both renewable generation and loads, as well as topology, can be
modeled 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 extended
to capture renewable power sources as well as loads. As mentioned in Chapter 1,
the grid integration of substantial amounts of renewable generation will significantly
change the dynamical properties of the power system, such as the manner in which
the system returns to a synchronous state after a disturbance [23,52]. As in Chapter
3, we will not directly address questions about regions of synchronous stability or
how disturbances propagate in the face of intermittent energy, but instead assume
the disturbances to be sufficiently small to ensure an asymptotically stable linear
system. We thus presume that the system will recover synchrony after a disturbance
event and evaluate the control effort in terms of the real power required to achieve
this synchrony. This real power loss during the transient is again quantified as the
square of the system’s H2 norm.
The new system will be modeled by adding power injections with first order dynamics to the existing oscillating system in order to create a coupled system, which
we will, by some abuse of terminology, refer to as one of coupled first and second
order oscillators. These types of complex oscillator networks have been studied for
a number of applications, see [13] for a survey along with applications in mechanical
and biological systems. In the context of power systems stability, they were first
analyzed by Bergen and Hill [5], who proposed a model which preserves network
structure and explicitly includes loads. This model is in contrast to the commonly
studied classical machine model introduced in Chapter 3 which incorporates load
buses into the lines, thus altering the topology. The loads of the Bergen-Hill model
were assumed to be frequency-dependent and thus described by first order dynamics. In this chapter, we use this model to represent renewable generation, such as
that from wind or solar power plants. We show that this usage to capture the
dynamics 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 offers many more complicated or high
order dynamical descriptions of asynchronous machines in general and wind power
generators in particular [14, 18, 46], the appealingly simple coupled first and second
order oscillator model has so far, to our knowledge, not been used to represent the
addition 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 also
simplify the stability analysis of the linear system. We will also present a result
on the generalized ratio between graph Laplacians similar to that of (3.10), and
show that a certain type of network augmentation used to combine different order
dynamics at the same node has no effect 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 different, for
networks of equal number of nodes they may all have the same H2 norm provided
that their damping coefficients are equal. This indicates that adding additional
renewable energy sources to a network will not increase the system losses if their
controllers can be adjusted to match the damping coefficients of the existing synchronous machines.
The remainder of this chapter is organized as follows. Section 4.1 introduces
the 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 for
its H2 norm, which is then compared to the results in Chapter 3. We present
numerical examples in Section 4.3 and summarize the chapter through a discussion
in Section 4.4.
4.1
Problem Formulation
In this section, the problem formulation of Section 3.1, will be extended to incorporate loads and asynchronous machines such as wind generators, using the machinery
introduced in [5]. We now consider a network of n0 nodes with m Æ n0 synchronous
generators of the kind modeled in Chapter 3. We will assume that the n0 ≠ m
remaining nodes are dominated by frequency-dependent first order dynamics, representing loads, asynchronous generation or controlled power electronics. We will
let the system at steady state be subject to a disturbance input and define the
resistive losses that arise in the system as it returns to synchrony as the output of
the system.
The remainder of this section is organized as follows. First, a means of augmenting the network to allow for different dynamics at the same bus will be introduced.
Then, the first order oscillator model of asynchronous machines, i.e. loads or renewable generation, will be presented and justified. We will then derive the state-space
description of the system, briefly discuss the disturbance input and finally define an
output of the system that is analogous to the definition in Section 3.1.
4.1. PROBLEM FORMULATION
4.1.1
41
Network Model
Consider a network of n0 nodes (buses) connected by the set E0 of edges (lines) and
let m of these nodes each be attached to a synchronous generator. Now, we augment
the system by introducing m ficticious buses associated with these generators and
connect them by purely reactive, i.e. lossless, lines, representing the generator’s
internal reactance, the so-called transient reactance. Call the set of these m lossless
lines used to augment the system Eaug . Figure 4.1 shows an example of how a system
is augmented in this manner.
We now have n = n0 + m buses in the system. Let each of these have the voltage
magnitude |Vi | and voltage phase angle ◊i . Without loss of generality, the buses will
be numbered such that buses 1, ..., n0 ≠m are without synchronous generation, nodes
n0 ≠ m + 1, ..., n0 are the the original generator buses, connected in rising order to
nodes n0 + 1, ..., n0 + m = n, which are the fictitious generator buses. See also
Table 4.1.
In all of the derivations that follow, and unless otherwise stated, the fictitious
generator bus n will be the grounded node. This means that we let ◊n © 0 and all
other phase angles and frequency deviations in the system will be measured with
respect 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-zero
load must be co-located with every synchronous generator in the pre-augmented system, which is what we will assume without loss of generality. To model a bus with
only a synchronous generator, the corresponding node does not require augmentation and its transient reactance can be absorbed into surrounding lines.
Table 4.1: Node numbering for the augmented network.
Indices
Set
1, . . . , n0 ≠ m
Nwind
n0 ≠ m + 1, ..., n0
Nload
n0 +1, ..., n0 +m = n,
Naug
n
4.1.2
-
Node type
Buses without synchronous generation. Assumed
to be dominated by renewable generation.
Generator/load buses of pre-augmented system,
load buses in augmented network.
Fictitious generator buses, connected to nodes
n0 ≠ m + 1, ..., n0 in rising order.
Grounded node (infinite bus)
Model of Asynchronous Machines
We will consider a system where the dynamics at the n0 ≠ m buses without synchronous generation are dominated by asynchronous machines, such as wind power
induction generators, which inject a frequency-dependent power to the grid. We
model these as:
0
Pwind,i = Pwind,i
≠ Di Êi
(4.1)
42
CHAPTER 4. LOSSES IN RENEWABLE ENERGY INTEGRATED SYSTEMS
G
3
G
+
jx Õ
+
5
3
r4
Õ 2
jx 1
r2
+
jx Õ
23
r 12
45
1
r34 + jxÕ34 4
r25 + jxÕ25
2
5
G
(a) Original network
G
1
7
jx47
4
r34 + jx34
+
jx
5
+
2
r4
3
jx 1
r2
+
jx
23
r 12
45
6
jx36
3
G
r25 + jx25
2
5
jx58
8
G
(b) Augmented network
Figure 4.1: Example of a 5 bus network with 3 generators (a), and the corresponding
augmented network (b) in which every generator/load bus is replaced by a load bus
and a fictitious generator bus. The lines in Eaug = {E36 , E47 , E58 } are the purely
reactive lines that model the transient reactances of the generators, which in (a) are
absorbed into xÕij . The arrows directed towards buses 1 and 2 symbolize a power
injection by an asynchronous generation source.
4.1. PROBLEM FORMULATION
43
0
’i œ Nwind where Pwind,i
is the constant steady-state input, Di > 0 is the frequency
dependence 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 real
power loads and is one of the models that has been used since to represent induction
motors drawing power from the grid [36]. Although induction machines are in
most cases modeled as a higher order system using voltage, current, power and
frequency as state variables, it is not generally clear how these models can be used to
analyze the transient stability of the overall power system. The model (4.1) captures
the relationship between the active power and frequency in induction machines by
modelling it as a linear function of the slip (see Section 2.4). Since this model is in
terms of the same states as the classical machine model, it is also useful for studies
related to synchronous stability. An asynchronous induction generator works in the
same way as an induction motor, see Section 2.4, and an induction motor model
such as (4.1) can therefore be used to represent fixed speed wind turbines, that use
cage type induction generators.
For the increasingly installed variable speed wind farms, that use more advanced
control systems, or other power sources such as photovoltaics, a DC/AC power
converter (inverter) is used as an interface towards the grid. A means of controlling
this converter’s frequency to emulate a synchronous machine is by the droop control
law
Êi = Ê ú ≠ ki (Pe,i ≠ Piú ),
(4.2)
where Pe,i is the active power demand at bus i and Piú is the constant power injected
to the grid when operated at the rated frequency Ê ú . The perameter ki is the socalled droop coefficient and is subject to design. In [50] it is shown that there is an
exact 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 types
of renewable generation in power systems.
At the buses in Nload , which are the generator buses of the pre-augmented
system, we assume that there is a certain power load drawn from the bus, with the
same dynamics as in (4.1):
0
Pload,i = Pload,i
+ Di Êi .
(4.3)
These will differ from the asynchronous generators in the size of the parameter D
and will enter the bus power balance with a negative sign to signify that they are
drawing power from the system.
4.1.3
System Dynamics
The oscillator dynamics at the synchronous generator buses are still given by the
second order swing equation (2.1). This together with (4.1) and (4.3) give that the
1
Multiply (4.2) by k1i and rearrange to obtain Pe,i = Piú + k1i Ê ú ≠ k1i Êi . Since Piú and Ê ú
are rated, constant quantities, we can set Piú + k1i Ê ú = Pi0 , and by identfying k1i = Di we have
retrieved (4.1).
44
CHAPTER 4. LOSSES IN RENEWABLE ENERGY INTEGRATED SYSTEMS
dynamics at each bus i œ {1, . . . n} can be described by
0
0
0
Mi ◊¨i + —i ◊˙i + Di ◊˙i = Pm,i
+ Pwind,i
≠ Pload,i
≠ Pe,i ,
0
where all parameters are non-negative and Di = 0, Mi , —i , Pm,i
> 0 only for
0
0
i œ Naug , Pwind,i > 0 only for i œ Nwind and Pload,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’s
n-dimensional susceptance and conductance matrices LB and LG be given by (2.5)(2.6). As in Section 3.1, we then define the reduced, or grounded, Laplacians L̃B
and L̃G , 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 asynchronous machine nodes separately as
Ë
DÊD = ≠ In0
Ë
È
0 (L̃B ◊ ≠ P 0 )
(4.4)
È
MÊ̇G + BÊG = ≠ 0 Im≠1 (L̃B ◊ ≠ P 0 ),
(4.5)
0 + P0
0
where P 0 = diag{Pi0 } with Pi0 = Pm,i
wind,i ≠ Pload,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
C
D
C
D
I
0
◊˙ = n0 ÊD +
Ê = T1 Ê D + T2 Ê G .
0
Im≠1 G
(4.6)
Now, from (4.4), we get:
T1 ÊD = ≠T1 D≠1 T1T (L̃B ◊ ≠ P 0 )
which allows the state ÊD to be eliminated. To simplify the notation, we can define
TD := T1 D≠1 T1T and finally write the state equations for ◊ and ÊG as:
C
D
C
d ◊
≠TD L̃B
T2
=
≠M≠1 T2T L̃B ≠M≠1 B
dt ÊG
DC
D
C
D
◊
TD
+
w,
ÊG
M≠1 T2T
(4.7)
where we have lumped the constant power P 0 into the disturbance input w. By
slight abuse of notation, we will in the following let the formulation (4.7) represent
deviations from a steady-state operating point.
4.1. PROBLEM FORMULATION
4.1.4
45
System Inputs
As in Chapter 3, we have here let the system states represent deviations from a
steady state operating point and in doing so lumped the constant power input or
output Pi0 at each bus i into the disturbance input wi . Even though P 0 originally
has both positive and negative quantities in its entries:
S
T
0
Pwind,1
..
.
W
X
W
X
W
X
W
X
0
W Pwind,n
X
0 ≠m
W
X
W≠Pload,n0 ≠m+1 X
W
X
W
X
..
P0 = W
X,
.
W
X
W ≠P
X
W
X
load,n0
W
X
0
W
X
Pm,n
0 +1
W
X
..
W
X
U
V
.
0
Pm,n
0 +m≠1
the disturbance w is a zero-mean stochastic variable which represents deviations
from 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 coefficients
Di for i = 1, . . . , n0 and Mi for i = n0 + 1, . . . , n ≠ 1 respectively. However, for the
sake of brevity, that case is not considered in this chapter.
4.1.5
Performance Metric
To evaluate the performance of the system (4.7), we will as in Chapter 3 measure
the control effort needed to drive the system to a synchronous state2 . This control
effort is defined as the real power flowing between the nodes due to the phase angle
differences arising from disturbances and is the same as the resistive power losses
in the system during the transient phase.
Under the assumptions that phase angle differences are small and that the system has a flat voltage profile, the resistive power loss over an edge Eij œ E = E0 fiEaug
is Pijloss = gij |◊i ≠ ◊j |2 and thus the total losses in the network are:
Ploss =
ÿ
i≥j
gij |◊i ≠ ◊j |2 .
(4.8)
This is equivalent to the quadratic form Ploss = ◊ú L̃G ◊. We can therefore define the
(n ≠ 1)-dimensional output y of the system (4.7) as:
y=
2
Ë
1/2
L̃G
0
È
C
D
◊
,
ÊG
(4.9)
Since the system is reduced and all states are measured against node n, the synchronous state
is given by ◊ © 0.
46
CHAPTER 4. LOSSES IN RENEWABLE ENERGY INTEGRATED SYSTEMS
1/2
by which Ploss = y ú y. Since L̃G is positive semidefinite, we let L̃G be the unique
positive semidefinite square-root of L̃G . Note that over the purely reactive lines
that connect the fictitious generator buses, no real power will flow and yi © 0 for
i œ Naug .
Remark 4.2 The positive semidefiniteness of L̃G is not immediately obvious, since
the graph underlying LG is not connected (recall the lines connecting the fictitious
nodes being purely reactive). By this construction, the m last rows and columns of
LG and the m ≠ 1 last rows and columns of L̃G will be zero. However, the (n0 ◊ n0 )
submatrix with the non-zero entries of LG and L̃G will still be a graph Laplacian
describing the connected network between the n0 nodes in the pre-augmented network. For L0G , the properties listed in Section 2.2.3 all hold. L̃G will therefore have
m zero eigenvalues and n0 ≠ 1 positive ones, and remains positive semidefinite.
We can now define the system (4.7) and (4.9) as the input-output mapping H
from w to y:
C
d ◊
dt ÊG
D
y
=
C
≠TD L̃B
T2
≠M≠1 T2T L̃B ≠M≠1 B
=: AÂ + Bw
=
Ë
1/2
L̃G
=: CÂ
4.2
È
0
C
◊
ÊG
DC
D
C
D
◊
TD
+
w
ÊG
M≠1 T2T
D
(4.10)
Input-Output Analysis
In this section, we will first show that the linear system (4.10) is asymptotically
stable in order to ensure that its H2 norm exists. We will then derive an expression
for this norm and finally discuss upper and lower bounds on its value, as well as limits
where 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 with
augmented systems such as those of Figure 4.1b. A theorem relating these systems
is presented, which describes the cases in which the augmentation procedure does
not affect the H2 norm and the associated power losses.
4.2.1
Stability
We will follow the arguments in [5] to show that the system (4.10) is asymptotically
stable. Consider the Lyapunov function candidate
1
1 T
V (◊, ÊG ) = ◊T L̃B ◊ + ÊG
MÊG .
2
2
4.2. INPUT-OUTPUT ANALYSIS
47
Clearly, V (0, 0) = 0. L̃B is positive definite, since it is the reduced Laplacian of a
complete graph (see e.g. [40]), and M > 0, thus V (◊, ÊG ) > 0, ’◊, ÊG ”= (0, 0).
The derivative of V evaluated along the state trajectories, will after some algebraic
operations be given by
T
V̇ (◊, ÊG ) = ≠ÊG
BÊG ≠ ◊T L̃B TDT L̃B ◊
which is non-positive for all ◊, ÊG , since B > 0 and L̃B TDT L̃B Ø 0. For global
asymptotic stability, we also require V̇ (◊, ÊG ) © 0 … (◊, ÊG ) © (0, 0). Clearly, V̇ ©
0 requires ÊG © 0 and TDT L̃B ◊ © 0. The latter is equivalent Ëto T1T L̃BÈ ◊ © 0. By (4.7),
however, ÊG © 0 implies ≠M≠1 T2T L̃B ◊ © 0. Now, if T = T1 T2 = In0 +m≠1 , the
two give T L̃B ◊ © 0. Since T is the identity matrix and L̃B is positive definite, ◊ © 0.
We therefore conclude that the last criterion holds and all Lyapunov’s conditions
for 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 can
be calculated by evaluating (2.11), where the observability Gramian X is given by
the Lyapunov equation (2.12). If we partition X œ C(n+m≠2)◊(n+m≠2) into four
submatrices as
C
D
X 11 X 0
X=
,
X 0ú X 22
where X 11 œ C(n≠1)◊(n≠1) , X 0 œ C(n≠1)◊(m≠1) and X 22 œ C(m≠1)◊(m≠1) , the Lyapunov equation translates into the following three linearly independent equations:
L̃B TD X 11 + L̃B T2 M≠1 X 0ú + X 11 TD L̃B + X 0 M≠1 T2T L̃B =
0
≠1
L̃B TD X ≠ L̃B T2 M
T2T X 0
≠M
≠1
X
BX
22
22
11
0
≠1
22
≠1
+ X T2 ≠ X M
0ú
+ X T2 ≠ X M
B =
B =
L̃G
(4.11a)
0
(4.11b)
0.
(4.11c)
Evaluating (2.11) gives
||H||2H2 = tr(TD2 X 11 ) + tr(M≠2 X 22 ).
(4.12)
Assuming uniform dampings and inertia, i.e. M = M Im≠1 , D = D̄In0 and B =
—Im≠1 , (4.12) can be simplified to:
||H||2H2 =
1
— 1
tr(L̃≠1
) 2 tr(X 22 ),
B L̃G ) + (1 ≠
2D̄
D̄ M
where X 22 can be evaluated using equations (4.11a)-(4.11c).
(4.13)
48
4.2.3
CHAPTER 4. LOSSES IN RENEWABLE ENERGY INTEGRATED SYSTEMS
Properties of the Augmented Network Laplacians
Before we proceed to discuss the H2 norm of the system (4.10) and how the result
(4.13) relates to the results of Chapter 3, we will present some observations regarding
what we term the generalized graph Laplacian ratio tr(L̃≠1
B L̃G ) in the case where
the underlying graph is that of an augmented network.
The graph underlying LB is connected and L̃B is thus non-singular (otherwise
≠1
L̃B would not be defined). The graph underlying LG is, as discussed in Remark 4.2,
not connected and L̃G is therefore singular. Despite this fact, Lemma 3.3 still holds,
since its proof does not put any requirements on LG other than its being a graph
Laplacian. This observation will be used to prove the main result of this section.
The notion that tr(L̃≠1
B L̃G ) is a generalized ratio between the conductance and
susceptance matrices is further strengthened by the following result, which shows
that, for our augmented network, the purely reactive edges (with a zero conductance
to susceptance ratio) do not contribute to the generalized Laplacian ratio. Thus, the
susceptances of the lines connecting the augmented, fictitious nodes are irrelevant
and only the main network, connecting nodes 1, . . . , n0 , influence tr(L̃≠1
B L̃G ). This
idea is formalized through the following theorem:
Theorem 4.1 Consider a network of n0 nodes connected by the set E0 = {Eij } of
edges with weights bij > 0 and gij > 0. Denote the Laplacians associated with
the suseptance and conductance matrices for these weighted graphs as L0B and L0G
respectively.
Consider m Æ n0 of these nodes, which can without loss of generality be renumbered n0 ≠ m + 1, . . . , n0 , and divide each of these nodes into two nodes to create m
augmented nodes. Call the set of edges connecting these augmented nodes Eaug , and
let the weights be bij > 0 and gij = 0, for Eij œ Eaug . Let the full graph Laplacians of
the augmented network with n0 + m = n nodes and the set E = E0 fi Eaug of edges be
denoted as LB and LG respectively and their grounded versions, with the nth rows
and columns deleted be called L̃B and L̃G . Then
†
0† 0
tr(L̃≠1
B L̃G ) = tr(LB LG ) = tr(LB LG ).
Proof: See Appendix B.
In the case where all conductance to susceptance ratios (equivalently resistance
to reactance ratios) of all lines (edges) are equal, the generalized Laplacian ratio
scales 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 œ
E0 be equal in terms of their resistance to reactance-ratio, i.e.:
–ij =
rij
gij
=
= –.
xij
bij
4.2. INPUT-OUTPUT ANALYSIS
Then
49
tr(L̃≠1
B L̃G ) = –(n0 ≠ 1).
(4.14)
0† 0
0
0
Proof: We have tr(L̃≠1
B L̃G ) = tr(LB LG ). In this case, LG = –LG , and by
0† 0
0
≠1
0
0
≠1
0
Lemma 3.3: tr(LB LG ) = tr((L̃B ) L̃G ) = tr((L̃B ) –L̃B ) = tr(–In0 ≠1 ) = –(n0 ≠
1).
Remark 4.3 Note that these Theorem 4.1 and Corollary 4.2 allow for the edges
Eij 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 gij ”= 0 for Eij œ Eaug , the generalized Laplacian ratio would be:
0† 0
tr(L̃≠1
B L̃G ) = tr(LB LG ) +
ÿ
–ij ,
Eij œEaug
ij
where –ij = 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.
g
4.2.4
Relation to Previous Results
From (4.12) we see that in the case — = D̄, the H2 norm of the system H reduces
to
1
||H||2H2 =
tr(L̃≠1
(4.15)
B L̃G ).
2D̄
The frequency dependence coefficient D in (4.1) and (4.3) characterizing the asynchronous generators and power loads can be interpreted as a type of damping,
analogous to the dampings — of the synchronous generators. However, since wind
generators and loads are generally made out of smaller machines with less inertia
than traditional generators it is more physically meaningful to assume — Ø D. We
can then upper bound (4.12) by
||H||2H2 Æ
1
tr(L̃≠1
B L̃G ).
2D̄
(4.16)
Even for non-uniform asynchronous dampings, a choice of D̄ = mini=1,...,n0 Di =
Dmin 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 the
reference). This would correspond to a network of only first order dynamics, i.e.,
a system without synchronous generation in our modeling framework. Although
such a system may not be realistic because a power system requires inertia, the
resulting 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
50
CHAPTER 4. LOSSES IN RENEWABLE ENERGY INTEGRATED SYSTEMS
exact representation of droop-controlled inverters in a microgrid, e.g. a system with
distributed renewable generation in a low-voltage network.
To look further into this simpler model, let L0B and L0G be the graph Laplacians
for this network of n0 nodes and L̃0B and L̃0G the corresponding grounded Laplacians,
with the kth rows and columns removed. Let all oscillators have the same damping
coefficient D̄. We can then write the input-output mapping H̃1st with first order
dynamics as:
d˜
1
1
◊ = ≠ L̃0B ◊˜ + I w̃
dt
D̄
D̄
0 1/2 ˜
ỹ = (L̃G ) ◊,
(4.17)
where ◊˜ is the state vector of the n0 ≠ 1 bus angles of the reduced system. It is
then easy to show that ||H̃1st ||2H2 = 21D̄ tr((L̃0B )≠1 L̃0G ), which by Lemma 3.3 and
Theorem 4.1 is equivalent to (4.15).
By considering D̄ to be a surrogate for the damping coefficient —, and provided
the network remains unchanged, (4.15) is also the norm of a system with n0 second
order oscillators as derived in Chapter 3 (the system (4.10) reduces to the second
order model if n0 = m and the network is not augmented, see Remark 4.1). And,
according to the main result of this section, this H2 norm is also that of a network
of n0 first order oscillators with dampings D̄, augmented with m second order
oscillators with dampings — = D̄. The remarkable result that the m additional
oscillators, while affecting the dynamics, do not change the norm again follows.
We can conclude that if the same uniform parameters are used, the H2 norm and
therefore the resistive losses will be the same in a same-sized network, regardless of
whether the first order (4.17), the second order (3.6) or the combined model (4.10)
is used. Any differences between the models in terms of the resistive losses will thus
essentially depend on the damping parameters of the different types of generators.
These differences, as well as any loss-related synergies arising in the combined model
due to coupling of the parameters, will be explored through numerical examples in
Section 4.3.
We should point out that these results do not claim the models themselves to
be equivalent. The transient responses of the respective systems are substantially
different and for a stability analysis, the model order and parameters should be
chosen with care. What we term the price of synchrony and the network’s ability
to synchronize are two different issues, and it is only in terms of the H2 norm that
the different order models are equivalent.
Remark 4.5 In analogy to Lemma 3.2, it can be shown that for H̃1st in (4.17),
||H̃1st ||2H2 = ||H1st ||2H2 , where H1st would be the non-grounded system with the
full Laplacians. The same holds for H in (4.10); if —i = Di = D̄ ’i = 1, ..., n a
non-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 oscillators
if the nth node is grounded. It must then however be noted that the norm of the
grounded system is not in general equivalent to that of the non-grounded one.
4.3. CASE STUDIES
4.3
51
Case Studies
The result (4.13) indicates that the resistive losses in the network of coupled first
and second order oscillators depends on the sub-network of first order oscillators
with damping coefficients D̄, and increasingly on the second order oscillators, the
synchronous generators, as the damping coefficient — increases. In this section, we
present a brief case study to illustrate how the second term of (4.13) affects the H2
norm. We will also, as in Section 3.4.4, discuss how the location of synchronous and
asynchronous generators of different dampings affect the transient resistive losses,
in the case of non-uniform asynchronous dampings.
4.3.1
Increased Synchronous Damping
We now present an example of how the resistive losses in the network depicted in
Figure 4.1b scale with the synchronous damping — and the transient reactances in
Eaug .
To simplify the analysis, we assign to all edges Eij œ E0 the same impedances
zij = 0.05 + j0.25 and to all asynchronous machines the same frequency dependence
5
coefficient D̄ = 2fif
, where f0 = 60 Hz. We let the three synchronous generators
0
20
have inertia M = 2fif0 and vary their damping — as multiples of D̄. Figure 4.2 then
shows how this increase in synchronous damping affects the resistive losses in the
network for three cases with the transient reactances xij œ {0.01, 0.05, 0.1}, for all
Eij œ Eaug .
We note that while larger synchronous dampings reduce the synchronization
losses of the system, the marginal effect of the synchronous damping is decreasing.
The example indicates that, in accordance with the bound (4.16), it is the size of
the asynchronous dampings that drives the size of the losses. The losses also depend
on the transient reactances, where a smaller reactance implies smaller losses. An
intuitive explanation for this is that a larger susceptance allows for more power to
flow, which lets the larger damping of the synchronous generator attenuate more of
the remaining system’s oscillations. Note that the transient reactances are however
irrelevant when — = D̄. In that case, the losses are given by (4.14): ||H||2H2 =
–
(n0 ≠ 1) = 2·0.25 · 4 = 30.16.
2D̄
2fi60
4.3.2
Effects of Generator Placement
We will now consider a similar situation as in the example of Section 3.4.4 and
examine how the placement of generator/load buses and asynchronous generation
buses at highly and weakly interconnected nodes respectively affect the losses. Such
a study is of course highly parameter-dependent, but we will assume that the frequency dependence coefficient of the renewable generation sources Dwind are equal
to the synchronous dampings — (this may be achieved in a real system by appropriately designing the droop-control, see Section 4.1.2), and that the loads have a
very small frequency dependence Dload .
52
CHAPTER 4. LOSSES IN RENEWABLE ENERGY INTEGRATED SYSTEMS
||H||22
32
xaug = 0.01
30
xaug = 0.05
28
xaug = 0.1
26
24
22
20
18
1
2
3
4
5
6
7
8
/D
Figure 4.2: The (squared) H2 norm as given by (4.13) for the system depicted in
Figure 4.1b with increasingly large synchronous dampings and for different transient
reactances of the synchronous generators.
For this purpose, consider the network topology of Figure 3.1 and let node 1
be the grounded node. Then assume that three of the six remaining nodes are
10
20
1
synchronous generator/load buses with — = 2fif
, M = 2fif
and Dload = 2fif
.
0
0
0
The transient reactances are assumed to be xaug = 0.05. The remaining nodes
10
are dominated by renewable generation with Dwind = 2fif
. All lines in E0 have
equal impedances: zij = 0.1 + j0.6. We then compare the cases where (a) the
generator/load buses are placed at the nodes with the highest degrees (nodes 2, 6
and 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 the
lowest degrees (nodes 7, 3 and 5) and the renewable buses are placed at the nodes
with the highest degrees (nodes 2, 6 and 4).
The low dampings of the load buses do not have a significant contribution to
the attenuation of oscillations in the synchronization transient, and the co-located
synchronous generator dampings only have a limited effect on the losses (as seen
in Section 4.3.1). Therefore, allowing the generator/load buses to be most interconnected as in (a) increases the losses compared to the alternative (b) in which
the better damped renewable generation buses are placed at those nodes. The H2
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 different from the examples in Section 3.4,
in particular in the way the phase angles oscillate. The almost non-existent oscillations in the system 4.10 are due to the fact that what we have termed “first order
oscillators” are in reality not oscillators and will simply asymptotically dampen out
any initial disturbance. This may have practical consequences in real systems; an
increased integration of renewable energy with dynamics captured by (4.1) may
potentially reduce oscillations and thus wear-and-tear in synchronous generators.
53
2
2
1
1
0
θ
θ
4.4. DISCUSSION
−1
1
2
3
4
5
6
7
8
9
−2
0
10
1
1
0.5
0.5
G
ωG
−2
0
0
−1
ω
0
−0.5
−1
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
Time
6
7
8
9
10
0
−0.5
1
2
3
4
5
Time
6
7
(a) Configuration (a)
8
9
10
−1
0
(b) Configuration (b)
Figure 4.3: Simulation of the network in Figure 3.1 with (a) three generator/load
buses located at the three nodes with the largest degrees and three renewable generator 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 difficult
to ascertain from the figure.
4.4
Discussion
In this chapter, we have extended the model of Chapter 3 to include both loads and
asynchronous generation, representative of renewable energy sources, by adding
so-called first order oscillator dynamics to the existing model. The usage of the
first order dynamical model to capture power injections from renewable generation
sources is justified both by its common use as a model for induction machines and
its equivalence with the droop control law used to control the frequency of power
converters. This new combined first and second order model is deemed better suited
to characterize oscillations in power networks with highly heterogenous generation
than the classical machine model.
The resistive losses incurred in returning this system of coupled first and second order oscillators to a synchronous state was evaluated using the system’s H2
norm. The main result of this chapter is that this H2 norm is indifferent to the new
dynamics. That is, provided damping coefficients are uniform, the H2 norm of a
network with N second order oscillators derived in Chapter 3 is the same as that of
a network of N first order oscillators and also as that of N first order osccillators
augmented with m Æ N second order oscillators. All these models are thus equivalent in terms of the H2 norm, and any differences can be attributed to different
damping parameters. This result indicates that adding renewable energy sources
to the network, either at existing load buses or as a replacement of synchronous
generation, will not increase the transient resistive losses, provided their controllers
can be adjusted to match the dampings of the original synchronous machines.
However, there is a large and very significant difference in the transient behaviours of the systems with different order models. These behaviours are illus-
54
CHAPTER 4. LOSSES IN RENEWABLE ENERGY INTEGRATED SYSTEMS
trated by simulations found in Sections 3.4 and 4.3. Even for uniform damping
parameters, the models are by no means equivalent, and both model order and
parameters should be chosen with care if studying synchronous stability. What we
term the price of synchrony and the ability to synchronize are different issues, and
it 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 resistive
losses are the same for same-sized networks of first- and/or second order oscillators
with uniform dampings is the fact that the purely reactive lines used to augment
the network, in order to combine different dynamics at the same node, do not affect
the generalized Laplacian ratio tr(L†B LG ). It may not seem remarkable that zero
resistance lines do not contribute to the resistive losses, but the generators at these
lines are included in the dynamics and could therefore be expected to influence
the input-output behaviour of the system. In conclusion, for systems with uniform
dampings, it is not the number of oscillators or their order, but the number of nodes
and introduced disturbances, that affect the transient resistive losses.
If the damping parameters are not uniform, we can conclude through bounds
on the H2 norm and numerical examples, that large synchronous dampings may
reduce transient resistive losses, but that this effect is bounded, and decreases with
increasing transient reactances. Instead, the asynchronous dampings (or frequency
dependence parameters) have a large influence on the transient losses. Since these
may be subject to design through the droop coefficient, used to control the output
frequency of e.g. modern wind farms or photovoltaics, the losses can be reduced by
appropriate adjustment of the controllers. In these cases, our examples show that it
is particularly advantageous to increase the damping at highly interconnected nodes
in the network.
Chapter 5
Conclusions and Directions for Future
Work
In this thesis, we have formulated two linear dynamical power systems models; a
classical machine model of coupled swing equations describing the dynamics of the
synchronous generators that dominate current power systems, as well as a network
preserving model for a heterogenous renewable energy integrated system. We have
applied distributed disturbances to each of these networks and quantified the resistive power losses that they incur in regaining or maintaining a synchronous state.
These transient losses, which we have termed the “Price of Synchrony”, have been
evaluated through the H2 norm of the linear dynamical systems with the disturbance input and the resistive losses as the output.
By considering networks of identical synchronous generators, we have shown that
the transient losses depend on the generator dampings and a quantity that can be
interpreted as a generalized ratio between the conductance and susceptance matrices
of the power network. They also grow linearly with the network size, i.e., the number
of nodes. However, one of our main result shows that these transient power losses
are independent of the network connectivity. This conclusion is in contrast to typical
power systems stability notions, which imply that well interconnected networks have
better syncronous stability properties than loosely connected networks. Therefore,
given two networks of equal size, their ability to synchronize may differ, but the
control effort required to reach synchrony will be the same.
A similar conclusion holds for the heterogenous networks in which mixed first
and second order dynamics describe the oscillations. Our results therefore indicate
that although the transient behaviours of the two considered models are fundamentally different, they may incur the same transient power losses, if the damping
coefficients are uniform.
For an envisioned future power system with high levels of renewable energy
sources, these results have several consequences. In general, renewables cause larger
and more frequent disturbances that will induce synchronization transients more frequently. 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’s
power networks. Since the transient power losses scale linearly with the network
size, and are larger in the low-voltage distribution grids where much of this integration is expected to take place, the magnitude and importance of these losses can
be expected to increase. Our results also imply that efforts to improve the synchronous stability properties by increasing the grid connectivity will not alleviate
this effect. On the other hand, the losses also depend solely on the dampings of
the generators and synchronous machines. This indicates that although renewable,
asynchronous generators will alter the power system dynamics, they will not necessarily increase the transient power losses, if their dampings can be adjusted to
match the existing synchronous generators. Since controlled power inverters are required for grid-integration of many renewable sources, the damping is often a design
parameter. Therefore, these asynchronous machines can even be used to improve
the system’s damping and decrease transient power losses. Such a strategy is shown
to 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 insight to the synchronization transients of power systems, the results are only valid
for small-signal disturbances. The applicability of the linear power flow approximation in low-voltage grids is also disputed. A simulation study of the non-linear
power system dynamics is therefore an important direction for future work. Further
directions of future study also include comparing simulations of the simple renewable energy integrated power system dynamics proposed here to higher order models
found in literature. There are also remaining open questions related to the linear
systems that have been discussed in this thesis. For example, it was shown that the
H2 norm is indifferent to whether first order, second order or mixed dynamics are
modeled, but the HΠnorm was not studied. While the physical meaning of this
norm would no longer be the total transient power losses, its evaluation may give
additional insights regarding the transients of the different systems.
In a more general context, the coupled oscillators considered in this thesis model
a type of consensus dynamics. Similar formulations arise e.g. in multirobotic or biological systems and in vehicle formulation problems, where the outputs can be
defined so that the input-output H2 norm is a meaningful measure of coherence
or robustness. For example in a standard first order consensus problem subject to
disturbances, where the output is defined as an unweighted Laplacian, the H2 norm
represents the variance of the steady-state local error and is again given by a generalized Laplacian ratio. In this work, we have explored this generalized Laplacian
ratio analytically by proving its invariance with respect to network grounding and
augmentation. We have also evaluated bounds on its value and studied the special
case where the Laplacians are simultaneously diagonalizable so that the generalized
ratio reduces to a ratio of eigenvalues. Further work will also include formalizing
some of these concepts for general networked control problems and investigating the
extent to which the conclusions drawn here transfer to such systems.
Appendix A
Appendices to Chapter 3
A.1
Proof of Lemma 3.2
Consider the following state transformation of the system (3.6):
C D
◊
Ê
=:
C
0
U
U
0
DC
D
◊Õ
,
ÊÕ
where U is the unitary matrix which diagonalizes LB , i.e., U ú LB U = B =
B
B
diag{0, ⁄2 , ..., ⁄N }, where 0 = ⁄B
1 Æ ⁄2 Æ ...⁄N are the eigenvalues of LB . We
have assumed, without loss of generality, that U = [ Ô1N 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 = U ú w and
y Õ = U ú y to obtain an, in terms of the norm, equivalent system
d
dt
C
◊Õ
ÊÕ
D
yÕ
Now, observe that
=
C
0
I
1
≠M
DC
—
I
B ≠M
C D
Ë
È
1
◊Õ
= U úL 2 U 0
G
ÊÕ
◊Õ
ÊÕ
D
S
+
T
0 ··· 0
W ..
X
ú
U LG U = U . L̂G
V.
0
C
0
1
MI
D
wÕ
.
(A.1)
(A.2)
so that the first rows and columns of both U ú LG U and B are zero. We thus have
q
qN
Õ
Ô1
that the states ◊1Õ = Ô1N N
i=1 ◊i and Ê1 = N
i=1 Êi satisfy the dynamics:
◊˙1Õ = Ê1Õ
—
1 Õ
Ê˙1Õ = ≠ Ê1Õ +
w
M
M 1
y1Õ = 0,
57
58
APPENDIX A. APPENDICES TO CHAPTER 3
which reveals that this mode, which we call H1Õ and that corresponds to the single
zero eigenvalue of LB , is unobservable from the output. The remaining eigenvalues
of the system (A.1) lie strictly in the left half of the complex plane due to LB ’s
positive semidefiniteness and it follows that the input-output transfer function from
wÕ to y Õ is stable and has finite H2 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 H1Õ and Ĥ. We take L̂G as
B
B
the Hermitian positive definite submatrix in (A.2) and define ˆ B = diag{⁄B
2 , ⁄3 , ..., ⁄N }
and write the input-output mapping Ĥ as:
d
dt
C D
◊ˆ
Ê̂
=
C
0
I
—
1 ˆ
≠M
I
BC ≠
DM
Ë 1
È ◊ˆ
ŷ = L̂ 2 0
G
Ê̂
DC D
◊ˆ
Ê̂
+
C
0
1
MI
D
ŵ
(A.3)
d ˜
„ = A„˜ + B w̃; ỹ = C„.
or … dt
Note that the systems Ĥ1 and H̃ are completely decoupled and we therefore
have that ||H||2H2 = ||H1Õ ||2H2 + ||Ĥ||2H2 = ||Ĥ||2H2 .
The H2 norm can then be calculated in perfect analogy to the derivations in
Section 3.2.2 and we obtain that
1
tr( ˆ ≠1
(A.4)
ÎHÎ2H2 =
B L̂G ).
2—
Now, we show that the result of Lemma 3.1 can be written in terms of the state
transformed matrices ˆ B and L̂G . Define the N ◊ (N ≠ 1) and the (N ≠ 1) ◊ N
matrices R and P by:
S
0
W
R=U
···
IN ≠1
0
T
S
0
Ik≠1
X
W
V,P = U
0
≠1
IN ≠k
T
X
V,
where k is the index of the grounded node and ≠1 is the (N ≠ 1) ◊ 1 vector with
all entries equal to ≠1. By this design, ˆ B = Rú B R, L̂G = Rú U ú LG U R and
LB/G = P ú L̃B/G P . Further, to simplify notation, we define the (N ≠ 1) ◊ (N ≠ 1)
non-singular matrix V = P U R. Then we can write
≠1 ≠1
tr(L̃≠1
L̃B (V ú )≠1 V ú L̃G ),
B L̃G ) = tr(V V
since V V ≠1 = (V ú )≠1 V ú = I. By the cyclic properties of the trace:
ú ≠1 ú
≠1 L̃≠1 (V ú )≠1 V ú L̃ V )
tr(V V ≠1 L̃≠1
B (V ) V L̃G ) = tr(V
B
G
= tr((V ú L̃B V )≠1 V ú L̃G V ).
But V ú L̃B V = Rú U ú P ú L̃B P U R = ˆ B and V ú L̃G V = Rú U ú P ú L̃G P U R = L̂G .
Hence,
ˆ ≠1
tr(L̃≠1
B L̃G ) = tr( B L̂G ).
A.2. PROOF OF LEMMA 3.3
59
In conclusion,
ÎHÎ2H2 =
1
1
2
tr( ˆ ≠1
tr(L̃≠1
B L̂G ) =
B L̃G ) = ÎH̃ÎH2 ,
2—
2—
which proves the Lemma.
A.2
Proof of Lemma 3.3
ˆ ≠1
By the proof of Lemma 3.2, we have that tr(L̃≠1
B L̃G ) = tr( B L̂G ). Now,
tr( ˆ ≠1
B L̂G ) = tr
AC
0
0
0
ˆ ≠1 L̂
B
G
DB
= tr
By definition, see e.g. [27], U ú L†B U = diag{0,
AC
0
0
0
D
ú
B
ˆ ≠1 U LG U .
B
1
, ..., ⁄1B },
⁄B
2
N
which makes the above
equivalent to: tr(U ú L†B U U ú LG U ) = tr(U ú L†B LG U ). But since the trace is unitarily
invariant, it follows that
†
tr( ˆ ≠1
B L̂G ) = tr(LB LG ),
which concludes the proof.
A.3
H2 Norm With Simultaneously Diagonalizable
Laplacians
In cases where the Laplacians LG and LB are simultaneously diagonalizable, the
H2 norm can be expressed directly in terms of the Laplacian eigenvalues. In the
simplest and most relevant of these cases, the two Laplacians are multiples of each
other, and the system can be analyzed in terms of the system lines’ resistance to
reactance ratio, as is done in Section 3.2.3.
By definition, see e.g. [29], two matrices A and B are simultaneously diagonalizable if there exists a single nonsingular matrix S such that both S ≠1 AS and S ≠1 BS
are 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 both
B
ú
LB and LG , i.e., U ú LB U = B = diag{0, ⁄B
G =
2 , ..., ⁄N } and U LG U =
G
G
diag{0,
⁄
,
...,
⁄
}.
Without
loss
of
generality,
we
have
here
assumed
that
U =
2
N
Ë Ô
È
1/ N 1 u2 · · · uN , where the ui are the eigenvectors corresponding to the
G
eigenvalues ⁄B
i , ⁄i respectively. Since we are considering Hermitian matrices, the
eigenvectors ui are orthogonal, which is why U is unitary. Then, one definition of the
B
Moore-Penrose pseudo inverse reveals that U ú L†B U = diag{0, 1/⁄B
2 , ..., 1/⁄N } [27].
60
APPENDIX A. APPENDICES TO CHAPTER 3
Now consider the result from Theorem 3.4. Since the trace is unitarily invariant,
the squared H2 norm for the special case of simultaneously diagonalizable Laplacians
is:
ÎHÎ2H2
=
=
=
1
1
tr(L†B LG ) =
tr(U ú L†B U U ú LG U )
2—
2—
S
TS
T
0
0
W
XW
X
1/⁄B
⁄G
W
XW
X
1
2
2
XW
X)
tr(W
.
.
W
X
W
X
..
..
2— U
VU
V
G
1/⁄B
⁄
N
N
(A.5)
N
1 ÿ
⁄G
i
.
2— i=2 ⁄B
i
Case: Equal Line Ratios
The result considered in Section 3.2.3 is derived from the above as follows. If
LG = –LB , then trivially LB LG = LB –LB = –LB LB = LG LB , i.e., the Laplacians
commute. Using (A.5), we then obtain:
ÎHÎ2H2 =
N
1 ÿ
–⁄B
–
i
=
(N ≠ 1),
B
2— i=2 ⁄i
2—
(A.6)
The General Case
Despite a lack of direct relevance for power systems applications, general cases of
simultaneously diagonalizable Laplacians are discussed here for the sake of completeness.
The matrices X which commute with the Laplacian LG and which are thus
simultaneously diagonalizable with LG 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 ≠Q
being a zero matrix instead of what is normally assumed to be a negative definite
one (note that LG = LúG ).
Given a certain conductance matrix LG , any solutions X̄ of (A.7) for which it
also holds:
i. X̄ Hermitian and
ii. X̄ has the same nonzero elements as LG ,
would be a candidate for the susceptance matrix LB such that LG and LB are
simultaneously diagonalizable (assumption ii. can be relaxed if we need not assume
the 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 of
the existence of candidates X̄ for general graph Laplacians lies outside the focus of
this thesis. It is however an interesting algebraic problem, and we can present one
result in this context:
Lemma A.1 Consider a complete graph with n nodes and let L1 be a Laplacian in
which every edge Eij carries the same weight “. Let L2 be a weighted Laplacian with
weights wij = wji for every edge Eij in the same n-node complete graph. Under this
configuration, L1 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 entries
equal to 1.)
A.4
Proof of Corollary 3.6
Without loss of generality, let the grounded node k be node N and let M̃ =
diag{M1 , ..., MN ≠1 } and B̃ = diag{—1 , ..., —N ≠1 }. The reduced system H̃1 can then
be written as
S
◊˜
T
S
TS
0
0
W
X
W
0
0
d W ◊N +1 X
W
W
X = W≠M̃≠1 L̃
0
U
B
dt U Ễ V
bN,N +1
0
≠ MN +1
ÊN +1
C
ỹ
yN +1
D
=
C
1/2
L̃G
0
Ô
0
gN,N +1
IN
0
0
W 0
0
1 X
XW
W
≠M̃≠1 B̃
0 X
V UM̃≠1
—N +1
0
0
≠ MN +1
S
D
◊˜
T
0
0
0
1
MN +1
W
X
0 0 W ◊N +1 X
W
X.
0 0 U Ễ V
ÊN +1
T
C
D
X
w̃
X
X
V wN +1
(A.8)
Let the input-otput mapping HN +1 be the SISO subsystem of (A.8):
C
d ◊N +1
dt ÊN +1
D
=
yN +1 =
C
0
1
N,N +1
≠M
N +1
≠ MNN+1
+1
ËÔ
b
—
DC
C
D
È ◊
N +1
gN,N +1 0
D
◊N +1
+
ÊN +1
C
0
1
MN +1
D
wN +1
ÊN +1
From (A.8), it is clear that the systems H̃0 and HN +1 are entirely decoupled and
since we can write H̃1 = diag{H̃0 , H̃1 },
||H̃1 ||2H2 = ||H0 ||2H2 + ||HN +1 ||2H2 .
62
APPENDIX A. APPENDICES TO CHAPTER 3
Now, the H2 norm of HN +1 can be calculated in scalar analogy to the derivation in
Section 3.2.2 and is found to be:
||HN +1 ||2H2 =
which concludes the proof.
1
gN,N +1
–N,N +1
=
,
2—N +1 bN,N +1
2—N +1
Appendix B
Appendices to Chapter 4
B.1
Proof of Theorem 4.1
†
By Lemma 3.3, tr(L̃≠1
B L̃G ) = tr(LB LG ). The Lemma is applicable since it requires
no other properties of LG 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 L̃B,n0 and L̃G,n0 . It then holds:
tr(L†B LG ) = tr(L̃≠1
B,n0 L̃G,n0 ).
Without loss of generality, we assume that the numbering of the nodes is such that
nodes n0 ≠ m + 1, ..., n0 are each connected in rising order to one of the nodes
n0 + 1, ..., n0 + m, like in the example in Figure 4.1b. We can then consider the
transformation matrix
S
T
In0 ≠m 0
0
W
X
Im 0 V ,
V =U 0
0
Im Im
for which it holds that
C
D
L0 0
V LB V = B
.
0 B
ú
where B = diag{bn0 ≠m+1,n0 +1 , ..., bn0 ,n0 +m }. We also observe that
C
D
L0 0
V LG V = G
= LG .
0 0
ú
Continue to define the deleted transformation matrix Ṽ by the matrix that arises
when deleting row and column n0 from V . Then
C
D
C
D
L˜0 0
L˜0 0
Ṽ L̃B,n0 Ṽ = B
, Ṽ ú L̃G,n0 Ṽ = G
,
0 B
0 0
ú
where L˜0B and L˜0G are the reduced versions of the Laplacians L0B and L0G , with node
n0 grounded.
63
64
APPENDIX B. APPENDICES TO CHAPTER 4
It is evident from its structure that V is non-singular and its deleted version,
Ṽ , has the same property. We can therefore write:
≠1 ≠1
tr(L̃≠1
L̃B,n0 (Ṽ ú )≠1 Ṽ ú L̃G,n0 ),
B,n0 L̃G,n0 ) = tr(Ṽ Ṽ
since Ṽ Ṽ ≠1 = (Ṽ ú )≠1 Ṽ ú = In0 +m≠1 . By the cyclic properties of the trace,
ú ≠1 ú
≠1 ≠1
tr(Ṽ Ṽ ≠1 L̃≠1
L̃B,n0 (Ṽ ú )≠1 Ṽ ú L̃G,n0 Ṽ )
B,n0 (Ṽ ) Ṽ L̃G,n0 ) = tr(Ṽ
= tr((Ṽ ú L̃B,n0 Ṽ )≠1 Ṽ ú L̃G,n0 Ṽ ).
It holds that1
(Ṽ L̃B,n0 Ṽ )
ú
≠1
C
D
˜0 ≠1
0 .
= LB
0
B ≠1
Then
C
≠1
L˜0B
0
tr(L̃≠1
L̃
)
=
tr(
B,n0 G,n0
0
B ≠1
DC
D
≠1
L˜0G 0
) = tr(L˜0B L˜0G ) + tr(B ≠1 0)
0 0
≠1
= tr(L˜0B L˜0G ).
Now, again by Lemma 3.3:
≠1
0
tr(L˜0B L˜0G ) = tr(L0†
B LG )
with concludes the proof.
1
5
In0 ≠1
0
This can be shown by:
6
0
= In0 +m≠1 .
Im
5
≠1
L˜0B
0
0
B ≠1
65
L˜0B
0
0
B
6
=
5
≠1
L˜0B L˜0B
0
0
B ≠1 B
6
=
List of Figures
1.1
Energy mix in the Nordic countries in 2010 and 2050. . . . . . . . . . .
2.1
Principle of the electromechanical equilibrium in a synchronous generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mechanical analogy to the power system dynamics. . . . . . . . . . . .
A network of n robots. Source: http://users.ece.gatech.edu/magnus/
projects.html . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A 4-pole synchronous generator. Source: http://www.user.tu-berlin.de/
h.gevrek/ordner/ilse/wind/wind5e.html . . . . . . . . . . . . . . . . . .
Principles of fixed-speed and doubly fed wind generators. Source: http://
lejpt.academicdirect.org/A12/083_094.htm . . . . . . . . . . . . . . . .
2.2
2.3
2.4
2.5
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
4.1
4.2
4.3
Example of a network with N = 7 nodes. . . . . . . . . . . . . . . . . .
Mechanical analogy to grounded power system dynamics. . . . . . . . .
Resistance to reactance ratios r/x for the IEEE 14, 30, 57 and 118 bus
benchmark cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H2 norms for IEEE benchmark systems with increasingly varying line
ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of a 4 node radial and complete graph . . . . . . . . . . . . . .
H2 norms for growing radial and complete networks. . . . . . . . . . . .
Simulation of 7 bus system with “light” and “heavy” generator added.
Simulation of a system with non-uniform dampings subject to different
generator placements. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
8
10
12
16
17
20
24
28
32
33
34
35
36
Example of network augmentation. . . . . . . . . . . . . . . . . . . . . . 42
H2 norms for system with nonuniform synchronous and asynchronous
dampings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Simulation of the renewable energy integrated system in two configurations. 53
65
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